Related papers: Quantitative classification of potential Navier-St…
Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D…
In this paper, we mainly prove the existence of the minimal blow-up initial data in critical Fourier-Herz space $F\dot{B}^{2-{\frac3p}}_{p,q}(\RR^3)$ with $1<p\leq\infty$ and $1\leq q<\infty$ for the three dimensional incompressible…
Recently Qi S. Zhang provides examples of solutions to the Navier-Stokes equations which, under suitable hypothesis, blow up in finite time. He considers axially symmetric solutions in a cylinder $D\,$ under appropriate boundary conditions…
In this paper we give optimal lower bounds for the blow-up rate of the $\dot{H}^{s}\left(\mathbb{T}^3\right)$-norm, $\frac{1}{2}<s<\frac{5}{2}$, of a putative singular solution of the Navier-Stokes equations, and we also present an…
We study the singularity formation of a quasi-exact 1D model proposed by Hou-Li in \cite{hou2008dynamic}. This model is based on an approximation of the axisymmetric Navier-Stokes equations in the $r$ direction. The solution of the 1D model…
We consider the 3D stochastic Navier-Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates in for the energy error with…
We study the low-energy solutions to the 3D compressible Navier-Stokes-Poisson equations. We first obtain the existence of smooth solutions with small $L^2$-norm and essentially bounded densities. No smallness assumption is imposed on the…
We construct weak solutions to the Navier-Stokes inequality, $$ u\cdot \left(\partial_t u -\nu \Delta u + (u\cdot \nabla) u +\nabla p \right) \leq 0 $$ in $\mathbb{R}^3$, which blow up at a single point $(x_0,T_0)$ or on a set $S \times…
Under assumption that $T^{\ast}$ is the maximal time of existence of smooth solution of the 3D Navier-Stokes equations in the Sobolev space $H^{s}$, we establish lower bounds for the blow-up rate of the type$\ \left( T^{\ast }-t\right)…
In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from \cite{Isab}, we…
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…
In this work we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant…
In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the…
We consider some complex-valued solutions of the Navier-Stokes equations in $R^{3}$ for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of…
We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier-Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is \textit{a priori} estimate…
The potential failure of energy equality for a solution $u$ of the Euler or Navier-Stokes equations can be quantified using a so-called `energy measure': the weak-$*$ limit of the measures $|u(t)|^2\,\mbox{d}x$ as $t$ approaches the first…
This paper introduces a novel class of initial data for which the three-dimensional incompressible Navier--Stokes equations yield unique global-in-time solutions. Building on a logarithmically improved regularity criterion, we impose a…
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 prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The…
In the note, various scenarios of potential Type II blowups of suitable weak solutions to the Navier-Stokes equations are studied. It is shown, that under some assumptions, such type of blowups cannot happen. In this case, corresponding…