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.

The weak well-posedness of strong damping wave equations defined by fractal Laplacians is proved by using Galerkin method. These fractal Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. In general, the structure of self-similar measures with overlap are complicated and intractable. However, some important information about the structure of the three measures above can be obtained. We make use of these information to set up a framework for one-dimensional measures to discretize the equations, and use the finite element and central difference methods to obtain numerical approximations of the weak solutions. We also show that the numerical solutions converge to the actual solution and obtain the rate of convergence.

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.

As a continuation of [28], the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on the velocity field and the perfect conducting boundary condition is given for the magnetic field. Under an assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a L8 estimate on the error by multi-scale analysis.