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In the present work we prove the interior Hölder continuity of the gradient matrix of any weak solution of equations, which describes the motion of non-Newtonian fluid in two dimensions, restricting ourself to the shear thinning case .

In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if $limsu{p}_{R\to 0⁺}1/R{\int }_{Q_R(x₀,t₀)}|curlu\times u/|u\left|\right|²dxdt\le {\epsilon }_{*}$ for a sufficiently small ${\epsilon }_{*}>0$.

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 $u$.

We prove the interior Hölder continuity of weak solutions to parabolic systems $$\frac{\partial u^j}{\partial t}-D_{\alpha }{a}_j^{\alpha }(x,t,u,\nabla u)=0\text{in}Q(j=1,...,N)$$ ($Q=\Omega \times (0,T),\Omega \subset {ℝ}^2$), where the coefficients $a_j^{\alpha }(x,t,u,\xi )$ are measurable in $x$, Hölder continuous in $t$ and Lipschitz continuous in $u$ and $\xi$.

We study a linear system of equations arising from fluid motion around a moving rigid body, where rotation is included. Originally, the coordinate system is attached to the fluid, which means that the domain is changing with respect to time. To get a problem in the fixed domain, the problem is rewritten in the coordinate system attached to the body. The aim of the present paper is the proof of the existence of a strong solution in a weighted Lebesgue space. In particular, we prove the existence...