Le Floch, Philippe Explicit formula for scalar nonlinear conservation laws with boundary condition. (English) Zbl 0679.35065 Math. Methods Appl. Sci. 10, No. 3, 265-287 (1988). 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 Cited in 1 ReviewCited in 29 Documents MSC: 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 Keywords:scalar conservation law; boundary value problem; existence and uniqueness; free boundary value problem; variational inequality PDF BibTeX XML Cite \textit{P. Le Floch}, Math. Methods Appl. Sci. 10, No. 3, 265--287 (1988; Zbl 0679.35065) Full Text: DOI References: [1] Bardos, Comm. P.D.E. 4 pp 1017– (1979) [2] Keyfitz, Comm. P.A.M. 24 pp 125– (1971) [3] Kruskov, Math. USSR Sbornik 10 pp 217– (1970) [4] Lax, Comm. P.A.M. 10 pp 537– (1957) [5] Lax, S.I.A.M. 13 (1973) [6] Le Floch, Trans. A.M.S. [7] Oleinik, A.M.S. Transl. 2 pp 95– (1964) [8] Schonbek, J. Math. Anal. 15 pp 1125– (1984) [9] Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, 1983, p. 258. · Zbl 0508.35002 · doi:10.1007/978-1-4684-0152-3 [10] Linear and Non-linear Waves, New York, Wiley Interscience, 1974. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.