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End date: 2024 × Start date: 2021 × End date: 2024 × Start date: 2020 × DDC: 510 × Type: article × Subject: Navier–Stokes equations × WGL Institute: WIAS ×

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    On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations
    (Basel : Springer, 2020) Eiter, Thomas; Kyed, Mads; Shibata, Yoshihiro
    This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal Lp–Lq regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of R-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the R-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (R-solvers and their application to periodic Lp estimates, Preprint in 2019) for the Lp boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper. © 2020, The Author(s).
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    Analysis of improved Nernst–Planck–Poisson models of compressible isothermal electrolytes
    (Cham (ZG) : Springer International Publishing AG, 2020) 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 elastic deformation of the medium, that induces an inherent coupling of mass and momentum transport, is taken into account. The model consists of convection–diffusion–reaction 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, cross-diffusion phenomena must occur, and the mobility matrix (Onsager matrix) has a non-trivial kernel. In this paper, we establish the existence of a global-in-time weak solution, allowing for a general structure of the mobility tensor and for chemical reactions with fast nonlinear 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.