Generalized solutions and hydrostatic approximation of the Euler equations

Solutions to the Euler equations on a 3D domain D 3 (typically the unit cube or the periodic unit cube) can be formally obtained by minimizing the action of an incompressible fluid moving inside D 3 between two given configurations. When these two configurations are very close to each other, classical solutions do exist, as shown by Ebin and Marsden. However, Shnirelman found a class of data (essentially 2D in the sense that they trivially depend on the vertical coordinate) for which there cannot be any classical minimizer. For such data, generalized solutions can be shown to exist, as a substitute for classical solutions. These generalized solutions have unusual features that look highly unphysical (in particular, different fluid parcels can cross at the same point and at the same time), but the pressure field, which does not depend on the vertical coordinate, is well and uniquely defined. In the present paper, we show that these generalized solutions are actually quite conventional in the sense they obey, up to a suitable change of variable, a well-known variant (widely used for geophysical flows) of the 3D Euler equations, for which the vertical acceleration is neglected according to the so-called hydrostatic approximation.