Quantitative compactness estimates for Hamilton-Jacobi equations

Abstract

We study quantitative compactness estimates in ${\mathbf{W}}_{loc}^{1,1}$ for the map $S_t$, $t>0$ that associates to every given initial data $u_0\in Lip({\mathbb{R}}^N)$ the corresponding solution $S_t{u}_0$ of a Hamilton-Jacobi equation $$u_t+H\left({\nabla }_{/!x}u\right)=0,t\ge 0,x\in {\mathbb{R}}^N,$$ with a uniformly convex Hamiltonian $H=H(p)$. We provide upper and lower estimates 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. Estimates of this type are inspired by a question posed by P.D. Lax within the context of conservation laws, andcould provide a measure of the order of "resolution" of a numerical method implemented for this equation.

31 pages, 1 figure

References in corpus (1)

• On compactness estimates for hyperbolic systems of conservation laws