Stokes operator and stability of stationary Navier-Stokes flows in infinite cylindrical domains

• Main Author: Ri, Myong Hwan
• Format: Others
• Language: English; en
• Published: 2006
• Online Access: https://tuprints.ulb.tu-darmstadt.de/686/1/Diss_Ri.pdf; Ri, Myong Hwan <http://tuprints.ulb.tu-darmstadt.de/view/person/Ri=3AMyong_Hwan=3A=3A.html> (2006): Stokes operator and stability of stationary Navier-Stokes flows in infinite cylindrical domains.Darmstadt, Technische Universität, [Online-Edition: http://elib.tu-darmstadt.de/diss/000686 <http://elib.tu-darmstadt.de/diss/000686> <official_url>],[Ph.D. Thesis]

infinite cylindrical domains have been attracting great attention due to its theoretical and practical significance. However, in most cases, stationary Navier-Stokes problems were dealt with whereas instationary Navier-Stokes problems have been less studied. The Lq-approach to instationary Navier-Stokes problems is very important and convenient to analyze existence, uniqueness as well as strong energy inequality and partial regularity for solutions; to this end, the study of the Stokes operator is fundamental. The aim of this dissertation is to get resolvent estimates, maximal regularity and boundedness of H∞-calculus of Stokes operators in infinite cylindrical domains and to apply them to the stability of stationary Navier-Stokes flows in infinite cylindrical domains. We start with a Stokes resolvent system in an infinite straight cylinder. The Stokes resolvent system on an infinite straight cylinder is reduced by the (one-dimensional) partial Fourier transform along the axis of the cylinder to a parametrized Stokes system with the Fourier variable as a parameter. Using the Fourier multiplier theory in weighted spaces we get estimates for the parametrized Stokes system with bound constants independent of parameters. Based on these estimates resolvent estimate and maximal regularity of the Stokes operator in weighted Lebesgue spaces on an infinite straight cylinder are shown using the techniques of operator-valued Fourier multiplier theory and Schauder decomposition in Banach spaces with UMD property. Next we consider the Stokes operator in general infinite cylinders with several exits to infinity. A resolvent estimate of the Stokes operator in Lq-space is obtained using cut-off techniques based on the result of generalized Stokes resolvent system in an infinite straight cylinder. In particular, the Stokes operator is shown to generate a bounded and exponentially decaying analytic semigroup in any Lq-space on a general infinite cylinder. Moreover, it is proved that the Stokes operator admits a bounded H∞-calculus in any Lq-space on an infinite cylinder with several exits to infinity. As an application of the obtained properties of the Stokes operator we study stability of stationary Navier-Stokes flows in infinite cylindrical domains. First, existence and uniqueness for stationary Navier-Stokes systems in infinite cylinders are shown. Then the exponential stability of the stationary Navier-Stokes flow is proved based on Lr-Lq estimates of the perturbed Stokes semigroup.