Optimal stability results and nonlinear duality for L∞ entropy and L1 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:(⋆)$\int |S_t{u}_0-S_t{v}_0|{\phi }_{0}dx\le \int |u_0-v_0|G_t{\phi }_{0}dx,\forall {\phi }_0\ge 0,\forall u_0,\forall v_0,$ where $S_t$ is the entropy solution semigroup of the anisotropic degenerate parabolic equation${\partial }_{t}u+\mathrm{div}F\left(u\right)=\mathrm{div}\left(A\right(u\left)Du\right),$ and where we look for the smallest semigroup $G_t$ satisfying (⋆). 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${\partial }_t\phi ={\sup }_{\xi }\left\{F^{\prime }\right(\xi )\cdot D\phi +\text{tr}(A\left(\xi \right)D^2\phi \left)\right\}.$ 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 φ. 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 $L_{\text{int}}^{\infty }$. 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].