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| Christiane Tretter (U. Bern) | WPI Seminar room, 8th floor Fak.Math. Univ. Wien | Wed, 11. Nov 26, 10:00 |
| Essential numerical ranges | ||
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| Anton Arnold (TU Wien) | WPI Seminar room, 8th floor Fak.Math. Univ. Wien | Wed, 11. Nov 26, 10:45 |
| Short- and long-time behavior in evolution equations: the role of the hypocoercivity index | ||
Evolution equations with a system matrix or generator that is not coercive are called hypocoercive, if they still exhibit uniform exponential decay towards the steady state. Typical examples are kinetic Fokker-Planck equations, BGK equations, or even ODE systems. The "index of hypocoercivity" is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it describes the interplay between these two parts. In a general Hilbert space setting, we show that this index characterizes the polynomial decay of the propagator norm for short time. We also show that bounded generators always have a finite index. We illustrate these concepts for the Lorentz kinetic equation on a torus. Using a modal decomposition reveals that this kinetic model has hypocoercivity index 1 (while coercive generators have index 0). Discrete time analogues of the above systems (obtained via the mid-point rule in time) are contractive, but typically not strictly contractive. For this setting we introduce "hypocontractivity" and an "index of hypocontractivity" and discuss their close connection to the continuous time evolution equations. This talk is based on joint work with F. Achleitner, E. Carlen, S. Egger, E. Nigsch, and V. Mehrmann. | ||
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| Stephan Egger (TU Wien) | WPI Seminar room, 8th floor Fak.Math. Univ. Wien | Wed, 11. Nov 26, 14:15 |
| Spectral analysis of the linearized spatially inhomogeneous Vicsek-BGK equation | ||
The Vicsek-BGK equation is a kinetic equation modelling collective phenomena such as swarming and collective motion. The equilibria of the spatially homogeneous equation and their stability properties are well understood, while only partial results are available for the spatially inhomogeneous equation. In this work, we extend the existing results by performing a spectral analysis of the linearized spatially inhomogeneous Vicsek-BGK operator. In particular, we establish the surprising result that the spatially inhomogeneous model exhibits a bifurcation that is absent in the spatially homogeneous setting. Moreover, we show that, for a large class of equilibria, their stability properties differ between the spatially homogeneous and spatially inhomogeneous equations. | ||
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| Marco Roschkowski (Bergische U. Wuppertal) | WPI Seminar room, 8th floor Fak.Math. Univ. Wien | Wed, 11. Nov 26, 15:30 |
| Open discussion: Generalizing the Kalman rank condition to the unbounded case | ||
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| Stéphane Mischler (U. Paris Dauphine) | WPI Seminar room, 8th floor Fak.Math. Univ. Wien | Thu, 12. Nov 26, 10:00 |
| Longtime asymptotic for kinetic equations without confinement | ||
I will present some more or less recent contributions to the understanding of the longtime asymptotic for kinetic (as relaxation and Kinetic Fokker-Planck) equations set in the whole space without confinement mechanism. The general message is that the behavior of solutions is similar to the behavior of solutions to the heat equation. The proof is based on Hypocoercivity estimates together with interpolation / Hilbert expansion / Doblin-Harris arguments. | ||
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| Eduard Nigsch (TU Wien) | WPI Seminar room, 8th floor Fak.Math. Univ. Wien | Thu, 12. Nov 26, 10:45 |
| Hypocoercivity for nonlinear semi-dissipative Hamiltonian systems | ||
We introduce and analyze the notion of hypocoercivity index for real nonlinear semi-dissipative Hamiltonian systems. We construct a nonlinear staircase form in analogy to the linear case and examine how the decay behavior of solutions is related to the hypocoercivity index and the staircase form in this setting. | ||
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| Christian Schmeiser (U. Wien) | WPI Seminar room, 8th floor Fak.Math. Univ. Wien | Fri, 13. Nov 26, 10:00 |
| First thoughts on a non-instantaneous collision version of the Kac model | ||
We formulate a one-dimensional kinetic model with non-instantaneous binary collisions, where during a collision the pair of velocities is rotated in the plane. The formal instantaneous limit is the Boltzmann-Kac equation. Several other formal properties will be discussed, such as Gaussian equilibria, entropy dissipation, and the dynamics of moments. Even for the spatially homogeneous model a hypocoercivity issue arises, since the collision dynamics is of transport type (joint work with Laura Kanzler). | ||
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