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Explicit formula for scalar nonlinear conservation laws with boundary condition. (English) Zbl 0679.35065
The author proves existence and uniqueness theorem for the entropy weak solution of nonlinear scalar hyperbolic conservation law \(u_ t+f(u)_ x=0\), \(x>0\), \(t>0\) with initial and boundary conditions of the form \(u(x,0)=u_ 0(x)\), and \([u(0,t)=u_ 1(t)\) or \(f'(u(0,t))\leq 0\) and \(f(u(0,t))\geq f(u_ 1(t))]\). The function f is assumed to be strictly convex.
The piecewise continuous solution is obtained by means of certain explicit formula, which generalizes a result of Lax. In particular, a free boundary value problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup due to Keyfitz. The weighted Burger’s equation \((x^ au)_ t+(x^ au^ 2/2)_ x=0\) is also studied.
Reviewer: A.Doktor

35L65 Hyperbolic conservation laws
35Q99 Partial differential equations of mathematical physics and other areas of application
35R35 Free boundary problems for PDEs
35L60 First-order nonlinear hyperbolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
Full Text: DOI
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