TY - JOUR

T1 - On the piecewise smoothness of entropy solutions to scalar conservation laws

AU - Tadmor, Eitan

AU - Tassa, Tamir

N1 - Funding Information:
Research was supported in part by the Basic Research Foundation of the Israeli Academy of Sciences and Humanities. Additional support for the first author was provided by ONR Contract number N-00014-91-J-1096 and by SSF Grant number DMS91-03104.

PY - 1993/1/1

Y1 - 1993/1/1

N2 - The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well known that such entropy solutions consist of at most countable number of C1-smooth regions. We obtain new upper, bounds on the higher order derivatives of the entropy solution in any one of its (C1-smoothness regions. These bounds enable us to measure the high order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cn-semi norm - localized to the smooth part of the entropy solution, and we show that the entropy solution is stable with respect to this norm. We also address the question regarding the number of C1-smoothness pieces; we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of piecewise smoothness which is assumed — sometime implicitly — in many finite-dimensional computations for such discontinuous problems.

AB - The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well known that such entropy solutions consist of at most countable number of C1-smooth regions. We obtain new upper, bounds on the higher order derivatives of the entropy solution in any one of its (C1-smoothness regions. These bounds enable us to measure the high order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cn-semi norm - localized to the smooth part of the entropy solution, and we show that the entropy solution is stable with respect to this norm. We also address the question regarding the number of C1-smoothness pieces; we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of piecewise smoothness which is assumed — sometime implicitly — in many finite-dimensional computations for such discontinuous problems.

UR - http://www.scopus.com/inward/record.url?scp=0001400260&partnerID=8YFLogxK

U2 - 10.1080/03605309308820988

DO - 10.1080/03605309308820988

M3 - Article

AN - SCOPUS:0001400260

VL - 18

SP - 1631

EP - 1652

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 9-10

ER -