Asymptotics for multifractal conservation laws

Piotr Biler; Grzegorz Karch; Wojbor Woyczynski

Studia Mathematica (1999)

  • Volume: 135, Issue: 3, page 231-252
  • ISSN: 0039-3223

Abstract

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We study asymptotic behavior of solutions to multifractal Burgers-type equation u t + f ( u ) x = A u , where the operator A is a linear combination of fractional powers of the second derivative - ∂ 2 / ∂ x 2 and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the L p -norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.

How to cite

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Biler, Piotr, Karch, Grzegorz, and Woyczynski, Wojbor. "Asymptotics for multifractal conservation laws." Studia Mathematica 135.3 (1999): 231-252. <http://eudml.org/doc/216653>.

@article{Biler1999,
abstract = {We study asymptotic behavior of solutions to multifractal Burgers-type equation $u_t + f(u)_x = Au$, where the operator A is a linear combination of fractional powers of the second derivative $-∂^2/ ∂ x^2$ and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the $L^p$-norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.},
author = {Biler, Piotr, Karch, Grzegorz, Woyczynski, Wojbor},
journal = {Studia Mathematica},
keywords = {generalized Burgers equation; fractal diffusion; asymptotics of solutions; decay rates of the -norms; multifractal Burgers-type equation},
language = {eng},
number = {3},
pages = {231-252},
title = {Asymptotics for multifractal conservation laws},
url = {http://eudml.org/doc/216653},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Biler, Piotr
AU - Karch, Grzegorz
AU - Woyczynski, Wojbor
TI - Asymptotics for multifractal conservation laws
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 3
SP - 231
EP - 252
AB - We study asymptotic behavior of solutions to multifractal Burgers-type equation $u_t + f(u)_x = Au$, where the operator A is a linear combination of fractional powers of the second derivative $-∂^2/ ∂ x^2$ and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the $L^p$-norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.
LA - eng
KW - generalized Burgers equation; fractal diffusion; asymptotics of solutions; decay rates of the -norms; multifractal Burgers-type equation
UR - http://eudml.org/doc/216653
ER -

References

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  1. [1] C. Bardos, P. Penel, U. Frisch and P. L. Sulem, Modified dissipativity for a nonlinear evolution equation arising in turbulence, Arch. Rational Mech. Anal. 71 (1979), 237-256. Zbl0421.35037
  2. [2] P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998), 9-46. Zbl0911.35100
  3. [3] P. Biler, Interacting particle approximation for nonlocal quadratic evolution problems, submitted. Zbl0985.60091
  4. [4] P. Biler and W. A. Woyczynski, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., to appear. Zbl0940.35035
  5. [5] J. L. Bona, K. S. Promislow and C. E. Wayne, Higher-order asymptotics of decaying solutions of some nonlinear, dispersive, dissipative wave equations, Nonlinearity 8 (1995), 1179-1206. Zbl0837.35130
  6. [6] J. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974. Zbl0302.60048
  7. [7] A. Carpio, Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal. 27 (1996), 449-475. Zbl0845.76019
  8. [8] B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693. Zbl0762.35110
  9. [9] D. B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal. 27 (1996), 708-724. Zbl0852.35123
  10. [10] J. Duoandikoetxea et E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I 315 (1992), 693-698. Zbl0755.45019
  11. [11] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in ℝ N , J. Funct. Anal. 100 (1991), 119-161. Zbl0762.35011
  12. [12] T. Funaki and W. A. Woyczynski, Interacting particle approximation for fractal Burgers equation, in: Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis 1943-1995, I. Karatzas, B. S. Rajput and M. S. Taqqu (eds.), Birkhäuser, Boston, 1998, 141-166. Zbl0931.60087
  13. [13] G. Karch, Large-time behavior of solutions to nonlinear wave equations: higher-order asymptotics, submitted. Zbl0947.35141
  14. [14] M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892. Zbl1101.82329
  15. [15] T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math. 21 (1984), 113-132. Zbl0535.60063
  16. [16] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1988. 
  17. [17] J. A. Mann, Jr. and W. A. Woyczynski, Rough surfaces generated by nonlinear transport, invited paper, Symposium on Non-linear Diffusion, TMS International Meeting, September 1997. 
  18. [18] S. A. Molchanov, D. Surgailis and W. A. Woyczynski, The large-scale structure of the Universe and quasi-Voronoi tessellation of shock fronts in forced Burgers turbulence in ℝ d , Ann. Appl. Probab. 7 (1997), 200-228. Zbl0895.60066
  19. [19] P. I. Naumkin and I. A. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves, Transl. Math. Monographs 133, Amer. Math. Soc., Providence, 1994. Zbl0802.35002
  20. [20] A. I. Saichev and W. A. Woyczynski, Distributions in the Physical and Engineering Sciences, Vol. 1, Distributional and Fractal Calculus, Integral Transforms and Wavelets, Birkhäuser, Boston, 1997. Zbl0880.46028
  21. [21] A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997), 753-764. Zbl0933.37029
  22. [22] J.-C. Saut, Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. Pures Appl. 58 (1979), 21-61. Zbl0449.35083
  23. [23] M. E. Schonbek, Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations 5 (1980), 449-473. Zbl0476.35012
  24. [24] M. F. Shlesinger, G. M. Zaslavsky and U. Frisch (eds.), Lévy Flights and Related Topics in Physics, Lecture Notes in Phys. 450, Springer, Berlin, 1995. 
  25. [25] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer, Berlin, 1994. Zbl0807.35002
  26. [26] D. W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrsch. Verw. Gebiete 32 (1975), 209-244. Zbl0292.60122
  27. [27] N. Sugimoto, "Generalized" Burgers equations and fractional calculus, in: Nonlinear Wave Motion, A. Jeffrey (ed.), Longman Sci., Harlow, 1989, 162-179. 
  28. [28] W. A. Woyczynski, Computing with Brownian and Lévy α-stable path integrals, in: 9th 'Aha Huliko'a Hawaiian Winter Workshop "Monte Carlo Simulations in Oceanography" (Hawaii, 1997), P. Müller and D. Henderson (eds.), SOEST, 1997, 91-100. 
  29. [29] W. A. Woyczynski, Burgers-KPZ Turbulence-Göttingen Lectures, Lecture Notes in Math. 1700, Springer, Berlin, 1998. 
  30. [30] E. Zuazua, Weakly nonlinear large time behavior in scalar convection-diffusion equations, Differential Integral Equations 6 (1993), 1481-1491. Zbl0805.35054

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