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19
Pointwise semigroup methods and stability of viscous shock waves
 Indiana Univ. Math. J
, 1998
"... Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator abou ..."
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Cited by 63 (32 self)
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Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ([Sat], [K.2], resp.), each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We overcome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing earlier work of Howard [H.1], Kapitula [K.12], and Zeng [Ze,LZe]. These techniques allow us to do \hard " analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications
Nonequilibrium statistical mechanics of strongly anharmonic chains of oscillators
 Comm. Math. Phys
"... We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with ..."
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Cited by 44 (11 self)
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We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hörmander’s theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.
Renewal theory and computable convergence rates for geometrically ergodic Markov chains
, 2003
"... We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start f ..."
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Cited by 39 (0 self)
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We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition toward a “small set. ” The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the firstentrance– lastexit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory. 1. Introduction. Let {Xn:n ≥ 0
Minimization of Electrostatic Free Energy and the PoissonBoltzmann Equation for Molecular Solvation with Implicit Solvent
, 2008
"... In an implicitsolvent description of the solvation of charged molecules (solutes), the electrostatic free energy is a functional of the ionic concentrations in the solvent. The charge density is determined by such concentrations together with the point charges of the solute atoms, and the electrost ..."
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Cited by 12 (9 self)
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In an implicitsolvent description of the solvation of charged molecules (solutes), the electrostatic free energy is a functional of the ionic concentrations in the solvent. The charge density is determined by such concentrations together with the point charges of the solute atoms, and the electrostatic potential is determined by the Poisson equation with a variable dielectric coefficient. Such a free energy functional is considered in this work for both the case of point ions and that of ions with a finite size. It is proved for each case that there exists a unique set of equilibrium concentrations that minimize the free energy and that are given by the corresponding Boltzmann distributions through the equilibrium electrostatic potential. Such distributions are found to depend on the boundary data for the Poisson equation. Proofs are also given for the existence and uniqueness of the boundaryvalue problem of the resulting PoissonBoltzmann equation that determines the equilibrium electrostatic potential. Finally, the equivalence of two different forms of such a boundaryvalue problem is proved. 2000 Mathematics Subject Class: 35J, 35Q, 49S, 82D, 92C. Key words and phrases: implicit solvent, electrostatic free energy, ionic concentrations, electrostatic potentials, the PoissonBoltzmann equation, variational methods, nonlinear elliptic interface problems.
COMPLETE ELECTRODE MODEL OF ELECTRICAL IMPEDANCE TOMOGRAPHY: APPROXIMATION PROPERTIES AND CHARACTERIZATION OF INCLUSIONS
, 2004
"... In electrical impedance tomography one tries to recover the spatial admittance distribution inside a body from boundary measurements. In theoretical considerations it is usually assumed that the boundary data consists of the NeumanntoDirichlet map; when conducting realworld measurements, the obt ..."
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Cited by 4 (1 self)
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In electrical impedance tomography one tries to recover the spatial admittance distribution inside a body from boundary measurements. In theoretical considerations it is usually assumed that the boundary data consists of the NeumanntoDirichlet map; when conducting realworld measurements, the obtainable data is a linear finitedimensional operator mapping electrode currents onto electrode potentials. In this paper it is shown that when using the complete electrode model to handle electrode measurements, the corresponding currenttovoltage map can be seen as a discrete approximation of the traditional NeumanntoDirichlet operator. This approximating link is utilized further in the special case of constant background conductivity with inhomogeneities: It is demonstrated how inclusions with strictly higher or lower conductivities can be characterized by the limit behavior of the range of a boundary operator, determined through electrode measurements, when the electrodes get infinitely small and cover all of the object boundary.
A framework for modeling subgrid effects for twophase flows
"... In this paper, we study upscaling for twophase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effect. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equati ..."
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Cited by 4 (4 self)
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In this paper, we study upscaling for twophase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effect. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the twophase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale twophase flows in practical applications. 1
Harmonic Calculus On Fractals  A Measure Geometric Approach I
 II, Trans. Amer. Math. Soc
"... Differentiation of functions w.r.t. finite atomless measures with compact support on the real line is introduced. The related harmonic calculus is similar to that of the classical Lebesgue case. As an application we obtain the Weyl exponent for the spectral asymptotics of the Laplacians w.r.t. linea ..."
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Cited by 3 (3 self)
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Differentiation of functions w.r.t. finite atomless measures with compact support on the real line is introduced. The related harmonic calculus is similar to that of the classical Lebesgue case. As an application we obtain the Weyl exponent for the spectral asymptotics of the Laplacians w.r.t. linear Cantor type measures with arbitrary weights. Keywords: Measure geometric Laplacian, Green's function, selfsimilar measure 2000 Mathematics Subject Classification. Primary 28A80; Secondary 35P20 1 Introduction In related mathematical literature one can find different ways of defining a Laplacian on fractal sets, for example, the probabilistic approach (\Delta as the infinitesimal generator of a Brownian motion which is the limit of renormalized random walks on certain "prefractals", see [6], [12], [2], [1], [13]) or the analytic approach (\Delta as the limit of difference operators defined for functions on the "construction steps" of the fractal, see [10], [11]). In order to avoid depe...
THE NONRELATIVISTIC LIMIT IN RADIATION HYDRODYNAMICS: I. WEAK ENTROPY SOLUTIONS FOR A MODEL PROBLEM
"... Abstract. This paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions. The system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integraltype source term to a family of radiation transport equations. We ..."
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Cited by 3 (0 self)
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Abstract. This paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions. The system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integraltype source term to a family of radiation transport equations. We first show existence of entropy solutions to Cauchy problems of the model system in the framework of functions of bounded variation. This is done by using differences schemes and discrete ordinates. Then we establish strong convergence of the entropy solutions, indexed with c, as c goes to infinity. The limit function satisfies a scalar integrodifferential equation. 1.
Stochastic control for distributed systems with applications to wireless communications
, 2003
"... This thesis investigates control and optimization of distributed stochastic systems motivated by current wireless applications. In wireless communication systems, power control is important at the user level in order to minimize energy requirements and to maintain communication Quality of Service (Q ..."
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Cited by 2 (2 self)
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This thesis investigates control and optimization of distributed stochastic systems motivated by current wireless applications. In wireless communication systems, power control is important at the user level in order to minimize energy requirements and to maintain communication Quality of Service (QoS) in the face of user mobility and fading channel variability. Clever power allocation provides an efficient means to overcome in the uplink the socalled nearfar effect, in which nearby users with higher received powers at the base station may overwhelm signal transmission of far away users with lower received powers, and to compensate for the random fluctuations of received power due to combined shadowing and possibly fast fading (multipath interference) effects. With the wireless uplink power control problem for dynamic lognormal shadow fading channels as an initial paradigm, a class of stochastic control problems is formulated which includes a fading channel model and a power adjustment model. For optimization of such a system, a cost function is proposed which reflects the QoS requirements of mobile users in wireless systems. For the resulting stochastic control