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Author: Druet, Pierre-Etienne × End date: 2019 × Start date: 2010 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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    Analysis of improved Nernst-Planck-Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Druet, Pierre-Etienne
    We continue our investigations of the improved NernstPlanckPoisson model introduced in [DGM13]. In the paper [DDGG16] the analysis relies on the hypothesis that the mobility matrix has maximal rank under the constraint of mass conservation (rank N-1 for a mixture of N species). In this paper we allow for the case that the positive eigenvalues of the mobility matrix tend to zero along with the partial mass densities of certain species. In this approach the mobility matrix has a variable rank between zero and N-1 according to the number of locally available species. We set up a concept of weak solution able to deal with this scenario, showing in particular how to extend the fundamental notion of differences of chemical potentials that supports the modelling and the analysis in [DDGG16]. We prove the global-in-time existence in this solution class.
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    Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Druet, Pierre-Etienne
    We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.