Global well-posedness to the two-dimensional incompressible vorticity equation in the half plane

Abstract

This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity $\omega_0\in W^{k,p}(\R^{2}_+)$ with $k\geq3$ and $1<p<2$, it is shown that the two-dimensional incompressible vorticity equation admits a unique solution $\omega\in C([0,T];W^{k,p}(\R^{2}_+))$ for any $T>0$. An elementary and self-contained proof is presented and delicate estimates of the velocity and its derivatives are obtained in this paper. It should be emphasized that the uniform estimate on $\int^t_0\|u(\tau)\|_{W^{1,\infty}(\R^2_+)}d\tau$ is required to complete the global regularity of the solution. To do that, the double exponential growth in time of the gradient of the vorticity in the half plane is established and applied. This is different from the proof of global well-posedness of the Euler velocity equations in the Sobolev spaces, in which a Kato-type or logarithmic-type estimate of the gradient of the velocity is enough to close the energy estimates.