Related papers: Global regularity for the hyperdissipative Navier-…
We investigate the formation of singularities in the incompressible Navier-Stokes equations in $d\geq 2$ dimensions with a fractional Laplacian $|\nabla |^\alpha$. We derive analytically a sufficient but not necessary condition for…
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 the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial…
We prove that the steady--state Navier--Stokes problem in a plane Lipschitz domain $\Omega$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $\a \in L^2(\partial\Omega)$ satisfies ${1\over…
First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The…
This paper considers solutions $u_\alpha$ of the three-dimensional Navier--Stokes equations on the periodic domains $Q_\alpha:=(-\alpha,\alpha)^3$ as the domain size $\alpha\to\infty$, and compares them to solutions of the same equations on…
Consider the Cauchy problem of incompressible Navier-Stokes equations in $\mathbb{R}^3$ with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the…
In this paper, we study the global regularity of strong solution to the Cauchy problem of 3D incompressible Navier-Stokes equations with large data and non-zero force. We prove that the strong solution exists globally for $\nabla u\in…
The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for…
In a recent article, J.-Y. Chemin, I. Gallagher and M. Paicu obtained a class of large initial data generating a global smooth solution to the three dimensional, incompressible Navier-Stokes equations. This data varies slowly in the…
We extend the Caffarelli-Kohn-Nirenberg type partial regularity theory for the steady $5$-dimensional fractional Navier-Stokes equations with external force to the hyperdissipative setting. In our argument we use the methods of Colombo-De…
We consider the global Cauchy problem for the generalized incompressible Navier- Stokes system in 3D whole space $$ u_t+u\cdot\nabla u+\nabla p=\mathcal{A}_h u, $$ \begin{equation}\label{main0} \nabla\cdot u=0, \end{equation} $$…
In this paper, we consider the model of 3D incompressible Navier-Stokes equations and 2D supercritical Surface Quasi-Geostrophic equations with time oscillation in the nonlinear term. We obtain that there exists global smooth solution of…
We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t \theta + \mathcal{R}^\perp \theta \cdot \nabla \theta + \Lambda^\gamma \theta = 0, \qquad \theta(\cdot,0) =\theta_0 \] on…
This paper investigates the existence and regularity of strong solutions to the incompressible Navier-Stokes equations within a bounded domain $\Omega \subset \mathbb{R}^3$, subject to the boundary condition $(u\cdot \vec{n})|_{\partial…
In this paper we study the periodic Navier--Stokes equation. From the periodic Navier--Stokes equation and the linear equation $\partial_t u = \nu\Delta u + \mathbb{P} [v\nabla u]$ we derive the corresponding equations for the time…
In geophysical flows such as large-scale ocean dynamics, the vertical viscosity is often much smaller than the horizontal viscosity. This anisotropy makes it natural to ask whether solutions of the full anisotropic compressible…
We study the existence and stability of non-trivial steady-state solutions to the two-dimensional incompressible Navier-Stokes equations in an annular domain $\Omega = B(0,b) \setminus \overline{B(0,a)}$ with radii $b>a>0$.The outer…
In this paper, we generalize the main results of [1] and [31] to Lorentz spaces, using a simple procedure. The main results are the following. Let $n\geq 3$ and let $u$ be a Leray-Hopf solution to the $n$-dimensional Navier-Stokes equations…
In this paper we prove that the Navier-Stokes initial value problem (1) has a unique smooth local strong solution and if the following condition are satisfied (1) and is H\"older continuous about on, (2) The initial value