L1 and L∞ intermediate asymptotics for scalar conservation laws

Asymptotic Analysis

Volumn 41, Issue 3-4, 2005, Pages 189-213

  Dolbeault, Jean ^a   Escobedo, Miguel ^b  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITY OF THE BASQUE COUNTRY UPV EHU   (Spain)

Author keywords

Asymptotics; Entropy; Graph convergence; N waves; Rankine Hugoniot condition; Scalar conservation laws; Self similar solutions; Shocks; Time dependent rescaling; Uniform convergence; Weighted L1 norm

Indexed keywords

ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; ENTROPY; GRAPH THEORY; SET THEORY; THEOREM PROVING;

ASYMPTOTICS; GRAPH CONVERGENCE; N-WAVES; RANKINE-HUGONIOT CONDITION; SCALAR CONSERVATION LAWS; SELF-SIMILAR SOLUTIONS; SHOCKS; TIME-DEPENDENT RESCALING; UNIFORM CONVERGENCE; WEIGHTED L1 NORM;

CONVERGENCE OF NUMERICAL METHODS;

EID: 14844347570     PISSN: 09217134     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (17)

1. A. Arnold, P. Markovich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations 26 (2001), 43-100.
2. M. Del Pino and J. Dolbeault, Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the p-Laplacian, C. R. Acad. Sci. Paris Sér. I Math. 334(5) (2002), 349-440.
3. M. Del Pino and J. Dolbeault, Asymptotic behaviour of nonlinear diffusions, Math. Res. Lett. 10(4) (2003), 551-557.
4. p-Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl. 293(2) (2004), 375-388.
5. J. Dolbeault and G. Rein, Time-dependent rescalings and Lyapunov functionals for the Vlasov-Poisson and Euler-Poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics, Math. Models Methods Appl. Sci. 11(3) (2001), 407-432.
6. M. Escobedo, J. Vazquez and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124(1) (1993), 43-65.
7. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998.
8. S. Kamin, The asymptotic behaviour of the solutions of the filtration equation, Israel J. Math 14 (1973), 76-87.
9. Y.-J. Kim, Asymptotic behaviour of solutions to scalar conservation laws and optimal convergence to N-waves, J. Differential Equations 192(1) (2003), 202-224.
10. S.N. Kruzkov, The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the first order with several variables, Soviet Math. Dokl. 5 (1964), 493-496; Translation from Dokl. Akad. Nauk SSSR 155 (1964), 743-746.
11. S.N. Kruzkov, The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the first order with several variables, Soviet Math. Dokl. 5 (1964), 493-496; Translation from Dokl. Akad. Nauk SSSR 155 (1964), 743-746.
12. P.D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10 (1957), 537-566.
13. T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), 419-441.
14. T.-P. Liu and T. Yang, A new entropy functional for a scalar conservation law, Commun. Pure Appl. Math. 52(11) (1999), 1427-1442.
15. D. Serre, Systems of Conservation Laws - 1. Hyperbolicity, Entropies, Shock Waves - 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems, Cambridge Univ. Press, 1999 and 2000.
16. J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 258, Springer-Verlag, New York, 1994.
17. E. Tadmor and T. Tassa, On the piecewise smoothness of entropy solutions to scalar conservation laws, Comm. Partial Differential Equations 18(9-10) (1993), 1631-1652.