Transport equations with nonlocal diffusion and applications to Hamilton-Jacobi equations

We investigate regularity and a priori estimates for Fokker-Planck and Hamilton-Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $s\in(1/2,1)$. As for Fokker-Planck equations, we establish integrability estimates under a fractional version of the Aronson-Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans' nonlinear adjoint method we prove new integral, sup-norm and H\"older estimates for weak and strong solutions to fractional Hamilton-Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calder\'on-Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo-Nirenberg inequalities, we deduce optimal $L^q$-regularity for fractional Hamilton-Jacobi equations.


Introduction
In this paper, we analyze regularity properties of transport equations of Fokker-Planck-type and Hamilton-Jacobi equations with fractional diffusion driven by a fractional power of the Laplacian, (−Δ) , with subcritical order ∈ ( 1 2 , 1). In particular, we address well-posedness, parabolic Bessel regularity and integrability estimates for solutions to (backward) fractional Fokker-Planck equations of the form − ( , ) + (−Δ) ( , ) + div( ( , ) ( , )) = 0 in := T × (0, ) , where the nonlocal diffusion operator is defined on the flat torus T ≡ R \Z [83], under "rough" integrability conditions on the velocity field, mainly when either ∈ Q ( P ) (cf (3) below) or ∈ ( ), > 1, without requiring a control on its divergence. Our second aim is to apply the results for the above transport-diffusion equation to obtain a priori gradient estimates for strong solutions and regularization effects for weak solutions of fractional Hamilton-Jacobi equations with subcritical diffusion of the form where ∈ ( ) for some > 1 and ( , ) ∼ | | , > 1, i.e. has superlinear gradient growth.
Following the approach in [37,38], we first obtain Sobolev-type regularity for solutions to (1). This level of regularity is crucial to derive new integral, sup-norm and Hölder estimates for solutions to (2) by means of the nonlinear adjoint method introduced by L.C. Evans [46,45]. These results are then combined with Gagliardo-Nirenberg interpolation inequalities and maximal regularity in Lebesgue spaces for fractional heat equations to obtain optimal regularity in Lebesgue spaces for (2). This approach to deduce a priori estimates for nonlinear problems has been inspired by [7] (see also [17,16] for later contributions), where semi-linear equations with quadratic growth in the gradient have been studied. These interpolation methods have been also employed in e.g. [76] (see also the references therein) and recently revived in [51,50,38] in the context of Mean Field Games [66,65]. In particular, our results extend those in [37,38] in the fractional framework for ∈ (1/2, 1), both for (1) and (2).
As announced, in order to study the regularity properties of (1) we need to extend well-known results for linear viscous equations with unbounded coefficients to the fractional framework. In the viscous case, the first works date back to [62,9,10] for linear and quasi-linear problems, see also [20] for the case of measurable ingredients. Within this framework, well-posedness and integrability estimates are well-established when ∈ Q ( P ) with Q , P satisfying the so-called Aronson-Serrin interpolated condition. We emphasize that this assumption on the drift has been shown to be sharp to get integrability estimates in Lebesgue classes, at least in the viscous case, cf [19]. As far as we know, nonlocal heat equations with unbounded coefficients have been treated in [68], see also [56] for gradient perturbations of the fractional Laplacian and the nonlinear analysis carried out in [2,3] in the context of (fractional) Kardar-Parisi-Zhang models. The well-posedness and integrability results we present here are new when the velocity field satisfies a fractional version of the above Aronson-Serrin condition, i.e. ∈ Q ( P ) with Q , P fulfilling 2 P + 1 In particular, we mention that in this setting we are not able to cover the equality in (3) neither for the well-posedness nor for integrability estimates of solutions of (1), and this remains at this stage an open problem. The second step in the analysis of (1) concerns fractional Bessel regularity estimates of solutions to (1) when ∈ ( ) for some > 1, i.e. in terms of the crossed term which is widely analyzed for classical viscous Fokker-Planck equations in [77,81]. In particular, we prove that for some suitable > 1 one has the estimate for ′ in some range determined in terms of the regularity ′ of the terminal data. This bound is obtained by duality, following [77,37,38], via a maximal regularity estimate for nonlocal equations with divergence-type terms of the form These estimates are fundamental to study regularity properties for PDEs arising in Mean Field Games, cf [41,37,81,82], see also [23] for the time-fractional framework. We remark that when = − ( , ), (1) becomes the adjoint equation to (2) and bounds on the quantity (4) are natural for the mean field equations by duality [81,36,37].
Owing to these results for (1), we deduce sup-norm, integral and Hölder estimates for solutions to (2) with ∈ . These bounds are obtained by duality and exploit the aforementioned Bessel regularity properties of solutions to Fokker-Planck equations. To our knowledge, these estimates for fractional Hamilton-Jacobi equations have not yet been investigated in the literature, especially in the context of unbounded coefficients in scales. Furthermore, with respect to Hölder estimates, we provide a Hölder's regularization effect when ≥ 2 . Finally, we are able to partially prove optimal -regularity as addressed in [39,38] for elliptic and evolutive equations respectively, driven by the Laplacian. This means, within our context, a control on , (−Δ) , | | ∈ in terms of ∈ for an appropriate range of the integrability exponent >¯ , see Section 5.5.1 for details on the threshold¯ . As a result, we find that (2) behaves in terms of regularity as the fractional heat equation for appropriate values of , despite the presence of the nonlinear coercive term = ( , ) ∼ | | . By letting → 1 we recover the same results of the viscous case, but we produce only partial a priori estimates in the supercritical regime > 2 .
In the viscous stationary case, the proof has been given refining the integral Bernstein method, which, however, does not seem the right path to treat both fractional and time-dependent problems like (2).
We recall that maximal -regularity properties of Calderón-Zygmund-type are well-known for general abstract linear evolution equations, see e.g. [53,64,73], and are recalled in Lemma 5.16 below, while in the case = 1 2 a result for nonlocal equations with drift terms can be found in [96].
More precisely, our strategy consists in regarding (2) as a perturbation of a fractional heat equation where ( , ) ∼ | | . Then, maximal regularity for linear nonlocal problems, cf [53], applied to the above equation yields The second step relies on applying fractional Gagliardo-Nirenberg inequalities involving integral norms of the form , (1− ) ∞ ( ) for some ∈ (1, ∞], ∈ (0, 1) such that < 1 when < 2 , and those involving Hölder norms , for ≥ 2 , cf Lemmas 3.5 and 3.6 and [80,78]. This scheme allows to show that maximal -regularity for (2) occurs for strong solutions when ∈ ( ), > We refer to Remark 5.4 and Remark 5.17 for further comments on the restrictions for . At this stage, we do not know neither if our restrictions when > 2 are sharp nor counterexamples to the maximal regularity below the threshold +2 (2 −1) ′ when ≤ 2 . However, by letting → 1 our results agree with those obtained in the local case [39,38]. We believe that our duality approach to obtain integral and Hölder bounds, together with the maximal regularity results, can be adapted to the stationary counterpart of (1) and (2), leading to new a priori estimates. This will be matter of a future research. Finally, we remark in passing that, in the classical viscous case, the purely quadratic regime = 2 can be addressed using the Hopf-Cole transform, cf [28,Lemma 4.2]. Here, however, when = 2 it is not known whether there exists a fractional analogue of that transformation which allows to reduce (2) into a simpler fractional PDE. Thus, even the natural (critical) growth case becomes not trivial to analyze.
We now recall some related results for (1) and (2). As for fractional Fokker-Planck equations, when ∈ ∞ or some control on the divergence is assumed, we refer to [32] for stationary problems and to [36] for the evolutive case. Instead, the viscous case is well-known, even under weaker assumptions on the velocity field [62,77,20,19,41,37,81,38]. As for Hamilton-Jacobi equations, Hölder's regularity results have been largely investigated for parabolic problems in the borderline cases = 0 and = 1. For first order and second order degenerate problems, we refer to [26,24,33,29], while we mention [62,63] for the uniformly parabolic case. Recently, Hölder estimates for second order degenerate problems with unbounded right-hand side have been analyzed in [31], see also [93] for PDEs driven by the Laplacian, via De Giorgi's techniques. Hölder, integral and sup-norm estimates for the parabolic problem have been already addressed in the paper [38] for the viscous case = 1, and we recover those results by letting → 1. As for integrability estimates, we refer to [27] for the degenerate case and [38] for the viscous problem. Hölder's regularity of fully nonlinear nonlocal equations with super-quadratic first-order terms has been treated in [30], where the regularity stems from the coercivity of rather than the ellipticity. Hölder's regularization effect of solutions to fractional Hamilton-Jacobi equations with first order terms having at most critical growth = 2 has been observed by L. Silvestre in [86]. In this case, the author has also obtained Hölder bounds in the fractional supercritical regime 2 < < 2 + imposing some smallness conditions on the data. More recently, a regularization effect when = 1/2 has been investigated in [55] under a smallness condition on the initial datum in Besov scales. Instead, the literature on Lipschitz regularity is huge. The conservation of Lipschitz regularity (i.e. with (0) ∈ 1,∞ ) for every ∈ (0, 1) and a smoothing effect when ∈ (1/2, 1) go back to [44] (see also [54,57]). Besides, Lipschitz and further regularity for nonlocal Hamilton-Jacobi equations has been investigated in the case of critical diffusion = 1/2 by L. Silvestre in [87]. Gradient regularity for viscosity solutions of coercive fractional Hamilton-Jacobi equations has been widely analyzed using viscosity solutions' techniques. In [13] the authors have analyzed Lipschitz regularity of solutions via the Ishii-Lions method when is bounded (which requires the restriction < 2 , as for the classical viscous case = 1) and via a weak version of the Bernstein method in the periodic setting [14], where ∈ 1,∞ in the space variable and > 1, even for more general integro-differential operators than fractional powers of the Laplacian. We finally mention that fractional Hamilton-Jacobi-type PDEs and regularity issues have been recently investigated in the framework of periodic homogenization problems [11]. As for the stationary counterpart of (2) with unbounded terms in Lebesgue scales, we mention [2,1]. Related results for Hamilton-Jacobi equations, even degenerate, can be found in [39,70,12,15] and the references therein. Other regularity estimates for space-fractional Fokker-Planck equations, also combined with Hamilton-Jacobi equations in the context of Mean Field Games, can be found in [36,32], while for advection equations with fractional diffusion we refer the reader to [88,89].
Outline. Section 2 presents a list of the main results and the assumptions used throughout the paper. Section 3 is devoted to introduce the main functional spaces and related embedding properties. Section 4 concerns the analysis of the well-posedness, Bessel regularity and integrability estimates for fractional Fokker-Planck equations, while Section 5 comprises the applications to regularity issues for equations of Hamilton-Jacobi-type with nonlocal diffusion. Appendix A collects some properties for advection equations with fractional diffusion.

Assumptions and main results
Throughout all the manuscript we will assume ∈ (1/2, 1) unless otherwise stated. Our first main results concern the fractional Fokker-Planck equation (1): in the first one, is assumed to belong to mixed Lebesgue classes in the fractional Aronson-Serrin zone, while in the second result parabolic Bessel regularity is studied in terms of the crossed quantity (4). More precisely, in this section for ∈ R we deal with anisotropic spaces of the form where (T ) is the space of Bessel potentials on the torus. We refer to Section 3.2 for additional properties of these spaces.
We will assume the following additional assumption, referring to Appendix A for further discussions on its validity.
As for (2), we suppose that ( , ) is 1 (T × R ), convex in and has polynomial growth in the gradient entry, i.e. there exist constants > 1 and > 0 such that for every ∈ T , ∈ R . Moreover, we suppose without loss of generality that ≥ 0. Recall that the Lagrangian : T × R → R, ( , ) := sup { · − ( , )}, namely the Legendre transform of in the -variable, is well defined by the superlinear behavior of ( , ·). Moreover, by convexity of ( , ·), The following properties of are standard (see, e.g. [25]): for some > 0, for all ∈ R . Concerning the case ≥ 2 and when dealing with Hölder regularity, we will impose some additional space regularity, i.e for ∈ (0, 1) to be determined, for all , ∈ T and ∈ R . A prototype example of satisfying (H) is Note that whenever ℎ ∈ (T ), this model Hamiltonian satisfies also ( ). Unless otherwise stated, in the next results we will always assume that (I) holds to have the full well-posedness of the adjoint problem. However, our approach via duality makes use only of the existence of positive solutions to (1). Then, our first results for (2) concern sup-norm and integral estimate for strong solutions as in Definition 5.3 when ∈ , obtained via the nonlinear adjoint method via the strategy already implemented in [37,38].
Owing to a similar approach and following the scheme of [38], spatial Hölder's regularity estimates for weak energy solutions to fractional Hamilton-Jacobi equations with unbounded right-hand side are provided. with , , as follows: (2), then ∈ ∞ (0, ; (T )). In particular, there exists a positive constant 2 depending on , 0 (T ) , with , as in (C).
We remark that so that we find the same thresholds in [38].
Note that so that we recover the same thresholds in [38]. We finally point out that our results, and in particular the Hölder bounds, apply to the so-called fractional Kardar-Parisi-Zhang equations, see [57,3], of the form where satisfies (H). In other terms, the sign in front of is not important since = − solves (2) with ( , ) = ( , − ).

Functional spaces 3.1 Stationary spaces: definitions and useful results
We denote by (T ) the space of all measurable and periodic functions belonging to (R ) endowed with the norm · = · ( (0,1) ) . Let be a nonnegative integer. We denote by , (T ) the space of (T ) functions with distributional derivatives in (T ) up to order . For ∈ R and ∈ (1, ∞), the space of Bessel potentials (T ) comprises those distributions verifying the integrability condition We denote the norm in (T ) as The proof of the latter equivalence is given in [36,Remark 2.3]. Let us also remark that when = is a nonnegative integer, , is isomorphic to , see e.g. [36,Remark 2.3]. Moreover, it can be seen that the operator ( − Δ) 2 maps isometrically + in for any , ∈ R, see again [36, Remark 2.3] for the proof. We further recall that another characterization of spaces of Bessel potentials can be given via complex interpolation methods, namely where [ , ] stands for the complex interpolation space among the Banach spaces ( , ), see e.g. [73,18] for a complete account. Let now ∈ (0, 1) and 1 ≤ , ≤ ∞. The Besov space (T ) consists of all functions ∈ (T ) such that the norm is finite. When = = ∞ and ∈ (0, 1) we have ∞∞ (T ) ≃ (T ) (cf [84, Section 3.5.4 p. 168-169]), i.e. the classical Hölder space, which is endowed with the equivalent norm where dist( , ) is the geodesic distance among , ∈ T . When = and is not an integer, one has (T ) ≃ , (T ), where , (T ) is the classical Sobolev-Slobodeckii scale in the periodic setting, see [73, p.13]. When = ∞ and is finite, the space ∞ (T ) ≃ , (T ) is known as Nikol'skii space [79] and the aforementioned norm is interpreted in the usual sense via where ( , ) , stands for the real interpolation space of the interpolation couple of Banach spaces ( , ), with equivalence of the respective norms, see e.g. [73, Example 1.10], [67,Theorem 17.24]. We also denote by (T ) the periodic Triebel-Lizorkin scale, and refer to [84, Section 3.5.2] for its definition. We recall some standard embeddings we will use in the sequel among the aforementioned spaces.

Lemma 3.3.
We have the following inclusions for ∈ R.
Proof. The result on R is proven in [ We now recall the following compactness result.
Proof. When = ∈ N this is the classical Rellich-Kondrachov theorem [67]. We restrict to consider ∈ (0, 1). When = 2 the result can be deduced by the fact that 2 (T ) ≃ ,2 (T ), and by classical compactness properties of real interpolation spaces, cf [95, Section 1.16.4] applied with = ( 2 (T ), 1,2 (T )) ,2 , 0 = = 2 (T ), 1 = 1,2 (T ), using the compactness of 1,2 (T ) onto 2 (T ) that give the compact embedding of 2 onto 2 . Then, the compact embedding of 2 onto , 1 < < 2 /( −2 ) follows by interpolation as we will show in the case ≠ 2 below. The general case can be handled as follows. It is well-known that 1, (T ) is compactly embedded onto (T ) for all such that 1 < < − by Rellich-Kondrachov Theorem, and hence the identity map : 1, (T ) → (T ), ( ) = is compact. Moreover, is also continuous from (T ) onto itself. Thus, one first recalls that Besov spaces can be defined via real interpolation as follows (T ) = ( (T ), 1, (T )) , . Then, one may compose continuous embeddings from Lemma 3.3-(i) and -(ii) with compact embeddings for fractional Sobolev spaces , (T ) onto (T ) as obtained in [6]. The latter can be deduced in turn via compactness results for real interpolation spaces [95, Section 1. 16.4] as in the case = 2. Therefore, we have the compact embedding of (T ) onto (T ). We now take a bounded sequence in (T ). Therefore, one can extract a subsequence converging strongly in (T ). By interpolation, for every < < − , there exists ∈ (0, 1) such that , which is in turn continuously embedded onto − (T ) by Lemma 3.1-(iii). Then we have the strong convergence in with as above, as desired.
We first recall the following Gagliardo-Nirenberg inequality Then, there exists a constant depending on , , , , and ∈ (0, 1) such that Proof. Inequality (8)  We now provide a Gagliardo-Nirenberg interpolation inequality involving Hölder and Bessel potential scales.
On the other hand, owing to the embedding 1 1 ↩→ 1 , together with the fact that ∞∞ ≃ , we conclude and ≠ 2 − .
, ( ) stands for the Gagliardo seminorm. As a consequence, when = T , for ∈ (1/2, 1) there exists 2 > 0 such that Proof. The first inequality can be found in [8,Proposition B.11], [67,Chapter 17]. The second one follows from the first and the inclusions among Bessel and Sobolev-Slobodeckii spaces
. A slight modification of that proof allows even to prove the endpoint case = ( +2 ) +2 − , which we provide below. Here, we distinguish the cases 1 < ≤ 2 and 2 < < ∞ in view of the inclusions stated in Lemma 3.3. To prove the first case 1 < ≤ 2, we note that for any Therefore, for all where we used that for 1 < ≤ 2, −2 / , is embedded onto −2 / (cf Lemma 3.3-(i)). Then, the last inequality is less than or equal to where, in the second inequality we used the embedding in Lemma 3.9 while, in the last one, Young's inequality. As for the case ≥ 2, we interpolate in the Sobolev-Slobodeckii scale. In particular, one uses that , can be obtained by real interpolation among , and −2 / , . Moreover, , is continuously embedded in + / − / , in view of Lemma 3.2-(iii). Hence, for a.e. , Then, for all verifying where we used that is embedded onto , when ≥ 2 (see Lemma 3.3). At this stage, one has to use the maximal regularity embedding in Lemma 3.9 to get We now recall a maximal regularity theorem for fractional heat equations. Consider the problem We have the following result for strong solutions to (10), i.e. ∈ H 2 , the equation is solved a.e. and (0) is meant in the sense of traces.
where > 0 depends on , , , (but remains bounded for bounded values of ).
Proof. The proof is a consequence of well-known results for abstract evolution equations when = 2 , see e.g. [53]. The general case can be handled using the isometry of the Bessel operator as in [60], and it is proved in [34]. In particular, in [34] the proof is provided for stochastic PDEs, which makes necessary the restriction > 2. However, for standard PDEs one simply requires > 1, as it can be seen in [34, Lemma 3.2 and Lemma 3.4].
The last part of the section is devoted to present a Sobolev embedding theorem for the parabolic Bessel potential class H 2 −1 with traces on the hyperplane = 0 in 1 . This can be regarded as a nonlocal counterpart of [37,Proposition A.2]. The result is given via the above interpolation theory arguments, although a different proof can be done as in [37, Appendix A] via duality.
where the constant depends on , , ′ , , but remains bounded for bounded values of .
Throughout this section we will assume that ∈ −1 (T ), ≥ 0, and We further observe that since > 1/2 we have ∈ H 2 −1 2 and H 2 −1 2 ↩→ 2( +2 ) +2−2 ↩→ 2 ↩→ 1 and hence ( ) ∈ 1 (T ) for a.e. . Therefore, by using ≡ 1 as a test function one obtains ∫ T ( ) = 1 for ∈ (0, ). Therefore, when looking the equation at small scales, for ∈ (1/2, 1) one has to check the effect of the scaling on the Lebesgue norm of the velocity filed. In such case, the subcritical space turns out to be the mixed space Q ( P ) when the exponents P ≥ /(2 − 1) and Q ≥ 2 /(2 − 1) fulfill the condition which can be seen as the fractional counterpart of the classical Aronson-Serrin interpolated condition for viscous problems with unbounded coefficients [62,19,20] mentioned in the introduction. This condition allows to give a distributional sense to the transport term. Indeed, for ∈ H 1 2 , ∈ H 2 −1 2 and P = Q , we have by Hölder's inequality Classical Fokker-Planck equations with low regularity assumptions on the drift have been studied in [81,77,21,37] and references therein.

Existence and integrability estimates
We premise the following auxiliary result that allows to deduce positivity and uniqueness for the solution to (12).
Proof. Let be a weak solution to the problem Then, by duality we immediately get (17).
We now present the main result of this section. Note that our approach is based on maximal regularity arguments, which is a different strategy compared to [20]. Note that M[ ; 0] = 0 by standard results for fractional heat equations. We first show that it is welldefined. We start with the case P = Q (whence condition (13) becomes P > +2 2 −1 ). By parabolic Calderón-Zygmund regularity theory (cf Theorem 3.12) we have Now, note that We then argue by interpolation, exploit the embedding of +2−2 ( ) in Lemma 3.10 and the fact that ∈ 1 ( ) to show, applying also Young's inequality, for some ∈ (0, 1), > 0. Then, for = 1/2 we have (12) and the a priori estimate (18) carry through uniformly on ∈ [0, 1]. Thus, we obtain the existence of a constant > 0 depending only on the data (namely P ( ) , , , ) such that We finally show that the map M is compact. Let be a bounded sequence in H 2 −1 2 ( ) and let = M[ ; ] with ( ) = . Since | | ∈ 2 ( ) we have that div( ) ∈ H −1 2 ( ) and hence by Theorem 3.12 we deduce ∈ H 2 −1 2 ( ). By the compactness of H 2 −1 2 onto 2 ( ) (cf Lemma 3.11), which is ensured by the restriction > 1/2, we have that, along a subsequence, converges strongly in 2 ( ) to and (−Δ) −1/2 converges weakly to (−Δ) −1/2 in 2 ( ). Moreover, solves the same problem as given the couple ( , ). We use (−Δ) −1 ( − ) ∈ H 1 2 ( ) as admissible test function in the weak formulation of the equation satisfied by , together with the fact that Since | | ∈ 2 ( ) and (−Δ) −1/2 converges weakly to (−Δ) −1/2 in 2 ( ) the first term on the right-hand side of the above inequality converges to 0. Similarly, since ∈ H −1 2 ( ) and exploiting again the weak convergence of (−Δ) −1/2 in 2 ( ) the third term goes to 0. Similar motivations provide the convergence of the second term. This shows that (−Δ) −1/2 converges strongly to (−Δ) −1/2 in 2 ( ). Finally, to show the strong convergence of to in H −1 2 ( ) we argue by duality. For every which yields the strong convergence of to in H −1 2 ( ) in view of the previous claims. The general case P ≠ Q can be dealt with similarly. Indeed, in the borderline case Q = ∞, we observe that 2 ( ) ≤ | | ∞ (0, ; P (T )) .
Step 2. A priori estimates via Duhamel's formula. The proof we are going to present can be made rigorous by regularization (cf [82, Lemma 2.3]), using Duhamel's formula for the regularized PDE and then passing to the limit. The approach is inspired by [19] and it has been also recently implemented in [40,Lemma A.3] to get estimates in mixed Lebesgue scales and in [35,Lemma A.3]. We claim that there exists * ∈ (0, ] independently of ∈ (T ) such that for some 2 > 0. Set˜ (·, ) := (·, − ) for all ∈ [0, ] and use Duhamel's formula to represent the solution of the (forward) equation as where we applied the decay estimates of the fractional heat semigroup among spaces of Bessel potentials (cf [36]). We then use Hölder's inequality to bound the right-hand side of the last inequality with In particular, the above integral term is well-posed provided that which is indeed satisfied precisely when Hence ˜ ∞ (0, ; (T )) ≤ (T ) + which gives ˜ ∞ (0, ; (T )) ≤ 2 (T ) by taking and hence the validity of the estimate on [0, * ]. Note that * does not depend on (T ) and hence one can iterate the argument to get the estimate in [0, ] as in [19].
Step 3. Positivity and uniqueness. Positivity and uniqueness follows exploiting Lemma 4.4. In particular, if 1 , 2 are two solutions of (12), by (17)   by assuming a smallness condition on Q ( P ) , since interpolation inequalities are no longer available (cf [91] for the elliptic viscous case).

Parabolic Bessel regularity
We finally describe further regularity results that rely on the information ∈ ( ) for some > 1, that will be used in the forthcoming sections. We start with the following maximal -regularity result for PDEs with divergence-type terms and terminal data in 1 . The method of proof we present below has been already used in [37,41,77]. Proposition 4.6. Let be a (non-negative) weak solution to (12) and Then, there exists > 0, depending on ′ , , , such that Note that here does not depend on ∈ (0, ]. Proof. Let be smooth, the general case follows by an approximation argument. Let be a smooth test function vanishing at the initial time (·, 0) = 0. The strategy follows the proof of [37,Proposition 2.4] and it is based on duality arguments. A different proof of the result will be provided in the next Remark 4.7, see also Theorem 4.8. Using the weak formulation of (32), we write for as above Let > 0 and = be the solution to the forward fractional heat equation By maximal -regularity, we get We take = (−Δ) − 1 2 in the weak formulation (21) to see that after integrating by parts and let → 0 to conclude The estimate on ∈ ′ ( ) follows by using that of (−Δ) − 1 2 ′ ( ) and the fractional Poincaré-Wirtinger inequality in Lemma 3.8. The estimate on the time derivative can be obtained by duality. Indeed, for any ∈ (0, ; 1 (T )) we have where we used that 1, ≃ 1 and Hölder's inequality. By exploiting Sobolev embedding for fractional Sobolev spaces in Lemma 3.2, one immediately obtains that Indeed, by [84, Section 3.5.4] we have ( , (T )) ′ = − , ′ (T ) and thus by definition we get where the last inequality is a consequence of the embedding 2 / ′ −2 +1, (T ) ↩→ (T ) (cf Lemma 3.2-(ii)) when (2 / ′ − 2 + 1) > , that is > + 2 or, in other words, when ′ satisfies (19). This highlights that the range of ′ is imposed by the heat part of the equation.
The next results asserts fractional Bessel regularity of the fractional Fokker-Planck equation when the trace belongs to some suitable Lebesgue class. A different proof has been proposed in [38, Proposition 2.2]. (12), ∈ ′ (T ) and either

Proposition 4.8. Let be a (non-negative) weak solution to
Then, there exists > 0, depending on ′ , , , such that Proof. We can proceed as in Proposition 4.6, except for the treatment of the term involving . We modify (22) as for < + 2 , and for any ∈ (1, ∞) when = + 2 .
As a consequence, the above results yield the proof of Theorem 2.2.
Proof of Theorem 2.2. We use Proposition 4.6 and Proposition 4.8, depending on the range of ′ , and the generalized Hölder's inequality to conclude for > ′ satisfying 1 Then, by Young's inequality, for all > 0 One can verify that the identity ′ = 1 + +2 (2 −1) and (25) yield Indeed, (25) gives and then the definition of ′ in (2.2) yields the conclusion. The continuous embedding of H 2 −1 ′ ( ) in ( ) stated in Lemma 3.13 then implies Hence, the term ( ) can be absorbed by the left hand side of (27) by choosing = (2 1 ) −1 , thus providing the assertion.  Proof. The first estimate is a consequence of Remark 4.9 and Theorem 2.2 applied with 1 together with the continuous embedding of H 2 −1 ′ onto ′ ( ). The second one exploits also the +2 +(2 −1) ′ (T )).

On the notions of solutions
We first provide the following notion of weak solution to (2) we will need to discuss Hölder's regularization effects for (2). See [37] for a similar definition in the viscous case = 1.

Definition 5.1. We say that
i) is a local weak solution to (2) if for all 0 < < and for some P > and for all 0 < < ≤ , ∈ H 2 −1 .

Further estimates for the adjoint variable via duality
First, we prove a simple representation formula for (2) by duality with the adjoint problem where ∈ ∞ (T ), ≥ 0 with 1 = 1.
Lemma 5.6. Let be a local weak solution to (2). Assume that is a weak solution to (32). Then, for all We are now ready to prove the crossed integrability bound on with respect to the density . Proof. Rearrange the representation formula (33) to get, for 0 < 1 < < , Fix to be determined such that We use the bounds on the Lagrangian and Hölder's inequality to get By Lemma 3.10 we have .
Finally, the right-hand side can be absorbed in the left-hand side when ′ < ′ , i.e.
The crossed integrability of against the adjoint variable finally provides the ′ regularity of (−Δ) −1/2 . The next result extends [37,Corollary 3.4] to the fractional framework.
Corollary 5.9. Let be a local weak solution to (2) and be a weak solution to (12). Let¯ be such that .

Then, there exists a positive constant such that
where depends in particular on , Remark 5.10. Note that condition on¯ can be rewritten as Proof. Since¯ ′ < +2 +2 −1 , (2.2) applies (with =¯ ), yielding If ′ = ′ , use Proposition 5.7 to conclude. Otherwise, if ′ < ′ use Young's inequality first to control

Sup-norm and integral estimates
We are now ready to prove the sup-norm estimate for global weak solutions to fractional Hamilton-Jacobi equations in terms of 0 (T ) .
Step 1. We first prove that for all ∈ (0, ) and ∈ T , ∈ [1, ∞). For > 0 consider the weak nonnegative solution to the following backward problem for the fractional heat equation We apply Hölder's inequality to the second term of the right-hand side of the above equality, the fact that ≥ 0 and the fact that the fractional heat equation preserves the norms, i.e. ( ) ′ (T ) ≤ 1 for all ∈ [0, ], to get, after sending → ∞, Step 2. To prove the bound on the negative part, consider for > 0 the solution = to the adjoint problem First, by Corollary 4.10 we have with not depending on . We note again that in view of the parabolic Kato's inequality − is a weak subsolution to Owing to the representation formula we get .
Remark 5.12. We underline that knowing a well-posedness result, together with integrability estimates, for the adjoint equation (12) when ∈ +2 2 −1 , would allow to give an estimate of ∈ −1 2 − , when < 2 under finer properties of the data, see e.g. [38,Remark 3.6]. This would be also consistent with results in [74]. However, such properties of the adjoint variable in the borderline case ∈ +2 2 −1 , are at this stage an open problem, cf [62,20,19] for the case = 1.

Hölder regularity results
We are now in position to prove Hölder bounds for solutions to the fractional Hamilton-Jacobi equation (2) as in Theorem 2.5.
Step 2. We now estimate all the right hand side terms of (48). We emphasize that constants , 1 , . . . are not going to depend on , , ℎ, . Proposition 5.7 and Corollary 5.9 applied with = for all ∈ T , ∈ R , ℎ > 0. Thus, (·, ) is Hölder continuous, and Remark 5.13. Using the same scheme of Theorem 2.5, we believe one can even handle the case < 2 by considering appropriate weak solutions (not continuous on the whole cylinder ) to (2).

An overview of the results in the viscous case
Let us first consider the following viscous problem where is an unbounded source term belonging to a suitable Lebesgue space . In [71,72] P.-L. Lions proposed the following conjecture: Conjecture 5.15. Let ∈ (Ω), > 1, for some and > 1. Then, every solution to (50) satisfies the a priori estimate 2 + | | ≤ ( , , , ) .
Moreover, the estimate is false when ≤ / ′ .
A byproduct of this statement is a maximal -regularity for solutions to (50). In other words, (50) behaves, in terms of regularity, as the Poisson equation (cf [48, Theorem 9.9]) under the regime (51) of the integrability exponent of the right-hand side . Conjecture 5.15 completes the results in [70] for the subcritical range of the integrability of the forcing term, where it is shown a Lipschitz regularity result when ∈ , > , and every > 1, obtained via an integral Bernstein method. A proof of Conjecture 5.15 has been proposed in [39] appropriately modifying the Bernstein method, while an extension to the parabolic viscous framework has been already provided in [38]. More precisely, in [38] it is proved that maximal regularity for viscous (2) occurs for strong solutions when ∈ ( ) with Note that the threshold = ( +2) ( −1)/ can be regarded as a parabolic analogue to the one in (51). We also refer to [40] for more recent maximal regularity results for viscous ( = 1) problems with quadratic gradient growth and right-hand sides in mixed Lebesgue scales, and to [58] (and the references therein) for some maximal regularity properties for fully nonlinear second order uniformly parabolic problems in the context of viscosity solutions.

The fractional case
We first recall the following Calderón-Zygmund regularity result for the fractional heat equation with unbounded potential.
Remark 5.17. The restriction < 1− is not new in the literature. Under the assumption ∈ +2 +1 , 1− it has been recently proved existence of solutions for fractional Kardar-Parisi-Zhang equations with righthand side in under suitable restrictions on for problems posed on bounded 1,1 domains of R .

A Nonlocal equations with drift terms
We begin with the following comparison principle for (5), whose proof follows the strategy implemented in [47]. By weak solutions to (5) we mean a function in the space H 1 2 ( ) satisfying the equation in the sense of distributions.
We then note that by Hölder's inequality .
Remark A.3. The proof of the above result remains the same if one adds a forcing term ∈ 2 , . Remark A.4. The existence of a suitable weak solution when ∈ P ( ), P > +2 2 −1 , has been obtained in [56,Example 3 p.335].