Prandtl方程在Gevrey空间的适定性

Gevrey well-posedness for the Prandtl equation

Prandtl方程属于退化型方程(组),其中非局部项有一阶切向导数的损失,这是Prandtl方程典型的退化特征.证明Prandtl方程适定性理论的关键之处在于如何克服导数损失,目前主要有两个框架.第一个框架对初始值有结构性假设限制,在Oleinik单调性条件下,通过Crocco坐标变换或者采用速度方程和旋度方程之间的消去机制,来克服导数损失,进而在Sobolev空间中建立其适定性理论.第二个框架对初始值有解析正则性要求,从而可以采用抽象Cauchy-Kovalevskaya定理证明Prandtl方程在解析空间中的适定性.本文主要讨论在没有任何结构性假设条件下,如何降低初始值的解析正则性,在较弱的Gevrey空间证明Prandtl方程的适定性,侧重介绍如何结合抽象Cauchy-Kovalevskaya定理以及消去机制,在Gevrey框架下刻画切向导数的损失.

The Prandtl equation is a degenerate type equation(or system),and the typical feature is that the loss of derivatives occurs in a non-local term.The main difficulty to investigate the well-posedness property is to overcome the loss of derivatives.There are two main settings for the well-posedness theory.The first one refers to the Sobolev space,which needs Oleinik monotonicity assumption,so that the loss of derivative is overcome by using Crocco transformation or cancellation mechanism.The second framework imposes the analytic regularity on the initial data so that the abstract Cauchy-Kovalevskaya may apply.In this survey,we reduce the analytic regularity of the initial data and prove the well-posedness in Gevrey setting without any structural assumptions.The main tool is the combination of the abstract Cauchy-Kovalevskaya theory and cancellation mechanism.