On the H s Theory of Hydrostatic Euler Equations

Abstract

In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H s solutions under the local Rayleigh condition. This extends Brenier's (Nonlinearity 12 (3):495-512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak-strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H s a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.

Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface

“…We consider the three-dimensional hydrostatic Navier-Stokes system [24] describing a free surface gravitational flow moving over a bottom topography z b (x, y). For free surface flows, the hydrostatic assumption consists in neglecting the vertical acceleration, see [25,26,27,28] for justifications of such hydrostatic models.…”

“…In order to obtain(22) we multiply(27) by g(h + z b ) − |u α | 2 /2 and (19) by u α then we perform simple manipulations. More precisely, the momentun equation along the x axis multiplied by u α gives…”

Continuous data assimilation for the 3D primitive equations of the ocean

“…The global existence of strong solutions with initial data in H 1 was proven by Cao and Titi in the classic paper [30]. For other works on the primitive equations, we refer the readers to [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]60] as well as [48][49][50][51][52][53] for the inviscid case.…”

Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

“…Therefore, we will consider in this paper the inviscid primitive equations without the Coriolis rotation term, and we will show that for certain class of smooth initial data if their corresponding smooth solutions exist then they will develop a singularity (blowup) in finite time. For results concerning the short time existence and uniqueness of the inviscid primitive equations see, for example, [6,18,23,27] and references therein. Notably, it is unknown of whether the rotation term in the inviscid primitive equations, in particular for large values of R, plays a stabilizing mechanism by preventing the formation of singularity as in the case of Burgers equations [1,21], or by extending the life of span of the solution and postponing the blowup as in the case of the three-dimensional Euler equations [2,3,4,5,12,15]; this is a subject of ongoing and future research.…”