Related papers: Self-improving bounds for the Navier-Stokes equati…

In this paper, we study the quantitative regularity and blowup criteria for classical solutions to the three-dimensional incompressible Navier-Stokes equations in a critical Besov space framework. Specifically, we consider solutions $u\in…

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…

We obtain an improved blow-up criterion for solutions of the Navier-Stokes equations in critical Besov spaces. If a mild solution $u$ has maximal existence time $T^* < \infty$, then the non-endpoint critical Besov norms must become infinite…

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…

We establish some interior regularity criterions of suitable weak solutions for 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…

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…

We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$, $p > 3$. These solutions satisfy a certain…

This paper is concerned with quantitative estimates for the Navier-Stokes equations. First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound…

In this paper, we prove a quantitative regularity theorem and a blow-up criterion of classical solutions for the three-dimensional Navier-Stokes equations. By adapting the strategy developed by Tao in [20], we obtain an explicit blow-up…

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 revisit the regularity theory of Escauriaza, Seregin, and \v{S}ver\'ak for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical $L^3_x(\mathbf{R}^3)$ norm. By replacing all invocations of…

We give new a priori assumptions on weak solutions of the Navier-Stokes equation so as to be able to conclude that they are smooth. The regularity criteria are given in terms of mixed radial-angular weighted Lebesgue space norms.

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 are concerned with the problem,originated from Seregin [18,19,20], what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3d Naiver-Stokes equations. We prove some interior regularity criteria, in…

We improve previous known lower bounds for Sobolev norms of potential blow-up solutions to the three-dimensional Navier-Stokes equations in $\dot{H}^\frac{3}{2}$. We also present an alternate proof for the lower bound for the…

We present some new regularity criteria for ``suitable weak solutions'' of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are H\"older continuous up to the boundary provided that the…

In this paper we develop new methods to obtain regularity criteria for the three-dimensional Navier-Stokes equations in terms of dynamically restricted endpoint critical norms: the critical Lebesgue norm in general or the critical weak…

We formulate a new criterion for regularity of a suitable weak solution v to the Navier-Stokes equations at the space-time point (x_0,t_0). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of…

We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space $\dot B^{-1+\frac 3p}_{p,q}(\mathbb{R}^3)$, with $3 <p,q< \infty$, gives rise to a strong solution with a singularity at a finite…

In this note, boundary Type I blowups of suitable weak solutions to the Navier-Stokes equations are discussed. In particular, it has been shown that, under certain assumptions, the existence of non-trivial mild bounded ancient solutions in…