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 $0\leq \alpha\leq 2$ 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 $0\leq\alpha\leq 2$. For subcritical $1<\alpha\leq 2$ 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 $1<\alpha\leq 2$. 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 $\rho_0\geq0$, the solution exists globally and converges to steady state.