Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions

Abstract

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality: ˆ |Stu0−Stv0|ϕ0dx≤ ˆ |u0−v0|Gtϕ0dx, ∀ϕ0≥0,∀u0,∀v0, ( ) where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation ∂tu+divF(u)=div(A(u)Du), and where we look for the smallest semigroup Gt satisfying ( ). This amounts to finding an optimal weighted L1contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation ∂tϕ=supξ{F(ξ)·Dϕ+tr(A(ξ)D2ϕ)}. Since weightedL1 contraction results are mainly used for possibly nonintegrable L∞solutions u, the natural spaces behind this duality are L∞for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions ϕ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it – a subspace of L1 called L∞ int. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].