We studied coupled systems of the Fokker--Planck equation and the Navier--Stokes equation modeling the Hookean and the finitely extensible nonlinear elastic (FENE)-type polymeric flows. We proved the continuous embedding and compact embedding theorems in weighted spaces that naturally arise from related entropy estimates. These embedding estimates are shown to be sharp. For the Hookean polymeric system with a center-of-mass diffusion and a superlinear spring potential, we proved the existence of a global weak solution. Moreover, we were able to tackle the FENE model with $L^2$ initial data for the polymer density instead of the $L^\infty$ counterpart in the literature.

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 $U_{\lambda,x_0}(x)$.

In this paper, we introduce a random particle blob method for the Keller-Segel equation (with dimension ) and establish a rigorous convergence analysis.

Nous considérons un modèle provenant de la biologie, composé d03équations de chimiotactisme couplées aux équations de fluide visqueux incompressible par le transport et le for04age externe. L03existence globale des solutions du problème de Cauchy est étudiée sous certaines conditions. Précisément, pour le système chimiotactisme–Navier–Stokes en deux dimensions d03espace, nous obtenons l03existence globale pour des données grandes. En trois dimensions d03espace, nous démontrons l03existence globale des solutions faibles pour le système chimiotactisme–Stokes avec une diffusion non-linéaire de la densité des cellules.

We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.