arXiv subjects
Publications and source records attributed to Dalibor Pražák.
We consider a reaction diffusion equation, where the lower-order nonlinearity allows for a non-smooth part, described by a maximal monotone relation. We show that the global attractor has finite fractal dimension, and also obtain explicit upper estimates in terms of the data.
We consider evolutionary Stokes system, coupled with the so-called dynamic slip boundary condition, in the simple geometry of a $d$-dimensional half-space. Using the standard technique of the Fourier transform in tangential directions, we obtain an explicit formula for the resolvent. We then deduce estimates for both the weak (i.e. $W^{1,p}$) and strong (hence $W^{2,p}$) solutions, which are optimal in terms of the data belonging to appropriate negative Sobolev or fractional Besov space. In the latter case $L^p$-integrability of the pressure gradient is included. We allow for solutions with non-zero divergence, thus preparing the way for extensions to general domains. As a by-product, we show that the system generates an analytic semigroup in $L^p(Ω)\times L^p(\partial Ω)$. Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of $\mathcal{H}^{\infty}$-calculus are implicitly present; but we stay away from the concept of $R$-boundedness and related heavy functional analytic machinery.
We consider incompressible Navier-Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition.
We investigate the classical problem of motion of a mathematical pendulum with an oscillating pivot. This simple mechanical setting is frequently used as the prime example of a system exhibiting the parametric resonance phenomenon, which manifests itself by surprising stabilisation/destabilisation effects. In the classical case the pivot oscillations are described by a cosine wave, and the corresponding stability analysis requires one to investigate the behaviour of solutions to the Mathieu equation. This is not a straightforward procedure, and it does not lead to exact and simple analytical results expressed in terms of elementary functions. Consequently, the explanation of the parametric resonance phenomenon can be in this case obscured by the relatively involved technical calculations. We show that the stability analysis is much easier if one considers the pivot motion described by a non-smooth function -- a triangular or a nearly rectangular wave. The non-smooth pivot motion leads to the presence of singularities (Dirac distributions) in the corresponding Mathieu type equation, which seemingly further complicates the analysis. Fortunately, this is only a minor technical difficulty. Once the mathematical setting for the non-smooth forcing is settled down, the corresponding stability diagram is indeed straightforward to obtain, and the stability boundaries are, unlike in the classical case, given in terms of simple analytical formulae involving only elementary functions.
We deal with the flows of non-Newtonian fluids in three dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index $p\ge 11/5$ we establish regularity properties of a solution with respect to time variable. Consequently, we can use this better information for showing the uniqueness of the solution provided that the initial data are good enough for all power--law indexes $p\ge 11/5$. Such a result was available for $p\ge 12/5$ and therefore the paper fills the gap and extends the uniqueness result to the whole range of $p$'s for which the energy equality holds.
We consider a nonlinear reaction-diffusion equation settled on the whole euclidean space. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L^2. Then we adapt the short trajectory method to establish the existence of the global attractor and, if the space dimension is at most 3, we also find an upper bound of its Kolmogorov's entropy.