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Wed, 08 Feb 2017 11:53:33 +0200
Wed, 08 Feb 2017 11:53:33 +0200

The NavierStokes equations for elliptic quasicomplexes
https://publishup.unipotsdam.de/opus4ubp/frontdoor/index/index/docId/39849
The classical NavierStokes equations of hydrodynamics are usually written in terms of vector analysis. More promising is the formulation of these equations in the language of differential forms of degree one. In this way the study of NavierStokes equations includes the analysis of the de Rham complex. In particular, the Hodge theory for the de Rham complex enables one to eliminate the pressure from the equations. The NavierStokes equations constitute a parabolic system with a nonlinear term which makes sense only for oneforms. A simpler model of dynamics of incompressible viscous fluid is given by Burgers' equation. This work is aimed at the study of invariant structure of the NavierStokes equations which is closely related to the algebraic structure of the de Rham complex at step 1. To this end we introduce NavierStokes equations related to any elliptic quasicomplex of first order differential operators. These equations are quite similar to the classical NavierStokes equations including generalised velocity and pressure vectors. Elimination of the pressure from the generalised NavierStokes equations gives a good motivation for the study of the Neumann problem after Spencer for elliptic quasicomplexes. Such a study is also included in the work.We start this work by discussion of LamÃ© equations within the context of elliptic quasicomplexes on compact manifolds with boundary. The nonstationary LamÃ© equations form a hyperbolic system. However, the study of the first mixed problem for them gives a good experience to attack the linearised NavierStokes equations. On this base we describe a class of nonlinear perturbations of the NavierStokes equations, for which the solvability results still hold.
Azal Jaafar Musa Mera
doctoralthesis
https://publishup.unipotsdam.de/opus4ubp/frontdoor/index/index/docId/39849
Wed, 02 Aug 2017 11:53:33 +0200

Deformation quantisation and boundary value problems
https://publishup.unipotsdam.de/opus4ubp/frontdoor/index/index/docId/7715
We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
Boris Fedosov; Nikolai Nikolaevich Tarkhanov
preprint
https://publishup.unipotsdam.de/opus4ubp/frontdoor/index/index/docId/7715
Tue, 26 May 2015 16:09:59 +0200