Nonclassical Shocks and Kinetic Relations: Strictly Hyperbolic Systems

We consider strictly hyperbolic systems of conservation laws whose characteristic fields are not genuinely nonlinear, and we introduce a framework for the nonclassical shocks generated by diffusive or diffusive-dispersive approximations. A nonclassical shock does not fulfill the Liu entropy criterion and turns out to be undercompressive. We study the Riemann problem in the class of solutions satisfying a single entropy inequality, the only such constraint available for general diffusive-dispersive approximations. Each non-genuinely nonlinear characteristic field admits a two-dimensional wave set, instead of the classical one-dimensional wave curve. In specific applications, these wave sets are narrow and resemble the classical curves. We find that even in strictly hyperbolic systems, nonclassical shocks with arbitrarily small amplitudes occur. The Riemann problem can be solved uniquely using nonclassical shocks, provided an additional constraint is imposed: we stipulate that the entropy dissipation across...

[1]  R. Cohen,et al.  Nonlinear evolution of parallel‐propagating hydromagnetic waves , 1974 .

[2]  James Glimm.  Nonlinear Waves: Overview and Problems , 1991 .

[3]  L. Hsiao.  Uniqueness of admissible solutions of the Riemann problem for a system of conservation laws of mixed type , 1990 .

[4]  Tai-Ping Liu,et al.  NONLINEAR STABILITY AND INSTABILITY OF OVERCOMPRESSIVE SHOCK WAVES , 1993 .

[5]  Dan Marchesin,et al.  Transitional waves for conservation laws , 1990 .

[6]  M. Shearer,et al.  The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  Peter D. Lax,et al.  The zero dispersion limit, a deterministic analogue of turbulence , 1991 .

[8]  C. Conley,et al.  On the Structure of Magnetohydrodynamic Shock Waves. , 1974 .

[9]  B. Hayes,et al.  Measure solutions to a strictly hyperbolic system of conservation laws , 1996 .

[10]  Dan Marchesin,et al.  A global formalism for nonlinear waves in conservation laws , 1992 .

[11]  Stephen Schecter,et al.  Undercompressive shocks for nonstrictly hyperbolic conservation laws , 1991 .

[12]  Tai-Ping Liu,et al.  On a nonstrictly hyperbolic system of conservation laws , 1985 .

[13]  Stephen Schecter,et al.  Structurally Stable Riemann Solutions , 1996 .

[14]  J. K. Knowles,et al.  Kinetic relations and the propagation of phase boundaries in solids , 1991 .

[15]  Thomas Y. Hou,et al.  A Level-Set Approach to the Computation of Twinning and Phase-Transition Dynamics , 1999 .

[16]  Tai-Ping Liu.  Admissible solutions of hyperbolic conservation laws , 1981 .

[17]  C. Dafermos.  The entropy rate admissibility criterion for solutions of hyperbolic conservation laws , 1973 .

[18]  C. David Levermore,et al.  The small dispersion limit of the Korteweg‐de Vries equation. III , 1983 .

[19]  Kevin Zumbrun,et al.  On nonlinear stability of general undercompressive viscous shock waves , 1995 .

[20]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[21]  B. McKinney,et al.  Traveling Wave Solutions of the Modified Korteweg-deVries-Burgers Equation , 1995 .

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

[23]  M. Slemrod.  Admissibility criteria for propagating phase boundaries in a van der Waals fluid , 1983 .

[24]  L. Truskinovskii,et al.  Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium☆ , 1987 .

[25]  Lev Truskinovsky,et al.  Kinks versus Shocks , 1993 .

[26]  Michael Shearer,et al.  The Riemann problem for a class of conservation laws of mixed type , 1982 .

[27]  C. Wu,et al.  Formation, structure, and stability of MHD intermediate shocks , 1990 .

[28]  Luc Tartar,et al.  The Compensated Compactness Method Applied to Systems of Conservation Laws , 1983 .

[29]  R. J. DiPerna.  Convergence of approximate solutions to conservation laws , 1983 .

[30]  C. Wu,et al.  The small amplitude magnetohydrodynamic Riemann problem , 1993 .

[31]  D. G. Schaeffer,et al.  Solution of the riemann problem for a prototype 2×2 system of non-strictly hyperbolic conservation laws , 1987 .

[32]  Zhouping Xin,et al.  Stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws , 1992 .

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

[34]  J. K. Knowles,et al.  Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids , 1991 .

[35]  Barbara Lee Keyfitz,et al.  A system of non-strictly hyperbolic conservation laws arising in elasticity theory , 1980 .

[36]  Tai-Ping Liu.  The Riemann problem for general 2×2 conservation laws , 1974 .

[37]  Heinrich Freistühler,et al.  Nonlinear stability of overcompresive shock waves in a rotationally invariant system of viscous conservation laws , 1993 .

[38]  B. Hayes,et al.  Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws , 1997 .

[39]  B. Hayes,et al.  Nonclassical Shocks and Kinetic Relations: Finite Difference Schemes , 1998 .

[40]  R. Menikoff,et al.  The Riemann problem for fluid flow of real materials , 1989 .

[41]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[42]  Haitao Fan,et al.  The Riemann Problem for Systems of Conservation Laws of Mixed Type , 1993 .

[43]  E. Isaacson,et al.  Nonlinear resonance in systems of conservation laws , 1992 .

[44]  Kevin Zumbrun,et al.  Nonlinear stability of an undercompressive shock for complex Burgers equation , 1995 .

[45]  Gui-Qiang G. Chen.  Hyperbolic systems of conservation laws with a symmetry , 1991 .

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

[47]  J. K. Hunter,et al.  Rotationally invariant hyperbolic waves , 1990 .

[48]  C. Dafermos.  Admissible Wave Fans in Nonlinear Hyperbolic Systems , 1989 .

[49]  조의연,et al.  Conservation laws with vanishing nonlinear diffusion and dispersion 1 1 This work was partially carr , 1999, 0711.0411.

[50]  B. Keyfitz,et al.  A geometric theory of conservation laws which change type , 1995 .