Global existence and spatial analyticity for a nonlocal flux with fractional diffusion

Abstract

In this paper, we study a one dimensional nonlinear equation with diffusion $-\nu (-{\partial }_{xx}{)}^{\frac{\alpha }{2}}$ for $0\le \alpha \le 2$ and $\nu >0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(\mathbb{R})\cap H^{1/2}(\mathbb{R})$ when $0\le \alpha \le 2$. For subcritical $1<\alpha \le 2$ and critical case $\alpha =1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<\alpha \le 2$. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case $\alpha =1$, if the initial data ${\rho }_0$ satisfies $-\nu <\inf {\rho }_0<0$, we use the characteristics methods for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If ${\rho }_0\ge 0$, the solution exists globally and converges to steady state.

Replace of previous version arXiv:2008.08860v1