论文标题
Lagrangian submanifolds的几何形状的半神经方程的解决方案和奇异性
Solutions and Singularities of the Semigeostrophic Equations via the Geometry of Lagrangian Submanifolds
论文作者
论文摘要
使用Monge-Ampère几何形状，我们研究了三个维度的一类非线性Monge-ampère方程的单数结构，这是在地球物理流体动力学中产生的。我们通过检查诱导度量在cotangent束的拉格朗日亚曼菲尔德（Lagrangian Submanifolds）的作用来扩展关于Monge-Ampère几何形状的早期研究。特别是，我们表明该度量的签名用作Monge-ampère方程的分类，而奇异性和椭圆形的透明跃迁则由度量标准的变性揭示出来。该理论是通过应用于半神经方程的示例解决方案来说明的。
Using Monge-Ampère geometry, we study the singular structure of a class of nonlinear Monge-Ampère equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Ampère geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge-Ampère equation, while singularities and elliptic-hyperbolic transitions are revealed by the degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations.
