商品描述
This monograph provides a rigorous analysis of a wide range of stationary (steady state) boundary value problems for elliptic systems of Stokes and Navier-Stokes type, as encountered in fluid dynamics. Addressing Dirichlet, Neumann, Robin, mixed, and transmission problems in both the isotropic and anisotropic cases, it makes systematic use of the notion of relaxed ellipticity recently introduced by the authors. The problems are treated in Lipschitz domains in the Euclidean setting as well as in compact Riemannian manifolds and in manifolds with cylindrical ends (non-compact manifolds), with given data in a variety of spaces - Lebesgue, standard or weighted Sobolev, Bessel potential, and Besov. A detailed and comprehensive study is provided of the main mathematical properties of boundary value problems related to the Navier-Stokes equations with variable coefficients, such as existence, uniqueness, and regularity of solutions. These are considered in bounded, periodic, and also unbounded domains, in the Euclidean setting as well as on manifolds (compact, or non-compact). The included results represent the authors' contributions to the field of stationary Stokes, Navier-Stokes, and related equations, the main novelty being the analysis of the related boundary problems with anisotropic variable coefficients and on manifolds. The book is aimed at researchers, graduate and advanced undergraduate mathematics students, physicists, and computational engineers interested in mathematical fluid mechanics, partial differential equations, and geometric analysis. The prerequisites include the basics of partial differential equations, the variational approach and function spaces; some sections need the fundamentals of integral equations, the theory of Riemannian manifolds, and fixed-point techniques.
商品描述(中文翻譯)
這本專著對於流體力學中遇到的斯托克斯（Stokes）和納維-斯托克斯（Navier-Stokes）類型的各種靜態（穩態）邊界值問題進行了嚴謹的分析。針對狄利克雷（Dirichlet）、諾伊曼（Neumann）、羅賓（Robin）、混合及傳輸問題，無論是在各向同性還是各向異性情況下，均系統性地利用了作者最近提出的放鬆橢圓性概念。這些問題在歐幾里得空間的利普希茨（Lipschitz）區域以及在緊緻黎曼流形和具有圓柱端的流形（非緊緻流形）中處理，並在各種空間中給定數據，包括勒貝格（Lebesgue）、標準或加權索博列夫（Sobolev）、貝塞爾潛能（Bessel potential）和貝索夫（Besov）空間。對於與變係數納維-斯托克斯方程相關的邊界值問題的主要數學性質，如解的存在性、唯一性和正則性，提供了詳細而全面的研究。這些問題考慮在有界、周期性以及無界區域中，無論是在歐幾里得空間還是在流形上（緊緻或非緊緻）。所包含的結果代表了作者對靜態斯托克斯、納維-斯托克斯及相關方程領域的貢獻，主要的新穎之處在於對具有各向異性變係數的相關邊界問題及流形的分析。
本書的目標讀者為研究人員、研究生及高年級本科數學學生、物理學家和對數學流體力學、偏微分方程及幾何分析感興趣的計算工程師。先修知識包括偏微分方程的基礎、變分方法和函數空間；某些章節需要積分方程的基本知識、黎曼流形理論及不動點技術。
作者簡介
Mirela Kohr is a Full Professor in the Department of Mathematics at Babeş-Bolyai University, Cluj-Napoca, Romania. She has an extensive list of publications on various areas of fluid mechanics, PDEs (especially Navier-Stokes), complex analysis, and geometric analysis. She has been the principal investigator on six research grants, an invited speaker to many conferences, and has made several research visits to the University of Toronto, and various universities in Italy, Germany, the UK, Japan and France. Sergey E. Mikhailov is a Full Professor in Applied Mathematics and Analysis at Brunel University, London, UK. He has numerous publications on the analysis and numerics of boundary integral equations, partial differential (especially Navier-Stokes) equations and theoretical solid mechanics. He graduated from the Moscow Institute of Physics and Technology and worked in Moscow before moving to the University of Stuttgart, Germany, in 1993, as a Humboldt Fellow, and then to the UK. Victor Nistor is a Full Professor in the Department of Mathematics at Lorraine University, Metz, France. He has an extensive list of publications covering various areas of partial differential equations, geometric analysis, applied mathematics, and operator algebras. He was an NSF Young Investigator and a Sloan Fellow. Wolfgang L. Wendland studied Mathematics and Fluid Mechanics at TU Berlin. He is Professor Emeritus Dr.-Ing. at University of Stuttgart, where he was a Full Professor of Applied Mathematics (1986-2005). He was a Full Professor of Mathematics at TH Darmstadt (1970-1986), Uni del Chair Prof. at the University of Delaware, USA (1973-1974) and J.G. Herder Professor at Babeş-Bolyai University in Cluj, Romania (2005 and 2007). He has numerous publications on analysis and numerical methods for PDEs and boundary integral equations.
作者簡介(中文翻譯)
Mirela Kohr 是羅馬尼亞克盧日-納波卡的巴貝什-博利亞大學數學系的全職教授。她在流體力學、偏微分方程（特別是 Navier-Stokes）、複雜分析和幾何分析等多個領域擁有廣泛的出版物。她是六個研究計畫的主要研究者，曾受邀在多個會議上演講，並多次前往多倫多大學以及意大利、德國、英國、日本和法國的各大學進行研究訪問。Sergey E. Mikhailov 是英國倫敦布魯內爾大學應用數學與分析的全職教授。他在邊界積分方程、偏微分方程（特別是 Navier-Stokes）和理論固體力學的分析與數值方面有許多出版物。他畢業於莫斯科物理技術學院，並在莫斯科工作，之後於1993年作為洪堡學者前往德國斯圖加特大學，然後移居英國。Victor Nistor 是法國梅斯的洛林大學數學系的全職教授。他在偏微分方程、幾何分析、應用數學和算子代數等多個領域擁有廣泛的出版物。他曾是美國國家科學基金會的青年研究者和斯隆獎學者。Wolfgang L. Wendland 在柏林工業大學學習數學和流體力學。他是斯圖加特大學的名譽教授 Dr.-Ing.，曾於1986年至2005年擔任應用數學的全職教授。他曾於1970年至1986年擔任達姆施塔特應用科技大學的全職數學教授，並於1973年至1974年擔任美國德拉瓦大學的特聘教授，以及於2005年和2007年擔任羅馬尼亞巴貝什-博利亞大學的 J.G. Herder 教授。他在偏微分方程和邊界積分方程的分析與數值方法方面有許多出版物。
