Analysis of complex and non-smooth flows

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The proposal comprises four projects addressing some fundamental problems in mathematical fluid dynamics as well as the development of functional analytic methods and tools for their treatment.The four projects are:(1) Multiplication and Nemytskij operators in anisotropic function spaces;(2) Heterogeneous catalysis;(3) Stokes equations on domains with edges and vertices;(4) Dynamic contact lines.Project (1) yields an important tool for the treatment of nonlinear, i.p. quasilinear, problems. The outcome obtained in (1) will intensively be used in projects (2) and (4). In fact, by the lack of smoothness and integrability for the treatment of the contact line models in (4) in the Lp-setting it is crucial to have the optimal value for p which can only be provided by the detailed analysis on multiplication of anisotropic function spaces given in (1). Even though theory on domains with edges and vertices for classical elliptic and parabolic PDE is well-developed, corresponding results for the Stokes equations are very rare. Particularly, for the instationary Stokes equations only very few results exist. It is the objective to provide such kind of results by Project (3). They represent the key ingredient for an analytical approach to models describing contact line movement as considered in (4). In connection with very recent developments on the topic we are quite confident that we can achieve fundamental progress in these directions.Projects (1) and (3) also include basic tools for an approach to further complex flows. As a further objective these tools are utilized in Project (2) for the treatment of models describing heterogeneous catalysis. Summarizing, the main objective of the proposal is to derive substantial progress in the analysis of non-smooth flows, i.p. dynamic contact lines, and of complex flows such as models describing heterogeneous catalysis.