Optimal stability results and nonlinear duality for $L^\infty$ entropy and $L^1$ viscosity solutions

TL;DR

This paper establishes a duality relation between entropy and viscosity solutions of nonlinear PDEs, providing optimal contraction estimates and a new $L^ ext{int}_ ext{infty}$ framework, with implications for anisotropic degenerate parabolic equations.

Contribution

It introduces a rigorous duality between entropy and viscosity solutions, identifying the minimal semigroup satisfying a key inequality and developing an $L^ ext{int}_ ext{infty}$ theory for viscosity solutions.

Findings

Derived a nonlinear dual inequality relating entropy and viscosity solutions.

Identified the viscosity solution semigroup as the minimal one satisfying the dual inequality.

Extended domain of dependence estimates to second order anisotropic degenerate parabolic PDEs.

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| \varphi_0 \mathrm{d}x\leq \int |u_0-v_0| G_t \varphi_0 \mathrm{d}x, \quad \forall \varphi_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 \varphi = \mathrm{sup}_\xi \{F'(\xi) \cdot D \varphi+\mathrm{tr}(A(\xi) D^2\varphi)\}.\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 $\varphi$. 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 $L^\infty_{\mathrm{int}}$. 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].