Related papers: On possible time singular points and eventual regu…

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $\alpha$ in the nonlinear term in the following…

We prove a local-in-time regularity criterion for the 3D Navier-Stokes equations. In particular, it follows from the criterion that the Hausdorff dimension of possible singular times of Leray-Hopf weak solutions $u\in L^r_t…

We consider solutions of the Navier-Stokes equation with fractional dissipation of order $\alpha\geq 1$. We show that for any divergence-free initial datum $u_0$ such that $||u_0||_{H^{\delta}} \leq M$, where $M$ is arbitrarily large and…

We prove that for the Navier Stokes equation with dissipation $(-\Delta)^{\alpha}$, where $1<\alpha<{5/4}$, and smooth initial data, the Hausdorff dimension of the singular set at time of first blow up is at most $5-4\alpha$. This unifies…

We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian $(-\Delta )^s$ for $s\in (\frac34,1)$, and we provide a new bootstrapping scheme that makes it possible to…

In this paper, we consider suitable weak solutions of incompressible Navier--Stokes equations in four spatial dimensions. We prove that the two-dimensional time-space Hausdorff measure of the set of singular points is equal to zero.

In this note, we investigate partial regularity of weak solutions of the three dimensional chemotaxis-Navier-Stokes equations, and obtain the $\frac53$-dimensional Hausdorff measure of the possible singular set is vanishing at the first…

We consider a family of Leray-$\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\theta$, of the Navier-Stokes equations. In particular, they share…

By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force $f \in L_{\text{loc}}^q \cap L^{3/2}, q>3$ in general open subdomains of $\mathbb{R}^6$.…

We prove that the spacetime singular set of any suitable Leray-Hopf solution of the surface quasigeostrophic equation with fractional dissipation of order $0< \alpha < \frac{1}{2}$ has Hausdorff dimension at most $\frac{1}{2\alpha^2}\,.$…

We consider suitable weak solutions of the incompressible Navier--Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of…

In this paper, we establish temporal decay for a weak solution $u(x,t)$ (with initial data $u_0$) of the Navier-Stokes equations with supercritical fractional dissipation $\alpha \in (0,\frac{5}{4})$ in $L^2(\mathbb{R}^3)$ and…

We show that if $u$ is a Leray-Hopf weak solution to the incompressible Navier--Stokes equations with hyperdissipation $\alpha \in (1,5/4)$ then there exists a set $S\subset \mathbb{R}^3$ such that $u$ remains bounded outside of $S$ at each…

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokes equations. Firstly, we prove that the upper box dimension of the potential…

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…

We study partial regularity of weak solutions of the 3D valued non-stationary Hall magnetohydrodynamics equations on $ \Bbb R^2$. In particular we prove the existence of a weak solution whose set of possible singularities has the space-time…

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($\alpha< 1/2$) dissipation $(-\Delta)^\alpha$ : If a Leray-Hopf weak solution is H\"{o}lder continuous $\theta\in C^\delta({\mathbb…

Consider steady-state weak solutions to the incompressible Navier-Stokes equations in six spatial dimensions. We prove that the 2D Hausdorff measure of the set of singular points is equal to zero. This problem was mentioned in 1988 by…

We consider the behaviour of weak solutions of the unforced three-dimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ($\|u\|^2=\int|u|^2$) but infinite enstrophy ($\|Du\|^2=\int|{\rm curl}…

In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $\dot{H}^{-\alpha}(\mathbb{R}^{3})$ or $\dot{H}^{-\alpha}(\mathbb{T}^{3})$ with…