The Stokes equations in spaces of bounded functions and spaces of bounded mean oscillation (Q2827293)

The purpose of the present book (thesis) is to study the Stokes equations NEWLINE\[NEWLINE v_t-\Delta v + \nabla p = 0, \;\;div\, v =0 \;\text{ in } \;\Omega\times(0,T); \;\;v=0 \;\text{ on } \;\partial \Omega \times (0,T); \;\;v(0)=v_0, NEWLINE\]NEWLINE i.e., the linearization of the Navier-Stokes equations. The author summarises in Chapter 1 (Introduction) his main results as compared to the existing ones in the literature related to this subject. In Chapter 2 he reminds the main known results on the Stokes equations in spaces of bounded functions. Chapter 3 is devoted to the stationary Stokes equations in layer domains on spaces of bounded and integrable functions. Chapter 4 is about bounds on the Stokes semigroup in exterior domains on spaces of bounded functions. In Chapter 5 the author introduces a new kind of spaces of bounded mean oscillation (BMO) and considers the Stokes equations within these spaces. BMO-spaces are generalizations of \(L^{\infty}\) which can be succesfully used to overcome technical difficulties. In Chapter 6 the author generalizes the theory of K. Abe and Y. Giga on the Stokes equations on spaces of bounded functions in admissible domains to the more general BMO-type spaces. In particular, the author derives the analyticity of the Stokes semigroup on BMO-type functions. Finally, as an application of the theory of the Stokes semigroup on BMO-type spaces, it is shown (in Chapter 7) that the Helmholtz projection is not necessary for obtaining analyticity results on the Stokes semigroup.NEWLINENEWLINEThis thesis is an excellent addition to the literature related to the Stokes equations and I warmly recommend it to interested researchers.