Related papers: Quantitative bounds for critically bounded solutio…

We consider regular solutions to the Navier-Stokes equation and provide an extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative regularity Besov scale, with regularity arbitrarly close to -1. Our results rely on…

We generalize two results in the Navier-Stokes regularity theory whose proofs rely on `zooming in' on a presumed singularity to the local setting near a curved portion $\Gamma \subset \partial\Omega$ of the boundary. Suppose that $u$ is a…

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on…

We point out some criteria that imply regularity of axisymmetric solutions to Navier-Stokes equations. We show that boundedness of $\|{v_{r}}/{\sqrt{r^3}}\|_{L_2({\rm R}^3\times (0,T))}$ as well as boundedness of…

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 $\mathbb{R}^2$.…

Considering initial data in $\dot{H}^s$, with $\frac{1}{2} \textless{} s \textless{} \frac{3}{2}$, this paper is devoted to the study of possible blowing-up Navier-Stokes solutions such that $(T*(u\_{0}) -t)^{\frac{1}{2} (s- \frac{1}{2})}…

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 prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we…

Let us consider an initial data $v_0$ for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$,…

We consider the compressible Navier-Stokes system in three dimensions with general inflow-outflow boundary conditions, meaning that we prescribe a boundary velocity which has non-zero normal component and accordingly the density is…

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…

This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted…

In this paper we continue to develop an alternative viewpoint on recent studies of Navier-Stokes regularity in critical spaces, a program which was started in the recent work by C. Kenig and the second author (Ann Inst H Poincar\'e Anal Non…

This paper is concerned with geometric regularity criteria for the Navier-Stokes equations in $\mathbb{R}^3_{+}\times (0,T)$ with no-slip boundary condition, with the assumption that the solution satisfies the `ODE blow-up rate' Type I…

This paper focuses on the regularity of the Navier-Stokes equations in critical space. Let $ u(x,t) $ and $ p(x,t) $ denote suitable weak solution of the Navier-Stokes equations in $Q_T=\mathbb{R}^3\times(-T, 0)$. We prove that if $u(x,t)$…

The first goal of our paper is to give a new type of regularity criterion for solutions $u$ to Navier-Stokes equation in terms of some supercritical function space condition $u \in L^{\infty}(L^{\alpha ,*})$ (with…

This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $\epsilon$ regularity criterion, we show that if the mild solution…

Global existence of strong solutions to the three-dimensional incompressible Navier-Stokes equations remains an open problem. A posteriori existence results offer a way to rigorously verify the existence of strong solutions by ruling out…

We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and \v{S}ver\'ak [J\v{S}14], is a central tool in two of the authors' recent work on…

Let us consider an initial data $v_0$ for the classical 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$, then for any…