On zero mass solutions of viscous conservation laws

Communications in Partial Differential Equations

Volumn 27, Issue 9-10, 2002, Pages 2071-2100

  Karch, Grzegorz ^a   Schonbek, Maria Elena ^b  

^a UNIVERSITY OF WROC AW   (Poland)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Cauchy problem; Convection diffusion equation; Large time behavior of solutions; Self similar solutions

Indexed keywords

EID: 0036908891     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1081/PDE-120016137     Document Type: Article

References (37)

1. Aguirre, J.; Escobedo, M.; Zuazua, E. Self-Similar Solutions of a Convection Diffusion Equation and Related Semilinear Problems. Comm. Partial Differential Equations 1990, 15, 139-157.
2. Biler, P.; Karch, G.; Woyczyński, W.A. Multifractal and Lévy Conservation Laws. C.R. Acad. Sci. Paris Ser. I 2000, 330, 343-348.
3. Biler, P.; Karch, G.; Woyczyński, W.A. Asymptotics for Conservation Laws Involving Lévy Diffusion Generators. Studia Mathematica 2001, 148, 171-192.
4. Biler, P.; Karch, G.; Woyczyński, W.A. Critical Nonlinearity Exponent and Self-Similar Asymptotics for Lévy Conservation Laws. Ann. Inst. H. Poincaré, Anal. non Linéaire 2001, 18, 613-637.
5. Cannone, M. Ondelettes, Paraproduits et Navier-Stokes; Diderot Editeur, Arts et Sciences: Amsterdam, Paris, New York, 1995.
6. Cannone, M. A Generalization of a Theorem by Kato on Navier-Stokes Equations. Rev. Mat. Iberoamericana 1997, 13, 515-541.
7. 3. Comm. Partial Differential Equations 1996, 21, 179-194.
8. Carlen, E.A.; Loss, M. Optimal Smoothing and Decay Estimates for Viscously Damped Conservation Laws, with Applications to the 2-D Navier-Stokes Equation. Duke Math. J. 1996, 86, 135-157.
9. I.-Liang Chern; Tai-Ping Liu. Convergence to Diffusion Waves of Solutions for Viscous Conservation Laws. Comm. Math. Phys. 1987, 110, 503-517.
10. Duoandikoetxea, J.; Zuazua, E. Moments, Masses de Dirac et Décomposition de Fonctions. C.R. Acad. Sci. Paris Sér. I 1992, 315, 693-698.
11. N with Asymptotically Constant Diffusion. Comm. Partial Differential Equations 1999, 24, 1283-1340.
12. N with Periodic Coefficients. J. Differential Equations 2000, 167, 275-315.
13. N. J. Funct. Anal. 1991, 100, 119-161.
14. Escobedo, M.; Vázquez, J.L.; Zuazua, E. Asymptotic Behavior and Source-Type Solutions for a Diffusion-Convection Equation. Arch. Rat. Mech. Anal. 1993, 124, 43-65.
15. Escobedo, M.; Vázquez, J.L.; Zuazua, E. A Diffusion Convection Equation in Several Space Dimensions. Indiana Univ. Math. J. 1993, 42, 1413-1440.
16. 1-Stability of Constant States of Degenerate Convection-Diffusion Equations. Asymptot. Anal. 1999, 19, 267-288.
17. 1 Stability of Shock Waves in Scalar Viscous Conservation Laws. Comm. Pure Appl. Math. 1998, 51, 291-301.
18. Hoff, D.; Zumbrun, K. Multidimensional Diffusion Waves for the Navier-Stokes Equations of Compressible Flow. Indiana Univ. Math. J. 1995, 44, 603-676.
19. xx. Comm. Pure Appl. Math. 1950, 3, 201-230.
20. Jeffrey, A.; Zhao, H. Global Existence and Optimal Temporal Decay Estimates for Systems of Parabolic Conservation Laws. II. The Multidimensional Case. J. Math. Anal. Appl. 1998, 217, 597-623.
21. Karch, G. Scaling in Nonlinear Parabolic Equations. J. Math. Anal. Appl. 1999, 234, 534-558.
22. Karch, G. Self-Similar Large Time Behavior of Solutions to Kortewegde Vries-Burgers Equation. Nonlinear Analysis 1999, 35, 199-219.
23. Karch, G. Large-Time Behavior of Solutions to Non-Linear Wave Equations: Higher-Order Asymptotics. Math. Methods Appl. Sci. 1999, 22, 1671-1697.
24. Kawashima, S. Large-Time Behavior of Solutions to Hyperbolic-Parabolic Systems of Conservation Laws and Applications. Proc. Royal Soc. Edinburgh 1987, 106A, 169-194.
25. Mei, M. Large-Time Behavior of Solution for Generalized Benjamin-Bona-Mahony-Burgers Equations. J. Differential Equations 1999, 158, 314-340.
26. N. Funkc. Ekv. 1998, 41, 383-434.
27. Miyakawa, T. Hardy Spaces of Solenoidal Vector Fields, with Applications to the Navier-Stokes Equations. Kyushu J. Math. 1996, 50, 1-64.
28. n. Ann. Inst. H. Poincaré, Analyse non Linéaire 1996, 13, 319-336.
29. Schonbek, M.E. Decay of Solutions to Parabolic Conservation Laws. Comm. Partial Differential Equations 1980, 5, 449-473.
30. Schonbek, M.E. Uniform Decay Rates for Parabolic Conservation Laws. Nonlinear Anal. T. M. A. 1986, 10, 943-956.
31. Schonbek, M.E. Large Time Behavior of Solutions to the Navier-Stokes Equations. Comm. Partial Differential Equations 1986, 11, 733-763.
32. Schonbek, M.E.; Süli, E. Decay of Total Variation and Hardy Norms of Solutions to Parabolic Conservation Laws. Nonlinear Anal. 2001, 45, 515-528.
33. 1 Pour les lois de Conservation Scalaires Visqueuses. C. R. Acad. Sci. Paris Sér. I Math. 1996, 323, 359-363.
34. Stein, E.M. Singular Integrals and Differentiability Properties of Functions; Princeton University Press: Princeton, New Jersey, 1970.
35. Stein, E.M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals; Princeton University Press: Princeton, New Jersey, 1993.
36. N. J. London Math. Soc. 1987, 35, 303 -313.
37. Zuazua, E. Weakly Nonlinear Large Time Behavior in Scalar Convection-Diffusion Equations. Differential Integral Equations 1993, 6, 1481-1491.