Delort’s vortex-sheet problem

Existence of weak 2D Euler solutions for sign-changing vortex-sheet data

A manuscript proposing global weak flows from arbitrary finite signed vorticity in H⁻¹, with a strong initial velocity trace and integrable vorticity at every positive time.

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The problem

Delort’s discussion

Delort proved global weak existence for compactly supported vorticity measures of one sign in H⁻¹(ℝ²) and made the following remark in his 1990 École Polytechnique seminar talk:

Let us conclude with a remark on the positivity assumption imposed on the initial data ω₀. One may ask whether the vortex-sheet problem admits a solution with no assumption on the initial vorticity other than that it be a compactly supported measure belonging to H⁻¹(ℝ²).

Texte original

Terminons par une remarque concernant l’hypothèse de positivité faite sur les données de Cauchy ω₀. On peut se demander si le problème des nappes de tourbillon admet une solution sans autre condition sur le tourbillon initial que d’être une mesure à support compact appartenant à H⁻¹(ℝ²).

English translation and French original from Delort’s discussion of the sign condition.

The manuscript’s main claim

No restriction on the initial sign

Theorem

For every finite signed Radon measure ω₀ ∈ H⁻¹(ℝ²), there are infinitely many distinct global weak velocity solutions u with initial value $u_{0} = \nabla^{⊥} Δ^{- 1} ω_{0}$ , which satisfy

u - u_{0} \in C ([0, \infty); L^{2}), (u - u_{0}) (0) = 0 .

The vorticity ω = curl u satisfies ω(0) = ω₀ and the following regularity properties:

$ω \in C ([0, \infty); H^{- 1}) \cap C ((0, \infty); L^{1}) .$
ω(t) is weakly continuous as a finite signed Radon measure, i.e.

$t \mapsto \int_{ℝ^{2}} φ d ω (t) \in C ([0, \infty)) for every φ \in C_{0} (ℝ^{2}) .$
Moreover, if ω₀ ∈ L¹, one also has ω ∈ C([0, ∞); L¹).

Remark

The approach uses convex integration in the style of Bruè–Colombo–Kumar: starting from an Euler–Reynolds subsolution, one iteratively reduces the Reynolds stress through the time-averaged effect of moving dipole perturbations. The perturbations remain summable in velocity L² and vorticity L¹.

The result concerns weak velocity solutions. It does not assert an energy inequality, preservation of the sign of vorticity, propagation of higher L^p norms, or renormalized transport.

The paper

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Global weak Euler flows with finite signed initial vorticity

AI-assisted manuscript · Subsequently polished with ChatGPT · 25 pages

This is the current, revised manuscript. The original output and the material from the research run are available on the source page.

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Team

Our purpose

We want to examine the mathematical ideas behind this work, understand their scope and limitations, and explore related open problems and possible extensions.

This page brings together the AI-assisted manuscript, its source material, and our efforts to interpret and explain it. We hope to collect comments, questions, and expositions that can be read alongside the original argument. We aim for transparency and to share the relevant AI interactions whenever feasible. Over time, we hope to complement the AI-assisted work with human expositions.

Invitation

Everyone is invited to collaborate and discuss. If you would like to examine an argument, point out a gap, contribute an exposition, or exchange ideas, please contact josef.demmel@fau.de. Questions, corrections, and comments are very welcome.