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
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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.
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- Pierangelo Marcati and Bruno Rubino, Hyperbolic to parabolic relaxation theory for quasilinear first order systems, J. Differential Equations 162 (2000), no. 2, 359–399. MR 1751710, DOI https://doi.org/10.1006/jdeq.1999.3676
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R. J. DiPerna. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal., 82:27–70, 1983.
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.
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.
F. Huang and R. Pan. Convergence rate for compressible Euler equations with damping and vacuum. Arch. Rational Mech. Anal., 166:359–376, 2003.
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.
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.
P. L. Lions and G. Toscani. Diffusive limit for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoamericana, 13:473–513, 1997.
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.
P. Marcati, A. Milani, and P. Secchi. Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system. Manuscripta Math., 60:49–69, 1988.
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.
P. Marcati and B. Rubino. Hyperbolic to parabolic relaxation theory for quasilinear first order systems. J. Differential Equations, 162:359–399, 2000.
F. Murat. Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5:489–507, 1978.
B. Rubino. Porous media flow as the limit of a nonstrictly hyperbolic system of conservation laws. Comm. Partial Differential Equations, 21:1–21, 1996.
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.
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.
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.
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