Abstract
We prove the existence of steady space quasi-periodic stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel R×[-1,1]. These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash–Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin’s cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.
Original language | English (US) |
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Article number | 81 |
Journal | Archive for Rational Mechanics and Analysis |
Volume | 248 |
Issue number | 5 |
DOIs | |
State | Published - Oct 2024 |
Keywords
- 35Q31
- 37K55
- 76B03
ASJC Scopus subject areas
- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering