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The Extra Vanishing Structure and Nonlinear Stability of Multi-Dimensional Rarefaction Waves: The Geometric Weighted Energy Estimates

Haoran He, Qichen He

Abstract

We study the resolution of discontinuous singularities in gas dynamics via multi-dimensional rarefaction waves. While the mechanism is well-understood in one spatial dimension, the rigorous construction in higher dimensions has remained a challenging open problem since Majda's proposal, primarily due to the characteristic nature of rarefaction fronts which leads to derivative losses in linearized estimates. In this paper, we establish the nonlinear stability of multi-dimensional rarefaction waves for the compressible Euler equations with ideal gas law. We prove that for initial data being small perturbations of the planar rarefaction wave in $H^s$ ($s > s_c$), there exists a unique global solution that converges asymptotically to the background rarefaction wave as $t \to \infty$. Our proof relies on a novel Geometric Weighted Energy Method (GWEM), which yields stable energy estimates without loss of derivatives in standard Sobolev spaces, overcoming the limitations of previous Nash-Moser schemes. A key ingredient is a detailed geometric description of the rarefaction wave fronts via the acoustical metric, where we identify a hidden extra vanishing structure in the top-order derivatives of the characteristic speed. This is the first paper in a series, providing the crucial a priori energy bounds. The existence of solutions and applications to the multi-dimensional Riemann problem will be addressed in the forthcoming companion paper.

The Extra Vanishing Structure and Nonlinear Stability of Multi-Dimensional Rarefaction Waves: The Geometric Weighted Energy Estimates

Abstract

We study the resolution of discontinuous singularities in gas dynamics via multi-dimensional rarefaction waves. While the mechanism is well-understood in one spatial dimension, the rigorous construction in higher dimensions has remained a challenging open problem since Majda's proposal, primarily due to the characteristic nature of rarefaction fronts which leads to derivative losses in linearized estimates. In this paper, we establish the nonlinear stability of multi-dimensional rarefaction waves for the compressible Euler equations with ideal gas law. We prove that for initial data being small perturbations of the planar rarefaction wave in (), there exists a unique global solution that converges asymptotically to the background rarefaction wave as . Our proof relies on a novel Geometric Weighted Energy Method (GWEM), which yields stable energy estimates without loss of derivatives in standard Sobolev spaces, overcoming the limitations of previous Nash-Moser schemes. A key ingredient is a detailed geometric description of the rarefaction wave fronts via the acoustical metric, where we identify a hidden extra vanishing structure in the top-order derivatives of the characteristic speed. This is the first paper in a series, providing the crucial a priori energy bounds. The existence of solutions and applications to the multi-dimensional Riemann problem will be addressed in the forthcoming companion paper.
Paper Structure (107 sections, 50 theorems, 246 equations, 5 figures, 12 tables)

This paper contains 107 sections, 50 theorems, 246 equations, 5 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Background: Resolution of Singularities in Compressible Fluid Dynamics
  3. The One-Dimensional Theory
  4. The Multi-Dimensional Challenge
  5. Previous Works and Major Obstacles
  6. The Majda Program and Its Limitation for Rarefaction Waves
  7. Alinhac's Breakthrough and the Nash-Moser Scheme
  8. Recent Developments and Remaining Gaps
  9. Comparison with Concurrent Works
  10. Main Results
  11. Functional Setting and Notations
  12. Statement of Main Theorems
  13. Strategy of the Proof
  14. Step 1: Geometric Setup and Acoustical Coordinates
  15. Step 2: The Linearized Equations and Derivative Loss
  16. ...and 92 more sections

Key Result

Theorem 1.1

Let $n \geq 2$ and $\gamma > 1$. Consider the compressible Euler equations eq:euler_full with initial data where $\tilde{U}_0 \in H^s(\mathbb{R}^n)$ with $s > \frac{n}{2} + 1$. There exists a constant $\epsilon_0 > 0$ depending only on $n$, $\gamma$, and the background rarefaction wave $\bar{U}$, such that if then there exists a unique global classical solution $U(t,x)$ defined for all $t \in [0

Figures (5)

  • Figure 1: Spacetime diagram in $(t,x)$ coordinates for nonlinear acoustic waves. The blue curves represent outgoing characteristics $\mathcal{C}_u$ (level sets of the eikonal function $u$), which are visibly curved due to nonlinear wave speed variation. The red dashed curve is the sonic line ($\mu=0$), a degenerate characteristic $\mathcal{C}_{u_*}$. The shaded orange region $\mathcal{D}_{t,u}$ is bounded by the initial slice $\Sigma_0$, final slice $\Sigma_t$, and the characteristic $\mathcal{C}_u$. The purple arrow indicates the outward normal boundary flux across $\mathcal{C}_u$, crucial for energy estimates.
  • Figure 2: The geometry of the rarefaction wave. The characteristic hypersurfaces $\mathcal{C}_u$ emanate from the origin. The lapse function $\mu$ vanishes at the sonic line ($u = u_-$).
  • Figure 3: Geometry of the rarefaction wave in the acoustical coordinates $(t, u, \vartheta)$. The characteristic hypersurfaces $\mathcal{C}_u$ (blue lines) emanate from the origin, filling the Rarefaction Fan. The left boundary ($u=u_-$) is the Sonic Line, where the lapse function $\mu$ vanishes. The horizontal dashed line represents a time slice $\Sigma_t$, whose intersection with $\mathcal{C}_u$ defines the spheres $S_{t,u}$. The null vectors $L$ and $\underline{L}$ are tangent to the outgoing and incoming characteristics, respectively.
  • Figure 4: Profile of the background rarefaction wave. The solution is constant outside the fan $[\lambda_1(U_-), \lambda_1(U_+)]$ and varies smoothly inside. The lapse function $\mu$ vanishes at the left edge of the fan.
  • Figure 5: Geometric comparison: rarefaction waves (expansion, $\text{tr}\chi \sim \mu$) vs shock formation (contraction, $\text{tr}\chi \sim -1/\mu$).

Theorems & Definitions (137)

  • Theorem 1.1: Global Existence and Uniqueness
  • Remark 1.1
  • Theorem 1.2: Uniform Energy Bounds
  • Remark 1.2
  • Theorem 1.3: Asymptotic Stability
  • Remark 1.3
  • Corollary 1.1: Multi-Dimensional Riemann Problem
  • Remark 1.4
  • Definition 2.1: Acoustical Metric
  • Remark 2.1
  • ...and 127 more