[Paper Review] On the Well-Posedness of a Fractional Stokes-Transport System

Dimitri Cobb|arXiv (Cornell University)|Jan 25, 2023

Stability and Controllability of Differential Equations29 references4 citations

TL;DR

This paper establishes the well-posedness of a fractional Stokes-Transport system modeling sedimentation in a fluid with generalized viscosity governed by the fractional Laplacian $(-\Delta)^{\alpha/2}$, proving global weak solutions for $\alpha > 0$, local existence and uniqueness for $\alpha \geq 0$, global existence and uniqueness for $\alpha \geq 1$, and deriving a directional blow-up criterion and lower bound for lifespan when $0 \leq \alpha < 1$. The results provide a continuous framework bridging the classical Stokes ($\alpha=2$) and inviscid porous media ($\alpha=0$) equations.

ABSTRACT

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 $(- Δ)^{α/2}$, with $α= 2$ corresponding to the case of a normal viscous fluid, and $α= 0$ reducing the problem to the Inviscid Incompressible Porous Media equation. For each value of $α\in [0, d]$, we prove various results related to well-posedness in critical function spaces, such as the existence of global weak solutions (for $α&gt; 0$), local existence and uniqueness (for $α\geq 0$), global existence and uniqueness (for $α\geq 1$), as well as study the lifespan of local solutions (for $0 \leq α&lt; 1$). In particular, we show that gravity stratification leads to a directional blow-up criterion for local solutions (for $α\in [0, 1[$) and find a lower bound for the lifespan of solutions which depends on the value of the dissipation parameter $α\in [0, 1[$.

Motivation & Objective

• To analyze the existence, uniqueness, and lifespan of solutions to a fractional Stokes-Transport system with viscosity parameterized by $\alpha \in [0,d]$.
• To bridge the gap between the classical Stokes equation ($\alpha=2$) and the inviscid porous media equation ($\alpha=0$) by studying intermediate values of $\alpha$.
• To clarify the role of fractional dissipation in enabling well-posedness, particularly in the regime where $\alpha < 1$ where classical methods fail.
• To derive a directional blow-up criterion and a lower bound for the lifespan of local solutions when $0 \leq \alpha < 1$, under gravity stratification.

Proposed method

• The system is formulated as a coupled active scalar equation: $\partial_t \rho + u \cdot \nabla \rho = 0$, $(-\Delta)^{\alpha/2} u + \nabla \pi = \rho g$, $\text{div}(u) = 0$, with $g = e_d$.
• The velocity field is expressed via the Leray projection: $u = (-\Delta)^{-\alpha/2} \mathbb{P}(\rho g)$, reducing the system to a transport equation with a non-local velocity law.
• Critical function spaces, particularly homogeneous and inhomogeneous Besov spaces $B^{s}_{p,r}$ and $\dot{B}^{s}_{p,r}$, are used to analyze regularity and well-posedness.
• The transport equation is analyzed using paradifferential calculus and commutator estimates in the framework of Chemins' and Danchin's theory of Besov spaces.
• Lifespan estimates are derived using logarithmic interpolation and regularity loss control for $\log$-Lipschitz velocity fields.
• A directional blow-up criterion is established for $\alpha \in [0,1)$, showing that blow-up can only occur in the direction of gravity due to stratification.

Experimental results

Research questions

• RQ1 For which values of $\alpha \in [0,d]$ does the fractional Stokes-Transport system admit global weak solutions?
• RQ2 What is the lifespan of local solutions when $0 \leq \alpha < 1$, and can a lower bound be established?
• RQ3 Does gravity stratification lead to a directional blow-up criterion in the case $\alpha \in [0,1)$?
• RQ4 How does the regularity of the velocity field affect the lifespan and uniqueness of solutions in the $\alpha < 1$ regime?
• RQ5 Can the well-posedness theory for $\alpha \in (0,1)$ be extended to global existence and uniqueness, and what regularity thresholds are required?

Key findings

• Global weak solutions exist for all $\alpha > 0$ in critical Besov spaces, extending known results for the classical Stokes system.
• Local existence and uniqueness of solutions hold for all $\alpha \geq 0$, with solutions in $C^0_T(B^s_{p,r})$ or $C^0_w(T; B^s_{p,r})$ depending on the regularity index.
• Global existence and uniqueness are established for $\alpha \geq 1$, indicating that sufficient dissipation regularizes the system.
• For $0 \leq \alpha < 1$, a directional blow-up criterion is proven: blow-up can only occur in the direction of gravity, due to the stratified nature of the forcing.
• A lower bound for the lifespan of local solutions is derived, which depends explicitly on the dissipation parameter $\alpha \in [0,1)$, showing that higher $\alpha$ leads to longer-lived solutions.
• The solution norm exhibits controlled regularity loss when the velocity field is $\log$-Lipschitz, with exponential dependence on the $LL^q_\alpha$-norm of the velocity gradient.