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Quantitative compactness estimates for Hamilton-Jacobi equations

Abstract : We study quantitative compactness estimates in $\mathbf{W}^{1,1}_{loc}$ 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 $$ u_t+H\big(\nabla_{/!x} u\big)=0\,, \qquad t\geq 0,\quad x\in \mathbb{R}^N, $$ with a uniformly convex Hamiltonian $H=H(p)$. We provide upper and lower estimates of order $1/\varepsilon^N$ on the the Kolmogorov $\varepsilon$-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.
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https://hal.inria.fr/hal-01098258
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Submitted on : Tuesday, December 23, 2014 - 2:37:12 PM
Last modification on : Monday, October 5, 2020 - 11:22:01 AM
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  • HAL Id : hal-01098258, version 1
  • ARXIV : 1403.4556

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Fabio Ancona, Piermarco Cannarsa, Khai T. Nguyen. Quantitative compactness estimates for Hamilton-Jacobi equations. 2014. ⟨hal-01098258⟩

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