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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This paper deals with a degenerate diffusion Patlak–Keller–Segel system in $n\geq 3$ dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent $m=2n/(n+2)$, which is smaller than the usual exponent $m^{*}=2-2/n$ used in other studies. With the exponent $m=2n/(n+2)$, the associated free energy is conformal invariant, and there is a family of stationary solutions $U_{\lambda,x_0}(x)=C(\frac{\lambda} {\lambda^2+|x-x_0|^2})^{\frac{n+2}{2}}$ $\forall \lambda>0$, $x_0\in {\mathbb R}^n$. For radially symmetric solutions, we prove that if the initial data are strictly below $U_{\lambda,0}(x)$ for some $\lambda$, then the solution vanishes in $L^1_{loc}$ as $t\to\infty$; if the initial data are strictly above $U_{\lambda,0}(x)$ for some $\lambda$, then the solution either blows up at a finite time or has a mass concentration at $r=0$ as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the $L^m$ norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow-up of the solution if the $L^m$ norm for initial data is larger than the $L^m$ norm of $U_{\lambda,x_0}(x)$, which is constant independent of $\lambda$ and $x_0$, and the free energy of initial data is smaller than that of $U_{\lambda,x_0}(x)$.

We establish two nonlinear compactness theorems in Lp(0,T;B) with hypothesis on time translations, which are nonlinear counterparts of two results by Simon (1987) [1]. The first theorem sharpens a result by Maitre (2003) [10] and is important in the study of doubly nonlinear elliptic–parabolic equations. Based on this theorem, we then obtain a time translation counterpart of a result by Dubinskii˘ (1965) [5], which is supposed to be useful in the study of some nonlinear kinetic equations (e.g. the FENE-type bead–spring chains model).

In this paper we study the Euler-Poincar\'{e} equations in $\Bbb R^N$. We prove local existence of weak solutions in $W^{2,p}(\Bbb R^N),$ $p>N$, and local existence of unique classical solutions in $H^k (\Bbb R^N)$, $k>N/2+3$, as well as a blow-up criterion. For the zero dispersion equation($\alpha=0$) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as $\alpha\to0$, provided that the limiting solution belongs to $C([0, T);H^k(\Bbb R^N))$ with $k>N/2 +3$. For the {\em stationary weak solutions} of the Euler-Poincar\'{e} equations we prove a Liouville type theorem. Namely, for $\alpha>0$ any weak solution $\mathbf{u}\in H^1(\Bbb R^N)$ is $\mathbf{u}=0$; for $\alpha=0$ any weak solution $\mathbf{u}\in L^2(\Bbb R^N)$ is $\mathbf{u}=0$.