On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains

We study integral operators related to a regularized version of the classical Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincar\'e-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincar\'e-type operators) and with full Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by $C^\infty$ functions.Comment: 23 page

The maximum principle for the equation of the continuity of a compressible medium

From the text (translated from the Russian): "Two-sided estimates for the density rho (x,t) in terms of the initial state rho _0(x) and the flow v(x,t), having the character of a maximum principle, are useful when solving initial-boundary value problems of the dynamics of a compressible continuum. For a bounded domain Omega subset {bf R}^n, ngeq 2, we consider both classical and generalized solutions of the problem (1) partial rho /partial t+{rm div}(rho v)=0, xin Omega , t>0, (2) rho vert _{t=0}=rho _0(x), (v,nu )vert _{partial Omega }=0, for a given flow v(x,t) with a condition for the nonpenetration of partial Omega , where nu is the unit normal to partial Omega . We present conditions on the flow v(x,t), sufficient for a maximum principle to hold, whose accuracy we illustrate with examples of flows for which the maximum principle makes no sense, namely: either the density rho (x,t) vanishes (i.e., an expanding bubble forms in the continuum) or a generalized solution of problem (1), (2) does not even exist.

Some mathematical models of hurricane dynamics

Summary: "An initial-boundary value problem for the Navier-Stokes equations with a time-dependent discontinuity in the type of boundary conditions is proposed to describe both ocean dynamics under the influence of a hurricane and the dynamics of a hurricane over solid ground. Tools for solving the stated problems are given, including the case of a compressible medium.