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This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. We also introduce the free boundary problem for the compressible primitive equations.

In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new “almost” Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson’s equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.

We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set$K$defined on the boundary of the unit ball.

We consider the kinetic Fokker-Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a gen- eral force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve results about classical force to general force case. Our result also improve the rate of convergence for the Fitzhugh-Nagumo equation from non-quantitative to quantitative explicit rate.