Analysis of open systems in continuum thermodynamics

Abstract

The models describing processes in continuum (thermo)dynamics are still a fer- tile field of research because of its multitude of unresolved questions, especially when the interaction with the surroundings is present. The unifying theme of this thesis is an inves- tigation of an incompressible Navier-Stokes-like system subjected to the so-called dynamic slip boundary condition. Our main focus lies in studying the long-time behaviour of its solutions, in particular, the existence of a finite-dimensional global attractor. As a necessary tool, various results on a higher regularity are established along the journey. 1

Modely popisujı ́cı ́ procesy v (termo)dynamice kontinua jsou nada ́le plod- nou oblastı ́ vy ́zkumu, jelikoz ̌ vyvola ́vajı ́ celou pleja ́du dosud nezodpove ̌zeny ́ch ota ́zek a to obzvla 's ̌te ̌ v situacı ́ch, kdy v nich docha ́zı ́ k interakci s okolı ́m. Zastr ̌es ̌ujı ́cı ́m te ́mate ́m te ́to pra ́ce je zkouma ́nı ́ nestlac ̌itelny ́ch zobecne ̌ny ́ch Navierovy ́ch-Stokesovy ́ch rovnic s takzvanou dynamickou okrajovou podmı ́nkou. V prve ́ r ̌ade ̌ se zame ̌r ̌ujeme na studium chova ́nı ́ jejich r ̌es ̌enı ́ pro velke ́ c ̌asy, pr ̌edevs ̌ı ́m se jedna ́ o existenci konec ̌ne ̌-dimenziona ́lnı ́ho globa ́lnı ́ho atraktoru. Coby nezbytny ́ na 'stroj take ́ v pru ̊be ̌hu odvodı ́me rozlic ̌ne ́ vy 'sledky o regularite ̌. 1