In this talk we present recent results on the Hall-MHD system. We consider the incompressible MHD-Hall equations in $\mathbb{R}^3$.

$\partial_tu +u \cdot u + \nabla u+\nabla p = \left ( \nabla \times B \right )\times B +\nu \nabla u,$
$\nabla \cdot u =0, \nabla \cdot B =0, $
$\partial_tB - \nabla \times \left (u \times B\right ) + \nabla \times \left (\left (\nabla \times B\right )\times B \right ) = \mu \nabla B,$
$u\left (x,0 \right )=u_0\left (x\right ) ; B\left (x,0 \right )=B_0\left (x\right ).$

Here $u=\left (u_1, u_2, u_3 \right ) = u \left (x,t \right ) $ is the velocity of the charged fluid, $B=\left (B_1, B_2, B_3 \right ) $ the magnetic field induced by the motion of the charged fluid, $p=p \left (x,t \right )$ the pressure of the fluid. The positive constants $\nu$ and $\mu$ are the viscosity and the resistivity coefficients. Compared with the usual viscous incompressible MHD system, the above system contains the extra term $\nabla \times \left (\left (\nabla \times B\right )\times B \right ) $ , which is the so called Hall term. This term is important when the magnetic shear is large, where the magnetic reconnection happens. On the other hand, in the case of laminar ows where the shear is weak, one ignores the Hall term, and the system reduces to the usual MHD. Compared to the case of the usual MHD the history of the fully rigorous mathematical study of the Cauchy problem for the Hall-MHD system is very short. The global existence of weak solutions in the periodic domain is done in [1] by a Galerkin approximation. The global existence in the whole domain in $\mathbb{R}^3$ as well as the local well-posedness of smooth solution is proved in [2], where the global existence of smooth solution for small initial data is also established. A refined form of the blow-up criteria and small data global existence is obtained in [3]. Temporal decay estimateof the global small solutions is deduced in [4]. In the case of zero resistivity we present finite time blow-up result for the solutions obtained in [5]. We note that this is quite rare case, as far as the authors know, where the blow-up result for the incompressible flows is proved.[-]

In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of discoveries and works which have gone in several directions. Among them the most notable is the recent proof of Phil Isett of a long-standing conjecture of Lars Onsager in the theory of turbulent flows. In a joint work with László, Tristan Buckmaster and Vlad Vicol we improve Isett's theorem to show the existence of dissipative solutions of the incompressible Euler equations below the Onsager's threshold.[-]

We first summarize the derivation of viscoelastic (rate-type) fluids with stress diffusion that generates the models that are compatible with the second law of thermodynamics and where no approximation/reduction takes place. The approach is based on the concept of natural configuration that splits the total response between the current and initial configuration into the purely elastic and dissipative part. Then we restrict ourselves to the class of fluids where elastic response is purely spherical. For such class of fluids we then provide a mathematical theory that, in particular, includes the long-time and large-data existence of weak solution for suitable initial and boundary value problems. This is a joint work with Miroslav Bulicek, Vit Prusa and Endre Suli.[-]

Consider the motion of a viscous incompressible fluid in a 3D exterior domain $D$ when a rigid body $\mathbb R^3\setminus D$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near $t=s$ as well as generation of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ was shown by Hansel and Rhandi [1] under reasonable conditions. In this presentation we develop the $L^q$-$L^r$ decay estimates of the evolution operator $T(t,s)$ as $(t-s)\to\infty$ and then apply them to the Navier-Stokes initial value problem.[-]

In this talk we shall present some results concerning global smooth solutions to the three-dimensional Navier-Stokes equations set in the whole space $(\mathbb{R}^3)$ :
$\partial_tu+u\cdot \nabla u-\Delta u = -\nabla p$, div $u=0$
We shall more particularly be interested in the geometry of the set $\mathcal{G}$ of initial data giving rise to a global smooth solution.
The question we shall address is the following: given an initial data $u_0$ in $\mathcal{G}$ and a sequence of divergence free vector fields converging towards $u_0$ in the sense of distributions, is the sequence itself in $\mathcal{G}$ ? The related question of strong stability was studied in [1] and [2] some years ago; the weak stability result is a recent work, joint with H. Bahouri and J.-Y. Chemin (see [3]-[4]). As we shall explain, it is necessary to restrict the study to sequences converging weakly up to rescaling (under the natural rescaling of the equation). Then weak stability can be proved, using profile decompositions in the spirit of P. Gerard's work [5], in an anisotropic context.[-]

The purpose of the talk will be the proof of the following result for the homogeneous incompressible Navier-Stokes system in dimension three: given an initial data $v_0$ with vorticity $\Omega_0= \nabla \times v_0$ in $L^{\tfrac{3}{2}}$ (which implies that $v_0$ belongs to the Sobolev space $H^{\tfrac{1}{2}}$ ), we prove that the solution $v$ given by the classical Fujita-Kato theorem blows up in a finite time $T^*$ only if, for any $p$ in ]4,6[ and any unit vector $e$ in $\mathbb{R}^3$ ; there holds
$\int_{0}^{T^*}\left \| v(t)\cdot e\right \|^p_{\frac{1}{2}+\frac{2}{p}}dt=\infty $.
We remark that all these quantities are scaling invariant under the scaling transformation of Navier-Stokes system.[-]

It is well known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence.
In recent joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov's h-principle.
In this lecture we explain this connection and outline recent progress towards Onsager's conjecture.[-]