Related papers: Small $\dot B^{-1}_{\infty,\infty}$ implies regula…

In this paper, we prove that a local weak solution to the $d$-dimensional incompressible Navier-Stokes equations ($d \geq 2$) can be constructed by taking the hydrodynamic limit of a velocity-discretized Boltzmann equation with a simplified…

The paper deals with the regularity criterion for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution $(u,\theta )$ becomes regular provided that…

In the classic work of Beale-Kato-Majda ({[}2{]}) for the Euler equations in $\mathbb{R^{\mathrm{3}}}$, regularity of a solution throughout a given interval $[0,T_{*}]$ is obtained provided that the curl $\omega$ satisfies $\omega\in…

The asymptotic stability is one of the classical problems in the field of mathematical analysis of fluid mechanics. In $\mathbb{R}^n$ with $n \geq 3$, it is easily proved by the standard argument that if the given small external force…

For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on…

This paper studies the stability of a stationary solution of the Navier-Stokes system with a constant velocity at infinity in an exterior domain. More precisely, this paper considers the stability of the Navier-Stokes system governing the…

This paper studies the dynamics of two incompressible immiscible fluids in 2D modeled by the inhomogeneous Navier-Stokes equations. We prove that if initially the viscosity contrast is small then there is global-in-time regularity. This…

We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic…

Let us consider an initial data $v_0$ for the classical 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 $T^\star$, then for any…

We establish some interior regularity criterions of suitable weak solutions for the 3-D Navier-Stokes equations, which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve…

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 prove several Liouville type results for stationary solutions of the $d$-dimensional compressible Navier-Stokes equations. In particular, we show that when the dimension $d \geqslant 4$, the natural requirements $\rho \in L^{\infty}…

We give a new concise proof of a certain one-scale epsilon regularity criterion using weak-strong uniqueness for solutions of the Navier-Stokes equations with non-zero boundary conditions. It is inspired by an analogous approach for the…

We study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time…

This paper is dedicated to the construction of a pseudo-norm, for which small shock profiles of the barotropic Navier-Stokes equation have a contraction property. This contraction property holds in the class of any large 1D weak solutions…

In a plane polygon $P$ with straight sides, we prove analytic regularity of the Leray-Hopf solution of the stationary, viscous, and incompressible Navier-Stokes equations. We assume small data, analytic volume force and no-slip boundary…

Norm inflation implies certain discontinuous dependence of the solution on the initial value. The well-posedness of the mild solution means the existence and uniqueness of the fixed points of the corresponding integral equation. For ${\rm…

In the paper, a new {\it slightly supercritical} condition, providing {\it local} regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local…

We study the regularity criteria for weak solutions to the $3D$ incompressible Navier--Stokes equations in terms of the geometry of vortex structures, taking into account the boundary effects. A boundary regularity theorem is proved on…

We investigate the global regularity problem for the three-dimensional incompressible Navier-Stokes equations restricted to axisymmetric flows in a finite cylinder $D = \{(r,\theta,x_3): 0 \le r \le 1, 0 \le \theta < 2\pi, 0 \le x_3 \le…