JournalsaihpcVol. 18, No. 5pp. 613–637

Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

  • Wojbor A Woyczyński

    Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, OH 44106-7054, USA

  • Piotr Biler

    Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

  • Grzegorz Karch

    Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
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Abstract

Nonlocal conservation laws of the form

where −L is the generator of a Lévy semigroup on L1(ℝn), are encountered in continuum mechanics as model equations with anomalous diffusion. They are generalizations of the classical Burgers equation. We study the critical case when the diffusion and nonlinear terms are balanced, e.g. L∼(−Δ)α/2, 1<α<2, f(s)∼s|s|r−1, r=1+(α−1)/n. The results include decay rates of solutions and their genuinely nonlinear asymptotic behavior as time t tends to infinity, determined by self-similar source solutions.

Cite this article

Wojbor A Woyczyński, Piotr Biler, Grzegorz Karch, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 5, pp. 613–637

DOI 10.1016/S0294-1449(01)00080-4