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Effective velocity and $L^\infty$-based well-posedness for incompressible fluids with odd viscosity

Francesco Fanelli, Alexis F. Vasseur

Abstract

The present paper is concerned with the well-posedness theory for non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. Differently from previous works, we consider here the full odd viscosity tensor. Similarly to the work of Bresch and Desjardins in compressible fluid mechanics, we identify the presence of an effective velocity in the system, linking the velocity field of the fluid and the gradient of a suitable function of the density. By use of this effective velocity, we propose a new formulation of the original system of equations, thus highlighting a strong similarity with the equations of the ideal magnetohydrodynamics. By taking advantage of the new formulation of the equations, we establish a local in time well-posedness theory in Besov spaces based on $L^\infty$ and prove a lower bound for the lifespan of the solutions implying ``asymptotically global'' existence: in the regime of small initial density variations, $ρ_0-1= O(\varepsilon)$ for small $\varepsilon>0$, the corresponding solution is defined up to some time $T_\varepsilon>0$ satisfying the property $T_\varepsilon\,\longrightarrow\,+\infty$ when $\varepsilon\to0^+$.

Effective velocity and $L^\infty$-based well-posedness for incompressible fluids with odd viscosity

Abstract

The present paper is concerned with the well-posedness theory for non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. Differently from previous works, we consider here the full odd viscosity tensor. Similarly to the work of Bresch and Desjardins in compressible fluid mechanics, we identify the presence of an effective velocity in the system, linking the velocity field of the fluid and the gradient of a suitable function of the density. By use of this effective velocity, we propose a new formulation of the original system of equations, thus highlighting a strong similarity with the equations of the ideal magnetohydrodynamics. By taking advantage of the new formulation of the equations, we establish a local in time well-posedness theory in Besov spaces based on and prove a lower bound for the lifespan of the solutions implying ``asymptotically global'' existence: in the regime of small initial density variations, for small , the corresponding solution is defined up to some time satisfying the property when .
Paper Structure (28 sections, 13 theorems, 191 equations)

This paper contains 28 sections, 13 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. The system of equations
  3. Previous results
  4. Contents of the paper
  5. Well-posedness for the full system in Sobolev spaces
  6. Reformulation and well-posedness in endpoint spaces
  7. Role of the effective velocity
  8. BD-entropy structure
  9. Effective transport à la Brenner
  10. The final reformulated system
  11. Well-posedness in a $L^\infty$-based functional framework
  12. A priori estimates
  13. Propagation of $L^p$ norms
  14. High frequency analysis
  15. Estimates of the non-linear terms
  16. ...and 13 more sections

Key Result

Theorem 2.1

Let $s>2$. Take an initial datum $(\rho_0,u_0)\in L^\infty(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2)$ and assume that In addition, assume that $\rho_0-1\in H^{s+1}(\mathbb{R}^2)$. Then, there exists $0<T=T(\rho_0,u_0)\leq +\infty$ and a unique solution $(\rho,u,\nabla \Pi)$ to system eq:odd on $[0,T]\times\mathbb{R}^2$ such that:

Theorems & Definitions (19)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Proposition 5.3
  • ...and 9 more