×

zbMATH — the first resource for mathematics

Fractal Burgers equations. (English) Zbl 0911.35100
The theory of generalized Burgers-type equations with a fractional power of the Laplacian in the principal part and with a general algebraic nonlinearity is presented. Such equations naturally appear in continuum mechanics. The given results include existence, uniqueness, regularity, asymptotic behavior of solutions to the Cauchy problem and a construction of self-similar solutions. The role of critical fractal exponents is also discussed. The results presented in the paper are widely compared with those obtained by others authors.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
26A33 Fractional derivatives and integrals
35B40 Asymptotic behavior of solutions to PDEs
76F20 Dynamical systems approach to turbulence
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Avellaneda, M., Statistical properties of shocks in Burgers turbulence, Comm. math. phys., 172, 13-38, (1995) · Zbl 0844.35144
[3] Avrin, J.D., The generalized Burgers’ equation and the navier – stokes equation inrnwith singular initial data, Proc. amer. math. soc., 101, 29-40, (1987) · Zbl 0633.35038
[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, 237-256, (1979) · Zbl 0421.35037
[5] Bertini, L.; Cancrini, N.; Jona-Lasinio, G., The stochastic Burgers equation, Comm. math. phys., 165, 211-232, (1994) · Zbl 0807.60062
[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, 275-282, (1984) · Zbl 0561.35064
[7] Biler, P., The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia math., 114, 181-205, (1995) · Zbl 0829.35044
[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 · Zbl 0860.60038
[9] Burgers, J., The nonlinear diffusion equation, (1974), DordrechtAmsterdam · Zbl 0302.60048
[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, 241-263, (1990) · Zbl 0714.60048
[12] Dix, D.B., Nonuniqueness and uniqueness in the initial-value problem for Burgers’ equation, SIAM J. math. anal., 27, 708-724, (1996) · Zbl 0852.35123
[13] Frisch, U.; Lesieur, M.; Brissaud, A., Markovian random coupling model for turbulence, J. fluid mech., 65, 145-152, (1974) · Zbl 0285.76021
[14] Funaki, T.; Surgailis, D.; Woyczynski, W.A., Gibbs-Cox random fields and Burgers’ turbulence, Ann. appl. prob., 5, 701-735, (1995) · Zbl 0838.60016
[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 · Zbl 0860.76002
[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 · Zbl 0511.46029
[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, 119-141, (1994) · Zbl 0804.35158
[19] Kardar, M.; Parisi, G.; Zhang, Y.-C., Dynamic scaling of growing interfaces, Phys. rev. lett., 56, 889-892, (1986) · Zbl 1101.82329
[20] Komatsu, T., On the martingale problem for generators of stable processes with perturbations, Osaka J. math., 21, 113-132, (1984) · Zbl 0535.60063
[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, 959-1014, (1994) · Zbl 0803.35068
[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, 229-238, (1996) · Zbl 0866.35147
[24] Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[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, 209-226, (1995) · Zbl 0818.60046
[27] Molchanov, S.A.; Surgailis, D.; Woyczynski, W.A., Large-scale structure of the universe and the quasi-Voronoi tessellation structure of shock fronts in forced Burgers turbulence inrd, Ann. appl. prob., 7, 200-228, (1997) · Zbl 0895.60066
[28] Saichev, A.S.; Woyczynski, W.A., Advection of passive and reactive tracers in multidimensional Burgers’ velocity field, Physica D, 100, 119-141, (1997) · Zbl 0890.35122
[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 · Zbl 0880.46028
[30] Saut, J.-C., Sur quelques généralisations de l’équation de korteweg – de Vries, J. math. pures appl., 58, 21-61, (1979) · Zbl 0449.35083
[31] Schonbek, M.E., Decay of solutions to parabolic conservation laws, Comm. partial differential equations, 5, 449-473, (1980) · Zbl 0476.35012
[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, 601-621, (1992) · Zbl 0755.60105
[34] Smoller, J., Shock waves and reaction – diffusion equations, (1994), Springer-Verlag Berlin · Zbl 0807.35002
[35] Stroock, D.W., Diffusion processes associated with Lévy generators, Z. wahr. verw. geb., 32, 209-244, (1975) · Zbl 0292.60122
[36] Sugimoto, N.; Kakutani, T., Generalized Burgers equation for nonlinear viscoelastic waves, Wave motion, 7, 447-458, (1985) · Zbl 0588.73046
[37] Sugimoto, N., “generalized” Burgers equations and fractional calculus, (), 162-179 · Zbl 0674.35079
[38] Sugimoto, N., Burgers equation with a fractional derivative: hereditary effects on nonlinear acoustic waves, J. fluid mech., 225, 631-653, (1991) · Zbl 0721.76011
[39] Sugimoto, N., Propagation of nonlinear acoustic waves in a tunnel with an array of Helmholtz resonators, J. fluid mech., 244, 55-78, (1992) · Zbl 0760.76085
[40] Sznitman, A.S., A propagation of chaos result for Burgers’ equation, Probab. theory related fields, 71, 581-613, (1986) · Zbl 0597.60055
[41] Taylor, M., Analysis on Morrey spaces and applications to navier – stokes and other evolution equations, Comm. partial differential equations, 17, 1407-1456, (1992) · Zbl 0771.35047
[42] Triebel, H., Theory of function spaces, I and II, Monographs in mathematics, 78 and 84, (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, 325-356, (1994)
[44] Weissler, F.B., Local existence and nonexistence for semilinear parabolic equations inL^p, Indiana univ. math. J., 29, 79-102, (1980) · Zbl 0443.35034
[45] Zaslavsky, G.M., Fractional kinetic equations for Hamiltonian chaos, Physica D, 76, 110-122, (1994) · Zbl 1194.37163
[46] Zaslavsky, G.M.; Abdullaev, S.S., Scaling properties and anomalous transport of particles inside the stochastic layer, Phys. rev. E, 51, 3901-3910, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.