Maximal $$L^q$$-Regularity for Parabolic Hamilton–Jacobi Equations and Applications to Mean Field Games

In this paper we investigate maximal $$L^q$$ L q -regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided.

Hölder regularity for porous medium systems

We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. $$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$ ∂ t u - div ( m | u | m - 1 D u ) = 0 , m > 0 . As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.

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 $$s\in (1/2,1)$$ s ∈ ( 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 Hölder 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 Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

Continuity of solutions to a nonlinear fractional diffusion equation

We study a parabolic equation for the fractional p-Laplacian of order s, for $$p\ge 2$$ p ≥ 2 and $$0<s<1$$ 0 < s < 1 . We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of Moser’s technique.

Hölder estimates for magnetic Schrödinger semigroups in $${\mathbb {R}}^{d}$$ from mirror coupling

We use the mirror coupling of Brownian motion to show that under a $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) -dependent Kato-type assumption on the possibly nonsmooth electromagnetic potential, the corresponding magnetic Schrödinger semigroup in $${\mathbb {R}}^d$$ R d has a global $$L^{p}$$ L p -to-$$C^{0,\beta }$$ C 0 , β Hölder smoothing property for all $$p\in [1,\infty ]$$ p ∈ [ 1 , ∞ ] ; in particular, his all eigenfunctions are uniformly $$\beta $$ β -Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly $$\beta $$ β -Hölder continuous under weak $$L^q$$ L q -assumptions on the magnetic potential.

VARIATION ON A THEME BY KISELEV AND NAZAROV: HÖLDER ESTIMATES FOR NONLOCAL TRANSPORT-DIFFUSION, ALONG A NON-DIVERGENCE-FREE BMO FIELD

We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator and a general advection field in of bounded mean oscillation, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by Silvestre, our advection field does not need to be bounded. A similar result can be obtained in the supercritical case if the advection field is Hölder continuous. Our proof is inspired by Kiselev and Nazarov and is based on the dual evolution technique. The idea is to propagate an atom property (i.e., localisation and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.