We prove the existence and uniqueness of global strong solutions to the Cauchy problem for 3D incompressible MHD equations with nonlinear damping terms. Moreover, the preliminary L² decay for weak solutions is also established.

In this short note we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation and the behavior of the direction of the velocity $u/\left|u\right|$. It is shown that the control of $\mathrm{Div}(u/|u\left|\right)$ in a suitable $L_t^p\left(L_x^q\right)$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very...

We prove a regularity criterion for a nonhomogeneous incompressible Ginzburg-Landau-Navier-Stokes system with the Coulomb gauge in ${ℝ}^3$. It is proved that if the velocity field in the Besov space satisfies some integral property, then the solution keeps its smoothness.

Existence of a global attractor for the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe has been shown already. In this paper we prove the higher regularity of the attractor.

Some results on regularity of weak solutions to the Navier-Stokes equations published recently in [3] follow easily from a classical theorem on compact operators. Further, weak solutions of the Navier-Stokes equations in the space $L^2(0,T,W^{1,3}{\left(𝛺\right)}^3)$ are regular.

We consider the axisymmetric Navier-Stokes equations with non-zero swirl component. By invoking the Hardy-Sobolev interpolation inequality, Hardy inequality and the theory of $*A_{\beta }$ (1 < β < ∞) weights, we establish regularity criteria involving $u^r$, ${\omega }^z$ or ${\omega }^{\theta }$ in some weighted Lebesgue spaces. This improves many previous results.

We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that $$|{\omega }^r(x,t)|+|{\omega }^z(r,t)|\le \frac{C}{r^{10}},0<r\le \frac{1}{2}.$$ By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ${\omega }^r$, ${\omega }^z$ and ${\omega }^{\theta }/r$ on different hollow cylinders, we are able to improve it and obtain $$|{\omega }^r(x,t)|+|{\omega }^z(r,t)|\le \frac{C\left|\ln r\right|}{r^{17/2}},0<r\le \frac{1}{2}.$$

We study the Cauchy problem for the MHD system, and provide two regularity conditions involving horizontal components (or their gradients) in Besov spaces. This improves previous results.

Nous démontrons dans cet article que le système MHD tridimensionnel à densité et viscosité variables est localement bien posé lorsque $({{\rho }_0}^{-1}-1,u_0,B_0)\in {\dot{B}}_{p1}^{\frac{3}{p}}\left({ℝ}^3\right)\times {\dot{B}}_{p1}^{\frac{3}{p}-1}\left({ℝ}^3\right)\times {\dot{B}}_{p1}^{\frac{3}{p}-1}\left({ℝ}^3\right),$ pour $p\in ]1,3]$ et la densité initiale est proche d’une constante strictement positive. Nous démontrons également un résultat d’existence et d’unicité dans l’espace de Sobolev $H^{\frac{3}{2}+\alpha }\left({ℝ}^3\right)\times H^{\frac{3}{2}-1+\alpha }\left({ℝ}^3\right)\times H^{\frac{3}{2}-1+\alpha }\left({ℝ}^3\right)$ pour $\alpha \&gt;0,$ sans aucune condition de petitesse sur la densité.

La compréhension du passage des équations de la mécanique des fluides compressibles aux équations incompressibles a fait de grands progrès ces vingt dernières années. L’objectif de cet exposé est de présenter l’évolution des méthodes mathématiques mises en œuvre pour étudier ce passage à la limite, depuis les travaux de S. Klainerman et A. Majda dans les années quatre–vingts, jusqu’à ceux récents de G. Métivier et S. Schochet (pour les équations non isentropiques). Suivant les conditions initiales...