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Pré-Publication, Document De Travail Année : 2024

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} \renewcommand*{\theHequation}{notag.\theequation} where $S_t$ is the entropy solution semigroup of the anisotropic degenerate parabolic equation \begin{equation*} %\label{a} %\tag{$a$} \partial_t u+\textup{div} F(u)=\textup{div} (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 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 $L^\infty_{\textup{int}}$. 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, {\it J. Differ. Equ.,} 2018].
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Dates et versions

hal-01945687 , version 1 (05-12-2018)
hal-01945687 , version 2 (17-12-2019)
hal-01945687 , version 3 (26-04-2023)
hal-01945687 , version 4 (12-04-2024)

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Paternité - Pas d'utilisation commerciale

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Nathaël Alibaud, Jørgen Endal, Espen Robstad Jakobsen. Optimal stability results and nonlinear duality for $L^\infty$ entropy and $L^1$ viscosity solutions. 2024. ⟨hal-01945687v4⟩
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