Optimal regularizing effect for scalar conservation laws

François Golse; Benoît Perthame

doi:10.4171/RMI/765

Optimal regularizing effect for scalar conservation laws

François Golse
, Benoît Perthame

Sorbonne Université

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. We prove that all such solutions belong to the Besov space B ^{1/3 3}_{∞, loc}. Since C. de Lellis and M. Westdickenberg [11] have proved the existence of such solutions that do not belong to B^s,p_{q, loc} if either s > 1/max(p, 3) or s = 1/3 and 1 ≤ q < p < 3 or s = 1/p with p ≥ 3 and q < ∞, this regularizing effect is optimal. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space.

Original language	English
Pages (from-to)	1477-1504
Number of pages	28
Journal	Revista Matematica Iberoamericana
Volume	29
Issue number	4
DOIs	https://doi.org/10.4171/RMI/765
Publication status	Published - 1 Jan 2013

Keywords

Compensated compactness
Kinetic formulation
Regularizing effect
Scalar conservation law

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AU - Perthame, Benoît

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N2 - We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. We prove that all such solutions belong to the Besov space B 1/3 3∞, loc. Since C. de Lellis and M. Westdickenberg [11] have proved the existence of such solutions that do not belong to Bs,pq, loc if either s > 1/max(p, 3) or s = 1/3 and 1 ≤ q < p < 3 or s = 1/p with p ≥ 3 and q < ∞, this regularizing effect is optimal. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space.

AB - We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. We prove that all such solutions belong to the Besov space B 1/3 3∞, loc. Since C. de Lellis and M. Westdickenberg [11] have proved the existence of such solutions that do not belong to Bs,pq, loc if either s > 1/max(p, 3) or s = 1/3 and 1 ≤ q < p < 3 or s = 1/p with p ≥ 3 and q < ∞, this regularizing effect is optimal. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space.

KW - Compensated compactness

KW - Kinetic formulation

KW - Regularizing effect

KW - Scalar conservation law

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EP - 1504

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

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Optimal regularizing effect for scalar conservation laws

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