Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms

Abstract

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.

On the Problem of Maximal $$L^q$$-regularity for Viscous Hamilton–Jacobi Equations

“…Finally, our technique does not apply to the parabolic counterpart of (M). In this direction, some results based on rather different duality methods developed in [12] to get Lipschitz regularity, have been obtained in [13]. 2 will denote the partial derivative in the i-th direction, the gradient, and the Hessian operator respectively.…”

Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

“…In [12], they are studied in connection with space-fractional PDEs using arguments inspired by [28,29]. These spaces are natural in the context of parabolic PDEs and the corresponding L p theory is crucial in the study of parabolic regularity properties even for equations with divergence-type terms (see [5,11,13,38,41]).…”

“…In this section, we prove gradient bound for classical solution to (16). The method implemented here is based on the so-called nonlinear adjoint method (see [11,22] and [12, Proposition 3.6]), that is on testing the Hamilton-Jacobi equation against the (classical) solution to…”

“…For the latter argument, a crucial step is an estimate of crossed quantity |Du| γ ρ dxdt (6) which is accomplished by means of the aforementioned nonlinear adjoint method. We remark that bounds of type (6) play an important role in Mean Field Games theory and stochastic control ( [12,22,41], Sobolev regularity for the Fokker-Planck equation ( [5,38,13]), Hamilton-Jacobi equations [11]. We refer to Remark 5.6 for a discussion about the restriction β ∈ ( 1 2 , 1) in Theorem1.1.…”

“…In the study of (1), we are mainly motivated by problems arising in Mean Field Games theory for subdiffusion processes (see [7]), where typically H = H(x, p) behaves like |p| γ , γ > 1 in p and V is a so-called regularizing coupling [8]. Starting with [30], quasi-linear equations of the form (1) with standard time derivative have been extensively studied in literature and several results are available depending on growth conditions on H with respect to the gradient variable (see [11,22,41] and references therein). Recently, a theory of weak solutions (in viscosity sense), for the Hamilton-Jacobi equation (1) has been investigated in [21,50].…”