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Low Mach number limit of the full Navier-Stokes equations

T. Alazard

TL;DR

The paper studies the low Mach number limit for the full Navier-Stokes equations, incorporating non-isentropic effects and thermal conduction, and allows large temperature variations and general initial data. Using a fluctuation-based change of variables $p = (1/\varepsilon) \log(P/\bar P)$ and $\theta = \log(\mathcal{T}/\bar{\mathcal{T}})$ along with a frequency-localized energy method, the authors establish uniform local-in-time well-posedness with bounds independent of $\varepsilon$, $\mu$, $\kappa$. As $\varepsilon \to 0$, they prove convergence to a limit mixed hyperbolic/parabolic system, with $\varepsilon^{-1}(P^\varepsilon - \bar P) \to 0$ locally in $L^2$ and $(v^\varepsilon, \mathcal{T}^\varepsilon) \to (v, \theta)$ solving the limit equations; the proof relies on acoustic energy decay and a Metivier–Schochet framework. The analysis combines high- and low-frequency Fourier techniques (including Littlewood-Paley localization) and $L^2$ energy estimates for the linearized system, together with a non-isotropic slow-fast decomposition to justify the limit and connect to Lions' formal computations.

Abstract

The low Mach number limit for classical solutions to the full Navier Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are accounted. In particular we consider general initial data. The equations leads to a singular problem, depending on a small scaling parameter, whose linearized is not uniformly well-posed. Yet, it is proved that the solutions exist and are uniformly bounded for a time interval which is independent of the Mach number Ma in (0,1], the Reynolds number Re in [1,+\infty] and the Peclet number Pe in [1,+\infty]. Based on uniform estimates in Sobolev spaces, and using a Theorem of G. Metivier and S. Schochet, we next prove that the large terms converge locally strongly to zero. It allows us to rigorously justify the well-known formal computations described in the introduction of the book of P.-L. Lions.

Low Mach number limit of the full Navier-Stokes equations

TL;DR

The paper studies the low Mach number limit for the full Navier-Stokes equations, incorporating non-isentropic effects and thermal conduction, and allows large temperature variations and general initial data. Using a fluctuation-based change of variables and along with a frequency-localized energy method, the authors establish uniform local-in-time well-posedness with bounds independent of , , . As , they prove convergence to a limit mixed hyperbolic/parabolic system, with locally in and solving the limit equations; the proof relies on acoustic energy decay and a Metivier–Schochet framework. The analysis combines high- and low-frequency Fourier techniques (including Littlewood-Paley localization) and energy estimates for the linearized system, together with a non-isotropic slow-fast decomposition to justify the limit and connect to Lions' formal computations.

Abstract

The low Mach number limit for classical solutions to the full Navier Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are accounted. In particular we consider general initial data. The equations leads to a singular problem, depending on a small scaling parameter, whose linearized is not uniformly well-posed. Yet, it is proved that the solutions exist and are uniformly bounded for a time interval which is independent of the Mach number Ma in (0,1], the Reynolds number Re in [1,+\infty] and the Peclet number Pe in [1,+\infty]. Based on uniform estimates in Sobolev spaces, and using a Theorem of G. Metivier and S. Schochet, we next prove that the large terms converge locally strongly to zero. It allows us to rigorously justify the well-known formal computations described in the introduction of the book of P.-L. Lions.

Paper Structure

This paper contains 36 sections, 45 theorems, 399 equations.

Table of Contents

  1. Introduction
  2. Setting the problem up
  3. Uniform stability
  4. The low Mach number limit
  5. Main results
  6. Structural assumptions
  7. Uniform stability result
  8. Convergence toward the solution of the limit system
  9. Changes of variables
  10. Changes of variables
  11. Localization in the frequency space
  12. Preliminaries
  13. A product estimate
  14. A Friedrichs' Lemma
  15. Energy estimates for the linearized system
  16. ...and 21 more sections

Key Result

Theorem 1.2

Let d\geqslant 1 and \mathbb{D} denotes either the whole space \mathbb{R}^{d} or the torus \mathbb{T}^{d}. Consider an integer s>1+d/2. For all positive \underline{P}, \underline{\mathcal{T}} and M_{0}, there is a positive time T such that for all a=(\varepsilon,\mu,\kappa)\in A and all initial data the Cauchy problem for system:ANSF--system:CI has a unique solution (P^{a},v^{a},\mathcal{T}^{a}) s

Theorems & Definitions (101)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 91 more