A cheap Caffarelli—Kohn—Nirenberg inequality for the Navier—Stokes equation with hyper-dissipation

By Nets Hawk Katz, Natavsa Pavlovi'c. 2001

174 citations recorded by the source corpus.

Abstract. We prove that for the Navier—Stokes equation with dissipation $ (-\Delta)^\alpha $ where 1 < α < 5 /4, and smooth initial data, the Hausdorff dimension of the singular set at time of first blow up is at most 5 — 4α. This unifies two directions from which one might approach the problem of global solvability, though it provides no direct progress on either.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/247cc878ec90715f181aa8aa87e67d11273e751e

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

The Generalized Caffarelli‐Kohn‐Nirenberg Theorem for the Hyperdissipative Navier‐Stokes System (2017, background): “In [12] Katz and Pavlović gave a first version of the Caffarelli–Kohn–Nirenberg theorem in the range (3).”
Blow up and regularity for fractal Burgers equation (2008, background): “Recently, there has been increased interest in models involving fractional dissipation, in particular Navier-Stokes (see [13]) and surface quasi-geostrophic equations (see e.”
Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations (2005, background): “More precisely, a nonuniqueness result would represent such an insulting paradox to classical determinism, that the introduction of a more sophisticated model for the study of the motion of a viscous fluid would certainly be justified [39,42,84,119].”
Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations (2014, background): “1) has a unique global smooth solution for any prescribed smooth initial data (also see [6,16,18]).”
About the Regularized Navier–Stokes Equations (2003, unknown): “1”
Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces Ḃ∞,∞−(2β−1)(Rn) (2012, methodology): “1), we refer the readers to Dong and Li [24], Katz and Pavlović [31], Li and Zhai [25], Wu [32], and Zhai [33].”
Blow-up in finite time for the dyadic model of the Navier-Stokes equations (2006, unknown): “This model, motivat ed by [13], is an infinite system of nonlinear ODEs that describes evolutions of wavelet coef fici nts.”
Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation (2009, background): “This hyper-dissipative variant of the Navier-Stokes equation becomes subcritical for α > d+2 4 (and critical for α = d+2 4 ), and it is known that global regularity can be recovered in these cases; see [5] for further discussion.”
Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014, background): “Such an approach succeeds, for instance, if the initial data u0 are sufficiently small 5 in a suitable critical norm (see [29] for an essentially optimal result in this direction), or if the dissipative operator −Δ is replaced by a hyperdissipative operator (−Δ) for some α ≥ 5/4 (see [25]) or with v”