Partial Differential Equations in Fluid Mechanics

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.

Preface Charles L. Fefferman, James C. Robinson and José L. Rodrigo; 1. Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier–Stokes equations Claude Bardos; 2. Time-periodic flow of a viscous liquid past a body Giovanni P. Galdi and Mads Kyed; 3. The Rayleigh–Taylor instability in buoyancy-driven variable density turbulence John D. Gibbon, Pooja Rao and Colm-Cille P. Caulfield; 4. On localization and quantitative uniqueness for elliptic partial differential equations Guher Camliyurt, Igor Kukavica and Fei Wang; 5. Quasi-invariance for the Navier–Stokes equations Koji Ohkitani; 6. Leray's fundamental work on the Navier–Stokes equations: a modern review of 'Sur le mouvement d'un liquide visqueux emplissant l'espace' Wojciech S. Ozanski and Benjamin C. Pooley; 7. Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations Reimund Rautmann; 8. Energy conservation in the 3D Euler equations on T2 x R+ James C. Robinson, José L. Rodrigo and Jack W. D. Skipper; 9. Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure Chuong V. Tran and Xinwei Yu; 10. A direct approach to Gevrey regularity on the half-space Igor Kukavica and Vlad Vicol; 11. Weak-strong uniqueness in fluid dynamics Emil Wiedemann.

Charles L. Fefferman is the Herbert Jones Professor in the Mathematics Department at Princeton University, New Jersey. He was awarded the Fields Medal in 1978.
James C. Robinson is a Professor of Mathematics at the University of Warwick. He is also a Royal Society University Research Fellow and an EPSRC Leadership Fellow.
José L. Rodrigo is a Professor of Mathematics at the University of Warwick, and has been awarded an ERC Consolidator Grant.