1 Introduction↩︎

We consider the evolution of a passive tracer in a bounded domain which is advected by a regular Hamiltonian flow with closed streamlines in the large Péclet number \(\mathrm{Pe}=LU/\kappa\) regime, where \(L, U\) are the characteristic lenght and velocity scales and \(\kappa\) is the diffusivity coefficient. This is a paradigmatic example of a wide range of physical, chemical and biological processes and can be thought of as a toy model for some hydrodynamic stability problems around laminar flows. For instance, in geophysical applications there are persistent lone eddies which are likely to homogenize tracers along their streamlines before standard diffusive processes take over rhines1983rapidly?. When \(\mathrm{Pe}\gg 1\), it is often observed that the averaging along the streamlines of the flow happen on a time-scale \(T_a\) much shorter than the diffusive time-scale \(T_d=L^2/\nu\), a phenomenon which is linked to the effective diffusivity studied in homogenization theory e.g. MR1265233?. This separation in time-scales, also known as dissipation enhancement, is related to the intricate interplay between diffusion and advection. In earlier investigations in the applied literature the precise scalings of the averaging time \(T_a\) in terms of \(\mathrm{Pe}\) have been debated rhines1983rapidly?, but it is now clear that they heavily depend on specific properties of the advecting velocity field. From a mathematical point of view, the picture is well-understood for radial and shear flows. In these cases, \(T_a\) is related to the behavior around critical points (or the absence of them) for the velocity profile. The great advantage of radial and shear flows is that the average along the streamlines (average in the angular or shearing direction) commutes with the diffusion, meaning that the dynamics of the average is decoupled from the rest of the solution. Therefore, it is enough to quantify the decay rates of the solution minus its streamline average to estimate the separation of time-scales. This fact is crucially exploited in all the quantifications of \(T_a\) for shear and radial flows, obtained through spectral methods Vukadinovic15?, Gallay:2021aa?, Colombo21?, Feng23?, He22?, hypocoercivity Bedrossian17?, CZD20? and the quantitative version of Hörmander’s classical theory Albritton22?.

On the other hand, for more general Hamiltonian flows, this particular structure is lost and less is known. An insightful quantitative homogenization-type result was obtained in kumaresan2018advection?, Vukadinovic21? for a class of Hamiltonians. They were able to show that there is a time \(T_*\) with \(T_a\ll T_*\ll T_d\) for which the tracer is indeed averaged along the streamlines in the \(\mathrm{Pe}\to \infty\) limit, where the time \(T_a\) is determined by the behavior around critical points of the period function (the analogue of the velocity profile in shear and radial flows). This can be thought of as a form of dissipation enhancement, but the quantitative control they obtain is not uniform in time. We comment more about this result at the end of this introduction, where we will compare it with what we obtain in the present study.

The enhanced dissipation effect is much faster when the flow is turbulent instead of laminar, but we keep the discussion short and somewhat informal given the purposes of this paper. In particular, we say that a flow is turbulent if it has ‘complicated trajectories’, such as mixing flows constantin2008diffusion?. In these cases, there are results showing dissipation enhancement both in the deterministic elgindi2023optimal?, CZDE20?, constantin2008diffusion? and in the stochastic setting Bedrossina21Blum?. Finally, also the anomalous dissipation results TDTEGIIJ22?, CCS22?, AV23?, EL23?, BBS23? obtained recently can be interpreted as an extreme form of dissipation enhancement.

1.1 Main results↩︎

In this paper, we aim at giving a different perspective on Hamiltonian flows containing only elliptic points, naturally arising from perturbations of radial flows or in regions near elliptic points of general Hamiltonians. Let \(M\subset \mathbb{R}^2\) be a bounded domain with smooth boundary \(\partial M\). Let \(H \colon M \to \mathbb{R}\) be a \(C^3\) function such that \(H\) is constant on \(\partial M\). Define \(u = \nabla^{\perp} H = (-{\partial}_{x_2},{\partial}_{x_1}) H\) and let \(\rho:[0,\infty)\times M\to \mathbb{R}\) be the solution to the advection diffusion equation \[\begin{align} \label{eq:advdiff} \begin{cases}{\partial}_t \rho+u\cdot\nabla \rho=\nu \Delta \rho, \qquad x \in M, \, t> 0,\\ \rho|_{t=0}=\rho^{in}, \qquad \rho|_{\partial M}=0 \text{ (or }\partial_n \rho|_{\partial M}=0). \end{cases} \end{align}\tag{1}\] We consider either Dirichlet (or homogeneous Neumann) boundary conditions. Notice that \(u\) is tangent to the boundary (since \(H\) is constant on \(\partial M\)) and we have already properly adimensionalized the equations so that \(\nu:=\mathrm{Pe}^{-1}\). We discuss more about precise hypotheses on \(H\) in the sequel, but an essential assumption is that the level sets \(\{H=h\}\) are Jordan curves foliating \(M\).

To understand the evolution of \(\rho\), as we explained above, it is natural to introduce the average along the streamlines, corresponding to an integration over the level sets of \(H\). For the class of Hamiltonians we consider here, as we show in Section 2, this is equivalent to performing the orthogonal projection in \(L^2(M)\) onto the kernel of the transport operator \(u\cdot \nabla\). For instance, all the functions of the type \(F(H)\) are in \(\mathrm{Ker}(u\cdot \nabla)\). Thus, we split a function \(f\in L^2(M)\) as \[f:=P_0f+P_\perp f=f_0+f_\perp, \qquad P_0 f \in \mathrm{Ker}(u\cdot \nabla).\] This is also the natural splitting suggested by the inviscid problem, where \(P_0 \rho^{in}\) is known to be, in the cases considered here, the weak limit in \(L^2\) of the evolution of 1 with \(\nu=0\) (quantitative decay rates of this convergence are related to mixing properties of \(u\)). Projecting the equation 1 , one has \[\begin{align} \label{eq:PperpP0} \begin{cases} {\partial}_t\rho_\perp +u\cdot\nabla \rho_{\perp}=\nu P_\perp\Delta \rho_\perp+\nu P_\perp\Delta \rho_0,\\ \rho_\perp|_{t=0}=\rho_{\perp}^{in}, \end{cases} \qquad \qquad \begin{cases} {\partial}_t\rho_0 =\nu P_0\Delta \rho_0+\nu P_0 \Delta \rho_\perp,\\ \rho_0|_{t=0}=\rho_{0}^{in}, \end{cases} \end{align}\tag{2}\] with the same Dirichlet or Neumann boundary conditions and where we have used the fact that \(P_0\) commutes with \(u\cdot \nabla\), as is precisely shown in Section 2. In general, the commutator \([P_\perp,\Delta]=[\Delta,P_0]\neq0\) (which instead is zero for shear or radial flows) and therefore the dynamics of \(\rho_\perp\) and \(\rho_0\) are coupled.

Our goal is to characterize the long-time behavior of \(\rho_\perp\). A first guess, suggested by the behavior of shear and radial flows, would be that \(\rho_\perp\) decays on a sub-diffusive time-scale depending on the properties of \(u\). For instance \[\left\lVert \rho_{\perp}(t) \right\rVert_{L^2}=\left\lVert (\rho - \rho_{0})(t) \right\rVert_{L^2}\lesssim{\rm e}^{-\lambda_\nu t} \|\rho^{in}-\rho^{in}_0\|_{L^2}, \qquad \text{ with} \quad \nu/\lambda_\nu \to 0 \quad\text{as}\quad\nu \to 0,\] where \(\lambda_\nu\) is called enhanced dissipation rate. To achieve this bound, a possibility is that the forcing term \(\nu P_\perp\Delta\rho_0\) decays at least at the same rate, meaning that the interplay between the commutator \([P_\perp,\Delta]\) and the dynamics of \(\rho_0\) result in some dissipation enhancing property of this forcing term, which can be quite challenging to quantify though. On the other hand, since for \(\nu=0\) the \(\rho_0\) is conserved in time, one can hope that \(\nu P_{\perp}\Delta \rho_0\) generates at worst a small term that one can subtract from the dynamics of \(\rho_\perp\) and still get decay. This is still a delicate statement to formalize since higher order derivatives are involved, which in general can cost inverse powers of \(\nu\) due to the formation of small scales. However, in account of the reasoning above, our first objective is to obtain the enhanced dissipation of \(\rho_\perp\) up to a ‘small’ correction \(g_{\mathrm{corr}}\), namely \[\label{eq:goal1} \left\lVert (\rho - \rho_{0} - g_{\mathrm{corr}})(t) \right\rVert_{L^2}\lesssim{\rm e}^{-\lambda_\nu t} \|\rho^{in}-\rho^{in}_0\|_{L^2}, \qquad \text{ with} \quad \nu/\lambda_\nu \to 0 \quad\text{as}\quad\nu \to 0.\tag{3}\] There is also a weaker result which would still be indicative of some dissipation enhancement generated by the flow. When \(\rho_0^{in}=0\), all the errors remain so small that at least on a sub-diffusive time \(T_\nu=O(\lambda_{\nu}^{-1})\) one is able to prove \[\label{bd:disssub} \left\lVert (\rho - \rho_{0})(T_\nu) \right\rVert_{L^2} \leq (1-c_*)\|\rho^{in} \|_{L^2}, \qquad \text{ with } c_*\in (0,1).\tag{4}\] Namely, the \(L^2\) norm is reduced at times \(T_\nu\ll T_d=O(\nu^{-1})\), that is an instance of dissipation enhancement. In most of the cases the averaging time \(T_a\) is in fact identified directly with \(T_\nu\). For shear, radial and relaxation-enhancing flows¹ the bound 4 would automatically imply the exponential decay with rate \(\lambda_\nu\) since \(\rho_0(t)=0\) for all times if it is zero initially, see for instance Feng19?, CZDE20?. Unfortunately, since \(\rho_0\) is not conserved in our case, we cannot hope to iterate the estimate 4 in a straightforward way, but still a bound like 4 is an indication of dissipation enhancement.

Making the heuristic arguments above rigorous seems complicated in general, especially due to the lack of a precise understanding of the dynamics of \(\rho_0\). In fact, we believe that studying quantitatively the evolution of \(\rho_0\) is an interesting subject by itself. As an example, we also show that \(\rho_0\) can be created and remain quantitatively lower bounded on a short-time interval even when starting from \(\rho_0^{in}=0\). In Proposition 18 we exhibit an example concentrated close to a hyperbolic point whereas in Proposition 19 we exploit an elliptic point.

Regarding our goals for \(\rho_\perp\), in Corollary 1 we show that bounds as in 4 holds true for Hamiltonians in the class [HamiltonianClassAm], precisely defined in Definition 4. To have \(H\) in [HamiltonianClassAm] we essentially require the following:

• Regularity, that is \(H\in C^3\).

• There is one non-degenerate elliptic critical point \(x_0\), i.e. \(\nabla H(x_0)=0\) and \(\mathrm{det}(D^2 H)(x_0)>0\).

To prove the stronger result encoded in 3 , we need to further assume a more precise control on \(H\), roughly speaking:

• The modulus of the velocity at any point can be quantitatively controlled with its average along the streamline.

We will denote this class as 5 where \(\varepsilon\) essentially quantifies how close the Hamiltonian is to being a radial flow. As we show in Section 5, any perturbation of a radial Hamiltonian of the form \[\begin{align} \label{intro:Hamiltonian} H_\epsilon(r,\varphi)=H(r)+\epsilon f(r,\varphi), \qquad r,\varphi \text{ are the standard polar coordinates,} \end{align}\tag{5}\] is in the class 5 with \(\varepsilon\simeq \epsilon\). Moreover, this behavior is typical of some non-degenerate Hamiltonians close to elliptic points.²

The index \(m\) in both classes, is instead related to the enhanced dissipation rate \(\lambda_\nu\), which takes into account the difference in velocity along different streamlines. This can be quantified by looking at the period and frequency functions defined respectively as \[\label{def:period} T(h):=\oint_{\{H=q(h)\}} \frac{{\rm d}\ell}{|\nabla H|}, \qquad \Omega(h):=\frac{1}{T(h)}.\tag{6}\] where \(q\) is some non-constant function related to the behavior of \(H\) around its critical point (typically \(q(h)=h^2\)). Then, an Hamiltonian in class [HamiltonianClassAm], which includes the class 5, is such that \[\begin{align} |\Omega'(h)|\simeq h^{m-1}, \qquad \text{for } \quad |h|\ll 1 \end{align}\] where we are assuming, without loss of generality, that the only critical point of \(H\) is located at \(h=0\).

In the class of Hamiltonians introduced above, by analogy with what is known for shear or radial flows, we expect that the (largest) averaging time \(T_a\) is inversely proportional to \[\label{def:lambdanu} \lambda_\nu:=\nu^{\frac{m}{m+2}}.\tag{7}\] Our first main result is the following.

Let us give a few remarks about the results contained in the theorem above.

1.2 Model problems↩︎

As mentioned in Remark 1, a key point in the proof of Theorem 1 is the study of pseudospectral properties of the operator \[\label{def:Lperp} \mathcal{L}_\perp=u\cdot \nabla-\nu P_\perp\Delta P_\perp,\tag{12}\] with Dirichlet or Neumann boundary conditions, and \(\mathcal{L}_{\perp} : D(\mathcal{L}_\perp)\subset L^2_\perp(M)\to L^2_\perp(M)\) with \[\label{def:L2perp} L^2_{\perp}(M):= \{ f\in L^2(M) \, : \, P_0 f=0\}.\tag{13}\] In particular, we aim at proving enhanced dissipation for the model problem \[\label{eq:model} \begin{cases} {\partial}_tg +\mathcal{L}_{\perp}g=0, \qquad x \in M, \, t> 0,\\ g|_{t=0}=\rho_{\perp}^{in}, \qquad g|_{\partial M}=0 \text{ (or }\partial_n g|_{\partial M}=0). \end{cases}\tag{14}\] Our second main result, which holds for the more general class [HamiltonianClassAm], is the following:

The precise decay estimate on \(g\) follows by the lower bound on the pseudospectral abscissa that we prove in Proposition 14, which is one of the main challenges of this paper. Moreover, the solution \(g\) to the model problem 14 is behind the definition of our correction \(g_{\mathrm{corr}}\). Indeed, as we show in Section 3, we will define \[g_{\mathrm{corr}}(t):=(\rho-\rho_0-g)(t).\] A consequence of Theorem 2 is a property analogous to 4 for Hamiltonians in the class [HamiltonianClassAm].

Naively, one would like to iterate the result above and conclude the dissipation enhancement property on the original solution for the larger class [HamiltonianClassAm]. However, since \(\rho_0\) is not conserved, an iteration would require a quantitative control of its evolution. Thus, we also need to introduce the natural model problem for \(\rho_0\), which is given by \[\label{eq:model00} \begin{cases} {\partial}_t\eta=\nu P_0\Delta P_0 \eta:=-\mathcal{L}_0\eta , \qquad x \in M, \, t> 0,\\ \eta|_{t=0}=\rho_{0}^{in}, \qquad \eta|_{\partial M}=0 \text{ (or }\partial_n \eta|_{\partial M}=0). \end{cases}\tag{16}\] This is the leading order approximation of \(\rho_0\) in the asymptotic expansion we perform in the proof of Theorem 1, in the sense that \(\|\rho_0-\eta\|_{L^2}\lesssim \varepsilon\|\rho^{in}\|_{L^2}\).

Finally, by assuming a priori an upper bound on \(\|\rho_0-\eta\|_{L^2}\), we are also able to prove the following conditional result.

The result in ?? is a homogenization type result towards the effective dynamics dictated by the solution to our model problem 16 . Thus, we have a very precise control on the evolution of \(\rho\) once assumption ?? is verified, which however we believe it can be quite challenging to prove.

1.3 Main difficulties and discussion↩︎

Let us first highlight the main difficulties by comparison with the known case of radial flows. For radial flows (included in the \(\varepsilon=0\) case), the pseudospectral abscissa of \(\mathcal{L}_\perp\) can be estimated by the one of the operators \(ik \Omega(r)-\nu ({\partial}_{rr}+r^{-1}{\partial}_r-k^2/r^2)\) with \(k\in \mathbb{Z}\setminus\{0\}\), see Gallay:2021aa?. This can be easily seen by passing to polar coordinates and taking the angular Fourier transform. The decoupling in \(k\) is a property crucially related to the geometry of the streamlines, thanks to which the operator \(\mathcal{L}_\perp\) leaves invariant the space of \(k\)-fold symmetric functions (in the angular direction). The decoupling in \(k\) is fundamentally used to study pseudospectral bounds on a given finite number of isolated thickened level sets, e.g. a \(\delta\)-neighbourhood of \(\{|\Omega(r)-\lambda/k|\lesssim \delta^m\}\), and cleverly split the domain in two regions, in which the following happens:

1. In the thin sets, smallness can be gained via standard Poincaré type inequalities.

2. Outside of the thin sets, the transport operator can be cleverly used to get a good upper bound.

The combination of the two points above lead to the sharp estimate on the pseudospectral abscissa after optimizing the size of the thin sets.

The natural connection between Hamiltonian and radial flows is given by an action-angle change of variables \((x,y)\to(h,\theta)\). This transforms the operator \(u\cdot \nabla\) in \(\Omega(h){\partial}_\theta\) and hence \(ik\Omega(h)\) after a Fourier transform in the angle variable. The commutator \([P_k , \Delta]\), where \(P_k\) denotes the projection on the \(k\)-th Fourier mode along the angular variable, is usually non zero. Therefore, the diffusion operator \(P_k \Delta\) does not decouple angular frequencies, meaning that the dynamics of the \(k\)-th Fourier mode is affected by the dynamics of all the other Fourier modes. As a consequence, a \(k\)-by-\(k\) analysis is challenging. However, the commutator \([P_\perp, \Delta]\) can be uniformly controlled in terms of \(\varepsilon\) (related to the \(\epsilon\) in 5 as explained above). Then, in our case we cannot think of \(k\) as a simple rescaling of the spectral parameter \(\lambda\) and instead we have to carefully redefine an arbitrary large number of thickened level sets (of cardinality approximately \(|\lambda| \in \mathbb{R}_+\)), avoiding their overlap as well. Here we crucially use the non-degeneracy assumption on the Hamiltonian, without which our sets are not mutually disjoint.

We then need to quantify precisely the relation between thin sets in action-angle variables and their size in standard Cartesian coordinates, where the geometry of the streamlines play a fundamental role. This is precisely done in Section 2, where we introduce our rescaled action-angle variables that allow us to prove useful Poincaré type inequalities (that can be of independent interest). In particular, we scale properly our change of variables to avoid potential degeneracies when taking derivatives in the action direction. This is crucial to directly link the size of a thin sets in both set of coordinates and treat the action variable in a more uniform way. This idea can be technically useful in other contexts and we believe it is one of the main reasons why in the radial case it is more convenient to work in polar coordinates instead of a canonical action-angle change of variables (whose Jacobian is \(1\) instead of \(r\)). Finally, to perform the desired estimates outside the thin sets, we use a Fourier multiplier method inspired by Gallay:2021aa?, LiPseudo20?, gallay2019estimations?, but here the multiplier is implicitly defined in rescaled action-angle variables. Again, the definition of the rescaled action-angle variables simplifies the estimate for derivatives of the Fourier multipliers.

We conclude this introduction by comparing our result and the ones obtained by Vukadinovic and Kumaresan in kumaresan2018advection?, Vukadinovic21?, which also establish dissipation enhancement (or homogenization) properties for more general Hamiltonians. In both cases, an approximating problem exhibiting dissipation enhancement is introduced and used to deduce quantitative estimates on 1 . The first main difference is the operator replacing the full diffusion operator \(\Delta\), which for us is \(P_\perp\Delta P_\perp\). On the other hand, in kumaresan2018advection?, Vukadinovic21? they introduce an ‘averaged’ version of the Laplacian, denoted by \(\langle\Delta\rangle\), which is related to the effective diffusion equation found by Friedlin and Wentzell Freidlin94?. The way \(\langle\Delta \rangle\) is precisely defined is technical, but it can be easily explained informally. Indeed, let \((J,\theta)\) be the canonical action-angle coordinates associated to the Hamiltonian \(H\). Then, one has \(\Delta \to A_H(J,\theta){\partial}_{JJ}+B_H(J,\theta){\partial}_{J\theta}+C_H(J,\theta){\partial}_{\theta\theta}\). Denoting with \(\langle \cdot \rangle\) the averaging in \(\theta\) (which is related to our \(P_0\)), the averaged diffusion operator is defined as \[\langle \Delta \rangle := \langle A_H\rangle(J){\partial}_{J\theta}+\langle B_H \rangle(J){\partial}_{J\theta}+\langle C_H \rangle(J){\partial}_{\theta\theta}.\] The great advantage of the operator above is that it decouples angular frequencies, meaning that we can reduce everything to a one-dimensional problem as in the radial flow case discussed above. Moreover, the analysis of the model problem can be done for a general class of Hamiltonians with restrictive assumptions on the initial data (ruling out our initial data in the examples in Proposition 18 and Proposition 19). For instance, the cellular flow in \(\mathbb{T}^2\) defined as \(H=\sin(x)\sin(y)\) is included in their analysis and a dissipation enhancement rate \(\lambda_\nu=\nu^{1/2}\) can be proved for their model problem.

The proof in kumaresan2018advection?, Vukadinovic21? is based on the hypocoercivity method, which crucially requires a particular hypothesis on the initial data that exclude something concentrated close to the hyperbolic point. More precisely, the so-called ‘\(\gamma\)-terms’ in the hypocoercive energy functional require that \[\left\lVert \Omega' \rho^{in} \right\rVert_{L^2}\leq C,\] where \(\Omega'\) is the derivative of the frequency function whereas \(C\) is a constant independent of \(\nu\). Now, close to an hyperbolic point at \(h=0\), by the estimates in kumaresan2018advection?, Vukadinovic21? we know that \(|\Omega'|(h) \simeq 1/|h(\ln(h))^2|\). Therefore, the weight \(\Omega'\) degenerates in presence of hyperbolic points and we cannot consider data really concentrated close to it. The initial data proposed in Proposition 18 does not satisfy the desired bound with \(C\) independent of \(\nu\), where we show a pathologic behaviour not included in kumaresan2018advection?, Vukadinovic21?.

Regarding the use of the model problem to deduce bounds for solutions to the advection–diffusion equation 1 , the authors in kumaresan2018advection?, Vukadinovic21? assume higher order derivatives bounds independent on \(\nu\) on the initial data (see for instance Vukadinovic21?) and need to restrict themselves to the case \(m\in \{0,1\}\)³. This is related to the fact that all the errors include derivatives whose bounds cannot be uniform in \(\nu\). Thus, there is a delicate balance of powers of \(\nu\). For instance, for \(m=1\) (the case of the cellular flow) one error⁴ is bounded by \(\nu^{\frac{14}{30}}\sqrt{t}\), which is large for any \(t>\nu^{-\frac{28}{30}}\) (and exploding as \(\nu\to 0\) in diffusive time scales). Therefore, one drawback of the approximation with \(\langle \Delta\rangle\) is that it is not clear how to extend it to \(m\geq 2\) and how to control the errors uniformly in time.

Plan of the paper↩︎

In Section 2 we introduce the main technical tools needed in the rest of the paper. Here, we introduce a version of action-angle coordinates and we present Poincaré type inequalities in thin sets adapted to the shape of the streamlines. In Section 3, by assuming the result in Theorem 2, we prove Theorem 1 and Corollary 1. In Section 4 we present the pseudospectral bounds for the operator \(\mathcal{L}_\perp\), and consequently prove Theorem 2. In Section 5 we present some examples, where we first show that perturbations of radial flows are in the class 5. Then, we show the results related to the dynamics of \(\rho_0\) we mentioned in the discussion above. We finally include some concluding remarks in Section 6.

2 Toolbox and hypotheses on the Hamiltonians↩︎

In this section, we first introduce the main technical tools and notation used in the rest of the paper. This include a coordinate change which simplifies computations in comparison to the standard action-angle coordinates when computing spectral estimates. With this notation at hand, we can finally state the main assumptions for the Hamiltonian \(H\). Then, we prove useful results regarding Fourier multipliers which will be used in the pseudo-spectral estimates. Finally, we derive Poincaré inequalities in thin streamline sets.

2.1 Rescaled action-angle variables↩︎

When studying the problem 1 , the standard procedure is to consider the canonical action-angle variables, e.g. Vukadinovic21?, which is a coordinate change (in general non-global) adapted to the level-curves of the Hamiltonian. For an Hamiltonian whose closed streamlines foliate the domain \(M\) up to a zero measure set, it is well known that the action variable is the area enclosed within the streamline(s) \(\{ H=s \}\)⁵. For our purposes, we find it more convenient to work in a set of coordinates which resembles more the polar coordinates and are particularly adapted to the behavior near elliptic points of the Hamiltonian. We refer to this set of coordinates as rescaled action-angle variables and the goal of this section is to introduce them. The technical advantage is to avoid carrying over the area function and it will be easier to quantify precisely derivatives in the new reference frame.

To define rescaled action-angle variables we need a function \(q\) defined on the image of \(H\) which is well-adapted to the behaviour of the Hamiltonian around critical points. For the classes of Hamiltonians in this study, we select \(q\) in the following manner. Let \(M \subset \mathbb{R}^2\) be a bounded domain with smooth boundary. Let \(H \in C^2(\overline{M})\) be an Hamiltonian with only one elliptic and non-degenerate critical point \(x_0\). Set \(h_0 = H(x_0)\) and \(I = H(M \setminus \{ x_0 \} )\). We then define the function \(q\) as \[q(h) = (h - h_0)^2.\] We remark that the function \(q\) is tailored to the Taylor expansion of the Hamiltonian around the elliptic point. In particular, since the elliptic point is non-degenerate, \(q\) is selected to be quadratic. Other choices are possible and without doubt useful in different contexts. For degenerate elliptic points, selecting a higher order monomial depending on the degeneracy seems suit the problem while for Hamiltonians with several critical points a polynomial is required.

Let us now show that this change of coordinates exists under appropriate conditions on the period function.

The rescaled action-angle change of coordinates satisfies some useful properties we state below.

For the purpose of stating the assumptions on the Hamiltonian, we also introduce the notion of streamline average.

Note that, by definition \(\langle f \rangle\) is constant along streamlines and hence independent on the choice of rescaled action-angle variables.

2.2 Fourier series in angle variable↩︎

One of the main advantages of having an angle coordinate is the possibility to perform the Fourier transform along the streamlines. Indeed, recall that the mapping \(\Phi\) defined in the previous subsection provides a change of variables from \(\mathbb{T}\times I\) to \(M\). We denote with \(\mathcal{F}\) the Fourier transform on \(\mathbb{T}\) and, given \(f \in L^2(M)\), \(h\in I\), and \(k\in \mathbb{Z}\), we introduce the following notation \[\widehat{f \circ \Phi}(k,h) \mathrel{\vcenter{:}}= \mathcal{F}[(f \circ \Phi)(\cdot, h)](k)=\int_{\mathbb{T}}{\rm e}^{2 \pi ik \theta}(f \circ \Phi)(\theta, h) {\rm d}\theta.\] With this notation, Parseval’s identity reads as \[\label{eq:ParsevalsIdentity} \int_{\mathbb{T}} (f_1 \circ \Phi)(\theta, h) \overline{(f_2 \circ \Phi)(\theta, h)} \, d \theta = \sum_{k \in \mathbb{Z}} \widehat{(f_1 \circ \Phi)}(k, h) \overline{\widehat{(f_2 \circ \Phi)}(k, h)}\tag{29}\] for all functions \(f_1, f_2 \in L^2(M)\) and a.e. \(h \in I\). Then, in account of 23 , notice that \[(u\cdot \nabla f)\circ \Phi= (\nabla^\perp H\cdot \nabla f)\circ \Phi=\frac{1}{T(h)} {\partial}_\theta (f\circ \Phi).\] Thus, recalling the definition of the frequency function (see Definition 1), we get \[\mathcal{F}((u\cdot \nabla f)\circ \Phi)(k,h)= \frac{ik}{T(h)} \widehat{(f\circ \Phi)}(k,h)=ik\Omega(h) \widehat{(f\circ \Phi)}(k,h).\] In other words, the operator \(u\cdot \nabla\) becomes \(\Omega {\partial}_\theta\) in the new reference frame. The projection onto the \(0\)-th angular mode, defined as \[\begin{align} (P_0 f \circ \Phi)(\theta, h) = \int_\mathbb{T}(f \circ \Phi)(\theta, h) {\rm d}\theta \end{align}\] is equivalent to the projection onto the kernel of \(u\cdot\nabla\) and average along the streamlines. In terms of the Fourier series notation introduced above, we have \[\begin{align} \label{def:P0Fourier} P_0(\widehat{ f \circ \Phi})(k,h)= \begin{cases} \widehat{f \circ \Phi}(0,h) &\text{if k = 0,} \\ 0 &\text{if k \neq 0,} \end{cases} \qquad P_\perp =I-P_0. \end{align}\tag{30}\] Thanks to the Fourier series characterization, we see that \(P_0\), and therefore \(P_\perp\), commutes with \(u\cdot \nabla\).

We also need to use more general Fourier multipliers, for which we use the following notation. We say that \(\mathcal{W}\colon L^2(M) \to L^2(M)\) is a Fourier multiplier with with Fourier symbol \(w(k,h):\mathbb{Z}\times I\to \mathbb{R}\) if \[\label{def:W} \widehat{((\mathcal{W}f) \circ \Phi)}(k, h) = w(k, h) \widehat{f \circ \Phi}(k, h).\tag{31}\] For instance, \(P_0\) is a Fourier multiplier with symbol \(\mathbb{1}_{k=0}(h)\).

2.3 Hypotheses on the Hamiltonian↩︎

We are now ready to introduce the two classes of Hamiltonians for which we prove Theorems 1 and 2 and Corollary 1. We start with the least restrictive class for which we can prove Theorem 2 and Corollary 1.

For Hamiltonians belonging to this class, it is not hard to show the following result using the fact that the elliptic point is non-degenerate. The proof simply follows by Taylor expansion around the elliptic point.

2.4 Fourier multipliers and derivatives↩︎

In the sequel, we need a precise control of the interplay between derivatives and the Fourier multipliers defined in 31 . Indeed, the operators \(\nabla\) and \(\mathcal{W}\) do not commute for a general Hamiltonian (as opposed to the shear or radial flow case). However, for Hamiltonians in the classes introduced above we are able to prove the following.

2.5 Poincaré-type inequalities on thin sets↩︎

One of the key ingredients for the subsequent pseudospectral estimates, are Poincaré-type inequalities on thin neighbourhoods of streamlines. These are needed to quantitatively estimate where one can gain something from the transport operator, as we explain more precisely in Section 4. The prototypical bound we aim at proving is the following, which is what one needs in the case of a radial Hamiltonian: \[\int_{\{R_1\leq |x|\leq R_2\}}|f|^2(x){\rm d}x\leq 2(R_2-R_1)\left\lVert f \right\rVert_{L^2}\left\lVert \nabla f \right\rVert.\] A proof of this bound can be found in Gallay:2021aa?, and notice that one can effectively gain smallness parameters whenever \(|R_2-R_1|\ll 1\). However, to the best of our knowledge, there are no such results that can be applied as a black box in our context. Therefore, in this section we prove in detail the estimates we need, some of which might be of independent interest.

With the notation introduced in Section 2, we denote a rectangle in \(\mathbb{T}\times I\) as \[\label{def:Box} Q_{\eta, \gamma}(\overline{\theta}, \overline{h}) \mathrel{\vcenter{:}}= \{ (\theta, h) \in \mathbb{T}\times I : |\theta - \overline{\theta}| < \eta, |h - \overline{h}| < \gamma \} \subset \mathbb{T}\times I.\tag{43}\] In account of the periodicity of the domain, when \(\eta=1\) there is no need to specify \(\overline{\theta}\) and we simply denote \[\label{def:Box1} Q_{ \gamma}(\overline{h})= Q_{1, \gamma}(0, \overline{h})\tag{44}\] Thanks to the map \(\Phi\) in 22 , we define the stream-adapted rectangles as \[\label{def:cBox} \mathcal{Q}_{\eta,\gamma}(\bar{\theta},\bar{h}):=\Phi(Q_{\eta, \gamma}(\overline{\theta}, \overline{h})).\tag{45}\] Notice that \(\mathcal{Q}_{\cdot}(\cdot)\) is a collection of ‘curly’ rectangles around the streamlines \(\{H= q(\bar{h}) \}\). In all the definitions above, we refer to \((\overline{\theta}, \overline{h})\) as the center of the rectangle or stream-adapted rectangle. Whenever it is clear from the context, we will choose not to write \((\overline{\theta}, \overline{h})\) to lighten the notation.

The following result is the main result of this section.

The inequality 47 quantitatively exploits the smallness in the streamwise (or angular) direction, as can be seen by the factor \(\eta\) multiplying the second term in 47 . Similarly, the inequality 46 uses the thinness in the streakwise (or action) direction. The first term in both inequalities containing the factor \(1/N\) is related to the need of removing the average over a given direction in the domain, and in fact in 48 is not present. This will still be sufficient for our purposes because we can use \(1/N\) as a smallness parameter to absorb that error with the \(L^2\) norm in the full domain.

3 The model problem as the leading order approximation for \(\rho_\perp\)↩︎

In this section, we aim at showing that our model problem 14 , namely, \[\begin{cases} {\partial}_tg +u\cdot \nabla g=\nu P_\perp\Delta P_\perp g, \qquad x \in M, \, t> 0,\\ g|_{t=0}=\rho_{\perp}^{in} \qquad g|_{\partial M}=0, \text{ or }\partial_n g|_{\partial M}=0, \end{cases}\] can be thought as the leading order approximation describing the dynamics of \(\rho_\perp\), which solves 2 . In this section we assume the result in Theorem 2, namely we need the bound 15 on \(g\). With this bound at hand, we first prove first prove Theorem 1 and then Corollary 1 and Proposition 4.

3.1 Proof of Theorem 1↩︎

First of all, we observe that since \(\nabla \cdot u=0\) and we have Dirichlet or Neumann boundary conditions on \(\rho\), a standard energy estimate for 1 give us \[\frac{1}{2}\frac{{\rm d}}{{\rm d}t}\left\lVert \rho \right\rVert^2_{L^2}=-\nu \left\lVert \nabla \rho \right\rVert^2_{L^2}\leq -\frac{\nu}{C_P} \left\lVert \rho \right\rVert^2,\] where we used the Poincaré inequality (which holds true since we have \(\int_M\rho^{in}=0\) and the domain \(M\) is bounded). With the notation in Section 2.2, from the inequality above we get \[\label{bd:heatdecay} \left\lVert \rho(t) \right\rVert_{L^2}^2=\left\lVert \rho_0(t) \right\rVert^2_{L^2}+\left\lVert \rho_\perp(t) \right\rVert^2_{L^2}\leq {\rm e}^{-2(\nu/C_P ) t}\left\lVert \rho^{in} \right\rVert_{L^2}^2.\tag{51}\] When \(\rho_{0}^{in}\neq 0\), the bound above readily implies the decay on time-scales \(O(\nu^{-1})\) for \(\rho_0\) stated in 10 . Notice that when \(\rho_0^{in}=0\) we have to gain an extra factor of \(\varepsilon\), that does not follow by the standard energy estimate above.

To improve the bounds on \(\rho_\perp\) (and \(\rho_0\) when \(\rho_0^{in}=0\)), we use that \(H\) belongs to the class 5 to perform an asymptotic expansion in \(\varepsilon\). We recall that \(\varepsilon\) is the smallness parameter quantifying how far we are from being “radial", more precisely we need the hypotheses [item:ClassPCond4] and [item:ClassPCond5] in Definition [def:ClassHamiltoniansP]. We want to construct solutions to 2 as \[\label{eq:expansion} \rho_\perp=:g+\sum_{n=1}^{+\infty}\varepsilon^{n}\rho^{(n)}_{\perp}=:g+g_{\mathrm{corr}}, \qquad \rho_0=\eta+\sum_{n=1}^{+\infty}\varepsilon^{n}\rho^{(n)}_{0},\tag{52}\] where \(g\) solves 14 , \(\eta\) is the solution to 16 . We define the approximations as follows⁶: denote \(\rho_\perp^{(0)}:=g\) and \(\rho_0^{(0)}=\eta\).

• For \(n\geq 1\), we define \(\rho^{(n)}_\perp\) to be the solution of \[\label{pb:rhoperpn} \begin{cases} \displaystyle {\partial}_t\rho^{(n)}_\perp +u\cdot\nabla \rho^{(n)}_\perp=\nu P_\perp\Delta \rho^{(n)}_\perp+\frac{\nu}{\varepsilon}P_\perp\Delta \rho^{(n-1)}_0,\\ \rho^{(n)}_\perp|_{t=0}=0. \end{cases}\tag{53}\]

• For \(n\geq 1\) we define \(\rho^{(n)}_0\) to be the solution of \[\label{pb:rho0n} \begin{cases} \displaystyle {\partial}_t\rho^{(n)}_0 =\nu P_0\Delta \rho^{(n)}_0+\frac{\nu}{\varepsilon}P_0\Delta \rho^{(n-1)}_\perp,\\ \rho^{(n)}_0|_{t=0}=0, \end{cases}\tag{54}\]

With the definitions above, if the series in 52 is well defined, it is not difficult to see that indeed \(\rho_\perp,\rho_0\) solve 2 . In particular, we aim at proving that \(\rho^{(n)}_{\iota}\), with \(\iota\in\{\perp,0\}\), remain sufficiently small in a suitable functional space. This would imply the desired convergence of the series and smallness of \(g_{\mathrm{corr}}\). Hence, the decay on the time-scale \(O(\lambda_\nu)\) in 8 is coming from \(g\) and Theorem 2, whereas the other error is related to the bounds on \(\rho^{(n)}_\iota\). Moreover, we also need to show that \(g_{\mathrm{corr}}\) decays at least on \(O(\nu^{-1})\) time-scales, but this will be a simple consequence of the Poincaré inequality.

To obtain the desired bounds on \(\rho_\iota^{(n)}\) we heavily rely on the assumptions in Definition [def:ClassHamiltoniansP]. We are able to prove the following.

Before proving this proposition, let us show that Theorem 1 is a direct consequence of Proposition 11 and Theorem 2.

It thus remain to prove Proposition 11.

3.2 Proof of Corollary 1↩︎

The proof of this corollary is based on a standard comparison argument as done, for instance, in CZDE20?. Indeed, let \(T_\nu=C_* \nu^{-\frac{m}{m+2}}\) with \(C_*\) to be specified later. Then, let \(\tilde{c}_*\in (0,1)\) be a fixed constant. If \[2\nu \int_0^{T_\nu}\left\lVert \nabla \rho(\tau) \right\rVert^2_{L^2}{\rm d}\tau\geq \tilde{c}_*\|{\rho^{in}_\perp}\|^2_{L^2},\] by the standard energy estimate on 1 , since \(\rho^{in}=\rho^{in}_\perp\), we get \[\begin{align} \left\lVert \rho(T_\nu) \right\rVert^2_{L^2}=\|{\rho^{in}_\perp}\|^2_{L^2}-2\nu \int_0^{T_\nu}\left\lVert \nabla \rho(\tau) \right\rVert^2_{L^2}{\rm d}\tau\leq (1-\tilde{c}_*)\|{\rho^{in}_\perp}\|^2_{L^2}. \end{align}\] By the properties of \(P_0\) , we have \[\|\rho_\perp(T_\nu)\|^2_{L^2}\leq\left\lVert \rho(T_\nu) \right\rVert^2_{L^2}\leq (1-\tilde{c}_*)\|{\rho^{in}_\perp}\|^2_{L^2},\] thus proving the desired result in this case. On the other hand, consider now the case \[\label{bd:hypcstar} 2\nu \int_0^{T_\nu}\left\lVert \nabla \rho(\tau) \right\rVert^2_{L^2}{\rm d}\tau< \tilde{c}_*\|{\rho^{in}_\perp}\|^2_{L^2}.\tag{66}\] Let \(g\) be the solution to 14 . Then, by the linearity of the equations considered, we notice that \(\rho_\perp-g\) solves \[{\partial}_t(\rho_\perp-g)+u\cdot \nabla(\rho_\perp-g)=\nu P_\perp\Delta (\rho_\perp-g) +\nu P_\perp\Delta P_0\rho,\] with zero initial data. Therefore, multiplying by \(\rho_\perp-g\) and integrating in space, we deduce that \[\frac{1}{2} \frac{{\rm d}}{{\rm d}t}\|\rho_\perp-g\|^2_{L^2}+\nu \|\nabla (\rho_\perp-g)\|^2_{L^2}=-\nu \left\langle \nabla (P_0\rho), \nabla (\rho_\perp-g) \right\rangle_{L^2}.\] Applying the Cauchy-Schwarz and Young’s inequality, and integrating in time, we get \[\label{bd:energy} \|(\rho_\perp-g)(T_\nu)\|^2_{L^2}+\nu \int_0^{T_\nu}\|\nabla (\rho_\perp-g)(\tau)\|^2_{L^2}{\rm d}\tau\leq \nu \int_0^{T_\nu}\|\nabla (P_0 \rho)(\tau)\|^2_{L^2}{\rm d}\tau.\tag{67}\] Applying Lemma 4 and in particular 33 with \(w=\mathbb{1}_{k=0}\), we know that for an Hamiltonian in class [HamiltonianClassAm] we have \[\begin{align} \|\nabla (P_0 \rho)(\tau)\|^2_{L^2}\leq C_H \|\nabla \rho(\tau)\|_{L^2}^2 \end{align}\] for a suitable constant \(C_H>0\) depending only the Hamiltonian \(H\). Combining the bound above with the energy estimate 67 and the condition 66 , we see that \[\begin{align} \|(\rho_\perp-g)(T_\nu)\|^2_{L^2}\leq \tilde{c}_* C_H\|\rho^{in}\|^2_{L^2}. \end{align}\] Finally, thanks to the bounds on \(g\) in 15 we see that \[\begin{align} \|\rho_\perp(T_\nu)\|^2_{L^2}\leq2( \|g(T_\nu)\|^2_{L^2}+\|(\rho_\perp-g)(T_\nu)\|^2_{L^2})\leq 2({\rm e}^{-\delta_* C_*+\pi/2}+\tilde{c}_* C_H)\|\rho^{in}\|^2_{L^2}. \end{align}\] Choosing \(C_*\) sufficiently large and \(\tilde{c}_*\) sufficiently small, we get \[\begin{align} \|\rho_\perp(T_\nu)\|^2_{L^2}\leq \frac{1}{2}\|\rho^{in}\|^2_{L^2}. \end{align}\] Therefore, we conclude the proof of the corollary by choosing \(c_*\) such that \(1-c_*=\max\{ 1/2, 1-\tilde{c}_*\}\). 0◻

3.3 Proof of Proposition 4↩︎

The proof is based on iteration of the previous corollary, using crucially the assumption on the uniform estimate of \(\| (\rho_0 - \eta) (t) \|_{L^2}\leq \nu^\alpha \|\rho^{in}\|_{L^2}\). Let \(T_{\nu}=C_* \nu^{-\frac{m}{m+2}}\) with \(C_*\) sufficiently large and \(\tilde{c}_*\in (0,1/2)\) sufficiently small as in the proof of Corollary 1. In the iteration, we assume \((n+1) T_{\nu} \leq \nu^{-1}\), where the case \(n=0\) is the previous corollary, and divide the proof into two distinct cases.

Case 1: Assume

\[\label{eq:CondCase1} 2\nu \int_{n T_{\nu}}^{(n+1) T_{\nu}}\left\lVert \nabla \rho(\tau) \right\rVert^2_{L^2}{\rm d}\tau\geq \tilde{c}_* \| \rho_\perp (n T_{\nu} )\|^2_{L^2}.\tag{68}\] By properties of the projection \(P_0\) and the fact that \(P_\perp \eta=0\), we get \[\begin{align} \| (\rho & - \eta) ((n+1) T_{\nu})\|^2_{L^2} =\| \rho_\perp ((n+1) T_{\nu}) \|^2_{L^2} + \left\lVert (\rho_0 - \eta )((n+1)T_{\nu}) \right\rVert_{L^2}^2 \\ \label{id:pop37} & = \| \rho ((n+1 )T_{\nu}) \|_{L^2}^2 - \left\lVert \rho_0 ((n+1)T_{\nu}) \right\rVert_{L^2}^2 + \left\lVert (\rho_0 - \eta) ((n+1)T_{\nu}) \right\rVert_{L^2}^2. \end{align}\tag{69}\] Using the assumption, we can bound the third term above by \(\nu^{2\alpha} \| \rho^{in}\|_{L^2}^2\). By standard energy estimates on 1 , thanks to 68 , the first term can be bounded by \[\begin{align} \| \rho (n T_{\nu}) \|_{L^2}^2 - \tilde{c}_*\| \rho_\perp (n T_{\nu}) \|_{L^2}^2 &= (1- \tilde{c}_*) \| \rho_\perp (n T_{\nu}) \|_{L^2}^2 + \| \rho_0 (n T_{\nu}) \|_{L^2}^2 \\ &= (1- \tilde{c}_*) \| (\rho- \rho_0) (n T_{\nu}) \|_{L^2}^2 + \| \rho_0 (n T_{\nu}) \|_{L^2}^2 \\ &\leq (1- \tilde{c}_*) \| (\rho - \eta) (n T_{\nu}) \|_{L^2}^2 +\| \eta (n T_{\nu}) \|_{L^2}^2 \\ & \quad + 2 \| (\eta - \rho_0) (n T_{\nu}) \|_{L^2}^2 \\ & \quad + 4 \| (\eta - \rho_0)(n T_{\nu}) \|_{L^2} ( \| \eta (n T_\nu) \|_{L^2} + \| (\rho - \eta) ( n T_\nu) \|_{L^2} ) \end{align}\] where in the last inequality we sum and subtract \(\eta\) in both terms and perform trivial bounds by expanding the squares. Having that both \(\rho\) and \(\eta\) are decreasing and thus bounded by the intial data, in view of the assumption on \(\rho_0-\eta\) we get \[\begin{align} \| \rho ((n+1 )T_{\nu}) \|_{L^2}^2 & \leq (1- \tilde{c}_*) \| (\rho- \eta )(n T_{\nu}) \|_{L^2}^2 +\| \eta (n T_{\nu}) \|_{L^2}^2 + 10 \nu^\alpha \| \rho^{in} \|_{L^2}^2. \end{align}\] Analogously, for the term involving \(\left\lVert \rho_0 ((n+1)T_{\nu}) \right\rVert_{L^2}^2\) we get \[\begin{align} \left\lVert \rho_0 ((n+1)T_{\nu}) \right\rVert_{L^2}^2 & \geq \left\lVert \eta ((n+1) T_\nu) \right\rVert_{L^2}^2 - 2 \| (\rho_0 - \eta)((n+1)T_{\nu}) \|_{L^2} \| \eta ((n+1)T_{\nu}) \|_{L^2} \\ & \geq \left\lVert \eta ((n+1) T_\nu) \right\rVert_{L^2}^2 - 2 \nu^\alpha \| \rho^{in} \|_{L^2}^2. \end{align}\] Thus, all in all, we deduce that \[\begin{align} \| (\rho & - \eta) ((n+1) T_{\nu})\|^2_{L^2} \\ & \leq (1 - \tilde{c}_* ) \| (\rho - \eta) (n T_{\nu}) \|_{L^2}^2 + \left\lVert \eta ( nT_{\nu}) \right\rVert_{L^2}^2 - \left\lVert \eta ((n+1)T_{\nu}) \right\rVert_{L^2}^2 + 10 \nu^\alpha \| \rho^{in}\|_{L^2}^2. \end{align}\] Then, since \(\tau \mapsto \| \nabla \eta(\tau) \|_{L^2}\) is decreasing (this can be proved by testing 16 by \(P_0 \Delta P_0 \eta\) and using that \(P_0 \eta = \eta\)), we notice that \[\begin{align} \| \eta ((n+1) T_{\nu}) - \eta ( n T_{\nu}) \|_{L^2}^2 & \leq \nu \int_{nT_{\nu}}^{(n+1) T_{\nu}} \| \nabla \eta (n T_{\nu}) \|_{L^2}^2 ds \\ & \leq C_* \nu^{\frac{2}{m+2}} \| \nabla \rho^{in} \|_{L^2}^2 \leq C C_* \nu^{ \frac{2}{m+2}} \| \rho^{in} \|_{L^2}^2 \, \end{align}\] Therefore, from the previous bound we conclude that there exists a constant \(C_0\) such that \[\begin{align} \label{eq:bound:iterate} \| (\rho - \eta) ((n+1) T_{\nu})\|^2_{L^2} \leq (1 - \tilde{c}_* ) \| (\rho - \eta) (n T_{\nu}) \|_{L^2}^2 +( C_0 \nu^{\frac{2}{m+2}} + 10 \nu^\alpha) \| \rho^{in} \|_{L^2}^2 \,. \end{align}\tag{70}\]
Case 2: Assume \[2\nu \int_{n T_{\nu}}^{(n+1) T_{\nu}}\left\lVert \nabla \rho(\tau) \right\rVert^2_{L^2}{\rm d}\tau < \tilde{c}_* \| \rho_\perp (n T_{\nu} )\|^2_{L^2}.\] We get \[\begin{align} \|(\rho_\perp-g)((n+1)T_{\nu})\|^2_{L^2} & \leq \nu \int_{n T_{\nu}}^{(n+1)T_{\nu}}\|\nabla (P_0\rho)(\tau)\|^2_{L^2}{\rm d}\tau \\ & \leq C_H \nu \int_{n T_{\nu}}^{(n+1)T_{\nu}} \|\nabla \rho(\tau)\|^2_{L^2}{\rm d}\tau \\ & \leq C_H \tilde{c}_* \| \rho_\perp (n T_{\nu}) \|_{L^2}^2 \end{align}\] where where \(g\) is the solution to the model problem 14 initialised at time \(n T_{\nu}\) with initial data \(\rho_{\perp}(n T_{\nu})\). From this, we conclude that \[\begin{align} \|\rho_\perp((n+1)T_{\nu})\|^2_{L^2}& \leq2( \|g((n+1)T_{\nu})\|^2_{L^2}+\|(\rho_\perp-g)((n+1)T_{\nu})\|^2_{L^2}) \\ & \leq 2({\rm e}^{-\delta_* C_*+\pi/2}+\tilde{c}_* C_H)\|(\rho - \rho_0) (n T_{\nu}) \|^2_{L^2}. \end{align}\] and by choosing \(C_*\) large enough and \(c_*\) sufficiently small (exactly as the ones in the proof of Corollary 1), we have \[\| (\rho - \rho_0 )((n +1)T_{\nu}) \|_{L^2}^2 \leq \frac{1}{2} \|(\rho - \rho_0) (n T_{\nu}) \|^2_{L^2}\,.\] Therefore, using the assumption, summing and subtracting \(\eta\) we get \[\| (\rho - \eta) ((n +1)T_{\nu}) \|_{L^2}^2 \leq \frac{1}{2} \| (\rho - \eta) (n T_{\nu}) \|_{L^2}^2 + 6 \nu^\alpha \| \rho^{in} \|_{L^2}^2 \,.\]

Finally, in both cases, we are able to prove the bound 70 . Iterating this bound we get \[\begin{align} \label{bd:iterativefinal} \| (\rho - \eta) ((n+1) T_{\nu})\|^2_{L^2} & \leq (1- \tilde{c}_*)^{n+1} \| \rho^{in}_{\perp} \|_{L^2}^2 + ( C_0 \nu^{\frac{2}{m+2}} + 10 \nu^\alpha) \| \rho^{in} \|_{L^2}^2 \sum_{j=0}^{n} (1- \tilde{c}_*)^{j}. \end{align}\tag{71}\] To conclude the bound, we need to control \((\rho-\eta)(t)\) for \(t\in (nT_\nu,(n+1)T_\nu)\), where it is not hard to see that it can be controlled by \(\|(\rho-\eta)(nT_\nu)\|^2\) up to an error of size \(\max\{\nu^{\frac{2}{m+2}},\nu^\alpha\}\|\rho^{in}\|^2\). To see this, we can repeat computations similar to the Case 1. The starting identity 69 is replaced by \[\begin{align} \notag \| (\rho - \eta) (t)\|^2_{L^2} =\| \rho (t) \|_{L^2}^2 - \left\lVert \rho_0 (t) \right\rVert_{L^2}^2 + \left\lVert (\rho_0 - \eta) (t) \right\rVert_{L^2}^2. \end{align}\] We can then bound the first term by \(\|\rho(nT_\nu)\|_{L^2}^2\) and repeat the computations in Case 1 with \(\tilde{c}_*=0\) to get \[\begin{align} \| \rho (t) \|_{L^2}^2 & \leq \| (\rho- \eta )(n T_{\nu}) \|_{L^2}^2 +\| \eta (n T_{\nu}) \|_{L^2}^2 + 10 \nu^\alpha \| \rho^{in} \|_{L^2}^2. \end{align}\] For the remaining terms we argue exactly as in Case 1 and, using 71 , we finally see that there exists \(C_1>0\) such that \[\| (\rho - \eta) (t )\|^2_{L^2} \leq (1- \tilde{c}_*)^{n} \| \rho^{in} \|_{L^2}^2 + C_1 ( \nu^{\frac{2}{m+2}} + \nu^\alpha ) \| \rho^{in} \|_{L^2}^2\] for any \(t\in (nT_{\nu},(n+1)T_{\nu})\). Therefore, recalling that \(T_{\nu} = C_* \nu^{- \frac{m}{m+2}} = C_* \lambda_\nu^{-1}\) there exists \(\delta_*\) such that \[\| (\rho - \eta) (t )\|^2_{L^2} \lesssim \left ( \exp (- \delta_* \lambda_\nu t) + \nu^{\frac{2}{m+2}} + \nu^\alpha \right ) \| \rho^{in} \|_{L^2}^2\] for any \(t \leq \nu^{-1}\). 0◻

4 Pseudo spectral bounds on the model problem↩︎

In this section, we prove Theorem 2, which quantifies the enhanced dissipation effect for the model problem 14 . The proof of the bound 15 is in fact a direct consequence of the Gearhart-Prüss type theorem proved by Wei in WeiDiffusion19? (see also Helffer:2010aa?, HelfferImproving21?, Gallagher09?), which we state here for completness.

Therefore, the proof of Theorem 2 is reduced to an estimate of the pseudo spectral abscissa of the operator \[\label{tmqzcgpy} \mathcal{L}_{\perp}=u\cdot \nabla -\nu P_\perp \Delta P_\perp,\tag{74}\] when \(X=L^2_\perp(M)\). Depending on the boundary conditions we have, the domain of this operator will be \[D(\mathcal{L}_\perp)=\{f \in H^2(M) \, ;\, {\partial}_n f|_{\partial M} =0\} \qquad \text{ or } \qquad D(\mathcal{L}_\perp)=\{f \in H^2(M) \, ;\, f|_{\partial M} =0\}.\] First of all, we need the following

We are now in the position of applying Theorem 3 to the operator \(\mathcal{L}_\perp\), meaning that we need to estimate the pseudo spectral abscissa defined in 73 . The main result in this section is the following.

The proof strategy of this proposition is inspired by what is done for shear and radial flows Gallay:2021aa?. However, here there are major differences that give rise to several technical challenges to overcome. One major obstacle is the fact that we cannot perform a \(k\)-by-\(k\) analysis since the operator \(P_\perp\Delta P_\perp\) does not decouple in frequencies. The decoupling in frequencies is helpful since one can replace \(\lambda\) by \(\lambda/k\) and study a simpler one dimensional problem. On the other hand, here we need to be much more careful since we will need to sum over all \(k\)’s, and to ensure summability we have to exploit specific properties of the Hamiltonians we consider.

The rest of this section is dedicated to the proof of Proposition 14, which we have to split in different cases that needs to be treated separately.

4.1 Proof of Proposition 14↩︎

By the definition 73 , the goal is to prove \[\label{bd:pseudolambda} \left\lVert (\mathcal{L}_\perp-i\lambda)f \right\rVert_{L^2_\perp}\geq \delta_* \lambda_\nu\left\lVert f \right\rVert_{L^2_\perp}, \qquad \text{ for all } \lambda \in \mathbb{R}.\tag{76}\] In the sequel, we will often omit the \(\perp\) and \(L^2_\perp\) subscript to ease the notation, and we denote \[\begin{align} &\mathcal{L}_\lambda:=\mathcal{L}_\perp-i\lambda=(u\cdot \nabla -i\lambda)-\nu P_\perp\Delta P_\perp,\\ &\left\langle f_1,f_2 \right\rangle:=\left\langle f_1,f_2 \right\rangle_{L^2(M)}=\int_M f_1 f_2 {\rm d}x, \qquad \text{for all } f_1,f_2\in L^2_\perp(M). \end{align}\]

A first rough upper bound on the pseudo spectral abscissa is readily obtained from the dissipative part of the operator. Namely, since \(P_\perp f=f\) for all \(f\in L^2_{\perp}\), integrating by parts we get \[{\rm Re}(\left\langle \mathcal{L}_{\lambda} f,f \right\rangle)=-\nu \left\langle \Delta P_\perp f, P_\perp f \right\rangle=-\nu \left\langle \Delta f, f \right\rangle=\nu \left\lVert \nabla f \right\rVert^2\] Thus, from the Cauchy-Schwarz inequality we have \[\label{bd:ReL0} \nu \left\lVert \nabla f \right\rVert^2=|{\rm Re}(\left\langle \mathcal{L}_{\lambda} f,f \right\rangle)|\leq \left\lVert \mathcal{L}_{\lambda} f \right\rVert\left\lVert f \right\rVert.\tag{77}\] Since \(P_0 f=0\), we can apply Poincaré’s inequality (see also 48 ) above and obtain \[\label{bd:ReL} \left\lVert f \right\rVert^2\leq C_P\left\lVert \nabla f \right\rVert^2\qquad \Longrightarrow \qquad \left\lVert \mathcal{L}_{\lambda} f \right\rVert\geq \frac{\nu}{C_P} \left\lVert f \right\rVert,\tag{78}\] which is the standard spectral gap one expects from the dissipative operator \(-\nu P_\perp \Delta P_\perp\). To improve this bound, we have to crucially exploit the advection, which in the estimate above has not been used. With the notation introduced in Section 2.2, we know that \[\mathcal{F}(u\cdot \nabla -i \lambda)=ik (\Omega(h)-\lambda/k).\] Since we are working on \(L^2_\perp\), we have \(k\neq 0\). Thus, we can hope to gain information from this part of the operator \(\mathcal{L}_\lambda\) whenever \(\Omega(h)-\lambda/k\) is bounded from below. We need to introduce some sets and distinguish different cases to account for this. The general idea is to find a thin set \(\mathcal{E}_{\lambda,\delta}\) concentrated where \(|\Omega(h)-\lambda/k|\ll1\) for all \(k\) and split \[\label{eq:split} \int_M |f|^2 {\rm d}x=\int_{M\setminus \mathcal{E}_{\lambda,\delta}} |f|^2 {\rm d}x+\int_{\mathcal{E}_{\lambda,\delta}} |f|^2 {\rm d}x.\tag{79}\] For the integral concentrated on \(\mathcal{E}_{\lambda,\delta}\), we exploit the “thinness/smallness" of this set through the inequalities in Lemma 5. On the set \(M\setminus \mathcal{E}_{\lambda,\delta}\) we exploit the advection via the Fourier multiplier method used in WeiDiffusion19?, Gallay:2021aa?, LiPseudo20? for instance, suitably adapted to our case where we cannot handle each angular mode separately.

In the following, we recall that \(m\in \mathbb{N}\) is the order of the critical point of the period function \(\Omega(h)\), see Definition 4. Then, we fix two constants \[0<\delta=\delta(\nu)\ll 1, \qquad 0<\gamma_0\ll 1\] which will be specified later in the proof.

We divide the proof on two cases, depending on how large \(\lambda\) is.

4.1.1 Case \(\boldsymbol{| \lambda | \leq \gamma_0\delta^{-1}}\)↩︎

To perform the splitting argument in 79 , we have to first introduce the thin sets and the main Fourier multiplier which will allows us to exploit properties of the advection operator. After proving some key properties of these sets, we can proceed with the pseudospectral bounds.

\(\circ\) Thin sets. Given \(\lambda\), \(k\neq0\) and \(0<\delta\ll 1\), we define the thickened level sets \[\label{def:Eklambda} E_{k, \lambda, \delta} \mathrel{\vcenter{:}}= \left\{ (\theta, h) \in \mathbb{T}\times I : \left|\Omega(h) - \frac{\lambda}{k} \right| < \delta^m \right\}, \qquad \mathcal{E}_{k,\lambda,\delta}=\Phi( E_{k, \lambda, \delta}),\tag{80}\] where we are following the notation introduced in Section 2. We use bold symbols to denote a \(\delta\)-neighbourhood of the sets above, namely \[\label{def:Eklambdathick} \boldsymbol{E}_{k, \lambda, \delta} =\{h \in I \, : \, \mathrm{dist}(h,E_{k,\lambda,\delta})<\delta\}, \qquad \boldsymbol{\mathcal{E}}_{k, \lambda, \delta}=\Phi( \boldsymbol{E}_{k, \lambda, \delta}).\tag{81}\] Notice that, with the stream-adapted rectangles defined in 44 , we also know \[\boldsymbol{E}_{k, \lambda, \delta}=\bigcup_{\{h'\in E_{k,\lambda,\delta}\}} Q_{\delta}(h'),\] where clearly one can extract a finite covering with rectangles thanks to the compactness of the set \(E_{k,\lambda,\delta}\). For technical reasons, it is convenient to directly isolate also a set around the critical point without having any dependence on \(k\). Hence, since we are assuming that the critical point is at \(h=0\), following the notation 44 , given \(\mathsf{C}_0\) sufficiently large to be specified afterwards, we define ⁸ \[\label{def:Bdelta} Q_{\mathsf{C}_0\delta} =Q_\delta(0)= \{ (\theta, h) \in \mathbb{T}\times I : |h| \leq \mathsf{C}_0 \delta \}, \qquad \mathcal{Q}_{\mathsf{C}_0\delta}=\Phi( Q_{\mathsf{C}_0\delta}).\tag{82}\] Finally, we choose the set for the splitting argument in 79 as \[\label{def:Elambda} \mathcal{E}_{\lambda,\delta} \mathrel{\vcenter{:}}= \mathcal{Q}_{\mathsf{C}_0\delta} \cup \left ( \bigcup_{k \in \mathbb{Z}\setminus \{ 0\}} \boldsymbol{\mathcal{E}}_{k, \lambda, \delta} \cap \mathcal{Q}_{\mathsf{C}_0\delta}^c \right ), \qquad E_{\lambda,\delta}=\Phi^{-1}(\mathcal{E}_{\lambda,\delta}).\tag{83}\] These sets satisfies the following properties.

\(\circ\) Bounds close to level sets: with the sets \(\mathcal{E}_{\lambda,\delta}\) 83 and Lemma 7 at hand, we are ready to estimate the second term in 79 . In particular, we claim that there exists a universal constant \(C_l>0\) such that \[\label{bd:claimE} \int_{\mathcal{E}_{\lambda,\delta} } |f|^2 \, dx \leq C_l\frac{\delta^2}{\nu}\| \mathcal{L}_{\lambda} f \|\left\lVert f \right\rVert+ \frac{1}{2} \| f \|^2 \,.\tag{90}\] To prove this claim, appealing to 48 and Lemma 3, we first notice that \[\int_{\mathcal{Q}_\delta } |f|^2 \, dx\leq \left\lVert T (\nabla H\circ \Phi) \right\rVert_{L^\infty(Q_\delta)}^2\left\lVert \nabla f \right\rVert_{L^2(\mathcal{Q_\delta})}^2\lesssim \delta^2\left\lVert \nabla f \right\rVert^2.\] Using 77 , we get \[\label{bd:Bdelta} \int_{\mathcal{Q}_\delta } |f|^2 \, dx\leq \tilde{C} \frac{\delta^2}{\nu}\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert\tag{91}\] for some universal constant \(\tilde{C}\). This bound is consistent with the claim 90 .

Let us now consider the bound on the set \(\boldsymbol{\mathcal{E}}_{k,\lambda,\delta}\cap Q_{\delta}^c\), where we want to apply the Poincaré-type inequality across the streamlines 46 . In particular, thanks to 84 , we observe that \[\begin{align} \int_{\boldsymbol{\mathcal{E}}_{k,\lambda,\delta}\cap \mathcal{Q}_{\delta}^c}|f|^2{\rm d}x&= \int_{\boldsymbol{E}_{k,\lambda,\delta}\cap Q_{\delta}^c}|f\circ\Phi|^2q'(h)T(h){\rm d}h{\rm d}\theta\\ &\leq \int_{ Q_{C_*\delta}(h_{k})}|f\circ\Phi|^2q'(h)T(h){\rm d}h{\rm d}\theta=\left\lVert f \right\rVert^2_{L^2(\mathcal{Q}_{C_*\delta}(h_{k}))}, \end{align}\] where \(h_k=h_{k,\lambda,\delta}\). Therefore, applying 46 with \(\eta=1\), \(\gamma=C_*\delta\) and \(K=32\), we get \[\begin{align} \int_{\boldsymbol{\mathcal{E}}_{k,\lambda,\delta}\cap \mathcal{Q}_{\delta}^c}|f|^2{\rm d}x\leq \, &\frac{1}{4} \left\lVert f \right\rVert^2_{L^2(\mathcal{Q}_{32C_*\delta}(h_{k}))}\\ \notag& +32C_*\delta \left\lVert f \right\rVert_{L^2(\mathcal{Q}_{32C_*\delta}(h_{k}))}\left\lVert \nabla f \right\rVert_{L^2(\mathcal{Q}_{32C_*\delta}(h_{k}))}\left\lVert {\partial}_h\Phi \right\rVert_{L^\infty(Q_{32C_*\delta}(h_{k}))}\\ \label{bd:Ek} & +16C_*\delta \left\lVert f \right\rVert_{L^2(\mathcal{Q}_{32C_*\delta}(h_{k}))}^2\left\lVert [\log(q'T)]' \right\rVert_{L^\infty(Q_{32C_*\delta}(h_{k}))}. \end{align}\tag{92}\] Thanks to the choice of the rescaled action angle variables we know that \({\partial}_h \Phi \in L^\infty\). To bound the last term in 92 , since \(h_k\in \boldsymbol{\mathcal{E}}_{k,\lambda,\delta} \cap Q_{\delta}^c\), we know that \[\begin{align} \left\lVert [\log(q'T)]' \right\rVert_{L^\infty(Q_{32C_*\delta}(h_{k}))}&\leq \left\lVert \left|\frac{T'}{T}\right|+\frac{q''}{q'} \right\rVert_{L^\infty(Q_{32C_*\delta}(h_{k}))}\\ \label{bd:log}&\lesssim \left\lVert h^{-1} \right\rVert_{L^\infty(Q_{32C_*\delta}(h_{k}))}\leq \frac{\tilde{C}_0}{\mathsf{C}_0\delta}, \end{align}\tag{93}\] where in the last inequality we used \(q(h)=h^2\), that \(T'/T\) bounded and \(\mathsf{C}_0\) is the constant involved in the definition 82 . Hence, taking \(\mathsf{C}_0\) sufficiently large we infer \[\begin{align} \label{bd:Ek1} \int_{\boldsymbol{\mathcal{E}}_{k,\lambda,\delta}\cap \mathcal{Q}_{\delta}^c}|f|^2{\rm d}x\leq \frac{3}{8} \left\lVert f \right\rVert^2_{L^2(\mathcal{Q}_{32C_*\delta}(h_{k}))}+\tilde{C}_1 \delta \left\lVert f \right\rVert_{L^2(\mathcal{Q}_{32C_*\delta}(h_{k}))}\left\lVert \nabla f \right\rVert_{L^2(\mathcal{Q}_{32C_*\delta}(h_{k}))}. \end{align}\tag{94}\] Using the bound above, in account of 85 , we can consider the union over \(k\) and prove that \[\begin{align} \int_{\cup_{k\in \mathbb{Z}\setminus \{0\}}\boldsymbol{\mathcal{E}}_{k,\lambda,\delta}\cap \mathcal{Q}_{\delta}^c}|f|^2{\rm d}x\leq \frac{3}{8}\left\lVert f \right\rVert^2+\tilde{C}_1\delta \left\lVert f \right\rVert\left\lVert \nabla f \right\rVert. \end{align}\] Using again 77 , we conclude that \[\begin{align} \int_{\cup_{k\in \mathbb{Z}\setminus \{0\}}\boldsymbol{\mathcal{E}}_{k,\lambda,\delta}\cap \mathcal{Q}_{\delta}^c}|f|^2{\rm d}x\leq \frac{1}{2}\left\lVert f \right\rVert^2+\tilde{C}_3\frac{\delta^2}{\nu} \left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert. \end{align}\] Combining the bound above with 91 , the claim 90 is proved.

\(\circ\) Bounds away from the level sets: Let us introduce the key Fourier multiplier needed to exploit the effects of the transport operator. We define \(\mathcal{X}\colon L^2(M) \to L^2(M)\) to be an operator with Fourier symbol \(\chi(k,h)\) (with the Fourier notation introduced in Section 2.2), namely \[\label{def:chi} \widehat{((\mathcal{X}f) \circ \Phi)}(k, h) = \chi(k, h) \widehat{f \circ \Phi}(k, h).\tag{95}\] We choose \(\chi \colon \mathbb{Z}\times I\to \mathbb{R}\) to be a function (which can be explicitly constructed) satisfying the following properties:

1. \(\chi ( k, h) \operatorname{sign}\left( k\Omega(h) - {\lambda} \right) \geq 0\) for any \(h\) and \(k \neq 0\).

2. \(\left\lVert \chi \right\rVert_{L^\infty}\leq 1\) and \(\left\lVert {\partial}_h\chi \right\rVert_{L^\infty} \lesssim \delta^{-1}\).

3. \(\chi(k, h) = \operatorname{sign}\left( k\Omega(h) - {\lambda} \right)\) for any \(h \in \boldsymbol{E}_{k, \lambda, \delta}^c\) and for any \(k \neq 0\).

4. \(\chi (0 , \cdot) \equiv 0\).

The key properties of \(\mathcal{X}\) are encoded in its definition. In particular, we see that we can apply the case a) in Lemma 4 with \(N=\delta^{-1}\).

We can now proceed with the proof. Integrating by parts, we have \[\begin{align} \label{eq:proof-identity} {\rm Im}\langle \mathcal{L}_{ \lambda} f , \mathcal{X}f \rangle = {\rm Im}\langle u \cdot \nabla f - i \lambda f, \mathcal{X}f \rangle+ \nu {\rm Im}\langle \nabla f, \nabla (\mathcal{X}f) \rangle:=\mathcal{I}_1+\mathcal{I}_2. \end{align}\tag{96}\] To control \(\mathcal{I}_1\), we exploit the change of variables introduced in Section 2.1 and the Fourier expansion in Section 2.2. In particular, we have \[\begin{align} \mathcal{I}_1 &= {\rm Im}\int_I \sum_{k\neq 0} i(k\Omega(h)-\lambda)\chi(k,h)|\widehat{f\circ \Phi}|^2(k,h) q(h)T(h){\rm d}h\\ &\geq\int_{E_{\lambda,\delta}^c} \sum_{k\neq 0} |k|\left|\Omega(h)-\frac{\lambda}{k}\right||\widehat{f\circ \Phi}|^2(k,h) q(h)T(h){\rm d}h\\ \label{bd:lowI10} &\geq \delta^m \int_{M\setminus{\mathcal{E}}_{\lambda,\delta}} |f|^2{\rm d}x, \end{align}\tag{97}\] where in the last inequality we used that \(h\notin E_{k,\lambda,\delta}\) (see 80 ) and we undid the change of variables.

To control \(\mathcal{I}_2\), by Cauchy-Schwarz and case a) in Lemma 4, we have \[\label{bd:I2} |\mathcal{I}_2|\leq \nu \left\lVert \nabla f \right\rVert\left\lVert \nabla (\mathcal{X}f) \right\rVert\lesssim \nu\left\lVert \nabla f \right\rVert^2+\frac{\nu}{\delta}\left\lVert \nabla f \right\rVert\left\lVert f \right\rVert\tag{98}\] Since \(|\chi|\leq 1\), we also know that \[\label{bd:claimW} \left\lVert \mathcal{X}f \right\rVert\leq \left\lVert f \right\rVert.\tag{99}\] Hence, combining 96 , 97 and 98 we get \[\begin{align} \int_{M\setminus \mathcal{E}_{\lambda,\delta}} |f|^2{\rm d}x&\leq \frac{1}{\delta^{m}} \mathcal{I}_1\leq \frac{1}{\delta^{m}}(\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert \mathcal{X}f \right\rVert+|\mathcal{I}_2|)\\ &\lesssim \frac{1}{\delta^{m}}\left(\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert+\nu \left\lVert \nabla f \right\rVert^2+\frac{\nu}{\delta}\left\lVert \nabla f \right\rVert \left\lVert f \right\rVert\right),\\ &\leq \frac{1}{\delta^{m}}\left(\tilde{C}_4\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert+\tilde{C}_5\delta_0^{-(2+m)}\nu \left\lVert \nabla f \right\rVert^2+\frac{\delta_0^{2+m}}{4}\frac{\nu}{\delta^2} \left\lVert f \right\rVert^2\right)\\ \label{bd:M-E}&\leq \frac{1}{\delta^m}\left(\tilde{C}_6\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert+\frac{\delta_0^{2+m}}{4}\frac{\nu}{\delta^2}\left\lVert f \right\rVert^2\right), \end{align}\tag{100}\] where in the last inequality we used 77 and \(\delta_0\) is a small fixed constant (with \(\tilde{C}_6\) depending on \(\delta_0\)).

\(\circ\) Concluding the proof. It remains to optimize the choice of \(\delta\) in order to conclude the proof. From 100 , we can only hope to absorb the last term in that inequality if we choose \(\delta\) such that \[\label{def:choicedelta} \frac{\nu}{\delta^2}\simeq \delta^m\qquad \Longrightarrow \qquad \delta:=\delta_* \nu^{\frac{1}{m+2}}.\tag{101}\] Therefore, combining 100 with 90 and the choice of \(\delta\), we finally arrive at \[\begin{align} \left\lVert f \right\rVert^2&=\int_{M\setminus{\mathcal{E}}_{\lambda,\delta}} |f|^2{\rm d}x+\int_{{\mathcal{E}}_{\lambda,\delta}} |f|^2{\rm d}x\leq \tilde{C}_7 \nu^{-\frac{m}{m+2}}\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert+\frac{3}{4}\left\lVert f \right\rVert^2, \end{align}\] which implies \[\left\lVert \mathcal{L}_\lambda f \right\rVert\geq \frac{1}{4\tilde{C}_7} \nu^{\frac{m}{m+2}} \left\lVert f \right\rVert\] whence proving 76 for \(0<\delta_*:=1/(4\tilde{C}_7)\ll1\) which can be explicitly computed.

4.1.2 Case \(\boldsymbol{| \lambda | > \gamma_0\delta^{-1}}\)↩︎

In this regime, we are heuristically studying the high-\(k\)-frequency case separately. Indeed, whenever \(|\Omega(h)-\lambda/k|\ll 1\) and \(\Omega(h)\leq \overline{C}\), we necessarily have \(|\lambda/k|\lesssim 1\) and therefore \(|k|\gtrsim \delta^{-1}\). With our choice of \(\delta\) in 101 , we would then have \(|k|\gg \nu^{-\frac{1}{m+2}}\), which is a situation where the dissipation is expected to be much stronger.

From a practical point of view, we simply change the definition of 80 as follows: \[\label{def:Eklambdahigh} E_{k, \lambda, \delta} \mathrel{\vcenter{:}}= \left\{ (\theta, h) \in \mathbb{T}\times I : \left|\Omega(h) - \frac{\lambda}{k} \right| < \frac{\delta^{m-1}}{\mathsf{C}_1|k|} \right\}, \qquad \mathcal{E}_{k,\lambda,\delta}=\Phi( E_{k, \lambda, \delta}),\tag{102}\] where \(\mathsf{C}_1>0\) is a sufficiently large constant specified below. Then, instead of using a \(\delta\)-neighbourhood of the set above, we use a \(|k|^{-1}\)-neighbourhood, namely \[\label{def:Eklambdathickhigh} \boldsymbol{E}_{k, \lambda, \delta} =\{h \in I \, : \, \mathrm{dist}(h,E_{k,\lambda,\delta})<|k|^{-1}\}, \qquad \boldsymbol{\mathcal{E}}_{k, \lambda, \delta}=\Phi( \boldsymbol{E}_{k, \lambda, \delta}).\tag{103}\] The set \(Q_{\mathsf{C}_0\delta}\) in 82 remains the same whereas 83 is redefined accordingly. We have the following.

For the Fourier multiplier, we define \(\mathcal{X}\) as in 95 , but we choose \(\chi\) satisfying the properties [chi:inequality], [chi-sign], [chi-zero] and we replace [chi-derivative] with

1. \(\left\lVert \chi \right\rVert_{L^\infty}\leq 1\) and \(\left\lVert {\partial}_h\chi \right\rVert_{L^\infty} \lesssim |k|\).

This is because now we are working with a \(|k|^{-1}\)-neighbourhood of \(E_{k,\lambda,\delta}\). Hence, we are in case b) of Lemma 4, meaning that we will control \(\|\nabla (\chi f)\|\) with \(\|\nabla f\|\). This apparent loss of optimality is compensated by the fact that now the thin sets are of size \(\delta^{m-1}/|k|\ll \delta^{m}\) for all \(|k|\) sufficiently large.

We are now in the position of concluding the proof. Thanks to the properties in Lemma 8 and the fact that \(|k|\simeq |\lambda|\gtrsim \delta^{-1}\), we can repeat the proof of 90 and show that there exists a universal constant \(C_{l_1}>0\) such that \[\label{bd:claimE1} \int_{\mathcal{E}_{\lambda,\delta} } |f|^2 \, dx \leq C_{l_1}\frac{\delta^2}{\nu}\| \mathcal{L}_{\lambda} f \|\left\lVert f \right\rVert+ \frac{1}{2} \| f \|^2 \,.\tag{107}\] For the bounds away from the level sets, we consider the splitting in 96 but we now control \(\mathcal{I}_1\) as follows: \[\begin{align} \mathcal{I}_1 &={\rm Im}\left\langle (u\cdot \nabla-i\lambda)f,\mathcal{X}f \right\rangle= {\rm Im}\int_I \sum_{k\neq 0} i(k\Omega(h)-\lambda)\chi(k,h)|\widehat{f\circ \Phi}|^2(k,h) q(h)T(h){\rm d}h\\ &\geq\int_{E_{\lambda,\delta}^c} \sum_{k\neq 0} |k|\left|\Omega(h)-\frac{\lambda}{k}\right||\widehat{f\circ \Phi}|^2(k,h) q(h)T(h){\rm d}h\\ \label{bd:lowI1} &\geq \delta^{m-1} \int_{M\setminus{\mathcal{E}}_{\lambda,\delta}} |f|^2{\rm d}x. \end{align}\tag{108}\] Therefore, similarly as done before in the case \(| \lambda | \leq \gamma_0 \delta^{-1}\), using the identity 96 , the bound of the Fourier multiplier \(\mathcal{X}\) 34 in Lemma 4 and the identity 77 , we get \[\begin{align} \label{bd:M-E1} \int_{M\setminus \mathcal{E}_{\lambda,\delta}} |f|^2{\rm d}x&\leq \frac{1}{\delta^{m-1}} \tilde{C}_{6_1}\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert. \end{align}\tag{109}\] Combining together 90 and 109 with the choice made before \(\delta = \nu^{\frac{1}{m+2}}\), we get \[\begin{align} \left\lVert f \right\rVert^2\leq \tilde{C}_{7_1}(\delta \nu^{-\frac{m}{m+2}} + \nu^{-\frac{m}{m+2}})\left\lVert \mathcal{L}_\lambda f \right\rVert\left\lVert f \right\rVert+\frac{3}{4}\left\lVert f \right\rVert^2, \end{align}\] which implies \[\left\lVert \mathcal{L}_\lambda f \right\rVert\geq \frac{1}{8\tilde{C}_{7_1}} \nu^{\frac{m}{m+2}} \left\lVert f \right\rVert.\] Therefore the bound 76 holds true with \(0<\delta_*:=1/\max\{4\tilde{C}_7,8\tilde{C}_{7_1})\}\ll1\), whence concluding the proof of Proposition 14. 0◻

5 Examples↩︎

In this section, we firstly prove a proposition that allows to consider a general class of Hamiltonians satisfying our assumptions of Definition 4 and Definition [def:ClassHamiltoniansP]. Then, we provide two examples where the average along the streamline has a non-trivial evolution, implying in particular that the commutator \([\Delta, P_0]\) is non-trivial.

5.1 Perturbation of non-degenerate radial Hamiltonians↩︎

5.2 On the evolution of \(\rho_0\) and the commutator \([P_0,\Delta]\)↩︎

We firstly provide an example where the solution of the advection-diffusion equation starts with an initial data with zero average along the streamline, and, in finite time, the average along the streamlines nearly matches the mass of the initial data as measured in \(L^1\). This example exploits the hyperbolic point of the cellular flow and highlights the observation that, in a short time span, non-trivial phenomena can occur due to differences in width between two streamlines (this phenomenon is ruled out by our assumptions [item:ClassACond4], [item:ClassPCond4]).

We now provide a second example where we show that the the commutator \([P_0, \Delta]\) can really affect the \(L^2\) dynamics in a non-trivial way. In fact, in the previous example we cannot have a lower bound in \(L^2\) due to the unboundedness of the period near the hyperbolic point. Here, the analysis is performed near a non-degenerate elliptic point where the period is bounded.

In the example we construct, we have an Hamiltonian not complying with the assumptions in Definition 4. In particular, there are stagnation regions where \(\nabla H=0\) (the Hamiltonian is constant in a portion of the domain). The proof does not rely on these stagnation regions and can be adapted to avoid this issue. However, this requires some more technicalities, as the use of Feynman-Kac formula or the heat kernel in more general bounded domains instead of \(\mathbb{T}^2\). The main purpose of this example is investiganting the behavior of \(\rho_0\) when we are localized around elliptic points, and we prefer to keep the proof lighter at the price of having an Hamiltonian not belonging to the class [HamiltonianClassA].

6 Concluding remarks↩︎

In this section we list some future directions based on our approach. The aim is to provide a better understanding of the enhanced dissipation effect for general autonomous Hamiltonian flows.

On unbounded period↩︎

In the proof of the pseudospectral bounds for the model problem in Section 4, we crucially used that the period is bounded. This is useful to prove that the sets \(\{ E_{k, \lambda, \delta} \}_{k \in \mathbb{N}}\) introduced in 80 and 102 are disjoint (and well separated). In particular, the fact that \(\Omega(h) \simeq 1\) is needed to conclude that \[\frac{| \lambda |}{|k|} \sim 1,\] as used in 89 and in 106 . If the period is unbounded, meaning that \(\Omega \to 0\) (which happens if for instance \(H(r)=r^{2+m}\) with \(m \geq 1\)), we can modify our proof and define \[E_{k, \lambda, \delta} \mathrel{\vcenter{:}}= \left\{ (\theta, h) \in \mathbb{T}\times I : \left|\Omega(h) - \frac{\lambda}{k} \right| < \varphi(\delta, k) \right\},\] with a \(\varphi\) to be determined. Similarly to 88 , it can be shown that \[| \Omega (h_k) - \Omega (h_{k'})| \sim \frac{| \lambda | }{|k|} \frac{|k - k'|}{|k '|} - 2 \varphi (\delta, k) \,.\] Cutting a \(\delta\)-neighbourhood near the elliptic point as in 82 , and assuming the behaviour \(\Omega (h) \sim h^m\), we get \[\frac{| \lambda | }{|k|} \gtrsim \delta^m.\] This implies that we need \(\varphi(\delta, k) \sim \delta^m/|k|\) to ensure that the sets are disjoint. Essentially, instead of taking \(\delta^m\) for small \(|k|\) or \(\delta^{m-1}/|k|\) for larger \(|k|\), it is enough to always use \(\delta^m/|k|\). However, repeating our proof with this definition, one obtains the pseudospectral bound \[\left\lVert \mathcal{L}_\lambda f \right\rVert \gtrsim \nu^{\frac{m+1}{m+3}} \left\lVert f \right\rVert,\] where \(\lambda_\nu=\nu^{\frac{m+1}{m+3}}\) is the rate expected for a critical point of order \(m+1\) instead of the order \(m\) we are assuming. Indeed, for the radial flow \(H(r)=r^{2+m}\) with \(m \geq 1\), this is not the optimal bound, which would be given by \(\nu^{\frac{m}{m+2}}\) Gallay:2021aa?. We believe that this mismatch is due to the fact that we are treating all the \(k\)-s at once, and a more careful analysis is needed to handle unbounded periods in an optimal way.

For instance, one might try to do \(k\)-by-\(k\) estimates by studying the operator \(P_k\Delta P_k\). However, the commutator estimates and the comparison with the original equation 1 will be way more challenging.

On hyperbolic points↩︎

It may be possible to get a pseudospectral bound of the model problem 14 when the Hamiltonian is allowed to have hyperbolic points (a similar statement as Theorem 2, but it is not clear what would be the optimal \(\lambda_\nu\) with this method). To treat this case there are two technical problems to take into account:

• Understand the optimal bound of \(\| D_x X \|_{L^\infty ((0, T(h)) \times \Omega )}\), which is then used to bound \(\| \partial_h \Phi \|_{L^\infty}\) in the Poincaré inequality Lemma 5 (for estimates away from hyperbolic points one would loose \(|\log(\nu)|\) factors). Moreover, one would need to adapt the rescaled action angle variables in Section 2.1, as to optimise the parameters in the Poincaré inequalities. A preliminary computation shows that this is possible for the cellular flow, but with a rate that is not consistent with the \(\nu^{1/2}\) expected from the behavior around the elliptic point, see Brue:2022aa? as well.

• The gradient of the Fourier multiplier operator introduced in 95 and bounded in Lemma 4 degenerates, since 35 is not true near the hyperbolic point. Therefore, the proof of Corollary 1 fails in this context, since it relies on property [item:ClassACond4], which is not true near the hyperbolic point. Hence, it is not clear how to relate our model problem 14 to the advection diffusion equation solution near the hyperbolic point (i.e. the solution to 1 is close in \(L^2\) to the solution of the model problem 14 in subdiffusive time scales).

Averaged in time commutator estimate↩︎

It would be interesting to understand if the estimate 9 in Theorem 1 holds with \(\varepsilon\) replaced by \(a(t, \nu)\) depending on time and \(\nu >0\) such that \(a(t, \nu) \leq a(s, \nu)\) for any \(\nu >0\), \(t \geq s\) and \[\lim_{\nu \to 0} \, \lim_{t \to T_\nu } \, a (t, \nu) =0 \,,\] where \(T_\nu\) is such that \[\lim_{\nu \to 0} \, \nu T_\nu=0 \,.\] This corresponds to proving that the solution of the advection-diffusion is converging to the average along the streamlines on subdiffusive times. We remark that our example in Proposition 19 provides a non-trivial evolution for \(\rho_0\) for short times \(t \leq 1\), which indicates a dissipation of \(\rho_\perp\). However, their dynamics is coupled and it is not clear how to control their interactions directly.

On initial data concentrated near elliptic points↩︎

Consider an Hamiltonian having at least one non-degenerate elliptic point and containing also hyperbolic points. We believe it would be interesting to find extra assumptions on the initial data of 1 , such as being supported close to a non-degenerate elliptic point, so that we can still effectively compare our model problem to the original equation 1 . The goal would be to apply our pseudospectral bound in Theorem 2 in a suitable bounded domain contaning the non-degenerate elliptic point, with the aim of proving a result in the spirit of Theorem 1.

Propagating smallness of the streamline average↩︎

As shown in Proposition 4, quantitative bounds on \(\rho_0\) will automatically improve the bounds on \(\rho\). We believe that, starting with \(\rho_0^{in}=0\), it would be interesting to obtain uniform bounds of the type \(\|\rho_0(t)\|_{L^2}\lesssim \nu^q\) for some \(q>0\) and \(0<t<\nu^{-p}\) with \(1/\lambda_\nu<p\leq 1\). Depending on the specific values of \(q\) and \(p\), we can run the proof Proposition 4 and eventually obtain the \(\|\rho(t)\|_{L^2}\lesssim \max\{\nu^{q/2},\nu^{1/(m+2)},{\rm e}^{-\delta_*\lambda_\nu t/2}\}\). However, the only quantitative information we have of this type, is that for Hamiltonians in the class 5 \(\rho_0\) remains of size \(\varepsilon\). To improve this last bound even for perturbations of radial flows, it seems that a much deeper understanding of more general properties is needed. For instance, one would like to understand the averaging properties of the stochastic flow associated to a random perturbation of a given Hamiltonian flow.

Acknowledgements↩︎

The research of MD, MS, CJ was supported by the Swiss State Secretariat for Education, Research and lnnovation (SERI) under contract number MB22.00034 through the project TENSE. MD was acknowledge support by the GNAMPA-INdAM. The authors thank Maria Colombo for useful discussion about the problem.

Conflict of interest↩︎

The authors have no competing interests to declare that are relevant to the content of this article.

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1. As defined in constantin2008diffusion?.↩︎

2. For the cellular flow \(H(x,y)=\sin(x)\sin(y)\), the parameter \(\varepsilon\) indicates how close one is to one of the elliptic points, say \((\pi/2,\pi/2)\). The streamlines are indeed smooth deformation of circles close to the elliptic point, but the behavior of the flow cannot be considered close to radial when one gets too close to the separatrices associated to the hyperbolic point.↩︎

3. \(m\) is the order of vanishing, see for instance [item:ClassACondOnDerivative].↩︎

4. See the bound for the term \(c_{c1}^{(2)}\) in the proof of Theorem 22 in Vukadinovic21?.↩︎

5. For instance, given a radial Hamiltonian \(H=H(r)\), the canonical action-angle variables are the coordinates \(x=\sqrt{2r}\cos(\theta), y=\sqrt{2r}\sin(\theta)\).↩︎

6. without specifying the domain and boundary conditions, which are the same ones of \(g\)↩︎

7. An operator \(\mathcal{A}\) is maximally accretive if \(\|(a I+\mathcal{A})f\|_X\geq a \|f\|_X\) for all \(a\geq 0\) and \(a_0I+\mathcal{A}\) is surjective for some \(a_0>0\).↩︎

8. The definition of this set is related to the specific choice of \(q(h)\), namely one needs \(|q'(h)|\lesssim\delta\). In the non-degenerate setting, we have \(q(h)=h^2\) and therefore the sets are equivalent (up to constants).↩︎