Differential and Integral Equations

An equivalent definition of renormalized entropy solutions for scalar conservation laws

Kazuo Kobayasi and Satoru Takagi

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Abstract

We introduce a new notion of renormalized dissipative solutions for a scalar conservation law $u_{t}+\mathrm{div}\, {\mathrm{\mathbf{F}}}(u)=f$ with locally Lipschitz ${\mathrm{\mathbf{F}}}$ and $L^{1}$ data, and prove the equivalence of such solutions and renormalized entropy solutions in the sense of Benilan et al. The structure of renormalized dissipative solutions is more useful in dealing with relaxation systems than the renormalized entropy scheme. As an example, we apply our result to contractive relaxation systems in merely an $L^{1}$ setting and construct a renormalized dissipative solution via relaxation.

Article information

Source
Differential Integral Equations, Volume 18, Number 1 (2005), 19-33.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060234

Mathematical Reviews number (MathSciNet)
MR2105337

Zentralblatt MATH identifier
1212.35299

Subjects
Primary: 35L65: Conservation laws
Secondary: 35D05 35L45: Initial value problems for first-order hyperbolic systems 35L60: Nonlinear first-order hyperbolic equations 47N20: Applications to differential and integral equations

Citation

Kobayasi, Kazuo; Takagi, Satoru. An equivalent definition of renormalized entropy solutions for scalar conservation laws. Differential Integral Equations 18 (2005), no. 1, 19--33. https://projecteuclid.org/euclid.die/1356060234


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