Remarks on sparseness and regularity of Navier-Stokes solutions

Article Mathematics, Applied

A regularity criterion for 3D NSE in 'dynamically restricted' local Morrey spaces

Zoran Grujic et al.

Summary: By imposing a smallness condition, it is ensured that a local-in-time strong solution to the 3D Navier-Stokes equations remains regular on a specific time interval, preventing finite-time blow up.

APPLICABLE ANALYSIS (2022)

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Article Mathematics

Finite-time singularity formation for C1,α solutions to the incompressible Euler equations on R3

Tarek M. Elgindi

Summary: The 3D incompressible Euler equation is locally well-posed for velocity fields with Ho center dot lder continuous gradient and suitable decay at infinity. However, it is shown that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

ANNALS OF MATHEMATICS (2021)

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Article Multidisciplinary Sciences

Geometry of turbulent dissipation and the Navier-Stokes regularity problem

Janet Rafner et al.

Summary: The recently introduced mathematical framework based on a suitably defined 'scale of sparseness' of the regions of intense vorticity has shown promise in closing the scaling gap in the super-critical Navier-Stokes regularity problem. Numerical simulations confirm the framework's ability to detect the onset of dissipation and provide further evidence that ongoing mathematical efforts may succeed in reducing the scaling gap.

SCIENTIFIC REPORTS (2021)

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Article Mathematics

A survey of geometric constraints on the blowup of solutions of the Navier-Stokes equation

Evan Miller

Summary: This article discusses regularity criteria for the Navier-Stokes equation that provide geometric constraints on possible finite-time blowup, as well as the physical significance of these criteria.

JOURNAL OF ELLIPTIC AND PARABOLIC EQUATIONS (2021)

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Article Physics, Mathematical

Quantitative Regularity for the Navier-Stokes Equations Via Spatial Concentration

Tobias Barker et al.

Summary: This paper focuses on quantitative estimates for the Navier-Stokes equations, exploring the relation of quantitative bounds to the behavior of critical norms near potential singularities, and the conclusion that smooth finite-energy solutions do not blow up under certain conditions. By developing a new proof strategy and utilizing local-in-space smoothing results and Carleman inequalities, the technology introduced here allows for a quantitative bound on the number of singular points in specific blow-up scenarios.

COMMUNICATIONS IN MATHEMATICAL PHYSICS (2021)

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Article Mathematics