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

Nathaël Alibaud (1, 2) , Jørgen Endal (3) , Espen Robstad Jakobsen (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 ENSMM - Ecole Nationale Supérieure de Mécanique et des Microtechniques
3 NTNU - Norwegian University of Science and Technology [Trondheim]

This paper has three contribution: (i) nonstandard and optimal contraction results for $L^1$ viscosity solutions of the Hamilton-Jacobi-Bellman equation \begin{equation*} \partial_t \varphi=\sup_\xi \{b(\xi) \cdot D \varphi+{\rm tr}(a(\xi) D^2\varphi)\}, \end{equation*} (ii) nonstandard and optimal contraction results for $L^\infty$ entropy solutions of the anisotropic degenerate parabolic equation \begin{equation*} \partial_t u+{\rm div} F(u)={\rm div} (A(u) D u), \end{equation*} and (iii) rigorous identification of a new duality relation between these notions of generalized solutions. More precisely, we obtain a quasicontraction principle for the first equation in the weakest possible $L^1$ type Banach setting where stability holds in general. We rigorously identify this topology and show that our contraction estimate is optimal for model HJB equations. For the second equation, we obtain a weighted $L^1$ contraction principle for possibly nonintegrable $L^\infty$ solutions. Here we identify the optimal weight in general. It is interestingly a solution of the first equation, and it is the precise formulation of this result that leads to the above mentioned duality.

Mots clés: Hamilton-Jacobi-Bellman equations; degenerate parabolic equations; conservation laws; $L^1$ viscosity solutions; $L^\infty$ entropy solutions; quantitative stability; contraction principle; quasicontraction principle; weighted $L^1$ contraction principle; duality

Domaines: Equations aux dérivées partielles [math.AP]

Nathaël Alibaud, Jørgen Endal, Espen Robstad Jakobsen. Optimal and dual stability results for $L^1$ viscosity and $L^\infty$ entropy solutions. 2019. ⟨hal-01945687v2⟩