Navier–Stokes Equations: An Introduction with Applications

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Springer, Apr 12, 2016 - Mathematics - 390 pages
This volume is devoted to the study of the Navier–Stokes equations, providing a comprehensive reference for a range of applications: from advanced undergraduate students to engineers and professional mathematicians involved in research on fluid mechanics, dynamical systems, and mathematical modeling. Equipped with only a basic knowledge of calculus, functional analysis, and partial differential equations, the reader is introduced to the concept and applications of the Navier–Stokes equations through a series of fully self-contained chapters. Including lively illustrations that complement and elucidate the text, and a collection of exercises at the end of each chapter, this book is an indispensable, accessible, classroom-tested tool for teaching and understanding the Navier–Stokes equations.
Incompressible Navier–Stokes equations describe the dynamic motion (flow) of incompressible fluid, the unknowns being the velocity and pressure as functions of location (space) and time variables. A solution to these equations predicts the behavior of the fluid, assuming knowledge of its initial and boundary states. These equations are one of the most important models of mathematical physics: although they have been a subject of vivid research for more than 150 years, there are still many open problems due to the nature of nonlinearity present in the equations. The nonlinear convective term present in the equations leads to phenomena such as eddy flows and turbulence. In particular, the question of solution regularity for three-dimensional problem was appointed by Clay Institute as one of the Millennium Problems, the key problems in modern mathematics. The problem remains challenging and fascinating for mathematicians, and the applications of the Navier–Stokes equations range from aerodynamics (drag and lift forces), to the design of watercraft and hydroelectric power plants, to medical applications such as modeling the flow of blood in the circulatory system.
 

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Contents

1 Introduction and Summary
1
2 Equations of Classical Hydrodynamics
11
3 Mathematical Preliminaries
38
4 Stationary Solutions of the NavierStokes Equations
83
5 Stationary Solutions of the NavierStokes Equations with Friction
94
6 Stationary Flows in Narrow Films and the Reynolds Equation
111
7 Autonomous TwoDimensional NavierStokes Equations
143
8 Invariant Measures and Statistical Solutions
169
11 Nonautonomous NavierStokes Equations and PullbackAttractors
251
12 Pullback Attractors and Statistical Solutions
276
13 Pullback Attractors and Shear Flows
297
14 Trajectory Attractors and Feedback Boundary Control in Contact Problems
317
15 Evolutionary Systems and the NavierStokes Equations
337
16 Attractors for Multivalued Processes in Contact Problems
358
References
377
Index
387

9 Global Attractors and a Lubrication Problem
182
10 Exponential Attractors in Contact Problems
207

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About the author (2016)

Grzegorz Łukaszewicz is Professor of Mathematical Physics at the University of Warsaw. His research interests include partial differential equations, hydrodynamics, dynamical systems.

Piotr Kalita works at the Faculty of Mathematics and Computer Science at Jagiellonian University in Krakow, Poland. His research interests include global dynamics for evolutionary partial differential equations, inequalities, and inclusions;
modeling in mathematical physics; and numerical methods for evolutionary problems."

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