#### Filter Results:

- Full text PDF available (70)

#### Publication Year

1988

2017

- This year (1)
- Last 5 years (20)
- Last 10 years (42)

#### Publication Type

#### Co-author

#### Journals and Conferences

#### Brain Region

#### Method

#### Organism

Learn More

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. Theâ€¦ (More)

- Alexander Mielke
- 1997

We develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations. In particular, it enables us to detect sideband instabilities. We treat in all detail the stability question of roll solutions in the two{dimensional Swift{Hohenberg equation and derive a condition on the amplitude and the waveâ€¦ (More)

- Alexander Mielke
- 2009

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given inâ€¦ (More)

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizersâ€¦ (More)

- Alexander Mielke, Aida M. Timofte
- SIAM J. Math. Analysis
- 2007

This paper is devoted to the homogenization for a class of rate-independent systems described by the energetic formulation. The associated nonlinear partial differential system has periodically oscillating coefficients, but has the form of a standard evolutionary variational inequality. Thus, the model applies to standard linearized elastoplasticity withâ€¦ (More)

- Alexander Mielke
- 2002

We consider rate-independent models which are defined via two functionals: the time-dependent energy-storage functional I : [0, T ]Ã—X â†’ [0,âˆž] and the dissipation distance D : X Ã— X â†’ [0,âˆž]. A function z : [0, T ] â†’ X is called a solution of the energetic model, if for all 0 â‰¤ s < t â‰¤ T we have stability: I(t, z(t)) â‰¤ I(t, zÌƒ) +D(z(t), zÌƒ) for all zÌƒ âˆˆ X;â€¦ (More)

We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffithâ€™s local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of thisâ€¦ (More)

We are interested in the long{time behavior of nonlinear parabolic PDEs deened on unbounded cylindrical domains. For dissipative systems deened on bounded domains, the long{time behavior can often be described by the dynamics in their nite{dimensional attractors. For systems deened on the innnite line, very little is known at present, since the lack ofâ€¦ (More)

Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extendedâ€¦ (More)

- Alexander Mielke
- 2005

We consider the dynamics of infinite harmonic lattices in the limit of the lattice distance Îµ tending to 0. We allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by Fourier transform and linear algebra. Our aim is to derive macroscopic continuum limit equations for Îµ â†’ 0. For the weak limit ofâ€¦ (More)