Asymptotics for conservation laws involving Lévy diffusion generators (Q2773399)

The aim of the paper is to study the large time behavior of solutions of the Cauchy problem NEWLINE\[NEWLINE(1)\quad u_t+Lu+ \nabla\cdot f(u)=0,\qquad (2)\quad u(x,0)= u_0(x),NEWLINE\]NEWLINE where \(x\in\mathbb{R}^n\), \(t\geq 0\), \(u:\mathbb{R}^n \times \mathbb{R}^+ \to\mathbb{R}\), \(f:\mathbb{R} \to\mathbb{R}^n\) is a nonlinear term and \(u_0\) is in \(L^1 (\mathbb{R}^n)\). The pseudo differential operator \(-L\) is the generator of a symmetric, positivity-preserving semigroup \(e^{-tL}\), \(t>0\) on \(L^1(\mathbb{R}^n)\). Equations of the form (1) are called the Lévy conservation laws. Their prototype is the classical one-dimensional Burgers equation \(u_t-u_{xx}+ (u^2)_x =0\), being a model for physical phenomena where shock formation plays an important role.NEWLINENEWLINENEWLINEThe paper under review generalizes earlier results of the authors [Stud. Math. 135, 231-252 (1999; Zbl 0931.35015)] concerning the one-dimensional model equations called the multifractal conservation laws. It is organized as follows: Section 2 recalls needed facts from the theory of linear Lévy semigroups. In Section 3 the local-in-time solution of (1), (2) is obtained via the Banach contraction theorem, this solution is extended in the sequel to a global solution using suitable a priori estimate. The unicity and regularity of this solution is proved and its time decay is established. Sections 4 and 5 deal with two consecutive terms of the asymptotics of solutions of (1).NEWLINENEWLINENEWLINEThe results of this paper have been announced in [C. R. Acad Sci., Paris, Sér. I, Math. 330, 343-348 (2000; Zbl 0945.35015)].