Global Existence and Asymptotic Limits of Weak Solutions of the Bipolar Hydrodynamic Model for Semiconductors

For the case of the adiabatic exponents being larger than , we establish the global existence of entropy weak solutions of the Cauchy problem to the bipolar hydrodynamic model for semiconductors. Using the theory of compensated compactness, we hence give finally a complete answer on the related existence problems with the -law pressure relation. A new kind of singular limit of the modified entropy weak solution is discussed. To some extent, the limit of this sort can provide some information about the uniform boundedness of the scaled solution sequences. The quasineutral-relaxation limit of the entropy weak solutions is also investigated.     

A Quasi-Neutral Limit in a Hydrodynamic Model for Charged Fluids

 The combined quasineutral and relaxation time limit for a bipolar hydrodynamic model is considered. The resulting limit problem is a nonlinear diffusion equation describing a neutral fluid. We make use of various entropy functions and the related entropy productions in order to obtain strong enough uniform bounds. The necessary strong convergence of the densities is obtained by using a generalized version of the “div-curl” Lemma and monotonicity methods. Received September 27, 2001; in revised form February 25, 2002     

Global Existence and Exponential Stability of Smooth Solutions to a Full Hydrodynamic Model to Semiconductors

 We study a full hydrodynamic semiconductor model in multi-space dimension. The global existence of smooth solutions is established and the exponential stability of the solutions as is investigated.     

Smooth singular flows in dimension 2 with the minimal self-joining property

It is proved that some velocity changes in flows on the torus determined by quasi-periodic Hamiltonians on : where α₁/α₂ is an irrational number with bounded partial quotients, lead to singular flows on with an ergodic component having a minimal set of self-joinings. Authors’ address: K. Frączek and M. Lemańczyk, Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland Research partially supported by KBN grant 1 P03A and by Marie Curie “Transfer of Knowledge” program, project MTKD-CT-2005-030042 (TODEQ) 03826.     

Convergence to Global Equilibrium for Spatially Inhomogeneous Kinetic Models of Non-Micro-Reversible Processes

We study the long-time asymptotics of linear kinetic models with periodic boundary conditions or in a rectangular box with specular reflection boundary conditions. An entropy dissipation approach is used to prove decay to the global equilibrium under some additional assumptions on the equilibrium distribution of the mass preserving scattering operator. We prove convergence at an algebraic rate depending on the smoothness of the solution. This result is compared to the optimal result derived by spectral methods in a simple one dimensional example.     

Ergodic Optimization for Renewal Type Shifts

We consider a class of countable Markov shifts and a locally H?lder potential φ. We prove that the existence of φ-optimal measures is closely related to the behaviour of the pressure function t → P(tφ). Using a Theorem by Sarig it is possible to prove that there exists a critical value t _c ∈ (0, ∞] such that for t < t _c the pressure is analytic and for t > t _c is linear. We prove that if t _c is finite, then there are no φ-optimal measures, and if it is infinite, then φ-optimal measures do exist. The author was partially supported by FCT/POCTI/FEDER and the grant SFRH/BPD/21927/2005.     

Existence and Exponential Decay of Solutions to a Quasilinear Thermoelastic Plate System

We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in , n ≤ 3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is shown that finite energy solutions decay exponentially to zero with the rate depending only on the (finite energy) size of initial conditions. The proofs are based on methods of weak compactness along with nonlocal partial differential operator multipliers which supply the sought after “recovery” inequalities. Regularity of solutions is also discussed by exploiting the underlying analyticity of the linearized semigroup along with a related maximal parabolic regularity [1, 16, 44]. The research of I. Lasiecka has been partially supported by DMS-NSF Grant Nr 0606882. S. Maad was supported by the Swedish Research Council and by the European Union under the Marie Curie Fellowship MEIF-CT-2005-024191.     

Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models

We prove asymptotic stability results for nonlinear bipolar drift-diffusion-Poisson Systems arising in semiconductor device modeling and plasma physics in one space dimension. In particular, we prove that, under certain structural assumptions on the external potentials and on the doping profile, all solutions match for large times with respect to all q-Wasserstein distances. We also prove exponential convergence to stationary solutions in relative entropy via the so called entropy dissipation (or Bakry-émery) method. Authors’ addresses: Marco Di Francesco, Sezione di Matematica per L’Ingegneria, Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, Piazzale E. Pontieri, 2, Monteluco di Roio, 67040 L’Aquila, Italy; Marcus Wunsch, Fakult?t für Mathematik, Universit?t Wien, Nordbergstra?e 15, A-1090 Wien, Austria     

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.Present address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal     

Entropies and Equilibria of Many-Particle Systems: An Essay on Recent Research

This essay is intended to present a fruitful collaboration which has developed among a group of people whose names are listed above: entropy methods have proved over the last years to be an efficient tool for the understanding of the qualitative properties of physically sound models, for accurate numerics and for a more mathematical understanding of nonlinear PDEs. The goal of this essay is to sketch the historical development of the concept of entropy in connection with PDEs of continuum mechanics, to present recent results which have been obtained by the members of the group and to emphasize the most striking achievements of this research. The presentation is by no way an exhaustive review of the methods and results involving the entropy, not even in the field of PDEs. Many other researchers in and outside Europe have contributed to the development of this field, including – but not only – in collaboration with some of the people of the group. However, it can be claimed that this group had a leading role over the recent years and this essay is intended to explain how this occurred.     

Diffusion Limit of a Non-Linear Kinetic Model without the Detailed Balance Principle

 The aim of this paper is the rigorous derivation of a non-linear drift-diffusion model from the Boltzmann equation in a semiconductor device. Collisions are taken into account through the non-linear Pauli operator, without assuming any relation concerning the cross-section (such as the so-called detailed balance principle). The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The same program is applied to the derivation of a non-linear SHE (Spherical Harmonics Expansion) model, when elastic collisions are considered, and is rigorously carried through in the case of zero electric field. (Received 5 October 2000; in revised form 3 September 2001)     

Thermodynamic formalism of transcendental entire maps of finite singular type

We build a version of a thermodynamic formalism for maps of the form f(z) = ∑ _{j = 0} ^{p + q} a _j e ^(j−p)z where p, q > 0 and . We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J _f ^r ) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J _f ^r is the radial Julia set. Partially supported by NSF Grant DMS 0100078. Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean FONDECYT Grant No. 11060280.     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

 This paper describes the cutting sequences of geodesic flow on the modular surface with respect to the standard fundamental domain of . The cutting sequence for a vertical geodesic is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain ℱ up to an isometry of the hyperbolic plane . We give conversion methods between the cutting sequence for the vertical geodesic , the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first symbols when given as input the first symbols of the ACF expansion, which takes time and space . (Received 11 August 2000; in revised form 18 April 2001)     

Parabolic Implosion and Julia-Lavaurs Sets in the Exponential Family

We deal with all the maps from the exponential family f _ε(z) = (e ⁻¹ + ε)exp(z), with ε ≥ 0. Let h _ε = HD(J _r,ε) be the Hausdorff dimension of the radial Julia sets J _r,ε. Observing the phenomenon of parabolic implosion, it is shown that the function ε ↦ h _ε is not continuous from the right. The research of the first author was supported in part by the NSF Grant DMS 0100078.     

Bounded Positive Solutions of Schrödinger Equations in Two-Dimensional Exterior Domains

We prove under quite general assumptions the existence of a bounded positive solution to the semilinear Schrödinger equation in a two-dimensional exterior domain.     

Rotating Charge Coupled to the Maxwell Field: Scattering Theory and Adiabatic Limit

We consider a spinning charge coupled to the Maxwell field. Through the appropriate symmetry in the initial conditions the charge remains at rest. We establish that any time-dependent finite energy solution converges to a sum of a soliton wave and an outgoing free wave. The convergence holds in global energy norm. Under a small constant external magnetic field the soliton manifold is stable in local energy seminorms and the evolution of the angular velocity is guided by an effective finite-dimensional dynamics. The proof uses a non-autonomous integral inequality method.Supported partly by the Wittgenstein 2000 Award of Peter Markowich, funded by the Austrian Science Foundation (FWF), research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).On leave Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia. Supported partly by Max Planck Institute for the Mathematics in Sciences (Leipzig) and the Austrian Science Foundation (FWF) START Project (Y-137-TEC) of Norbert Mauser.