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 ( − ∂ x x ) α 2 for 0 ≤ α ≤ 2 and ν > 0. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space L 1 ( R ) ∩ H 1 / 2 ( R ) when 0 ≤ α ≤ 2. For the subcritical case 1 < α ≤ 2 and critical case α = 1, we obtain the global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrapping method to improve the regularity of mild solutions in the Bessel potential spaces for the subcritical case 1 < α ≤ 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 ρ0 satisfies −ν < inf ρ0 < 0, we use the method of characteristics for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If ρ0 ≥ 0, the solution exists globally and converges to steady state.