Weak and Measure-Valued Solutions to Evolutionary PDEs by Malek J. – A Comprehensive Guide for Graduate Students and Res

Introduction

Weak and Measure-Valued Solutions to Evolutionary PDEs by Malek J. is a rigorous and advanced mathematical text that delves deep into the theory of nonlinear evolutionary partial differential equations. Published by CRC Press, this hardcover edition is indispensable for researchers, graduate students, and professionals in applied mathematics, physics, and engineering. The book offers a systematic treatment of weak and measure-valued solutions, providing essential tools to tackle complex problems in fluid dynamics and continuum mechanics, with particular emphasis on non-Newtonian fluids.

Book Overview

This volume presents a comprehensive framework for understanding evolutionary PDEs beyond classical solutions. It bridges abstract functional analysis with concrete physical models, making it a vital resource for those working on nonlinear phenomena. The author develops measure-valued solutions as a powerful concept to handle singularities and oscillations arising in fluid flow, elasticity, and biological systems. By focusing on existence, uniqueness, and stability, the book equips readers with modern analytical techniques necessary for cutting-edge research.

Key Highlights

• Rigorous Foundation: Builds a solid mathematical basis for weak and measure-valued solution theories.
• Non-Newtonian Fluids: Extensive analysis of complex fluid models, including viscoelastic and shear-thinning behaviour.
• Interdisciplinary Relevance: Applications in physics, biology, and mechanical engineering are clearly illustrated.
• Self-Contained Approach: Includes necessary preliminaries in functional analysis and measure theory for smooth reading.
• Advanced Problem Sets: Thought-provoking exercises at the end of each chapter reinforce learning.

Inside the Book

The book is structured into well-organised chapters that progressively build complexity. It begins with a review of Sobolev spaces, weak convergence, and Young measures, then moves to evolutionary equations and their weak formulations. Later chapters focus on measure-valued solutions, including DiPerna-Lions techniques and renormalised solutions. Detailed proofs and examples clarify abstract concepts, while sections on Navier-Stokes equations and power-law fluids connect theory to real-world problems. The final chapters explore applications in biological tissue modelling and industrial fluid mechanics.

Key Topics

• Weak solutions for parabolic and hyperbolic PDEs
• Measure-valued solutions and Young measures
• Existence and compactness methods
• Non-Newtonian fluid models
• Renormalised solutions for transport equations
• Stability and long-time behaviour
• Applications in biomechanics and engineering

Reader Benefits

Readers gain a deep, mathematically rigorous understanding of how to handle PDEs that lack smooth solutions. The techniques presented are directly applicable to research in fluid dynamics, material science, and mathematical biology. The book also enhances problem-solving skills through careful reasoning and step-by-step derivations. Indian students pursuing advanced degrees in mathematics or engineering will find this text particularly valuable for thesis work and competitive examinations like GATE and NET.

Learning Outcomes

After studying this book, readers will be able to formulate weak and measure-valued solutions for a wide class of evolutionary PDEs, apply compactness arguments to prove existence, and analyse the qualitative behaviour of solutions. They will also develop proficiency in using Young measures to capture microstructural oscillations and will be prepared to read contemporary research literature in nonlinear PDEs.

Who Should Read

This book is ideal for postgraduate students and researchers in pure and applied mathematics, especially those specialising in PDEs, fluid dynamics, or continuum mechanics. It is also highly relevant for physicists and engineers working on computational fluid dynamics, polymer processing, or biological flow problems. Faculty members teaching advanced courses on nonlinear PDEs will find it an excellent reference.

About the Author

Malek J. is a distinguished mathematician known for his fundamental contributions to the theory of nonlinear partial differential equations and non-Newtonian fluid mechanics. With decades of teaching and research experience, he has authored numerous influential papers and books. His work has shaped modern approaches to weak and measure-valued solutions, making him a leading authority in the field.

About the Publisher

CRC Press is a premier academic publisher renowned for high-quality textbooks and monographs in science, technology, engineering, and mathematics. With a legacy spanning over a century, CRC Press is trusted by scholars worldwide for rigorous editorial standards and authoritative content. This hardcover edition reflects their commitment to excellence in mathematical literature.

Conclusion

Weak and Measure-Valued Solutions to Evolutionary PDEs is an essential acquisition for any serious mathematics library. It offers a clear, rigorous, and application-oriented treatment of advanced topics that are central to modern PDE theory. Whether you are a student embarking on research or an experienced professional seeking deeper insights, this book will serve as a reliable companion. Order your copy from Bookshops.in today and add a cornerstone text to your collection.