arXiv subjects
Publications and source records attributed to Michael Zelina.
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 a 2D infinite channel domain with an incompressible fluid satisfying the so-called dynamic slip boundary condition on the (part of the) boundary. Introducing an exhaustion by a sequence of bounded sub-domains of the whole channel we show that the unique weak solution is strong. We then construct the global attractor and find an explicit upper bound of its fractal dimension with regard to the physical parameters. This result is compatible with the analogous estimate in the case of the Dirichlet boundary condition.
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 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 $L^p(\Omega)\times L^p(\partial \Omega)$. 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.