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

In this paper, we study a one dimensional nonlinear equation with diffusion $- u(-partial_{xx})^{frac{alpha}{2}}$ for $0leq alphaleq 2$ and $ u>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 $0leqalphaleq 2$. For subcritical $1<alphaleq 2$ 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 $1<alphaleq 2$. 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 $- u<infrho_0<0$, 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 $rho_0geq0$, the solution exists globally and converges to steady state.