Nonlocal quadratic evolution problems

Piotr Biler; Wojbor Woyczyński

Banach Center Publications (2000)

  • Volume: 52, Issue: 1, page 11-24
  • ISSN: 0137-6934

Abstract

top
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.

How to cite

top

Biler, Piotr, and Woyczyński, Wojbor. "Nonlocal quadratic evolution problems." Banach Center Publications 52.1 (2000): 11-24. <http://eudml.org/doc/209050>.

@article{Biler2000,
abstract = {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.},
author = {Biler, Piotr, Woyczyński, Wojbor},
journal = {Banach Center Publications},
keywords = {self-similar solutions; fractal anomalous diffusion; asymptotic behavior of solutions; nonlinear nonlocal parabolic equations; evolution of density of mutually interacting particles; fractional power of the Laplacian; global in time existence versus finite time blow-up; Monte Carlo approximation schemes},
language = {eng},
number = {1},
pages = {11-24},
title = {Nonlocal quadratic evolution problems},
url = {http://eudml.org/doc/209050},
volume = {52},
year = {2000},
}

TY - JOUR
AU - Biler, Piotr
AU - Woyczyński, Wojbor
TI - Nonlocal quadratic evolution problems
JO - Banach Center Publications
PY - 2000
VL - 52
IS - 1
SP - 11
EP - 24
AB - 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.
LA - eng
KW - self-similar solutions; fractal anomalous diffusion; asymptotic behavior of solutions; nonlinear nonlocal parabolic equations; evolution of density of mutually interacting particles; fractional power of the Laplacian; global in time existence versus finite time blow-up; Monte Carlo approximation schemes
UR - http://eudml.org/doc/209050
ER -

References

top
  1. [BPFS] C. Bardos, P. Penel, U. Frisch and P. L. Sulem, Modified dissipativity for a non-linear evolution equation arising in turbulence, Arch. Rat. Mech. Anal. 71 (1979), 237-256. Zbl0421.35037
  2. [B1] P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles, III, Colloq. Math. 68 (1995), 229-239. Zbl0836.35076
  3. [B2] P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181-205. Zbl0829.35044
  4. [B3] P. Biler, Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743. Zbl0913.35021
  5. [BFW1] P. Biler, T. Funaki and W. A. Woyczyński, Fractal Burgers equations, J. Diff. Eq. 148 (1998), 9-46. Zbl0911.35100
  6. [BFW2] P. Biler, T. Funaki and W. A. Woyczyński, Interacting particle approximation for nonlocal quadratic evolution problems, Probab. Math. Statist. 19 (1999), 321-340. Zbl0985.60091
  7. [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. Zbl0814.35054
  8. [BN] P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math. 66 (1993), 131-145. Zbl0818.35046
  9. [BW] P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math. 59 (1999), 845-869. Zbl0940.35035
  10. [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. Zbl0860.60038
  11. [C] M. Cannone, Ondelettes, paraproduits et Navier-Stokes, Diderot Editeur; Arts et Sciences, Paris, 1995. 
  12. [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. Zbl0931.60087
  13. [GMO] Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rat. Mech. Anal. 104 (1988), 223-250. Zbl0666.76052
  14. [G] J. Goodman, Convergence of the random vortex method, Comm. Pure Appl. Math. 40 (1987), 189-220. Zbl0635.35077
  15. [GK] E. Gutkin and M. Kac, Propagation of chaos and the Burgers equation, SIAM J. Appl. Math. 43 (1983), 971-980. Zbl0554.35104
  16. [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. Zbl0909.35071
  17. [KO] S. Kotani and H. Osada, Propagation of chaos for Burgers' equation, J. Math. Soc. Japan 37 (1985), 275-294. Zbl0603.60072
  18. [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. 
  19. [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. 
  20. [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. 
  21. [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. 
  22. [W] W. A. Woyczyński, Burgers-KPZ Turbulence, Göttingen Lectures, Lecture Notes in Math. 1700, Springer-Verlag, Berlin, 1998. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.