This book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(.,.)-Navier-Stokes equations describing the motion of an incompressible electro-rheological fluid. Due to the variable dependence of the power-law index p(.,.) in this system, the classical weak existence analysis based on the pseudo-monotone operator theory in the framework of Bochner-Lebesgue spaces is not applicable. As a substitute for Bochner-Lebesgue spaces, variable Bochner-Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute, the theory of pseudo-monotone operators is extended to unsteady problems with variable exponents, leading to the weak solvability of the unsteady p(.,.)-Navier-Stokes equations under general assumptions.
Aimed primarily at graduate readers, the book develops the material step-by-step, starting with the basics of PDE theory andnon-linear functional analysis. The concise introductions at the beginning of each chapter, together with illustrative examples, graphics, detailed derivations of all results and a short summary of the functional analytic prerequisites, will ease newcomers into the subject.
- 1. Introduction. - 2. Preliminaries. -
Part I Main Part. - 3. Variable Bochner-Lebesgue Spaces. - 4. Solenoidal Variable Bochner-Lebesgue Spaces. - 5. Existence Theory for Lipschitz Domains. -
Part II Extensions. - 6. Pressure Reconstruction. - 7. Existence Theory for Irregular Domains. - 8. Existence Theory for p- < 2. - 9. Appendix.
"This book is essentially based on the author's doctoral thesis ... . The book also contains an appendix and references. ... The book could be used by graduate students and researchers working on such problems." (Gheorghe Morosanu, zbMATH 1526.35002, 2024)
