math.AP — Analysis of PDEs

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

Contributed by Umberto Pappalettera, Massimo Sorella

For every finite $p>2$, we construct one real compactly supported zero-circulation initial vorticity in $L^1(\mathbb R^2)\cap L^p(\mathbb R^2)$ whose unforced Navier--Stokes evolutions have two distinct strong vanishing-viscosity accumulation paths. Both limits are renormalized unforced Euler solutions with the same strong initial trace. The construction uses a fixed infinite concentric stack, a smooth unstable fourfold profile, actual positive scalar envelopes, and a finite nonlinear expansion retaining the interacting harmonic histories. Discounted source history is controlled through the complete prospective receiver stop, and exact-cap normalization yields a depth-uniform outgoing reset. Protected first crossings, phase degree and a sparse phase assignment then separate the two limits. The analytical preliminaries, finite-sector spectral scope, source-reference comparison and uniform parameter order are included in the proof.