Multi-dimensional scalar conservation laws with unbounded integrable initial data

Abstract : We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the decay of the solution as time increases, in relation with the nonlinearity. The main ingredient is our recent theory of divergence-free positive symmetric tensor. We apply in particular the so-called compensated integrability to a tensor which generalizes the one that L. Tartar used in one space dimension. It allows us to establish a Strichartz-like inequality, in a quasilinear context. This program is carried out in details for a multi-dimensional version of the Burgers equation.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-01837034
Contributor : Denis Serre <>
Submitted on : Thursday, July 26, 2018 - 10:28:49 AM
Last modification on : Tuesday, November 19, 2019 - 10:40:19 AM
Long-term archiving on : Saturday, October 27, 2018 - 12:47:42 PM

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  • HAL Id : ensl-01837034, version 2
  • ARXIV : 1807.10474

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Denis Serre. Multi-dimensional scalar conservation laws with unbounded integrable initial data. 2018. ⟨ensl-01837034v2⟩

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