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2000 | 52 | 1 | 11-24
Tytuł artykułu

Nonlocal quadratic evolution problems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered. The inertial type nonlinearity is quadratic and nonlocal while the diffusive term, also nonlocal, is anomalous and fractal, i.e., represented by a fractional power of the Laplacian. Conditions for global in time existence versus finite time blow-up are studied. Self-similar solutions are constructed for certain homogeneous initial data. Monte Carlo approximation schemes by interacting particle systems are also mentioned.
Rocznik
Tom
52
Numer
1
Strony
11-24
Opis fizyczny
Daty
wydano
2000
Twórcy
autor
  • Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • Department of Statistics and Center for Stochastic and Chaotic Processes, in Science and Technology, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A.
Bibliografia
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  • [B1] P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles, III, Colloq. Math. 68 (1995), 229-239.
  • [B2] P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181-205.
  • [B3] P. Biler, Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743.
  • [BFW1] P. Biler, T. Funaki and W. A. Woyczyński, Fractal Burgers equations, J. Diff. Eq. 148 (1998), 9-46.
  • [BFW2] P. Biler, T. Funaki and W. A. Woyczyński, Interacting particle approximation for nonlocal quadratic evolution problems, Probab. Math. Statist. 19 (1999), 321-340.
  • [BHN] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis T.M.A. 23 (1994), 1189-1209.
  • [BN] P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math. 66 (1993), 131-145.
  • [BW] P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math. 59 (1999), 845-869.
  • [BT] M. Bossy and 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.
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  • [FW] T. Funaki and W. A. Woyczyński, Interacting particle approximation for fractal Burgers equation, in: Stochastic Processes and related topics, A volume in memory of S. Cambanis, I. Karatzas, B. Rajput and M. Taqqu, Eds., Birkhäuser, Boston 1998, 141-166.
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  • [HMV] M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity 10 (1997), 1739-1754.
  • [KO] S. Kotani and H. Osada, Propagation of chaos for Burgers' equation, J. Math. Soc. Japan 37 (1985), 275-294.
  • [McK] H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, in: Stochastic differential equations VII, Lecture Series in Differential Equations, Catholic University, Washington D.C., 1967, 177-194.
  • [O] H. Osada, Propagation of chaos for two dimensional Navier-Stokes equation, Proc. Japan Ac. 62A (1986), 8-11, and Taniguchi Symp. PMMP, Katata, 1985, 303-334.
  • [SZF] M. F. Shlesinger, G. M. Zaslavsky and U. Frisch, Eds., Lévy Flights and Related Topics in Physics, Lecture Notes in Phys. 450, Springer-Verlag, Berlin, 1995.
  • [Sz] A. S. Sznitman, Topics in propagation of chaos, in: École d'été de St. Flour, XIX - 1989, Lecture Notes in Math. 1464, Springer-Verlag, Berlin, 1991, 166-251.
  • [W] W. A. Woyczyński, Burgers-KPZ Turbulence, Göttingen Lectures, Lecture Notes in Math. 1700, Springer-Verlag, Berlin, 1998.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv52z1p11bwm
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