Partial Differential Equations in Fluid Mechanics

Charles L. Fefferman, James C. Robinson, José L. Rodrigo

• Publisher: Cambridge University Press
• Year: 2018
• Print ISBN: 9781108460965
• eBook ISBN: 9781316997031
• Topic: Mathématiques
• Subtopic: Équations différentielles

The Euler and Navier–Stokes equations are the fundamental mathematical models of fluid mechanics, and their study remains central in the modern theory of partial differential equations. This volume of articles, derived from the workshop 'PDEs in Fluid Mechanics' held at the University of Warwick in 2016, serves to consolidate, survey and further advance research in this area. It contains reviews of recent progress and classical results, as well as cutting-edge research articles. Topics include Onsager's conjecture for energy conservation in the Euler equations, weak-strong uniqueness in fluid models and several chapters address the Navier–Stokes equations directly; in particular, a retelling of Leray's formative 1934 paper in modern mathematical language. The book also covers more general PDE methods with applications in fluid mechanics and beyond. This collection will serve as a helpful overview of current research for graduate students new to the area and for more established researchers.

Table of contents

• Cover
• Series information
• Title page
• Copyright information
• Table of contents
• List of contributors
• Preface
• 1 Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier–Stokes equations
• 2 Time-periodic flow of a viscous liquid past a body
• 3 The Rayleigh–Taylor instability in buoyancy-driven variable density turbulence
• 4 On localization and quantitative uniqueness for elliptic partial differential equations
• 5 Quasi-invariance for the Navier–Stokes equations
• 6 Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace”
• 7 Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations
• 8 Energy conservation in the 3D Eulerequations on T[sup(2)] × R[sub(+)]
• 9 Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure
• 10 A direct approach to Gevrey regularity on the half-space
• 11 Weak-Strong Uniqueness in Fluid Dynamics