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A new graduate textbook in partial differential equations by Professor Emeritus David Hoff has been published by the American Mathematical Society as part of its Mathematical Surveys and Monographs series.How does heat flow? How does water move? How does air flow around a wing? These questions are often modelled by systems of partial differential equations (PDEs). A subtlety is that while the equations involve derivatives of functions, in many situations the solutions to the equations can become irregular, having no derivatives. You can see this in the vortices that form when pulling a canoe paddle through a river, or in the turbulence in a plume of smoke from a candle. Mathematically, this is dealt with through careful construction of suitable spaces of functions in which to model the physical system.The book collects material previously spread through disparate sources, and presents a coherent general theory. From the publisher:“The text presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of PDEs with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces…This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. “The book is available for order at https://bookstore.ams.org/surv-251 .
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