December 08, 2025
We prove spatial Schauder estimates – optimal regularity estimates in Hölder spaces – and well-posedness results for mild and classical solutions of viscous Hamilton–Jacobi equations with subcritical nonlocal and mixed local-nonlocal diffusions in \(\mathbb{R}^d\). Our results hold under mild assumptions on the nonlocal/mixed operators and Hamiltonians. The Laplacian, fractional Laplacians, nonsymmetric, spectrally one-sided, and strongly anisotropic integral operators, as well as sums of such operators are covered. We observe an interplay between the regularity of the initial data and the growth of the Hamiltonian in the gradient, and give results for the two canonical cases: (i) Lipschitz initial data and general Hamiltonians that are Hölder in space and merely locally Lipschitz in the gradient, and (ii) Hölder initial data and Hamiltonians that are Hölder in space and locally Lipschitz with power growth in the gradient. We compute explicit blow-up rates for \(C^1\) and higher order Hölder norms as \(t\to 0\). The results include short and long time existence of mild solutions, optimal regularity in Hölder spaces and corresponding Schauder a priori estimates, and that spatially smooth mild solutions are regular in time and pointwise classical solutions.
We investigate well-posedness and Schauder regularity estimates for the initial value problem for the nonlocal viscous Hamilton–Jacobi equation: \[\begin{align}\label{eq:hjb} \left\{ \begin{aligned} \partial_t u -\mathcal{L}u-H(t,x,Du)&=0, \qquad &(t,x)&\in (0,T]\times \mathbb{R}^d, \\ u(0,x)&=u_0(x), \qquad &x &\in \mathbb{R}^d, \end{aligned} \right. \end{align}\tag{1}\] where \(\mathcal{L}\) is the generator of a Lévy process defined by \[\begin{align} \label{eq:L} \mathcal{L}\varphi(x) &= \mathop{\mathrm{div}}(AD\varphi(x)) + \int_{\mathbb{R}^d}\big[\varphi(x+z)-\varphi(x)-\mathbf{1}_{|z|<1}D\varphi(x)\cdot z\big]\, d\mu(z),\quad x\in \mathbb{R}^d, \end{align}\tag{2}\] \(A\) is a nonnegative definite matrix, and \(\mu\geq 0\) is a Borel measure satisfying \(\int_{\mathbb{R}^d}(1 \wedge |z|^2) \;d\mu(z) < \infty\), i.e. \(\mu\) is a Lévy measure. In other words, \(\mathcal{L}\) is a Lévy operator with the Lévy triplet \((A,0,\mu)\). In general, Lévy operators can also have a constant drift term \(B Du\) – then the Lévy triplet would be \((A,B,\mu)\). Assuming \(B=0\) does not affect the generality, since \(B Du\) can be absorbed into the Hamiltonian \(H\).
Throughout the paper we assume that the nonpositive operator \(\mathcal{L}\) is subcritical, i.e. of order \(\alpha\in(1,2]\), a condition we express in the language of the heat kernel of \(\mathcal{L}\) in assumption [NDa] below. This condition is satisfied by a large class of local, nonlocal, and mixed local-nonlocal Lévy operators, including fractional Laplacians and strongly anisotropic and nonsymmetric operators, see Example 1 below. We focus on existence and regularity results of Schauder type, where solutions gain exactly \(\alpha\) derivatives on the data \(H\) when measured in Hölder norms. We give results for short-time existence, Schauder estimates, and long-time existence, in cases with and without gradient blow-up as \(t\to 0\).
Main results. The main results of the paper are the short-time existence and optimal Schauder regularity (with blow-up rates) of solutions of 1 . We also show that the solutions are classical and we give their long-time existence. We refrain from giving the assumptions in the full form here, instead referring to the formulations of the results.
Short-time existence (Theorems 8 and 12). The proof of the short-time existence of mild solutions of 1 in is based on a Duhamel formulation and a fixed point argument in a subset of \(C((0,T]; C^1(\mathbb{R}^d))\). There is an interplay between the regularity of \(u_0\) and the growth of \(H=H(t,x,p)\) in \(p\) that results in two different cases:
(I) Lipschitz initial data and locally Lipschitz \(H\) in \(p\).
(II) Hölder initial data and \(H\) with explicit power-type growth in \(p\).
Case (I) is rather standard as the solutions have uniformly bounded gradient. Case (II) allows for solutions with gradient blow-up as \(t\to 0\), and to control it, the growth of the Hamiltonian needs to be compensated for by regularity of the initial condition. We refer to [assump:B1] for the range of the admissible Hölder exponents \(\delta\) of \(u_0\) with respect to the growth rate \(r\) of \(H\). We note that the short-time existence does not require any smoothness or integrability of \(H\) in \(x\) or \(t\), just continuity and boundedness.
Optimal Schauder regularity with blow-up rates (Theorems 15 and 19). We show that under condition [NDa], the solutions to 1 gain Hölder regularity of order \(\alpha\) over the regularity of \(H\) in \(x\). To this end we apply the “diagonal splitting” of the space-time cylinder previously used e.g. by Chaudru de Raynal, Menozzi and Priola [1] in the context of linear equations with drift and supercritical diffusion, i.e. \(\alpha\in(0,1)\). We note that [NDa] is slightly weaker than the assumption used in [1], which involves a certain moment condition on the heat kernel needed to handle the unbounded drift coefficient. In the case of unbounded gradients, the optimal Hölder norm blow-up rate is quite cumbersome to obtain – because of two singularities coming from the gradient and the heat kernel we need to apply a “doubly fractional” version of Grönwall’s inequality, see Lemma 33, and in some cases we also use a bootstrap argument.
Classical solutions and long-time existence (Theorem 24 and 26). Finally, we show that solutions are classical and that we have long-time existence under further assumptions on \(H\). If we assume Hölder regularity of \(H\) in \(x\), then the solutions are classical, which follows from the regularity results mentioned above. Subsequently, using results from viscosity solution theory, we achieve long-time existence in the case of \(x\)-Lipschitz \(H\) with explicit power-type growth conditions. More precisely, we get well-posedness and global Lipschitz bounds for viscosity solutions on the (long) time interval \([0,T]\), and conclude by uniqueness and previous results that this solution coincides locally with smooth mild/classical solutions.
Background. Viscous HJ equations arise as dynamical programming equations in optimal stochastic control problems and differential games. The unknown function \(u\) then represents the (upper or lower) value function for the optimally controlled process [2]–[4]. Our setting corresponds to the case of controlled drift in the presence of noise given by a pure-jump Lévy process \(X_t\) with generator \(\mathcal{L}\). For more details on this connection, see e.g. [5], [6]. For the classical case of Brownian noise, corresponding to 1 with \(\mathcal{L}\) replaced by \(\Delta\), we refer to [3], see also [7]. We also mention that the equation is closely connected with the theory of large deviations of stochastic processes [7]–[9], and the KPZ equation from physics [10], a stochastically forced version of the equation and an important example of an ill-posed problem that can be solved in Hairer’s theory of regularity structures [11].
Studies of viscous HJ equations were initiated almost half a century ago, with the pioneering works of Kružkov [12] and Crandall and Lions [13]. The topic has been very active, and a sample of various aspects of the research concerning HJ equations can be found in [14]–[22], but this list is far from complete. A strong incentive for further investigation of HJ equations came from the theory of mean field games [23], [24], which is a major motivation behind our work. Mean field games have also been studied for nonlocal operators [25]–[28], but the setting of classical solutions usually requires strong assumptions on data. Our work should facilitate analysis of mean field game systems in a lower regularity setting.
Let us present the literature more closely related to our methods and settings. We use general viscosity solution theory for nonlocal and mixed local-nonlocal operators as presented in Jakobsen and Karlsen [29]. There are many other references on this topic including [30]–[32]. For \(\mathcal{L}= -(-\Delta)^{\alpha/2}\) Imbert [33] established a comparison principle and Lipschitz estimates in the context of viscosity solutions, and in the case \(H = H(Du)\), showed that mild solutions are smooth classical solutions. Improvements and extensions to general nondegenerate fractional operators \(\mathcal{L}\) can be found in [26], [34], and elements of a first (suboptimal) Schauder theory is given in [34] – see also [35]. The setting of mild solutions and Duhamel formula that we use here was also present in e.g. [36]–[38]. Schauder estimates for linear parabolic equations involving quite general Lévy-type operators were given by Mikulevičius and Pragarauskas [39]. More recently, Dong, Jin and Zhang [40] gave Schauder estimates for fully nonlinear equations involving nonlocal operators with the Lévy measure comparable to the one of the fractional Laplacian. Our present Schauder estimates improve the regularity estimates in [26], [33], [34] in several ways: We obtain more (and maximal) regularity under the same assumptions, we give new results for Hamiltonians \(H\) of low regularity (measured in Hölder scales) and with much more general growth in the gradient, and we give new, more precise and explicit, a priori estimates for high order Hölder norms.
There are several recent works on fractional HJ equations with low-regularity data, admitting solutions with unbounded gradient. Mild solutions for the HJ equation on \(\mathbb{R}^d\) with \(H=H(Du)=|Du|^r\) with critical diffusion, i.e. \(\alpha=1\), were studied by Iwabuchi and Kawakami [41], for initial conditions in Besov spaces. Goffi [42] considered forcing terms in \(L^p\) and initial conditions which are continuous or are in Besov spaces on the torus, with power-type growth conditions on \(H\). For the (deterministic) fractional KPZ equation with a forcing term on bounded domains with homogeneous Dirichlet conditions, Abdellaoui, Peral, Primo, and Soria [43] gave the existence and nonexistence results under certain integrability assumptions on the initial condition and the forcing term, in relation to the power in the gradient term. Matioc and Walker [44] established a general semigroup framework for quasilinear equations in time-weighted spaces, tracing the blow-up of certain norms, but it does not allow for the scale of Hölder spaces. Here we also mention the earlier paper by Benachour and Laurençot [45], which addressed the case of local diffusion.
The paper is organized as follows. We begin by covering the relevant notation. In Section 2 we list the assumptions on \(\mathcal{L}\), give some examples, and establish regularizing effects of the heat kernel of \(\mathcal{L}\). Sections [sec::short95time95existence], 4 and 5 compose the main parts of the paper, covering the short-time existence, optimal regularity, and solutions being classical and long-time existence respectively. Finally, we discuss extensions and give closing remarks in Section 6. Some technical results used throughout the paper are collected in the Appendix 7.
Notation. Below, \(\mathbb{N} = \{1,2,\ldots\}\) and \(d\in \mathbb{N}\). As usual, for \(r\in \mathbb{R}\), the function \(\lceil r\rceil\) gives the smallest integer greater than or equal to \(r\) and \(\lfloor r\rfloor\) gives the largest integer less than or equal to \(r\).
Let \(\kappa=(k_1,\dots, k_d)\), \(k_i\in\mathbb{N}\cup\{0\}\), be a multi-index of order \(|\kappa|=k_1 + \dots + k_d\) and define the spatial derivative \(\partial^\kappa:= \frac{\partial^{|\kappa|}}{\partial x_1^{k_1}\partial x_2^{k_2} \dots \partial x_d^{k_d}}\). We will sometimes use the subscript to indicate that an operator acts with respect to the variable in the subscript. By \(D^m \varphi\), \(m\in \{0,1,2,\ldots\}\), we mean the tensor of all derivatives of order \(m\). For \(n\in\mathbb{N}\cup\{0\}\), let \(C_b^n(\mathbb{R}^d)\) be the space of functions with \(n\) continuous and bounded derivatives and norm \(\|\varphi \|_{C_b^n}=\sum_{j=0}^n \max_{|\kappa|=j}\|\partial^\kappa \varphi\|_\infty,\) where \(\|\cdot \|_\infty\) is the \(L^\infty\) (or \(C_b\)) norm. For \(\gamma=n+\beta\), where \(\beta\in(0,1)\), we define the Hölder space \[C_b^\gamma(\mathbb{R}^d) :=\{\varphi\in C_b^n(\mathbb{R}^d) \;: \;\|\varphi \|_{C_b^\gamma}<\infty \},\] where \[\|\varphi \|_{C_b^\gamma} := \|\varphi \|_{C_b^n}+\max_{|\kappa|=n}[\partial^\kappa\varphi]_\beta \qquad \text{and} \qquad [\varphi]_{\beta}:= \sup_{\substack{x, h\in \mathbb{R}^d \\ h \neq 0}}\frac{ |\varphi(x+h)-\varphi(x)|}{|h|^{\beta}}.\] Note that \([\cdot]_\beta\) defines a seminorm for all \(\beta \in (0,1]\) and that \([\varphi]_1\) is the Lipschitz constant of \(\varphi\). We also let \([\varphi]_0 = \|\varphi\|_\infty\). Norms, seminorms and operators act on the unspecified arguments of the function they are applied on, e.g. for a function \(\varphi\colon A\times B \rightarrow \mathbb{R}\) we will write \(\|\varphi\|_\infty := \sup_{(a,b)\in A\times B } |\varphi(a,b)|\), while \(\|\varphi(\cdot,b)\|_\infty :=\sup_{a\in A} |\varphi(a,b)|\), and so on.
In this section we introduce the heat kernel and the heat semigroup associated with the nonlocal diffusion operator \(\mathcal{L}\) defined in 2 , and prove optimal parabolic regularizing effect of the semigroup under general assumptions. Our results are equivalent to proving optimal regularity results for the fractional heat equation \(\partial_tv -\mathcal{L}v = f\). Taking the Fourier transform of 2 , we find that \(\mathcal{F}(\mathcal{L}\varphi)(\xi)=\widehat{\mathcal{L}}(\xi) \widehat{\varphi}(\xi)\), where the symbol \[\begin{align} \widehat{\mathcal{L}}(\xi)= \xi A \xi + \int_{\mathbb{R}^d}(1-e^{i \xi \cdot z}+i\xi \cdot z \mathbf{1}_{|z|<1})\;d\mu(z),\qquad \xi\in \mathbb{R}^d. \end{align}\] The heat kernel \(p_t\) of \(\mathcal{L}\) is then given by \[\begin{align} \mathcal{F}p_t(\xi)= (e^{-t\widehat{\mathcal{L}}(\cdot)})(\xi),\qquad \xi\in \mathbb{R}^d. \end{align}\] By the Lévy–Khintchine theorem [46] \(p_t\) are probability measures for \(t>0\), and they converge weakly to the Dirac delta as \(t\rightarrow 0\) [46]. We assume that the heat kernel is absolutely continuous with respect to the Lebesgue measure – this is contained in the assumption [NDa] below. With the convention that \(p_t(x)\, dx :=p_t(dx)\), we then have \((\partial_t -\mathcal{L})p_t =0\) in the classical sense. The semigroup associated with the heat kernel is \[P_t\varphi(x) := p_t*\varphi(x)=\int_{\mathbb{R}^d} p_t(x-y)\varphi(y) dy,\qquad t\geq 0,\;x\in \mathbb{R}^d.\] For \(\varphi \in C^2_c(\mathbb{R}^d)\), the operator \(\mathcal{L}\) coincides with the generator of \(P_t\) [47].
We will assume that the nonpositive operator \(\mathcal{L}\) is a non-degenerate operator of order \(\alpha\), a condition we formulate in terms of its heat kernel.
Remark 1. Let \(m\in\mathbb{N}\). Since \(D^mp_t=(Dp_{\frac{t}{m}})^{*m}\), by Young’s inequality for convolutions and [NDa], \[\begin{align} \label{NDam} \int_{\mathbb{R}^d} |D^mp_t(y)| dy \leq ( m^{\frac{1}{\alpha}} c_0)^mt ^{-\frac{m}{\alpha}}. \end{align}\tag{3}\] We will mostly use \(m=2\) (for short-time existence in Section 3) and \(m=\lceil \alpha+\beta\rceil\) where \(\beta\) is the Hölder continuity of the data given by [assump:H] or [assump:B3] (for Schauder regularity in Section 4).
Example 1. Operators \(\mathcal{L}\) in 2 satisfying [NDa], cf. [26].
Theorem 4.3 in [26] (see also [48]): If \(A=0\) and there is \(\alpha\in (1,2)\) such that \(\frac{d\mu}{dz}\approx |z|^{-d-\alpha}\) for \(|z|\leq 1\), then \(\mathcal{L}\) satisfies [NDa]. Here \(\mathcal{L}\) is purely nonlocal with the Lévy measure \(\mu\) absolutely continuous in \(|z|\leq1\), but no assumption on the tails (\(|z|>1\)). An example is the fractional Laplacian \(-(-\Delta)^{\alpha/2}\) with Lévy triplet \((0,0,c_{d,\alpha}|z|^{-d-\alpha}\, dz)\).
\(\mathcal{L}= -(-\partial^2_{x_1})^{\alpha_1/2}-(-\partial^2_{x_2})^{\alpha_2/2} -\ldots -(-\partial^2_{x_d})^{\alpha_d/2}\) for \(\alpha_i\in (1,2)\) satisfies [NDa] with \(\alpha = \min_i \alpha_i\). Here the corresponding Lévy measure \(\mu\) is not absolutely continuous.
The Riesz–Feller operator on \(\mathbb{R}\) corresponding to triplet \((0,0,|z|^{-1-\alpha}\boldsymbol{1}_{(0,\infty)}(z)\, dz)\) satisfies [NDa], see [49]. This corresponds to a spectrally one-sided process and a very non-symmetric \(\mathcal{L}\).
The generator of the CGMY process in Finance satisfies [NDa], see [26]. This is an example of a tempered and non-symmetric process and \(\mathcal{L}\).
Operators \(\mathcal{L}\) with strictly positive definite matrix \(A\) satisfy [NDa] with \(\alpha=2\). Here the principal part of \(\mathcal{L}\) is the non-degenerate second derivative term.
If \(\mathcal{L}\) satisfies [NDa] and \(\tilde{\mathcal{L}}\) is any Lévy operator 2 , then \(\mathcal{L}+ \tilde{\mathcal{L}}\) satisfies [NDa]. One example is the degenerate second order operator \(\tilde{\mathcal{L}}\phi = \mathop{\mathrm{div}}(AD\phi)\) where \(A\geq0\) is not invertible. Here there will be a gain of regularity of \(\alpha\) derivatives in all directions, but as we will see in Section 6.2, we gain 2 full derivatives in non-degenerate directions of \(A\).
We now show that the heat semigroup \(P_t\) has optimal smoothing properties in Hölder scales, in the sense that there is a gain of \(\alpha\) derivatives on the input data. We explicitly quantify the rates of blow-up for our Hölder estimates as \(t\to 0\).
Theorem 2. Assume [NDa], \(\beta \in [0,1]\), and \(\varphi \in C_b^{\beta}(\mathbb{R}^d)\). Then for \(k\in\mathbb{N}\), \(\gamma\in(0,1)\), and \(t\in(0,T]\), \[\begin{align} &(i)\;\; \big\| D^k P_t \varphi(\cdot) \big\|_\infty \leq c_{k,\beta,d}[\varphi]_{\beta}t^{-\frac{k-\beta}{\alpha}}, \\[0.2cm] & (ii) \;\; [D^kP_t\varphi(\cdot)]_{\gamma}\leq2^{1-\gamma}c_{k+1 ,\beta,d}[\varphi]_\beta t^{-\frac{k+\gamma -\beta}{\alpha}}, \end{align}\] where \(c_{k,\beta,d}= ( k^{\frac{1}{\alpha}} c_0)^{k-\beta}\) for \(\beta\in\{0,1\}\), \(c_{k,\beta,d}= ( k^{\frac{1}{\alpha}} c_0)^{k-\beta} C_{\beta,d}\) for \(\beta\in(0,1)\), and \(C_{\beta,d}\) is given by Theorem 31.
Note that when \(\gamma=\alpha+\beta- k\) and \(k< \alpha+\beta\), then the estimate in (ii) is reminiscent of standard semigroup results of the type \(\|\mathcal{L}P_t \varphi\|\leq Ct^{-1}\|\varphi\|\).
Proof. (i) Consider first \(\beta \in(0,1]\). Let \(m\in\mathbb{N}\cup\{0\}\) and note that by [NDa] (see Remark 1) (or \(\|p_t\|_{L^1}=1\) when \(m=0\)), \(x,x' \in \mathbb{R}^d\), \[\begin{align} |D^{m}P_t\varphi(x) - D^mP_t\varphi(x')| &\leq \int_{\mathbb{R}^d}|D^mp_t(y)||\varphi(x-y) - \varphi(x'-y)|\, dy\\ & \leq |x-x'|^\beta [\varphi]_{\beta} \|D^mp_t\|_{L^1} \leq |x-x'|^\beta [\varphi]_{\beta} ((m\, \vee \, 1)^{\frac{1}{\alpha}} c_0)^{m} t^{-\frac{m}{\alpha}}, \end{align}\] that is, \([D^{m}P_t\varphi]_{\beta} \leq ((m\, \vee \, 1)^{\frac{1}{\alpha}} c_0)^{m}[\varphi]_{\beta}t^{-\frac{m}{\alpha}}\). Since \(D^mp_t\in L^1(\mathbb{R}^d)\), we could interchange derivatives and integrals using standard arguments for convolutions. Recall that \(k\geq 1\). When \(\beta =1\), this yields \[\begin{align} \|D^k P_t \varphi\|_\infty= [D^{k-1}P_t\varphi]_{1}\leq ( k^{\frac{1}{\alpha}} c_0)^{k-1} [\varphi]_1 t^{-\frac{k-1}{\alpha}}, \end{align}\] while for \(\beta \in(0,1)\), by interpolation (Lemma 31), we get \[\begin{align} \|D^kP_t\varphi\|_{\infty} &\leq C_{\beta, d}[D^{k-1}P_t\varphi]_{\beta}^\beta [D^kP_t\varphi]_{\beta}^{1-\beta} \leq C_{\beta, d} ( k^{\frac{1}{\alpha}} c_0)^{k-\beta} [\varphi]_{\beta}t^{-\frac{k-\beta}{\alpha}} , \end{align}\] When \(\beta=0\), we directly get that \[\begin{align} |D^kP_t\varphi(x)| &\leq \int_{\mathbb{R}^d}|D^kp_t(y)||\varphi(x-y)|\, dy \leq \|\varphi\|_{\infty} ( k^{\frac{1}{\alpha}} c_0)^{k} t^{-\frac{k}{\alpha}}. \end{align}\]
Interpolation (Theorem 31 with \(\eta=1\)) and (i) yield the estimate. ◻
Next we look at the regularizing effect of the heat semigroup on a space-time convolution term coming from the Duhamel formula for heat equations with right-hand sides \(f\).
Theorem 3. Assume [NDa], \(\beta \in [0,1]\), \(\alpha+\beta\notin \{ 2,3\}\), \(\gamma \in [0,1)\), and for \(t \in (0,T]\), \(f(t,\cdot)\in C_b^{\beta}(\mathbb{R}^d)\), \(\|f(t,\cdot)\|_{C^\beta_b} = O(t^{-\gamma})\) as \(t\to 0\), and \[w(t,x):=\int_0^t P_{t-s}[f(s)](x)\, ds.\] Then for \(t\in(0,T]\), if \(k=1\) or \(k=\lfloor\alpha+\beta\rfloor\), \[\begin{align} &(i)\quad \|D^kw(t,\cdot) \|_\infty \leq c^{(k)}_{\gamma,\alpha,\beta} \sup_{s\in(0,T]} \|s^\gamma f(s,\cdot) \|_{C_b^\beta}t^{-\gamma}t^{\frac{\alpha+\beta-k}{\alpha}},\\ \intertext{and ifk = \lfloor \alpha+\beta\rfloorand \{\alpha+\beta\}=\alpha+\beta-\lfloor \alpha+\beta\rfloor we also have\smallskip} & (ii) \quad [D^{\lfloor \alpha+\beta\rfloor}w(t,\cdot)]_{\{\alpha+\beta\} } \leq \overline{c} _{\gamma,\alpha,\beta} \sup_{s\in(0,T]} \|s^{\gamma}f(s,\cdot) \|_{C_b^\beta}t^{-\gamma}, \end{align}\] with constants \(c^{(k)}_{\gamma,\alpha,\beta}\) and \(\overline{c}_{\gamma,\alpha,\beta}\) only depending on \(\alpha,\beta,\gamma,k\).
Remark 4. (a) By assumptions \(\alpha+\beta\in(1,3)\), so \(\lfloor \alpha+\beta\rfloor\) is either \(1\) or \(2\).
(b) We do not consider \(\alpha+\beta\in \mathbb{N}\) in this paper. It is known that for equations of the form \(\mathcal{L}u = f\) with \(\alpha+\beta\in \mathbb{N}\), it is possible that \(u\notin C^{\alpha+\beta}_b(\mathbb{R}^d)\) and even \(D^{\alpha+\beta-1}u\) is not Lipschitz, see [50]. Repeating the proof below in this case yields an additional logarithmic factor in the modulus of continuity of \(D^{\alpha+\beta-1}u\).
(c) From the proofs it follows that \[\begin{align} &c^{(k)}_{\gamma,\alpha,\beta}= c_{k,\beta,d} B(1-\gamma,\tfrac{\alpha+\beta-k}{\alpha}), \\ & \overline{c}_{\gamma,\alpha,\beta} = 4c_{ \lfloor \alpha+\beta\rfloor,\beta,d}\big(\tfrac 1{1-\gamma} + \tfrac{\alpha}{\{\alpha+\beta\} }\big)\vee 2\Big( c_{\lfloor \alpha+\beta\rfloor ,\beta,d}B(1-\gamma,\tfrac{\{\alpha+\beta\}}{\alpha}) + c_{\lfloor \alpha+\beta\rfloor+1,\beta,d}(\tfrac{\alpha}{ 1-\{\alpha+\beta\} }+\tfrac{1}{1-\gamma})\Big), \end{align}\] with \(c_{k,\beta,d}\) from Theorem 2 and \(B(s_1, s_2)=\int_0^1 x^{s_1-1}(1-x)^{s_2-1} \, dx\) for \(s_1,s_2>0\) – the beta function. For \(\gamma=0\) we have (slightly better) constants: \(c^{(k)}_{0,\alpha,\beta} = \tfrac{\alpha c_{k,\beta,d}}{\alpha+\beta-k}\), \(\overline{c}_{0,\alpha,\beta}= \alpha \big(\tfrac{c_{k+1,\beta,d}}{k+1-(\alpha+\beta)}+\tfrac{2c_{k,\beta,d}}{\alpha+\beta-k}\big)\).
The following proof is adapted from the arguments in [1].
Proof. (i) Let \(t\in (0,T]\), \(x\in\mathbb{R}^d\). Using Theorem 2 and setting \(z=\frac{s}{t}\), \[\begin{align} |D^kw(t,x)| &\leq \int_{0}^t s^{-\gamma}|D^kP_{t-s}[s^{\gamma} f(s)](x)|\, ds \\&\leq \sup_{s \in (0,T]} \|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} c_{k,\beta,d} \int_0^t s^{-\gamma}(t-s)^{-\frac{k-\beta}{\alpha}}\, ds \\ &= t^{\frac{\alpha+\beta-k}{\alpha}-\gamma} \sup_{s\in(0,T]} \|s^{\gamma} f(s,\cdot)\|_{C_b^\beta} c_{k,\beta,d} \int_0^1 z^{-\gamma}(1-z)^{-\frac{k-\beta}{\alpha}} \, dz. \end{align}\] Interchanging derivative and integral in the first line holds by the Dominated Convergence Theorem.
Consider first the case \(\alpha+\beta \in (1,2)\) and \(k=\lfloor\alpha+\beta\rfloor=1\). We use Theorem 2 repeatedly. Let \(x, x' \in \mathbb{R}^d\) and \[\Delta:=\Big|\int_{0}^tDP_{t-s}[f(s)](x)\, ds- \int_{0}^tDP_{t-s}[f(s)](x')\, ds\Big|.\] If \(|x-x'|^{\alpha}\geq t\), then \[\begin{align} \Delta & \leq 2\int_{0}^t\big\|DP_{t-s}[f(s)]\big\|_\infty ds \leq 2 c_{1,\beta,d} \sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \int_0^{t}s^{-\gamma}(t-s)^{-\frac{1-\beta}{\alpha}}ds \\ &\leq 2c_{1,\beta,d}\sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \Big[\big(\tfrac{t}{2} \big)^{-\frac{1-\beta}{\alpha}} \int_0^{\frac{t}{2}}s^{-\gamma}ds +\big(\tfrac{t}{2}\big)^{-\gamma} \int_{\frac{t}{2}}^t (t-s)^{-\frac{1-\beta}{\alpha}}ds \Big]\\ &=2c_{1,\beta,d}\sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \Big[\big(\tfrac{t}{2} \big)^{-\frac{1-\beta}{\alpha}} \tfrac{1}{1-\gamma}\big( \tfrac{t}{2}\big)^{1-\gamma} +\big(\tfrac{t}{2}\big)^{-\gamma} \tfrac{\alpha }{\alpha+\beta-1} \big(\tfrac{t}{2} \big)^{\frac{\alpha+\beta-1}{\alpha}}\Big] \\ &=t^{-\gamma}t^{\frac{\alpha+\beta-1}{\alpha}}\sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \;2^{\tfrac{1-\beta}{\alpha}+\gamma}\;c_{1,\beta,d} \Big(\tfrac{1}{1-\gamma} +\tfrac{\alpha}{\alpha+\beta-1} \Big) \\ &\leq t^{-\gamma}|x-x'|^{\alpha+\beta-1}\sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \;4\;c_{1,\beta,d} \Big(\tfrac{1}{1-\gamma} +\tfrac{\alpha}{\alpha+\beta-1} \Big). \end{align}\] When \(t_0:=|x-x'|^{\alpha}< t\), we split the integral as \(\int_0^t = \int_0^{t-t_0} +\int_{t-t_0}^t\) obtaining \(\Delta\leq \Delta_1+\Delta_2\). By two changes of variables, including \(z=\frac{s}{t_0}\), \[\begin{align} \Delta_2 & :=\Big| \int_{t-t_0}^t DP_{t-s}[f(s)](x)\, ds- \int_{t-t_0}^t DP_{t-s}[f(s)](x')\, ds\Big| \\ &\leq 2 \int_{0}^{t_0} \|DP_{s}[f(t-s)]\|_\infty\, ds \\ &\leq 2c_{1,\beta,d} \sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \int_0^{t_0}(t-s)^{-\gamma}s^{-\frac{1-\beta}{\alpha}}\,ds \\ &= t^{-\gamma} t_0^{\frac{\alpha+\beta-1}{\alpha}}2c_{1,\beta,d}\sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \int_0^{1}\big(1-\tfrac{t_0}{t}z\big)^{-\gamma}z^{-\frac{1-\beta}{\alpha}}dz \\ & \leq t^{-\gamma} |x-x'|^{\alpha+\beta-1}2c_{1,\beta,d} \sup_{s\in(0,T]}\|s^{\gamma} f(s,\cdot) \|_{C_b^\beta} \int_0^{1}(1-z)^{-\gamma}z^{-\frac{1-\beta}{\alpha}}dz. \end{align}\] Then we note that \[\begin{align} \Delta_1&:=\Big| \int_0^{t-t_0} DP_{t-s}[f(s)](x)\, ds- \int_0^{t-t_0} DP_{t-s}[f(s)](x')\, ds\Big| \\ &\leq |x-x'| \int_0^{t-t_0}\int_0^1 |D^2P_{t-s}[f(s)](x+\lambda(x-x'))|\,d\lambda\, ds \\ & \leq c_{2,\beta,d} \sup_{s\in(0,T]} \|s^{\gamma}f(s,\cdot) \|_{C_b^\beta} |x-x'|\: I_\gamma, \end{align}\] where \(I_\gamma:=\int_{t_0}^t (t-s)^{-\gamma} s^{-\frac{2-\beta}{\alpha}}\, ds\). For \(\gamma = 0\), \[\begin{align} I_0= \tfrac{\alpha}{2-(\alpha+\beta)} (t_0^{\frac{\alpha+\beta-2}{\alpha}}-t^{\frac{\alpha + \beta -2}{\alpha}})\leq \tfrac{\alpha}{2-(\alpha+\beta)}t_0^{\frac{\alpha+\beta-2}{\alpha}}. \end{align}\] For \(\gamma \in(0,1)\) we consider two cases. For \(t_0\geq \frac{t}{2}\), \[\begin{align} I_\gamma & \leq t_0^{-\frac{2-\beta}{\alpha}} \int_{\frac{t}{2}}^t (t-s)^{-\gamma} ds = t_0^{\frac{\beta-2}{\alpha}} \tfrac1{1-\gamma} \big(\tfrac t 2\big)^{1-\gamma} \leq \tfrac{1}{1-\gamma} \big(\tfrac t 2\big)^{-\gamma}t_0^{\frac{\alpha+\beta-2}{\alpha}}, \end{align}\] while for \(t_0< \frac{t}{2}\), \[\begin{align} I_\gamma &\leq \big(\tfrac{t}{2} \big)^{-\gamma}\int_{t_0}^{\frac{t}{2}} s^{-\frac{2-\beta}{\alpha}}\, ds +\big(\tfrac{t}{2} \big)^{-\frac{2-\beta}{\alpha}}\int_{\frac{t}{2}}^{t}(t-s)^{-\gamma} \, ds\\ &\leq \tfrac{\alpha}{2-\alpha-\beta}\big(\tfrac{t}{2} \big)^{-\gamma}t_0^{-\frac{2-\alpha-\beta}{\alpha}} + \tfrac{1}{1-\gamma}\big(\tfrac{t}{2} \big)^{-\frac{2-\alpha-\beta}{\alpha}-\gamma}\\ &\leq 2^\gamma\big(\tfrac{\alpha}{2-\alpha-\beta} + \tfrac{1}{1-\gamma}\big)t^{-\gamma}t_0^{-\frac{2-\alpha-\beta}{\alpha}}. \end{align}\] Since \(t_0=|x-x'|^\alpha\), in all cases we have shown that \[\Delta\leq \overline{c}_{\gamma,\alpha,\beta}\sup_{s\in(0,T]} \|s^{\gamma}f(s,\cdot) \|_{C_b^\beta}t^{-\gamma}|x-x'|^{\alpha+\beta-1},\] which concludes the proof for \(\alpha+\beta \in (1,2)\).
The case \(\alpha+\beta \in (2,3)\), \(k=\lfloor\alpha+\beta\rfloor=2\) is similar – we replace \(D\) with \(D^2\) in the definition of \(\Delta\). ◻
Remark 5. It is well known that the solutions to the inhomogeneous heat equation for \(\mathcal{L}\), \[\begin{align} \label{linhe} \left\{ \begin{aligned} \partial_t u -\mathcal{L}u&=f(t,x), \qquad &(t,x)&\in (0,T]\times \mathbb{R}^d, \\ u(0,x)&=u_0(x), \qquad &x &\in \mathbb{R}^d, \end{aligned} \right. \end{align}\tag{4}\] can be expressed as \(u(t,x) = P_t u_0(x) + \int_0^t P_{t-s}[f(s)](x)\, ds\). Theorems 2 and 3 immediately give Schauder estimates for 4 . One of our main goals is to use Theorems 2 and 3 to obtain Schauder regularity for the nonlinear HJ equation 1 in Section 4. The estimates for 4 can also be viewed as a special case of the results below.
In this section we prove short-time existence of mild solutions of the viscous HJ equation 1 . We consider two interesting cases separately: (I) a setting with Lipschitz \(u_0\) that allows for global in time Lipschitz solutions, and (II) a setting with Hölder \(u_0\) where the gradient blows up as \(t\to 0\). In case (I) (see Theorem 8) existence holds under rather general local Lipschitz conditions on the Hamiltonian \(H\), while in case (II) (see Theorem 12) there is an interplay between the blowup of the gradient and the growth in \(p\) of the Lipschitz bounds for \(H(t,x,p)\).
Definition 1. A function \(\varphi\in C((0,T]; C^1_b(\mathbb{R}^d))\) is a mild solution of 1 if \[\begin{align} \varphi(t,x)= P_t[u_0](x)+\int_0^t P_{t-s}\big[ H(s,\cdot,D\varphi(s,\cdot)) \big](x)\, ds,\;t\in (0,T],\;x\in \mathbb{R}^d. \label{eq:duhamel95map} \end{align}\tag{5}\]
We give a classical result on translation in time of the Duhamel map, presented as in [51]. The proof follows from the semigroup property and linearity of \(P_t\).
Lemma 6. Assume \(T>0\), \(g_0\in L^{\infty}(\mathbb{R}^d)\), \(g\colon (0,T]\times \mathbb{R}^d\to\mathbb{R}\) is measurable, \(g(s,\cdot)\in L^\infty(\mathbb{R}^d)\) for \(s\in(0,T]\), \(\sup_{x\in\mathbb{R}^d}\int_0^{T}|P_{T-s}[g(s,\cdot)](x)| \, ds <\infty\), and define \[\begin{align} \omega(t,x) = P_t[g_0](x)+\int_0^t P_{t-s}[g(s,\cdot)](x) \, ds, \quad t\in(0,T],\;x\in\mathbb{R}^d, \end{align}\] If \(t_0, \tau, t_0+\tau \in(0,T]\), then \[\begin{align} \omega(t_0+\tau,x) = P_{\tau}[\omega(t_0,\cdot)](x)+\int_{t_0}^{t_0+\tau}P_{t_0+\tau-s}[g(s,\cdot)](x) \, ds. \end{align}\]
Here we work under assumptions ensuring that the gradient of the solution, \(\|Du(t,\cdot)\|_{\infty}\), is bounded as \(t\to 0\):
\(H\colon [0,T]\times \mathbb{R}^d \times \mathbb{R}^d\to \mathbb{R}\) is continuous, and for each \(R>0\), there are constants \(H_0,L_R>0\) such that for \(t\in(0,T)\), \(x\in\mathbb{R}^d\), and \(p,q \in B(0,R)\), \[|H(t,x,0)|\leq H_0 \qquad\text{and}\qquad|H(t,x,p)-H(t,x,q)| \leq L_R|p-q|.\]
\(u_0\) is bounded and Lipschitz.
Remark 7. Under [assump:H2], the Hamiltonian \(H\) is locally Lipschitz in \(p\), uniformly in \((t,x)\). An example is \(H(t,x,p)=b(t,x)g(p)+f(t,x)\) where \(b,f\in C_b([0,T]\times\mathbb{R}^d)\) and \(g\) is \(C^1(\mathbb{R}^d)\) or locally Lipschitz.
We have the following short-time existence result.
Theorem 8 (Short time existence (I)). Assume [NDa], [assump:H2], and [assump:H1]. Then there is a \(T>0\) for which there exists a unique mild solution \(u\in C_b((0,T]; C_b^1(\mathbb{R}^d))\) of 1 .
Note that a mild solution is a fixed point of the map \(S\) defined by \[\begin{align} \tag{6} &S[\phi](x,t) = P_t[u_0](x)+\int_0^t P_{t-s}\big[ H(s,\cdot,D\phi(s,\cdot)) \big](x)\, ds,\quad t\in (0,T],\;x\in \mathbb{R}^d.\\ \intertext{We will prove existence by an application of Banach fixed point theorem in the set\smallskip} \tag{7} &X_A := \{\varphi \in C_b((0,T]; C_b^1(\mathbb{R}^d)) \;: \;\|\varphi\|_{X_A} \leq R_1:= \|u_0\|_{C_b^1}+1 \}, \\ \intertext{where} &\|\varphi \|_{X_A} := \sup_{t \in (0,T]} \|\varphi(t,\cdot) \|_{C_b^1} = \sup_{t \in (0,T]} \big( \|\varphi(t, \cdot)\|_\infty + \|D\varphi(t, \cdot) \|_\infty \big).\nonumber \end{align}\] The following result will be used to show that \(S\) is a contractive self-map on \(X_A\).
Lemma 9. Assume [NDa], [assump:H1], [assump:H2], and \(u,v\in X_A\).
(i) For \(t\in (0,T]\), \(x \in \mathbb{R}^d\), and \(k\in\{0,1\}\), \[\begin{align} \qquad&|D^kS[u](t,x)|\leq \|D^ku_0 \|_\infty + (L_{R_1}R_1+H_0) c_0^k \frac{T^{1-\frac{k}{\alpha}}}{1 - \frac{k}{\alpha}}.\\ &|D^kS[u](t,x)-D^kS[v](t,x)| \leq L_{R_1} \sup\limits_{t\in (0,T]} \|D^ku(t,\cdot)-D^kv(t,\cdot)\|_{\infty} c_0^k \frac{T^{1-\frac{k}{\alpha}}}{1 - \frac{k}{\alpha}}.\\ \intertext{\noindent (ii) \;For t\in (0,T], x \in \mathbb{R}^d, and \gamma\in(0,\alpha-1),} &[DS[u](t,\cdot)]_\gamma\leq\|Du_0\|_{\infty}\, 2^{1-\gamma}c_{2,1,d} \, t^{-\frac{\gamma}{\alpha}}+ 4 c_0 (L_{R_1}R_1+H_0)\tfrac{\alpha}{\alpha-(1+\gamma)} t^{\frac{\alpha-(1+\gamma)}{\alpha}}. \\ \intertext{\noindent (iii) \; The mapping t \mapsto S[u](t, \cdot) is continuous in the C_b^1-norm, i.e. for t\in(0,T),} &\|S[u](t+\tau,\cdot) - S[u](t,\cdot)\|_{C_b^1} \xrightarrow{\tau \rightarrow 0} 0. \end{align}\]
Now we can prove existence.
Proof of Theorem 8. If \[\begin{align} T \leq T_0:=\bigg(\frac{\alpha-1}{2\alpha c_0(L_{R_1}R_1+H_0)}\bigg)^{\frac{\alpha}{\alpha-1}} \;\wedge \;\frac{1}{2(L_{R_1}R_1+H_0)} ,\label{eq:intervallength} \end{align}\tag{8}\] then by Lemma 9 (i), there is \(C<1\) such that \[\|S[u]\|_{X_A} \leq R_1 \qquad\text{and}\qquad \|S[u]-S[v]\|_{X_A} \leq C\|u-v\|_{X_A}.\] By Lemma 9 (ii) and (iii) we therefore get that \(S[u]\in X_A\). The existence of a fixed point of \(S\) in \(X_A\) then follows by Banach’s fixed point theorem. ◻
It remains to prove Lemma 9.
Proof of Lemma 9. Note that by [assump:H2], \(|H(t,x,Du)| \leq L_{R_1}R_1+H_0\) for \(u \in X_A\).
\((i)\) By the definition of \(S\), Young’s inequality for convolutions, [assump:H2], \(\|p_t\|_{L^1}=1\), and [NDa], \[\begin{align} |D^kS[u](t,x)| & \leq |D^kP_t[u_0](x)|+ \int_0^t |D^kp_{t-s}* H( s,\cdot, Du(s,\cdot))(x)|\, ds \\ &\leq \|D^ku_0 \|_\infty + \int_0^t \|D^kp_{t-s}\|_{L^1} \|H(s,\cdot, Du(s,\cdot))\|_{\infty} \, ds \\ &\leq\|D^ku_0 \|_\infty +(L_{R_1}R_1+H_0)\cdot \begin{cases} T, & k=0, \\ c_0\frac{T^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}},& k=1. \end{cases} \end{align}\] Similar computations for the differences conclude the proof of \((i)\): \[\begin{align} \|D^k(S[u]-S[v])(t,\cdot)\|_{\infty} &\leq \int_0^t \sup_{x\in \mathbb{R}^d} \big| D^kp_{t-s}*\big(H(s,\cdot,Du)-H(s,\cdot,Dv)\big)(x) \big| \, ds \\ &\leq L_{R_1}\int_0^t \| Du(s,\cdot)-Dv(s,\cdot)\|_{\infty} c_0^k (t-s)^{-\frac{k}{\alpha}} \, ds. \end{align}\]
\((ii)\) Note that \[\begin{align} [DS[u](t,\cdot)]_\gamma \leq [ DP_t u_0 ]_{\gamma} + \int_0^t [g_{t-s}]_\gamma \, ds \end{align}\] for \(g_{t-s}(x):=\int_{\mathbb{R}^d} Dp_{t-s}(x-y) H(s,y,Du(s,y)) \, dy.\) By interpolation (Lemma 31), [assump:H2], and [NDa], \[\begin{align} [g_{t-s}]_{\gamma} & \leq 2^{1-\gamma} \|g_{t-s} \|_\infty^{1-\gamma} \|Dg_{t-s} \|_\infty^{\gamma} \leq 2^{1-\gamma} \|H(\cdot,\cdot,Du)\|_{\infty} \|Dp_{t-s}\|_{L^1}^{1-\gamma}\|D^2p_{t-s}\|_{L^1}^\gamma\\ &\leq 2^{1-\gamma} (L_{R_1}R_1+H_0) c_0^{1+\gamma}2^{\frac{2\gamma}{\alpha}} (t-s)^{-\frac{1+\gamma}{\alpha}}. \end{align}\] By Theorem 2 we have that \([ DP_t u_0 ]_{\gamma} \leq 2^{1-\gamma}c_{2,1,d}\|Du_0 \|_\infty t^{-\frac{\gamma}{\alpha}}\).
\((iii)\) We show that \((0,T]\ni t \mapsto S[u](t, \cdot)\) is continuous in the \(C_b^1\)-norm. Let \(\tau>0\) and \(t+\tau \in(0,T]\) and note that by Lemma 6, assumptions [assump:H2] and [NDa], and the fact that \(\|p_t\|_{L^1}=1\), \[\begin{align} &\big| D^kS[u](t+\tau, x) - D^kS[u](t, x) \big| \\ & \leq \underbrace{\big| p_\tau * D^kS[u](t,\cdot)(x) - D^kS[u](t, x)\big|}_{A_k} + \int_t^{t+\tau}| D^kp_{t+\tau-s}* H(s,\cdot,Du)(x)| \, ds \\ &\leq {A_k} + \big( L_{R_1}R_1 + H_0 \big)\cdot \begin{cases} \tau, &\quad k=0, \\ c_0 \frac{\tau^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}}, &\quad k=1. \end{cases} \end{align}\] To estimate \(A_k\), fix \(t\in(0,T)\), and let \(\eta_0 = 1\) and \(\eta_1 =\gamma\) such that \(1<1+\gamma<\alpha\). Given \(\epsilon>0\), pick \(\delta>0\), and \(\tau_0>0\) small enough, such that \[\begin{align} &\delta^{\eta_k} < \frac{\epsilon}{2[D^kS[u](t,\cdot)]_{\eta_k}} \quad \text{and} \quad \int_{\mathbb{R}^d \setminus B_\delta(0)}p_\tau(y) \, dy < \frac{\epsilon}{4 \|D^k S[u]\|_\infty}, \quad 0<\tau<\tau_0, \end{align}\] hold, which is possible since \(p_\tau\stackrel{\ast}{\rightharpoonup} \delta_0\) in \(C_b(\mathbb{R}^d)\) by [52] and part \((ii)\) above. Then, \[\label{eq:A95k95calculation} \begin{align} A_k &= \Big| \int_{\mathbb{R}^d}p_\tau(y) \big[D^kS[u](t,x-y) - D^kS[u](t, x)\big] \, dy \Big| \\ & \leq \int_{B_\delta(0)}p_\tau(y) [D^kS[u](t,\cdot)]_{\eta_k}|y|^{\eta_k} \, dy + 2 \|D^kS[u] \|_\infty\int_{\mathbb{R}^d \setminus B_\delta(0)}p_\tau(y) \, dy \\ & < [D^kS[u](t,\cdot)]_{\eta_k} \delta^{\eta_k} + \frac{\epsilon}{2} < \epsilon, \end{align}\tag{9}\] so since \(\epsilon>0\) was arbitrary, \(A_k \xrightarrow{\tau \rightarrow 0} 0\) uniformly in \(x\) for each fixed \(t\). As a result, \[\begin{align} \|S[u](t+\tau,\cdot) - S[u](t,\cdot)\|_{C_b^1} \xrightarrow{\tau \rightarrow 0} 0, \end{align}\] for each \(t\in(0,T)\). The proof for \(t=T\) is similar, with \(t-\tau\) instead of \(t+\tau\). ◻
Remark 10. When \(\alpha\in (1,2)\), the claim in Lemma 9 \((ii)\) holds also for \(\gamma=\alpha-1\). This follows from using Theorem 3 instead of interpolation on the integral term.
We consider cases where the solution \(u\) satisfies \(\|Du(t,\cdot)\|_{\infty} = O(t^{-\gamma})\text{ as } t \rightarrow 0\) for some \(\gamma>0\), using the following assumptions:
\(H: (0,T]\times \mathbb{R}^d \times \mathbb{R}^d\to \mathbb{R}\) is continuous, and there are \(r\geq1\), \(H_0,L>0\), such that for \(t\in(0,T)\) and \(x,p,q \in \mathbb{R}^d\), \[|H(t,x,0)|\leq H_0 \qquad\text{and}\qquad|H(t,x,p)-H(t,x,q)| \leq L(1+|p|^2+|q|^2)^{\frac{r-1}{2}}|p-q|.\]
\(u_0\in C_b^\delta(\mathbb{R}^d)\) for some \(\delta\in[0,1)\) satisfying \(\begin{cases} \delta = 0,\quad & r \in [1,\alpha),\\[0.1cm] \delta\in (0,1),\quad &r=\alpha,\\[0.1cm] \delta \in \big(\frac{r-\alpha}{r-1},1\big),\quad &r\in(\alpha,\infty). \end{cases}\)
Remark 11. (a) By [assump:B295new], \(|H(t,x,p)|\leq L(1+|p|^2)^{\frac{r}{2}} + H_0\) and \(r\) is the growth rate of \(H\) in \(p\). An example is \(H(t,x,p)=b(t,x)|p|^r+f(t,x)\) where \(b,f\in C_b(\mathbb{R}^d)\).
(b) Note the interplay between \(\delta\) and \(r\) in [assump:B295new] and [assump:B1]; higher \(r\) requires higher \(\delta\). Moreover, for any \(r\geq 1\), there are \(\delta\in[0,1)\) such that [assump:B1] and [assump:B295new] hold, and vice versa for any \(\delta\) there are \(r\)’s.
We now show the short-time existence of mild solutions.
Theorem 12 (Short time existence (II)). Assume [NDa], [assump:B295new], and [assump:B1]. Then there is \(T>0\) for which there exists a unique mild solution \(u\in C((0,T]; C_b^1(\mathbb{R}^d))\) of 1 , and \(\|Du(t,\cdot)\|_\infty = O(t^{-(1-\delta)/\alpha})\) as \(t\to 0\).
To prove this result, we use a fixed point argument on the space \(X_B\), defined for \(\gamma\in (0,1)\) as \[\begin{align} X_B &:= \{\phi\in C((0,T]; C_b^1(\mathbb{R}^d)): \|\phi\|_{X_B} \leq R_2\},\;\text{where}\\[0.1cm] \|\varphi \|_{X_B} &:= \sup_{t \in (0,T]} \Big( \|\varphi(t, \cdot)\|_\infty + t^{\gamma}\|D\varphi(t, \cdot) \|_\infty \Big). \end{align}\] The constant \(R_2\) can be made explicit by inspecting the computations below.
Lemma 13. Assume [NDa], [assump:B295new], [assump:B1], \(u,v\in X_B\), and \(S[u]\) is defined in 6 . If \[\begin{align} (i) \quad \gamma r < 1, \qquad (ii) \quad \frac{1-\delta}{\alpha}\leq \gamma, \qquad \text{and}\qquad (iii) \quad \frac{1}{\alpha}+\gamma(r-1)<1, \end{align}\] then there are constants \(C_1,C_2>0\) such that \[\begin{align}&\|S[u]\|_{X_B} \leq \|u_0\|_\infty+ c_{1,\delta,d}[u_0]_\delta T^{\gamma-\frac{1-\delta}{\alpha}}\\&+C_1 \Big(T+T^{\gamma+1-\frac{1}{\alpha}}+ L(T^{2\gamma}+R_2^2)^{\frac{r}{2}} \big( T^{1-\gamma r}+T^{1-\frac{1}{\alpha} -\gamma (r-1)} \big)\Big), \\[0.1cm] &\|S[u] - S[v]\|_{X_B} \leq \;C_2\|u - v\|_{X_B}(T^{2\gamma}+2R_2^2)^{\frac{r-1}{2}} \big(T^{1-\gamma r}+T^{1-\frac{1}{\alpha}-\gamma(r-1)} \big). \end{align}\]
Proof. (a) Let \(k=0,1\). By the definition of \(S\) we have \[\begin{align} |D^kS[u](t,x)| & \leq |D^kP_t[u_0](x)|+ \int_0^t |D^kp_{t-s}*H(s,\cdot,Du(s,\cdot))(x)|\, ds. \end{align}\] By [assump:B295new], [NDa], and since \(\|p_t\|_{L^1}=1\), \[\begin{align} &\int_0^t |D^kp_{t-s}*H(s,\cdot,Du(s,\cdot))(x)| \, ds \\ & \leq H_0 c_0^k\frac{t^{1-\frac{k}{\alpha}}}{1-\frac{k}{\alpha}} + Lc_0^k\Big(t^{2\gamma}+ \sup_{t\in(0,T]}\|t^{\gamma}Du(t,\cdot)\|_\infty^2\Big)^{\frac{r}{2}}\int_0^t (t-s)^{-\frac{k}{\alpha}}s^{-\gamma r} \, ds \\ &\leq H_0\cdot\begin{cases} t, & \;k=0,\\[0.2cm] c_0\frac{t^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}}, & \;k=1, \end{cases} \quad + \quad L\Big(T^{2\gamma}+R_2^2\Big)^{\frac{r}{2}}\cdot \begin{cases} \frac{t^{1-\gamma r}}{1-\gamma r}, & \;k=0,\\[0.2cm] t^{1-\frac{1}{\alpha}-\gamma r}c_0 \int_0^1 (1-z)^{-\frac{1}{\alpha}}z^{-\gamma r } \, dz , & \;k=1. \end{cases} \end{align}\] By [assump:B1], [NDa], \(\|p_t\|_{L^1}=1\), and Theorem 2, we have \[\begin{align} |P_t[u_0](x)|\leq\|u_0\|_\infty \qquad \text{and} \qquad \|DP_t[u_0]\|_\infty \leq c_{1,\delta, d} [u_0]_\delta t^{-\frac{1-\delta}{\alpha}}. \end{align}\] The result now follows by combining the estimates to compute \(\|S[u](t,\cdot)\|_\infty+t^\gamma\|DS[u](t,\cdot)\|_\infty\), and noting that by the assumptions on \(\alpha,\delta,\gamma,r\), the integral is finite and all the resulting powers of \(t\) are nonnegative.
(b) Let \(k=0,1\). By [assump:B295new] and the definition of \(S\), \[\begin{align} & |D^k(S[u]-S[v])(t,x)| \\ &\leq \int_0^t \|D^kp_{t-s} \|_{L^1} \| H(s, \cdot, Du(s,\cdot))-H(s, \cdot, Dv(s,\cdot)) \|_\infty \; ds \\ & \leq c_0^k L \int_0^t (t-s)^{-\frac{k}{\alpha}} \big(1+\| Du(s,\cdot)\|_\infty^{2}+\| Dv(s,\cdot)\|_\infty^{2} \big)^{\frac{r-1}{2}}\|Du(s,\cdot)-Dv(s,\cdot) \|_\infty \; ds \\ &\leq c_0^k L \|u-v\|_{X_B} \sup_{t\in(0,T]}\big(t^{2\gamma}+\| t^{\gamma}Du(t,\cdot)\|_\infty^{ 2}+\|t^\gamma Dv(t,\cdot)\|_\infty^{ 2} \big)^{\frac{r-1}{2}} \int_0^t (t-s)^{-\frac{k}{\alpha}}s^{-\gamma r } \, ds \\ &\leq L \|u-v\|_{X_B} \big(T^{2\gamma}+2R_2^2 \big)^{{\frac{r-1}{2}}} \cdot\begin{cases} \frac{t^{1-\gamma r}}{1-\gamma r}, &\quad k=0, \\[0.2cm] t^{1-\frac{1}{\alpha}-\gamma r}c_0 \int_0^1 (1-z)^{-\frac{1}{\alpha}}z^{-\gamma r} \; dz, &\quad k=1, \end{cases} \end{align}\] where we have done the change of variables \(z=\frac{s}{t}\) in the final line for \(k=1\). The result now follows by combining the estimates to compute \(\|(S[u]-S[v])(t,\cdot)\|_\infty+t^\gamma\|D(S[u]-S[v])(t,\cdot)\|_\infty\), and noting that by the assumptions on \(\alpha,\gamma,r\), the integral is finite and all the resulting powers of \(t\) are nonnegative. ◻
It follows that \(S\) is a contraction on \(X_B\) for small \(T\), if assumptions (i) – (iii) in Lemma 13 hold: \[\begin{align} (i) \quad \gamma < \frac{1}{r}, \qquad (ii) \quad \frac{1-\delta}{\alpha}\leq \gamma, \qquad (iii)\quad \gamma < \frac{1-\frac{1}{\alpha}}{r-1}. \end{align}\] The powers of \(T\) in Lemma 13 (b) are then positive and the \(T\)-factor can be made small by taking \(T\) small. The above conditions can be reformulated as [assump:B1]. Indeed, note that \[\begin{align} \frac{1}{r} \leq \frac{1-\frac{1}{\alpha}}{r-1}\quad \iff \quad r\leq \alpha. \end{align}\] Elementary calculations show that if [assump:B1] holds, then \((i)\), \((ii)\), and \((iii)\) are satisfied with \(\gamma = (1-\delta)/\alpha\).
Proof of Theorem 12. We use Banach’s fixed point theorem in \(X_B\) with \(R_2=\|u_0\|_\infty+ c_{1,\delta,d}[u_0]_\delta+1\). For small enough \(T\), \(\|S[u]\|_{X_B}\leq R_2\) for all \(u\in X_B\) by Lemma 13 (a). Moreover, \(S[u](t,\cdot)\in C^1_b(\mathbb{R}^d)\) and is continuous in the \(C^1_b(\mathbb{R}^d)\)-norm by Lemma 9 \((ii)\) and \((iii)\) and Lemma 6, since functions in \(X_B\) have uniformly bounded spatial gradients at \(t\in (\varepsilon,T]\) for any \(\varepsilon>0\). It follows that \(S\) maps \(X_B\) into \(X_B\). Contractivity for small \(T\) follows from Lemma 13 (b). ◻
Assuming mild solutions exist on some time interval \([0,T]\), in this section we show that they attain optimal \(C^{\alpha+\beta}_b\) Hölder regularity in \(x\), a gain of order \(\alpha\) (= the order of \(\mathcal{L}\)) over the \(C^\beta_b\) regularity in \(x\) of \(H(t,x,p)\) (see [assump:H] and [assump:B3] below). When \(\mathcal{L}\) is a local second order elliptic operator, this corresponds to classical Schauder regularity theory [53]. We are also interested in the blow-up rates as \(t\to0\) of the optimal Hölder norms, and therefore we consider separately the cases where the spatial gradients of solutions are uniformly bounded in time or not.
We assume Hölder regularity in \(x\) of \(H(t,x,p)\), locally uniformly in \(p\):
There exists \(\beta \in [0,1]\), such that for each \(R>0\) there is a constant \(M_R>0\) for which for all \(t\in[0,T]\), \(p\in B(0,R)\), and \(x,x' \in \mathbb{R}^d\) with \(|x-x'|<1\), \[|H(t, x, p)-H(t, x', p)| \leq M_R|x-x'|^\beta.\]
Remark 14. This condition is consistent with [assump:H2], and when \(\beta=0\) it follows from [assump:H2]. An example satisfying both [assump:H2] and [assump:H] is \(H(t,x,p)=b(t,x)g(p)+f(t,x)\) for \(b,f\in C_b([0,T];C^\beta(\mathbb{R}^d))\) and locally Lipschitz \(g\).
We now give the \(C^{\alpha+\beta}_b\) Schauder regularity result for solutions that are Lipschitz in \(x\), uniformly in \(t\).
Theorem 15 (Schauder regularity (I)). Assume [assump:H2], [assump:H1], [assump:H] for some \(\beta \in [0,1]\), [NDa] with \(\alpha\in (1,2]\), \(\alpha+\beta \notin \mathbb{N}\), \(T>0\), and \(u\in C_b((0,T],C^1_b(\mathbb{R}^d))\) is a mild solution of 1 . Then the following statements hold:
(i) \(u(t)\in C_b^{\alpha +\beta}(\mathbb{R}^d)\) for \(t\in (0,T]\),
(ii) \([D^{\lfloor \alpha+\beta \rfloor} u(t,\cdot)]_{\{\alpha+\beta\}} = O(t^{-\frac{\alpha+\beta-1}{\alpha}})\) as \(t \rightarrow 0\),where \(\{\alpha+\beta\}=\alpha+\beta-\lfloor \alpha+\beta \rfloor\).
(iii) If \(\alpha+\beta> 2\), then \(\| D^2u(t,\cdot)\|_\infty=O(t^{-\frac{1}{\alpha}})\) as \(t\rightarrow 0\).
Remark 16. This a priori result holds for arbitrary \(T>0\) provided a mild solution exists in \([0,T]\).
To prove Theorem 15 we first show an intermediate result on \(C^{1+\beta}_b\) regularity of \(u\). Then a bootstrap argument using Theorem 3 will give the full regularity.
Lemma 17. Assume [NDa] with \(\alpha\in (1,2]\), [assump:H2], [assump:H1], and [assump:H] holds with some \(\beta\in [0,1]\). If \(u\in C_b((0,T];C^1_b(\mathbb{R}^d))\) is a mild solution of 1 , then \[\sup_{t \in (0,T]}[t^{\frac{\beta}{\alpha}}Du(t,\cdot)]_{\beta}<\infty \qquad \text{and} \qquad \sup_{t \in (0,T]}[t^{\frac{\beta}{\alpha}}H(t,\cdot, Du(t,\cdot))]_{\beta}<\infty.\]
Proof. The result is immediate for \(\beta=0\) by [assump:H2], so consider \(\beta>0\). We begin as in the proof of Lemma 9 (ii), but now we exploit the regularity of \(H\). By the definition of a mild solution, \[\begin{align} \frac{|Du(t,x+h)-Du(t,x)|}{|h|^\beta} \leq \frac{|DP_tu_0(x+h)-DP_tu_0(x)|}{|h|^\beta}+\int_0^t \frac{|g_{t-s}(x+h)-g_{t-s}(x)|}{|h|^\beta}, \end{align}\] where \(g_{t-s}(x):=\int_{\mathbb{R}^d} Dp_{t-s}(y) H(s,x-y,Du(s,x-y)) \, dy\).
By Theorem 2, \([ DP_t u_0 ]_{\beta}\leq \|Du_0\|_{\infty}2^{1-\beta}c_{2,1,d}t^{-\frac{\beta}{\alpha}}\). Assumptions [assump:H2] and [assump:H] yield \[\begin{align} &|g_{t-s}(x+h)-g_{t-s}(x)|\\ &\leq \Big|\int_{\mathbb{R}^d} Dp_{t-s}(y)\big( H(s,x-y+h,Du(s,x-y+h))-H(s,x-y+h,Du(s,x-y))\big)\, dy \Big| \\ &\qquad + \Big|\int_{\mathbb{R}^d} Dp_{t-s}(y)\big( H(s,x-y+h,Du(s,x-y))-H(s,x-y,Du(s,x-y)) \big) \, dy \Big| \\ &\leq c_0(t-s)^{-\frac{1}{\alpha}}\bigg( L_{R_1}\sup_{x\in\mathbb{R}^d}|Du(s,x+h)-Du(s,x)| + \begin{cases} M_{R_1}|h|^\beta, &|h|<1,\\ 2(L_{R_1}R_1+H_0)|h|^\beta, &|h|\geq 1 \end{cases} \bigg), \end{align}\] where \(R_1=\|Du\|_\infty\). Let \(K_{R_1}:=\max \{M_{R_1}, 2(L_{R_1}R_1+H_0) \}\). Integrating in time, we see that \[\begin{align} \frac{1}{|h|^\beta}\int_0^t &|g_{t-s}(x+h)-g_{t-s}(x)| \; ds \\ &\leq c_0\frac{T^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}}K_{R_1} + c_0L_{R_1}\int_0^t(t-s)^{-\frac{1}{\alpha}} \sup_{x\in \mathbb{R}^d}\frac{|Du(s,x+h)-Du(s,x)|}{|h|^\beta} \, ds. \end{align}\] Consequently, by summing up the estimates, \[\begin{align} \label{eq:whdef} w_{h}(t)&:=\sup_{x \in \mathbb{R}^d} \frac{|Du(t,x+h)-Du(t,x)|}{|h|^\beta} \\ &\leq \|Du_0\|_{\infty}2^{1-\beta}c_{2,1,d}\,t^{-\frac{\beta}{\alpha}} +c_0 K_{R_1} \frac{T^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}} +c_0L_{R_1}\int_0^t(t-s)^{-\frac{1}{\alpha}} w_{h}(s) \, ds.\nonumber \end{align}\tag{10}\] We now want to use the generalized Grönwall inequality of Lemma 32. We indeed have that \(w_{h}(t)\) is locally integrable on \((0,T]\), since \[\begin{align} \label{eq:w95h95loc95int} 0 \leq w_{h}(t) = \sup_{x\in \mathbb{R}^d} \frac{|Du(t,x+h)-Du(t,x)|}{|h|^{{\beta}}} \leq \frac{2}{|h|^{{\beta}}}\|Du\|_{\infty} <\infty, \end{align}\tag{11}\] for each fixed \(h\). Hence, all the assumptions of Lemma 32 are satisfied and we get \[\begin{align} \label{eq:H95regularity95gronwall95calc} w_{h}(t) &\leq \Big(2^{-\beta} c_{2,1,d}\, \|Du_0\|_{\infty} + C_{1} T^{1-\frac{1}{\alpha}}\Big)t^{-\frac{{\beta}}{\alpha}} +c_0 K_{R_1}\frac{T^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}}+C_2T^{1-\frac{1}{\alpha}}, \end{align}\tag{12}\] where \(C_1 = C_1(T,\alpha,\beta,R_1,u_0)\) and \(C_2=C_2(T,\alpha,R_1)\). Importantly, the constants do not depend on \(h\). Thus, for each \(t\in (0,T]\), taking the supremum over \(|h|\) yields \[\begin{align} [Du(t)]_{\beta} < \infty, \end{align}\] so \(Du(t,\cdot)\in C_b^\beta(\mathbb{R}^d)\). Multiplying 12 by \(t^{\frac{\beta}{\alpha}}\), we get \(\sup_{t \in (0,T]}[t^{\frac{\beta}{\alpha}}Du(t,\cdot)]_{\beta}<\infty\).
The following computation completes the proof: If \(|x-x'|<1\), then \[\begin{align} &|H(t,x,Du(t,x))-H(t,x',Du(t,x'))| \\ &\leq |H(t,x,Du(t,x))-H(t,x',Du(t,x))|+|H(t,x',Du(t,x))-H(t,x',Du(t,x'))| \\ &\leq M_{R_1}|x-x'|^\beta+L_{R_1}|Du(t,x)-Du(t,x')| \\ &\leq (M_{R_1}+L_{R_1}[Du(t,\cdot)]_{\beta})|x-x'|^\beta, \end{align}\] whence we find that \[[H(t,\cdot,Du(t,\cdot))]_\beta \leq L_{R_1}[Du(t,\cdot)]_\beta + K_{R_1} < \infty,\] where we have used assumptions [assump:H2] and [assump:H], and that \(Du(t,\cdot)\in C_b^\beta(\mathbb{R}^d)\). ◻
Proof of Theorem 15. Differentiating both sides of \(u=S[u]\) for a fixed \(t\in(0,T]\), we get \[\begin{align} \label{eq:full95schauder95duhamel} Du(t,x) = P_t\big[Du_0\big](x)+\int_0^t DP_{t-s}\big[H(s,\cdot,Du(s,\cdot))\big](x)\, ds. \end{align}\tag{13}\] Let \(v(t,x): =\int_0^t s^{-\frac{\beta}{\alpha}} DP_{t-s}\big[s^{\frac{\beta}{\alpha}}H(s,\cdot,Du(s,\cdot))\big](x)\, ds.\)
Case 1: \(\alpha + \beta \in (1,2)\) and \(\{\alpha+\beta\}=\alpha+\beta-1\). By 13 , Lemma 17, Theorem 2 (ii), and Theorem 3 (ii), \[\begin{align}\label{eq::full95schauder95case95195calc} [Du(t,\cdot)]_{\alpha+\beta-1} &\leq [P_t[Du_0]]_{\alpha+\beta-1}+[v(t,\cdot)]_{\alpha+\beta-1} \\ &\leq \|Du_0\|_{\infty} 2^{2-(\alpha+\beta)}c_{2,1,d}\, t^{-\frac{\alpha+\beta-1}{\alpha}}\\ &\qquad \qquad +t^{-\frac{\beta}{\alpha}} \overline{c}_{\frac{\beta}{\alpha},\alpha,\beta} \sup_{t\in(0,T]}\|t^{\frac{\beta}{\alpha}}H(t,\cdot,Du(t,\cdot)) \|_{C^\beta_b} < \infty. \end{align}\tag{14}\]
Case 2: \(\alpha+\beta \in (2,3)\) and \(\{\alpha+\beta\}=\alpha+\beta-2\). Since \((t-s)^{-\frac{2-\beta}{\alpha}}=(t-s)^{-1+\frac{\alpha+\beta-2}{\alpha}}\) is integrable at \(s=t\), by Theorem 2 and the Dominated Convergence Theorem, \[\begin{align} D^2u(t,x) &= DP_t[Du_0](x)+\int_0^t D^2P_{t-s}\big[H(s,\cdot,Du(s,\cdot))\big](x)\, ds, \;\;t\in(0,T],\;x\in \mathbb{R}^d. \end{align}\] By Theorem 2 (i) (and \(c_{1,0,d}=c_0\)), Lemma 17, and Theorem 3 (i), we get that \[\begin{align} \label{eq::full95schauder95second95deriv95blowup} \|D^2 u(t)\|_\infty \leq t^{-\frac{1}{\alpha}}c_0\|Du_0\|_\infty+t^{-\frac{2-\alpha}{\alpha}} c^{(2)}_{\frac{\beta}{\alpha},\alpha,\beta} \sup_{t \in (0,T]} \|t^{\frac{\beta}{\alpha}}H(t,\cdot, Du(t,\cdot)) \|_{C_b^{\beta}}<\infty. \end{align}\tag{15}\] To get the blow-up rate for the optimal Hölder seminorm we do similar calculations as in Case 1: \[\begin{align} [D^2u(t,\cdot)]_{\alpha+\beta-2} \leq &\;\|Du_0\|_{\infty} 2^{3-(\alpha+\beta)} c_{3,1,d} \, t^{-\frac{\alpha+\beta-1}{\alpha}} \\ &\qquad +t^{-\frac{\beta}{\alpha}} \overline{c}_{\frac{\beta}{\alpha},\alpha,\beta} \sup_{t\in(0,T]}\|t^{\frac{\beta}{\alpha}}H(t,\cdot,Du(t ,\cdot)) \|_{C^\beta_b} < \infty. \qedhere \end{align}\] ◻
In order to handle blow-up rates for higher order Hölder norms with non-Lipschitz initial conditions we use an \(x\)-Hölder condition on \(H(t,x,p)\) with at most polynomial growth in \(p\) of order \(r\).
There exist \(\beta\in [0,1]\) and \(M>0\) such that for \(t\in(0,T]\) and \(x,x',p \in \mathbb{R}^d\) with \(|x-x'|<1\), \[|H(t, x, p)-H(t, x', p)| \leq M(1+|p|^2)^{\frac{r}{2}}|x-x'|^\beta,\] where \(r\) is defined in [assump:B295new].
Remark 18. This assumption is consistent with [assump:B295new], and when \(\beta=0\) it follows from [assump:B295new]. An example satisfying both [assump:B295new] and [assump:B3] is \(H(t,x,p)=b(t,x)|p|^r+f(t,x)\) for \(b,f\in C([0,T];C_b^\beta(\mathbb{R}^d))\).
We now give the \(C^{\alpha+\beta}_b\) Schauder regularity result for solutions with spatial gradients that blow up as \(t\to 0\).
Theorem 19 (Schauder regularity (II)). Assume [assump:B295new], [assump:B1], [assump:B3] for some \(\beta \in [0,1]\), [NDa] with \(\alpha\in (1,2]\), and \(\alpha+\beta \notin \mathbb{N}\), and \(u\in C((0,T],C^1_b(\mathbb{R}^d))\) is a mild solution of 1 . Then the following statements hold:
(i) \(u(t)\in C_b^{\alpha +\beta}(\mathbb{R}^d)\) for each \(t\in (0,T]\).
(ii) \([D^{ \lfloor\alpha+\beta\rfloor} u(t,\cdot)]_{\{\alpha+\beta\}} = O(t^{-\frac{\alpha+\beta-\delta}{\alpha} })\) as \(t \rightarrow 0\), where \(\{\alpha+\beta\}=\alpha+\beta-\lfloor\alpha+\beta\rfloor\).
(iii) If \(\alpha+\beta> 2\), then \(\| D^2u(t,\cdot)\|_\infty=O(t^{-\frac{2-\delta}{\alpha}})\) as \(t\rightarrow 0\).
Remark 20. Since solutions are assumed to be Lipschitz for \(t>0\), we can use Theorem 15 to see that part (i) holds even if we replace [assump:B3] by the weaker assumption [assump:H].
To prove Theorem 19 \((ii)\) and \((iii)\) we first show an intermediate result on the \(C^{1+\beta}_b\) blow-up rate as \(t\rightarrow 0\) of \(u\).
Lemma 21. Assume [NDa] with \(\alpha\in (1,2]\), [assump:B3] with \(\beta\in [0,1]\), [assump:B295new] with \(r\geq1\), [assump:B1] with \(\delta\in[0,1)\), and \(u\in C((0,T],C^1_b(\mathbb{R}^d))\) is a mild solution of 1 . Then \[\begin{align} \sup_{t \in (0,T]}[t^{\frac{1+\beta-\delta}{\alpha}}Du(t,\cdot)]_{\beta}<\infty\qquad\text{and}\qquad \sup_{t \in (0,T]}[t^{\frac{r(1-\delta)+\beta}{\alpha}}H(t,\cdot, Du(t,\cdot))]_{\beta}<\infty.\label{eq:Hbetareg} \end{align}\qquad{(1)}\]
The proof is given at the end of the section.
Proof of Theorem 19. \((i)\) \(C^{\alpha+\beta}\) regularity for \(t>0\) follows from Theorem 15 since \(u(t)\) by assumption is Lipschitz for \(t>0\). It remains to show the blow-up rates in \((ii)\) and \((iii)\).
Case 1: \(\alpha+\beta \in (1,2]\) and \(\{\alpha+\beta\}=\alpha+\beta-1\). Let \(0<\epsilon<T\), \(t\in (0,T-\epsilon]\). From 14 we have that \[\begin{align} [Du(t+\epsilon,\cdot)]_{\alpha+\beta-1} &\leq \|Du(\epsilon,\cdot)\|_{\infty} \,2^{2-(\alpha+\beta)}c_{2,1,d}\, t^{-\frac{\alpha+\beta-1}{\alpha}} \\ &\qquad +t^{-\frac{\beta}{\alpha}} \overline{c}_{\frac{\beta}{\alpha},\alpha,\beta} \sup_{s\in(0,T-\epsilon]}\|s^{\frac{\beta}{\alpha}}H(s+\epsilon,\cdot,Du(s+\epsilon,\cdot)) \|_{C^\beta_b} \\ &\leq \|Du(\epsilon,\cdot)\|_{\infty}\, 2^{2-(\alpha+\beta)}c_{2,1,d}\, t^{-\frac{\alpha+\beta-1}{\alpha}} \\ &\qquad +t^{-\frac{\beta}{\alpha}} \epsilon^{-\frac{r(1-\delta)}{\alpha}}\overline{c} _{\frac{\beta}{\alpha},\alpha,\beta} \sup_{s\in(0,T-\epsilon]}\|(s+\epsilon)^{\frac{r(1-\delta)+\beta}{\alpha}}H(s+\epsilon,\cdot,Du(s+\epsilon,\cdot)) \|_{C^\beta_b}. \end{align}\] Evaluating at \(t=\epsilon\) and noting that \(\alpha -\delta -r(1-\delta)\geq 0\) by [assump:B1], \[\begin{align} [Du(2\epsilon,\cdot)]_{\alpha+\beta-1} &\leq \epsilon^{-\frac{\alpha+\beta-\delta}{\alpha}} \sup_{s\in(0,T]}\|s^{\frac{1-\delta}{\alpha}} Du(s,\cdot)\|_{\infty}2^{2-(\alpha+\beta)}c_{2,1,d} \nonumber \\ & \qquad +\epsilon^{-\frac{r(1-\delta)+\beta}{\alpha}} \overline{c}_{\frac{\beta}{\alpha},\alpha,\beta} \sup_{s \in(0,T]}\|s^{\frac{r(1-\delta)+\beta}{\alpha}}H(s,\cdot,Du(s,\cdot)) \|_{C^\beta_b}\nonumber\\ &\leq \epsilon^{-\frac{\alpha+\beta-\delta}{\alpha}}\Big(\sup_{s\in(0,T]}\|s^{\frac{1-\delta}{\alpha}} Du(s,\cdot)\|_{\infty} 2^{2-(\alpha+\beta)}c_{2,1,d}\label{stestim}\\ &\qquad +T^{\frac{\alpha-\delta-r(1-\delta)}{\alpha}} \overline{c} _{\frac{\beta}{\alpha},\alpha,\beta} \sup_{s \in(0,T]}\|s^{\frac{r(1-\delta)+\beta}{\alpha}}H(s,\cdot,Du(s,\cdot)) \|_{C^\beta_b} \Big).\nonumber \end{align}\tag{16}\] By Theorem 12 and Lemma 21 the expression in the brackets is finite.
Case 2: \(\alpha+\beta \in (2,3)\) and \(\{\alpha+\beta\}=\alpha+\beta-2\). The Hölder estimate follows in a similar manner and yields the same blow-up rate as Case 1. The blow-up rate for \(\|D^2u(t,\cdot)\|_\infty\) can be shown in a similar way from 15 . ◻
We now prove Lemma 21. In the proof we have to distinguish between two cases, \(\delta\geq \beta\) and \(\delta<\beta\). The second case relies on a bootstrapping argument that uses the result from the first case and reuses arguments from the proofs of both the first case and Theorem 19 above.
Proof of Lemma 21. The case \(\beta = 0\) follow from Theorem 12 and assumption [assump:B295new], so assume \(\beta>0\).
Case 1: \(\delta\geq\beta\). By Theorem 2, \[\begin{align} \label{eq:Du0beta} [ DP_t u_0 ]_{\beta} &\leq 2^{1-\beta} c_{2,\delta,d} [u_0]_\delta t^{-\frac{1+\beta-\delta}{\alpha}}. \end{align}\tag{17}\] Let \(g_{t-s}(x):=\int_{\mathbb{R}^d} Dp_{t-s}(y) H(s,x-y,Du(s,x-y)) \, dy\), and define \[G:=\sup_{t\in (0,T]}\big(t^{2\frac{1-\delta}{\alpha}}+2\|t^{\frac{1-\delta}{\alpha}}Du(t,\cdot) \|_\infty^2\big),\] \(\widetilde{L}:= G^{\frac{r-1}{2}}L\), and \(K:=\max\big\{G^{\frac{r}{2}}M, H_0 T^{\frac{r(1-\delta)}{\alpha}}+LG^{\frac{r}{2}} \big\}\). By [assump:B295new], [assump:B3], and the bound \(\|Du(t,\cdot)\|_\infty=O(t^{-\frac{1-\delta}{\alpha}})\) from Theorem 12 (for \(t\) small) and the assumption \(u\in C((0,T],C^1_b(\mathbb{R}^d))\) (for \(t\) not small), \[\begin{align} &|g_{t-s}(x+h)-g_{t-s}(x)| \leq c_0(t-s)^{-\frac{1}{\alpha}}\Big(\widetilde{L} s^{-\frac{(r-1)(1-\delta)}{\alpha}} \sup_{x\in\mathbb{R}^d}|Du(s,x+h)-Du(s,x)| + K s^{-\frac{r(1-\delta)}{\alpha}} |h|^\beta \Big). \end{align}\] Integrating in time, with \(B:=B(1-\tfrac{r(1-\delta)}{\alpha},1-\tfrac1\alpha)\) (see Remark 4 (c)), yields \[\begin{align} \frac{1}{|h|^\beta}\int_0^t &|g_{t-s}(x+h)-g_{t-s}(x)| \; ds \leq c_0 KB t^{1-\frac{1}{\alpha}-\frac{r(1-\delta)}{\alpha}} \label{eq:gbeta}\\ &+ c_0 \widetilde{L} \int_0^t(t-s)^{-\frac{1}{\alpha}}s^{-\frac{(r-1)(1-\delta)}{\alpha}} \sup_{x\in \mathbb{R}^d}\frac{|Du(s,x+h)-Du(s,x)|}{|h|^\beta} \, ds.\nonumber \end{align}\tag{18}\] Since \(u\) is a mild solution of 1 , it follows from estimates 17 and 18 that \[\begin{align} &w_h(t) :=\sup_{x \in \mathbb{R}^d} \frac{|Du(t,x+h)-Du(t,x)|}{|h|^\beta} \\ & \leq t^{-\frac{1+\beta-\delta}{\alpha}} \big( 2^{1-\beta} c_{2,\delta,d} [u_0]_\delta + c_0BK T^{1+\frac{\beta-\delta}{\alpha}-\frac{r(1-\delta)}{\alpha}} \big) +c_0\widetilde{L}\int_0^t(t-s)^{-\frac{1}{\alpha}}s^{-\frac{(r-1)(1-\delta)}{\alpha}} w_h(s) \, ds. \end{align}\] Then since [assump:B1] and \(\delta\geq \beta\) hold, it follows from the Grönwall-type inequality in Lemma 33 with \(\overline{\alpha}=1-\frac{1+\beta-\delta}{\alpha}\), \(\overline{\beta}=1-\frac{1}{\alpha}\), \(\overline{\gamma}=1-\frac{(r-1)(1-\delta)}{\alpha}\) that \[\begin{align} \label{eq:H95regularity95gronwall95calc95case95B} w_h(t) \leq \overline{C} \, t^{-\frac{1+\beta-\delta}{\alpha}}, \end{align}\tag{19}\] for some constant \(\overline{C}>0\).
Multiplying 19 by \(t^{\frac{1+\beta-\delta}{\alpha}}\) and taking the supremum over \(|h|\), we see that \(\sup_{t \in (0,T]}[t^{\frac{1+\beta-\delta}{\alpha}}Du(t,\cdot)]_{\beta}<\infty\). The proof is complete by noting that by [assump:B295new], [assump:B3], and the bound \(\|Du(t,\cdot)\|_\infty=O(t^{-\frac{1-\delta}{\alpha}})\) from Theorem 12 (for \(t\) small) and the assumption \(u\in C((0,T],C^1_b(\mathbb{R}^d))\) (for \(t\) not small), \[\begin{align} \label{eq:H95Du95triangle95ineq} \begin{aligned} |H(t,x,&Du(t,x))-H(t,x',Du(t,x'))| \\ &\leq Kt^{-\frac{r(1-\delta)}{\alpha}}|x-x'|^\beta+\widetilde{L}t^{-\frac{(r-1)(1-\delta)}{\alpha}}|Du(t,x)-Du(t,x')| \\ &\leq t^{-\frac{r(1-\delta)+\beta}{\alpha}}\Big(KT^{\frac{\beta}{\alpha}}+\widetilde{L}\sup_{t\in (0,T]}[t^{\frac{1+\beta-\delta}{\alpha}}Du(t,\cdot)]_{\beta}\Big)|x-x'|^\beta. \end{aligned} \end{align}\tag{20}\]
\(0\leq\delta<\beta\). Here we use a combination of Grönwall and bootstrapping arguments.
Initial \(C^{1+\beta_0}\) estimate for some small \(\beta_0\in(0,\beta]\). We begin by proving that estimates ?? hold when \(\beta\) is replaced by \(\beta_0\). If \(\delta>0\) we take \(\beta_0=\delta\), and by Case 1 (with \(\beta=\beta_0\)) it follows that \[\sup_{t\in(0,T]} [t^{\frac{1}{\alpha}}Du(t,\cdot)]_{\beta_0} < \infty.\] If \(0=\delta<\beta\), then \(r\in[1,\alpha)\) by [assump:B1] and we take \(\beta_0\in(0,\alpha-r)\). We redo the proof of Case 1 with \(\beta=\beta_0\) and \(\delta=0\). By our choice of \(\beta_0\), we can again apply Grönwall (Lemma 33) and conclude that \[\sup_{t\in(0,T]} [t^{\frac{1+\beta_0}{\alpha}}Du(t,\cdot)]_{\beta_0} < \infty.\] In either case, since \(H(t,\cdot,p)\) is \(\gamma\)-Hölder regular for any \(\gamma\in(0,\beta]\) by [assump:B3], estimate 20 holds with \(\beta_0\) replacing \(\beta\), and hence \[\begin{align} \label{eq::blow95up95H95bootstrap} \sup_{t\in(0,T]}[t^{\frac{r(1-\delta)+\beta_0}{\alpha}}H(t,\cdot,Du(t,\cdot))]_{\beta_0} \leq KT^\frac{\beta_0}{\alpha}+\widetilde{L}\sup_{t\in(0,T]} [t^{\frac{1+\beta_0-\delta}{\alpha}}Du(t,\cdot)]_{\beta_0}<\infty. \end{align}\tag{21}\] We conclude that estimates ?? hold when \(\beta\) is replaced by \(\beta_0\).
Bootstrapping. In view of (i), we can now redo the proof of Theorem 19 with \(\beta_0\) replacing \(\beta\) to obtain \(C^{\alpha+\beta_0}\) regularity of \(u\) and corresponding blow-up estimate 16 : For \(\beta_1=\alpha+\beta_0-1\), \[\begin{align} \label{eq::blow95up95Du95bootstrap} \begin{aligned} [Du(t, \cdot)]_{\beta_1} &\leq t^{-\frac{1+\beta_1-\delta}{\alpha}} \Big(\sup_{s\in(0,T]}\|s^{\frac{1-\delta}{\alpha}} Du(s,\cdot)\|_{\infty} 2^{1-\beta_1}c_{2,1,d} \\ &\qquad +T^{\frac{\alpha-\delta-r(1-\delta)}{\alpha}} \overline{c} _{\frac{\beta_0}{\alpha},\alpha,\beta_0} \sup_{s \in(0,T]}\|s^{\frac{r(1-\delta)+\beta_0}{\alpha}}H(s,\cdot,Du(s,\cdot)) \|_{C^{\beta_0}_b} \Big), \end{aligned} \end{align}\tag{22}\] where the norms are finite by Lemma 12 and part (i). We can now update 21 , replacing \(\beta_0\) by \(\beta_1\), and conclude that estimates ?? hold when \(\beta\) is replaced by \(\beta_1\). Note the gain, \(\beta_1-\beta_0=\alpha-1>0\).
Iteration. For \(k=1,2,\dots,N-1\), we repeat part (ii) replacing \(\beta_k\) with \(\beta_{k+1}\), where \(\beta_{k}=k(\alpha-1)+\beta_0\). The result is \(C^{1+\beta_{k+1}}\) regularity and that estimates ?? hold when \(\beta\) is replaced by \(\beta_{k+1}\). Let \(N\) be smallest integer such that \((N+1)(\alpha-1)+\beta_0>\beta\). Once we reach \(k=N-1\), we do a final iteration with \(\beta_{N+1} =\beta\) to achieve \(C^{1+\beta}\) regularity and conclude the proof of ?? . ◻
We use the regularity results of Section 4 to prove that mild solutions of 1 are classical solutions. We first state a result for the linear case.
Lemma 22. Assume [NDa] and \(\phi\in L^\infty(\mathbb{R}^d)\). Then \(P_t\phi\in C^\infty_b(\mathbb{R}^d)\) for \(t>0\) and \[\begin{align} \partial_t P_t \phi(x) = \mathcal{L}P_t\phi(x),\qquad t>0,\;x\in \mathbb{R}^d. \end{align}\]
Proof. First note that since \(P_t\phi\) is an \(L^1-L^\infty\) convolution, it is uniformly continuous. By [NDa] and the dominated convergence theorem we further get that \(P_t\phi\in C^1_b(\mathbb{R}^d)\) and \(DP_{t}\phi(x) = P_{t/2}DP_{t/2}\phi(x)\). By a bootstrap argument we get that \(P_t\phi\in C^{\infty}_b(\mathbb{R}^d)\). By [54], the convolution operators \(P_t\) form a \(C_b\)-semigroup and by [54], \(C^2_b(\mathbb{R}^d)\) belongs to the domain of the \(C_b\)-generator of \(P_t\), which we denote by \(A\) (see [54]). Therefore, by [54] we have \[\begin{align} \partial_t P_t\phi = \partial_t P_{t/2}P_{t/2}\phi = AP_{t/2}P_{t/2}\phi = AP_t \phi. \end{align}\] It remains to prove that for \(C^2_b(\mathbb{R}^d)\) functions the operator \(A\) agrees pointwise with \(\mathcal{L}\). To this end, let \(\tau\) be a smooth cut-off function vanishing outside \(B(x,1)\). Then, \[\begin{align} AP_t\phi(x) = A[(P_t\phi)\tau](x) + A[(P_t\phi)(1-\tau)](x) = \mathcal{L}[(P_t\phi)\tau](x) + \mathcal{L}[(P_t\phi)(1-\tau)](x), \end{align}\] where the last equality follows from [47] for the first term and from [47] and the definition of \(A\) for the second term. ◻
When the maximal regularity of the mild solution is \(\alpha+\beta < 2\), we naturally need to restrict the maximal order of the operator \(\mathcal{L}\) to be less than \(\alpha+\beta\) for our problems to have classical solutions. We give such a condition in terms of the Levy triplet:
Remark 23. (a) Assumption [NDa] implies that the order of \(\mathcal{L}\) is at least \(\alpha\), while Assumption [assump:L2] implies that \(\mathcal{L}\) does not contain any terms of order \(\geq \sigma(<\alpha+\beta)\). E.g. if \(\beta=\frac{1}{2}\), then \(\mathcal{L}=-(-\delta^2_{x_1})^{\frac{1}{4}}-(-\Delta_{\mathbb{R}^d})^{\frac{1.1}{2}}-(-\delta^2_{x_2})^{\frac{3}{4}}\) satisfies [NDa] with \(\alpha=1.1\) and [assump:L2] for any \(\sigma\in(\frac{3}{2},1.1+\frac{1}{2})\).
(b) Let \(\mathcal{L}u = \mathop{\mathrm{div}}(ADu) -(-\Delta)^{\frac{\tilde{\alpha}}{2}}u\) with degenerate nontrivial \(A\) and \(\tilde{\alpha}\in (1,2)\). Then [NDa] holds with \(\alpha=\tilde{\alpha}<2\), but because of the second derivatives, \(\alpha+\beta>2\) is needed for solutions of 1 to be \(C^2\) and classical. We use [assump:L2] to exclude this case.
(c) Any Lévy operator \(\mathcal{L}\) satisfies \(\int_{\mathbb{R}^d}(1\wedge |z|^2)\, d\mu(x)<\infty\) and hence \(\|\mathcal{L}u\|_\infty \leq C\|u\|_{C^2_b(\mathbb{R}^d)}\). Assumption [assump:L2] implies \(\|\mathcal{L}u\|_\infty \leq C\|u\|_{C^{\sigma+}_b(\mathbb{R}^d)}\) for \(\sigma<2\).
Theorem 24. Assume the assumptions of Theorem 15 hold for \(\alpha\in (1,2]\), \(\beta \in (0, 1]\), \(\alpha + \beta \notin \mathbb{N}\), and either \(\alpha+\beta > 2\) or [assump:L2] holds. Then the mild solution \(u\) of 1 is a classical solution, meaning that it satisfies 1 pointwise and \(u(t,x)\to u_0(x)\) as \(t\to 0\) for \(x\in \mathbb{R}^d\).
Remark 25. The result also holds if \(u_0\in C^\delta_b(\mathbb{R}^d)\) under the assumptions of Theorem 19. Since the solution immediately becomes \(\alpha+\beta\)-Hölder, the proof is almost identical. The only difference is that to show the convergence to \(u_0\) we use the fact that the bound on \(\|H(s,\cdot,Du(s,\cdot))\|_\infty\) is integrable in \(s\) (in Case (I) it is bounded).
The following proof mirrors that of [33].
Proof. Consider the Duhamel representation of \(u\): \[\begin{align} u(t,x) = P_t\big[u_0\big](x)+\int_0^t P_{t-s}\big[H(s,\cdot,Du(s,\cdot))\big](x)\, ds. \end{align}\] By Theorem 15 \(Du\) is uniformly bounded, so \(H(s,x,Du(s,x))\) is also uniformly bounded. From this we immediately get that \(u(t,x)\to u_0(x)\) as \(t\to 0\) for all \(x\in \mathbb{R}^d\), so it suffices to verify that \(u\) satisfies 1 .
By Lemma 6 and Theorem 8 we can assume that \(u\in C([0,T],C^1_b(\mathbb{R}^d))\). By Lemma 22, \[\begin{align} \partial_t P_t\big[u_0\big](x) = \mathcal{L}P_t\big[u_0\big](x). \end{align}\] To simplify the notation, let \(f(t,x)=H(t,x,Du(t,x))\). It remains to show that \[\begin{align} \label{eq:classical95sol95remains95to95show} \partial_t \int_0^t P_{t-s}\big[f(s,\cdot)\big](x) \, ds = \mathcal{L}\int_0^t P_{t-s}\big[f(s,\cdot)\big](x) \, ds + f(t,x), \end{align}\tag{23}\] which formally follows from the Leibniz integral rule, but requires some work due to the singularity of the heat kernel at time zero. It suffices to show 23 for \(t \in [\delta_0, T]\), with \(\delta_0>0\) fixed. Let \(\delta \in (0,\delta_0/2)\), \[\begin{align} &\varphi(t,s,x) = P_{t-s}\big[f(s,\cdot)\big](x), \qquad (t,s,x) \in [\delta_0, T]\times(0,T)\times \mathbb{R}^d, \;s< t, \\ &\Phi_\delta(t,x)= \int_0^{t-\delta} \varphi(t,s,x) \, ds, \quad \Phi(t,x)=\int_0^t \varphi(t,s,x) \, ds, \qquad (t,x)\in [\delta_0, T] \times \mathbb{R}^d. \end{align}\] We will first show that \(\partial_t \Phi_\delta = \mathcal{L}\Phi_\delta\). Then, we prove that \(\Phi_\delta(\cdot,x)\) converges uniformly to \(\Phi(\cdot,x) + \varphi(\cdot,\cdot,x)\) on \([\delta_0,T]\) and that on the same interval \(\partial_t\Phi_\delta(\cdot,x)\) converge uniformly to \(\mathcal{L} \Phi(\cdot,x) + f(\cdot,x)\). This implies that \(\partial_t\Phi\) exists and is equal to \(\mathcal{L} \Phi + f\).
Let \(t_1,t_2\in(0,T]\), \(R>0\). By continuity of \(H\) in [assump:H2] there is a modulus of continuity \(\omega_R\) such that, \[\begin{align} |H(t_1,x,p) - H(t_2,x,p)| \leq \omega_R(t_1-t_2),\qquad x,p\in B(0,R), \end{align}\] and since \(u\in C([0,T],C^1_b(\mathbb{R}^d))\) by assumption, there is a modulus of continuity \(\tilde{\omega}\) such that \(|Du(t_1,y)-Du(t_2,y)|\leq \tilde{\omega}(t_2-t_1)\) for all \(y\in \mathbb{R}^d\). Take \(R_1=\max_{t\in[\delta_0,T]}\|Du(\cdot,t)\|_{L^\infty(\mathbb{R}^d)}\), then for all \(x \in B(0,R)\), \[\label{eq:f95locally95uniformly95cont} \begin{align} |f(t_2,x)-f(t_1,x)| &\leq |H(t_2,x,Du(t_2,x))-H(t_1,x,Du(t_2,x))| \\ &\qquad \qquad +|H(t_1,x,Du(t_2,x))-H(t_1,x,Du(t_1,x))| \\ &\leq \omega_{R}(t_2-t_1)+L_{R_1}\tilde{\omega}(t_2-t_1) =: \overline{\omega}_R(t_2-t_1). \end{align}\tag{24}\]
Step 1: The result holds for \(\Phi_\delta\). We claim that \[\begin{align} \label{eq:classical95sol95calc95leibniz} \partial_t \Phi_\delta(t,x) &= \varphi(t,t-\delta,x)+\int_0^{t-\delta}\partial_t\varphi(t,s,x) \, ds =\varphi(t,t-\delta,x) + \mathcal{L}\Phi_\delta(t,x). \end{align}\tag{25}\] The first equality is justified as follows: By splitting integrals and using the mean value theorem for integrals we can write, for some \(\lambda \in (0,1)\) depending on \(\tau\), \[\begin{align} \frac{\Phi_\delta(t+\tau,x)-\Phi_\delta(t,x)}{\tau} = \varphi(t+\tau, t-\delta +\lambda\tau, x) + \int_0^{t-\delta} \frac{\varphi(t+\tau,s,x)-\varphi(t,s,x)}{\tau} \, ds. \end{align}\] Here \(\varphi(t+\tau, t+\lambda\tau-\delta, x)\to\varphi(t, t-\delta, x)\) as \(\tau\to 0\): For any \(\epsilon>0\), we can take \(R\) large so that \(\underset{s\in [0,\delta]}{\sup}\int_{B(0,R)^c}p_s(y)\, dy < \epsilon\) [52], and then \[\begin{align} &|\varphi(t+\tau, t-\delta+\lambda\tau, x) - \varphi(t,t-\delta,x)|\\ &= \int_{\mathbb{R}^d} p_{(1-\lambda)\tau+\delta} (x-y) |f(t-\delta+\lambda\tau,y) - f(t-\delta,y)|\, dy +\, |(P_\delta - P_{(1-\lambda)\tau+\delta} ) f(t-\delta)( x)|\\ &\leq\, 2\epsilon \|f\|_{\infty} + \overline{\omega}_R(\lambda \tau) + |(P_{ (1-\lambda) \tau} - I)P_{\delta}f(t-\delta)( x)|\to 2\epsilon \|f\|_{\infty}\qquad\text{as}\qquad\tau\to 0. \end{align}\]
By Lemma 17, \(\sup_{t\in(0,T]}\|f(t,\cdot)\|_\infty < \infty\), so (in particular) \(\varphi(t,s,\cdot) \in C^2_b(\mathbb{R}^d)\) for all \(0<s<t\). For \(s\in (0,t-\delta)\), Lemma 22 and Theorem 2 (with \(\beta=0\)) then yield \[\begin{align} \label{eq::varphi95bounded95calc} |\partial_t \varphi(t,s,x)| &= |\mathcal{L}\varphi(t,s,x)| \lesssim \|\varphi(t,s,\cdot)\|_{C^2_b} \lesssim (t-s)^{-\frac{2}{\alpha}} \leq \delta^{-\frac{2}{\alpha}} < \infty. \end{align}\tag{26}\] Therefore, by the dominated convergence theorem, \[\begin{align} \lim_{\tau \rightarrow 0} \int_0^{t-\delta} \frac{\varphi(t+\tau,s,x)-\varphi(t,s,x)}{\tau} \, ds = \int_0^{t-\delta} \partial_t \varphi(t,s,x) \, ds. \end{align}\] The second equality of 25 then follows by the arguments related to 26 and Fubini’s theorem: \[\begin{align} \int_0^{t-\delta} \partial_t \varphi(t,s,x) \, ds= \int_0^{t-\delta} \mathcal{L}\varphi(t,s,x) \, ds = \mathcal{L}\Phi_\delta(t,x). \end{align}\] Step 2: \(\partial_t \Phi_\delta\) converges uniformly as \(\delta \rightarrow 0\). In the rest of the proof, let \(\sigma=2\) if \(\alpha+\beta > 2\), and \(\sigma\in(\alpha,\alpha+\beta)\) be given by [assump:L2] if \(\alpha+\beta<2\). Recall that \(\Phi(t,\cdot)\in C^{\alpha+\beta}_b(\mathbb{R}^d)\) by Theorem 15, so \(\mathcal{L}\Phi(t,x)\) is well-defined. Furthermore, Theorems 2 and 17 (and [assump:L2] if \(\alpha+\beta<2\)) yield \[\begin{align} &\int_0^t\int_{\mathbb{R}^d}|\varphi(t,s,x+z) - \varphi(t,s,x) - D\varphi(t,s,x)\cdot z\boldsymbol{1}_{B(0,1)}(z)|\, d\mu(z)\, ds \\ &\leq \int_0^t \|\phi(t,s,\cdot)\|_{C^{\sigma}_b} \int_{\mathbb{R}^d} (1\wedge|z|^\sigma)\, d\mu(z)\, ds\leq C\int_0^t (t-s)^{-\frac{\sigma-\beta}{\alpha}}s^{-\frac{\beta}{\alpha}}\, ds < \infty. \end{align}\] By this, Fubini’s theorem, and the dominated convergence theorem (if \(\mathcal{L}\) has a second order part), \(\mathcal{L}\Phi(t,x) = \int_0^{t} \mathcal{L}\varphi(t,s,x) \, ds\).
Consider now the right-hand side of 25 . Let \[M_{\delta_0} := \sup_{s\in (t-\delta,t)} \|f(s, \cdot) \|_{C_b^{\beta}} \leq \sup_{s\in (\delta_0/2,T]} \|f(s, \cdot) \|_{C_b^{\beta}} < \infty.\] Similar computations as above, using Theorem 2, Lemma 17, (and [assump:L2] if \(\alpha+\beta<2\)), show that \[\begin{align} |\mathcal{L}\Phi(t,x)-\mathcal{L}\Phi_\delta(t,x)| &\leq \int_{t-\delta}^t |\mathcal{L}\varphi(t,s,x)| \, ds \lesssim \int_{t-\delta}^t \| \varphi(t,s,\cdot) \|_{C_b^{ \sigma}} \\ & \lesssim M_{\delta_0}\int_{t-\delta}^t (t-s)^{-\frac{ \sigma-\beta}{\alpha}} \, ds \simeq M_{\delta_0} \delta^{\frac{\alpha+\beta- \sigma}{\alpha}} \xrightarrow{\delta \rightarrow 0} 0. \end{align}\] By adding and subtracting terms, we see that \[\begin{align} \varphi(t,t-\delta,x) =\int_{\mathbb{R}^d}p(\delta,x-y)(f(t-\delta,y)-f(t,y) ) \, dy + \int_{\mathbb{R}^d}p(\delta,x-y)f(t,y) \, dy. \end{align}\] Arguing as in 9 and using the \(\beta\)-Hölder continuity of \(f\) from Lemma 17, \[\begin{align} \Big|f(t,&x)-\int_{\mathbb{R}^d}p_\delta(x-y)f(t,y) \, dy \Big| \xrightarrow{\delta \rightarrow 0} 0, \end{align}\] and by 24 , \[\begin{align} \Big| \int_{\mathbb{R}^d} &p_\delta(x-y)(f(t-\delta,y)-f(t,y) ) \, dy \Big| \\ &\leq 2\|f \|_\infty \int_{\mathbb{R}^d \setminus B(x,1)} p_\delta(x-y) \, dy + \overline{\omega}_1(\delta)\int_{B(x,1)} p_\delta(x-y) \, dy \xrightarrow{\delta \rightarrow 0} 0, \end{align}\] where the first term tends to zero as \(\delta \rightarrow 0\) by [52].
Consequently, \(\partial_t\Phi_\delta\) converges uniformly to \(H(t,x,Du(t,x)) + \mathcal{L}\Phi(t,x)\). This, together with the fact that \(\Phi_\delta\) converges uniformly to \(\Phi\), \[\begin{align} |\Phi(t,x)-\Phi_\delta(t,x)| \leq M_{\delta_0}\int_{t-\delta}^t \|p(t-s)\|_{L^1} \, ds = M_{\delta_0} \delta \xrightarrow{\delta \rightarrow 0} 0, \end{align}\] implies that \(\partial_t \Phi\) exists and equals the limit as \(\delta \rightarrow 0\) of \(\partial_t \Phi_\delta\), i.e. 23 holds. ◻
Under additional assumptions on the Hamiltonian \(H\), global Lipschitz bounds on \(u\) have been shown in the literature. In such settings, we now prove long-time existence for mild and smooth solutions of 1 . The gradient estimates are usually obtained in the context of viscosity solutions, see e.g. [33] or [29], but they apply also to smooth solutions since smooth solutions are viscosity solutions and viscosity solutions are unique.
Theorem 26. Assume \(T>0\), [assump:H2], [assump:H1], [NDa] with \(\alpha\in (1,2]\), and that one of the following hold:
There exists an \(L>0\) such that \[\begin{align} |H(t,x,p)-H(t,y,p)|\leq L(1+|p|)|x-y|, \quad x,y,p\in\mathbb{R}^d, \;t\in (0,T]. \end{align}\]
There exists \(m>1\), \(\overline{K}, b_m, L_H>0\) and a modulus of continuity \(\zeta\) such that for all \(\mu \in (0,1)\), \(x,y, p, q \in \mathbb{R}^d\), \(|q|\leq 1\) and \(t\in(0,T]\) the following hold: \[\begin{align} &\mu H(t,x,\mu^{-1}p)-H(t,x,p) \geq (1-\mu)(b_m|p|^m-\overline{K}), \\ &H(t,y,p+q) -H(t,x,p) \leq L_H|x-y|(1+|p|^m)+\zeta(|q|)(1+|p|^{m-1}). \end{align}\]
Then there is a unique bounded viscosity solution \(u\) of 1 on \((0,T]\times \mathbb{R}^d\) and a constant \(M_T>0\) such that \(\|Du(t,\cdot) \|_\infty \leq M_T\). Furthermore, \(u\) is a mild and classical solution on \((0,T]\) and \(u(t,\cdot) \in C_b^{\alpha+1}(\mathbb{R}^d)\) if \(\alpha\in (1,2)\), while \(u(t,\cdot) \in C_b^{\alpha+1-\epsilon}(\mathbb{R}^d)\) for arbitrarily small \(\epsilon\) if \(\alpha=2\).
Remark 27. (a) When \((i)\) holds, we can take \(M_T = e^{2L T}\big(\tfrac{1}{2}L + \|Du_0 \|_\infty^2 \big)^{\frac{1}{2}}\) as in [26] (with \(f=0\), see also [33]). When \((ii)\) holds, \(M_T\) depends only on \(u_0\), \([u_0]_1\), \(T\), and \(\text{osc}_T(u):=\sup_{t\in (0,T]} \{\sup_{\mathbb{R}^d} u(t,\cdot) - \inf_{\mathbb{R}^d} u(t,\cdot) \}\) by [55].
(b) \(H\) is superlinear in case (ii). The first inequality is a coercivity condition that enables a weak Bernstein argument to be used (see the discussion in [55]). An example satisfying \((ii)\) is given in [56], \[\begin{align} H(x,p)=c(x)|p|^m+a(x)|p|^l, \qquad x,p \in \mathbb{R}^d, \end{align}\] with \(c, a\) bounded uniformly continuous in \(\mathbb{R}^d\), \(c(x)\geq \underline{c}>0\), \(m>1\), \(1\leq l <m\).
(c) Both \((i)\) and \((ii)\) assume (locally) \(x\)-Lipschitz \(H\), and hence [assump:H] holds and mild solutions are smooth (Schauder regular) by Theorem 15 and classical solutions by Theorem 24. Using the Ishii–Lions method and strong ellipticity of \(\mathcal{L}\), it is possible to obtain global Lipschitz bounds when \(H\) is just continuous but satisfy certain growth bounds. We refer to [57] for general results and examples, including a setting with \(H(t,x,Du) = c(x) |Du|^\alpha\), \(c\in C_b(\mathbb{R}^d)\), and \(\mu(dz) \approx |z|^{-d-\alpha}\, dz\) [57] Eq. (25).
Proof. Assume first \((i)\) holds. Existence, uniqueness, and Lipschitz bounds for bounded viscosity solutions are given by [26]. Our assumptions are the same, except the local Lipschitz continuity for \(H(t,x,\cdot)\) in [assump:H2] instead of their \(C^3\)-type condition \((A3)\). But the proofs are not affected, since higher derivatives of \(H\) are only needed to prove higher regularity of \(u\).
Then we consider case \((ii)\). Existence and uniqueness of a bounded viscosity solution and the Lipschitz bound are given by [55] respectively. Note that their (A) and (J) assumptions are trivially satisfied in our setting (corresponding to \(j(x,z)=z\) and \(A=0\) in [55]).
Let \(u\) be the viscosity solution from step 1, \(\widetilde{M}_T:=\sup_{t\in [0,T]} \|u(t,\cdot) \|_{C_b^1}<\infty\), and \(t_0\in [0,T]\). Then equation 1 with initial condition \(u(t_0,\cdot)\), has a mild solution \(v\) on \((t_0,t_0+\widetilde{T}_0)\) by Theorem 8, where \[\begin{align} \widetilde{T}_0:= \bigg(\frac{\alpha-1}{2\alpha c_0(L_{\widetilde{R}_1}\widetilde{R}_1+H_0)}\bigg)^{\frac{\alpha}{\alpha-1}} \;\wedge \;\frac{1}{2(L_{\widetilde{R}_1}\widetilde{R}_1+H_0)}\qquad \text{and} \qquad \widetilde{R}_1 = \widetilde{M}_T+1. \end{align}\] Since both (i) and (ii) imply that [assump:H] holds (with \(\beta=1\)), the mild solution \(v\) is also classical by Theorem 24 and \(C^{\alpha+1}_b\) in space by Theorem 15. Note that the existence time \(\widetilde{T}_0\) is independent of \(t_0\). Since classical solutions are viscosity solutions and viscosity solutions are unique, \(u=v\) on \((t_0,t_0+\widetilde{T}_0)\). Since \(t_0\in[0,T]\) is arbitrary, this means that \(u\) is a mild and classical solution satisfying the Schauder regularity for every \(t\in(0,T)\). ◻
In this section we discuss some extensions of our results and sketch their proofs.
With almost no change in the proofs we can get optimal Schauder estimates for Hamiltonians depending also on other quantities \(Qu\) of order not exceeding 1. More precisely, we let \(H = H(t,x,p,q)\) and assume: \[\begin{align} &|H(t,x',p',q') - H(t,x,p,q)| \leq C_R(1\wedge|x-x'|^\beta + |p-p'| + |q-q'|)\quad \textrm{and}\quad H(t,x,0,0)\leq C\tag{27}\\ &\qquad\qquad\textrm{for all}\quad t\in [0,T],\;x,x'\in \mathbb{R}^d,\;p,q,q'\in B(0,R),\nonumber \\[0.2cm] &\|Qu\|_\infty \leq C\|u \|_{C^1_b(\mathbb{R}^d)} \qquad \textrm{and}\qquad [Qu]_\sigma \leq C \|u\|_{C^{1+\sigma}_b}\quad\text{for all}\quad \sigma\in (0,\beta].\tag{28} \end{align}\] We then have the following generalization of Theorems 8 and 15 on short-time existence and Schauder regularity for the case of solutions with bounded gradients uniformly in time.
Theorem 28. Assume [assump:H1], 27 , 28 for some \(\beta\in [0,1]\), and [NDa] with \(\alpha\in (1,2]\) such that \(\alpha+\beta\notin \mathbb{N}\). Then, for sufficiently small \(T\), the equation \[\begin{align} \left\{ \begin{aligned} \partial_t u -\mathcal{L}u-H(t,x,Du,Qu)&=0, \qquad &(t,x)&\in (0,T]\times \mathbb{R}^d, \\ u(0,x)&=u_0(x), \qquad &x &\in \mathbb{R}^d, \end{aligned} \right. \end{align}\] has a mild solution \(u\in C_b((0,T],C^1_b(\mathbb{R}^d))\) and for all \(t\in (0,T]\) we have \(u(t)\in C^{\alpha+\beta}_b(\mathbb{R}^d)\).
Sketch of the proof. The short-time existence is done in almost the same way as in Theorem 8, in the set \(X_A\) defined in 7 . The only difference is that now to control the Hamiltonian term, we use 27 and the fact that \(\|Qu\|_\infty \leq \|u\|_{C^1_b(\mathbb{R}^d)}\).
For the \(C^{\alpha+\beta}_b(\mathbb{R}^d)\) estimate we also use a bootstrap argument – first establishing \(C^{1+\beta}_b(\mathbb{R}^d)\) regularity and then improving it as was done in Theorem 15. The difference now is that we use different arguments to get \(C^{1+\beta}_b(\mathbb{R}^d)\) regularity of \(u(t)\), as Grönwall’s inequality seems to impose additional restrictions on \(Q\): If \(\alpha > 1+\beta\) it follows immediately from (ii) in Lemma 9, while for \(\alpha\leq 1+\beta\) we only get \(u(t)\in C^{\alpha-\epsilon}_b(\mathbb{R}^d)\) from Lemma 9, but then \(H(t,x,Du,Qu)\) is \(\alpha-\epsilon-1\) Hölder regular by 27 and 28 and we can get \(1+\beta\) regularity by bootstrapping with the use of Theorem 3. Once we get that \(u(t)\in C^{1+\beta}_b(\mathbb{R}^d)\), the remainder of the proof is identical to that of Theorem 15. ◻
Example 2. (a) \(Qu = u\) satisfies the assumption 28 .
(b) Let \(\nu\) be a Lévy measure satisfying \(\int_{\mathbb{R}^d} (1 \wedge |z|)\, d\nu(z)< \infty\) and define \[\begin{align} Q u(x) = \int_{\mathbb{R}^d} (u(x+j(x,z)) - u(x))\, d\nu(z), \quad x\in \mathbb{R}^d, \end{align}\] where for \(x,y,z\in \mathbb{R}^d\), \[|j(x,z)-j(y,z)|\leq L(1\wedge |z|)|x-y|^\beta\qquad\text{and}\qquad|j(x,z)|\leq K(1\wedge |z|).\] Then a straightforward calculation shows that 28 holds: \[\begin{align} \|Q u\|_\infty &\leq C \|u\|_{C^1_b}\qquad\text{and}\qquad |Qu(x) - Qu(y)| \leq C \|u\|_{C^{1+\sigma}_b} |x-y|^\sigma,\quad \sigma \in (0,1],\;x,y\in \mathbb{R}^d. \end{align}\]
(c) Theorem 28 covers problems 1 when \(\mathcal{L}\) is replaced by an operator with modulated jumps, \[\begin{align} \mathcal{L}_{\boldsymbol{j}} u(x) = \lim\limits_{\epsilon\to 0}\int_{B(0,\epsilon)^c} (u(x) - u(x+\boldsymbol{j}(z)))\, d \mu(z),\quad x\in \mathbb{R}^d, \end{align}\] provided that \(\boldsymbol{j}\colon \mathbb{R}^d \to \mathbb{R}^d\) behaves well at the origin, e.g. if \[\begin{align} \label{eq:jcond}|\boldsymbol{j}(z) - z| \leq C|z|^2,\quad |z|\leq 1. \end{align}\tag{29}\] We simply write \(\mathcal{L}_{\boldsymbol{j}} = \mathcal{L}+ (\mathcal{L}_{\boldsymbol{j}} - \mathcal{L})\) and view this as \(\mathcal{L}\) plus a linear perturbation \(Q\), \[\begin{align} Qu(x):=(\mathcal{L}_{\boldsymbol{j}} - \mathcal{L})u(x) = \lim\limits_{\epsilon\to 0}\int_{B(0,\epsilon)^c} (u(x+z) - u(x+\boldsymbol{j}(z)))\, d \mu(z),\quad x\in \mathbb{R}^d. \end{align}\] Assuming 29 , it is easy to see that 28 holds, and by redefining \(H(t,x,Du) + Qu\) as \(H(t,x,Du,Qu)\), we see that also 27 holds.
Remark 29. It is now quite standard to extend Theorem 28 to Bellman/dynamic programming equations for optimal control problems for jump-diffusions of order less than one [5], [6], \[\begin{align} \left\{ \begin{aligned} \partial_t u -\mathcal{L}u-\sup_{\theta}\Big\{Q^\theta u + b^\theta(t,x)Du + c^\theta (t,x)u + f^\theta(t,x)\Big\}&=0, \qquad &(t,x)&\in (0,T]\times \mathbb{R}^d, \\ u(0,x)&=u_0(x), \qquad &x &\in \mathbb{R}^d, \end{aligned} \right. \end{align}\] where \(Q^\theta\) is defined as in (b) with \(j^\theta\) in place of \(j\), and we assume the following uniformly in \(\theta\): \[|j^\theta(x,z)|+\frac{|j^\theta(x,z)-j^\theta(y,z)|}{|x-y|} \leq K(1\wedge |z|),\] combined with (standard) uniform in \(\theta\) boundedness and Lipschitz conditions on \((b^\theta,c^\theta,f^\theta)\).
Our optimal Schauder regularity implies that under assumptions [assump:H] and [NDa] the solutions are \(C^{\alpha+\beta}_b(\mathbb{R}^d)\), in the sense that the total derivative of order \(\lfloor \alpha+\beta\rfloor\) is \(\{\alpha +\beta\}\)-Hölder regular. However, for some specific operators \(\mathcal{L}\) we can expect that some directional derivatives will have more regularity. This is the case e.g. when \(\mathcal{L}= (-\Delta)^{\alpha_1/2}_{x_1} + (-\Delta)^{\alpha_2/2}_{x_2}\) with \(1<\alpha_2\leq\alpha_1\leq 2\) and \(x_1\in \mathbb{R}^{d_1}\), \(x_2 \in \mathbb{R}^{d_2}\). The reason for this is that the heat kernel \(p_t\) of such an operator is a convolution of the heat kernels of the two fractional Laplacians: \(p_t^1\) and \(p_t^2\). Then, given a function \(f\in C^{\beta}_b(\mathbb{R}^{d_1+d_2})\), we have \(p_t \ast f = p_t^1\ast (p_t^2\ast f)\) and since \(\|p_t^2\ast f(\cdot,x_2)\|_{C^\beta_b(\mathbb{R}^{d_1})}\) are uniformly bounded for \(x_2\in \mathbb{R}^{d_2}\), the regularity in \(x_1\) only depends on \(p_t^1\). The following more general result holds true.
Theorem 30. Let \(d=d_1+d_2\) for some \(0\leq d_1,d_2\leq d\) and for \(x\in \mathbb{R}^d\) denote \(x=(x_1,x_2)\) where \(x_i\in\mathbb{R}^{d_i}\), \(i=1,2\). Assume [assump:H2], [assump:H1], and [assump:H] with \(\beta\in[0,1]\), and that \(\mathcal{L}^1\) and \(\mathcal{L}^2\) are Lévy operators on \(\mathbb{R}^{d_1}\) and \(\mathbb{R}^{d_2}\) respectively, which satisfy [NDa] with \(\alpha_1\) and \(\alpha_2\) respectively, with \(1<\alpha_2\leq\alpha_1\leq 2\). Define a Lévy operator on \(\mathbb{R}^d\) by \(\mathcal{L}= \mathcal{L}^1_{x_1} + \mathcal{L}^2_{x_2}\).
(i)For sufficiently small \(T\), there exists a mild solution \(u\in C_b((0,T];C^1_b(\mathbb{R}^d))\) of 1 .
(ii)If \(\alpha_2+\beta \notin \mathbb{N}\), then \(u(t)\in C^{\alpha_2+\beta}_b(\mathbb{R}^d)\) for each \(t\in (0,T]\).
(iii)If also \(\alpha_1+\beta \notin \mathbb{N}\), then for each \(t\in (0,T]\) we have \(u(t,\cdot,x_2)\in C^{\alpha_1+\beta}_b(\mathbb{R}^{d_1})\) uniformly for \(x_2\in \mathbb{R}^{d_2}\), that is, \(\sup_{x_2}[D_{x_1}^{ \lfloor\alpha_1+\beta\rfloor} u(t,\cdot,x_2)]_{\{\alpha_1+\beta\}} < \infty\) where \(\{\alpha+\beta\}=\alpha+\beta-\lfloor \alpha+\beta \rfloor\).
Part (iii) above states that we get more regularity in the direction of \(x_1\). The result could be further generalized – in some cases we could get more regularity in non-axial directions, for example by adding a one-dimensional diffusion in such direction. We refer to [26] for a related discussion.
A. Rutkowski was supported by the National Science Center (Poland) grant 2023/51/B/ST1/02209. E. R. Jakobsen and R. Ø. Lien received funding from the Research Council of Norway under Grant Agreement No. 325114 “IMod. Partial differential equations, statistics and data: An interdisciplinary approach to data-based modelling”.
In this section we collect some technical results used throughout the paper. The first is a Hölder-interpolation result.
Lemma 31 (Hölder interpolation). If \(g\in C_b^\eta(\mathbb{R}^d)\) and \(0<\gamma<\eta\leq 1\), then \[[g]_{\gamma} \leq 2^{1-\frac{\gamma}{\eta}}(\|g\|_{\infty})^{1-\frac{\gamma}{\eta}}([g]_{\eta})^{\frac{\gamma}{\eta}}.\] When \(\eta=1\) we also have \[\|Dg\|_{\infty} \leq C_{\gamma, d} [g]_{\gamma}^{\gamma}[Dg]_{\gamma}^{1-\gamma}.\]
Proof. The first inequality follows from a straightforward calculation: \[\begin{align} [g]_{\gamma} &\leq \sup_{\substack{x, h\in \mathbb{R}^d \\ h \neq 0}} (2\|g\|_{\infty})^{1-\frac{\gamma}{\eta}}\bigg( \frac{|g(x+h)-g(x)|}{|h|^\eta}\bigg)^{\frac{\gamma}{\eta}} = 2^{1-\frac{\gamma}{\eta}}\|g\|_{\infty}^{1-\frac{\gamma}{\eta}}[g ]_{\eta}^{\frac{\gamma}{\eta}}. \end{align}\] The second inequality follows from [58]. ◻
Another result we will need is a generalization of Grönwall’s inequality.
Lemma 32 (Generalized Grönwall inequality I). Assume \(a_0, a_{T_0}, c\geq 0\), \(\gamma, \zeta <1\), \(T_0>0\), and \(u(t)\) is a nonnegative and locally integrable function on \([0,T_0)\) satisfying \[\begin{align} u(t) \leq a_0 t^{-\gamma}+a_{T_0}+c\int_{0}^t (t-s)^{-\zeta}u(s)\, ds, \qquad t\in[0,T_0). \end{align}\] Then there are constants \(C_1, C_2 \geq 0\) depending only on \(T_0, \gamma, \zeta, c\), in particular they are independent of \(t\), such that for \(t \in [0,T_0)\) the following bound holds: \[u(t) \leq (a_0+C_1 T_0^{1-\zeta}) t^{-\gamma}+a_{T_0}+C_2 T_0^{1-\zeta}.\]
Lemma 33 (Generalized Grönwall inequality II). Assume \(T_0>0\), \(a, b\geq 0\), \(\overline{\alpha}, \overline{\beta}, \overline{\gamma}>0\) such that \(\overline{\nu} := \overline{\beta} + \overline{\gamma} - 1 > 0\) and \(\overline{\delta} := \overline{\alpha} + \overline{\gamma} - 1 > 0\), \(u(t)\) is nonnegative, \(t^{\overline{\gamma}-1}u(t)\) is locally integrable on \([0,T_0)\), and that \[\begin{align} u(t) &\leq a t^{\overline{\alpha} - 1} + b \int_0^t (t - s)^{\overline{\beta} - 1} s^{\overline{\gamma} - 1} u(s) \, ds \quad \text{for} \quad t\in (0, T_0]. \end{align}\] Then \[\begin{align} u(t) &\leq a t^{\overline{\alpha} - 1} \sum_{m=0}^{\infty} C_m' (b \Gamma(\overline{\beta}))^m t^{m \overline{\nu}} \quad \text{for} \quad t\in (0,T_0], \end{align}\] where \(C_0' = 1\), \(C_{m+1}' / C_m' = \dfrac{\Gamma(m \overline{\nu} + \overline{\delta})}{\Gamma(m \overline{\nu} + \overline{\delta} + \overline{\beta})}\), and the right hand series converges uniformly in \([0,T_0]\).
Proof. This is [60], except for the uniform convergence of the series. Since the series is an increasing function of \(t\), uniform convergence follows from the Weierstrass \(M\)-test if the following series converges: \[\begin{align} \sum_{m=0}^\infty M_m:=\sum_{m=0}^{\infty} C_m' (b \Gamma(\overline{\beta}))^m T_0^{m \overline{\nu}}. \end{align}\] By Stirling’s formula, \(\frac{M_{m+1}}{M_m}=b\Gamma(\overline{\beta})T_0^{\overline{\nu}}\frac{\Gamma(m \overline{\nu} + \overline{\delta})}{\Gamma(m \overline{\nu} + \overline{\delta} + \overline{\beta})}\sim (m\overline{\nu}+\overline{\delta})^{-\overline{\beta}}\to 0\) as \(m\to\infty\), so \(\sum_m{M_m}\) converges by the the ratio test. ◻