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Incompressible and fast rotation limits for 3D compressible rotating Euler system with general initial data

Mikihiro Fujii, Yang Li, Pengcheng Mu

TL;DR

This work analyzes the simultaneous low Mach and fast rotation limits for the 3D compressible rotating Euler equations in the whole space with ill-prepared data. By introducing an intermediate 2D system and employing Strichartz-type dispersion for fast waves, it proves local well-posedness with uniform estimates and separates dynamics into slow and fast components. The slow part is shown to converge to a 2D quasi-geostrophic type system, while the fast part decays at a rate $\delta^{1/q}$, thereby justifying the 2D QG limit in this setting. A by-product result establishes a rigorous link from the 2D inviscid rotating shallow water equations to the 2D QG equations. The findings address open questions (e.g., Ngo and Scrobogna) and provide a framework for singular limits in geophysical fluid models with mixed-dimensional data, combining intermediate systems, spectral decompositions, and dispersive estimates.

Abstract

This paper is concerned with the low Mach and Rossby number limits of $3$D compressible rotating Euler equations with ill-prepared initial data in the whole space. More precisely, the initial data is the sum of a $3$D part and a $2$D part. With the help of a suitable intermediate system, we perform this singular limit rigorously with the target system being a $2$D QG-type. This particularly gives an affirmative answer to the question raised by Ngo and Scrobogna [\emph{Discrete Contin. Dyn. Syst.}, 38 (2018), pp. 749-789]. As a by-product, our proof gives a rigorous justification from the $2$D inviscid rotating shallow water equations to the $2$D QG equations in whole space.

Incompressible and fast rotation limits for 3D compressible rotating Euler system with general initial data

TL;DR

This work analyzes the simultaneous low Mach and fast rotation limits for the 3D compressible rotating Euler equations in the whole space with ill-prepared data. By introducing an intermediate 2D system and employing Strichartz-type dispersion for fast waves, it proves local well-posedness with uniform estimates and separates dynamics into slow and fast components. The slow part is shown to converge to a 2D quasi-geostrophic type system, while the fast part decays at a rate , thereby justifying the 2D QG limit in this setting. A by-product result establishes a rigorous link from the 2D inviscid rotating shallow water equations to the 2D QG equations. The findings address open questions (e.g., Ngo and Scrobogna) and provide a framework for singular limits in geophysical fluid models with mixed-dimensional data, combining intermediate systems, spectral decompositions, and dispersive estimates.

Abstract

This paper is concerned with the low Mach and Rossby number limits of D compressible rotating Euler equations with ill-prepared initial data in the whole space. More precisely, the initial data is the sum of a D part and a D part. With the help of a suitable intermediate system, we perform this singular limit rigorously with the target system being a D QG-type. This particularly gives an affirmative answer to the question raised by Ngo and Scrobogna [\emph{Discrete Contin. Dyn. Syst.}, 38 (2018), pp. 749-789]. As a by-product, our proof gives a rigorous justification from the D inviscid rotating shallow water equations to the D QG equations in whole space.

Paper Structure

This paper contains 9 sections, 8 theorems, 90 equations.

Table of Contents

  1. Introduction and main results
  2. Background and motivation
  3. Main results
  4. Local well-posedness and dispersion of the 3D perturbed system
  5. Local well-posedness
  6. Strichartz estimates
  7. QG limit of 2D compressible rotating Euler equations
  8. Strichartz estimates for the fast wave
  9. Convergence of the slow part

Key Result

Lemma 1.1

Let $(b_0^L,u_{h,0}^L, u^L_{3,0})\in H^{m+3}(\mathbb{R}^2)$ with a nonnegative integer $m$. Then for any $\varepsilon,\delta>0$ satisfying lim, the system 11183 admits a unique solution $(a,w_h,w_3)\in C([0,T^*];H^{m+3}(\mathbb{R}^2))$ for some $T^*>0$, and satisfies where the time $T^*$ and the positive constant $C$ are independent of $\delta$.

Theorems & Definitions (9)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3