Motivated by a phenomenon of phase transition in a model of alignment of self-propelled particles, we obtain a kinetic mean-field equation which is nothing else than the Doi equation (also called Smoluchowski equation) with dipolar potential. In a self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global well-posedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady-states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, there is also a family of non-isotropic equilibria. We give a rigorous prove of convergence of the solution to a steady-state as time goes to infinity. In particular we show that in the supercritical case, the only initial conditions leading to the uniform distribution in large time are those with vanishing momentum. For any positive value of the noise parameter, and any initial condition, we give rates of convergence towards equilibrium, exponentially for both supercritical and subcritical cases and algebraically for the critical case.

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The time evolution of the distribution function for the charged particles in a dilute gas is governed by the VlasovPoissonBoltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible NavierStokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].

In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting $\varepsilon >0$ to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders $O(1)$ and $O(\varepsilon^\alpha)$, respectively, with $\alpha>2$, for which the limiting system is the primitive equations with only horizontal viscosity as $\varepsilon$ tends to zero. In particular we show that for ``well prepared" initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as $\varepsilon$ tends to zero, and that the convergence rate is of order $O\left(\varepsilon^\frac\beta2\right)$, where $\beta=\min\{\alpha-2,2\}$. Note that this result is different from the case $\alpha=2$ studied in [Li, J.; Titi, E.S.: \emph{The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., \textbf{124} \rm(2019), 30--58], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order $O\left(\varepsilon\right)$.

The compressible Euler equations, which govern the gas flow surrounding a solid ball with mass M and frictional damping in n dimensions, t+(u)= 0,(u) t+ (u u)+ P ()=-Mx/| x| n-2u, where , u, P and M are the density, velocity, pressure and mass of the gas, respectively, n 3 is the dimension of x, and > 0 is the frictional constant, are examined. The pressure is assumed to satisfy the law and 12 , K is a positive constant. The existence and non-existence of global smooth solutions are studied for the initial boundary problem of the Euler equations.