Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 18, Issue 5, 2001, Pages 613-637

  Biler, Piotr ^a   Karch, Grzegorz ^a   Woyczyński, Wojbor A ^b  

^a UNIVERSITY OF WROC AW   (Poland)

^b CASE WESTERN RESERVE UNIVERSITY   (United States)

Author keywords

Conservation laws; Generalized Burgers equation; Levy diffusion; Self similar asymptotics of solutions; Source solutions

Indexed keywords

EID: 0012120698     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0294-1449(01)00080-4     Document Type: Article

References (28)

1. Bardos C., Penel P., Frisch U., Sulem P.L., Modified dissipativity for a nonlinear evolution equation arising in turbulence, Arch. Rat. Mech. Anal. 71 (1979) 237-256.
2. Bertoin J., Lévy Processes, Cambridge University Press, 1996.
3. Biler P., Funaki T., Woyczynski W.A., Fractal Burgers equations, J. Differential Equations 148 (1998) 9-46.
4. Biler P., Funaki T., Woyczyński W.A., Interacting particle approximation for nonlocal quadratic evolution problems, Prob. Math. Stat. 19 (1999) 267-286.
5. Biler P., Karch G., Woyczynski W.A., Asymptotics for multifractal conservation laws, Studia Math. 135 (1999) 231-252.
6. Biler P., Karch G., Woyczyński W.A., Asymptotics for conservation laws involving Levy diffusion generators, preprint.
7. Biler P., Karch G., Woyczyński W.A., Multifractal and Levy conservation laws, C. R. Acad. Sci. Paris, Sér. I Math. 330 (2000) 343-348.
8. Biler P., Woyczyński W.A., Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math. 59 (1999) 845-869.
9. Carpio A., Asymptotic behavior for the vorticity equations in dimensions two and three, Comm. PDF 19 (1994) 827-872.
10. Carpio A., Large time behaviour in some convection-diffusion equations, Ann. Sc. Norm. Sup. Pisa, ser. IV, 23 (1996) 551-574.
11. Davies E.B., Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
12. Duoandikoetxea J., Zuazua E., Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris, Sér. I Math. 315 (1992) 693-698.
13. N with asymptotically constant diffusion, Comm. PDE 24 (1999) 1283-1340.
14. Escobedo M., Vázquez J.L., Zuazua E., Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rat. Mech. Anal. 124 (1993) 43-65.
15. Escobedo M., Vázquez J.L., Zuazua E., A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J. 42 (1993) 1413-1440.
16. N, J. Funct. Anal. 100 (1991) 119-161.
17. N, Asymptotic Analysis 20 (1999) 133-173.
18. Jacob N., Pseudo-Differential Operators and Markov Processes, Akademie Verlag, Berlin, 1996.
19. Karch G., Self-similar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Analysis 35 (1999) 199-219.
20. Komatsu T., Uniform estimates for fundamental solutions associated with non-local Dirichlet forms, Osaka J. Math. 32 (1995) 833-860.
21. Ladyženskaja O.A., Solonnikov V.A., Ural'ceva N.N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1988.
22. Mann J.A., Jr., Woyczynski W.A., Growing fractal interfaces in the presence of self-similar hopping surface diffusion, Physica A 291 (2001) 159-183.
23. Pȩkalski A., Sznajd-Weron K. (Eds.), Anomalous Diffusion. From Basics to Applications, Lecture Notes in Physics, Vol. 519, Springer-Verlag, Berlin, 1999.
24. Shlesinger M.F., Zaslavsky G.M., Frisch U. (Eds.), Levy Flights and Related Topics in Physics, Lecture Notes in Physics, Vol. 450, Springer-Verlag, Berlin, 1995.
25. p (0, T; B), Annali Mat. Pura Appl. 156 (1987) 65-96.
26. Stroock D.W., Diffusion processes associated with Levy generators, Z. Wahr. verw. Geb. 32 (1975) 209-244.
27. Woyczynski W.A., Burgers-KPZ Turbulence - Göttingen Lectures, Lecture Notes in Math., Vol. 1700, Springer-Verlag, Berlin, 1998.
28. Zuazua E., Weakly nonlinear large time behavior in scalar convection-diffusion equations, Differential Integral Equations 6 (1993) 1481-1491.