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Inhomogenous Navier--Stokes equations with unbounded density

Jean-Paul Adogbo, Piotr B. Mucha, Maja Szlenk

Abstract

In the current state of the art regarding the Navier--Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins' inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a $C^1$ velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from \cite{DanMu}, we conclude the uniqueness of the solution.

Inhomogenous Navier--Stokes equations with unbounded density

Abstract

In the current state of the art regarding the Navier--Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins' inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from \cite{DanMu}, we conclude the uniqueness of the solution.

Paper Structure

This paper contains 8 sections, 10 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. The Results
  3. Sobolev regularity
  4. Weighted estimates
  5. Shift of Integrability
  6. Existence and uniqueness of solutions
  7. Appendix A
  8. Properties of the space $\mathcal{L}$ and $\mathcal{Y}_0$.

Key Result

Theorem 2.1

Let $\Omega$ be a $C^2$ bounded subset of $\mathbb{R}^2$ or the torus $\mathbb{T}^2$. Assume that the initial data $(\rho_0,v_0)$ satisfy (see (def-Lclass)) Then there exists a unique solution $(\rho, v, \nabla P)$ to system (INS) with data $(\rho_0, v_0)$ fulfilling the conservation of momentum (conser:momen) (in the case $\Omega=\mathbb{T}^2$), the conservation of mass conser:mass, the balance

Theorems & Definitions (27)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 3.1
  • Remark 3.2
  • proof : Proof of Proposition \ref{['pro:sobo:est']}
  • Lemma 3.3
  • proof
  • ...and 17 more