Pith. sign in

REVIEW 6 cited by

Illposedness of incompressible fluids in supercritical Sobolev spaces

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2404.07813 v2 pith:GBWFI7W5 submitted 2024-04-11 math.AP

Xiaoyutao Luo This is my paper · ORCID
classification math.AP
keywords equationfracnormcaseeulerinflationnavier-stokessobolev
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
abstract

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0 < s < \frac{5}{2} $, we construct a $C^\infty_c$ initial velocity field with arbitrarily small $H^{s}$ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large $\dot{H}^{s}$ norm inflation almost instantaneously. In the viscous case, the same $\dot{H}^{s}$ norm inflation occurs in the 3D Navier-Stokes equation for $0< s < \frac{1}{2} $, where $s = \frac{1}{2}$ is scaling critical for this equation.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

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

  1. Low regularity Sobolev well-posedness for Vlasov--Poisson

    math.AP 2025-10 conditional novelty 8.0 of 10

    Local well-posedness for Vlasov–Poisson in H^s for s > n/2 − 1/4, n≥3, with compact velocity support, allowing unbounded initial data.

  2. Strong ill-posedness for the MHD system in the supercritical regime: inviscid and viscous

    math.AP 2026-08 conditional novelty 7.0 of 10

    Arbitrarily small smooth data for 3D incompressible MHD produce norm inflation in every supercritical Sobolev space (e.g., H^s, 0<s<5/2, for ideal MHD), with the velocity staying bounded in the ideal case.

  3. Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space

    math.AP 2026-07 conditional novelty 7.0 of 10

    The 2D Euler vorticity equation is shown to be globally well-posed in the endpoint Sobolev space W^{2,1}, and the 3D axisymmetric no-swirl case in W^{3,1}, closing the p=1 endpoint of the critical Sobolev scale.

  4. Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces

    math.AP 2026-07 conditional novelty 7.0 of 10

    Norm inflation is proven for 2D inviscid and fully dissipative Boussinesq systems in almost all supercritical Besov spaces, with the density, not the velocity, carrying the blow-up.

  5. Norm Inflation For The Critical SQG Equation

    math.AP 2025-12 conditional novelty 7.0 of 10

    Critical SQG has H1 norm inflation from large smooth data and small-data norm inflation in supercritical W^{β,p} spaces.

  6. Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation

    math.AP 2025-08 conditional novelty 6.0 of 10

    Arbitrarily small supercritical H^s vorticities for 3D Euler can have a unique classical solution whose Sobolev regularity drops continuously from s at the sharp rate (s-ct)/(1+ct).

Pith tools