On the Well-Posedness of a Fractional Stokes-Transport System

The purpose of this paper is to study the existence, uniqueness and lifespan of solutions for a fractional Stokes-Transport system. This problem should be understood as a model for sedimentation in a fluid where the viscosity law is given by a fractional Lapalce operator $(- \Delta)^{\alpha/2}$, with $\alpha = 2$ corresponding to the case of a normal viscous fluid, and $\alpha = 0$ reducing the problem to the Inviscid Incompressible Porous Media equation. For each value of $\alpha \in [0, d]$, we prove various results related to well-posedness in critical function spaces, such as the existence of global weak solutions (for $\alpha > 0$), local existence and uniqueness (for $\alpha \geq 0$), global existence and uniqueness (for $\alpha \geq 1$), as well as study the lifespan of local solutions (for $0 \leq \alpha < 1$). In particular, we show that gravity stratification leads to a directional blow-up criterion for local solutions (for $\alpha \in [0, 1[$) and find a lower bound for the lifespan of solutions which depends on the value of the dissipation parameter $\alpha \in [0, 1[$.