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Superlinear gradient growth for 2D Euler equation without boundary
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Superlinear gradient growth for 2D Euler equation without boundary
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We consider the vorticity gradient growth of solutions to the two-dimensional Euler equations in domains without boundary, namely in the torus $\mathbb{T}^{2}$ and the whole plane $\mathbb{R}^{2}$. In the torus, whenever we have a steady state $\omega^*$ that is orbitally stable up to a translation and has a saddle point, we construct ${\tilde{\omega}}_0 \in C^\infty(\mathbb{T}^2)$ that is arbitrarily close to $\omega^*$ in $L^2$, such that superlinear growth of the vorticity gradient occurs for an open set of smooth initial data around ${\tilde{\omega}}_0$. This seems to be the first superlinear growth result which holds for an open set of smooth initial data (and does not require any symmetry assumptions on the initial vorticity). Furthermore, we obtain the first superlinear growth result for smooth and compactly supported vorticity in the plane, using perturbations of the Lamb-Chaplygin dipole.
Forward citations
Cited by 6 Pith papers
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Growth of vorticity gradient for the Euler equation on the sphere
Vorticity gradients for the Euler equation on the sphere are bounded above by double-exponential growth in time, with this rate achieved by explicit symmetric constructions.
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Linear Stability of the Lamb-Chaplygin Dipole
Linear stability analysis of the Lamb-Chaplygin dipole fully classifies the spectrum and Jordan chains, showing growth only through two explicit mechanisms tied to circulation and zero-eigenvalue chains.
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Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$
The 2D anisotropic Navier-Stokes equations with horizontal fractional dissipation of order 2s are globally stable with algebraic decay for all 0 ≤ s < 1.
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Remarks on Linear Growth of Vorticity Gradients and Support Diameters for 2D Euler Flow in Half-Plane
In the odd symmetric half-plane setting, every compactly supported nonnegative initial vorticity admits an arbitrarily small smooth nonnegative perturbation that forces linear-in-time filamentation for the 2D Euler flow.
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Small scale creation in 2D gravity-capillary water waves with vorticity
Initial data is constructed for 2D gravity-capillary water waves with vorticity in an unbounded domain such that the L^∞ norm of the vorticity gradient grows at least double-exponentially during the solution lifespan.
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On the stability of Lamb-Chaplygin dipole for the 2D Euler equation
Spectral stability of the Lamb-Chaplygin dipole holds for the 2D Euler equation without symmetry conditions, with linear fluctuation bounds and velocity control under symmetry.
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