Explicit formula for scalar nonlinear conservation laws with boundary condition. (English) Zbl 0679.35065

Le Floch, Philippe

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

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

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.