On Vortex Alignment and the Boundedness of the L q -Norm of Vorticity in Incompressible Viscous Fluids

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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020

http://actams.wipm.ac.cn

ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE Lq -NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS∗

og,)

Siran LI (

Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston, Texas, 77251, USA Current Address: Department of Mathematics, New York University-Shanghai, office 1146, 1555 Century Avenue, Pudong, Shanghai 200122, China E-mail : [email protected] Abstract We show that the spatial Lq -norm (q > 5/3) of the vorticity of an incompressible viscous fluid in R3 remains bounded uniformly in time, provided that the direction of vorticity is H¨ older continuous in space, and that the space-time Lq -norm of vorticity is finite. The H¨ older index depends only on q. This serves as a variant of the classical result by Constantin– Fefferman (Direction of vorticity and the problem of global regularity for the Navier–Stokes equations, Indiana Univ. J. Math. 42 (1993), 775–789), and the related work by Gruji´c– Ruzmaikina (Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE, Indiana Univ. J. Math. 53 (2004), 1073–1080). Key words

Navier–Stokes equations; vorticity; regularity; vortex alignment; weak solution; strong solution; incompressible fluid

2010 MR Subject Classification

1

35Q30, 76N10, 76D05

Introduction In this note, we consider the Cauchy problem of incompressible Navier–Stokes equations: ∂t u + div u ⊗ u − ν∆u + ∇p = 0 div u = 0 u|t=0 = u0

in ]0, T ] × R3 ,

(1.1)

3

(1.2)

3

(1.3)

in ]0, T ] × R ,

on {0} × R .

The constant ν > 0 is the viscosity, u : R3 → R3 is the velocity, and p : R3 → R is the pressure of the fluid. The existence, uniqueness, and regularity of eqs. (1.1)–(1.3) has for some time been a central research topic of nonlinear partial differential equations; see Fefferman [13], Constantin–Foias [12], Seregin [22], and many other references. The vorticity ω := ∇ × u is an important quantity for fluid motion; its time evolution is determined by the vorticity equation, obtained via taking the curl of eq. (1.1), ∂t ω + (u · ∇)ω − ν∆ω = S · ω, ∗ Received

August 15, 2019; revised December 9, 2019.

(1.4)

No.6

S.R. Li: VORTEX ALIGNMENT AND Lq -NORM OF VORTICITY

1701

where S is the 3 × 3 matrix

∇u + ∇⊤ u . (1.5) 2 The alignment of the vorticity is closely related to the regularity of weak solutions to the Navier–Stokes equations. A celebrated result by Constantin–Fefferman [10] shows that, if the vorticity direction does not change too rapidly in the regions with high vorticity magnitude, then a weak solution is automatically strong. More precisely, denote  ϕ(t, x, y) := ∠ ω(t, x), ω(t, y) , (1.6) S :=

and if there are constants Λ and ρ > 0 such that

| sin ϕ(t, x, y)| ≤

|x − y| ρ

(1.7)

whenever |ω(t, x)|, |ω(t, y)| ≥ Λ, then a weak solution u on [0, T ] must be a classical solution on [0, T ]. Here, weak solutions are defined in the Leray–Hopf sense: u ∈ L∞ (0, T ; L2(R3 )) ∩ L2 (0, T ; H 1(R3 )) with the energy