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### Open Access Journals

DCDS-S

We study the
regularity of steady planar flow of fluids where the shearing stress
may depend on the symmetric part of the
velocity vector field and the pressure. For simplicity the periodic
boundary conditions are considered. Using Meyers estimates we show
that there exists a solution which is smooth. In the case where
it is allowed to test weak formulation of the problem with a weak
solution we prove regularity of all weak solutions.

DCDS

The equations describing planar motion of a homogeneous,
incompressible generalized Newtonian fluid are considered.
The stress tensor is given constitutively as
$\T=\nu(1+\mu|\Du|^2)^{\frac{p-2}2}\Du$, where $\Du$ is the
symmetric part of the velocity gradient.
The equations are complemented by periodic boundary conditions.

For the solution semigroup the Lyapunov exponents are computed using a slightly generalized form of the Lieb-Thirring inequality and consequently the fractal dimension of the global attractor is estimated for all $p\in(4/3,2]$.

For the solution semigroup the Lyapunov exponents are computed using a slightly generalized form of the Lieb-Thirring inequality and consequently the fractal dimension of the global attractor is estimated for all $p\in(4/3,2]$.

CPAA

The equations of an incompressible, homogeneous fluid
occupying a bounded domain in $\mathbb R^3$ are considered.

The stress tensor has a general polynomial dependence on the symmetric velocity gradient. The goal is to estimate the dimension of the global attractor in terms of relevant physical constants.

The stress tensor has a general polynomial dependence on the symmetric velocity gradient. The goal is to estimate the dimension of the global attractor in terms of relevant physical constants.

CPAA

We consider an evolutionary, non-degenerate, symmetric $p$-Laplacian. By

*symmetric*we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes a breakdown of the Uhlenbeck structure. We derive interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having our regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.## Year of publication

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