Remarks On Regularity Of Non-Constant Vortex Patches

In this note, we show that if the initial vorticity ! 0 is a C ff(\Omega 0 ) non-constant patch, i.e., ! 0 = $ 0 د\Omega 0 with $ 0 2 C ff and @\Omega 0 2 C 1+ff ; then the weak solution ! of 2D vorticity equation will remain in the form ! = $د\Omega t for all t ? 0; with $ 2 C ff and\Omega t = \Phi (\Omega 0 ; t) 2 C 1+ff ; where \Phi (x; t) is the particle trajectory that also belongs to C 1+ff(\Omega 0 ). 1 Introduction The vortex motion of an inviscid incompressible flow in R 2 is described by the vorticity equation ! t + u \Delta r! = 0; (1.1) where ! = ! (x; t) for x 2 R 2 ; t ? 0; is the vorticity, u is the fluid velocity field that can be recovered from ! through the Biot-Savart law: u (x; t) = Z R 3 K (x \Gamma y) ! (y; t) dx; (1.2) 1 Key words. Incompressible flow, Euler system, vorticity equation, vortex patches, Newtonian potentials, global solutions. 2 AMS subject classification. 35L45, 35Q35, 76 where K (x) = 1 2ß jxj 2 (\Gammax 2; ..