The aim of this paper is to derive a general model for reduced viscous and resistive Magnetohydrodynamics (MHD) and to study its mathematical structure. The model is established for arbitrary density profiles in the poloidal section of the toroidal geometry of Tokamaks. The existence of global weak solutions, on the one hand, and the stability of the fundamental mode around initial data, on the other hand, are investigated.

Nous prouvons que pour toute solution $u$ du problème de Kelvin–Helmholtz des nappes de tourbillons pour l’équation d’Euler bi-dimensionnelle, définie localement en temps, la courbe de saut de $u$ et la densité de tourbillon sont analytiques (sous une hypothèse de régularité Holderienne de la courbe de saut). Nous donnons également un résultat de régularité partielle de la trace de $u$ sur $t=0$ lorsque $u$ est définie sur un demi-interval $[O,T[$.

Nous prouvons que pour toute solution u du problème de Kelvin–Helmholtz des nappes de tourbillons pour l'équation d'Euler bi-dimensionnelle, définie localement en temps, la courbe de saut de u et la densité de tourbillon sont analytiques (sous une hypothèse de régularité Holderienne de la courbe de saut). Nous donnons également un résultat de régularité partielle de la trace de u sur t=0 lorsque u est définie sur un demi-interval [O,T[.

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow...

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 the note we are concerned with higher regularity and uniqueness of solutions to the stationary problem arising from the large eddy simulation of turbulent flows. The system of equations contains a nonlocal nonlinear term, which prevents straightforward application of a difference quotients method. The existence of weak solutions was shown in A. Świerczewska: Large eddy simulation. Existence of stationary solutions to the dynamical model, ZAMM, Z. Angew. Math. Mech. 85 (2005), 593–604 and P....

In the context of the weak solutions of the Navier-Stokes equations we study the regularity of the pressure and its derivatives in the space-time neighbourhood of regular points. We present some global and local conditions under which the regularity is further improved.

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T4$ configurations. We then use this to construct weak solutions to the unstable interface problem (the...

J. Q. Yang (2019) established a regularity criterion for the 3D shear thinning fluids in the whole space ${ℝ}^3$ via two velocity components. The goal of this short note is to extend this result in viewpoint of Lorentz space.

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.

This article recalls the results given by A. Dutrifoy, A. Majda and S. Schochet in [1] in which they prove an uniform estimate of the system as well as the convergence to a global solution of the long wave equations as the Froud number tends to zero. Then, we will prove the convergence with weaker hypothesis and show that the life span of the solutions tends to infinity as the Froud number tends to zero.