171 research outputs found
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 [C. Heinemann, C. Kraus: Existence results of weak
solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv.
Math. Sci. Appl. 21 (2011), 321--359]. Then we prove existence of weak
solutions by means of regularization, time-discretization and different
variational techniques
Complete damage in linear elastic materials - Modeling, weak formulation and existence results
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying Dirichlet boundary
conditions. The evolution of the system is described by a doubly nonlinear
differential inclusion for the damage process and a quasi-static balance
equation for the displacement field which are strongly nonlinearly coupled. In
our proposed model, the material may completely disintegrate which is
indispensable for a realistic modeling of damage processes in elastic
materials. Complete damage theories lead to several mathematical problems since
for instance coercivity properties of the free energy are lost and, therefore,
several difficulties arise. For the introduced complete damage model, we
propose a classical formulation and a corresponding suitable weak formulation
in an -framework. The main aim is to prove existence of weak solutions for
the introduced degenerating model. In addition, we show that the classical
differential inclusion can be regained from the notion of weak solutions under
certain regularity assumptions which is a novelty in the theory of complete
damage models of this type
A compressible mixture model with phase transition
We introduce a new thermodynamically consistent diffuse interface model of Allen--Cahn/Navier--Stokes type for multi-component flows with phase transitions and chemical reactions.
For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques.
We consider two scaling regimes, i.e.~a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized
Allen-Cahn/Euler system for mixtures with chemical
reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young--Laplace and a Stefan type law
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A solution of Braess' approximaiton problem on powers of the distance function
The polynomial approximation behaviour of the class of functions
Fs:R2(x0,y0)−>R,Fs(x,y)=((x−x0)2+(y−y0)2)(−s),sin(0,infty),
is studied in [Bra01]. There it is claimed that the obtained results can be embedded in a more general setting. This conjecture will be confirmed and complemented by a different approach than in [Bra01]. The key is to connect the approximation rate of F_s with its holomorphic continuability for which the classical Bernstein approximation theorem is linked with the convexity of best approximants. Approximation results of this kind also play a vital role in the numerical treatment of elliptic differential equations [Sau]
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Maximal convergence theorems for functions of squared modulus holomorphic type and various applications
In this paper we extend the theory of maximal convergence introduced by Walsh to functions of squared modulus holomorphic type. We introduce in accordance to the well-known complex maximal convergence number for holomorphic functions a real maximal convergence number for functions of squared modulus holomorphic type and prove several maximal convergence theorems. We achieve that the real maximal convergence number for F is always greater or equal than the complex maximal convergence number for g and equality occurs if L is a closed disk in R^2. Among other various applications of the resulting approximation estimates we show that for functions F of squared holomorphic type which have no zeros in a closed disk B_r the relation
limsupntoinftysqrt[n]En(Br,F)=limsupntoinftysqrt[n]En(partialBr,F)
is valid, where E_n is the polynomial approximation error
The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs--Thomson law
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end approximate solutions are constructed by means of variational functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formulation
Bernstein--Walsh type theorems for real analytic functions in several variables
The aim of this paper is to extend the classical maximal convergence
theory of Bernstein and Walsh for holomorphic functions in the complex plane to
real analytic functions in R^N.
In particular, we investigate the polynomial approximation behavior for
functions , of the type
, where g and h are holomorphic in a neighborhood of
a compact set . To this end the maximal convergence number
for continuous functions f defined on a compact set
S_c \subset \C^N is connected to a maximal
convergence number for continuous functions F defined on a
compact set
A solution of Braess' approximation problem on powers of the distance function
The polynomial approximation behaviour of the class of functions
F_s: R^2\(x_0, y_0 ) -> R, F_s(x,y) = ( (x-x_0)^2 + (y-y_0)^2 )^(-s), s \in (0, \infty),
is studied in [Bra01]. There it is claimed that the obtained results
can be embedded in a more general setting. This conjecture will be confirmed and complemented by
a different approach than in [Bra01]. The key is to connect
the approximation rate of F_s with its holomorphic
continuability for which the classical Bernstein approximation theorem
is linked with the convexity of best approximants.
Approximation results of this kind also play a vital role in
the numerical treatment of elliptic
differential equations [Sau]
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