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Low Mach number limit of the compressible Navier-Stokes system for large initial date with critical regularity on the torus

Sai Li

Abstract

We study the low Mach number limit of the compressible Navier-Stokes equations on the torus. For large initial data with critical regularity, we prove that solutions to the compressible Navier-Stokes system exist as long as the corresponding solutions to the incompressible Navier-Stokes system exist, provided that the Mach number is sufficiently small. Furthermore, we establish the convergence of solutions of the compressible system to those of the incompressible system as the Mach number tends to zero. Our approach combines high-medium-low frequency analysis of density and velocity with the solution filtering technique via acoustic wave groups. This work provides an affirmative answer to the problem posed by Danchin [Amer.J.Math.,124(2002),1153-1219]:"Does convergence hold for large data with critical regularity?"

Low Mach number limit of the compressible Navier-Stokes system for large initial date with critical regularity on the torus

Abstract

We study the low Mach number limit of the compressible Navier-Stokes equations on the torus. For large initial data with critical regularity, we prove that solutions to the compressible Navier-Stokes system exist as long as the corresponding solutions to the incompressible Navier-Stokes system exist, provided that the Mach number is sufficiently small. Furthermore, we establish the convergence of solutions of the compressible system to those of the incompressible system as the Mach number tends to zero. Our approach combines high-medium-low frequency analysis of density and velocity with the solution filtering technique via acoustic wave groups. This work provides an affirmative answer to the problem posed by Danchin [Amer.J.Math.,124(2002),1153-1219]:"Does convergence hold for large data with critical regularity?"
Paper Structure (13 sections, 13 theorems, 292 equations)

This paper contains 13 sections, 13 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Notations and the main theorem
  3. Notations
  4. The main theorem
  5. Proof strategy and difficulties
  6. Full demonstration of Theorem \ref{['main']}
  7. The decay properties of ${\boldsymbol V}^\varepsilon-{\boldsymbol V}$ and $\mathbb{P}{\boldsymbol u}^\varepsilon-\mathbf{v}$
  8. The proof of Proposition \ref{['pro1']}
  9. The proof of Proposition \ref{['pro2']}
  10. The relationship between $Y^{\zeta,\varepsilon}(T)$ and $X^\varepsilon(T)+P^\varepsilon(T)$, as well as estimates for $[a^\varepsilon,{\boldsymbol u}^\varepsilon]^{\zeta,\varepsilon}_{h,m}(T)$
  11. The proof of Proposition \ref{['pro3']}
  12. The proof of Proposition \ref{['pro4']}
  13. Appendix

Key Result

Theorem 2.1

Let $0<T_0\leq\infty$, $0<\vartheta<\frac{1}{2}$. Consider the initial data $(a_0,{\boldsymbol u}_0)$ satisfy cond1, with external force $f\in L^1_{T_0}(B^{\frac{d}{2}-1}_{2,1}(\mathbb{T}_b^d))$ such that Assume system equ3 with initial data $\mathbf{v}_0=\mathbb{P}{\boldsymbol u}_0$ and forcing term $h=\mathbb{P}f$ admits a solution $\mathbf{v}\in \left(\widetilde{C}_{T_0}(B^{\frac{d}{2}-1}_{2,1

Theorems & Definitions (17)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Lemma 5.1
  • ...and 7 more