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Low Mach number limit on perforated domains for the evolutionary Navier-Stokes-Fourier system

Danica Basarić, Nilasis Chaudhuri

Abstract

We consider the Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat-conducting fluid on a domain perforated by tiny holes. First, we identify a class of dissipative solutions to the Oberbeck-Boussinesq approximation as a low Mach number limit of the primitive system. Secondly, by proving the weak-strong uniqueness principle, we obtain strong convergence to the target system on the lifespan of the strong solution.

Low Mach number limit on perforated domains for the evolutionary Navier-Stokes-Fourier system

Abstract

We consider the Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat-conducting fluid on a domain perforated by tiny holes. First, we identify a class of dissipative solutions to the Oberbeck-Boussinesq approximation as a low Mach number limit of the primitive system. Secondly, by proving the weak-strong uniqueness principle, we obtain strong convergence to the target system on the lifespan of the strong solution.
Paper Structure (23 sections, 12 theorems, 171 equations, 2 figures)

This paper contains 23 sections, 12 theorems, 171 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Primitive system
  3. Perforated domain
  4. Constitutive relations
  5. Well-prepared initial data
  6. Target system
  7. Notation
  8. Concepts of solution and main result
  9. Weak solution
  10. Dissipative solution
  11. Main result
  12. Preparation
  13. Extension of functions
  14. Essential and residual parts
  15. Uniform bounds
  16. ...and 8 more sections

Key Result

Theorem 2.6

Let Moreover, let Then there exists a positive time $T^*$ such that, passing to suitable subsequences as the case may be, where $[\textbf{U}, \Theta]$ is the strong solution to the Oberbeck-Boussinesq system emanating from $[\textbf{U}_0, \Theta_0]= [\textbf{u}_0, \vartheta_{0}^{(1)}]$, with $\textbf{u}_0, \vartheta_{0}^{(1)}$ the weak limits appearing in i2, convergence initial temperatures, r

Figures (2)

  • Figure 1: An example of perforated domain
  • Figure 2: Perforated domains with $\varepsilon_1 > \varepsilon_2$

Theorems & Definitions (29)

  • Remark 1.1
  • Definition 2.1: Weak solution of the Navier--Stokes--Fourier system on perforated domains
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Dissipative solution of the Oberbeck-Boussinesq system
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 3.1
  • ...and 19 more