Global classical solutions to partially dissipative quasilinear hyperbolic systems with one weakly linearly degenerate characteristic

Abstract

For a kind of partially dissipative quasilinear hyperbolic systems without Shizuta-Kawashima condition, in which all the characteristics, except a weakly linearly degenerate one, are involved in the dissipation, the global existence of H ² classical solution to the Cauchy problem with small initial data is obtained.

Keywords

Global classical solution / Quasilinear hyperbolic system / Weak linear degeneracy / Partial dissipation

References

[1] Beauchard K., Zuazua E.. Large time asymptotics for partially dissipative hyperbolic systems. Arch. Rat. Mech. Anal., 2011, 199: 177-227

[2] Bianchini S., Hanouzet B., Natalini R.. Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math., 2007, 60: 1559-1622

[3] Hanouzet B., Natalini R.. Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rat. Mech. Anal., 2003, 169: 89-117

[4] Hsiao L., Li T. T.. Global smooth solution of Cauchy problems for a class of quasilinear hyperbolic systems. Chin. Ann. Math., 1983, 4B(1): 109-115

[5] John F.. Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math., 1974, 27: 377-405

[6] Li, T. T., Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson/John Wiley, New York, 1994.

[7] Li T. T., Qin T. H.. Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms. Chin. Ann. Math., 1985, 6B(2): 199-210

[8] Li, T. T. and Wang, L. B., Global Propagation of Regular Nonlinear Hyperbolic Waves, Progress in Nonlinear Differential Equations and Their Applications, 76, Birkhäuser, Boston, 2009.

[9] Li, T. T. and Yu, W. C., Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, Duke University, Durham, 1985.

[10] Li T. T., Zhou Y., Kong D. X.. Global classical solutions for general quasilinear hyperbolic systems with decay initial data. Nonlinear Anal. Theory Meth. Appl., 1997, 28: 1299-1332

[11] Liu C., Qu P.. Global classical solution to partially dissipative quasilinear hyperbolic systems. J. Math. Pures Appl., 2012, 97: 262-281

[12] Majda A.. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Applied Mathematical Sciences, 1984, New York: Springer-Verlag

[13] Mascia C., Natalini R.. On relaxation hyperbolic systems violating the Shizuta-Kawashima condition. Arch. Rat. Mech. Anal., 2010, 195: 729-762

[14] Shizuta Y., Kawashima S.. Systems of equations of hyperbolic-parabolic type with applications to discrete Boltzmann equation. Hokkaido Math. J., 1985, 14: 249-275

[15] Yong W. A.. Entropy and global existence for hyperbolic balance laws. Arch. Rat. Mech. Anal., 2004, 172: 247-266

[16] Zeng Y.. Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Rat. Mech. Anal., 1999, 150: 225-279

[17] Zeng Y.. Gas flows with several thermal nonequilibrium modes. Arch. Rat. Mech. Anal., 2010, 196: 191-225

[18] Zhou Y.. Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy. Chin. Ann. Math., 2004, 25B(1): 37-56

[19] Zhou Y.. Global classical solutions for partially dissipative quasilinear hyperbolic systems. Chin. Ann. Math., 2011, 32B(5): 771-780

[20] Zhou Y.. Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems. Math. Meth. Appl. Sci., 2009, 32: 1669-1680