×

An existence result for scalar conservation laws using measure valued solutions. (English) Zbl 0704.35022

The author considers measure valued solutions of the initial value problem \[ \partial_ tu+\sum^{d-1}_{i=1}\partial_{x_ i}f_ i(u)=0\text{ in } {\mathbb{R}}^{d-1}\times {\mathbb{R}}_+,\quad u(,0)=u_ 0\text{ on } {\mathbb{R}}^{d-1}. \] Here, \(u_ 0\in L^ 1({\mathbb{R}}^{d- 1})\cap L^ P({\mathbb{R}}^{d-1})\), \(1<p\leq \infty\), and \(f=(f_ 1,...,f_{d-1})\) is continuous and satisfies \[ f(\lambda)=O(1+| \lambda |^ q),\quad 0\leq q<p,\quad \limsup_{\lambda \to 0}\frac{| f(\lambda)-f(0)|}{| \lambda |^{\alpha}}<\infty,\quad \frac{d-2}{d-1}<\alpha \leq 1. \] An existence and uniqueness result is proved for solutions in \(L^ 1({\mathbb{R}}^ d_+)\cap L^ p({\mathbb{R}}^ d_+)\).
Reviewer: M.Shearer

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35L65 Hyperbolic conservation laws
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/BF02764657 · Zbl 0246.35018 · doi:10.1007/BF02764657
[2] DOI: 10.1007/BF00752112 · Zbl 0616.35055 · doi:10.1007/BF00752112
[3] DOI: 10.1007/BF01214424 · Zbl 0626.35059 · doi:10.1007/BF01214424
[4] Friedman, A. 1964. Prentice-Hall International Inc.
[5] DOI: 10.1070/SM1970v010n02ABEH002156 · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156
[6] Lax P.D., Contributions to Nonlinear Functional Analysis, ed.E.A. Zarantonello
[7] Szepessy, A. ”Measure valued solutions to scalar conservation”. · Zbl 0702.35155
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.