Related papers: Alternative Theorem of Navier-Stokes Equations in …

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 paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in $\mathbb{R}$, where the viscosity depends on the density in a super-linear power law(i.e.,…

A sufficient condition of regularity for solutions to the Navier-Stokes equations is proved. It generalizes the so-called $L_{3,\infty}$-case.

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…

Higher moments of the vorticity field $\Omega_{m}(t)$ in the form of $L^{2m}$-norms ($1 \leq m < \infty$) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier-Stokes equations on the domain…

We show that a Leray-Hopf weak solution to the 3D Navier-Stokes Cauchy problem belonging to the space $L^\infty(0,T; B^{-1}_{\infty,\infty}(\mathbb R^3))$ is regular in $(0,T]$. As a consequence, it follows that any Leray-Hopf weak solution…

The initial problem for the Navier-Stokes type equations over ${\mathbb R}^n \times [0,T]$, $n\geq 2$, with a positive time $T$ in the spatially periodic setting is considered. First, we prove that the problem induces an open injective…

In this paper we will prove that the vorticity belongs to L1(0; T ; L2(R3)) for the Cauchy problem of 3D incompressible Navier-Stokes equation, then the existence of a global smooth solution is obtained. Our approach is to construct a set…

The global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed in [7] more than fifteen years ago. However, whether (optimal) time-decay rates could be shown in…

For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^\infty$ norm at time $T_*>0$, while remaining smooth in space and time on $\mathbb…

Let us consider an initial data $v_0$ for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$,…

We establish the existence of a uniformly bounded $ C^\infty $ solution of the Navier-Stokes equations on $\mathbb{R}^3 x\ [0, \infty) $ without external forces or boundaries for a divergence free initial condition $ u_o \in \cap_m H^m $…

We prove that the Cauchy problem for the three dimensional Navier-Stokes equations is ill posed in $\dot{B}^{-1,\infty}_{\infty}$ in the sense that a ``norm inflation'' happens in finite time. More precisely, we show that initial data in…

We utilize undetermined coefficient method and an iterative method to construct the series solutions of the 3D Cauchy problem for a class of incompressible Navier-Stokes and Euler Equations. Then we can turn the Navier-Stokes Equations…

Let $u_0\in C_0^5 ( B_{R_0})$ be divergence-free and suppose that $u$ is a strong solution of the three-dimensional incompressible Navier-Stokes equations on $[0,T]$ in the whole space $\mathbb{R}^3$ such that $\| u \|_{L^\infty ((0,T);H^5…

In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in $\Bbb R^3$ under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth…

Liouville type of theorems play a key role in the blow-up approach to study the global regularity of the three-dimensional Navier-Stokes equations. In this paper, we will prove Liouville type of theorems to the 3-D axisymmetric…

We prove space-time decay estimates of suitable weak solutions to the Navier-Stokes Cauchy problem, corresponding to a given asymptotic behavior of the initial data of the same order of decay. We use two main tools. The first is a result…

In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in $\Bbb R^3$, which model non-Newtonian fluids, where the Laplacian term $\Delta u$ is replaced by the corresponding non linear operator…

In this paper, we mainly investigate the Cauchy problem for the incompressible Navier-Stokes equations in homogeneous Besov spaces $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq p<\infty,\ 1\leq r\leq \infty, \ d\geq 2$. Firstly, we prove the…