## Measure-valued solutions of scalar conservation laws with boundary conditions.(English)Zbl 0702.35155

The equation into consideration is the conservation law $(1)\quad u_ t+\sum^{d}_{j=1}f_ j(u)_{x_ j}=0 \text{ in } \Omega \times R_+$ with initial condition (2) u(,0)$$=u_ 0$$ on $$\Omega$$ and boundary condition $(3)\quad (sgn(u(\bar x,t)-k)-sgn a(\bar x,t)- k)\times (f(u(\bar x,t)-f(k))n(\bar x)\geq 0$ for all $$k\in R$$, $$(\bar x,t)\in \Gamma \times R_+$$. Here $$\Omega$$ is a bounded open set of $$R^ d$$ with smooth boundary $$\Gamma =\partial \Omega$$ and outward unit normal n, u: $$\Omega\times R_+\to R$$, $$f=(f_ 1,...,f_ d): R\to R^ d$$, $$u_ 0: \Omega \to R$$, a: $$\Gamma\times R_+\to R$$. The boundary $$\Gamma$$ of $$\Omega$$ is supposed to be smooth. Smoothness is provided also for $$f,u_ 0$$ and a. A measure-valued solution to problem (1)-(3) is defined. A uniqueness theorem is proved. The obtained result is used to prove convergence towards the unique solution for approximate solutions which are uniformly bounded in $$L_{\infty}$$, weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.
Reviewer: I.Ginchev

### MSC:

 35L65 Hyperbolic conservation laws 35A25 Other special methods applied to PDEs
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### References:

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