Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows
M. Bulíček P. Kaplický
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 41-50 doi: 10.3934/dcdss.2008.1.41
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.
keywords: \alpha}$-regularity pressure and shear rate dependent viscosity $C^{1 non-Newtonian fluids.
Lyapunov exponents and the dimension of the attractor for 2d shear-thinning incompressible flow
P. Kaplický Dalibor Pražák
Discrete & Continuous Dynamical Systems - A 2008, 20(4): 961-974 doi: 10.3934/dcds.2008.20.961
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 $p\in(4/3,2]$.
keywords: global attractor Lieb-Thirring inequality. Power-law fluids Lyapunov exponents fractal dimension shear-thinning fluids
The dimension of the attractor for the 3D flow of a non-Newtonian fluid
M. Bulíček F. Ettwein P. Kaplický Dalibor Pražák
Communications on Pure & Applied Analysis 2009, 8(5): 1503-1520 doi: 10.3934/cpaa.2009.8.1503
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.
keywords: fractal dimension. Non-Newtonian fluid global attractor
Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences
Jan Burczak P. Kaplický
Communications on Pure & Applied Analysis 2016, 15(6): 2401-2445 doi: 10.3934/cpaa.2016042
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.
keywords: Evolutionary systems of PDEs iteration in Nikolskii-Bochner spaces. symmetric $p$-Laplacian local (interior) regularity of time derivatives

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