Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak Solutions for a non-Newtonian Stokes-Transport System

Dimitri Cobb, Geoffrey Lacour

Abstract

In this article, we study a non-Newtonian Stokes-Transport system. This set of PDEs was introduced as a model for describing the behavior of a cloud of particles in suspension in a Stokes fluid, and is a nonlinear coupling between a hyperbolic equation (Transport) and a nonlinear elliptic equation (non-Newtonian Stokes), and as such can be considered as an active scalar equation. We prove the existence of global weak solutions with initial data in critical Lebesgue spaces. In order to overcome the difficulties introduced by the highly nonlinear aspect of this problem, we resort to a combination of DiPerna-Lions theory of transport equations and Minty's trick for elliptic equations.

Weak Solutions for a non-Newtonian Stokes-Transport System

Abstract

In this article, we study a non-Newtonian Stokes-Transport system. This set of PDEs was introduced as a model for describing the behavior of a cloud of particles in suspension in a Stokes fluid, and is a nonlinear coupling between a hyperbolic equation (Transport) and a nonlinear elliptic equation (non-Newtonian Stokes), and as such can be considered as an active scalar equation. We prove the existence of global weak solutions with initial data in critical Lebesgue spaces. In order to overcome the difficulties introduced by the highly nonlinear aspect of this problem, we resort to a combination of DiPerna-Lions theory of transport equations and Minty's trick for elliptic equations.
Paper Structure (21 sections, 21 theorems, 183 equations)

This paper contains 21 sections, 21 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Physics of the Problem
  3. Overview of Previous Results on Related Systems
  4. Stokes-Transport Equations
  5. Active Scalar Equations
  6. Non-Newtonian Fluids
  7. Main Result
  8. Principal Difficulties in the Problem
  9. Statement of the Main Result
  10. Overview of the Proof
  11. Definition of Weak Solutions
  12. A Priori Estimates
  13. Well-Posedness for the Stokes System
  14. Continuity of the Inverse Stokes Map
  15. Approximate Solutions
  16. ...and 6 more sections

Key Result

Theorem 2

We work in dimension $d \geq 2$. Consider $p \in ]1, + \infty[$ and a function $\nu \in C^{0,\overline{\gamma}}(\mathbb{R} \setminus \{ 0 \}) \cap L^\infty(\mathbb{R})$ such that $\nu(|r|) \geq \nu_* |r|^\gamma$ for all $|r| \leq 1$ and some fixed constants $\nu_*,\gamma > 0$, and setting $\overline Then, for any initial datum $\rho_0 \in L^q(\mathbb{T}^d)$ such that $1/\rho_0 \in L^\sigma(\mathbb

Theorems & Definitions (38)

  • Remark 1
  • Theorem 2
  • Definition 3
  • Proposition 4
  • Remark 5
  • Remark 6
  • proof : Proof (of Proposition \ref{['p:APriori']})
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 28 more