Finite time singularities in a 1D model of the quasi-geostrophic equation

被引：76
作者：
Chae, D
Córdoba, A
Córdoba, D
Fontelos, MA
机构：
[1] CSIC, Inst Matemat & Fis Fundamental, Madrid 28006, Spain
[2] Sungkyunkwan Univ, Dept Math, Suwon 440746, South Korea
[3] Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain
[4] Univ Rey Juan Carlos, Dept Matemat Aplicada, Madrid 28933, Spain
来源：
ADVANCES IN MATHEMATICS | 2005年 / 194卷 / 01期
关键词：
quasi-geostrophic equation; nonlocal flux; singularities; complex Burgers equation;
D O I：
10.1016/j.aim.2004.06.004
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
In this paper we study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence of solutions for small initial data. (c) 2004 Elsevier Inc. All rights reserved.
引用
收藏
页码：203 / 223
页数：21
相关论文
共 28 条
[1]   Analytic structure of two 1D-transport equations with nonlocal fluxes [J].
Baker, GR ;
Li, X ;
Morlet, AC .
PHYSICA D, 1996, 91 (04) :349-375
[2]   The quasi-geostrophic equation in the Triebel-Lizorkin spaces [J].
Chae, D .
NONLINEARITY, 2003, 16 (02) :479-495
[3]   Global well-posedness in the super-critical dissipative quasi-geostrophic equations [J].
Chae, D ;
Lee, J .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2003, 233 (02) :297-311
[4]   A SIMPLE ONE-DIMENSIONAL MODEL FOR THE 3-DIMENSIONAL VORTICITY EQUATION [J].
CONSTANTIN, P ;
LAX, PD ;
MAJDA, A .
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1985, 38 (06) :715-724
[5]   Energy spectrum of quasigeostrophic turbulence [J].
Constantin, P .
PHYSICAL REVIEW LETTERS, 2002, 89 (18)
[6]   FORMATION OF STRONG FRONTS IN THE 2-D QUASI-GEOSTROPHIC THERMAL ACTIVE SCALAR [J].
CONSTANTIN, P ;
MAJDA, AJ ;
TABAK, E .
NONLINEARITY, 1994, 7 (06) :1495-1533
[7]   On the critical dissipative quasi-geostrophic equation [J].
Constantin, P ;
Cordoba, D ;
Wu, JH .
INDIANA UNIVERSITY MATHEMATICS JOURNAL, 2001, 50 :97-107
[8]   Behavior of solutions of 2D quasi-geostrophic equations [J].
Constantin, P ;
Wu, JH .
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 1999, 30 (05) :937-948
[9]   A maximum principle applied to quasi-geostrophic equations [J].
Córdoba, A ;
Córdoba, D .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2004, 249 (03) :511-528
[10]   A pointwise estimate for fractionary derivatives with applications to partial differential equations [J].
Córdoba, A ;
Córdoba, D .
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 2003, 100 (26) :15316-15317
123