## Differential and Integral Equations

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

### An equivalent definition of renormalized entropy solutions for scalar conservation laws

Kazuo Kobayasi and Satoru Takagi

#### 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