Low volume-fraction microstructures in shape memory alloys
Dienstag, 20.10.15, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
I will report recent analytical results on variational models for microstructures in low-hysteresis shape memory alloys. It has been conjectured based on experimental findings, that the width of the thermal hysteresis in certain martensitic transformations is closely related to the crystallographic compatibility of the highly symmetric austenite phase and the martensitic variants (Zhang, James, Mueller, Acta mat. 57(15):4332-4352, 2009). Following this ansatz, I will focus on the singularly-perturbed two-well problem for almost compatible phases. This talk is partly based on joint works with Sergio Conti and Johannes Diermeier (both Bonn).
Integral operators and inequalities with weights
Dienstag, 27.10.15, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
Duality, regularity and uniqueness for \(BV\)-minimizers
Dienstag, 17.11.15, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
In my talk I will discuss similar convex variational integrals under a\nlinear growth condition. After a short introduction to the dual problem\nin the sense of convex analysis I will explain the duality relations\nbetween generalized minimizers and the dual solution. The duality\nrelations can be interpreted as mutual representation formulas, and in\nparticular they allow to deduce statements on uniqueness and regularity\nfor generalized minimizers. The results presented in this talk are based\non a joined project with Thomas Schmidt (Erlangen).
Rayleigh-Benard convection at finite Prandtl number: bounds on the Nusselt number.
Dienstag, 24.11.15, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
We consider Rayleigh-Benard convection at finite Prandtl number\nas modeled by the Boussinesq equation. We are interested in the scaling\nof the average upward heat transport, the Nusselt number, in \nterms of\nthe Rayleigh number, and the Prandtl number.\n\nIn this talk I present a rigorous upper bound for the Nusselt number reproducing \nboth physical\n scalings in some parameter regimes up to logarithms. \n\nThis is a joint work with Felix Otto and Antoine Choffrut.\n
Existenz und Eindeutigkeit der stochastischen Allen-Cahn-Gleichung
Dienstag, 12.1.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Symmetry breaking in indented elastic cones
Dienstag, 2.2.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys. Rev. Lett., 2004), we consider a variational plate model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by suitable Dirichlet boundary conditions at the center and the boundary of the sheet, we identify the energy scaling law in the von-Karman plate model. Specifically, we find that three different regimes arise with increasing indentation: initially the energetic cost of the logarithmic singularity dominates, then there is a linear response corresponding to a moderate deformation close to the boundary of the cone, and for larger indentation a localized inversion takes place in the central region.\nThen we show that for large enough indentations minimizers of the elastic energy cannot be radially symmetric. We do so by an explicit construction that achieves lower elastic energy than the minimum amount possible for radially symmetric deformations.\n\nJoint work with S. Conti (IAM Bonn) and I. Tobasco (CIMS New York)
Remarks on the convergence of pseudospectra
Dienstag, 9.2.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Pseudospectra help to understand general properties of non-normal matrices and operators, e.g. spectral instabilities or decay rates of associated semigroups.First we give a summary of basic notions and results. Then we discuss the question whether a linear operator can have constant resolvent norm on an open set and present several results that exclude this phenomenon. Finally we show the convergence of pseudospectra for operators acting in different Hilbert spaces and mention applications to Schroedinger operators.The talk is based mainly on joint work with Sabine Boegli
Absolute instability of spatially developing/temporally oscillating unbounded flows and media
Dienstag, 23.2.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We present our recent results on the absolute instability of spatially developing/temporally oscillating unbounded flows and media. In the treatment of temporally oscillating\nflows, the Floquet theory is applied. As an example of the application of the treatment a nonlinear Schrodinger equation is analyzed on absolute instability. The spatially\ndeveloping flows treated include localized flows and flows with the tails that decay algebraically sufficiently rapidly, when x goes to + or - infinity: Flows having the same limit state, when x goes to + or - infinity as well as those having different limit states, when x goes to infinity and\nx goes to - infinity, i.e. fronts, are considered. In the treatment, no restriction of the rate of\nvariability of the base state in the finite domain is imposed and no approximations are used. The initial-value stability problem is treated by using the Laplace transform.\nThe resulting boundary-value problem with spatially variable coecients is treated as a dynamical system by using the exact asymptotic expressions, when x goes to + or - infinity for\nthe fundamental matrix of the problem. In the non-localized case, the derivation of\nthe asymptotics of the fundamental matrix is based on the application of the Levinson theorem. The boundary-value problem is solved formally and a set of the dispersion relation functions, Dn(w); for the global normal modes, for the corresponding regions,\nin complex domain, n >= 1, is obtained, where w is a frequency (and a Laplace transform parameter).\nThe solution of the stability problem is given by an inverse Laplace transform\nof the solution of the boundary-value problem. By using this solution, the conditions for the absolute instability of the \now in each case considered are obtained in terms of\nthe global dispersion-relation functions, the dispersion-relation functions of the limit states at + or - infinity and the matrix-functions entering into the asymptotics of the fundamental matrix of the boundary-value problem. Since all the objects controlling the instabilities are essentially global properties of the \nflow, it is maintained that the concept\nof local stability cannot be consistently defined for the \nows treated. A procedure for computing the instabilities is suggested.