# Asymptotics for multifractal conservation laws

Piotr Biler; Grzegorz Karch; Wojbor Woyczynski

Studia Mathematica (1999)

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

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topBiler, 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

top- [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] P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998), 9-46. Zbl0911.35100
- [3] P. Biler, Interacting particle approximation for nonlocal quadratic evolution problems, submitted. Zbl0985.60091
- [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] 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] J. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974. Zbl0302.60048
- [7] A. Carpio, Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal. 27 (1996), 449-475. Zbl0845.76019
- [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] 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] 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] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in ${\mathbb{R}}^{N}$, J. Funct. Anal. 100 (1991), 119-161. Zbl0762.35011
- [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] G. Karch, Large-time behavior of solutions to nonlinear wave equations: higher-order asymptotics, submitted. Zbl0947.35141
- [14] M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892. Zbl1101.82329
- [15] T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math. 21 (1984), 113-132. Zbl0535.60063
- [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] 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] 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 ${\mathbb{R}}^{d}$, Ann. Appl. Probab. 7 (1997), 200-228. Zbl0895.60066
- [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] 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] A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997), 753-764. Zbl0933.37029
- [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] M. E. Schonbek, Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations 5 (1980), 449-473. Zbl0476.35012
- [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] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer, Berlin, 1994. Zbl0807.35002
- [26] D. W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrsch. Verw. Gebiete 32 (1975), 209-244. Zbl0292.60122
- [27] N. Sugimoto, "Generalized" Burgers equations and fractional calculus, in: Nonlinear Wave Motion, A. Jeffrey (ed.), Longman Sci., Harlow, 1989, 162-179.
- [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] W. A. Woyczynski, Burgers-KPZ Turbulence-Göttingen Lectures, Lecture Notes in Math. 1700, Springer, Berlin, 1998.
- [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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