Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Asymptotic behavior and strong convergence for hyperbolic systems of conservation laws with damping


Authors: Corrado Lattanzio and Bruno Rubino
Journal: Quart. Appl. Math. 62 (2004), 529-540
MSC: Primary 35L65; Secondary 35B40, 35L40, 76N99, 76S05
DOI: https://doi.org/10.1090/qam/2086044
MathSciNet review: MR2086044
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A local type estimate will be proved here for general $ 2 \times 2$ hyperbolic systems of conservation laws with strong dissipative term. Following the idea of [15], this result will be achieved by using as a preliminary step the convergence in the mean, which is an immediate consequence of the result of [11] obtained by using the compensated compactness theory.


References [Enhancements On Off] (What's this?)

  • [1] R. J. DiPerna. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal., 82:27-70, 1983. MR 684413
  • [2] L. Hsiao and T.-P. Liu. Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Physics, 143:599-605, 1992. MR 1145602
  • [3] L. Hsiao and T. Luo. Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping. Quart. Appl. Math., 56:173-189, 1998. MR 1604829
  • [4] F. Huang and R. Pan. Convergence rate for compressible Euler equations with damping and vacuum. Arch. Rational Mech. Anal., 166:359-376, 2003. MR 1961445
  • [5] C. Lattanzio. On the $ 3 - D$ bipolar isentropic Euler-Poisson model for semiconductors and the driftdiffusion limit. Math. Models Methods Appl. Sci., 10:351-360, 2000. MR 1753116
  • [6] C. Lattanzio and P. Marcati. The relaxation to the drift-diffusion system for the $ 3 - D$ isentropic Euler-Poisson model for semiconductors. Discrete Contin. Dynam. Systems, 5(2):449-455, 1999. MR 1665756
  • [7] P. L. Lions and G. Toscani. Diffusive limit for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoamericana, 13:473-513, 1997. MR 1617393
  • [8] P. Marcati and A. Milani. The one-dimensional Darcy's law as the limit of a compressible Euler flow. J. Differential Equations, 84:129-147, 1990. MR 1042662
  • [9] P. Marcati, A. Milani, and P. Secchi. Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system. Manuscripta Math., 60:49-69, 1988. MR 920759
  • [10] P. Marcati and R. Natalini. Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch. Rational Mech. Anal., 129:129-145, 1995. MR 1328473
  • [11] P. Marcati and B. Rubino. Hyperbolic to parabolic relaxation theory for quasilinear first order systems. J. Differential Equations, 162:359-399, 2000. MR 1751710
  • [12] F. Murat. Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5:489-507, 1978. MR 506997
  • [13] B. Rubino. Porous media flow as the limit of a nonstrictly hyperbolic system of conservation laws. Comm. Partial Differential Equations, 21:1-21, 1996. MR 1373762
  • [14] B. Rubino. Weak solutions to quasilinear wave equations of Klein-Gordon or sine-Gordon type and relaxation to the reaction-diffusion equations. NoDEA Nonlinear Differential Equations Appl., 4:439-457, 1997. MR 1485731
  • [15] D. Serre and L. Xsiao. Asymptotic behavior of large weak entropy solutions of the damped $ p$-system. J. Partial Diff. Eqs., 10:355-368, 1997. MR 1486716
  • [16] L. Tartar. Compensated compactness and applications to partial differential equations, volume IV of Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, pages 136-212. Res. Notes in Math. 39, Pitman, 1979. MR 584398

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35L65, 35B40, 35L40, 76N99, 76S05

Retrieve articles in all journals with MSC: 35L65, 35B40, 35L40, 76N99, 76S05


Additional Information

DOI: https://doi.org/10.1090/qam/2086044
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society