In 1904, Prandtl introduced a fundamental system derived from the incompressible Navier-Stokes equation with no-slip boundary condition to capture the behavior of fluid motion near the boundary when viscosity vanishes. Even though there are fruitful mathematical theories developed since the seminal works by Oleinik in 1960s, most of the well-posedness theories are limited to the two space dimensions under Oleinik's monotonicity condition except the classical work by Sammartino-Caflisch in 1998 in the framework of analytic functions and some recent work in Gevrey function spaces.

In addition to its early application in aerodynamics and later in various areas in fluid dynamics and engineering, Prandtl equation can be viewed as a typical example of partial differential equations with rich structure that includes mixed type and degeneracy in dissipation. Hence, it provides many challenging mathematical problems and most of them remain unsolved after more than one hundred years from its derivation.

In this talk, we will present the intrinsic structure of the Prandtl operator and it various forms and relation with other physical models.