Related papers: Typicality results for weak solutions of the incom…

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $u\in L_\infty^tL_{d}^x((0,T)\times {\mathbb R}^d)$ is a Leray-Hopf weak solution, then $u$ is smooth and unique in…

Let $r,s \in [2,\infty]$ and consider the Navier-Stokes equations on $\mathbb{R}^3$. We study the following two questions for suitable $s$-homogeneous Banach spaces $X \subset \mathcal{S}'$: does every $u_0 \in L^2_\sigma$ have a weak…

This paper addresses a question concerning the behaviour of a sequence of global solutions to the Navier-Stokes equations, with the corresponding sequence of smooth initial data being bounded in the (non-energy class) weak Lebesgue space…

We show that t^{3/4}|| u(.,t) ||_{sup} --> 0 as t --> infty for all (global) Leray solutions of the incompressible Navier-Stokes equations in R3. It is also shown that t || u(.,t) - v(.,t) ||_{sup} --> 0 as t --> infty, where v(.,t) is the…

We obtain a new inequality that holds for general Leray solutions of the incompressible Navier-Stokes equations in Rn (n <= 4). This recovers important results previously obtained by other authors regarding the time decay of solution…

In this paper, we prove a sharp nonuniqueness result for the incompressible Navier-Stokes equations in the periodic setting. In any dimension $d \geq 2$ and given any $ p<2$, we show the nonuniqueness of weak solutions in the class $L^{p}_t…

We prove that if $u$ is a suitable weak solution to the three dimensional Navier-Stokes equations from the space $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, then all scaled energy quantities of $u$ are bounded. As a consequence, it is…

In this paper, we consider the fractional Navier-Stokes equations. We extend a previous non-uniqueness result due to Cheskidov and Luo, found in [5], from Navier-Stokes to the fractional case, and from $L^1$-in-time, $W^{1,q}$-in-space…

In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connected bounded domain, either in 2D or in 3D, with a Navier…

We prove that if the local second-order structure function exponents in the inertial range remain positive uniformly in viscosity, then any spacetime $L^2$ weak limit of Leray--Hopf weak solutions of the Navier-Stokes equations on any…

We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type…

This is a translation from French of my paper [R. May, Extension d'une classe d'unicite pour les equations de Navier-Stokes, Ann. I. H. Poincar\'{e}-AN 27 (2010) 705-718. doi:10.1016/j.anihp.2009.11.007]. Q. Chen, C. Miao, and Z. Zhang…

Ruggeri's hyperbolic Navier-Stokes equations are shown to possess, for any equilibrium state, smooth solutions in arbitrarily small $L^\infty$ neigborhoods of the reference state that in finite time cease to be differentiable.

We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the $3D$ incompressible Navier-Stokes equations with nonlinear exponential damping term $a (e^{b |u|^{\bf 4}}-1)u$, ($a,b>0$).

We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incom-pressible Navier-Stokes equations with initial data in weighted L 2 spaces. Our principal result concerns the…

We study Liouville type of theorems for the Navier-Stokes and the Euler equations on $\Bbb R^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$ and $\int_{\Bbb…

We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus ${\mathbb T}^d$, assuming that the solutions have norms for Besov space $B^{\sigma,\infty}_3({\mathbb T}^d),$…

We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the $3D$ incompressible Navier-Stokes equations with the nonlinear exponential damping term $a (e^{b |u|^{\bf 2}}-1)u$, ($a,b>0$) studied…

We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty, 0] \times (\mathbb{R}^2 \times \mathbb{T}^1)$. This is a step forward to completely solve the…

We show the existence of (a class of) weak solutions to the three-dimensional stationary incompressible inhomogeneous Navier--Stokes equations with density-dependent viscosity coefficient in the axially symmetric case. Further symmetric…