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### Open Access Journals

CPAA

We prove existence, uniqueness and continuous dependence on initial
data of global weak solutions to the generalized large-scale
semigeostrophic equations with periodic boundary conditions. This
family of Hamiltonian balance models for rapidly rotating shallow
water includes the $L_1$ model derived by R. Salmon in 1985 and its
2006 generalization by the second author. The analysis is based on the
vorticity formulation of the models supplemented by a nonlinear
velocity-vorticity relation. The results are fundamentally due to the
conservation of potential vorticity. While classical solutions are
known to exist provided the initial potential vorticity is
positive---a condition which is already implicit in the formal
derivation of balance models, we can assert the existence of weak
solutions only under the slightly stronger assumption that the
potential vorticity is bounded below by $\sqrt{5}-2$ times the
equilibrium potential vorticity. The reason is that the
nonlinearities in the potential vorticity inversion are felt more
strongly when working in weaker function spaces. Another
manifestation of this effect is that point-vortex solutions are not
supported by the model even in the special case when the potential
vorticity inversion gains three derivatives in spaces of classical
functions.

DCDS

This paper presents a first rigorous study of the so-called
large-scale semigeostrophic equations which were first introduced by
R. Salmon in 1985 and later generalized by the first author. We show
that these models are Hamiltonian on the group of $H^s$
diffeomorphisms for $s>2$. Notably, in the Hamiltonian setting an
apparent topological restriction on the Coriolis parameter disappears.
We then derive the corresponding Hamiltonian formulation in Eulerian
variables via Poisson reduction and give a simple argument for the
existence of $H^s$ solutions locally in time.

CPAA

We compare the vorticity corresponding to a solution of the Lagrangian
averaged Euler equations on the plane to a solution of the Navier–Stokes equation with the same initial data, assuming that the averaged Euler
potential vorticity is in a certain Besov class of regularity. Then the averaged
Euler vorticity stays close to the Navier–Stokes vorticity for a short interval of
time as the respective smoothing parameters tend to zero with natural scaling.

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