An essential companion to M. Vishik’s groundbreaking work in fluid mechanics The incompressible Euler equations are a system of partial differential equations introduced by Leonhard Euler more than 250 years ago to describe the motion of an inviscid incompressible fluid. These equations can be derived from the classical conservations laws of mass and momentum under some very idealized assumptions. While they look simple compared to many other equations of mathematical physics, several fundamental mathematical questions about them are still unanswered. One is under which assumptions it can be rigorously proved that they determine the evolution of the fluid once we know its initial state and the forces acting on it. This book addresses a well-known case of this question in two space dimensions. Following the pioneering ideas of M. Vishik, the authors explain in detail the optimality of a celebrated theorem of V. Yudovich in the sixties, which states that, in the vorticity formulation, the solution is unique if the initial vorticity and the acting force are bounded. In particular, the authors show that Yudovich’s theorem cannot be generalized to the L^p setting.

Epitomises what is best in Arthurian scholarship today. ZEITSCHRIFT FUER ROMANISCHE PHILOLOGIE This latest issue of Arthurian Literaturecontinues the tradition of the journal, combining critical studies with editions of primary Arthurian texts. Varied in their linguistic and chronological coverage, the articles dealwith major areas of Arthurian studies, from early French romance through late medieval English chronicle to contemporary fiction. Topics include Beroul's Tristan, Tristan de Nanteuil, the Anglo-Norman Brut, and the Morte, while an edition of the text of an extrait of Chretien's Erec et Enide prepared by the eighteenth-century scholar La Curne de Sainte-Palaye offers important insights into both scholarship on Chretien, and our understanding of the Enlightenment. The volume is completed with an encyclopaedic treatment of Arthurian literature, art and film produced between 1995 and 1995, acting as an update to The New Arthurian Encyclopedia.Contributors: RICHARD ILLINGWORTH, JANE TAYLOR, CARLETON CARROLL, MARIA COLOMBO TIMELLI, RALUCA RADULESCU, JULIA MARVIN, NORRIS LACY, RAYMOND THOMPSON.

An essential companion to M. Vishik’s groundbreaking work in fluid mechanics The incompressible Euler equations are a system of partial differential equations introduced by Leonhard Euler more than 250 years ago to describe the motion of an inviscid incompressible fluid. These equations can be derived from the classical conservations laws of mass and momentum under some very idealized assumptions. While they look simple compared to many other equations of mathematical physics, several fundamental mathematical questions about them are still unanswered. One is under which assumptions it can be rigorously proved that they determine the evolution of the fluid once we know its initial state and the forces acting on it. This book addresses a well-known case of this question in two space dimensions. Following the pioneering ideas of M. Vishik, the authors explain in detail the optimality of a celebrated theorem of V. Yudovich in the sixties, which states that, in the vorticity formulation, the solution is unique if the initial vorticity and the acting force are bounded. In particular, the authors show that Yudovich’s theorem cannot be generalized to the L^p setting.

The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.

This book develops a timely analysis of the complex trends and transformations emerging in EU competition law in the current turbulent times. Repeated economic crises, the climate emergency, digitalisation, and geopolitical and democratic threats are all having profound societal and economic effects on the EU. In light of its fundamental role in the Treaties, EU competition law has been called upon to play an important role in responding to this state of 'turbulence'. This brings about significant governance and constitutional challenges, firstly by questioning how the governance of EU competition law is being transformed to respond and adapt. Secondly, these crisis-induced transformations probe the logic and constitutional limits of EU competition law within the framework of EU law. This collection brings together EU institutional and competition lawyers to reflect on the governance and constitutional challenges emerging from the post-modernisation evolution of EU competition law against the backdrop of the recent multiple crises in the EU. The essays focus on the substantive and procedural developments across the three main policy areas of EU competition law: antitrust, merger control and State aid. EU constitutional and competition lawyers will be interested in this important new collection.