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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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