Search Results
A coarse‐grained electrothermal model for organic semiconductor devices
We derive a coarse-grained model for the electrothermal interaction of organic semiconductors. The model combines stationary drift-diffusion- based electrothermal models with thermistor-type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the Gauss–Fermi integral and mobility functions depending on the temperature, charge-carrier density, and field strength, which is required for a proper description of organic devices.
An existence result for a class of nonlinear magnetorheological composites
We prove existence of a weak solution for a nonlinear, multi-physics, multi-scale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):1--19, '20). The hybrid model couples the Stokes' equation with the quasi-static Maxwell's equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is based on: i) the augmented variational formulation of Maxwell's equations, ii) the definition of a new function space for the magnetic induction and the proof of a Poincaré type inequality, iii) the Altman--Shinbrot fixed point theorem when the magnetic Reynold's number, Rm, is small.
On a diffuse interface model of tumor growth
We consider a diffuse interface model of tumor growth proposed by A. Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction φ nonlinearly coupled with a reaction-diffusion equation for ψ which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation functionp(φ) multiplied by the differences of the chemical potentials for φ and ψ. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of φ+ψ. Here we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.
On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics
We define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. This concept is shown to be equivalent to weak solutions with energy conservation. Via a standard Galerkin discretization, we prove the existence of energy-variational solutions and thus weak solutions in any space dimension for the Navier--Stokes equations. In the limit of vanishing viscosity the same assertions are deduced for the incompressible Euler system. Via the selection criterion of maximal dissipation we deduce well-posedness for these equations.
On the Regularity of Weak Solutions to Time-Periodic Navier–Stokes Equations in Exterior Domains
Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.
Weak solutions to lubrication systems describing the evolution of bilayer thin films
The existence of global nonnegative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquidsolid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.
Modeling and analysis of a phase field system for damage and phase separation processes in solids
In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.
Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains
The aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertial effects. To this end, we first present a suitable weak formulation in order to deal with such evolution inclusions. Then, existence of weak solutions is proven by utilizing time-discretization, H2-regularization and variational techniques.
Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects
In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [HK11]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.
Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints
Adaptive finite element methods for optimization problems for second order linear elliptic partial di erential equations subject to pointwise constraints on the ℓ2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint quali cation such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.