M. Çalık, M. Oliver, and S. Vasylkevych,
Global well-posedness for the
generalized large-scale semigeostrophic equations,
Arch. Ration. Mech. An. 207 (2013), 969-990.
Abstract:
We prove existence and uniqueness of global classical 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 L1 model
derived by R. Salmon in 1985 and its 2006 generalization by the second
author. The results are, under the physical restriction that the
initial potential vorticity is positive, as strong as those available
for the Euler equations of ideal fluid flow in two dimensions.
Moreover, we identify a special case in which the velocity field is
two derivatives smoother in Sobolev space as compared to the general
case.
Our results are based on careful estimates which show that, although
the potential vorticity inversion is nonlinear, bounds on the
potential vorticity inversion operator remain linear in derivatives of
the potential vorticity. This permits the adaptation of an argument
based on elliptic Lp theory, proposed by Yudovich in
1963 for proving existence and uniqueness of weak solutions for the
two-dimensional Euler equations, to our particular nonlinear
situation.
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