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 $-ν(-\partial_{xx})^{\fracα{2}}$ for $0\leq α\leq 2$ and $ν>0$. 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 $0\leqα\leq 2$. For subcritical $1<α\leq 2$ and critical case $α=1$, 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 $1<α\leq 2$. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case $α=1$, if the initial data $ρ_0$ satisfies $-ν<\infρ_0<0$, 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 $ρ_0\geq0$, the solution exists globally and converges to steady state.