The fully compressible semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordin
ates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.
We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial genera
lization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a $2times 2$ matrix Riemann-Hilbert problem formulated in terms of both the Dirichlet and the Neumann boundary values on the boundary of a semistrip. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of the semistrip and constant along the bounded side; in this particular case we show that the jump matrices of the above Riemann-Hilbert problem can be expressed explicitly in terms of the width of the semistrip and the constant value of the solution along the bounded side. This Riemann-Hilbert problem has a unique solution.
