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:(⋆)∫|Stu0−Stv0|φ0dx≤∫|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 L1 contraction 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 weighted L1 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 Lint∞. 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].