BLOWUP FOR SYSTEMS OF CONSERVATION LAWS

We give examples of finite time blowup in sup-norm and total variation for 3 × 3systems of strictly hyperbolic conservation laws. The exact solutions are explicitly constructed. In the case of sup-norm blowup we also provide an example where all other p-norms, 1 ≤ p < ∞, remain uniformly bounded. Finally we consider appropriate rescalings for the different types of blowup.

[1]  N. Risebro.  A front-tracking alternative to the random choice method , 1993 .

[2]  C. Cheverry.  Justification de l’optique géométrique non linéaire pour un système de lois de conservation , 1997 .

[3]  R. J. Diperna.  Existence in the large for quasilinear hyperbolic conservation laws , 1973 .

[4]  Guy Métivier,et al.  Resonant One Dimensional Nonlinear Geometric Optics , 1993 .

[5]  A. Jeffrey.  Breakdown of the solution to a completely exceptional system of hyperbolic equations , 1974 .

[6]  Steven Schochet,et al.  Sufficient conditions for local existence via Glimm's scheme for large BV data , 1991 .

[7]  Robin Young,et al.  The large time stability of sound waves , 1996 .

[8]  C. Cheverry,et al.  Systèmes de lois de conservation et stabilité BV , 1998 .

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

[10]  R. Young.  Exact solutions to degenerate conservation laws , 1999 .

[11]  D. Serre,et al.  Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation , 1987 .

[12]  Takaaki Nishida,et al.  Global solution for an initial boundary value problem of a quasilinear hyperbolic system , 1968 .

[13]  R. Rosales,et al.  Resonantly interacting weakly nonlinear hyperbolic waves in the presence of shocks: A single space variable in a homogeneous, time independent medium , 1986 .

[14]  S. Schochet.  Glmm's scheme for systems with almost–planar inreraction , 1991 .

[15]  A. Bressan,et al.  Uniqueness for discontinuous O.D.E. and conservation laws , 1997 .

[16]  J. Joly,et al.  A nonlinear instability for 3×3 systems of conservation laws , 1994 .

[17]  P. Lax,et al.  Decay of solutions of systems of nonlinear hyperbolic conservation laws , 1970 .

[18]  Takaaki Nishida,et al.  Solutions in the large for some nonlinear hyperbolic conservation laws , 1973 .

[19]  A. Bressan,et al.  Unique solutions of 2x2 conservation laws with large data , 1995 .

[20]  A. Bressan,et al.  The semigroup generated by a Temple class system with large data , 1997, Differential and Integral Equations.

[21]  A. Bressan.  Global solutions of systems of conservation laws by wave-front tracking , 1992 .

[22]  H. Jenssen,et al.  On the Front-Tracking Algorithm , 1998 .

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

[24]  H. Alber,et al.  Local existence of weak solutions to the quasi-linear wave equation for large initial values , 1985 .

[25]  Strongly Nonlinear Hyperbolic Waves , 1989 .

[26]  R. Young.  Sup-norm stability for Glimm's scheme , 1993 .

[27]  Tai-Ping Liu.  The deterministic version of the Glimm scheme , 1977 .

[28]  Helge Kristian Jenssen,et al.  Well-Posedness for a Class of 2_2 Conservation Laws with L Data , 1997 .

[29]  R. J. Diperna,et al.  Convergence of the viscosity method for isentropic gas dynamics , 1983 .