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Author: Heida, Martin × End date: 2023 × Start date: 2020 × End date: 2019 × Start date: 2019 × Has files: Yes × WGL Institute: WIAS ×

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    On a thermodynamic framework for developing boundary conditions for Korteweg fluids
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Souček, Ondřej; Heida, Martin; Málek, Josef
    We provide a derivation of several classes of boundary conditions for fluids of Korteweg-type using a simple and transparent thermodynamic approach that automatically guarentees that the derived boundary conditions are compatible with the second law of thermodynamics. The starting assumption of our approach is to describe the boundary of the domain as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of continuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is non-negative, we identify a new class of boundary conditions, which on one hand generalizes in a nontrivial manner the Navier's slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis.
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    A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Franchi, Bruno; Heida, Martin; Lorenzani, Silvia
    In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.
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    Stochastic homogenization of Lambda-convex gradient flows
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heida, Martin; Neukamm, Stefan; Varga, Mario
    In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.