Weak solutions for generalized large-scale semigeostrophic equations
Mahmut Çalik Marcel Oliver
Communications on Pure & Applied Analysis 2013, 12(2): 939-955 doi: 10.3934/cpaa.2013.12.939
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.
keywords: nonlinear vorticity inversion. global weak solutions Balance models
Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling
Marcel Oliver Sergiy Vasylkevych
Discrete & Continuous Dynamical Systems - A 2011, 31(3): 827-846 doi: 10.3934/dcds.2011.31.827
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.
keywords: balance models Semigeostrophic equations rotating shallow water Poisson reduction. diffeomorphism group
The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$
Marcel Oliver
Communications on Pure & Applied Analysis 2002, 1(2): 221-235 doi: 10.3934/cpaa.2002.1.221
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.
keywords: Navier–stokes equations Averaged Euler equations

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