Euler systems and Kolyvagin systems

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

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Geometric Euler systems for locally isotropic motives

Compositio Mathematica ◽

10.1112/s0010437x03000113 ◽

2004 ◽

Vol 140 (02) ◽

pp. 317-332 ◽

Cited By ~ 2

Author(s):

Tom Weston

Keyword(s):

Euler Systems

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Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

ANNALI DELL UNIVERSITA DI FERRARA ◽

10.1007/s11565-017-0296-9 ◽

2017 ◽

Vol 64 (2) ◽

pp. 335-360 ◽

Cited By ~ 1

Author(s):

Emmanuel Franck ◽

Laurent Gosse

Keyword(s):

Two Dimensional ◽

Euler Systems

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Stickelberger elements and Kolyvagin systems

Nagoya Mathematical Journal ◽

10.1017/s0027763000010345 ◽

2011 ◽

Vol 203 ◽

pp. 123-173

Author(s):

Kâzim Büyükboduk

Keyword(s):

General Framework ◽

Euler System ◽

Real Field ◽

Ideal Class ◽

The Core ◽

Ideal Class Groups ◽

Kolyvagin Systems ◽

Totally Real ◽

Totally Real Fields ◽

The Ideal

AbstractIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank isr >1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

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Truncated Euler Systems over Imaginary Quadratic Fields

Nagoya Mathematical Journal ◽

10.1017/s0027763000009727 ◽

2009 ◽

Vol 195 ◽

pp. 97-111

Author(s):

Soogil Seo

Keyword(s):

Class Group ◽

Quadratic Field ◽

Abelian Extension ◽

Iwasawa Theory ◽

Quadratic Fields ◽

Galois Module ◽

Imaginary Quadratic Fields ◽

Euler Systems ◽

Graded Module ◽

Elliptic Units

AbstractLet K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group ClF of F is equal to the order of the quotient UF/ElF of the group of global units UF by the group of elliptic units ElF of F. We introduce a filtration on UF/ElF made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

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