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End date: 2022 × Start date: 2022 × Start date: 2020 × DDC: 510 × Type: Article × Publisher: Basel : Springer × WGL Institute: WIAS ×

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    A rigorous derivation and energetics of a wave equation with fractional damping
    (Basel : Springer, 2021) Mielke, Alexander; Netz, Roland R.; Zendehroud, Sina
    We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water–air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy–dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally damped wave equation with a time derivative of order 3/2.
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    Well-posedness analysis of multicomponent incompressible flow models
    (Basel : Springer, 2021) Bothe, Dieter; Druet, Pierre-Etienne
    In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities stays constant. In this type of models, the velocity field in the Navier–Stokes equations is not solenoidal and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.
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    An asymptotic analysis for a generalized Cahn–Hilliard system with fractional operators
    (Basel : Springer, 2021) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), the same authors have studied viscous and nonviscous Cahn–Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers A2r and B2σ (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space L2(Ω), for some bounded and smooth domain Ω⊂R3, and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter σ appearing in the operator B2σ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of B appears.