Related papers: Numerical verification of regularity in the three-…
In this paper, we study the global regularity of strong solution to the Cauchy problem of 3D incompressible Navier-Stokes equations with large data and non-zero force. We prove that the strong solution exists globally for $\nabla u\in…
We consider the initial value problem for the Navier-Stokes equations over $R^{3} \times [0,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-valued functions on $R^{3}$ with functions on the…
In this short survey paper, we focus on some new developments in the study of the regularity or potential singularity formation for solutions of the 3D Navier-Stokes equations. Some of the motivating questions are: Are certain norms…
We consider the motion of incompressible viscous non-homogeneous fluid described by the Navier-Stokes equations in a bounded cylinder under boundary slip conditions. Assume that the third co-ordinate axis is the axis of the cylinder.…
In the classic work of Beale-Kato-Majda ({[}2{]}) for the Euler equations in $\mathbb{R^{\mathrm{3}}}$, regularity of a solution throughout a given interval $[0,T_{*}]$ is obtained provided that the curl $\omega$ satisfies $\omega\in…
The periodic $3D$ Navier-Stokes equations are analyzed in terms of dimensionless, scaled, $L^{2m}$-norms of vorticity $D_{m}$ ($1 \leq m < \infty$). The first in this hierarchy, $D_{1}$, is the global enstrophy. Three regimes naturally…
This paper establishes a complete framework for infinitely nested logarithmic improvements to regularity criteria for the three-dimensional incompressible Navier-Stokes equations. Building upon our previous works on logarithmically improved…
We consider a family of Leray-$\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\theta$, of the Navier-Stokes equations. In particular, they share…
In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the…
In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be…
In this article, we establish several almost critical regularity conditions such that the weak solutions of the 3D Navier-Stokes equations become regular, based on one component of the solutions, say $u_3$ and $\partial_3u_3$.
We prove that the density of the law of any finite dimensional projection of solutions of the Navier--Stokes equations with noise in dimension $3$ is H\"older continuous in time with values in the natural space $L^1$. When considered with…
In this small note we strengthen the classic result about the regularity time t* of arbitrary Leray solutions to the (incompressible) Navier-Stokes equations in Rn (n = 3, 4), which have the form: t* <= K_{3} nu^{-5} || u(.,0) ||_{L2}^{4}…
In the present paper, we prove a sufficient condition of local regularity for suitable weak solutions to the Navier-Stokes equations having axial symmetry. Our condition is an axially symmetric analog of the so-called $L_{3,\infty}$-case in…
Smooth solutions to the axi-symmetric Navier-Stokes equations obey the following maximum principle: $$\sup_{t\geq 0}\|rv^\theta(t, \cdot)\|_{L^\infty} \leq \|rv^\theta(0, \cdot)\|_{L^\infty}.$$ We prove that all solutions with initial data…
It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes and MHD equations are H\"older continuous near boundary provided that either $r^{-3}\int_{B_r^+}|u(x)|^3dx$ or $r^{-2}\int_{B_r^+}|\nabla…
We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty…
The paper is concerned with the regularity of weak solutions to the Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin condition in order to obtain regularity for t > 0. The most interesting aspect of the result is…
We study the global regularity, for all time and all initial data in $H^{1/2}$, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution…
The existence of global regular axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder is proved. The main step in the proof is a proof of the Holder continuity of the swirl. This gives a possibility to prove…