Related papers: Dissipative models generalizing the 2D Navier-Stok…
We show that any Leray-Hopf weak solution to the $d$-dimensional Navier-Stokes equations $(d\geq 3)$ with initial values $u_0\in H^{s}(\mathbb R^d)$, $s\geq -1+\frac{d}{2}$, belongs to $L^\infty(0,\infty; H^{s}(\mathbb R^d))$ and thus it is…
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…
The real Schur form (RSF) of a generic velocity gradient field $\nabla \mathbf{u}$ is exploited to expose the structures of flows, in particular our field decomposition resulting in two vorticities with only mutual linkage as the…
Motivated by the classification of solutions of harmonic functions, we investigate Liouville type theorems for the fractional Navier-Stokes equations in $\mathbb{R}^3$ under some conditions on the boundedness of fractional derivatives. We…
In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the…
In this article, we study the regularity theory for two linear equations that are important in fluid dynamics: the passive scalar equation for (time-varying) shear flows close to Couette in $\mathbb T \times [-1,1]$ with vanishing…
In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous…
Fluid configurations in three-dimensions, displaying a plausible decay of regularity in a finite time, are suitably built and examined. Vortex rings are the primary ingredients in this study. The full Navier-Stokes system is converted into…
Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if…
By applying the delicate \textit{a priori} estimates for the equations of $(\Phi,\Gamma)$, which is introduced in the previous work, we obtain some multi-scale regularity criteria of the swirl component $u^{\theta}$ for the 3D axisymmetric…
In this paper, we simplify and extend the results of \cite{GZ} to include the case in which $\Om =\R^3$. Let ${[L^2({\mathbb{R}}^3)]^3}$ be the Hilbert space of square integrable functions on ${\mathbb {R}}^3 $ and let ${\mathbb…
For two dimensional inhomogeneous Navier-Stokes of incompressible flows, with the assumption that the viscosity depends on the density but with a positive lower bound, using a partial regularity approach, in particular some enhanced decay…
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…
In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative of the velocity in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular…
The full compressible Navier-Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid is studied in a three-dimensional simply connected bounded domain with smooth boundary having a…
We investigate in the present paper the Navier-Stokes equations on quantum Euclidean spaces $\mathbb{R}^d_{\theta}$ with $\theta$ being a $d\times d$ antisymmetric matrix, which is a standard example of non-compact noncommutative manifolds.…
In this paper we show that the long time dynamics (the global attractor) of the 2D Navier-Stokes equation is embedded in the long time dynamics of an ordinary differential equation, named {\it determining form}, in a space of trajectories…
In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-\Delta)^{\alpha}$ and magnetic diffusion given by reducing about logarithmic diffusion…
We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions…
The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation…