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 $\mathbf{u}=0$.

nthispaperwewillpresentthelocalstabilityanalysisandlocalerrorestimatefor the local discontinuous Galerkin (LDG) method, when solving the time-dependent singularly perturbed problems in one dimensional space with a stationary outflow boundary layer. Based on a general framework on the local stability, we obtain the optimal error estimate out of the local subdomain, which is nearby the outflow boundary point and has the width of O(h log(1/h)), for the semi-discrete LDG scheme and the fully-discrete LDG scheme with the second order explicit Runge–Kutta time-marching. Here h is the maximum mesh length. The numerical experiments are given also.

We present a domain decomposition method on a semilinear hyperbolic system with multiple relaxation times. In the region where the relaxation time is small, an asymptotic equilibrium equation can be used for computational efficiency. An interface condition based on the sign of the characteristic speed at the interface is provided to couple the two systems in a domain decomposition setting. A rigorous analysis, based on the Laplace Transform, on the $ L^2$ error estimate is presented for the linear case, which shows how the error of the domain decomposition method depends on the smaller relaxation time, and the boundary and interface layer effects. The given convergence rate is optimal. We present a numerical implementation of this domain decomposition method, and give some numerical results in order to study the performance of this method.