Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents
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Beschreibung
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.
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.
Includes the first proof of the existence of weak solutions of the unsteady p(t,x)-Navier-Stokes equations Provides a comprehensive review of the rapidly expanding field of unsteady problems with variable >exponents Requires only a basic knowledge of functional analysis
Autor*in
Alex Kaltenbach
Themen in »Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents«
Existence of Weak Solutions Variable Exponent Lebesgue Spaces Variable Exponent Bochner-Lebesgue Spaces Pseudo-monotone Operator Theory Electrorheological Fluids Variable Exponent Sobolev Spaces
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“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 Moroşanu, zbMATH 1526.35002, 2024)
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