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Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity

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arxiv 2408.07934 v1 pith:3JAIREDA submitted 2024-08-15 math.AP

Elia Bruè , Maria Colombo , Anuj Kumar This is my paper · ORCID
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keywords convexeulerflexibilityinftyintegrationtimetwo-dimensionalvorticity
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abstract

We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $L^\infty L^2$ weak solutions with vorticity in $L^\infty L^p$ for some $p>1$, surpassing for the first time the critical scaling of the standard convex integration technique. To achieve this, we introduce several new ideas, including: (i) A new family of building blocks built from the Lamb-Chaplygin dipole. (ii) A new method to cancel the error based on time averages and non-periodic, spatially-anisotropic perturbations.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

    math.AP 2025-02 conditional novelty 8.0 of 10

    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.

  2. Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

    math.AP 2025-09 conditional novelty 7.0 of 10

    For 2D Navier-Stokes on a torus, non-uniqueness holds with velocity gradients in C([0,T], H^p) for every exponent 0 < p < 1, making p = 1 the sharp threshold between non-uniqueness and uniqueness in the vorticity path space.

  3. Asymmetric Self-similar Spiral Solutions of 2-D Incomressible Euler Equations

    math.AP 2025-07 conditional novelty 7.0 of 10

    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.

  4. Local Well-Posedness for Vlasov--Poisson with $L^{d+}$ Initial Density and Fractional Velocity Regularity

    math.AP 2026-07 accept novelty 6.0 of 10

    Vlasov–Poisson is locally well-posed in d≥2 for finite-mass data with weighted L^{p>d} velocity envelopes and arbitrarily small uniform velocity Hölder regularity.

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