Local regularity of axisymmetric solutions to the Navier–Stokes equations

Abstract

In the note, a new regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It is slightly supercritical. Keywords Navier-Stokes equations, axisymmetric solutions, local regularityData availability statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

A Note on Local Regularity of Axisymmetric Solutions to the Navier–Stokes Equations

“…One can show also that σ ∈ L ∞ (Q(R)) for any 0 < R < 1, see, for example, papers [19] and [17]. What actually has been proved in paper [18], see also the last section of the present paper, is as follows:…”

“…With the regards to the state of arts in the regularity theory of axially symmetric solutions to the Navier-Stokes equations, we could refer to the previous papers [17] and [18] of the author and especially to references therein. For example, one could mention the following very interesting papers: [6], [23], [10], [14], [16], [2], [21], [4], [19], [8], [15], [9], [3], [22], and [25].…”

“…In the submitted note, we study potential singularities of axially symmetric flows of viscous incompressible fluids. It has been shown in the paper [17] that an axisymmetric solution to the three-dimensional nonstationary Navier-Stokes equations with a bounded scale-invariant energy quantity is actually smooth, i.e., axially symmetric solutions do no exhibit Type I blowups (see the paper [17] for more definitions and explanations).…”

A Slightly Supercritical Condition of Regularity of Axisymmetric Solutions to the Navier–Stokes Equations

“…In the same way, as it has been done in [26] and [24], one can show that σ ∈ L ∞ (Q(R)) for any 0 < R < 1.…”

“…In this note, we continue to analyse potential singularities of axisymmetric solutions to the non-stationary Navier-Stokes equations. In the previous paper [24], it has been shown that an axially symmetric solution is smooth provided a certain scale-invariant energy quantity of the velocity field is bounded. By definition, a potential singularity with bounded scale-invariant energy quantities is called the Type I blowup.…”