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Compactness Estimates for Hamilton-Jacobi Equations Depending on Space
by Fabio Ancona   Piermarco Cannarsa   Khai T. Nguyen

Vol. 11 No. 1 (2016) P.63~P.113

ABSTRACT

We study quantitative estimates of compactness in ${\bf{W}}_{loc}^{1,1}$ 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 an Hamilton-Jacobi equation
\begin{equation*} u_{t} + H(x,\bigtriangledown_{x}u) = 0, \: \:t\geq 0, \:\: x \in \mathbb{R}^{N}, \end{equation*} with a convex and coercive Hamiltonian H = $H(x, p)$. We provide upper and lower bounds of order $1/\varepsilon^{N}$ on the the Kolmogorov "$\varepsilon$-entropy in ${\bf{W}}^{1,1}$ 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 [23], 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 for the above Hamilton-Jacobi equation is also fundamental to establish the lower bounds on the $\varepsilon$-entropy.

KEYWORDS
Hamilton-Jacobi equations, Kolmogorov entropy, compactness, controllability of Hamilton-Jacobi equations.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35F21, 49N60, 47B06, 54D30, 35Q93.

MILESTONES