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.
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Marco Cirant, Alessandro Goffi, Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 4, pp. 757–784DOI 10.1016/J.ANIHPC.2020.01.006