The Riemann problem for general conservation laws
Author:
Tai Ping Liu
Journal:
Trans. Amer. Math. Soc. 199 (1974), 89112
MSC:
Primary 35L65
MathSciNet review:
0367472
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The Riemann Problem for a system of hyperbolic conservation laws of form with arbitrary initial constant states is considered. We assume that . Let be the left (right) eigenvectors of for eigenvalues . Instead of assuming the usual convexity condition we assume that on disjoint union of dim manifolds in the plane. Oleinik's condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax's shock inequalities and, in case , the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E).
 [1]
James
Glimm, Solutions in the large for nonlinear hyperbolic systems of
equations, Comm. Pure Appl. Math. 18 (1965),
697–715. MR 0194770
(33 #2976)
 [2]
J.
A. Smoller and J.
L. Johnson, Global solutions for an extended class of hyperbolic
systems of conservation laws, Arch. Rational Mech. Anal.
32 (1969), 169–189. MR 0236527
(38 #4822)
 [3]
P.
D. Lax, Hyperbolic systems of conservation laws. II, Comm.
Pure Appl. Math. 10 (1957), 537–566. MR 0093653
(20 #176)
 [4]
Takaaki
Nishida, Global solution for an initial boundary value problem of a
quasilinear hyperbolic system, Proc. Japan Acad. 44
(1968), 642–646. MR 0236526
(38 #4821)
 [5]
Takaaki
Nishida and Joel
A. Smoller, Solutions in the large for some nonlinear hyperbolic
conservation laws, Comm. Pure Appl. Math. 26 (1973),
183–200. MR 0330789
(48 #9126)
 [6]
O.
A. Oleĭnik, On the uniqueness of the generalized solution of
the Cauchy problem for a nonlinear system of equations occurring in
mechanics, Uspehi Mat. Nauk (N.S.) 12 (1957),
no. 6(78), 169–176 (Russian). MR 0094543
(20 #1057)
 [7]
J.
A. Smoller, On the solution of the Riemann problem with general
step data for an extended class of hyperbolic systems, Michigan Math.
J. 16 (1969), 201–210. MR 0247283
(40 #552)
 [8]
J.
A. Smoller, A uniqueness theorem for Riemann problems, Arch.
Rational Mech. Anal. 33 (1969), 110–115. MR 0237961
(38 #6238)
 [1]
 J. Glimm, Solution in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697715. MR 33 #2976. MR 0194770 (33:2976)
 [2]
 J. A. Smoller and J. L. Johnson, Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 32 (1969), 169189. MR 38 #4822. MR 0236527 (38:4822)
 [3]
 P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537566. MR 20 #176. MR 0093653 (20:176)
 [4]
 T. Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad. 44 (1968), 642646. MR 38 #4821. MR 0236526 (38:4821)
 [5]
 T. Nishida and J. A. Smoller, Solution in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 25 (1973), 183200. MR 0330789 (48:9126)
 [6]
 O. A. Oleĭnik, On the uniqueness of the generalized solution of Cauchy problem for nonlinear system of equations occuring in mechanics, Uspehi Mat. Nauk 12 (1957), no. 6 (78), 169176. (Russian) MR 20 #1057. MR 0094543 (20:1057)
 [7]
 J. A. Smoller, On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems, Michigan Math. J. 16 (1969), 201210. MR 40 #552. MR 0247283 (40:552)
 [8]
 , A uniqueness theorem for Riemann problems, Arch. Rational Mech. Anal. 33 (1969), 110115. MR 38 #6238. MR 0237961 (38:6238)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
35L65
Retrieve articles in all journals
with MSC:
35L65
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403674721
PII:
S 00029947(1974)03674721
Keywords:
Conservation laws,
shocks ,
rarefaction waves ,
contact discontinuities,
Oleinik condition (E),
Lax shock inequalities (L),
shock speed
Article copyright:
© Copyright 1974
American Mathematical Society
