Related papers: Global regularity of Leray-Hopf weak solutions to …

We show that any Leray-Hopf weak solution to the $d$-dimensional Navier-Stokes equations $(d\geq 3)$ with initial values $u_0\in H^{s}(\mathbb R^d)$, $s\geq -1+\frac{d}{2}$, belongs to $L^\infty(0,\infty; H^{s}(\mathbb R^d))$ and thus it is…

We show that a Leray-Hopf weak solution to the 3D Navier-Stokes Cauchy problem belonging to the space $L^\infty(0,T; B^{-1}_{\infty,\infty}(\mathbb R^3))$ is regular in $(0,T]$. As a consequence, it follows that any Leray-Hopf weak solution…

If $u$ is a Leray-Hopf solution to the Navier-Stokes equations with the initial data in $ L^2(\mathbb{R}^3)$, then $u$ is regular.

In this paper, we investigate some priori estimates to provide the critical regularity criteria for incompressible Navier-Stokes equations on $\mathbb{R}^3$ and super critical surface quasi-geostrophic equations on $\mathbb{R}^2$.…

We show that if a Leray-Hopf solution $u$ to the 3D Navier-Stokes equation belongs to $C((0,T]; B^{-1}_{\infty,\infty})$ or its jumps in the $B^{-1}_{\infty,\infty}$-norm do not exceed a constant multiple of viscosity, then $u$ is regular…

We consider the interior regularity of Leray-Hopf solutions to Navier-Stokes equations on critical case L^2_w(0,T;L^\infty(R^3)). Particularly, an open problem proposed in [KK] was solved.

In this note we prove that each weak solution for the 3D Navier-Stokes system satisfies Leray-Hopf property. Moreover, each weak solution is rightly continuous in the standard phase space $H$ endowed with the strong convergence topology.

It is shown both locally and globally that $L_t^{\infty}(L_x^{3,q})$ solutions to the three-dimensional Navier-Stokes equations are regular provided $q\not=\infty$. Here $L_x^{3,q}$, $0<q\leq\infty$, is an increasing scale of Lorentz spaces…

Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $…

In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on $(0,T]$ provided that either the norm…

We investigate a regularity for weak solutions of the following generalized Leray equations \begin{equation*} (-\Delta)^{\alpha}V- \frac{2\alpha-1}{2\alpha}V+V\cdot\nabla V-\frac{1}{2\alpha}x\cdot \nabla V+\nabla P=0, \end{equation*} which…

We show that a suitable weak solution to the incompressible Navier-Stokes equations on ${\mathbb{R}^3\times(-1,1)}$ is regular on $\mathbb{R}^3\times(0,1]$ if $\partial_3 u $ belongs to $M^{2p/(2p-3),\alpha } ((-1,0);L^p (\mathbb{R}^3 ))$…

We characterize the Leray--Hopf solutions of the 2D inhomogeneous Navier--Stokes system that become strong for positive times. This characterization relies on the strong energy inequality and the regularity properties of the pressure. As an…

We prove a local-in-time regularity criterion for the 3D Navier-Stokes equations. In particular, it follows from the criterion that the Hausdorff dimension of possible singular times of Leray-Hopf weak solutions $u\in L^r_t…

We prove short time regularity of suitable weak solutions of 3D incompressible Navier-Stokes equations near a point where the initial data is locally in $L^3$. The result is applied to the regularity problems of solutions with uniformly…

Let $r,s \in [2,\infty]$ and consider the Navier-Stokes equations on $\mathbb{R}^3$. We study the following two questions for suitable $s$-homogeneous Banach spaces $X \subset \mathcal{S}'$: does every $u_0 \in L^2_\sigma$ have a weak…

In this paper we establish a Serrin type regularity criterion on the gradient of pressure in weak spaces for the Leray-Hopf weak solutions of the Navier-Stokes equations in R3.

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $u\in L_\infty^tL_{d}^x((0,T)\times {\mathbb R}^d)$ is a Leray-Hopf weak solution, then $u$ is smooth and unique in…

In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the…

In 2016, Seregin and \u{S}ver\'ak, conceived a notion of global in time solution (as well as proving existence of them) to the three dimensional Navier-Stokes equation with $L_3$ solenoidal initial data called 'global $L_3$ solutions'. A…