Interior Regularity Criteria Research Articles

On the Interior Regularity of Suitable Weak Solutions to the Navier–Stokes Equations

We obtain the interior regularity criteria for the vorticity of “suitable” weak solutions to the Navier–Stokes equations. We prove that if two components of a vorticiy belongs to in a neighborhood of an interior point with 3/p + 2/q ≤ 2 and 3/2 < p < ∞, then solution is regular near that point. We also show that if the direction field of the vorticity is in some Triebel–Lizorkin spaces and the vorticity magnitude satisfies an appropriate integrability condition in a neighborhood of a point, then solution is regular near that point.

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point $z$ if either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\leq 2$, $1\leq q\leq \infty$, or the $L^{p,q}_{x,t}$-norm of the vorticity with $3/p+2/q\leq 3$, $1 \leq q < \infty$, or the $L^{p,q}_{x,t}$-norm of the gradient of the vorticity with $3/p+2/q\leq 4$, $1 \leq q$, $1 \leq p$, is sufficiently small near $z$.

Interior regularity criterion via pressure on weak solutions to the Navier–Stokes equations

AbstractConsider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a bounded domain in ℝ3. We prove interior regularity for suitable weak solutions under some condition on the pressure in the class of scaling invariance. The notion of suitable weak solutions makes it possible to obtain better information around the singularities. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Interior regularity criteria in weak spaces for the Navier-Stokes equations

We study the interior regularity of weak solutions of the incompressible Navier-Stokes equations in Ω×(0,T), where Open image in new window and 0<T<∞. The local boundedness of a weak solution u is proved under the assumption that Open image in new window is sufficiently small for some (r,s) with Open image in new window and 3≤r<∞. Our result extends the well-known criteria of Serrin (1962), Struwe (1988) and Takahashi (1990) to the weak space-time spaces.

Morrey space techniquesapplied to the interior regularity problem of theNavier-Stokes equations

The incompressible and viscous fluid motion, governed by the evolutionaryNavier-Stokes equations, is studied focusing on the application of Morreyspace techniques to the interior regularity problem first posed by Serrin(1962Arch. Rational Mech. Anal. 9 187-95)and later by Struwe (1988 Commun. Pure Appl. Math. 41437-58). A new interior regularity criterion is derived within theframework of Morrey spaces, which contain Lp(Lq) spaces used in thepreviously quoted references.

An Interior Regularity Criterion for an Axially Symmetric Suitable Weak Solution to the Navier—Stokes Equations

We show that if v is an axially symmetric suitable weak solution to the Navier—Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg — see [2]) such that either $ v_{\rho} $ (the radial component of v) or $ v_{\theta} $ (the tangential component of v) has a higher regularity than is the regularity following from the definition of a weak solution in a sub-domain D of the time-space cylinder Q T then all components of v are regular in D.

On interior regularity criteria for weak solutions of the navier-stokes equations

We are concerned with the behavior of weak solutions of the Navier-Stokes equations near possible singularities. We shall show that if a weak solution is in some Lebesgue space or small in some Lorentz space locally, it does not blowup there. Our basic idea is to estimate integral formulas for vorticity which satisfies parabolic equations.