Asymptotic behavior for a hyperbolic conservation law with periodic initial data

✍ Scribed by Joseph G. Conlon

• Publisher: John Wiley and Sons
• Year: 1979
• Tongue: English
• Weight: 602 KB
• Volume: 32
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160320104

✦ Synopsis

In this paper we discuss asymptotic behavior for the hyperbolic conservation law with periodic initial data of period p. The function f is a smooth nonlinear function of u. In general, equation (1.1) does not have a continuous solution for all time. Shock curves appear after finite time and tend to dominate the solution for large time. If solutions of (1.1) were continuous, then there would be several conserved quantities associated with the equation. Since solutions contain shock curves, there is only one such quantity, This is the average value of u over a period.

The asymptotic behavior of (1.1) when f(u) is convex or concave was determined rather precisely by Lax [3]. He shows that, for all t > O , u(x, t ) satisfies an inequality where the constant A depends only on the function f(u). For almost all initial * Research supported by the United Kingdom Science Research Council, This work was completed while the author was a Visiting Member at the Courant Institute of Mathematical Sciences, New York University. Reproduction in whole or in part is permitted for any purpose of the United States Government. Communications on Pure and Applied Mathematics, Val, XXXII, 99-112 (1979) @ 1979 John Wiley & Sans, Inc.