[ 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | All | Home ]

Preprint 2006-031

On the piecewise smoothness of entropy solutions to scalar conservation laws for a large class of initial data

T. Tang, J.-H. Wang and Y.-C. Zhao

Abstract: In this paper, we prove that if the initial data do not belong to a certain subset of Ck, which is hitherto smallest in the sense of inclusion relation of sets, then the solutions of scalar conservation laws are piecewise smooth. In particular, our initial data allow centered compression waves, which is the case not covered by Dafermos (1974) and Schaeffer (1973; doi:10.1016/0001-8708(73)90018-2). More precisely, we are concerned with the structure of the solutions in the neighborhoods of the points at which only a Ck+1 shock generates, while there can be infinite number of intervals, the characteristics from each of these intervals will meet at a point in any small neighborhood. We give sufficient and almost necessary conditions of the initial data for a degenerate point at which a Ck+1 shock generates. It is also shown that there are finitely many shocks for smooth initial data (in the Schwartz space) except a certain subset of \mathscr{S}(R) of the first category. It should be pointed out that this subset is smaller than those used in previous works. We point out that Thom's theory of catastrophes (Thom 1972), which plays a key role in (Schaeffer 1973), can not be used to analyze the larger class of initial data considered in this work.

Available as PDF (264 Kbytes).
T. Tang,
J.-H. Wang,
Y.-C. Zhao,
Publishing information:
Submitted by:
; 2006-08-25.