“Kolmogorov considered his work on turbulence to be non-mathematical. He wanted to explain observed phenomena from first principles.”
– V. I. Arnol’d [1]

In this Chapter, we collect mathematical results which shed light on turbulent fluid motion. Our focus is on theorems that connect seemingly distinct features to one another, such as properties of energy dissipation to precise regularity properties of the flow field. Most of these results appear in the literature in some form, while others are small novelties. A brief outline is as follows: in §[sec:sec:e], we introduce the incompressible Euler equations and discuss well-posedness theory and conservation laws. In §[sec:nssec] we introduce Navier-Stokes, Reynolds number and anomalous dissipation (without the eventual aid of viscosity) in turbulence. In §[sec:onssec] we review Onsager’s ideas which relate anomalous dissipation to \(\frac{1}{3}\) differentiability of the flow field. In § [sec:K41sec], we discuss Kolmogorov’s celebrated 1941 theory, as well as rigorous versions of his \(\frac{4}{3}\) and \(\frac{4}{5}\) laws. In §[sec:landausec] we describe Landau’s objection to Kolmogorov’s theory based upon intermittency of the dissipation field, along with rigorous constraints. In §[sec:models], model problems (the Burgers equation and passive scalar transport) are studied in some depth, illustrating some of the features noted above. Finally §[sec:particles] collects results pertaining to motion of particles in turbulence. Some open questions are posed.

The Euler equations of ideal fluid motion

“When a continuous medium is deprived of its physical properties (elasticity, thermal and electrical conductivity, and so on) its property of occupying a definite position in space remains, as do elementary interactions through the mutual pressure of its parts, due to Aristotle’s principle that it is impossible for two bodies to occupy the same space. It is amazing that it is these elementary interactions that cause the most complicated effects, including turbulence…” – V. I. Yudovich [2]

A vector field \(u:\mathbb{R}\times M\to \mathbb{R}^d\) on a domain \(M\subset \mathbb{R}^d\) is called the velocity field of an ideal incompressible fluid if it satisfies the Euler equations \[\begin{align} \tag{1} \partial_t u + u\cdot\nabla u&= -\nabla p,\\\tag{2} \nabla \cdot u &=0,\\ u\cdot n|_{\partial M} &=0. \tag{3} \end{align}\] The unknown function \(p\), named the pressure, is a Lagrange multiplier, which enforces the condition that \(u\) remain divergence-free for all times (cf. 7 –8 ).

As written, 1 –3 require differentiable velocity fields. However, the notion of solution to the equations of motion can be generalized in more irregular regimes such as turbulence, which is a singular regime of rough and tumble flow. As a general rule, we will not attempt to give a precise definition of turbulence, but instead will enumerate a number of features that are associated to this phenomenon. For reasons that will become clear further on, we will refer to the situation where the velocity field is differentiable as a non-turbulent regime. It is a regime in which the Euler equations may give a definite prediction for the motion of the fluid – a notion that three-dimensional turbulence seems to defy. One such prediction concerns conserved quantities. Being variational in nature (arising due to Hamilton’s principle), this regime is subject to the conservation laws implied by symmetries of its action via Noether’s theorem [3]. Specifically:

where \([u,v]= u\cdot \nabla v- v\cdot \nabla u\) is the Lie bracket and \(v_0\) is an arbitrary smooth divergence-free vector field, tangent to the boundary \(\partial M\). Space translation/rotation invariance gives rise to conservation of linear and angular momenta, provided that the fluid vessel is invariant under those actions. Momentum conservation thus holds for a broad family of conservation laws, both finite and infinite dimensional. On the other hand, circulation conservation is a truly infinite dimensional phenomenon – the number of conservation laws is in correspondence with the infinitesimal generators of volume preserving diffeomorphisms. As noted by Kelvin, they, in fact, define ideal fluid motion [5], [6]. The circulation theorem 4 is often presented in a slightly different, derived form. To recover this, let \(\Gamma\) be a rectifiable loop, and \(v_0(x) \rightharpoonup \oint_\Gamma \delta(x-\cdot) {\rm d}\ell\), a singular loop. We see that \(v_0\) is distributionally divergence-free \(\langle v_0, \nabla \varphi\rangle_{L^2(M)} = \oint_\Gamma \nabla \varphi \cdot {\rm d}\ell=0\) and that the solution to the Lie transport is \(v(t,x) = \oint_{X_t(\Gamma)} \delta(x-\cdot) {\rm d}\ell\), where \(X_t\) is the flow generated by \(u\), 14 –15 . As such, the circulation theorem 4 takes the form that justifies its name: \[\frac{{\rm d}}{{\rm d}t} \oint_{X_t(\Gamma)} u(t, \cdot)\cdot {\rm d}\ell = 0 \qquad \text{for any rectifiable loop} \;\Gamma \subset M.\] By Stokes’ theorem, this line integral is equivalent to the flux of the curl of the velocity vector field \(u\), the vorticity \(\omega\), through any comoving bounding surface. Letting the loop become infinitesimal about any given point \(x\in M\), with arbitrary orientation, we recover the Lie advection equation for the vorticity. In two dimensions one can identify the vorticity with a scalar field \(\omega= \nabla^\perp \cdot u\) where \(\nabla^\perp = (-\partial_2,\partial_1)\) and in three dimensions with a vector field \(\omega= \nabla \times u\). These are transported by \(u\): \[\begin{align} {2}\tag{5} d=2: \qquad \partial_t\omega + u\cdot \nabla \omega &=0, \\ \tag{6} d=3: \;\; \qquad \partial_t\omega + [u,\omega] &= 0. \end{align}\] Notice that, in three dimensions, the circulation theorem in its primal form 4 contains conservation of helicity \(H[u] = \langle u, \omega\rangle_{L^2(M)}\), at least provided \(\omega_0\cdot \hat{n}|_{\partial M}=0\) (a Lie-propagated condition, since \(u\cdot \hat{n}|_{\partial M}=0\)) if boundary is present.

For all of the above deductions to hold, we require the solution to be sufficiently regular (classical). Let us understand when these may exist. The Cauchy problem for 1 –3 consists of prescribed data at some moment \(u|_{t=0}=u_0\in X\) as an element of an appropriate functional space \(X\). To understand what the space \(X\) should satisfy in order for 1 –3 to be a predictive dynamical system, we formally differentiate 1 : \[\begin{align} (\partial_t + u \cdot \nabla) \nabla u +(\nabla u)^2 &= -\nabla^2 p,\\ -\Delta p &= {\rm tr} (\nabla u)^2, \tag{7}\\ \partial_n p|_{\partial M} &= u\cdot \nabla \hat{n} \cdot u|_{\partial M}, \tag{8} \end{align}\] and observe that, to have a good local theory in the space \(X\), it should be the case that \[\label{estimate} \frac{{\rm d}}{{\rm d}t} \| \nabla u \|_{Y} \lesssim \| \nabla u \|_{Y} ^2,\tag{9}\] where the space \(Y\) is such that if \(u\in X\) then \(\nabla u\in Y\). For 9 to hold, \(Y\) should encode enough regularity, namely \(Y\) should be

For (1), Hölder spaces \(X= C^{k+1,\alpha}\), \(Y=C^{k,\alpha}\) with \(k\geq 0\), \(\alpha\in (0,1)\) or Sobolev spaces \(X=H^{s+1}\), \(Y= H^s\) with \(s>d/2\) work. For (2), one should avoid endpoints spaces like \(X= C^k\) or \(W^{k,p}\) with \(p=1\) and \(\infty\). As for (3), the transport equation is generally well-posed at this level of regularity, which encodes stronger-than-Lipschitz information on \(u\). With such a space, one can obtain the a priori estimate \[\|\nabla u(t)\|_Y \leq e^{C\int_0^t \|\nabla u(s)\|_{L^\infty} {\rm d}s }\|\nabla u_0\|_Y.\] If \(T_*>0\) is the maximal existence time, either \(T_*=\infty\) or \(T_*<\infty\) and \(\int_0^{T_*} \|\nabla u(s)\|_{L^\infty} {\rm d}s=\infty\). The classical result, due to Lichtenstein and Gunther near the turn of the century, is

Such solutions are called classical, since each term in the equation makes sense pointwise on their domain of definition. The key to our understanding of the Cauchy problem for the Euler equations is the fact that vorticity (the antisymmetric part of the velocity gradient), is transported geometrically as a two-form. In two-dimensions, the vorticity can be identified with a scalar field while in three dimensions, it is a vector field. As a consequence, it is long known that in two space dimensions the equations are globally predictive, namely \(T_*=\infty\) [9]. In three dimensions, it remained an open question until, nearly a century later, Elgindi and Elgindi-Ghoul-Masmoudi showed \(T\) may be finite:

In fact, a flurry of recent progress indicates that even infinitely smooth initial conditions become singular, at least in the case where there is a solid boundary [12], see discussion in [13]. Thus, turbulence – which we will argue lies outside the class of functions which result in a local theory – cannot be avoided with certainty, even if one starts from the non-turbulent regime. Of arguably more relevance is the fact that turbulence in real world scenarios should not be modeled as arising from regular initial conditions – rather it often results from initial conditions which are already much too rough for a predictive Cauchy problem to exist. We shall discuss this point in the subsequent section.

Navier-Stokes and the inviscid limit

“…It is amazing that it is these elementary interactions that cause the most complicated effects, including turbulence (viscosity of course plays an essential role in generating it)." – V. I. Yudovich [2]

Real fluids that are encountered in nature are not ideal. R.P. Feynman referred to ideal fluid motion as the flow of dry water, highlighting its somewhat absurd character. Real fluids, made of an extremely large but finite collection of molecules, have intrinsic stickiness arising from friction in the mutual interaction of their parts, making them wet. The material-specific, dimensional parameter that measures the strength of said effect is called viscosity. The simplest model for wet water is the Navier-Stokes equations: \[\begin{align} \tag{10} \partial_t u^\nu + u^\nu\cdot\nabla u^\nu&= -\nabla p^\nu+\nu \Delta u^\nu + f,\\\tag{11} \nabla \cdot u^\nu &=0, \end{align}\] where \(\nu>0\) represents the kinematic viscosity of the medium, and \(f\) an external force. Dry water corresponds to a material with \(\nu=0\) exactly, but no such exists. However, it turns out the many flows that we encounter in Nature are almost dry! To understand this point, note that the force of this new viscous effect depends on the state of motion. Indeed, if – for example – we non-dimensionalize the system using a characteristic velocity \(\mathsf{U}=\|u_0\|_{L^2}\), force \(\mathsf{F} := \|f\|_{L^2}\), length \(\mathsf{L}={\rm diam}(M)\) and time \(\mathsf{T}=\mathsf{L}/\mathsf{U}\), then the resulting non-dimensional equations now feature the Reynolds number \(\mathsf{Re}= \mathsf{U}\mathsf{L}/\nu\) and (modified) Grashof number \(\mathsf{Gr} :=\mathsf{F}/(\mathsf{U}^2/\mathsf{L})\)², two dimensionless parameters of the system \[\begin{align} \tag{12} \partial_t u^\mathsf{Re}+ u^\mathsf{Re}\cdot\nabla u^\mathsf{Re}&= -\nabla p^\mathsf{Re}+\tfrac{1}{\mathsf{Re}} \Delta u^\mathsf{Re}+ \mathsf{Gr} f ,\\\tag{13} \nabla \cdot u^\mathsf{Re}&=0. \end{align}\] The principle of hydrodynamic similarity is that flows with the same Reynolds and Grashof number behave the same way. These numbers may be varied independently, although we will consider only \(\mathsf{Gr}\equiv 1\) from now on (which is why we do not decorate the velocity as \(u^{\mathsf{Re},\mathsf{Gr}}\)). That is, it is the dimensionless combination, \(\mathsf{Re}\), rather than any individual feature that dictate the observed phenomena. Moreover, \(\mathsf{Re}\) can be varied by considering setups where various combinations of \(\mathsf{U},\mathsf{L}\) and \(\nu\) are altered. Most often, the scales \(\mathsf{U}\) and \(\mathsf{L}\) can greatly vary, and are responsible for producing very large Reynolds numbers in Nature and in the laboratory. For example, Figure 1 shows flow of water past a fixed cylinder – \(\mathsf{Re}\) is increasing because the inflow velocity is being systematically increased.

We notice a few points from the diagrams; as the Reynolds number is increased, the flow appears to simultaneously become rougher (including smaller and smaller scale motions), and also spottier (what can and will be termed intermittent). Reynolds numbers encountered in nature may be many orders of magnitude larger than those depicted. For instance, in the wake of a commercial airline, typical Reynolds numbers are around \(\mathsf{Re}\approx 10^{8}\), around \(10^{12}\) in a tropical storm, and much much larger in astrophysics. A clearly relevant abstraction is to continue the procedure indefinitely to probe the limit \(\mathsf{Re}\to \infty\), which we shall conflate with the inviscid limit \(\nu\to 0\) without much danger of confusing the reader. What will likely cause confusion is that it is this limit what we shall use to operationally define fully developed (as opposed to transitional) turbulence for the sake of our discussion. Appropriate caveats here should be made, since the limit \(\mathsf{Re}\to \infty\) may, in some situations, not look turbulent at all. For example, if Navier-Stokes is posed in periodic three-space and initialized with some smooth initial conditions, then for times within the period of local existence for Euler given by Theorem 1, Navier-Stokes converges to the classical Euler solution as \(\nu\to 0\) which we have already declared to be non-turbulent. The caveat is then that we are not in this good situation – namely, either there are physical boundaries that trigger singular behaviors, or that the solution is becoming non-smooth in some other way, either from being forced or seeded with rough objects or by their emergence in finite time from smooth ones (Theorem 2).

Formally, the limiting \(\mathsf{Re}\to \infty\) velocity \(u^\mathsf{Re}\to u^\infty\) solves the "Euler equation" \[\begin{align} \tag{14} ``\partial_t u^\infty + u^ \infty\cdot\nabla u^ \infty&= -\nabla p^ \infty" ,\\\tag{15} ``\nabla \cdot u^\infty &=0". \end{align}\] Thus, the motion of dry water is relevant, even universal, as a description! But in what sense? Clearly from Figure 1, this limit is plagued by singularities. In particular, \(u^\infty\) is expected to be well below continuously differentiable, so the Euler equation cannot be satisfied in a classical sense.

Before continuing, we issue a small caveat regarding our treatment of the Navier-Stokes fluid. Here-forth, we will assume that Navier-Stokes solutions do not suffer from any finite time singularities and are smooth enough to justify any manipulation performed. This is not essential for four reasons. First, all the mathematical analysis that we perform can be slightly modified to hold for Leray-Hopf weak solutions of Navier-Stokes, which always exist starting from any finite energy data [14]. I will not do this in these notes because it unnecessarily complicates the discussion, and also for the next three reasons. Second, it appears that Navier-Stokes singularities, if they exist, are non-generic and do not play a relevant role in the subject. Thirdly, the specific mechanism of dissipation does not, apparently, qualitatively alter the turbulent features that are our focus. Thus, even if the Navier-Stokes model develops singularities, models with stronger – and possibly more fundamental – dissipative operators will not. Finally, it has been argued that a more accurate description of non-ideal fluids is a finite dimensional stochastic dynamical system, and there is no such issue in these models, which arise only in the continuum idealization [15], [20]. Luckily, we are in good company in doing this as, according to Y. Sinai [21] "Kolmogorov was never seriously interested in the problem of existence and uniqueness of solutions of the Navier-Stokes system".

We now return to the issue of in what sense is the Euler equation satisfied? Apriori, the only known bound (which holds with or without boundaries) that holds uniformly in the viscosity, is the one from the energy – the \(L^\infty_tL^2_x\) norm of the velocity. Such weak control allows to deduce only weak convergence as \(\nu \to 0\). Due to the quadratic nonlinearity, these limits are only known to be solutions of Euler in a very weak sense, either measure valued [22], [23] or their alternatives such as Lion’s dissipative solutions [24]. However, in practice, a great deal more regularity than \(L^2\) is retained uniformly in the viscosity. But this is based on observations of the system, from either physical or numerical experiment. For example, by numerical experiments of so-called homogeneous isotropic turbulence on the periodic box. See the left panel of Figure 2. Given such a simulation of a Navier-Stokes solution \(u^\nu\), one can compute a measure of its fractional regularity via its (absolute) \(p\)th–order structure functions: \[\label{SP} S_p^{u^\nu}(\ell) := \fint_0^T \fint_M\fint_{\mathbb{S}^{d-1}} |u^\nu(x+\ell\hat{z},t) -u^\nu (x,t)|^p \;{\rm d}\sigma(\hat{z}) {\rm d}x {\rm d}t,\tag{16}\]

where \(\ell>0\) and \(\sigma\) is the standard surface measure on the sphere. These objects, in the turbulent regime, will be the primary subject of our investigations in these notes.

The following behavior of the \(p\)th–order structure functions is robustly observed: there exists a “dissipative scale" \(\ell_\nu/L\sim \nu^\gamma\) for some \(\gamma>0\) and numbers \(\zeta_p\in(0, p]\) such that \[\label{Spbeh} S_p^{u^\nu}(\ell) \sim \begin{cases} \nu^{-\gamma(p-\zeta_p)} \left(\tfrac{\ell}{L}\right)^p &\text{for} \;\;0 \leq \ell \ll \ell_\nu \\ \left(\tfrac{\ell}{L}\right)^{\zeta_p} & \text{for} \; \ell_\nu \ll \ell \ll L \end{cases},\qquad{(1)}\] where the implicit constants and \(\zeta_p\) are independent of viscosity. In principle, all constants may depend on the time window of observation \(T\) although in the "statistical steady state" this is not observed to be the case. The range of scales \(\ell\in [0,\ell_\nu)\) is termed the dissipative range, where diffusion becomes dominant and the vector field looks smooth. The subrange of \((\ell_\nu, L)\) where the scaling \(S_p^{u^\nu}(\ell) \sim \left({\ell}/{L}\right)^{\zeta_p}\) holds is called the inertial range, which is dominated by Euler-like behavior. See the right panel of Figure 2 for the behavior at fixed viscosity of moments \(p=2,4,6\), and Figure 3 for the behavior as viscosity decreases for \(p=2\). It is a primary goal of many interested in the”turbulence problem" to compute, based on a principled theory, what the spectrum of numbers \(\{\zeta_p\}_{p\in \mathbb{N}}\) are in a "generic turbulent setting". We shall comment more later about what can be said rigorously about their behavior. For now, we note that from the observed behavior ?? we may infer a uniform bound of \[\label{Spbnd} S_p^{u^\nu}(\ell) \lesssim \left(\tfrac{\ell}{L}\right)^{\zeta_p}\tag{17}\] for some \(\zeta_p>0\), with the implicit constant being uniform in viscosity. This behavior holds at least for moderately low moments \(p\), say \(p\) less than twelve as, in practice, moments higher than ten are difficult to stably compute. Together with a bound on \(L_t^pL^p_x\) norms, 17 gives a measure of compactness in \(L^p\). Indeed, such a vector field could be said to have \(\sigma_p:=\frac{\zeta_p}{p}\) derivatives in \(L^p\), and thus it sits in the “time-averaged" Besov space \(L^p(0,T; B_{p,\infty}^{\sigma_p}(M))\). The following equivalence is shown in [26]:

We remark that, this definition of the Besov space differs slightly from the conventional one in that the sup and time integral are ordinarily reversed. This space is slightly weaker, but in fact is more natural to measure and, as we will see from the example of the Burgers equation and Kraichnan model, also theoretically. We also remark that this result holds true locally for a bounded domain \(M\subset \mathbb{R}^d\), provided the increment does not leave the domain. We now sketch the proof of this fact, which is elementary.

We remark further that, for Navier-Stokes solutions, there is correspondence (at order \(p=2\)) between the absolute structure function \(S_p^{u^\nu}(\ell)\) and the longitudinal structure function, which is an object that is far more frequently computed in experiment: \[\label{longsp} S_{p,\|}^{u^\nu}(\ell) := \fint_0^T \fint_M\fint_{\mathbb{S}^{d-1}} \Big(\hat{z}\cdot(u^\nu(x+\ell\hat{z},t) -u^\nu (x,t))\Big)^p \;{\rm d}\sigma(\hat{z}) {\rm d}x {\rm d}t.\tag{19}\]

Specifically, it is proved that \(S_2^{u^\nu}(\ell)\lesssim \ell^{\zeta_2}\) if and only if \(S_{2,\|}^{u^\nu}(\ell)\lesssim \ell^{\zeta_2}\) [25]. This correspondence does not hold for a general vector field in \(u\in L^2_t B_{2,\infty}^{\zeta_2/2}\) and crucially makes use of a structural property of the fluid nonlinearity \(u\cdot \nabla u\) which holds for weak solutions. As such, all available measurements from experiment and numerical simulation (typically for longitudinal structure functions) of which the author is aware point to a uniform–in–Reynolds modulus of continuity in \(L^2\), as in Figure 3. This, in turn, provides \(L^2\) compactness, as we now describe.

Indeed, by the Fréchet–Kolmogorov compactness theorem together with the Aubin-Lions-Simon lemma (see [27] for a precise application), this bound suffices to ensure the existence of strong-\(L_{x,t}^p\) sub-sequential limits \(u^{\nu_n}\to u^\infty\). This was observed in a number of works [15], [28]–[31] and is summarized by

So, keeping in mind the observations, it is in this weak, distributional, sense ?? –?? that the ideal Euler equations emerge as the description of the infinite Reynolds number, zero viscosity, limit of non-ideal Navier-Stokes flows. Of course, such objects maybe very irregular in space – the compactness theorem tells that the limit object \(u^\infty\) has just shy of \(\sigma_p=\frac{\zeta_p}{p}\) derivatives in \(L^p\), but need not have more. Indeed, the behavior observed in Fig 3 indicates \(\zeta_2^\infty \approx 0.72\) and not better.

As a brief aside, one might wonder why it is reasonable to make an assumption about space regularity and not time. At first sight, it appears unbalanced. Interestingly, for any weak solution of the Euler equations, space regularity implies a similar type of temporal regularity (see e.g. [32], [33] and also [34]):

Thus, up to changing the integrability exponents, turbulent flows look the same in time as they do in space. Such a statement is important to build up our space-time picture of the structure of turbulent flow.

Returning to our discussion of Euler, the distributional notion of solution builds in two global conserved quantities: momentum ?? and mass ?? (the former holding provided the domain \(M\) has symmetry). However, at this level of (ir)regularity, we cannot run the computations of §[sec:sec:e] on such a weak solution to derive other conserved quantities.

Take the kinetic energy, for instance (let alone circulations, which are far more delicate [35], [36]). The object itself makes sense for weak solutions, a.e. in time (being that they are at least space-time square integrable), but we are not, in general, allowed to insert \(u^\infty\) as a test function into ?? to derive a balance of energy, let alone say that \[\label{onethird} ``(u^\infty\otimes u^\infty; \nabla u^\infty)_{L^2_{t,x}} =0"\tag{20}\] which would clearly be the case for any smooth, solenoidal \(u^\infty\). Given the form 20 , which is cubic in the velocity and has one derivative, one is already tempted to think that having \(\frac{1}{3}\) of a derivative in \(L^3\) should be relevant for conservation or lack-there-of. This we shall discuss in the next section, as well as the broader point that this failure of weak solutions to predict conservation of energy is a great merit of their descriptive power, rather than a weakness.

One standout features of the Navier-Stokes equations is that they dissipate kinetic energy rather than conserve it. The local form of the energy balance is \[\begin{align} \label{energybal} \partial_t \left(\tfrac{1}{2} |u^\nu|^2 \right) + \text{div}\left((\tfrac{1}{2} |u^\nu|^2+p^\nu) u^\nu - \nu \nabla \tfrac{1}{2} |u^\nu|^2\right) &= -\varepsilon^\nu[u^\nu] + u^\nu \cdot f \end{align}\tag{21}\] where we have introduced the local measure of energy dissipation \(\varepsilon^\nu[u^\nu]\) via \[\begin{align} \label{locdissform} \varepsilon^\nu[u^\nu] &:= \nu |\nabla u^\nu|^2. \end{align}\tag{22}\] Consequently, strong solutions enjoy a global kinetic energy balance \[\begin{align} \frac{{\rm d}}{{\rm d}t} \int_M \tfrac{1}{2} |u^\nu(x,t)|^2 {\rm d}x =- \int_M \varepsilon^\nu[u^\nu](x,t) {\rm d}x + \int_M u^\nu(x,t) \cdot f(x,t){\rm d}x. \end{align}\] What is the fate of this balance in the high Reynolds number limit \(\nu \to 0\)? In a non-turbulent scenario where, say, the Euler solution starting from the same initial data is classical, then \(\int_M \varepsilon^\nu[u^\nu](x,t) {\rm d}x\lesssim \nu\), and energy conservation is restored in the limit. However, a primary observed feature of turbulence is the violation of this expectation. Specifically, from numerous physical experiments and some numerical simulations (see e.g. Figure 4), it is observed that the turbulent regime displays anomalous dissipation \[\label{anomalousdiss} \liminf_{\nu \to 0} \fint_0^T \fint_M\varepsilon^\nu[u^\nu](x,t) {\rm d}t{\rm d}x >0.\tag{23}\]

This observation appears so robust and primal to our understanding of turbulence (as we will see), that it has been termed the zeroth law. Nevertheless, we remark here that it may not be that all regimes that could rightly be called "turbulent" behave in the same way, see [37] for a study of statistically stationary turbulence without walls. Its status for real flows where turbulence is generated by boundaries (say, flow past a grid or in the wake of an airplane) seems far more clear [38]–[40]. See [41]. It seems fitting to say that the zeroth law of turbulence is that there are zero laws of turbulence!

For now, let us adopt this observation as an empirical fact, and try to understand more of its root cause. Clearly, it requires that irregularities emerge in the limit \(\nu\to 0\). Specifically, \(\|\nabla u^\nu\|_{L^2_tL^2_x} \sim \tfrac{1}{\sqrt{\nu}}\) is required to sustain anomalous dissipation 23 from the formula 22 . But can we say more about what singularities must develop?

Let us begin by investigating the fate of the local energy balance for weak solutions. The balance law is cubic, so to pass to the limit in the sense of distributions we should have strong compactness in \(L^3_{t,x}\). But this, according to Theorem 4 results from a uniform bound on the third order structure functions \(S_3^{u^\nu}(\ell)\lesssim \ell^{\zeta_3}\). This is, again, readily observed in experiment and simulation. See, for instance, [37], [43]–[45]. Thus, under this hypothesis, \(u^\nu\to u\) strongly in \(L^3_{t,x}\) and we may pass to the limit on the left hand side of the energy balance 21 : \[\begin{align} \nonumber \partial_t \left(\tfrac{1}{2} |u^\nu|^2 \right) + \text{div}\left((\tfrac{1}{2} |u^\nu|^2+p^\nu) u^\nu - \nu \nabla \tfrac{1}{2} |u^\nu|^2\right) \quad \stackrel{\nu \to 0}{\longrightarrow} \quad \partial_t \left(\tfrac{1}{2} |u|^2 \right) + \text{div}\left((\tfrac{1}{2} |u|^2+p) u \right) \end{align}\] where the limit is interpreted in the sense of distributions, e.g. holding upon integration against a smooth test function. As such, from the energy balance 21 at finite viscosity, the right hand side must also converge in the sense of distributions \[\label{loclimdis} \varepsilon^\nu[u^\nu] \;\;\stackrel{\nu \to 0}{\longrightarrow} \;\;\varepsilon[u] =:\partial_t \left(\tfrac{1}{2} |u|^2 \right) + \text{div}\left((\tfrac{1}{2} |u|^2+p) u \right).\tag{24}\] Note that the distribution \(\varepsilon[u]\) inherits positivity from the approximation by \(\varepsilon^\nu[u^\nu]\) and thus is, in fact, a non-negative Radon measure. The phenomenon of anomalous dissipation can now be phrased directly on the inviscid weak solution \(u\) as \[\label{andisse} (\varepsilon[u] , \varphi)_{L^2_{t,x}} > 0 \qquad \text{for some smooth} \quad \varphi\geq 0.\tag{25}\] Such weak solutions are termed locally dissipative. For the reasons discussed above, it is reasonable to believe that such dissipative weak solutions \(u\) of the Euler equations ?? –?? satisfying 25 provide descriptions for turbulence in its Platonic form.

Onsager’s ideal turbulence

“My tentative limiting formula for the correlation function in isotropic turbulence is not so obvious that any one student could be expected to find it. However, it seemed very probable to me that somebody would have investigated the line of reasoning, which is not far fetched.”
L. Onsager in letter to C.C. Lin [46]

The result of the last section is that, under an observed uniform regularity assumption, the high Reynolds number limit of a Navier-Stokes solution is a locally dissipative weak solution of the Euler equation. Anomalous dissipation is the phenomenon that this local dissipation measure is non-trivial. Moreover, we see that it can be computed in two ways. First, it can be computed as an inviscid limit of the viscous local dissipation measure 22 . But, evidently, it can also be computed on the limiting weak solution itself, see 24 . As such, we now seek another way to compute the dissipation on the inviscid weak solution \(u\) which illuminates its ultimate cause. Recall from our earlier considerations around 20 that we expect having exactly \(\frac{1}{3}\) of a derivative in \(L^3_{t,x}\) to play a special role. The person to envisage irregular weak Euler solutions as describing turbulent dynamics, and to realize this precise regularity threshold was the Nobel Laureate Lars Onsager, see left panel of Figure 5. He wrote a now famous statement to this effect in his 1949 paper on Statistical Hydrodynamics [47]. There he stated that, in order for anomalous dissipation to take place, the fluid cannot satisfy a Hölderian condition of the form \[|u(x+r) - u(x)|\lesssim |r|^h, \;\; \text{for} \;\;h>1/3\] everywhere in the domain. In fact, Onsager knew much more than what he wrote, having effectively proved a strengthening of this statement in his unpublished notes, see right panel of Figure 5. See [46] for an excellent discussion of this history, before which this fact was apparently unknown. The first rigorous result in the literature towards Onsager’s statement was Eyink’s work [48], followed by the work of Constantin, E and Titi [49]. Duchon and Robert [50] framed the results in the context of the inviscid limit, as we have done here, and proved a stronger local statement. Their approach is the most similar to Onsager’s own unpublished work on the matter.

Following Onsager, we first obtain a point-split energy balance (working on \(M=\mathbb{T}^d\)):

An immediate consequence of this result is the following integrated balance for \(\nu,f=0\): \[\nonumber \frac{{\rm d}}{{\rm d}t} \int_{\mathbb{T}^d} \int_{\mathbb{R}^d} u(x)\cdot u(x+ z) {F}_\ell(|z|){\rm d}z {\rm d}x= - \frac{1}{2} \int_{\mathbb{T}^d} \int_{\mathbb{R}^d} \tfrac{z}{|z|}\cdot \delta_{ z} u(x,t) |\delta_{ z} u(x,t)|^2\, {F}_\ell'(|z|){\rm d}z {\rm d}x.\] where we have taken \(G(z) = F(|z|)\) to be spherically symmetric, and replace his isotropy assumption with integration over the fluid vessel (interpreting his overline as a spatial average, in this case over \(\mathbb{T}^d\)), then one immediately recovers Onsager’s formula for dimension \(d=3\) from his notes (left panel of Figure 5).

An immediate corollary of Proposition 6 captures the approach of Onsager and reproduces the result of Duchon–Robert [50] (see [55] for experimental confirmation):

Thus, we return to Onsager’s assertion about energy conservation. The first published results are due to Eyink [48] and Constantin, E and Titi [49]. The final result is

Theorem 11 proves a strengthened version of Onsager’s original 1949 assertion that the flow cannot possess more that \(\frac{1}{3}\) of a derivative and be consistent with anomalous dissipation. The "negative side" of this conjecture was finally proved by Isett [56] where he exhibited a weak solution of Euler with slightly less than \(\frac{1}{3}\) of a derivative that did not conserve energy (such solutions can, in fact, be dissipative [57]). This followed a remarkable series of works on the subject of constructing wild weak solutions via a technique called convex integration [58]–[61]. See the comprehensive review by Buckmaster and Vicol [62]. The union of these two sides, positive and negative, is termed Onsager’s conjecture. However, an important question about the excluded middle remains open (and therefore, so does "Onsager’s conjecture"):

We are mainly interested in the case without forcing \(f=0\), but it’s reasonable to consider it for \(f\in L^1(0,T;L^2(\mathbb{T}^d))\). Note, being finite energy \(u\in L^\infty_t L_x^2\), one can choose a representative \(u\in C_tL^2_w\) (namely, continuous in time, weakly in \(L^2_x\)). It is simple to see that this object satisfies the integrated energy balance for times \(0\leq s\leq t\leq T\): \[\tfrac{1}{2} \|u(t)\|_{L^2}^2 =\tfrac{1}{2} \|u(s)\|_{L^2}^2 - \int_s^t \int_{\mathbb{T}^d} {\varepsilon}[u(x,\tau)]{\rm d}x {\rm d}\tau + \int_s^t \int_{\mathbb{T}^d} u(x,\tau)\cdot f(x,\tau){\rm d}x {\rm d}\tau\] where \({\varepsilon}[u] =\lim_{\ell\rightarrow 0}{\varepsilon}_\ell[u]\) with \({\varepsilon}_\ell[u]\) defined by ?? . Moreover \(\int_{\mathbb{T}^d} {\varepsilon}[u(x,t)]{\rm d}x\) is \(L^1_t\) since \(u\in L^3_tB_{3,\infty}^{1/3}\)³. This means, that \(\int_s^t \int_{\mathbb{T}^d} {\varepsilon}[u(x,\tau)]{\rm d}x {\rm d}\tau\) is continuous in \(s\) and \(t\), giving zero mass to zero measure time intervals.

Question 12 is particularly important because these spaces, in particular \(u\in L^3_tB_{3,\infty}^{1/3}\), are natural candidates for the regularity of actual turbulent flows. We will discuss this further point in §[sec:K41sec]. For the moment, I remark that there is only one “critical space" (namely a space where the flux \({\varepsilon}_\ell[u]\) is bounded but does not vanish for general vector fields in that class) for which the answer to Question 12 is known. This critical space is \(u\in L^\infty_{t,x}\cap L^1_t BV\), in which De Rosa and Inversi showed weak solutions conserve energy [63] (see also [64]). This result is remarkable since \(L^\infty_{t,x}\cap L^1_t BV_x\subset L^3_t B_{3,\infty}^{1/3}\) sharply. Indeed, if \(\|u(\cdot+ r)- u(\cdot)\|_{L^1}\lesssim |r|\) and \(\|u\|_{L^\infty}<\infty\), then by interpolation \(\|u(\cdot+ r)-u(\cdot)\|_{L^p}\lesssim |r|^{1/p}\) for all \(p\geq 1\). Moreover for compressible models such as Burgers, shocks live in \(u\in L^\infty_{t,x}\cap L^1_t BV_x\) and are dissipative. As for Question 12, I would guess the answer is no, although recent work [63] as well as [37] brings into question some of the intuition behind this guess. This will be discussed around Conjecture 24.

To conclude our discussion, we state one additional identity concerning the viscous dissipation \({\varepsilon}^\nu[u^\nu]\) (it can also apply directly to the Duchon-Robert distribution \({\varepsilon}[u]\)). It was derived in [71] and is a local form of the integrated identity used in [30]. We will then apply it to give quick proofs of some known results.

We don’t prove Proposition 16 here – it is a computation (see [71]). The utility of the identity ?? is that \(\ell\) on the right hand side is an absolutely free parameter which can be optimized in a variety of ways. For example, let us prove the result of [30]:

Meanwhile there is evidence, which should be further investigated, that Corollary 17 is sharp and dissipation vanishes in periodic turbulence [37]. See Figure 7.

As one last application of the identity ?? for energy, we give a quick estimate of the regularity of the energy profile for weak solutions of Euler originally due to Isett [32].

Let us make some remarks. First, the above result gives another proof of Onsager’s conjecture by taking \(\sigma_3>1/3\) and concluding \(e(t)\) must be constant. Second, when \(u\) is in the natural critical class \(u\in L^3_t B^{1/3}_{3,\infty}\), we see that the energy profile is \(W^{1,1}(0,T)\). Being a function of a single variable, it is thus continuous in time. This excludes the possibility that the profile does not have bounded variation on any \(I \subset (0,T)\) – a wild feature that is common among all convex integration constructions, separating them from behavior forced in critical classes.

Finally, one implication of this result, pointed out by Isett, is that the quality of energy dissipation for Euler flows is stable under perturbation, in the class \(u\in L^\infty_t B^{1/3}_{3,\infty}\). Namely, in this class \(\|e'\|_{L^\infty} \lesssim \|u\|_{L^\infty_t B^{1/3}_{3,\infty}}^3\) and \(e'(t)\) varies continuously in \(L^\infty\) as \(u\) varies in the \(L^\infty_t B^\sigma_{3,\infty}\) topology, namely \(\|e'_1 - e'_2\|_{L^\infty} \lesssim \|u_1-u_2\|_{L^\infty_t B^{1/3}_{3,\infty}}\). This means that anomalous dissipation \(e'(t)\leq -{\varepsilon}<0\) is an open condition in the space of weak Euler solutions in the critical class endowed with the strong topology. By contrast, Isett conjectured that for weak solutions with regularity below \(\frac{1}{3}\) of a derivative, or with time integrability below \(q=\infty\), the energy profile should generically fail to be of bounded variation on every time interval [32], [73]. In particular, a dissipative solution could be perturbed in this class to produce a nearby solution whose energy profile fails to be monotonic on every time interval. A version of this conjecture was proved by De Rosa and Tione [74] – namely the set of weak Euler solutions with energy profile not of bounded variation in any open interval is Baire generic in an appropriate complete metric space which is a strict subset of all \(\alpha\)–Hölder Euler solutions with \(\alpha<1/3\).

This conjecture is clearly very interesting and aims to capture a mechanism, or rather a cause, for regularization in turbulence. However, it should be mentioned that any weak Euler solution which arises from a strong vanishing viscosity limit, regardless of its regularity, must have a non-increasing energy profile. Such solutions, of course, should be meager in the space of all weak Euler solutions. Thus, the physical relevance of the stability of anomalous dissipation in any critical space \(u\in L^p_t B^{1/3}_{p,\infty}\) for \(p\in[3,\infty)\) is not completely clear. We will issue Conjecture 24 below which is similar in spirit to the above, but based on the \(\frac{4}{5}\) law.

Kolmogorov’s 1941 theory

"I had heard several times Kolmogorov talking about turbulence and had always been given the impression that these were talks by a pure physicist. One could easily forget that Kolmogorov was a great mathematician. He could discuss concrete equations of state of real gases and liquids, the latest data of experiments, etc. When Kolmogorov was close to eighty I asked him about the history of his discoveries of the scaling laws. He gave me a very astonishing answer by saying that for half a year he studied the results of concrete measurements. In the late Sixties Kolmogorov undertook a trip on board a scientific ship participating in the experiments on oceanic turbulence. Kolmogorov was never seriously interested in the problem of existence and uniqueness of solutions of the Navier-Stokes system. He also considered his theory of turbulence as purely phenomenological and never believed that it would eventually have a mathematical framework."
– Y. Sinai [21]

Kolmogorov, in his celebrated 1941 works, used the balance associated to the quantity ?? , to argue that \({\varepsilon}_\ell^\|[u]\), which represents a flux of energy through scale \(\ell\), should be effectively constant over a long range of scales \(\ell_\nu \ll \ell \ll \ell_I\) called the inertial range. The "dissipative" scale \(\ell_\nu\) is where viscosity eventually removes the energy which cascades down, and the "integral" scale \(\ell_I\) is the scale of the domain, or where the force is applied. In this scale range, Kolmogorov argued the flux would balance the anomalous rate of energy dissipation \(\langle {\varepsilon}_\ell^\|[u^\nu] \rangle \approx \langle {\varepsilon}\rangle\), as in ?? . From ?? , this results in his \(\frac{4}{5}\) law \[\label{k45law} S_{3,\|}^{u^\nu}(\ell)\sim- \tfrac{4}{5} \langle {\varepsilon}\rangle \ell, \qquad \ell_\nu \ll \ell \ll \ell_I.\tag{33}\] Under the assumptions of statistical homogeneity, isotropy and no-intermittency, Kolmogorov used the above "law" to make predictions for any \(p\)th order structure functions \(S_{p,\|}^{u^\nu}(\ell)\) in the inertial range. This is often presented on strictly dimensional analysis grounds, assuming that \(\nu, \langle {\varepsilon}\rangle\) (and not \(\ell_I\)) are the only relevant parameters. Here I present a slightly different "derivation" of the same, which highlights the role of the assumptions concerning the dissipation rate, which we will reexamine in the next section. \[\begin{align} \nonumber S_{p,\|}^{u^\nu}(\ell)&:= \fint_0^T \fint_M\fint_{\mathbb{S}^{d-1}} (\hat{z}\cdot\delta_{\ell \hat{z}}u^\nu )^p \;{\rm d}\sigma(\hat{z}) {\rm d}x {\rm d}t\qquad\qquad\qquad \qquad\! \! \! \! \! \! \! \text{by definition \eqref{longsp}}\\ \nonumber &\approx \fint_0^T \fint_M \left(\fint_{\mathbb{S}^{d-1}} (\hat{z}\cdot\delta_{\ell \hat{z}}u^\nu )^3 \;{\rm d}\sigma(\hat{z}) \right)^{p/3}{\rm d}x {\rm d}t\qquad \qquad\; \text{by isotropy}\\ \nonumber &\approx \left(\tfrac{12}{d(d+2)}\right)^{p/3} \fint_0^T \fint_M \;( {\varepsilon}[u] \ell)^{p/3} {\rm d}x {\rm d}t \quad \;\quad \quad \quad \quad \quad \;\; \;\;\text{by identity \eqref{Dell45}/\eqref{k45law}} \\ \nonumber &\approx C_p\ell^{p/3}\left(\fint_0^T \fint_M{\varepsilon}[u] {\rm d}x {\rm d}t\right)^{p/3} \qquad \qquad \quad \quad \quad \quad \quad\;\! \;\text{by no "intermittency"} \\ &\sim C_p (\langle {\varepsilon}\rangle \ell)^{p/3} \qquad \qquad \qquad\qquad\qquad\qquad\qquad\qquad \qquad\text{}\nonumber \end{align}\] with the approximations holding in the range \(\ell_\nu \ll \ell \ll \ell_I\). That is \(\zeta_p^\| = \frac{p}{3}\) in ?? . This prediction is often referred to as the K41 theory of turbulence: \[\label{k41} S_{p,\|}^{u^\nu}(\ell) \sim C_p (\langle {\varepsilon}\rangle \ell)^{p/3} \qquad \text{for} \qquad \ell_\nu \ll \ell \ll \ell_I.\tag{34}\] As one can see, the various assumptions such as no-intermittency are used to justify commuting averages with non-linearities (powers). Such approximations would hold well if, for example, \(p\approx 3\) or if the dissipation measure \({\varepsilon}[u]\) does not vary too widely from point to point in space time. However, if \({\varepsilon}[u]\) charges some lower dimensional set, then the error made in commuting these operations can be great. We shall return to this point in the next section.

For now, one famous consequence for \(p=2\) – the \(\frac{2}{3}\) law – makes also a prediction of a \(-\frac{5}{3}\) decay of the time averaged \(\langle \cdot \rangle\) Fourier energy spectrum: \[\label{k41spec} S_{2,\|}^{u^\nu}(\ell) \sim \ell^{2/3} \qquad \iff \qquad E(k) := \sum_{|k'|=k}\langle |\hat{u}(k')|^2\rangle \sim k^{-5/3}.\tag{35}\] Incidentally, Kolmogorov was the "somebody" to have arrived first at Onsager’s formula for the correlation function in isotropic turbulence. See Figure 8 for Kolmogorov’s hand drawn prediction over experimental data! Figure 9 shows the strength of this prediction. The equivalence 35 can be proved as follows

An equivalence holds for near-power laws (as in [eklem]) as a consequence of the Wiener-Khinchin theorem [76].

Kolmogorov’s law 33 can, in fact, be made quite rigorous. Following Duchon-Robert [50], this was done by Eyink [53] and Novack [54] who show Kolmogorov’s law holds in a space-time local and deterministic sense, at least asymptotically as \(\ell \to 0\). They prove

The identity ?? is Kolmogorov’s \(\frac{4}{3}\) law, and ?? is the \(\frac{4}{5}\) law if these are interpreted as holding for asymptotically small scales. It should be remarked that they are purely deterministic statements, without need to appeal to any notion of statistical ensembles (commonly thought necessary). Moreover, they are local in space and time statements, illustrating the power of Kolmogorov’s prediction.

We do not prove Proposition 21 here, as it unfortunately involves some cumbersome manipulations very similar to those which give rise to ?? . Instead, we use Proposition 6 to establish a stronger result – the Kolmogorov \(\frac{4}{3}\) law in a finite range of scales \[\begin{align} \nonumber S_{3,\rm mix}^{u^\nu}(\ell) &:=\fint_0^T\fint_M \langle (\hat{z} \cdot \delta_{\ell \hat{z}}u^\nu) |\delta_{\ell \hat{z}}u^\nu|^2 \rangle_{\rm ang} {\rm d}x{\rm d}t \sim - \frac{4}{3} \langle {\varepsilon}\rangle \ell, \qquad \ell_\nu \ll \ell \ll \ell_I. \end{align}\] This identity follows by analyzing the balance ?? . As such, one can understand the result a defining the inertial range of scales to be the one in which \(S_{3,\rm mix}^{u^\nu}(\ell)\) or \(S_{3,\|}^{u^\nu}(\ell)\) exhibit power law scalings. The size of these scales \(\ell_\nu\) and \(\ell_I\) can be mathematically controlled, under very mild assumptions on the velocity field. Indeed, we prove

See Figure 10 for a numerical demonstration of the \(\frac{4}{3}\) and \(\frac{4}{5}\) laws over finite ranges. A stochastic version of this theorem was proved in [79], [80]. We remark that the information on the scales above can be sharpened with further regularity assumptions on the solution. This is done in the proof below.

With Theorem 22 in sight, it is tempting to make the following conjecture

The intuition behind this conjecture is the following. Assume anomalous dissipation occurs \(\langle \varepsilon \rangle >0\). Then, Theorem 22 says that \(\langle S_{3,\|}^{u^\nu}(\ell)\rangle /\ell\) is finite over a finite (inertial) scale range. This range extends to zero length as \(\nu\to 0\), and \(\langle S_{3,\|}^{u^\nu}(\ell)\rangle /\ell\) is non-vanishing as \(\ell \to 0\) . Now, for any vector field in \(L_t^3 B_{3,\infty}^{s}\) with \(s\geq 1/3\), the \(\langle S_{3,\rm mix}^{u^\nu}(\ell)\rangle /\ell\) is bounded uniform in \(\ell\) but if \(s>1/3\) then the ratio clearly vanishes as \(\ell\to 0\). Therefore we should have \(s\leq 1/3\). But "generic" vector fields with \(s<1/3\) should have \(|\langle S_{3,\|}^{u}(\ell)\rangle /\ell \to \infty|\) as \(\ell \to 0\), since \(\langle S_{3,\|}^{u}(\ell)\rangle /\ell\sim (\delta_\ell u)^3/\ell\sim \ell^{3s-1}\). Turbulence should place the fluid velocity in such a generic category, why not? It thus seems that \(L_t^3 B_{3,\infty}^{1/3}\) is the only reasonable space, at least in the Besov scale. See also [25].

Indeed, this conjecture ends up being true for some model problems. For the Burgers equation it is known by works [82]–[84]. We offer a proof here in Theorem 33, following [85]. For the Kraichnan model of scalar turbulence (scalar is transported by a Gaussian random velocity field with Hölder regularity \(\alpha\)), it has been proved in [26], [86] that the scalar field retains \(1-\alpha\) derivatives in \(L^2\), at least after averaging. Both these results follow from the same principle: the flux of energy is limited by the input of energy (initial conditions), meanwhile it takes a coercive or nearly coercive form and this limits the possible (ir)regularity of the solution uniform in viscosity.

Aside from the clearly shaky reasoning behind Conjecture 24, the recent numerical results obtained in [37] call into question some components of the intuition, namely the answer to Question 12 being negative. In particular, [37] gives evidence that the dissipation slowly vanishing in the inviscid limit. In this case, the information in the \(\frac{4}{5}\) law degenerates, and does not shed any particular light on the regularity of the limiting object. In fact, those numerics indicate the \(L^3\) Besov regularity tends towards a value larger that \(\frac{1}{3}\) asymptotically which, via Corollary 17, is consistent with the perceived rate of decay of energy dissipation. See Figure 7. This may be taken as an (admittedly very weak) point in favor of the answer to Question 12 being positive, since why would the turbulent solution land on any particular \(\sigma_3>1/3\)?

Landau’s objection and intermittency

"...the energy associated with large wave-numbers is very unevenly distributed in space. There appear to be isolated regions in which the large wave-numbers are ‘activated’, separated by regions of comparative quiescence." Batchelor and Townsend [87]

Let us return to Kolmogorov’s prediction 34 concerning the scaling laws for \[\label{Sppred} S_p^\| (\ell) \sim C_p \ell^{\zeta_p}, \qquad \ell_\nu \ll \ell \ll \ell_I.\tag{37}\] Kolmogorov predicted \(\zeta_p=p/3\), which agrees with the "rigorous" prediction from the \(\frac{4}{5}\) law (see Theorem 22) at order \(p=3\). How does it hold up? One can see in Figure 11 a plot of the exponents \(\zeta_p\) vs \(p\). Nearby \(p=3\), Kolmogorov’s prediction does remarkably well (in particular, at order \(p=2\) it is only very slightly an underestimate, see also Figure 9). However, for higher moments \(p \gg 3\) the data clearly does not agree with the linear-in-\(p\) prediction. Instead, \(\zeta_p\) appears to be some non-linear convex function of \(p\). Such behavior is termed intermittency since it necessitates non-uniform irregularities in space and time. The flow field is spotty, with some regions being markedly smoother than others, beautifully captured by the quote of Batchelor and Townsend. All this happening, in principle, on fractal sets of lower dimension.

Predicting \(\zeta_p\) has been one of the major goals of physicists working on turbulence [88]–[91]. Constructing solutions which display intermittency [92], [93], as well imposing fundamental constraints the exponents \(\zeta_p\), is a goal of mathematicians. We will focus on the issue of imposing constraints on \(\zeta_p\) here.

Where is the weak point in the argument we sketch for the behavior 37 ? As we have seen from Theorem 22, the application of the \(\frac{4}{5}\) law is quite justified. This theorem also, in part, justifies the constancy of the flux in the inertial range since \(\langle\varepsilon\rangle\) directly appears there. Well then, what is the evidence towards this non-intermittent dissipation postulate? As it turns out, this seems to be the biggest gap in Kolmogorov’s picture since in the typically 3D turbulence situation, it appears badly violated. Lev Landau was apparently the first to recognize this⁵, and pointed it out to Kolmogorov after a seminar in Kazan in 1942 (see discussion in Frisch [78] for a deeper historical analysis). In their text, Landau and Lifshitz [94].

"It might be thought that the possibility exists in principle of obtaining a universal formula, applicable to any turbulent flow, which should give \(S_2(\ell)\) for all distances \(\ell\) that are small compared to \(\ell_I\). In fact, however, there can be no such formula as we see from the following argument. The instantaneous value of \((\hat{z}\cdot \delta_{\ell\hat{z}} u)^2\) might, in principle, be expressed as a universal function of the dissipation \({\varepsilon}[u]\) at the instant considered. When we average these expressions, however, an important part will be played by the manner of variation of \({\varepsilon}[u]\) over times of order of the periods of the large eddies (with size \(\sim \ell_I\)), and this variation is different for different flows. The result of the averaging therefore cannot be universal."
– Landau and Lifshitz, Fluid Mechanics [94]

Landau’s text concerns the temporal variability of the energy dissipation rate, see Figure 12. More generally, the dissipation thought of as a measure on spacetime should be highly non-uniform/intermittent, giving positive mass to lower dimensional sets (think of, e.g. a shock wave). See Figure 13 for a rendering of volumes with extreme energy dissipation from numerical simulation of turbulence in a periodic box. The upshot of charging a lower dimensional set is that the dissipation measure must diverge at some spacetime locations. Let us reexamine Kolmogorov’s argument for 37 , in view of the rigorous Theorem 22. This theorem says that, in the sense of distributions, Kolmogorov’s \(\frac{4}{5}\) law holds. Specifically, it implies the following coarse grained identity should hold in the inertial range \(\ell_\nu \ll \ell \ll \ell_I\): \[\label{mol45} \frac{\overline{( \langle (\hat{z}\cdot\delta_{\ell \hat{z}}u^\nu )^3 \rangle_{\rm ang})}_{\ell}}{\ell}\approx - \frac{12}{d(d+2)} \overline{{\varepsilon}^\nu[u^\nu]}_\ell,\tag{38}\] where, recall \(\overline{f}_\ell = f* G_\ell\) for some space-time mollifier \(G_\ell\). Thus, repeating Kolmogorov’s argument slightly more precisely, we have \[\begin{align} \nonumber S_{p,\|}^{u^\nu}(\ell)&:= \fint_0^T \fint_M \langle (\hat{z}\cdot\delta_{\ell \hat{z}}u^\nu )^p \rangle_{\rm ang} {\rm d}x {\rm d}t\qquad\qquad\qquad \;\;\quad\! \! \! \text{by definition \eqref{longsp}}\\ \nonumber &\approx \fint_0^T \fint_M \langle (\hat{z}\cdot\delta_{\ell \hat{z}}u^\nu )^3 \rangle_{\rm ang}^{p/3} \;{\rm d}x {\rm d}t\qquad \qquad \qquad \;\; \;\text{by isotropy}\\ \nonumber &= \fint_0^T \fint_M\overline{ ((\hat{z}\cdot\delta_{\ell \hat{z}}u^\nu )^3 \rangle_{\rm ang}^{p/3}) }_\ell {\rm d}x {\rm d}t\\ \nonumber &\approx \fint_0^T \fint_M\overline{( \langle (\hat{z}\cdot\delta_{\ell \hat{z}}u^\nu )^3 \rangle_{\rm ang})}_{\ell}^{p/3}{\rm d}x {\rm d}t\\ \nonumber &\approx \fint_0^T \fint_M \;( \tfrac{12}{d(d+2)} \overline{{\varepsilon}[u^\nu]}_\ell \ell)^{p/3} {\rm d}x {\rm d}t \quad \;\quad\quad\quad \quad \;\; \; \text{by identity \eqref{Dell45}/\eqref{k45law}} \end{align}\] In the above, we have introduced a mollification so as to make use of the exact identity 38 in a more precise fashion. As a result, the mollified energy dissipation makes its way into the picture (this is also the basis for Kolmogorov’s own refinement of his theory from 1962 [89], which features ball averaged dissipation). See also Kraichnan [96]. At this point, Kolmgorov’s original argument relates the \(\frac{p}{3}\) moment of the dissipation simply to its mean. Instead, following Landau’s suggestion, a reasonable implication of the observed spottiness of the dissipation would be power-law divergences of higher moments \[\label{landaubw} \fint_0^T \fint_M \; \overline{{\varepsilon}[u]}_\ell ^{p} \;{\rm d}x {\rm d}t \sim \langle \varepsilon \rangle^{p} \left(\frac{\ell}{\ell_I}\right)^{-\alpha_{p}}.\tag{39}\] See, e.g., the discussion in [95], [97], [98]. Here, we must have \(\alpha_1=0\), and based on the above, we might expect \(\alpha_1\geq 0\) for \(p> 1\) and \(\alpha_p\leq 0\) for \(p\in[0,1)\) which relates to the size of the sets that \({\varepsilon}[u]\) charges, and the rate at which \({\varepsilon}[u]\) diverges. Indeed, consider the case of the dissipation being concentrated on a smooth, co-dimension one hypersurface. This essentially reduces to thinking of \({\varepsilon}[u](x,t)= \langle\varepsilon \rangle \delta_{x=0}\) in one dimension, so that \(\overline{{\varepsilon}[u]}_\ell (x,t) = \langle\varepsilon \rangle G_\ell(x)\) as an approximation to the identity on the real line \(\mathbb{R}\) at each time \(t\), where \(G\) is smooth, positive, even and \({\rm supp} \;\!G \subset [-\ell_I, \ell_I]\). Then \[\begin{align} \label{shockbd} \fint_0^T \fint_M\; \overline{{\varepsilon}[u]}_\ell^{p} \;{\rm d}x {\rm d}t &= \langle\varepsilon \rangle^{p} \fint_\mathbb{R} G_\ell^{p}(x) {\rm d}x =\|G\|_{L^{p}}^{p}\langle\varepsilon \rangle^{p} \left(\frac{\ell}{\ell_I}\right)^{-(p-1)}. \end{align}\tag{40}\] So \(\alpha_p=p-1\) in this case. Using 39 inside Kolmogorov’s argument, we arrive at \[\label{k41a} S_{p,\|}^{u^\nu}(\ell) \sim C_p (\langle {\varepsilon}\rangle \ell)^{p/3} \left(\tfrac{\ell}{\ell_I}\right)^{-\alpha_{p/3}} \sim \left(\tfrac{\ell}{\ell_I}\right)^{ \frac{p}{3} - \alpha_{p/3}} \qquad \text{for} \qquad \ell_\nu \ll \ell \ll \ell_I.\tag{41}\] This formula relates two unknown scalings for the structure function 37 , \(\zeta_p= \frac{p}{3} - \alpha_{p/3}\). For the dissipation charging codimension one sets (e.g. shocks) this predicts \(\zeta_p=1\), as \(\alpha_1=0\). This indeed holds, at least for \(p\geq 3\) in models like the Burgers equation (see, e.g. Theorem 33 and discussion herein) and compressible Euler [34], [99], [100].

We remark that the above argument can be made tighter by actually controlling the error made from exchanging mollification with the \(\frac{p}{3}\) power. Potentially, one could show that the error from doing this is the same, or lower, order in \(\ell\) than 41 by bootstrapping this information. On the other hand, I don’t see a clean way to quantify the error incurred in the first step of the argument which appeals to "isotropy".

We remark also that the longitudinal structure function \(S_{p,\|}^{u^\nu}(\ell)\) is not coercive in that it does not imply regularization in the Besov-type space introduced in Lemma 3. Nontrivial scaling exponents (or bounds) on absolute structure functions \(S_{p}^{u^\nu}(\ell)\) 16 are. This can be related to multifractal picture of the turbulence velocity field [101]–[103]. It is the absolute structure functions that will be the subject of the results to follow.

Following [71], [104], to relate dissipation with regularity and (absolute) structure functions, we take a slightly different perspective based on the distributional energy balance which holds for \(L^3\) weak solutions. Specifically, it will be based on the balance law in Proposition 16, which relates the dissipation measure to, effectively, increments of the weak solution.

To exploit this relation, we must quickly introduce a framework which allows us to study a class of lower dimensional measures on \(\mathbb{R}^d\) that allow measurement of fractals (sets with non-integer dimensions). Such measures are called Hausdorff measures, and they will be denoted by \(\mathcal{H}^\gamma\). The idea is then to say that a set \(A \subset \mathbb{R}^d\) has dimension \(\gamma\in [0,d]\) if \(0<\mathcal{H}^\gamma(A)<\infty\), regardless of how complicated the set might be geometrically. To do this, given any \(A\subset \mathbb{R}^N\) and \(\gamma \geq 0\), for any \(\delta >0\) we set \[\mathcal{H}^\gamma_\delta (A) := \inf \left\{ \sum_i r_i^\gamma \, :\, A\subset \bigcup_i B_{r_i}, \, r_i<\delta \text{ for all } i \right\}.\] Namely, it is a sort of \(\gamma\)–dimensional volume of an optimal covering of the set \(A\) by balls of radius no bigger than \(\delta\) but may be not uniform. Then, the \(\gamma\)-dimensional Hausdorff measure is defined as \[\nonumber \mathcal{H}^\gamma(A) := \sup_{\delta > 0} \mathcal{H}^{\gamma}_\delta(A).\] This is a non-negative Borel measure on \(\mathbb{R}^N\) and \(\mathcal{H}^d\) is equivalent to the \(d\)-dimensional Lebesgue measure for all \(d\in \mathbb{N}\), \(d\leq N\). The Hausdorff dimension is obtained as \[\nonumber \dim_{\mathcal{H}}A: = \inf \{ \gamma\geq 0\,:\,\mathcal{H}^\gamma(A) = 0 \}.\] A simpler notion of fractal dimension can be made by demanding that all the balls covering the set \(A\) in the definition \(\mathcal{H}^\gamma_\delta (A)\) should be of fixed radius \(\delta\). This results in the Minkowski or Box counting dimension. In particular, the \(\gamma\)-dimensional upper Minkowski content is \[\overline{\mathcal{M}}^{\gamma}(A) := \limsup_{r \to 0} \frac{\mathcal{H}^N([A]_r)}{r^{N-\gamma}} \qquad \text{with }[A]_r = \left\{ x \in \mathbb{R}^N \colon \text{dist}(x,A) < r\right\}.\] Then, the corresponding upper Minkowski dimension is given by \[\overline{\dim }_{\mathcal{M}} A := \inf \{\gamma \geq 0\,:\,\overline{ \mathcal{M}}^\gamma(A) = 0\}.\] The lower Minkowski content \(\underline{\mathcal{M}}^{\gamma}(A)\) is defined instead by a liminf, and this is used to define the lower Minkowski dimension \(\underline{\dim }_{\mathcal{M}} A\). Of course, there is the following relationship between the dimensions, \[\dim_{\mathcal{H}}A\leq \underline{\dim }_{\mathcal{M}} A\leq \overline{\dim }_{\mathcal{M}} A,\] with Hausdorff being the sharpest notion among them.

Our first result will be to give lower bounds on the smallest dimension of a set the dissipation measure can charge. It is a special case of results from [104].

Remark 29 (Implication for intermittency). Note that if \(\gamma\) is less than \(d + 1\), then for \(p > 3\) this bound implies the structure function exponent \(\zeta_p\) must be strictly below \(\frac{p}{3}\). In particular, non-trivial lower dimensional dissipation necessarily results in a quantitative downward deviation from the Besov \(\frac{1}{3}\) regularity for all \(p>3\). Recall also from Remark 27 that Meneveau and Sreenivasan estimated that the fractal dimension of the dissipation support to be \(\gamma \approx 3.87\). We now give an argument for this value based on Theorem 28. Suppose Theorem 28 is sharp locally⁷ at \(p=3\). Set then \[\zeta_p^* := \tfrac{p}{3} - \tfrac{2(d+1-\gamma)(p-3)p}{ 9p -3(d+1-\gamma) (p-3)},\] i.e. the right hand side in \(\eqref{eq:32intermittenc32viscos}\). In dimension \(d=3\), \[\tfrac{d \zeta_p^*}{dp}\Big|_{p=3}=\tfrac{2\gamma -5}{9}.\] Numerical simulations of incompressible turbulence [44] indicate \(\frac{d \zeta_p}{dp}\Big|_{p=3} = 0.303 \pm 6.4\times 10^{-4}\). This corresponds to \(\gamma = 3.85\), remarkably close to the observations of Meneveau and Sreenivasan. See Figure 14 for an inspection of numerical data and this bound.

Finally, we report one additional result from [71] which limits the rate at which moments of ball-averaged dissipation can diverge.

This bound follows from placing the distribution \({\varepsilon}[u]\) in an appropriate Besov space with negative regularity index, \(B_{p/3,\infty}^{\frac{2\sigma_p}{1-\sigma_p}-1}\), locally in space-time. Let us inspect the implications of this last result to our heuristic Kolmogorov/Landau argument at the start of the section. Conflating \(\frac{{\varepsilon}[u](B_\ell (x,t))}{|B_\ell(x,t)|}\) with \(\overline{{\varepsilon}[u]}_\ell\), we obtain a constraint on 39 , namely \[\nonumber \fint_0^T \fint_M \; \overline{{\varepsilon}[u]}_\ell ^{p} \;{\rm d}x {\rm d}t \gtrsim \langle \varepsilon \rangle^{p} \left(\frac{\ell}{\ell_I}\right)^{-\alpha_{p}} \quad \implies \quad \alpha_p\leq p\left(1 +\frac{3}{p}(d+1)- \frac{2\sigma_p}{1-\sigma_p}\right).\] In principle, such information can be used to further constrain the argument of Kolmogorov and Landau.

In fact, this quality of the dissipation measure was also studied by Meneveau and Sreenivasan. Based on a multifractal model [101], they given evidence for the prediction \[\fint_0^T \fint_M \; {\varepsilon}[u](B_\ell (x,t))^p \;{\rm d}x {\rm d}t \sim \langle \varepsilon \rangle^{p} \left(\frac{\ell}{\ell_I}\right)^{\tau_p}\] where the spectrum of exponents \(\tau_p\) are determined by the formula \(\tau_p = \inf_\alpha [\alpha p - f(\alpha)]\), which requires as input a spectrum of “dimensions" \(f(\alpha)\) \[\nonumber f(\alpha) = \text{codimension on the set of (x,t) such that {\varepsilon}[u](B_\ell (x,t))\sim \ell^\alpha as \ell\to 0}.\]

Let us conclude the section with some discussion of the results. All the results of this section assume the dissipation measure charges some lower dimensional set, and derives implications of this to the structure of the velocity field. Specifically, we show the velocity cannot be too regular for this to happen, and the regularity cannot be uniform in \(p\), at least for \(p>3\). This implies that some form of intermittency must occur. On the other hand, we do not have any way of deducing that the dissipation measure must charge lower dimensional sets from measurements of the velocity field itself. One must instead measure the dissipation field directly, as was done by Meneveau and Sreenivasan. However, the multifractal model picture for Frisch and Parisi [101] (see also lecture notes of Eyink [72] and review of DuBrulle [108]) predicts that the slope of the curve of \(\zeta_p\) at \(p=0\), namely \(h_*:= \frac{{\rm d}\zeta_p}{{\rm d}p}\Big|_{p=0}\) relates to the "most probable" Hölder exponent in the flow. Namely the one occupying a full space-time volume, with other local exponents \(h\neq h_*\) occupying fractal sets of lesser dimension. Since \(\zeta_p\) is a concave function of \(p\), if any value of \(\zeta_p>p/3\) for \(p\in (0,3)\), then it follows that this exponent \(h_*>1/3\). Thus, in light of Onsager’s theorem about \(\frac{1}{3}\) being the requisite singularity for dissipativity, one might guess that \({\varepsilon}[u]\) will charge lower dimensional if \(\zeta_p>p/3\) for \(p\in (0,3)\). We therefore end with the following

Lessons from model problems

"It might be worth while to investigate the properties of certain systems of mathematical equations, which, although much simpler in structure than the equations of hydrodynamics, nevertheless show features which can be considered as the analogues of typical properties of the hydrodynamic equations." J.M. Burgers [109]

In this section, we will give details on two model problems that shed light on a variety of aspects of the theory explained above. In these models, one can make simple, rigorous and unconditional statements. The first model is the Burgers equation, the second is passive scalar transport.

Burgers’ equation for pressureless gas

Consider the viscous Burgers equation [110] on \(x\in \mathbb{T}_L :=[0,L)\) with periodic boundary conditions for \(t\geq 0\) \[\begin{align} \label{Burgers} \partial_t u^\nu + u^\nu \partial_x u^\nu &= \nu \partial_x^2 u^\nu + f,\\ u|_{t=0}&=u_0.\nonumber \end{align}\tag{43}\]

This equation was introduced by Burgers as a simple model displaying a number of the salient features of turbulence, see [111], [112] for excellent surveys. It is also a model of one-dimensional pressureless gas dynamics, and its multidimensional analogue has been used to great effect in cosmology by Zeldovich [113]. In what follows, we aim to see

We tackle the first two points above at once, following the proof of Alinhac [114] (it becomes a statement about singularity formation as soon as \(u_0\) is in a strong enough class, having a local well-posedness theory, e.g. \(C^k\) with \(k\geq 1\)).

From the above, we know any weak solution must develop a discontinuity of some kind. In fact, entropy solutions of Burgers develop shocks which propagate along rectifiable curves in space time. What should we expect, then, about anomalous dissipation and scaling of structure functions? There is a simple exact weak solution of Burgers that can give us an expectation: the Khokhlov sawtooth solution on \(\mathbb{T}_L= [-L,L)\) and \(t>0\) \[\label{ksol} u(x,t) =\begin{cases} \frac{x+L}{t} & -L\leq x\leq 0 \\ \frac{x-L}{t} &\;\;0\leq x\leq L \end{cases}.\tag{44}\] This solution has one downwards jump \(\Delta u = u^--u^+= \frac{2L}{t}\), where \(u^\mp\) are the left/right traces of \(u\) at zero. One immediately sees this solution is dissipative; for \(t_0<t\) one has \[\frac{{\rm d}}{{\rm d}t} \frac{1}{2} \|u(t)\|_{L^2}^2 = -\frac{1}{3} \frac{L^2}{t^3} = - \frac{1}{12 } \frac{(\Delta u)^3}{L}<0.\] Indeed, the inviscid limit of the dissipation measure can be studied for this data (see, e.g. [116]) and it converges to a Dirac mass at the shock. \[\label{dissrelburg} {\varepsilon}^\nu[u^\nu] := \nu |\partial_x u^\nu|^2 \to \frac{1}{12 } \frac{(\Delta u)^3}{L}\delta_0, \qquad \langle {\varepsilon}\rangle := \langle {\varepsilon}^\nu[u^\nu] \rangle = \frac{1}{12 } \frac{(\Delta u)^3}{L}.\tag{45}\] Structure functions can also easily be computed \[\begin{align} \nonumber S_p(\ell) &:=\frac{1}{2L} \int_{-L}^L |u(x+\ell)-u(x)|^p {\rm d}x\\ \nonumber &= (1-\frac{\ell}{2L}) \left(\frac{\ell}{t}\right)^p + \frac{\ell}{2L} \left(\frac{2L+\ell}{t}\right)^p \sim_p (\Delta u)^p \begin{cases} \left(\frac{\ell}{2L}\right)^p & 0<p<1\\ \frac{\ell}{2L} & \;\phantom{0<}p>1 \end{cases}. \end{align}\] Using the relationship with total dissipation 45 , we find for \(p>1\) that \[S_p(\ell)\sim (\ell \langle {\varepsilon}\rangle)^{p/3} \left(\frac{\ell}{L} \right)^{-(\frac{p}{3}-1)}.\] This fully agrees with our heuristic prediction following Kolmogorov and Landau 40 –41 . In contrast with \(p\geq 1\), the behavior for \(p\in (0,1)\) does not fall into this prediction and in fact may not be fully universal in decaying Burgulence [117], [118]. The significance of \(p=1\) here is that it is the dimension of space, which limits the dimension of the set of shock points. For instance, for bounded data, the entropy dissipation is a (space-time) measure concentrated on countably many Lipschitz curves [119].

Let us prove some rigorous and general results in this direction, tackling the second two points at once. First we remark about two important apriori estimates for Burgers solutions. The most basic estimates for the solution \(u^\nu\) are derived from energy balance \[\label{lygaidwr} \nu \int_0^T \int_{\mathbb{T}} | \partial_x u^\nu |^2{\rm d}x{\rm d}t = \tfrac{1}{2} \|u_0\|_{L^2}^2- \tfrac{1}{2} \|u^\nu(\cdot,T)\|_{L^2}^2 + \int_0^T \int_{\mathbb{T}}u^\nu f {\rm d}x {\rm d}t,\tag{46}\] which confers \(L^2(0,T;H^1(\mathbb{T}))\) regularity to the solution, but not uniformly in the viscosity. The strongest uniform estimate at first sight is \(L^\infty(0,T;L^\infty(\mathbb{T}))\), following from the maximum principle \[\label{maximumprinciple} \sup_{t\in [0,T]}\|u^\nu(t)\|_{L^\infty(\mathbb{T})} \leq \|u_0\|_{L^\infty(\mathbb{T})} + T\|f\|_{L^\infty(\mathbb{T})}.\tag{47}\] In fact, these two bounds are essentially enough to prove global regularity of the viscous Burgers equation. Starting from any bounded data \(u_0\in L^\infty\), the solution immediately becomes real analytic, is unique, and exists for all times. So from now on, any manipulation with the viscous Burgers solution will be justified.

Our main result is a uniform fractional regularity of \(u^\nu\) related to the \(p\)th–order structure functions \[\nonumber S_p^\nu(\ell,T) :=\frac{1}{T} \int_0^T \int_{\mathbb{T}} |u^\nu(x+\ell,t) - u^\nu(x,t)|^p {\rm d}x {\rm d}t.\] As discussed at length in §[sec:nssec], for turbulent flows these objects develop scaling ranges \[\nonumber S_p^\nu(\ell,T) \sim |\ell|^{\zeta_p} \qquad \text{for} \quad \ell_\nu \ll \ell \ll L.\] Our aim is to establish uniform upper bounds \(S_p^\nu(\ell,T) \leq C_p|\ell|^{\zeta_p}\) for all \(|\ell|>0\).

Some remarks are in order. First, the estimate 49 clearly shows anomalous dissipation, since the decay of the structure function is uniform in \(\nu\). Indeed, assuming without loss of generality that the mean is trivial \(\mathsf{m}:=\fint_{\mathbb{T}} u^\nu(y,t){\rm d}y =0\), then \[\nonumber \|u^\nu(\cdot, t)\|_{L^2}^2 = \left\|u^\nu(\cdot, t)-\fint_{\mathbb{T}} u^\nu(y,t){\rm d}y \right\|_{L^2}^2 \leq \int_{\mathbb{T}^2} |u(x,t) - u(y,t)|^2 {\rm d}x {\rm d}y \lesssim \sup_{|r|\leq L} S_2^\nu(r,t).\] The rate of decay in time may not be sharp, as can be seen by inspection of the exact solution 44 . We remark that the origin of anomalous dissipation in this model can be interpreted as non-uniqueness of particle trajectories backwards in time [116] (more on this perspective at the end of the next subsection). Secondly, condition 48 is equivalent to \(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_0^T\|u^\nu(t)\|_{\dot{B}^{1/p}_{p,\infty}(\mathbb{T})}^p {\rm d}t \leq C_p,\) with the Besov semi-norm introduced in Lemma 3. As such, we have a self-regularization result for viscous solutions – from bounded data, they immediately enter into the space \(X=\bigcap_{p\geq 3} L^\infty(0,T; B^{1/p}_{p,\infty}(\mathbb{T}))\) uniformly in the viscosity. Entropic shocks solutions saturate this regularity in that they live in \(X\) and no better space within the Besov scale. Combining Theorems 33 and 32, a cartoon of the evolution of the inviscid limit emerges in the form of Figure 15.

The regularization result is well known [82]–[84]; in fact there is a stronger uniform estimate for the total variation \(L^\infty(0,T;(L^\infty\cap BV)(\mathbb{T}))\subset L^\infty(0,T;B_{p,\infty}^{1/p}(\mathbb{T}))\) for all \(p\geq 1\). Our primary interest is in giving a short, intrinsically physical space, argument based on the following principle: nonlinear ideal conserved quantities that may be anomalously dissipated limit the degree to which the solution may suffer irregularities. A bound of this type was established by Goldman, Josien and Otto [85] using a modified energy flux. For Burgers, there are infinitely many such quantities (any convex function of the solution), suggesting there should be infinitely many such bounds 48 , which we here show. These estimates show that \(p\)th-order absolute structure functions obey a uniform bound with \(\zeta_p=1\) for \(p\geq 3\). This fact underlies the rigidity of the so-called multifractal spectrum of anomalous exponents in Burgulence. For Navier-Stokes, the only known inviscid invariant that is dissipated is the kinetic energy. Its flux, related to the dissipation by the Kolmogorov \(\frac{4}{5}\)–law [120], is not coercive unlike the present setting. With an additional alignment hypothesis, the law does confer limited regularity [25]. Whether this is a generic feature of turbulence is open.

Finally, we remark that much of the analysis here could carry over to general one-dimensional conservation laws, even those with non-degenerate, nonlinear viscosity \[\nonumber \partial_t u^{{\varepsilon}} + \partial_x h(u^{\varepsilon}) = {\varepsilon}\partial_x (\nu(u^{\varepsilon}) \partial_x u^{\varepsilon}),\] provided, at least, that the Hamiltonian function \(h(u)\) is sufficiently close to quadratic. For simplicity, we choose to focus here on the viscous Burgers equation. We require the identity (similar to that appearing in [121]):

Obukhov, Corrsin, and Kraichnan’s scalar turbulence

Next we discuss passive advection and diffusion of scalar fields. Let \(M\subset \mathbb{R}^d\) be open. Given a divergence-free vector field \(u:(0,T)\times M\rightarrow \mathbb{R}^d\), consider the advection-diffusion equation governing the motion of a tracer dye or temperature \[\label{T} \partial_t \theta^\kappa +\text{div}( u \theta^\kappa ) =\kappa \Delta \theta^\kappa.\tag{51}\] Just as for the velocity field modeled by the incompressible Navier-Stokes equations, the tracer energy is decaying. The local form of the balance is \[\partial_t \left(\tfrac{1}{2}|{\theta}^\kappa|^2\right) +\text{div}\left(\tfrac{1}{2}|{\theta}^\kappa|^2u -\kappa \nabla \tfrac{1}{2}|{\theta}^\kappa|^2 \right)=-\chi^\kappa[\theta^\kappa]\] where we have introduced the local measure of scalar dissipation \(\chi^\kappa[\theta^\kappa]\) via \[\begin{align} \label{chilocdissform} \chi^\kappa[\theta^\kappa]&:= \kappa |\nabla \theta^\kappa|^2. \end{align}\tag{52}\] See Figure 17 for some snapshots of scalar motion in Navier-Stokes turbulence and the Kraichnan model. Just as in hydrodynamic turbulence, it is observed that, provided \(u\) is rough enough, anomalous diffusion occurs [122], [123]: \[\label{anomalousdiff} \liminf_{\kappa \to 0} \fint_0^T \fint_M\chi^\kappa[\theta^\kappa](x,t) {\rm d}t{\rm d}x >0.\tag{53}\]

This balance holds directly for non-diffusive weak solutions \(\theta\) upon replacing \(\chi^\kappa[\theta^\kappa]\) with \(\chi[\theta]\), as soon as \(u\in L^p_{x,t}\), \(\theta\in L^q_{x,t}\) with \(\frac{1}{p}+\frac{2}{q}\leq 1\). Note that, being a linear equation, weak solutions can be obtained from weak compactness of vanishing diffusivity approximations. Provided that the limit is achieved strongly, it follows that \(\chi[\theta]\) is actually a non-negative Radon measure.

Obukhov [124], followed by Corrsin [125], developed a theory for passive scalar turbulence.⁸ Obukhov’s motivation was to understand the entropy (and therefore temperature) cascade in the incompressible limit. Both derived Onsager-type predictions on the requisite degree of singularity required to see anomalous diffusion in this context. The result is, roughly, that if \(\sigma\) represents the fractional regularity of the velocity, the scalar cannot have regularity \(\beta\) greater than \(\frac{1-\sigma}{2}\). This was made rigorous by Eyink [132] and works of Constantin and Procaccia [133], [134]. See the discussion in [105]. Recently, there have been mathematical constructions of passive scalars exhibiting anomalous diffusion [105], [135]–[137], even some that nearly exhibit the sharpness of the Obukhov–Corrsin theory in Hölder spaces [138], [139]. As an example, one has

Observations indicate, however, that typically the scalar field will be highly intermittent (even more so than the fluid velocity). This motivates obtaining constraints on intermittency of the scalar field, much the same way we did for the fluid velocity. In fact, all the analysis we preformed for the Euler equations can be carried over to the setting of passive scalars and gives precise constraints on the relation between intermittency and dissipation measure. The main implications of our framework in this situation are

The proof of this result is given in [71]. It follows from a similar formula for the scalar anomaly as in Proposition 16.

This result immediately recovers the Obukhov–Corrsin and Eyink bounds, which say that non-trivial dissipation requires \(\beta_s\leq \frac{1-\sigma_p}{2}\) for all \(p\in [1,\infty]\). If more is known about the dissipation measure, the above result gives a substantial refinement. Numerical work on scalar advection by three dimensional turbulent velocity fields [140] suggests that exponents saturate \(\zeta_q^\theta:=q\beta_q \to 1.2\) as \(q\rightarrow \infty\), approximately, thus quite far from being monofractal. See Figure 19. We believe that constructions in the spirit of [105], [135]–[139] could be made to show the sharpness of this intermittent Obukhov–Corrsin theory. See, in particular, the recent work [141].

An approach which is alternate (and perhaps opposite) to making specific constructions is to study the behavior of the scalar in a random field. Robert Kraichnan forwarded this idea in his celebrated work [143], see Figure 20. Specifically, consider the stochastic advection by a Gaussian random field \(u:=u(t,x): \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}^d\), \(d \geq 2\), which is spatially colored and white-in-time correlated:\[\label{eq:Kraichnan} \partial_t \theta_t^\kappa + u \circ \nabla \theta_t^\kappa = \kappa \Delta \theta_t^\kappa,\tag{54}\] where the symbol \(\circ\) denotes that we are interpreting the stochastic integral in the Stratonovich sense. See [144]. The velocity field, being Gaussian, is completely prescribed by its mean (taken zero for simplicity) and covariance \[\begin{align} \mathbb{E}[ u(t,x) \otimes u(t',x') ] = (t \wedge t') C(x,x') , \quad C : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^{d \times d}. \end{align}\] To simplify the situation, one can consider the velocity field to be spatially homogeneous, i.e. \(C(x,x')= C(x-x')\), and non-degenerate, i.e. \(C(0)\) is a positive definite matrix. We also assume \(u\) is incompressible, although this is not really necessary for many of the results we will mention. When \(u\) is spatially smooth the analysis of 54 is classical [145], [146]. However, we would like a model for turbulent transport which requires \(u\) be spatially rough: just shy of \(\alpha\)–Hölder in space for \(\alpha\in (0,1)\) and not better. This amounts to considering a covariance matrix with the following small-distance asymptotic behavior \[\label{covassum} Q(z) := \langle |u(x)-u(x+z)|^2\rangle = 2(C(0) - C(z)) =O( | z |^{2 \alpha}).\tag{55}\]

This model and the phenomena associated to it have been studied in great detail by physicists. I reference to the excellent review [144], and report here only some recent results that make more precise some of these insights. First is the quality of anomalous diffusion, studied by Rowan [147] and after in [26]

Thus, anomalous dissipation is a general phenomenon in this model. In [26] we study also the local energy balance for 54 , showing that the solution \(\theta=\lim_{\kappa\to 0} \theta^\kappa\) (which is unique, for reasons special to the Kraichnan model) admits a local energy balance that is governed by the stochastic PDE \[\partial_t |\theta_t|^2 +u \circ \nabla | \theta_t |^2 = - \chi[\theta_t],\] where \(\chi[\theta]\) is a non-negative distribution. This is analogous to the Duchon-Robert distribution for the Euler equations [50] and describes exactly how fluctuations of \(\theta\) dissipate in space and time. It is shown that \(\chi[\theta]({\rm d}t, {\rm d}x) = \lim_{\kappa\to 0} \kappa |\nabla \theta_t^\kappa(x)|^2 {\rm d}t {\rm d}x,\) where the equality is interpreted in the sense of distributions, and the limit in probability. In view of this identification \(\chi[\theta]\) is, in fact, almost surely a non-negative Radon measure. Assuming that the velocity is divergence-free, we can derive an explicit formula for \(\mathbb{E}\chi[\theta]\).

Formula ?? is a manifestation of Yaglom’s law for scalar turbulence [148], and can be regarded as an analogue of Duchon-Robert’s result for the Kraichnan model. Upon integration, ?? implies (from the energy balance) that \[\begin{align} \label{eq:regularization95besov95intro} {c} \lim_{\ell \rightarrow 0} \int_0^t \oint_{\mathbb{S}^{d-1}} \frac{\mathbb{E}\| \delta_{\ell \hat{z}} \theta_s \|^2_{L^2_x}}{\ell^{2- 2\alpha}} \sigma ( \mathrm{d} \hat{z}) {\rm d}s = \|\theta_0\|_{L^2_x}^2 - \mathbb{E} \|\theta_t\|_{L^2_x}^2 \in (0,\infty). \end{align}\tag{56}\] From ?? , one immediately deduces that \[\begin{align} \label{OCthkr} \int_0^T\mathbb{E} \| \theta_t \|_{B^{(1 - \alpha)+}_{2,\infty}}^2{\rm d}t < \infty \quad \Rightarrow\quad \mathbb{E} [\| \theta_t \|_{L^2_x}^2] = \mathbb{E} [\| \theta_s \|_{L^2_x}^2] \quad \text{ for all } s, t \in [0,T]. \end{align}\tag{57}\] The threshold regularity exponent \(1-\alpha\) corresponds to the Obukhov-Corrsin regularity theory [124], [125] for the Kraichnan model, representing the minimal requirement for anomalous dissipation to take place. In fact solutions possess exactly the borderline regularity allowing dissipation of mean energy. Namely, the "Yaglom law" 56 shows (see also [149]) \[\label{regu} \int_0^T\mathbb{E} \| \theta_t \|_{B^{1 - \alpha}_{2,\infty}}^2{\rm d}t <\infty,\tag{58}\] with the Besov-like spaces introduced in Lemma 3. In view of Theorem 37 and 57 , the solution has sharply \(1-\alpha\) derivatives in this \(L^2\) based Besov space, at least in the mean sense, thereby establishing the analogue of Conjecture 24 for the Kraichnan model. This kind of regularization was established first in the work [86] by different means, and it shows the sharpness of the Obukhov-Corrsin regularity theory at the level of \(L^2\), while allowing for intermittency [96]. In fact, it is remarkable that in the Kraichnan model, an asymptotic approximation of the structure function scaling exponents \(S_p^\theta(\ell):= \mathbb{E} |\delta_\ell \theta|^p \sim \ell^{\zeta_p^\theta}\) have been computed [150] \[\zeta_p^\theta = p(1-\alpha) - \frac{p(p-2)}{d+2} \alpha +\mathcal{O}(\alpha^2).\] Intermittency vanishes as \(\alpha\to 0\), where the scalar field is expected to be Gaussian. It also vanishes at \(p=2\), where the exact Yaglom law holds. It would be very interesting to establish a rigorous understanding of these results.

In summary, in the case of the Kraichnan model we have found that anomalous diffusion is a mechanism for anomalous regularization. This is indeed the intuition behind Conjecture 24 for hydrodynamic turbulence. We now ask the question: what is the mechanism for anomalous dissipation? For scalar transport, we can answer this. It relates to non-uniqueness of particle trajectories of the velocity field

For a more precise statement and its proof, see [151], [152] and Rowan [147]. The discovery of the general phenomenon, however, is due to Bernard, Gawȩdzki and Kupiainen [153]. The basic idea is this. If one introduces stochastic trajectories \[\label{stochtraj} {\rm d}X_t(a) = u(X_t(a),t) {\rm d}t+ \sqrt{2\kappa} {\rm d}W_t,\tag{59}\] then, by the Feynman-Kac representation, the scalar field can be constructed via the inverse flow as an average \[\theta_t^\kappa(x) = \mathbb{E} [\theta_0 (X_t^{-1}(x))].\] The scalar dissipation is given by a Lagrangian fluctuation–dissipation relation [151], [152], [154] \[\kappa \int_0^T \|\nabla \theta_t^\kappa\|_{L^2}^2 {\rm d}t = \int_M {\rm Var} [\theta_0 (X_t^{-1}(x))]\;{\rm d}x.\] This relation makes clear the connection between non-uniqueness and anomalous diffusion. If anomalous diffusion occurs, the variance of the (fixed) initial conditions sampled by the backwards flow must remain non-trivial as \(\kappa\to 0\) for a positive measure set of \(x\in M\). This can only happen if the ideal particle trajectories passing through a positive measure of such points are non-unique.

In the Kraichnan model where the phenomenon was discovered, this process can be more explicitly studied. Indeed, for the homogeneous isotropic model, one can show that \(\rho_t(a,b) := |X_t(a) - X_t(b)|\) shares a law of the one-dimensional drift diffusion process \[{\rm d}\rho_t = b(\rho_t) {\rm d}t + \sqrt{2 \sigma(\rho_t)} {\rm d}B_t + \sqrt{2\kappa} {\rm d}W_t. \qquad \rho_0= |a-b|.\] Here \(B_t\) is the noise coming from the Kraichnan velocity field and \(W_t\) is an independent noise arising from modeling dissipation as in 59 . The following occurs

The latter case of non-uniqueness has been named spontaneous stochasticity [155]. These results were rigorously proved by Le Jan and Raimond [156], [157], and can be visualized in Figure 21 which shows an ensemble of noisy trajectories with small \(\kappa\) in a Lipschitz and Hölder velocity, respectively.

Taylor, Richardson and the motion of particles

We saw in the last section that a property of Lagrangian (or particle) trajectories: \[\dot{X}_t(a) =u(X_t(a),t) \qquad X_0(a)=a,\] namely spontaneous stochasticity, is responsible for the phenomenon of anomalous dissipation in some model problems (both for Burger [116] and passive scalars [151]–[153]).

Should we expect this phenomenon in real turbulence? One of the first investigations on particles motion in turbulence is due to G.I. Taylor [158] (1921), who studied how a single particle diffuses at long time. Taylor predicted that a single particle acts diffusively at long times with effective diffusivity \(D\), e.g. with the Lagrangian velocity \(v_t(a):= \dot{X}_t(a)\), \[\frac{\langle | X_t(a) - a|^2\rangle}{2t} \;\stackrel{t \to \infty}{\longrightarrow} \; D \quad \text{where}\quad D := \int_0^\infty\langle v_\tau \cdot v_0\rangle\; {\rm d}\tau,\] provided the latter limit exists. Here \(\langle\cdot \rangle\) denotes average over the label \(a\in \mathbb{T}^d\), and possibly over an ensemble of velocity fields. Taylor’s result holds under the assumption that the Lagrangian velocity is stationary in time after averaging, namely \(\langle v_t(a)\cdot v_{t'}(a)\rangle = \langle v_{t-t'}(a) \cdot v_{0}(a)\rangle\), which should be an integrable function of \(\tau=t-t'\). Stationarity of the Lagrangian velocity holds, for instance, provided the Eulerian velocity field is statistical stationary and homogeneous. See [159] and [160] for an example of an early rigorous statement. Thus, we take it that a single particle looks like Brownian motion at long times, albeit with a diffusivity which is only implicitly predicted.

What about multiple particles? In 1926, shortly after the work of Taylor, Lewis Fry Richardson investigated the dispersion of two-particles by a turbulent flow. Based on theoretical considerations and weather balloon data, Richardson predicted a \(t^3\)–law: \[\label{tcubed} \langle |X_t(a+r)- X_t(a)|^2\rangle \sim G \langle \varepsilon\rangle t^3,\tag{60}\]

for some constant \(G\) and where \(|r|\gg \ell_\nu\), namely the particles are separated in the inertial range. The striking thing about Richardson’s prediction is that the right hand side is independent of the initial separation of the particles! Indeed, the inertial range extends to all scales as \(\nu \to 0\), so that \(r=0\) can be taken. In that case, Richardson predicts an ensemble of particles burst from the same starting point and have mean-squared displacement growing cubically in time, a rate which is faster than simple ballistic separation of particles experiencing a constant relative velocity. In short, already in 1926, Richardson foresaw the shadow of spontaneous stochasticity.

Although Richardson arrived at his result by a different theoretical consideration, let us quickly give a rigorous bound on such dispersion assuming some regularity on the velocity field that is coherent with 60 assuming Kolmogorov’s 1941 picture of turbulence.

Note that, for long times the bound becomes independent of \(|r|\). Its asymptotic form takes the shape of a bound on the deviation of non-unique trajectories of a Hölder field \(u\) passing through the same point \(a\in M\): \[|X_t^{(1)} (a) - X_t^{(2)}(a)| \leq \left((1-\alpha) [u]_{L^\infty_t C^\alpha_x} t \right)^{\frac{1}{1-\alpha}}.\] Of course, there are simple examples saturating this upper bound (take \(u(x)={\rm sgn}(x) |x|^\alpha\) in one dimension), showing Proposition 40 to be sharp. See [162], [163] for a realization of this simple example in quantum systems, and a renormalization group perspective. The works [164], [165] give higher dimensional examples and investigates when a unique measure valued solution might be obtained. The works [166]–[169] concern the same behavior for infinite dimensional models, where these features robustly appear.

Recalling K41 phenomenology, \(\alpha=1/3\) and \([u]_{L^\infty_t C^{1/3}_x}= (\langle {\varepsilon}\rangle \ell_I )^{1/3}\) (by dimensional considerations, taking e.g. \(\ell_I= {\rm diam}(\mathbb{T}^d)\)). Since this field is ideally rough, understood as arising in the \(\nu\to 0\) limit, we may take \(r=0\) so that particles start at the same point. The bound is then exactly consistent with the behavior 60 .

Although Richardson’s prediction has proved difficult to conclusively observe, impressive partial confirmations have been made experimentally [170], [171] and numerically [172]–[175]. The following Figure 23, taken from [175], represents essentially the state of the art in our observations of Richardson’s law from numerical simulation data. The \(t^3\)–law is observed, as is the "forgetting" of the initial separations as expected from Richardson’s theory as well as the simple-minded bound in Proposition 40.

Another striking aspect of Richardson’s prediction 60 is that the net anomalous energy dissipation rate \(\langle {\varepsilon}\rangle\) appears. Recall that for the Burgers equation and passive scalars both, non-uniqueness of trajectories gave an origin or explanation of the observed anomalies. Richardson’s law seems to do this as well, but it’s theoretical foundation is lacking. Does \(\langle {\varepsilon}\rangle\) appear in his theory for purely dimensional reasons, and true particle dispersion is not directly related to dissipation?

As it turns out, this aspect of Richardson’s theory can be rigorously proved, although only for a very short time regime and not for the dispersion itself, but rather a relative forward/backwards dispersion. To state the result, we must introduce a regularization of the trajectories, namely the flow in a mollified field: \[\label{molflow} \frac{{\rm d}}{{\rm d}t} X_t^\ell(a) = \overline{u}_\ell ( X_t^\ell(a) ,t) , \qquad X_0^\ell(a) = a.\tag{61}\] Fix an standard mollifier \(\psi\in C_0^\infty(\mathbb{T}^d)\) with \({\rm supp} (\psi)\subseteq B_1(0)\) and \(\psi_R(r) = R^{-d} \psi(r/R)\) and denote \(\langle f(r) \rangle_R:= \int_{\mathbb{T}^d} f(r) \psi_R(r)dr\) for any \(f\in L^1( \mathbb{T}^d)\). The result is:

We won’t give the proof of this result here, it can be found in [52]. We just give the main idea behind it, which derives from the works of Ott-Mann and Gawȩdzki [144], [170], as well as Jucha et. al. [176]. Indeed, the Ott-Mann–Gawȩdzki relation is sometimes described as the "Lagrangian analog of the \(4/5\)–law". For inertial range separations \(r\), this relation states: \[\label{OttMann0} \left.\frac{1}{2} \frac{{\rm d}}{{\rm d}\tau} \Big\langle |\delta_r { {v}}(\tau;{ {x} },t) |^2\Big\rangle\right|_{\tau=0} \simeq -2 \langle \varepsilon \rangle\tag{62}\] with the Lagrangian velocity \(v(\tau,x;t):={{u}}({X}_{t,t+\tau} ({ {x} }),t+\tau)\) and \(\delta_r { {v}}(\tau;{ {x} },t):= v(\tau,x+r;t)-v(\tau,x;t)\). Standard derivations of the relationship assume spatial isotropy and the average must be either interpreted as over the spatial domain, or as a time/ensemble average provided the fields are homogenous ([52] gives a more robust interpreation/derivation holding for any weak solution). With the Ott-Mann-Gawȩdzki relation in hand, the relative mean–squared dispersion of Lagrangian tracers for short-times can be calculated using only local (in time) Eulerian quantities by Taylor expansion: \[\begin{align} \label{reldispFG} \Big\langle | \delta {X}_{t,t+\tau}({{r}}; { {x} })-&{{r}}|^2\Big\rangle \approx S_2^{{{u}}_t}({{r}}) \tau^2 - 2 \langle \varepsilon \rangle \tau^3 +{O} (\tau^4). \end{align}\tag{63}\] Inspecting the behavior of 63 under time reversal \(\tau\to -\tau\), one sees that the \({O}(\tau^2)\) term is invariant whereas the \({O}(\tau^3)\) term changes sign. Taking of the forwards and backward in time dispersion suggests (the global analogue of) the formula ?? .

Unfortunately, at high-Reynolds numbers, the realm of validity of the expansion 63 becomes vanishing small. In particular, the Taylor series expansion of the particle trajectories used to derive 63 is only guaranteed to converge in a neighborhood of times that is vanishingly small as \(\nu\to 0\) (although Frishman & Falkovich [177], [178] argue on theoretical grounds that a similar expansion for the difference may not). Thus, it is desirable to have an alternative Lagrangian measure of time–asymmetry that remains valid for arbitrarily large Reynolds numbers. This is what is accomplished in Theorem 41 by introducing an (arbitrary) mollification of the velocity field.

We give one final result on particles in (Hölder-continuous) turbulent velocity fields, concerning their smoothness in time.

In view of the above result on trajectory regularity, we ask

Note that trajectory regularity does not imply any kind of general regularity of the vector field \(u\). For instance, if \(u\) is a bounded shear flow, the profile can be very irregular but the trajectories are straight lines.

It is unfortunate that many of the rigorous results discussed in this section require Hölder continuity as opposed to Besov. For instance, given the observations do not support \(\frac{1}{3}\) Hölder continuity, we wonder if there is an analogous Besov statement for Lagrangian regularity. Namely, if \(u\in L_t^p (0,T;B_{p,\infty}^\sigma(\mathbb{T}^d))\), is it true that \(v =L^{p/2}(\mathbb{T}^d; B^{\frac{\sigma}{1-\sigma}}_{p/2,\infty}(0,T))\)? The difficulty in establishing this type of result is that, unlike the Hölder case, it seems necessary to bundle trajectories and the regularity in the label \(a\) can be very bad. It may be that is impossible to bundle trajectories in such as way as to define a measure-preserving flowmap \(X_t(a)\) that depends measurably on \(a\) (see [188] for an example). Could such regularity at least be preserved uniformly along solutions of Navier-Stokes?

Conclusion

We have not come much closer to developing the "first principled" theory of turbulence that Kolmogorov sought. However, as a result of decades of profound work and deep understanding from many great minds, we have presented a framework that allows us to paint a precise conjectural picture of certain features of turbulence. Simply by making precise statements, we can perceive and prove some fundamental constraints on the phenomenon. It is useful to have in mind some mathematically well defined features and their consequences, even if we don’t know what characterizes turbulence in totality, or even whether those features are universal to all phenomena one might call turbulent.

Though it seems far off, a more complete understanding of turbulence might come from a dynamical systems approach as suggested by Kolmogorov himself [1], and/or by embracing elements of probability theory in a more fundamental way. We end with Richard Feynman’s plan in this direction, which sounds quite modern.

Acknowledgements. These notes are based on a series of lectures delivered during the “Turbulence on the Banks of the Arno" school in Scuola Normale Superiore in Pisa, Italy, in April 2025, as well as the Summer School on Advances in Hyperbolic Balance Laws, Hirschegg, Austria, in September 2025. I am grateful to the organizers for the invitation and hospitality, and to the audience and participants for engaging discussions and questions. I would like to thank Tarek Elgindi and Vlad Vicol for their remarks and comments. I am particularly grateful to Luigi De Rosa and Gregory Eyink for their detailed comments but mainly for the many discussions on turbulence over the years. My work was supported by the NSF CAREER award #2235395, a Stony Brook University Trustee’s award as well as an Alfred P. Sloan Fellowship.

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