Gerhard Ströhmer

SYMMETRY OF INTEGRAL EQUATIONS ON BOUNDED DOMAINS

In this paper, we will investigate the symmetry of both domains and solutions of integral equations on bounded domains via the method of moving planes.

ABOUT A CERTAIN CLASS OF PARABOLIC-HYPERBOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS

We consider a class of parabolic-hyperbolic systems of differential equations that includes those of compressible viscous fluid flow. We prove the existence of solutions to the initial-boundary value problem for a short time, using analytic semigroup methods. As a byproduct we obtain that if the functions constituting the equation, but not the initial values, are infinitely differentiab1 e, then so are the solutions, expressed in Lagrange coordinates, as functions of time alone.

About the equations of non-stationary magneto-hydrodynamics

We obtain weak solutions for the equations of magneto-hydrodynamics which are regular for almost all times. If these solutions are small in a suitable sense they are strong for all times. The conductivity, and to a lesser extent the magnetic permeability, are allowed to vary discontinuously.

INSTABILE MINIMALFLÄCHEN MIT HALBFREIEM RAND

In this paper the existence of unstable minimal surfaces is concluded from the usual hypothesis for a class of semi-free boundary problems, using techniques previously developed by the author. AMS-Classification: 53A10, 49F10

Existence and stability theorems for abstract parabolic equations, and some of their applications

For a class of semi-abstract evolution equations for sections on vector bundles on a three-dimensional compact manifold we prove that for initial values with certain symmetries strong solutions exist for all times. In case these solutions become small after some time, strong solutions exist also for small perturbations of these initial values. Many systems from fluid mechanics are included in this class.

PARTIALLY REGULAR WEAK SOLUTIONS FOR A SYSTEM OF EQUATIONS MODELING LIQUID CRYSTAL MOTION

This paper shows that a system of equations modeling liquid crystal flow introduced by F. H. Lin and studied by F. H. Lin and C. Liu can also be treated using the theory of abstract parabolic equations. The results thus obtained are similar to those of F. H. Lin and C. Liu.

About the eigenfunctions of the curl-operator

For simply connected domains we give an elementary proof that curl is selfadjoint and its eigenfunctions are continuously differentiable up to the boundary

About conormal and oblique derivative problems

Abstract We prove some estimates and regularity results for oblique derivative problems on Reifenberg flat domains with not very smooth coefficients and vectorfields by transforming these problems into conormal ones.

Asymptotic estimates for a semigroup related to compressible viscous fluid flow

The paper is related to the question of stability for the motionless spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. It considers a perturbation of the linearization of the governing equations of this problem, taking a step in the derivation of estimates which will allow us to prove non-linear stability of the equilibria. The perturbed operator, like the linearization considered earlier, generates an analytic semigroup, which allows us to derive asymptotic estimates as t → ∞.

A remark about a Galerkin method

It is shown that the approximating equations whose existence is required in the author’s previous work on partially regular weak solutions can be constructed without any additional assumption about the equation itself. This leads to a variation of a Galerkin method. In the paper [1] the existence of partially regular weak solutions of an abstract parabolic equation of the form (EQ) Ut +AU = F (U, t) is proved under the assumption that there exists a family of approximating equations