Vanishing viscosity and nonuniqueness

Non-unique vanishing-viscosity solutions of 2D Euler with vorticity in L^p

A manuscript claiming two distinct weak Euler solutions arising as vanishing-viscosity limits from the same compactly supported initial vorticity, for every 2 < p < ∞.

Read the paper Explore the main result Source material

The manuscript’s main claim

Two distinct vanishing-viscosity limits

This manuscript contains a claim of the following result:

Theorem

For every p ∈ (2, ∞), there exists a compactly supported vorticity $ω_{in} \in L^{p} (ℝ^{2})$ and a sequence of viscosities {ν_n}_{n ∈ ℕ} with ν_n ↓ 0 as n → ∞, such that the associated solutions of the 2D Navier-Stokes equations in vorticity form

{\begin{array}{lcl} \partial_{t} ω_{ν_{n}} + u_{ν_{n}} \cdot \nabla ω_{ν_{n}} & = & ν_{n} Δ ω_{ν_{n}}, \\ ω_{ν_{n}} |_{t = 0} & = & ω_{in}, \end{array}

satisfy

ω_{ν_{2 n}} \to ω_{e},

ω_{ν_{2 n + 1}} \to ω_{o},

ω_{e} \neq ω_{o} .

The convergences hold in the space $C ([0, T]; L^{p} (ℝ^{2}))$ for some T ∈ (0, ∞), and ω_e, ω_o are two distinct weak solutions of the 2D Euler equations with initial vorticity ω_in.

The paper

Read the full argument

Non-unique vanishing-viscosity selection for planar Euler

ChatGPT 6.1 Sol · Prompters: Umberto Pappalettera and Massimo Sorella

The revised manuscript is available below. The source material brings together the original chat, prompts, manuscript versions, and the complete source ZIP with its SHA-256 checksum.

Open the paper Download PDF Source material & ZIP

Our purpose

We are sharing the manuscript here because we would like the mathematics behind this result to be prioritized over a counterproductive race to upload AI-generated proofs to arXiv.

We are organizing a reading group to study the claimed proof in detail, identify and deeply understand its main underlying ideas, and communicate them in a form more accessible to the mathematical community, for instance as a set of lecture notes.