Partial Differential Equations (PDEs) are fundamental in fields such as physics and engineering, underpinning our understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics. They also arise in areas like differential geometry and the calculus of variations.
This book focuses on recent investigations of PDEs in Sobolev and analytic spaces. It consists of twelve chapters, starting with foundational definitions and results on linear, metric, normed, and Banach spaces, which are essential for introducing weak solutions to PDEs. Subsequent chapters cover topics such as Lebesgue integration, Lp spaces, distributions, Fourier transforms, Sobolev and Bourgain spaces, and various types of KdV equations. Advanced topics include higher order dispersive equations, local and global well-posedness, and specific classes of Kadomtsev-Petviashvili equations.
This book is intended for specialists like mathematicians, physicists, engineers, and biologists. It can serve as a graduate-level textbook and a reference for multiple disciplines.
Contents:
- Preliminaries
- Lebesgue Integration
- The Lp Spaces
- Distributions: The Fourier Transform
- Sobolev Spaces: Analytic Spaces
- Original Method for the KdV Equation in Hs(?)
- Fifth-Order Shallow Water Equation
- Higher-Order Nonlinear Dispersive Equation
- Kadomtsev-Petviashvili in Analytic Spaces
- Generalized Kadomtsev-Petviashvili I Equation
- Coupled System of KDV Equations in Gevrey Spaces
- System of Generalized KdV Equations
Readership: This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines. It is suitable for researchers in PDEs, mathematics, physics, biology, chemistry and informatics.
