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End date: 2018 × Start date: 2016 × Subject: Navier-Stokes equations × WGL Institute: WIAS ×

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Now showing 1 - 6 of 6
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    Strong solutions to nonlocal 2D Cahn-Hilliard-Navier-Stokes systems with nonconstant viscosity, degenerate mobility and singular potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Frigeri, Sergio; Gal, Ciprian G.; Grasselli, Maurizio; Sprekels, Jürgen
    We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.
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    Analysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part III: Compactness and convergence
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, Clemens
    We consider an improved Nernst-Planck-Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convectiondiffusionreaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigations of [DDGG17a, DDGG17b], we prove the compactness of the solution vector, and existence and convergence for the approximation schemes. We point at simple structural PDE arguments as an adequate substitute to the AubinLions compactness Lemma and its generalisations: These familiar techniques attain their limit in the context of our model in which the relationship between time derivatives (transport) and diffusion gradients is highly non linear.
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    Analysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, Clemens
    We consider an improved NernstPlanckPoisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convectiondiffusionreaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigation of [DDGG17a], we derive for thermodynamically consistent approximation schemes the natural uniform estimates associated with the dissipations. Our results essentially improve our former study [DDGG16], in particular the a priori estimates concerning the relative chemical potentials.
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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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    Analysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, Clemens
    We consider an improved NernstPlanckPoisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convectiondiffusionreaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a globalintime weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.
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    Existence of weak solutions for improved Nernst-Planck-Poisson models of compressible reacting electrolytes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, Clemens
    We consider an improved Nernst-Planck-Poisson model for compressible electrolytes first proposed by Dreyer et al. in 2013. The model takes into account the elastic deformation of the medium. In particular, large pressure contributions near electrochemical interfaces induce an inherent coupling of mass and momentum transport. The model consists of convection-diffusion-reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity and the Poisson equation for the electrical potential. Cross-diffusion phenomena occur due to the principle of mass conservation. Moreover, the diffusion matrix (mobility matrix) has a zero eigenvalue, meaning that the system is degenerate parabolic. In this paper we establish the existence of a global-in-time weak solution for the full model, allowing for cross-diffusion and an arbitrary number of chemical reactions in the bulk and on the active boundary.