L1-Theory of Scalar Conservation Law with Continuous Flux Function

✍ Scribed by Boris P. Andreianov; Philippe Bénilan; Stanislav N. Kruzhkov

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 221 KB
• Volume: 171
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3445

✦ Synopsis

Uniqueness of a generalized entropy solution (g.e.s.) to the Cauchy problem for N-dimensional scalar conservation laws u t +div x ,(u)= g, u(0, } )= f with continuous flux function , is still an open problem. For data ( f, g) vanishing at infinity, we show that there exist a maximal and a minimal g.e.s. to the Cauchy problem and to the associated stationary problem u+div x ,(u)= f. In the case of L 1 data, using the nonlinear semigroup theory, we prove that there is uniqueness for all data of a g.e.s. to the Cauchy problem if and only if there is uniqueness for all data of a g.e.s. to the related stationary problem. Applying this result and an induction argument on the dimension N, we extend uniqueness results of Be nilan, Kruzhkov (1996, Nonlinear Differential Equations Appl. 3, 395 419) for flux having some monotonicity properties.