Uniqueness via the adjoint problems for systems of conservation laws

We prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws that includes in particular the p-system of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy Condition, or WEC for short, introduced in this paper. The main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution. For the proof of uniqueness, we derive an existence result (respectively a uniqueness result) for the backward (respectively forward) adjoint problem associated with the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients. © 1993 John Wiley & Sons, Inc.

[1]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[2]  M. Lighthill.  Supersonic Flow and Shock Waves , 1949, Nature.

[3]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[4]  B. L. Rozhdestvenskii,et al.  DISCONTINUOUS SOLUTIONS OF HYPERBOLIC SYSTEMS OF QUASILINEAR EQUATIONS , 1960 .

[5]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .

[6]  A. I. Vol'pert.  THE SPACES BV AND QUASILINEAR EQUATIONS , 1967 .

[7]  J. Smoller.  A uniqueness theorem for Riemann problems , 1969 .

[8]  A. Hurd.  A uniqueness theorem for second order quasilinear hyperbolic equations , 1970 .

[9]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[10]  P. Lax.  Shock Waves and Entropy , 1971 .

[11]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[12]  The continuous dependence on the initial conditions of the generalized solutions of the gas-dynamic system of equations , 1974 .

[13]  Tai-Ping Liu,et al.  Uniqueness of weak solutions of the Cauchy problem for general 2 × 2 conservation laws , 1976 .

[14]  R. J. Diperna.  Uniqueness of Solutions to Hyperbolic Conservation Laws. , 1978 .

[15]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[16]  Constantine M. Dafermos,et al.  Hyperbolic Systems of Conservation Laws , 1983 .

[17]  D. Hoff.  The sharp form of Oleĭnik’s entropy condition in several space variables , 1983 .

[18]  Tatsien Li,et al.  Boundary value problems for quasilinear hyperbolic systems , 1985 .

[19]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[20]  David H. Wagner,et al.  Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions , 1987 .

[21]  Yann Brenier,et al.  The discrete one-sided Lipschitz condition for convex scalar conservation laws , 1988 .

[22]  Randall J. LeVeque,et al.  A geometric approach to high resolution TVD schemes , 1988 .

[23]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[24]  Philippe Le Floch,et al.  An existence and uniqueness result for two nonstrictly hyperbolic systems , 1990 .

[25]  C. Dafermos,et al.  Generalized characteristics uniqueness and regularity of solutions in a hyperbolic system of conservation laws , 1991 .

[26]  E. Tadmor.  Local error estimates for discontinuous solutions of nonlinear hyperbolic equations , 1991 .

[27]  Tai-Ping Liu,et al.  Existence theory for nonlinear hyperbolic systems in nonconservative form , 1993 .

[28]  G. D. Maso,et al.  Definition and weak stability of nonconservative products , 1995 .