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 \(\alpha \in [0, 1[\).