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Quantitative Compactness Estimates for Hamilton–Jacobi Equations

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Abstract

We study quantitative compactness estimates in \({\mathbf{W}^{1,1}_{{\rm loc}}}\) for the map \({S_t}\), \({t > 0}\) that is associated with the given initial data \({u_0\in {\rm Lip} (\mathbb{R}^N)}\) for 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 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 Lax (Course on Hyperbolic Systems of Conservation Laws. XXVII Scuola Estiva di Fisica Matematica, Ravello, 2002) within the context of conservation laws, and could provide a measure of the order of “resolution” of a numerical method implemented for this equation.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Fabio Ancona

  2. Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientica 1, 00133, Rome, Italy

    Piermarco Cannarsa

  3. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Khai T. Nguyen

Authors
  1. Fabio Ancona
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  2. Piermarco Cannarsa
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  3. Khai T. Nguyen
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Correspondence to Fabio Ancona.

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Communicated by C. Dafermos

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Ancona, F., Cannarsa, P. & Nguyen, K.T. Quantitative Compactness Estimates for Hamilton–Jacobi Equations. Arch Rational Mech Anal 219, 793–828 (2016). https://doi.org/10.1007/s00205-015-0907-5

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