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Traveling waves near shear flows for the inhomogeneous Euler equations with non-constant density

Qi Zhao, Weiren Zhao

TL;DR

The paper analyzes traveling waves for the 2D inhomogeneous Euler system in a finite channel with non-constant density. It shows, via a detailed spectral and bifurcation framework, that nontrivial traveling waves exist near monotone shear flows in low-regularity Sobolev spaces, indicating failure of inviscid damping at these levels; simultaneously, it proves high-regularity nonexistence under spectral assumptions on the distorted Rayleigh operator. The approach combines a Long-type reformulation, Crandall–Rabinowitz local bifurcation, and careful construction of perturbed shear/density profiles to induce cat's-eye structures, while sharp resolvent-type estimates preclude such traveling waves in higher regularity when no eigenvalues are present. The results highlight the delicate interplay between density stratification, spectral properties of the linearized operator, and nonlinear stability phenomena in inhomogeneous flows, with implications for understanding transition patterns and damping in stratified fluids.

Abstract

We investigate the existence and nonexistence of traveling wave solutions near monotonic shear flows with non-constant background density for the two-dimensional inhomogeneous Euler equations in a finite channel. For any small $τ>0$, first, we construct nontrivial traveling waves with velocity and density in $H^{5/2-τ}$ and $H^{3/2-τ}$, respectively, showing that inviscid damping fails at these regularities. Second, when the distorted Rayleigh operator has no eigenvalues, we prove that such traveling wave solutions cannot exist in higher regularity spaces ($H^{5/2+τ}$ for velocity and $H^{3/2+τ}$ for density).

Traveling waves near shear flows for the inhomogeneous Euler equations with non-constant density

TL;DR

The paper analyzes traveling waves for the 2D inhomogeneous Euler system in a finite channel with non-constant density. It shows, via a detailed spectral and bifurcation framework, that nontrivial traveling waves exist near monotone shear flows in low-regularity Sobolev spaces, indicating failure of inviscid damping at these levels; simultaneously, it proves high-regularity nonexistence under spectral assumptions on the distorted Rayleigh operator. The approach combines a Long-type reformulation, Crandall–Rabinowitz local bifurcation, and careful construction of perturbed shear/density profiles to induce cat's-eye structures, while sharp resolvent-type estimates preclude such traveling waves in higher regularity when no eigenvalues are present. The results highlight the delicate interplay between density stratification, spectral properties of the linearized operator, and nonlinear stability phenomena in inhomogeneous flows, with implications for understanding transition patterns and damping in stratified fluids.

Abstract

We investigate the existence and nonexistence of traveling wave solutions near monotonic shear flows with non-constant background density for the two-dimensional inhomogeneous Euler equations in a finite channel. For any small , first, we construct nontrivial traveling waves with velocity and density in and , respectively, showing that inviscid damping fails at these regularities. Second, when the distorted Rayleigh operator has no eigenvalues, we prove that such traveling wave solutions cannot exist in higher regularity spaces ( for velocity and for density).
Paper Structure (10 sections, 12 theorems, 185 equations)

This paper contains 10 sections, 12 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Idea of proof
  4. Nonexistence of the traveling wave solution in high regularity
  5. The existence of cat's eye structure
  6. Perturbation of the velocity and density
  7. The convergence of the perturbed distorted Rayleigh operator
  8. The Cat's-eyes structure near the modified shear flow
  9. Non-existence of the traveling wave solutions in $H^{\frac{3}{2}+\tau}$
  10. Appendix

Key Result

Theorem 1.1

Let $u, \rho \in C^3(-1, 1)$ satisfy For any $\tau > 0$, there exists $\epsilon_0 >0$ such that for all $0 <\epsilon < \epsilon_0$, there is a steady non-sheared flow $\mathbf{u}_\epsilon(x, y)$ and $\rho_\epsilon (x, y)$ (with $c = 0$) to Euler equations incompressible Euler system and

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 14 more