Optimal stability results and nonlinear duality for $L^\infty$ entropy and $L^1$ viscosity solutions
Résumé
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 {\it nonlinear dual inequality:}
\begin{equation}\label{star}
\tag{$\star$}
\int |S_t u_0-S_t v_0| \varphi_0 \dd x\leq \int |u_0-v_0| G_t \varphi_0 \dd x, \quad \forall \varphi_0 \geq 0, \forall u_0, \forall v_0,
\end{equation}
where $S_t$ is the entropy solution semigroup of the anisotropic degenerate parabolic equation
\begin{equation*}
%\label{a}
%\tag{$a$}
\partial_t u+\diver F(u)=\diver (A(u) D u),
\end{equation*}
and where we look for the smallest semigroup $G_t$ satisfying \eqref{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*}
%\label{b}
%\tag{$b$}
\partial_t \varphi=\sup\nolimits _\xi \{F'(\xi) \cdot D \varphi+\tr(A(\xi) D^2\varphi)\}.
\end{equation*}
Since weigthed $L^1$ 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 $L^\infty_\INT$.
A consequence of our results is a new and optimal domain of dependence 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, {\it J. Differ. Equ.,} 2018].
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