Compactness estimates for Hamilton-Jacobi equations depending on space

Abstract

We study quantitative estimates of compactness in ${\mathbf{W}}_{loc}^{1,1}$ for the map $S_t$, $t>0$ that associates to every given initial data $u_0\in \mathrm{Lip}({\mathbb{R}}^N)$ the corresponding solution $S_t{u}_0$ of a Hamilton-Jacobi equation $$u_t+H(x,{\nabla }_{x}u)=0,t\ge 0,x\in {\mathbb{R}}^N,$$ with a convex and coercive Hamiltonian $H=H(x,p)$. We provide upper and lower bounds of order $1/{\epsilon }^N$ on the the Kolmogorov $\epsilon$-entropy in ${\mathbf{W}}^{1,1}$ of the image through the map $S_t$ of sets of bounded, compactly supported initial data. Quantitative estimates of compactness, as suggested by P.D. Lax, could provide a measure of the order of "resolution" and of "complexity" of a numerical method implemented for this equation. We establish these estimates deriving accurate a-priori bounds on the Lipschitz, semiconcavity and semiconvexity constant of a viscosity solution when the initial data is semiconvex. The derivation of a small time controllability result is also fundamental to establish the lower bounds on the $\epsilon$-entropy.

36 pages. arXiv admin note: text overlap with arXiv:1403.4556