Asymptotic behaviour for a partially diffusive relaxation system
Authors:
Miguel Escobedo and Philippe Laurençot
Journal:
Quart. Appl. Math. 61 (2003), 495-512
MSC:
Primary 35K55; Secondary 35B40, 35L60
DOI:
https://doi.org/10.1090/qam/1999834
MathSciNet review:
MR1999834
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The time asymptotics of nonnegative and integrable solutions to a partially diffusive relaxation system is investigated. Under suitable assumptions on the relaxation term, the convergence to a self-similar source type solution, either of the heat equation or of the viscous Burgers equation, is proved. The proof relies on optimal decay rates and classical scaling arguments.
- Haïm Brézis and Avner Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9) 62 (1983), no. 1, 73–97. MR 700049
- I-Liang Chern, Long-time effect of relaxation for hyperbolic conservation laws, Comm. Math. Phys. 172 (1995), no. 1, 39–55. MR 1346371
- Michael G. Crandall and Luc Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), no. 3, 385–390. MR 553381, DOI https://doi.org/10.1090/S0002-9939-1980-0553381-X
- Miguel Escobedo, Juan Luis Vázquez, and Enrike Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124 (1993), no. 1, 43–65. MR 1233647, DOI https://doi.org/10.1007/BF00392203
- Miguel Escobedo and Enrike Zuazua, Large time behavior for convection-diffusion equations in ${\bf R}^N$, J. Funct. Anal. 100 (1991), no. 1, 119–161. MR 1124296, DOI https://doi.org/10.1016/0022-1236%2891%2990105-E
- R. E. Grundy, C. J. van Duijn, and C. N. Dawson, Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media: the fast reaction case, Quart. J. Mech. Appl. Math. 47 (1994), no. 1, 69–106. MR 1261838, DOI https://doi.org/10.1093/qjmam/47.1.69
- Markos A. Katsoulakis and Athanasios E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22 (1997), no. 1-2, 195–233. MR 1434144, DOI https://doi.org/10.1080/03605309708821261
- V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
- A. Kurganov and E. Tadmor, Stiff systems of hyperbolic conservation laws: convergence and error estimates, SIAM J. Math. Anal. 28 (1997), no. 6, 1446–1456. MR 1474223, DOI https://doi.org/10.1137/S0036141096301488
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
- Ph. Laurençot and F. Simondon, Long-time behaviour for porous medium equations with convection, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 2, 315–336. MR 1621331, DOI https://doi.org/10.1017/S0308210500012816
- Hailiang Liu and Roberto Natalini, Long-time diffusive behavior of solutions to a hyperbolic relaxation system, Asymptot. Anal. 25 (2001), no. 1, 21–38. MR 1814988
- Tai-Ping Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), no. 1, 153–175. MR 872145
- Tai-Ping Liu and Michel Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), no. 3, 419–441. MR 735207, DOI https://doi.org/10.1016/0022-0396%2884%2990096-2
- Roberto Natalini, Recent results on hyperbolic relaxation problems, Analysis of systems of conservation laws (Aachen, 1997) Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 99, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 128–198. MR 1679940
- G. Reyes, Asymptotic behaviour in convection-diffusion processes, Nonlinear Anal. 37 (1999), no. 3, Ser. A: Theory Methods, 301–318. MR 1694393, DOI https://doi.org/10.1016/S0362-546X%2898%2900048-0
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- Aslak Tveito and Ragnar Winther, On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28 (1997), no. 1, 136–161. MR 1427731, DOI https://doi.org/10.1137/S0036141094263755
- Juan Luis Vázquez, Asymptotic behaviour of nonlinear parabolic equations. Anomalous exponents, Degenerate diffusions (Minneapolis, MN, 1991) IMA Vol. Math. Appl., vol. 47, Springer, New York, 1993, pp. 215–228. MR 1246350, DOI https://doi.org/10.1007/978-1-4612-0885-3_15
- Enrique Zuazua, Weakly nonlinear large time behavior in scalar convection-diffusion equations, Differential Integral Equations 6 (1993), no. 6, 1481–1491. MR 1235206
H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pures Appl. 62 (1983), 73–97.
I.-L. Chern, Long-time effect of relaxation for hyperbolic conservation laws, Comm. Math. Phys. 172 (1995), 39–55.
M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), 385–390.
M. Escobedo, J. L. Vazquez and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124 (1993), 43–65.
M. Escobedo and E. Zuazua, Large time behaviour for convection-diffusion equations in ${\mathbb {R}^{n}}$, J. Funct. Anal. 100 (1991), 119–161.
R. E. Grundy, C. J. van Duijn, and C. N. Dawson, Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media: The fast reaction case, Q. Jl Mech. Appl. Math. 47 (1994), 69–106.
M. A. Katsoulakis and A. E. Tzavaras, Contractive Relaxation Systems and the Scalar Multi-dimensional Conservation Law, Comm. P.D.E. 22 (1997), 195–233.
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM Res. Appl. Math. 36, Masson/Wiley, Paris/Chichester, 1994.
A. Kurganov and E. Tadmor, Stiff systems of hyperbolic conservation laws: Convergence and error estimates, SIAM J. Math. Anal. 28 (1997), 1446–1456.
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, Amer. Math. Soc., Providence, 1968.
Ph. Laurençot and F. Simondon, Long-time behaviour for porous medium equations with convection, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 315–336.
H. Liu and R. Natalini, Long-time diffusive behaviour of solutions to a hyperbolic relaxation system, Asymptot. Anal. 25 (2001), 21–38.
T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153–175.
T.-P. Liu and M. Pierre, Source-solutions and asymptotic behaviour in conservation laws, J. Differential Equations 51 (1984), 419–441.
R. Natalini, Recent results on hyperbolic relaxation problems, in “Analysis of Systems of Conservation Laws,” Chapman & Hall/CRC, Boca Raton, 1999, pp. 128–198.
G. Reyes, Asymptotic behaviour in convection-diffusion processes, Nonlinear Anal. 37 (1999), 301–318.
J. Simon, Compact sets in the space ${L^p}\left ( 0, T; B \right )$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.
A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of conservation laws including a relaxation term, SIAM J. Math. Anal. 28 (1997), 136–161.
J. L. Vazquez, Asymptotic behaviour of nonlinear parabolic equations. Anomalous exponents, in “Degenerate Diffusions,” W. M. Ni, L. A. Peletier & J. L. Vazquez (eds.), IMA Vol. Math. Appl. 47, Springer, New York, 1993, pp. 215–228.
E. Zuazua, Weakly nonlinear large time behavior in scalar convection-diffusion equations, Differential Integral Equations 6 (1993), 1481–1491.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35K55,
35B40,
35L60
Retrieve articles in all journals
with MSC:
35K55,
35B40,
35L60
Additional Information
Article copyright:
© Copyright 2003
American Mathematical Society
