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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:
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