Smooth 2D Euler steady states with multiple critical points can be perturbed to make vorticity non-functional in the stream function, yielding isolated branches of stable states unlike the analytic case.
Smooth nonradial stationary Euler flows on the plane with compact support
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abstract
We prove the existence of nonradial classical solutions to the 2D incompressible Euler equations with compact support. More precisely, for any positive integer $k$, we construct compactly supported stationary Euler flows of class $C^k(\mathbb{R}^2)$ which are not locally radial. The proof uses a degree-theory-based bifurcation argument which hinges on three key ingredients: a novel approach to stationary Euler flows through elliptic equations with non-autonomous nonlinearities; a set of sharp regularity estimates for the linearized operator, which involves a potential that blows up as the inverse square of the distance to the boundary of the support; and overcoming a serious problem of loss of derivatives by the introduction of anisotropic weighted functional spaces between which the linearized operator is Fredholm.
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math.AP 3years
2026 3roles
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Vanishing viscosity selects constant-vorticity flows in bounded domains and shear flows in strips as the only possible limits for 2D steady Euler equations.
Non-trivial m-fold doubly-connected stationary vortex patches are proven to exist for the quasi-geostrophic shallow-water equations via bifurcation from annuli.
citing papers explorer
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On the flexibility of 2D Euler steady states
Smooth 2D Euler steady states with multiple critical points can be perturbed to make vorticity non-functional in the stream function, yielding isolated branches of stable states unlike the analytic case.
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A selection principle for 2D steady Euler flows via the vanishing viscosity limit
Vanishing viscosity selects constant-vorticity flows in bounded domains and shear flows in strips as the only possible limits for 2D steady Euler equations.
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On stationary Quasi-Geostrophic Shallow-Water flows
Non-trivial m-fold doubly-connected stationary vortex patches are proven to exist for the quasi-geostrophic shallow-water equations via bifurcation from annuli.