Optimal stability results and nonlinear duality for $L^{\infty }$ entropy and $L^1$ 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: \begin{equation}\int |S_t u_0-S_t v_0| φ_0 \mathrm{d}x\leq \int |u_0-v_0| G_t φ_0 \mathrm{d}x, \quad \forall φ_0 \geq 0, \forall u_0, \forall v_0, \qquad(\star)\end{equation} where $S_t$ is the entropy solution semigroup of the anisotropic degenerate parabolic equation \begin{equation*} \partial_t u+\mathrm{div} F(u) = \mathrm{div} (A(u) D u),\end{equation*} and where we look for the smallest semigroup $G_t$ satisfying ($\star$). This amounts to finding an optimal weighted $L^1$ contraction estimate for $S_t$. Our main result is that $G_t$ is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation\begin{equation*} \partial_t φ= \mathrm{sup}_ξ\{F'(ξ) \cdot D φ+\mathrm{tr}(A(ξ) D^2φ)\}.\end{equation*} Since weighted $L^1$ contraction results are mainly used for possibly nonintegrable $L^{\infty }$ solutions $u$, the natural spaces behind this duality are $L^{\infty }$ for $S_t$ and $L^1$ for $G_t$. We therefore develop a corresponding $L^1$ theory for viscosity solutions $\ddot{I}†$. But $L^1$ itself is too large for well-posedness, and we rigorously identify the weakest $L^1$ type Banach setting where we can have it -- a subspace of $L^1$ called $L_{\mathrm{int}}^{\infty }$. 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].

Version 4 is an update according to the referees suggestions. Most of the changes concerns the structure of the paper, but there are also some additional results on nonlinear duality. To appear in ''Journal de Math{é}matiques Pures et Appliqu{é}es''