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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
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References:

[1] DOI: 10.1007/BF02764657 · Zbl 0246.35018
[2] DOI: 10.1007/BF00752112 · Zbl 0616.35055
[3] DOI: 10.1007/BF01214424 · Zbl 0626.35059
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[5] DOI: 10.1070/SM1970v010n02ABEH002156 · Zbl 0215.16203
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[7] Szepessy, A. ”Measure valued solutions to scalar conservation”. · Zbl 0702.35155
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