We consider mixtures of compressible viscous fluids consisting of two miscible species. In contrast to the theory of non-homogeneous incompressible fluids where one has only one velocity field, here we have two densities and two velocity fields assigned to each species of the fluid. We obtain global classical solutions for quasi-stationary Stokes-like system with interaction term.

In this paper we consider weak solutions $𝐮:\Omega \to {ℝ}^d$ to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain $\Omega \subset {ℝ}^d$ ($d=2$ or $d=3$). For the critical case $q=\frac{3d}{d+2}$ we prove the higher integrability of $\nabla 𝐮$ which forms the basis for applying the method of differences in order to get fractional differentiability of $\nabla 𝐮$. From this we show the existence of second order weak derivatives of $u$.

The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, $2p<n$ for a solution of a degenerate nonlinear elliptic equation in a domain $B(x_0,1)\setminus F$, $F\subset B(x_0,d)=\{x\in {ℝ}^n|x_0-x|<d\}$, $d<\frac{1}{2}$, under the boundary-value conditions $u(x,k)=k$ for $x\in \partial F$, $u(x,k)=0$ for $x\in \partial B(x_0,1)$ and where $0<\rho \le dist(x,F)$, $w\left(x\right)$ is a weighted function from some Muckenhoupt class, and $ca{p}_{p,w}\left(F\right)$, $w\left(B\right(x,\rho \left)\right)$ are weighted capacity and measure of the corresponding sets.

We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem $$\begin{array}{ccc}\hfill -div\left(a(x,u)\right|{|}^{p-2}\nabla u)=& {\lambda b(x,u)\left|u\right|}^{p-2}u\text{in}\Omega ,u=\hfill & \hfill 0\text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^{\infty }\left(\Omega \right)$. The main tool is the investigation of the associated homogeneous eigenvalue problem...

On the complement of the unit disk $B$ we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field $u$ is equal to zero provided ${u|}_{\partial B}=0$ and ${lim}_{\left|x\right|\to \infty }{\left|x\right|}^{1/3}\left|u\left(x\right)\right|=0$ uniformly. For slow flows the latter condition can be replaced by ${lim}_{\left|x\right|\to \infty }\left|u\left(x\right)\right|=0$ uniformly. In particular, these results hold for the classical Navier-Stokes case.