Gevrey well-posedness for the Prandtl equation

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.