The Steady Navier-Stokes System: Basics of the Theory and the Leray Problem

This book provides a successful solution to one of the central problems of mathematical fluid mechanics: the Leray’s problem on existence of a solution to the boundary value problem for the stationary Navier--Stokes system in bounded domains under sole condition of zero total flux. This marks the culmination of the authors’ work over the past few years on this under-explored topic within the study of the Navier--Stokes equations. This book will be the first major work on the Navier--Stokes equations to explore Leray’s problem in detail. The results are presented with detailed proofs, as are the history of the problem and the previous approaches to finding a solution to it. In addition, for the reader’s convenience and for the self-sufficiency of the text, the foundations of the mathematical theory for incompressible fluid flows described by the steady state Stokes and Navier--Stokes systems are presented. For researchers in this active area, this book will be a valuable resource.

Mikhail Korobkov (born 17.09.1976) is an experienced researcher on the interaction of real analysis (＂fine properties of functions＂) with nonlinear PDEs. He gained fame in mathematics after his joint papers with Jean Bourgain (Fields medal prize 1994, Princeton Institute for Advanced Study), and Jan Kristensen (University of Oxford), where they obtained the Morse--Sard type theorems for the sharp case of Sobolev mappings. M.Korobkov started his work at Novosibirsk State University and Sobolev Institute of Mathematics, had visiting positions in Italy and Lithuania, and now he has the full professor position at Fudan University, one of the oldest and most famous universities in China. M.Korobkov has more than 50 research papers, he participated in many international mathematical conferences with plenary talks, and he was also invited to present a series of plenary talks at school-conferences in various countries -- in Europe, in the USA, and in Japan.

Konstantin Pileckas got his Ph.D. from the Leningrad Branch of Steklov Mathematical Institute (LOMI) under the supervision of Prof. V.A.Solonnikov in 1982. In 1994 he received a habilitation from the University of Paderborn, Germany. From 1997 until 2008 he was head of the Department of Differential Equations at the Institute of Mathematics and Informatics in Vilnius. Since 2008 he is head of the Department of Differential Equations at Vilnius University. He is a member of the Editorial Board of several scientific Journals. K.Pileckas was twice awarded by Lithuanian National Science Prize. He is a member of the Lithuanian Academy of Sciences.

Remigio Russo has been full professor of Rational Mechanics from 1990 at the University ``Federico II’’ of Naples and from 1993 to 2021 at the University of Campania ``L. Vanvitelli’’. He was a visiting professor in several Universities (Kyoto, Bayreuth, Herriott-Watt of Edimburgh, Vilnius, etc.). His scientific interest has been essentially connected with well posedness problems (existence, uniqueness, continuous dependence upon data and stability) as well as qualitative properties of solutions to differential systems of mathematical physics, like the system of linear elasticity and viscous fluid (Stokes and Navier--Stokes systems). R. Russo has (co) authored over 60 original research papers. Since 1999 is a member of the Academy of Sciences of Naples.