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Markus Kunze

Publications and source records attributed to Markus Kunze.

46 records · Page 3Linked to original sources
arXiv2009

Continuity and Equicontinuity of Transition Semigroups

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on $E$, every transition semigroup on $C_b(E)$ which is continuous with respect to the strict topology $β_0$ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of $β_0$-continuous semigroups on $C_b(E)$ and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be $β_0$-continuous.

math.FA↗
arXiv2008

Gravitational collapse and the formation of black holes for the spherically symmetric Einstein-Vlasov system

We review results on the spherically symmetric, asymptotically flat Einstein-Vlasov system. We focus on a recent result where we found explicit conditions on the initial data which guarantee the formation of a black hole in the evolution. Among these data there are data such that the corresponding solutions exist globally in Schwarzschild coordinates. We put these results into a more general context, and we include arguments which show that the spacetimes we obtain satisfy the weak cosmic censorship conjecture and contain a black hole in the sense of suitable mathematical definitions of these concepts which are available in the literature.

gr-qc↗
arXiv2008

The formation of black holes in spherically symmetric gravitational collapse

We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system. We find explicit conditions on the initial data, with ADM mass M, such that the resulting spacetime has the following properties: there is a family of radially outgoing null geodesics where the area radius r along each geodesic is bounded by 2M, the timelike lines $r=c\in [0,2M]$ are incomplete, and for r>2M the metric converges asymptotically to the Schwarzschild metric with mass M. The initial data that we construct guarantee the formation of a black hole in the evolution. We also give examples of such initial data with the additional property that the solutions exist for all $r\geq 0$ and all Schwarzschild time, i.e., we obtain global existence in Schwarzschild coordinates in situations where the initial data are not small. Some of our results are also established for the Einstein equations coupled to a general matter model characterized by conditions on the matter quantities.

gr-qc↗
arXiv2006

Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter

We prove a new global existence result for the asymptotically flat, spherically symmetric Einstein-Vlasov system which describes in the framework of general relativity an ensemble of particles which interact by gravity. The data are such that initially all the particles are moving radially outward and that this property can be bootstrapped. The resulting non-vacuum spacetime is future geodesically complete.

gr-qc↗
arXiv2004

The Darwin Approximation of the Relativistic Vlasov-Maxwell System

We study the relativistic Vlasov-Maxwell system which describes large systems of particles interacting by means of their collectively generated forces. If the speed of light $c$ is considered as a parameter then it is known that in the Newtonian limit $c\to\infty$ the Vlasov-Poisson system is obtained. In this paper we determine the next order approximate system, which in the case of individual particles usually is called the Darwin approximation.

math-ph↗
arXiv2001

Simplified models of electromagnetic and gravitational radiation damping

In previous work the authors analysed the global properties of an approximate model of radiation damping for charged particles. This work is put into context and related to the original motivation of understanding approximations used in the study of gravitational radiation damping. It is examined to what extent the results obtained previously depend on the particular model chosen. Comparisons are made with other models for gravitational and electromagnetic fields. The relation of the kinetic model for which theorems were proved to certain many-particle models with radiation damping is exhibited.

gr-qc↗
arXiv2001

Post-Coulombian Dynamics at Order 1.5

We study the dynamics of N charges interacting with the Maxwell field. If their initial velocities are small compared to the velocity of light, c, then in lowest order their motion is governed by the static Coulomb Lagrangian. We investigate higher order corrections with an explicit control on the error terms. The Darwin correction, order (v/c)^2, has been proved previously. In this contribution we obtain the dissipative corrections due to radiation damping, which are of order (v/c)^3 relative to the Coulomb dynamics. If all particles have the same charge-to-mass ratio, the dissipation would vanish at that order.

math-ph↗
arXiv2000

The Vlasov-Poisson system with radiation damping

We set up and analyze a model of radiation damping within the framework of continuum mechanics, inspired by a model of post-Newtonian hydrodynamics due to Blanchet, Damour and Schaefer. In order to simplify the problem as much as possible we replace the gravitational field by the electromagnetic field and the fluid by kinetic theory. We prove that the resulting system has a well-posed Cauchy problem globally in time for general initial data and in all solutions the fields decay to zero at late times. In particular, this means that the model is free from the runaway solutions which frequently occur in descriptions of radiation reaction.

math-ph↗
arXiv2000

Slow Motion of Charges Interacting Through the Maxwell Field

We study the Abraham model for $N$ charges interacting with the Maxwell field. On the scale of the charge diameter, $R_ϕ$, the charges are a distance $\eps^{-1}R_ϕ$ apart and have a velocity $\sqrt{\eps} c$ with $\eps$ a small dimensionless parameter. We follow the motion of the charges over times of the order $\eps^{-3/2}R_ϕ/c$ and prove that on this time scale their motion is well approximated by the Darwin Lagrangian. The mass is renormalized. The interaction is dominated by the instantaneous Coulomb forces, which are of the order $\eps^{2}$. The magnetic fields and first order retardation generate the Darwin correction of the order $\eps^{3}$. Radiation damping would be of the order $\eps^{7/2}$.

math-ph↗