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## Studia Mathematica

1999 | 135 | 3 | 231-252
Tytuł artykułu

### Asymptotics for multifractal conservation laws

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
231-252
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-09-24
Twórcy
autor
• Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
autor
• Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
autor
• Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, Ohio 44106-7054 U.S.A.
Bibliografia
• [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.
• [2] P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998), 9-46.
• [3] P. Biler, Interacting particle approximation for nonlocal quadratic evolution problems, submitted.
• [4] P. Biler and W. A. Woyczynski, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., to appear.
• [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.
• [6] J. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974.
• [7] A. Carpio, Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal. 27 (1996), 449-475.
• [8] B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693.
• [9] D. B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal. 27 (1996), 708-724.
• [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.
• [11] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $ℝ^N$, J. Funct. Anal. 100 (1991), 119-161.
• [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.
• [13] G. Karch, Large-time behavior of solutions to nonlinear wave equations: higher-order asymptotics, submitted.
• [14] M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892.
• [15] T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math. 21 (1984), 113-132.
• [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 $ℝ^d$, Ann. Appl. Probab. 7 (1997), 200-228.
• [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.
• [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.
• [21] A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997), 753-764.
• [22] J.-C. Saut, Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. Pures Appl. 58 (1979), 21-61.
• [23] M. E. Schonbek, Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations 5 (1980), 449-473.
• [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.
• [26] D. W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrsch. Verw. Gebiete 32 (1975), 209-244.
• [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.
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