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Singular limits for non-isentropic compressible rotating fluids

Yajia Yu, Chenxi Su, Ming Lu

Abstract

In this article, we study the singular limit of non-isentropic compressible rotating fluids. We incorporate the capillary effect into both the $α=1$ and $α=0$ cases, and investigate the Navier-Stokes-Korteweg equations involving the terms of low Mach number, low Rossby number and high Reynolds number. When $α=1$, the dispersion estimate of the acoustic wave equation is derived by Rage's theorem. When $α=0$, we obtain the convergence results by error estimate. Moreover, we obtain that the three dimensions compressible Navier-Stokes-Korteweg equations converge to the two dimensions incompressible Euler equations.

Singular limits for non-isentropic compressible rotating fluids

Abstract

In this article, we study the singular limit of non-isentropic compressible rotating fluids. We incorporate the capillary effect into both the and cases, and investigate the Navier-Stokes-Korteweg equations involving the terms of low Mach number, low Rossby number and high Reynolds number. When , the dispersion estimate of the acoustic wave equation is derived by Rage's theorem. When , we obtain the convergence results by error estimate. Moreover, we obtain that the three dimensions compressible Navier-Stokes-Korteweg equations converge to the two dimensions incompressible Euler equations.
Paper Structure (8 sections, 10 theorems, 106 equations)

This paper contains 8 sections, 10 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Limit problem of capillary effect when $\alpha=1$
  4. Uniform bounds
  5. Acoustic analysis
  6. Identifying the limit system
  7. Limit problem of capillary effect when $\alpha=0$
  8. Error estimate

Key Result

Theorem 1

ou2023low (Uniform estimates for ill-prepared initial data) Let $\Omega\subset R^3$ be a domain with smooth boundary $\partial\Omega$ and $\varepsilon\in(0,1]$ is an arbitrary constant. Suppose that the initial data $(q^\varepsilon_0,\mathbf u^\varepsilon_0,\theta^\varepsilon_0)\in H^3(\Omega)$ sati

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7: RAGE theorem
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 1 more