Interior Regularity Criteria Research Articles

On the interior regularity criteria for the viscoelastic fluid system with damping

Abstract We consider a system of PDEs that model a viscoelastic fluid with damping mechanism. In ℝ 3 {\mathbb{R}^{3}} , we construct some new local energy bounds that enable us to improve several ϵ-regularity criteria for the Caffarelli–Kohn–Nirenberg theorem for weak solutions of this system.

Remarks on interior regularity criteria without pressure for the Navier-Stokes equations

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the LtpLxq(Q1) norm of the velocity is sufficiently small, where 1≤2p+3q<2 and 2≤p≤∞. It improves the recent result of p,q>2 by Kwon [15], and also generalizes Chae-Wolf's Lt∞Lx32+ criterion [3].

On the interior regularity criterion for suitable weak solutions to the 3d magneto-hydrodynamics equations in terms of the velocity gradient

On the interior regularity criterion for suitable weak solutions to the 3d magneto-hydrodynamics equations in terms of the velocity gradient

Local Regularity Criteria in Terms of One Velocity Component for the Navier–Stokes Equations

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier–Stokes equations in three dimensions. It is shown that the velocity is regular near a point z if its scaled $$L^p_tL^q_x$$ -norm of some quantities related to the velocity field is finite and the scaled $$L^p_tL^q_x$$ -norm of one velocity component is sufficiently small near z.

Interior and boundary regularity criteria for the 6D steady Navier-Stokes equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫B1|u(x)|3dx+∫B1|f(x)|qdx or ∫B1|∇u(x)|2dx+∫B1|∇u(x)|2dx(∫B1|u(x)|dx)2+∫B1|f(x)|qdx with q>3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫B1+|u(x)|3dx+∫B1+|f(x)|3dx or ∫B1+|∇u(x)|2dx+∫B1+|f(x)|3dx is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8], Indiana Univ. Math. J., 2012), Dong-Gu ([7], J. Funct. Anal., 2014), and Liu-Wang ([27], J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary.

On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components

We present some interior regularity criteria of the 3-D Navier-Stokes equations involving two components of the velocity. These results in particular imply that if the solution is singular at one point, then at least two components of the velocity have to blow up at the same point.

Partial regularity of the 3-D inhomogeneous Navier-Stokes equations

In this paper, we first introduce the concept of suitable weak solutions for the inhomogeneous Navier-Stokes equations. In the case when the initial density is close to a positive constant, by combining local energy inequality, Sobolev embedding, pressure estimate and blow-up analysis, we establish interior regularity criterion of suitable weak solutions. Finally, we show that the 1-D Hausdorff measure of the set of possible singular points of suitable weak solutions vanishes by applying the interior regularity criterion.

On the interior regularity criteria for liquid crystal flows

In this paper, we provide some new regularity criteria for suitable weak solutions to the 3D nematic liquid crystal flow in terms of the velocity field. More precisely, we prove that the suitable weak solution (u,d) is regular provided that either the scaled Lp,q norm of u, or the scaled Lp,q norm of ∇×u is sufficiently small.

Interior regularity criterion for incompressible Ericksen-Leslie system

An interior regularity criterion of suitable weak solutions is formulated for the Ericksen-Leslie system of liquid crystals. Such a criterion is point-wise, with respect to some appropriate norm of velocity u and the gradient of d, and it can be viewed as a sort of simply sufficient condition on the local regularity of suitable weak solutions.

Blow-up of critical norms for the 3-D Navier-Stokes equations

Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3\times [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3＜p,q＜\infty$ and $u_h\in L^{\infty}(0,T; {\rm BMO}^{-1}(\R^3))$ with $u_h(T)\in {\rm VMO}^{-1}(\R^3)$, then $u$ can be extended beyond $T$. This result generalizes the recent result proved by Gallagher et al. (2016), which requires $u\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$. Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.

Refined regularity class of suitable weak solutions to the 3D magnetohydrodynamics equations with an application

By means of blow-up method and the special structure of the 3D viscous magnetohydrodynamics equations, we derive some interior regularity criteria in terms of horizontal part of the velocity with sufficiently small local scaled norm and both the vertical part of the velocity and the magnetic field with bounded local scaled norm for the suitable weak solutions to this system. As an application, this allows us to improve the previous limiting case for the regularity criterion about the MHD equations.

Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations

In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in [Formula: see text] for [Formula: see text]. In comparison with the work of the 3D fractional Navier–Stokes equations obtained by Tang and Yu in [Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations, Comm. Math. Phys. 334 (2015) 1455–1482], our results include their endpoint case [Formula: see text] and the external force belongs to a more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations.

An Anisotropic Partial Regularity Criterion for the Navier–Stokes Equations

In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier--Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$.

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L 2 norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.

Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations

This work is concerned with the partial regularity of the suitable weak solutions to the Boussinesq equations in \(\mathbb {R}^{n}\) where \(n=3,\,4\). By means of the De Giorgi iteration method developed in Vasseur (Nonlinear Differ Equ Appl 14(5–6):753–785, 2007), Wang, Wu (J Differ Equ 256(3):1224–1249, 2014), we obtain that \(n-2\) dimensional parabolic Hausdorff measure of the possible singular points set of the suitable weak solutions to this system is zero. Particularly, we obtain some interior regularity criteria only in terms of the scaled mixed norm of velocity for the suitable weak solutions to the Boussinesq equations, which implies that the potential singular points may only stem from the velocity field.

Partial regularity criteria for suitable weak solutions of the three‐dimensional liquid crystals flow

In this paper, we establish some new interior regularity criteria for suitable weak solutions of the liquid crystals flow in terms of the smallness of the scaled Lp,q‐norm of the velocity field or the vorticity, which extends the results by Scheffer in [Communications in Mathematical Physics 1980; 73:1–42]. Copyright © 2016 John Wiley &amp; Sons, Ltd.

On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

We establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-Sverak’s criterion. We also show that if a weak solution u satisfies $$\left\| {u( \cdot ,t)} \right\|_{L^p } \leqslant C( - t)^{(3 - p)/2p} $$ for some 3 < p < ∞, then the number of singular points is finite.

On the Interior Regularity Criteria for Suitable Weak Solutions of the Magnetohydrodynamics Equations

We present new interior regularity criteria for suitable weak solutions of the magneto-hydrodynamics (MHD) equations in terms of the velocity field. The result means that the velocity field plays a more important role than the magnetic field in the local regularity theory of the MHD equations. This also gives a positive answer to the problem proposed by Kang and Lee in [J. Differential Equations, 247 (2009), pp. 2310--2330].

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69 (1990), 237–254.

Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled L x , t p , q -norm of the velocity with 1 ⩽ 3 / p + 2 / q ⩽ 2 , 1 ⩽ q ⩽ ∞ is sufficiently small near z and if the scaled L x , t l , m -norm of the magnetic field with 1 ⩽ 3 / l + 2 / m ⩽ 2 , 1 ⩽ m ⩽ ∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.