tailieunhanh - Đề tài " Formation of singularities for a transport equation with nonlocal velocity "

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time. 1. Introduction In this paper we study the nature of the solutions to the following class of equations () | Annals of Mathematics Formation of singularities for a transport equation with nonlocal velocity By Antonio C ordoha Diego C ordoha and Marco A. Fontelos Annals of Mathematics 162 2005 1377-1389 Formation of singularities for a transport equation with nonlocal velocity By Antonio Córdoba Diego Córdoba and Marco a. Fontelos Abstract We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time. 1. Introduction In this paper we study the nature of the solutions to the following class of equations ỡt - HỠ 0x -VẢaỡ x G R where H0 is the Hilbert transform defined by H0 1PV Ị y dy n J x - y V is a real positive number 0 a 2 and Ảa0 A 120. This equation represents the simplest case of a transport equation with a nonlocal velocity and with a viscous term involving powers of the laplacian. It is well known that the equivalent equation with a local velocity v 0 known as Burgers equation may develop shock-type singularities in finite time when V 0 whereas the solutions remain smooth at all times if V 0 and a 2. Therefore a natural question to pose is whether the solutions to become singular in finite time or not depending on a and V. In fact this question has been previously considered in the literature motivated by the strong analogy with some important equations appearing in fluid mechanics such as the 3D Euler incompressible vorticity equation and the Birkhoff-Rott equation modelling the evolution of a vortex sheet where a crucial mathematical difficulty Partially supported by BFM2002-02269 grant. Partially supported by BFM2002-02042 grant. Partially supported by BFM2002-02042 grant. 1378 ANTONIO CORDOBA DIEGO CORDOBA AND MARCO A. FONTELOS lies in the nonlocality of the velocity. Since the fundamental problem concerning both 3D Euler and Birkhoff-Rott equations is the .