Navier-Stokes Flow Around a Rotating Obstacle: Mathematical Analysis of its Asymptotic Behavior (Atlantis Briefs in Differential Equations (3))

The book provides a comprehensive, detailed and self-contained treatment of the fundamental mathematical properties of problems arising from the motion of viscous incompressible fluids around rotating obstacles. It offers a new approach to this type of problems. We derive the fundamental solution of the steady case and we give pointwise estimates of velocity and its gradient (first and second one). Each chapter is preceded by a thorough discussion of the investigated problems, along with their motivation and the strategy used to solve them. The book will be useful to researchers and graduate students in mathematics, in particular mathematical fluid mechanics and differential equations. Preface 7 Contents 8 1 Introduction 10 2 Formulation of the Problem 13 2.1 Notations, Definitions and Auxiliary Results 14 3 Fundamental Solution of the Evolution Problem 16 3.1 Fundamental Solution of the Non-steady ``Rotating'' Oseen Problem 16 3.2 Basic Properties of the Fundamental Solution 23 3.3 Further Properties of Γjk 25 3.3.1 Technical Lemmas 25 3.3.2 Pointwise Estimates of Γjk 25 4 Fundamental Solution of the Stationary Problem 32 5 Representation Formula 46 5.1 Heuristic Approach 46 5.2 Mathematical Preliminaries to the Representation Formula 48 5.3 Derivation of the Representation Formula 51 6 Asymptotic Behavior 56 6.1 Some Volume Potentials 56 6.2 Asymptotic Profile 59 6.3 Representation Formula for the Navier--Stokes System 76 6.4 Asymptotic Profile of the Gradient of the Velocity Field 77 6.5 Decay Estimates of the Second Derivatives of the Velocity 79 7 Leray Solution 82 7.1 Introduction 82 7.2 Auxiliary Results 83 7.3 Representation Formula for the Leray solution 84 7.4 Asymptotic Profile of the Linear Case 87 7.5 Representation Formula for the Nonlinear Case 89 8 Latest Results 91 8.1 Statement of the Main Result 93 References 96 Index 99 Front Matter....Pages i-x Introduction....Pages 1-3 Formulation of the Problem....Pages 5-7 Fundamental Solution of the Evolution Problem....Pages 9-24 Fundamental Solution of the Stationary Problem....Pages 25-38 Representation Formula....Pages 39-48 Asymptotic Behavior....Pages 49-74 Leray Solution....Pages 75-83 Latest Results....Pages 85-89 Back Matter....Pages 91-96