Fractal Burgers Equations

Journal of Differential Equations

Volumn 148, Issue 1, 1998, Pages 9-46

  Biler, Piotr ^a   Funaki, Tadahisa ^b   Woyczynski, Wojbor A ^c  

^a UNIVERSITY OF WROC AW   (Poland)

^b UNIVERSITY OF TOKYO   (Japan)

^c CASE WESTERN RESERVE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002015456     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1998.3458     Document Type: Article

References (46)

1. Adams R. A. Sobolev Spaces. 1975;Academic Press, New York.
2. Avellaneda M. Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys. 172:1995;13-38.
3. nwith singular initial data. Proc. Amer. Math. Soc. 101:1987;29-40.
4. 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.
5. Bertini L., Cancrini N., Jona-Lasinio G. The stochastic Burgers equation. Comm. Math. Phys. 165:1994;211-232.
6. Biler P. Asymptotic behaviour in time of solutions to some equations generalizing the Korteweg-de Vries-Burgers equation. Bull. Pol. Acad. Sci., Mathematics. 32:1984;275-282.
7. Biler P. The Cauchy problem and self-similar solutions for a nonlinear parabolic equation. Studia Math. 114:1995;181-205.
8. M. Bossy, D. Talay, Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation, Ann. Appl. Prob. 6, 1996, 818, 861.
9. Burgers J. The Nonlinear Diffusion Equation. 1974;DordrechtAmsterdam.
10. Cannone M. Ondelettes, paraproduits et Navier-Stokes. 1995;Diderot Editeur; Arts et Sciences, Paris.
11. Dawson D. A., Gorostiza L. G. Generalized solutions of a class of nuclear-space-valued stochastic evolution equations. Appl. Math. Optim. 22:1990;241-263.
12. Dix D. B. Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation. SIAM J. Math. Anal. 27:1996;708-724.
13. Frisch U., Lesieur M., Brissaud A. Markovian random coupling model for turbulence. J. Fluid Mech. 65:1974;145-152.
14. Funaki T., Surgailis D., Woyczynski W. A. Gibbs-Cox random fields and Burgers' turbulence. Ann. Appl. Prob. 5:1995;701-735.
15. T. Funaki, W. A. Woyczynski, Interacting particle approximation for fractal Burgers equation, Stochastic Processes and Related Topics, A Volume in Memory of Stamatis Cambanis, Birkhäuser-Boston, 1998.
16. Gurbatov S., Malakhov A., Saichev A. Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles. 1991;Univ. Press, Manchester.
17. Henry D. B. How to remember the Sobolev inequalities. Differential Equations, Saotilde; Paulo 1981. Lecture Notes in Mathematics. 957:1982;Springer, Berlin. p. 97-109.
18. Holden H., Lindström T., Øksendal B., Ubøe J., Zhang T.-S. The Burgers equation with noisy force and the stochastic heat equation. Comm. Partial Differential Equations. 19:1994;119-141.
19. Kardar M., Parisi G., Zhang Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56:1986;889-892.
20. Komatsu T. On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21:1984;113-132.
21. Kozono H., Yamazaki M. Semilinear heat equation and Navier-Stokes equation with distributions in new functional spaces as initial data. Comm. Partial Differential Equations. 19:1994;959-1014.
22. Ladyženskaja O. A., Solonnikov V. A., Ural'ceva N. N. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc. 1988;Providence.
23. Leonenko N. N., Parkhomenko V. N., Woyczynski W. A. Spectral properties of the scaling limit solutions of the Burgers equation with singular data. Random Operators and Stochastic Eq. 4:1996;229-238.
24. Lions J.-L. Quelques méthodes de résolution des problèmes aux limites non linéaires. 1969;Dunod, Paris.
25. Mikhlin S., Prössdorf S. Singular Integral Operators. 1986;Springer-Verlag, Berlin.
26. Molchanov S. A., Surgailis D., Woyczynski W. A. Hyperbolic asymptotics in Burgers' turbulence and extremal processes. Comm. Math. Phys. 168:1995;209-226.
27. d. Ann. Appl. Prob. 7:1997;200-228.
28. Saichev A. S., Woyczynski W. A. Advection of passive and reactive tracers in multidimensional Burgers' velocity field. Physica D. 100:1997;119-141.
29. Saichev A. S., Woyczynski W. A. Distributions in the Physical and Engineering Sciences. Distributional and Fractal Calculus, Integral Transforms and Wavelets. 1997;Birkhäuser, Boston.
30. Saut J.-C. Sur quelques généralisations de l'équation de Korteweg-de Vries. J. Math. pures appl. 58:1979;21-61.
31. Schonbek M. E. Decay of solutions to parabolic conservation laws. Comm. Partial Differential Equations. 5:1980;449-473.
32. Shlesinger M. F., Zaslavsky G. M., Frisch U. Lévy Flights and Related Topics in Physics. Lecture Notes in Physics. 450:1995;Springer-Verlag, Berlin.
33. Sinai Ya. G. Statistics of shocks in solutions of inviscid Burgers' equation. Comm. Math. Phys. 148:1992;601-621.
34. Smoller J. Shock Waves and Reaction-Diffusion Equations. 1994;Springer-Verlag, Berlin.
35. Stroock D. W. Diffusion processes associated with Lévy generators. Z. Wahr. Verw. Geb. 32:1975;209-244.
36. Sugimoto N., Kakutani T. Generalized Burgers equation for nonlinear viscoelastic waves. Wave Motion. 7:1985;447-458.
37. Sugimoto N. "Generalized" Burgers equations and fractional calculus. Jeffrey A. Nonlinear Wave Motion. 1989;162-179 Longman Scientific, Harlow.
38. Sugimoto N. Burgers equation with a fractional derivative: Hereditary effects on nonlinear acoustic waves. J. Fluid Mech. 225:1991;631-653.
39. Sugimoto N. Propagation of nonlinear acoustic waves in a tunnel with an array of Helmholtz resonators. J. Fluid Mech. 244:1992;55-78.
40. Sznitman A. S. A propagation of chaos result for Burgers' equation. Probab. Theory Related Fields. 71:1986;581-613.
41. Taylor M. Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. Partial Differential Equations. 17:1992;1407-1456.
42. Triebel H. Theory of Function Spaces, I and II. Monographs in Mathematics. 78 and 84:1983 and 1992;Birkhäuser, Basel.
43. Vergassola M., Dubrulle B. D., Frisch U., Nullez A. Burgers equation, devil's staircases and mass distribution for the large-scale structure. Astroy. and Astrophys. 289:1994;325-356.
44. p. Indiana Univ. Math. J. 29:1980;79-102.
45. Zaslavsky G. M. Fractional kinetic equations for Hamiltonian chaos. Physica D. 76:1994;110-122.
46. Zaslavsky G. M., Abdullaev S. S. Scaling properties and anomalous transport of particles inside the stochastic layer. Phys. Rev. E. 51:1995;3901-3910.