Nonlinear Stability of Ekman Boundary Layers in Rotation Stratified Fluids
Title | Nonlinear Stability of Ekman Boundary Layers in Rotation Stratified Fluids PDF eBook |
Author | Hajime Koba |
Publisher | |
Pages | 132 |
Release | 2014-10-03 |
Genre | Fluid mechanics |
ISBN | 9781470414856 |
A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This book constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. The author calls such stationary solutions Ekman layers. This book shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, the author discusses the uniqueness of weak solutions and computes the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. The author also shows that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.
Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids
Title | Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids PDF eBook |
Author | Hajime Koba |
Publisher | American Mathematical Soc. |
Pages | 142 |
Release | 2014-03-05 |
Genre | Mathematics |
ISBN | 0821891332 |
A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This book constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. The author calls such stationary solutions Ekman layers. This book shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, the author discusses the uniqueness of weak solutions and computes the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. The author also shows that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.
Mathematical Analysis of the Navier-Stokes Equations
Title | Mathematical Analysis of the Navier-Stokes Equations PDF eBook |
Author | Matthias Hieber |
Publisher | Springer Nature |
Pages | 471 |
Release | 2020-04-28 |
Genre | Mathematics |
ISBN | 3030362264 |
This book collects together a unique set of articles dedicated to several fundamental aspects of the Navier–Stokes equations. As is well known, understanding the mathematical properties of these equations, along with their physical interpretation, constitutes one of the most challenging questions of applied mathematics. Indeed, the Navier-Stokes equations feature among the Clay Mathematics Institute's seven Millennium Prize Problems (existence of global in time, regular solutions corresponding to initial data of unrestricted magnitude). The text comprises three extensive contributions covering the following topics: (1) Operator-Valued H∞-calculus, R-boundedness, Fourier multipliers and maximal Lp-regularity theory for a large, abstract class of quasi-linear evolution problems with applications to Navier–Stokes equations and other fluid model equations; (2) Classical existence, uniqueness and regularity theorems of solutions to the Navier–Stokes initial-value problem, along with space-time partial regularity and investigation of the smoothness of the Lagrangean flow map; and (3) A complete mathematical theory of R-boundedness and maximal regularity with applications to free boundary problems for the Navier–Stokes equations with and without surface tension. Offering a general mathematical framework that could be used to study fluid problems and, more generally, a wide class of abstract evolution equations, this volume is aimed at graduate students and researchers who want to become acquainted with fundamental problems related to the Navier–Stokes equations.
A Homology Theory for Smale Spaces
Title | A Homology Theory for Smale Spaces PDF eBook |
Author | Ian F. Putnam |
Publisher | American Mathematical Soc. |
Pages | 136 |
Release | 2014-09-29 |
Genre | Mathematics |
ISBN | 1470409097 |
The author develops a homology theory for Smale spaces, which include the basics sets for an Axiom A diffeomorphism. It is based on two ingredients. The first is an improved version of Bowen's result that every such system is the image of a shift of finite type under a finite-to-one factor map. The second is Krieger's dimension group invariant for shifts of finite type. He proves a Lefschetz formula which relates the number of periodic points of the system for a given period to trace data from the action of the dynamics on the homology groups. The existence of such a theory was proposed by Bowen in the 1970s.
A Geometric Theory for Hypergraph Matching
Title | A Geometric Theory for Hypergraph Matching PDF eBook |
Author | Peter Keevash |
Publisher | American Mathematical Soc. |
Pages | 108 |
Release | 2014-12-20 |
Genre | Mathematics |
ISBN | 1470409658 |
The authors develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: `space barriers' from convex geometry, and `divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, they introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. They determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, their main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, the authors apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in -graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemerédi Theorem. Here they prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical they defer it to a subsequent paper.
Quaternionic Contact Einstein Structures and the Quaternionic Contact Yamabe Problem
Title | Quaternionic Contact Einstein Structures and the Quaternionic Contact Yamabe Problem PDF eBook |
Author | A. L. Carey |
Publisher | American Mathematical Soc. |
Pages | 94 |
Release | 2014-08-12 |
Genre | Mathematics |
ISBN | 0821898434 |
A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolds. All conformal transformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing torsion of the Biquard connection are explicitly described. A "3-Hamiltonian form" of infinitesimal conformal automorphisms of quaternionic contact structures is presented.
Turbulence in Rotating, Stratified and Electrically Conducting Fluids
Title | Turbulence in Rotating, Stratified and Electrically Conducting Fluids PDF eBook |
Author | P. A. Davidson |
Publisher | Cambridge University Press |
Pages | 701 |
Release | 2013-09-12 |
Genre | Science |
ISBN | 1107434343 |
There are two recurring themes in astrophysical and geophysical fluid mechanics: waves and turbulence. This book investigates how turbulence responds to rotation, stratification or magnetic fields, identifying common themes, where they exist, as well as the essential differences which inevitably arise between different classes of flow. The discussion is developed from first principles, making the book suitable for graduate students as well as professional researchers. The author focuses first on the fundamentals and then progresses to such topics as the atmospheric boundary layer, turbulence in the upper atmosphere, turbulence in the core of the earth, zonal winds in the giant planets, turbulence within the interior of the sun, the solar wind, and turbulent flows in accretion discs. The book will appeal to engineers, geophysicists, astrophysicists and applied mathematicians who are interested in naturally occurring turbulent flows.