Related papers: Localized smoothing and concentration for the Navi…

We show that if $v$ is a smooth suitable weak solution to the Navier-Stokes equations on $B(0,4)\times (0,T_*)$, that possesses a singular point $(x_0,T_*)\in B(0,4)\times \{T_*\}$, then for all $\delta>0$ sufficiently small one necessarily…

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with $(-1)$-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are…

In this paper we consider smooth solutions of the Navier--Stokes equations with a linear dependence on the spatial variable. We reduce the evolution of these solutions to a matrix ODE, and show that there are such solutions that blowup in…

We consider some complex-valued solutions of the Navier-Stokes equations in $R^{3}$ for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of…

In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth $H^1$, Schwartz, or…

In the paper, we have introduced the notion of mild bounded ancient solutions to the Navier-Stokes equations in a half space. They play a certain role in understanding whether or not solutions to the initial boundary value problem for the…

A forced solution $v$ of the axially symmetric Navier-Stokes equation in a finite cylinder $D$ with suitable boundary condition is constructed. The forcing term is in the super critical space $L^q_t L^1_x$ for all $q>1$. The velocity is in…

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 $u_h\in L^{\infty}(0,T; BMO^{-1}(\R^3))$…

We consider the 3D stochastic Navier-Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates in for the energy error with…

This article examines the smoothness of the solution to the Navier-Stokes equation from a novel perspective. Here, the existence of the smoother solution relative to x and to the time t was shown only for a finite time. Moreover, for each…

In this work we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant…

We address the local well-posedness for the stochastic Navier-Stokes system with multiplicative cylindrical noise in the whole space. More specifically, we prove that there exists a unique local strong solution to the system in…

We prove quantitative regularity and blowup theorems for the incompressible Navier-Stokes equations in $\mathbb R^d$, $d\geq4$ when the solution lies in the critical space $L_t^\infty L_x^d$. Explicit subcritical bounds on the solution are…

In this survey article, we will discuss some regularity criteria for the Navier--Stokes equation that provide geometric constraints on any possible finite-time blowup. We will also discuss the physical significance of such regularity…

Non-smooth Leray-Hopf solutions of the Navier-Stokes equation are constructed. The construction occurs in a finite periodic cube T3. Entropy production maximizing solutions with turbulent initial data are selected. The proof of finite time…

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…

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…

In the paper, a new {\it slightly supercritical} condition, providing {\it local} regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local…

For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^n)}\gtrsim_{\varphi,\epsilon,(p\vee…

We study the pointwise decay properties of solutions to the incompressible Navier-Stokes equations, both in the space and time variables. It is well known that generic global solutions on $\mathbb{R}^n$ do not decay faster at infinity than…