TY - JOUR

T1 - On the piecewise smoothness of entropy solutions to scalar conservation laws for a larger class of initial data

AU - TANG, Tao

AU - Wang, Jinghua

AU - Zhao, Yinchuan

N1 - Funding Information:
The research of this project was partially supported by CERG Grants of Hong Kong Research Grant Council, FRG grants of Hong Kong Baptist University, and International Research Team of Complex Systems of Chinese Academy of Sciences. The second author was supported by National Natural Foundation of China under contract 10371124 and 10671116.

PY - 2007/9

Y1 - 2007/9

N2 - We prove that if the initial data do not belong to a certain subset of Ck, then the solutions of scalar conservation laws are piecewise Ck smooth. In particular, our initial data allow centered compression waves, which was the case not covered by Dafermos (1974) and Schaeffer (1973). More precisely, we are concerned with the structure of the solutions in some neighborhood of the point at which only a Ck+1 shock is generated. It is also shown that there are finitely many shocks for smooth initial data (in the Schwartz space) except for a certain subset of ℒ(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, which plays a key role in Schaeffer (1973), cannot be used to analyze the larger class of initial data considered in this paper.

AB - We prove that if the initial data do not belong to a certain subset of Ck, then the solutions of scalar conservation laws are piecewise Ck smooth. In particular, our initial data allow centered compression waves, which was the case not covered by Dafermos (1974) and Schaeffer (1973). More precisely, we are concerned with the structure of the solutions in some neighborhood of the point at which only a Ck+1 shock is generated. It is also shown that there are finitely many shocks for smooth initial data (in the Schwartz space) except for a certain subset of ℒ(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, which plays a key role in Schaeffer (1973), cannot be used to analyze the larger class of initial data considered in this paper.

KW - A set of first category

KW - Conservation laws

KW - Piecewise smooth solutions

UR - http://www.scopus.com/inward/record.url?scp=34447271811&partnerID=8YFLogxK

U2 - 10.1142/S0219891607001185

DO - 10.1142/S0219891607001185

M3 - Article

AN - SCOPUS:34447271811

VL - 4

SP - 369

EP - 389

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

SN - 0219-8916

IS - 3

ER -