| Abstract: |
| As a variant of Prandtl boundary layer equations, the hydrostatic Euler equations describe the leading-order behavior of ideal flows passing through narrow domains. In this talk, we will discuss recent results about steady solutions to the hydrostatic Euler equations in nozzles. When expressed in terms of stream function formulation, the steady hydrostatic Euler equations become a degenerate elliptic equation, so the classical estimates for uniformly elliptic equations cannot directly apply. One of the key ingredients for the mathematical analysis is a new transformation that combines a change of variable and Euler-Lagrange transformation. With the aid of this new transformation, the solutions in the new coordinates enjoy explicit representations so that the regularity of steady solutions with respect to the horizontal variable can be gained in a clear way. |
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