Lyapunov exponents and the dimension of the attractor for 2d shear-thinning incompressible flow
P. Kaplický Dalibor Pražák
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
Attractors and entropy bounds for a nonlinear RDEs with distributed delay in unbounded domains
Dalibor Pražák Jakub Slavík
A nonlinear reaction-diffusion problem with a general, both spatially and delay distributed reaction term is considered in an unbounded domain $\mathbb{R}^N$. The existence of a unique weak solution is proved. A locally compact attractor together with entropy bound is also established.
keywords: Kolmogorov's $\varepsilon$-enthropy. unbounded domain attractor Nonlinear reaction-diffusion equation distributed delay
The dimension of the attractor for the 3D flow of a non-Newtonian fluid
M. Bulíček F. Ettwein P. Kaplický Dalibor Pražák
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
On the dimension of the attractor for the wave equation with nonlinear damping
Dalibor Pražák
We give an explicit estimate of the fractal dimension of the global attractor to the wave equation with nolinear damping. The nonlinearities are smooth functions of certain polynomial growth. As a by-product we estimate the dimension of the exponential attractor for the time $\tau$ solution operator provided that $\tau$ is sufficiently large. The main tool used in the proof is the so-called method of the trajectories.
keywords: nonlinear damping. global attractor fractal dimension wave equation exponential attractor
A stabilizing effect of a high-frequency driving force on the motion of a viscous, compressible, and heat conducting fluid
Eduard Feireisl Dalibor Pražák
We study the impact of an oscillating external force on the motion of a viscous, compressible, and heat conducting fluid. Assuming that the frequency of oscillations increases sufficiently fast as the time goes to infinity, the solutions are shown to stabilize to a spatially homogeneous static state.
keywords: heat conducting fluid high-frequency oscillations Compressible fluid stabilization.
Exponential attractor for the delayed logistic equation with a nonlinear diffusion
Dalibor Pražák
We study the generalized logistic equation where the feedback is captured by the time convolution with a nonnegative measure and the diffusion is the laplacian plus the p-laplacian with $p >= 2$. We prove that the equation has an exponential attractor provided that the solutions are asymptotically bounded.
keywords: nonlinear dissipation bounded delay. Exponential attractor logistic equation
On the dimension of the attractor for a class of fluids with pressure dependent viscosities
M. Bulíček Josef Málek Dalibor Pražák
We consider two-dimensional flows of an incompressible non Newtonian fluid where the departure from the Navier-Stokes fluid is due to the viscosity depending on both the rate of deformation and the pressure. We assume that the resulting extra-stress is uniformly elliptic and its derivative with respect to pressure is bounded in a proper manner. Considering just the spatially-periodic setting, one can prove the global existence and uniqueness of the strong solution. Using the so-called method of trajectories, we also prove the existence of an exponential attractor and estimate its fractal dimension in terms of the data of the equation.
keywords: fractal dimension exponential attractor pressure-dependent viscosity Non-newtonian fluid
Semilinear damped wave equation in locally uniform spaces
Martin Michálek Dalibor Pražák Jakub Slavík

We study a damped wave equation with a nonlinear damping in the locally uniform spaces and prove well-posedness and existence of a locally compact attractor. An upper bound on the Kolmogorov's $\varepsilon$-entropy is also established using the method of trajectories.

keywords: Damped wave equations nonlinear damping unbounded domains locally compact attractor Kolmogorov's entropy

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