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End date: 2023 × End date: 2018 × Start date: 2010 × Type: Text × Subject: existence × WGL Institute: WIAS ×

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Now showing 1 - 8 of 8
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    Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Lasarzik, Robert
    In this article, we prove the existence of measure-valued solutions to the EricksenLeslie system equipped with the OseenFrank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measure-valued solutions for vanishing regularization. Additionally, it is shown that the measure-valued solution fulfills an energy inequality.
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    Existence of weak solutions for the Cahn-Hilliard reaction model including elastic effects and damage
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Kraus, Christiane; Roggensack, Arne
    In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries. Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the coupled PDE system.
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    A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Sprekels, Jürgen; Valdinoci, Enrico
    In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the sth power of a positive definite operator having a discrete spectrum in R+. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter s. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter s serves as the control parameter that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new class of identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter s also the domain of definition, and thus the underlying function space, of the fractional operator changes.
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    Nonlocal isoperimetric problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Castro, Agnese di; Novaga, Matteo; Ruffini, Berardo; Valdinoci, Enrico
    We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of fractional perimeters. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers.
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    A temperature-dependent phase-field model for phase separation and damage
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Heinemann, Christian; Kraus, Christiane; Rocca, Elisabetta; Rossi, Riccarda
    In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [21, 22]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of entropic weak solutions, resorting to a solvability concept first introduced in [10] in the framework of Fourier-Navier-Stokes systems and then recently employed in [9, 38] for the study of PDE systems for phase transition and damage. Our global-intime existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.
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    Analysis of electronic models for solar cells including energy resolved defect densities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Glitzky, Annegret
    We introduce an electronic model for solar cells including energy resolved defect densities. The resulting drift-diffusion model corresponds to a generalized van Roosbroeck system with additional source terms coupled with ODEs containing space and energy as parameters for all defect densities. The system has to be considered in heterostructures and with mixed boundary conditions from device simulation. We give a weak formulation of the problem. If the boundary data and the sources are compatible with thermodynamic equilibrium the free energy along solutions decays monotonously. In other cases it may be increasing, but we estimate its growth. We establish boundedness and uniqueness results and prove the existence of a weak solution. This is done by considering a regularized problem, showing its solvability and the boundedness of its solutions independent of the regularization level.
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    An electronic model for solar cells including active interfaces and energy resolved defect densities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Glitzky, Annegret
    We introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generation-recombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level.
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    Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Patterson, Robert I.A.
    Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of d-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, textitStoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.