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On global dynamics of $3$-D irrotational compressible fluids

Qian Wang

TL;DR

The work addresses the global-in-time behavior of 3D irrotational, isentropic compressible Euler flow from broad, non-symmetric exterior data by formulating the problem on the acoustical spacetime with metric ${\mathbf g}$. The authors develop a robust geometric framework, identifying intrinsic null forms and decomposing commutators to overcome the lack of a standard null condition, and they implement a weighted-energy bootstrap to propagate $H^4$-regular solutions in the exterior region. A central achievement is the construction of a global exterior solution in $H^4$ and the demonstration that, for a broad class of data satisfying expansion-type conditions, rarefaction occurs at null infinity, manifested by ${\mathbf b}^{-1}\to 0$ along outgoing null geodesics. The analysis hinges on a hierarchical amplitude classification, a novel commutator decomposition, and a carefully designed energy hierarchy that balances large transversal derivatives with tangential controls, enabling decay estimates stronger than those of free waves. The results illuminate the global dynamics of compressible fluids in three dimensions, revealing a robust rarefaction mechanism without small-total-energy assumptions and offering precise geometric-analytic tools for related quasilinear wave-fluid systems.

Abstract

We consider global-in-time evolution of irrotational, isentropic, compressible Euler flow in $3$-D, for a broad class of $H^4$ classical Cauchy data without assuming symmetry, prescribed on an annulus surrounded by a constant state in the exterior. By giving a sufficient expansion condition on the initial data and using the nonlinear structure of the compressible Euler equations, we show that the decay rate of the first order transversal derivative of the normalized density is better than that of the same derivative of a free wave, provided that the perturbation arising from the tangential derivatives can be properly controlled for all $t$ by using a bootstrap argument. Building on this critical analysis, we construct global exterior solutions in $H^4$ for the broad class of data, with a rather general subclass forming rarefaction at null infinity. Our result does not require smallness on the transversal derivatives of classical data, thus applies to data with a total energy of any size.

On global dynamics of $3$-D irrotational compressible fluids

TL;DR

The work addresses the global-in-time behavior of 3D irrotational, isentropic compressible Euler flow from broad, non-symmetric exterior data by formulating the problem on the acoustical spacetime with metric . The authors develop a robust geometric framework, identifying intrinsic null forms and decomposing commutators to overcome the lack of a standard null condition, and they implement a weighted-energy bootstrap to propagate -regular solutions in the exterior region. A central achievement is the construction of a global exterior solution in and the demonstration that, for a broad class of data satisfying expansion-type conditions, rarefaction occurs at null infinity, manifested by along outgoing null geodesics. The analysis hinges on a hierarchical amplitude classification, a novel commutator decomposition, and a carefully designed energy hierarchy that balances large transversal derivatives with tangential controls, enabling decay estimates stronger than those of free waves. The results illuminate the global dynamics of compressible fluids in three dimensions, revealing a robust rarefaction mechanism without small-total-energy assumptions and offering precise geometric-analytic tools for related quasilinear wave-fluid systems.

Abstract

We consider global-in-time evolution of irrotational, isentropic, compressible Euler flow in -D, for a broad class of classical Cauchy data without assuming symmetry, prescribed on an annulus surrounded by a constant state in the exterior. By giving a sufficient expansion condition on the initial data and using the nonlinear structure of the compressible Euler equations, we show that the decay rate of the first order transversal derivative of the normalized density is better than that of the same derivative of a free wave, provided that the perturbation arising from the tangential derivatives can be properly controlled for all by using a bootstrap argument. Building on this critical analysis, we construct global exterior solutions in for the broad class of data, with a rather general subclass forming rarefaction at null infinity. Our result does not require smallness on the transversal derivatives of classical data, thus applies to data with a total energy of any size.
Paper Structure (71 sections, 109 theorems, 1906 equations, 2 figures)

This paper contains 71 sections, 109 theorems, 1906 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic set-up and the main result
  3. Motivation
  4. A heuristic example
  5. Remarks on the assumptions in Theorem \ref{['mainthm1']}
  6. The model argument
  7. Global exterior solution
  8. Rarefaction at null infinity
  9. Main steps of the proof
  10. Key technicalities for controlling the weighted energies
  11. Establish the hierarchy of amplitude
  12. New decompositions of commutators between tangential vector fields and the wave operator
  13. Energy hierarchy
  14. Organization
  15. Geometric set-up and the geometric null conditions
  16. ...and 56 more sections

Key Result

Theorem 1.3

(Main theorem) Let $\Phi$ be $(v^i, \varrho)$ in the equation system (4.10.1.19)-(4.10.2.19). Let the initial data of $\Phi$ satisfy (1.12.1.22) and (rarefied). Suppose where $X$ represents all the vector fields in $\{S, \Omega\}$, and ${{\textbf{$\varepsilon$}}}_0$ is sufficiently small, depending on $\mathcal{A}_0, c_*$. It yields the following rough bound for $q_0$, There hold the following r

Figures (2)

  • Figure 1: Shock formation
  • Figure 2: Formation of rarefaction

Theorems & Definitions (228)

  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 2.1
  • Corollary 2.2
  • Lemma 2.3: Crucial decomposition for second fundamental forms
  • Remark 2.4
  • proof
  • ...and 218 more