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Scalar conservation laws with general boundary condition and continuous flux function. (English) Zbl 1103.35069

Summary: We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: \(u_t+\text{div}\,\Phi(u) =f\) on \(Q=(0,T)\times\Omega\), \(u(0,\cdot)= u_0\) on \(\Omega\) and “\(u=a\) on some part of the boundary \((0,T) \times\partial\Omega\).” Existence and uniqueness of the entropy solution is established for any \(\Phi\in C(\mathbb{R};\mathbb{R}^N)\), \(u_0\in L^\infty (\Omega)\), \(f\in L^\infty(Q)\), \(a\in L^\infty((0,T)\times\partial \Omega)\). In the \(L^1\)-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.

MSC:

35L65 Hyperbolic conservation laws
35L50 Initial-boundary value problems for first-order hyperbolic systems
35D05 Existence of generalized solutions of PDE (MSC2000)
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