Lancelot Leung, Temple University
Abstract: The hydrostatic Euler equations describe the leading-order behavior of the incompressible Euler equations in narrow domains, in which the horizontal length scale is much larger than the other scales. In this seminar, we will discuss some recent results about steady solutions to the hydrostatic Euler equations in nozzle domains. Unlike the incompressible Euler equations, the stream function for the hydrostatic Euler equations satisfies a degenerate elliptic equation. As a result, classical estimates from the study of uniformly elliptic equations cannot be applied directly. To analyze the degenerate elliptic equation, a new transformation, which combines a change of variables and a Euler–Lagrange transformation, is introduced. With the aid of this transformation, the solutions in the new coordinates admit explicit representations, allowing the regularity of the steady solutions with respect to the horizontal variable to be obtained in a clear manner. This is a joint work with Tak Kwong Wong (HKU) and Chunjing Xie (SJTU).