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
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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.
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI https://doi.org/10.1007/BF00251724
- Ling Hsiao and Tai-Ping Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143 (1992), no. 3, 599–605. MR 1145602
- L. Hsiao and Tao Luo, Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping, Quart. Appl. Math. 56 (1998), no. 1, 173–189. MR 1604829, DOI https://doi.org/10.1090/qam/1604829
- Feimin Huang and Ronghua Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal. 166 (2003), no. 4, 359–376. MR 1961445, DOI https://doi.org/10.1007/s00205-002-0234-5
- Corrado Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci. 10 (2000), no. 3, 351–360. MR 1753116, DOI https://doi.org/10.1142/S0218202500000215
- Corrado Lattanzio and Pierangelo Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dynam. Systems 5 (1999), no. 2, 449–455. MR 1665756, DOI https://doi.org/10.3934/dcds.1999.5.449
- Pierre Louis Lions and Giuseppe Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana 13 (1997), no. 3, 473–513. MR 1617393, DOI https://doi.org/10.4171/RMI/228
- Pierangelo Marcati and Albert Milani, The one-dimensional Darcy’s law as the limit of a compressible Euler flow, J. Differential Equations 84 (1990), no. 1, 129–147. MR 1042662, DOI https://doi.org/10.1016/0022-0396%2890%2990130-H
- Pierangelo Marcati, Albert J. Milani, and Paolo Secchi, Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system, Manuscripta Math. 60 (1988), no. 1, 49–69. MR 920759, DOI https://doi.org/10.1007/BF01168147
- Pierangelo Marcati and Roberto Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal. 129 (1995), no. 2, 129–145. MR 1328473, DOI https://doi.org/10.1007/BF00379918
- 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
- François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997
- Bruno Rubino, Porous media flow as the limit of a nonstrictly hyperbolic system of conservation laws, Comm. Partial Differential Equations 21 (1996), no. 1-2, 1–21. MR 1373762, DOI https://doi.org/10.1080/03605309608821172
- Bruno Rubino, Weak solutions to quasilinear wave equations of Klein-Gordon or sine-Gordon type and relaxation to reaction-diffusion equations, NoDEA Nonlinear Differential Equations Appl. 4 (1997), no. 4, 439–457. MR 1485731, DOI https://doi.org/10.1007/s000300050024
- Denis Serre and Ling Xiao, Asymptotic behavior of large weak entropy solutions of the damped $P$-system, J. Partial Differential Equations 10 (1997), no. 4, 355–368. MR 1486716
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
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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