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Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations
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Building on an approach introduced by Golovkin in the '60s, we show that nonuniqueness in some forced PDEs is a direct consequence of the existence of a self-similar linearly unstable eigenvalue: the key point is a clever choice of the forcing term removing complicated nonlinear interactions. We use this method to give a short and self-contained proof of nonuniqueness in 2D perfect fluids, first obtained in Vishik's groundbreaking result. In particular, we present a direct construction of a forced self-similar unstable vortex, where we treat perturbatively the self-similar operator in a new and more quantitative way.
Forward citations
Cited by 2 Pith papers
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Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation
Non-uniqueness with forcing for alpha-SQG is established across the full supercritical Sobolev range s < alpha + 2/p, using new smooth compactly supported unstable vortices.
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Asymmetric Self-similar Spiral Solutions of 2-D Incomressible Euler Equations
For self-similar exponent mu > 1, any sufficiently small Fourier-weighted perturbation of a radial vortex produces a genuine asymmetric algebraic spiral weak solution of the 2-D Euler equations.
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