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search.filters.applied.f.access: openAccess × End date: 2022 × Start date: 2020 × Start date: 2000 × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: existence ×

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    Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hömberg, Dietmar; Lasarzik, Robert
    In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions.
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    Analysis of a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Lasarzik, Robert
    In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak-strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.
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    Leray--Hopf solutions to a viscoelastic fluid model with nonsmooth stress-strain relation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eiter, Thomas; Hopf, Katharina; Mielke, Alexander
    We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba--Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.