Decay of solutions to a new Hall–MHD system in R3

This paper discusses the large-time behavior of solutions for a new Hall–MHD system in ${\mathbb{R}}^3$. Using the Fourier splitting method, we establish the upper bound of the time-decay rate in $L^2({\mathbb{R}}^3)$ for weak solutions.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].     

Emergent dynamics of Cucker–Smale particles under the effects of random communication and incompressible fluids

We study the dynamics of infinitely many Cucker–Smale (C–S) flocking particles under the interplay of random communication and incompressible fluids. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the incompressible Navier–Stokes (N–S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present the global existence of weak and strong solutions in ${\mathbb{R}}^d$$(d=2,3)$. Under the extra regularity assumptions of the initial data, the unique solvability of strong solutions is also established in ${\mathbb{R}}^2$. In a large coupling regime and periodic spatial domain ${\mathbb{T}}^2:={\mathbb{R}}^2/{\mathbb{Z}}^2$, we show that the velocities of C–S particles and fluids are asymptotically aligned to two constant velocities which may be different.     

Global low-energy weak solution and large-time behavior for the compressible flow of liquid crystals

In this paper, we consider the weak solution of the simplified Ericksen–Leslie system modeling compressible nematic liquid crystal flows in ${\mathbb{R}}^3$. When the initial data are of small energy and initial density is positive and essentially bounded, we prove the existence of a global weak solution in ${\mathbb{R}}^3$. The large-time behavior of a global weak solution is also established.     

Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball

We establish new results for the radial nonlinear wave and Schrödinger equations on the ball in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from Burq and Tzvetkov (2008) [8], [9] and Tzvetkov (2006, 2008) [10], [11].     

Global weak solutions to the Vlasov–Poisson–BGK system for initial data in Lp(R3×R3)

We prove the existence of a global nonnegative weak solution to the Cauchy problem of the Vlasov–Poisson–BGK system for initial datum having finite mass and energy and belonging to $L^p({\mathbb{R}}^3\times {\mathbb{R}}^3)$ with $p>3$.