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.[-]

The purpose of this talk is to present two 1d congestion models: a soft congestion model with a singular pressure, and a hard congestion model in which the dynamic is different in the congested and non-congested zone (incompressible vs. compressible dynamic). The hard congested model is the limit of the soft one as the parameter within the singular presure vanishes.
For each model, we prove the existence of traveling waves, and we study their stability. This is a joint work with Charlotte Perrin.[-]

Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn's Hexagon, the huge cloud pattern at the level of Saturn's north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray's solutions of the Navier–Stokes equations. Our analysis also makes evidence of the isotropic character of the energy density of the fluid for sufficiently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.[-]

In this talk, I will present recent results on solutions to a one-dimensional Euler system coupling compressible and incompressible phases. With this original fluid system we intend to model congestion (or saturation) phenomena in heterogeneous flows (mixtures, wave-structure interactions, collective motion, etc.). Here the compressible-incompressible model will be seen as the limit of a fully compressible Euler system endowed with a singular pressure law. The goal of the talk is to present theoretical results concerning this singular limit. This is a joint work with Roberta Bianchini.[-]