相关论文: An alternative approach to regularity for the Navi…

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 is the second in a series of papers where we analyze the incompressible Navier-Stokes equations in H\"older spaces. We obtain, to our knowledge, the very first genuinely super-critical regularity criterion for this system of equations…

It is known that uniqueness of mild solutions to the incompressible Navier-Stokes equations holds in the critical class $C([0,T);L^n(\mathbb{R}^n))$ for $n \geqslant 3$. In this paper, we prove that this result is sharp in the sense that…

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We…

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…

A modified version of the three dimensional Navier-Stokes equations is considered with periodic boundary conditions. A bounded constant delay is introduced into the convective term, that produces a regularizing effect on the solution. In…

This paper addresses the numerical solution of the two-dimensional Navier--Stokes (NS) equations with nonsmooth initial data in the $L^2$ space, which is the critical space for the two-dimensional NS equations to be well-posed. In this…

The goal of this paper is twofold. First, we give a simple proof that sufficiently sparse Navier--Stokes solutions do not develop singularities. This provides an alternative to the approach of \cite{Grujic2013}, which is based on…

This paper extends the weak solution theory for the 3D Navier-Stokes equations of Barker, Seregin and Sverak from a critical setting to a supercritical setting making sure to include a useful a priori energy bound as well as a statement…

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…

H.-O. Bae and H.J. Choe, in a 1997 paper, established a regularity criteria for the incompressible Navier-Stokes equations in the whole space $\R^3$ based on two velocity components. Recently, one of the present authors extended this result…

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…

We show that for any given solenoidal initial data in $L^2$ and any solenoidal external force in $L_{\text{loc}}^q \bigcap L^{3/2}$ with $q>3$, there exist partially regular weak solutions of the Navier-Stokes equations in $\R^4 \times…

This paper deals with the uniqueness of mild solutions to the forced or unforced Navier-Stokes equations in the whole space. It is known that the uniqueness of mild solutions to the unforced Navier-Stokes equations holds in…

We are concerned with bilinear estimates and uniqueness of mild solutions for the Navier-Stokes equations in critical spaces. For that, we construct general settings in which estimates for the bilinear term of the mild formulation hold true…

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…

In this paper we show the non-uniqueness of mild solutions to the two-dimensional forced Navier-Stokes equations in the half space under the noslip boundary condition, following the program established by Albritton, Bru{\'e}, and Colombo in…

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 are concerned with the barotropic compressible Navier-Stokes equations on the real line. Our primary goal is to establish the global well-posedness in a critical regularity framework in the case where the initial data are small…

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 $u(x,t)$…