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Alexander A. Kiselev

William T. Laprade Distinguished Professor of Mathematics
Mathematics

Selected Publications


A simple reaction-diffusion system as a possible model for the origin of chemotaxis.

Journal Article Journal of biological dynamics · December 2023 Chemotaxis is a directed cell movement in response to external chemical stimuli. In this paper, we propose a simple model for the origin of chemotaxis - namely how a directed movement in response to an external chemical signal may occur based on purely rea ... Full text Cite

Illposedness of C2 Vortex Patches

Journal Article Archive for Rational Mechanics and Analysis · June 1, 2023 It is well known that vortex patches are wellposed in C1,α if 0 < α< 1 . In this paper, we prove the illposedness of C2 vortex patches. The setup is to consider the vortex patches in Sobolev spaces W2,p where the curvature of the boundary is Lp integrable. ... Full text Open Access Cite

On Nonexistence of Splash Singularities for the α -SQG Patches

Journal Article Journal of Nonlinear Science · April 1, 2023 In this paper, we consider patch solutions to the α-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo and Stra ... Full text Cite

Small Scale Formations in the Incompressible Porous Media Equation

Journal Article Archive for Rational Mechanics and Analysis · February 1, 2023 We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stra ... Full text Cite

Chemotaxis and reactions in biology

Journal Article Journal of the European Mathematical Society · January 1, 2023 Chemotaxis plays a crucial role in a variety of processes in biology and ecology. Quite often it acts to improve efficiency of biological reactions. One example is the immune system signalling, where infected tissues release chemokines attracting monocytes ... Full text Cite

The Flow of Polynomial Roots Under Differentiation

Journal Article Annals of PDE · December 1, 2022 The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics ... Full text Cite

Random Search in Fluid Flow Aided by Chemotaxis.

Journal Article Bulletin of mathematical biology · June 2022 In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, ... Full text Cite

Hitting time of Brownian motion subject to shear flow

Journal Article Involve · January 1, 2022 The 2-dimensional motion of a particle subject to Brownian motion and ambient shear flow transportation is considered. Numerical experiments are carried out to explore the relation between the shear strength, box size, and the particle’s expected first hit ... Full text Open Access Cite

GLOBAL REGULARITY FOR A NONLOCAL PDE DESCRIBING EVOLUTION OF POLYNOMIAL ROOTS UNDER DIFFERENTIATION

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2022 In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Steinerberger [Proc. Amer. Math. Soc., 147 (2019), pp. 4733-4744] to model dynamics of roots of polynomials under differentiation. This partial differential eq ... Full text Cite

On nonexistence of splash singularities for the $α$-SQG patches

Journal Article · November 26, 2021 In this paper, we consider patch solutions to the $\alpha$-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo a ... Link to item Cite

Boundary layer models of the Hou-Luo scenario

Journal Article Journal of Differential Equations · October 15, 2021 Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou [16] proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is ax ... Full text Cite

Random search in fluid flow aided by chemotaxis

Journal Article · July 6, 2021 In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, ... Open Access Link to item Cite

Chemotactic Reaction Enhancement in One Dimension

Journal Article · March 26, 2021 Chemotaxis, the directional locomotion of cells towards a source of a chemical gradient, is an integral part of many biological processes - for example, bacteria motion, single-cell or multicellular organisms development, immune response, etc. Chemotaxis d ... Link to item Cite

Small scale formations in the incompressible porous media equation

Journal Article · February 9, 2021 We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stra ... Link to item Cite

A simple reaction-diffusion system as a possible model for the origin of chemotaxis.

Journal Article Journal of biological dynamics · December 2023 Chemotaxis is a directed cell movement in response to external chemical stimuli. In this paper, we propose a simple model for the origin of chemotaxis - namely how a directed movement in response to an external chemical signal may occur based on purely rea ... Full text Cite

Illposedness of C2 Vortex Patches

Journal Article Archive for Rational Mechanics and Analysis · June 1, 2023 It is well known that vortex patches are wellposed in C1,α if 0 < α< 1 . In this paper, we prove the illposedness of C2 vortex patches. The setup is to consider the vortex patches in Sobolev spaces W2,p where the curvature of the boundary is Lp integrable. ... Full text Open Access Cite

On Nonexistence of Splash Singularities for the α -SQG Patches

Journal Article Journal of Nonlinear Science · April 1, 2023 In this paper, we consider patch solutions to the α-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo and Stra ... Full text Cite

Small Scale Formations in the Incompressible Porous Media Equation

Journal Article Archive for Rational Mechanics and Analysis · February 1, 2023 We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stra ... Full text Cite

Chemotaxis and reactions in biology

Journal Article Journal of the European Mathematical Society · January 1, 2023 Chemotaxis plays a crucial role in a variety of processes in biology and ecology. Quite often it acts to improve efficiency of biological reactions. One example is the immune system signalling, where infected tissues release chemokines attracting monocytes ... Full text Cite

The Flow of Polynomial Roots Under Differentiation

Journal Article Annals of PDE · December 1, 2022 The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics ... Full text Cite

Random Search in Fluid Flow Aided by Chemotaxis.

Journal Article Bulletin of mathematical biology · June 2022 In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, ... Full text Cite

Hitting time of Brownian motion subject to shear flow

Journal Article Involve · January 1, 2022 The 2-dimensional motion of a particle subject to Brownian motion and ambient shear flow transportation is considered. Numerical experiments are carried out to explore the relation between the shear strength, box size, and the particle’s expected first hit ... Full text Open Access Cite

GLOBAL REGULARITY FOR A NONLOCAL PDE DESCRIBING EVOLUTION OF POLYNOMIAL ROOTS UNDER DIFFERENTIATION

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2022 In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Steinerberger [Proc. Amer. Math. Soc., 147 (2019), pp. 4733-4744] to model dynamics of roots of polynomials under differentiation. This partial differential eq ... Full text Cite

On nonexistence of splash singularities for the $α$-SQG patches

Journal Article · November 26, 2021 In this paper, we consider patch solutions to the $\alpha$-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo a ... Link to item Cite

Boundary layer models of the Hou-Luo scenario

Journal Article Journal of Differential Equations · October 15, 2021 Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou [16] proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is ax ... Full text Cite

Random search in fluid flow aided by chemotaxis

Journal Article · July 6, 2021 In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, ... Open Access Link to item Cite

Chemotactic Reaction Enhancement in One Dimension

Journal Article · March 26, 2021 Chemotaxis, the directional locomotion of cells towards a source of a chemical gradient, is an integral part of many biological processes - for example, bacteria motion, single-cell or multicellular organisms development, immune response, etc. Chemotaxis d ... Link to item Cite

Small scale formations in the incompressible porous media equation

Journal Article · February 9, 2021 We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stra ... Link to item Cite

Small-scale creation for solutions of the sqg equation

Journal Article Duke Mathematical Journal · January 1, 2021 We construct examples of solutions to the conservative surface quasigeostrophic (SQG) equation that must either exhibit infinite-in-Time growth of derivatives or blow up in finite time. ... Full text Cite

The Flow of Polynomial Roots Under Differentiation

Journal Article · December 16, 2020 The question about the behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Stefan Steinerberger to ... Link to item Cite

Chemotaxis and Reactions in Biology

Journal Article · April 14, 2020 Chemotaxis plays a crucial role in a variety of processes in biology and ecology. Quite often it acts to improve efficiency of biological reactions. One example is the immune system signalling, where infected tissues release chemokines attracting monocytes ... Link to item Cite

Small Scale Creation in Active Scalars

Chapter · January 1, 2020 The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a sp ... Full text Cite

Small Scale Creation in Active Scalars

Conference PROGRESS IN MATHEMATICAL FLUID DYNAMICS · 2020 Full text Cite

Global regularity and fast small-scale formation for Euler patch equation in a smooth domain

Journal Article Communications in Partial Differential Equations · April 3, 2019 It is well known that the Euler vortex patch in R 2 will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this article, we study Euler vortex patches in a general smooth bo ... Full text Cite

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

Journal Article Journal of Nonlinear Science · December 1, 2018 The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbol ... Full text Cite

Special Issue Editorial: Small Scales and Singularity Formation in Fluid Dynamics

Journal Article Journal of Nonlinear Science · December 1, 2018 Full text Cite

Global Regularity for the Fractional Euler Alignment System

Journal Article Archive for Rational Mechanics and Analysis · April 1, 2018 We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is ... Full text Open Access Cite

Finite time blow up in the hyperbolic Boussinesq system

Journal Article Advances in Mathematics · February 5, 2018 In recent work of Luo and Hou [10], a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a ... Full text Cite

Global regularity for 1D eulerian dynamics with singular interaction forces

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2018 The Euler-Poisson-alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set of agents interacting through mutual attraction/repulsion as well as alignment forces. We consider one-dimensional EPA system with ... Full text Open Access Cite

On the Finite-Time Blowup of a One-Dimensional Model for the Three-Dimensional Axisymmetric Euler Equations

Journal Article Communications on Pure and Applied Mathematics · November 1, 2017 In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at ... Full text Cite

Local Regularity for the Modified SQG Patch Equation

Journal Article Communications on Pure and Applied Mathematics · July 1, 2017 We study the patch dynamics on the whole plane and on the half-plane for a family of active scalars called modified surface quasi-geostrophic (SQG) equations. These involve a parameter α that appears in the power of the kernel in their Biot-Savart laws and ... Full text Cite

Suppression of Chemotactic Explosion by Mixing

Journal Article Archive for Rational Mechanics and Analysis · November 1, 2016 Chemotaxis plays a crucial role in a variety of processes in biology and ecology. In many instances, processes involving chemical attraction take place in fluids. One of the most studied PDE models of chemotaxis is given by the Keller–Segel equation, which ... Full text Cite

A distinguished mathematical physicist Boris S. Pavlov

Journal Article Nanosystems: Physics, Chemistry, Mathematics · October 25, 2016 Full text Cite

Finite time singularity for the modified SQG patch equation

Journal Article Annals of Mathematics · January 1, 2016 It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law that is one derivative less regular than in the Euler case, and the ques ... Full text Cite

Blow up for the 2D Euler equation on some bounded domains

Journal Article Journal of Differential Equations · October 5, 2015 We find a smooth solution of the 2D Euler equation on a bounded domain which exists and is unique in a natural class locally in time, but blows up in finite time in the sense of its vorticity losing continuity. The domain's boundary is smooth except at two ... Full text Cite

Finite Time Blow Up for a 1D Model of 2D Boussinesq System

Journal Article Communications in Mathematical Physics · March 1, 2015 The 2D conservative Boussinesq system describes inviscid, incompressible, buoyant fluid flow in a gravity field. The possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics. We consider a 1D mode ... Full text Cite

On the Finite-Time Blowup of a 1D Model for the 3D Axisymmetric Euler Equations

Journal Article · July 17, 2014 In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a ... Link to item Cite

Small scale creation for solutions of the incompressible two-dimensional Euler equation

Journal Article Annals of Mathematics · January 1, 2014 We construct an initial data for the two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp - the double exponential growth is the fastest po ... Full text Cite

Global well-posedness of slightly supercritical active scalar equations

Journal Article Analysis and PDE · January 1, 2014 The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasigeostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly l ... Full text Cite

Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows

Journal Article Nonlinearity · January 1, 2014 Consider a diffusion-free passive scalar θ being mixed by an incompressible flow u on the torus d. Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field. Our main result shows that the mix-norm ... Full text Cite

A simple energy pump for the surface quasi-geostrophic equation

Journal Article Nonlinear Partial Differential Equations: The Abel Symposium 2010 · December 1, 2012 We consider the question of growth of high order Sobolev norms of solutions of the conservative surface quasi-geostrophic equation. We show that if s > 0 is large then for every given A there exists initial data with a norm that is small in Hs such that th ... Full text Cite

Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case

Journal Article Journal of Mathematical Physics · November 27, 2012 Many phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. In continuation of our work [A. Kiselev and L. Ryzhik, ... Full text Cite

Global well-posedness for a slightly supercritical surface quasi-geostrophic equation

Journal Article Nonlinearity · May 1, 2012 We use a non-local maximum principle to prove the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation. By this we mean that the velocity field u is obtained from the active scalar by a Fourier multiplier wit ... Full text Cite

Biomixing by Chemotaxis and Enhancement of Biological Reactions

Journal Article Communications in Partial Differential Equations · February 1, 2012 Many phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. However, to the best of our knowledge, the question of ... Full text Cite

Nonlocal maximum principles for active scalars

Journal Article Advances in Mathematics · August 1, 2011 Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler, Burgers, and 2D surface quasi-geostrophic equations. Many questions about regularity and properties of solutions of these equations ... Full text Cite

Variation on a theme of caffarelli and vasseur

Journal Article Journal of Mathematical Sciences · March 1, 2010 Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur have shown that a certain class of weak solutions to the drift diffusion equation with initial data in L2 gain Ḧolder continuity, provided that the BMO norm of the drift velocity is bounded u ... Full text Cite

Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation

Journal Article Nonlinearity · February 1, 2010 We consider the surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove the global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involvin ... Full text Cite

Regularity and blow up for active scalars

Journal Article Mathematical Modelling of Natural Phenomena · January 1, 2010 We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discus ... Full text Cite

The explosion problem in a flow

Journal Article Journal d'Analyse Mathematique · January 1, 2010 We consider the explosion problem in an incompressible flow introduced in [5]. We use a novel Lp - L∞ estimate for elliptic advection-diffusion problems to show that the explosion threshold obeys a positive lower bound which is uniform in the advecting flo ... Full text Cite

Absolutely continuous spectrum of discrete Schrödinger operators with slowly oscillating potentials

Journal Article Mathematische Nachrichten · April 1, 2009 We show that when a potential bn of a discrete Schrödinger operator, defined on l2(Z{double-struck}+), slowly oscillates satisfying the conditions bn ∈ l∞ and ∂bn = bn+1 - bn ∈ lp, p < 2, then all solutions of the equation Ju = Eu are bounded near infinity ... Full text Cite

Some recent results on the critical surface quasi-geostrophic equation: A review

Conference HYPERBOLIC PROBLEMS: THEORY, NUMERICS AND APPLICATIONS, PART 1 · January 1, 2009 Link to item Cite

Relaxation enhancement by time-periodic flows

Journal Article Indiana University Mathematics Journal · December 16, 2008 We study enhancement of diffusive mixing by fast incompressible time-periodic flows. The class of relaxation-enhancing flows that are especially efficient in speeding up mixing has been introduced in [2]. The relaxation-enhancing property of a flow has bee ... Full text Cite

Blow up and regularity for fractal burgers equation

Journal Article Dynamics of Partial Differential Equations · January 1, 2008 The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian α < 1/2, and globa ... Full text Cite

Diusion and mixing in fluid flow

Journal Article Annals of Mathematics · January 1, 2008 We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrari ... Full text Cite

Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Journal Article Inventiones Mathematicae · March 1, 2007 We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity. © 2006 Springer-Verlag. ... Full text Cite

Spectral properties of schrodinger operators with decaying potentials

Journal Article SPECTRAL THEORY AND MATHEMATICAL PHYSICS: A FESTSCHRIFT IN HONOR OF BARRY SIMON'S 60TH BIRTHDAY · January 1, 2007 Link to item Cite

Quenching of combustion by shear flows

Journal Article Duke Mathematical Journal · March 15, 2006 We consider a model describing premixed combustion in the presence of fluid flow: a reaction-diffusion equation with passive advection and ignition-type nonlinearity. What kinds of velocity profiles are capable of quenching (suppressing) any given flame, p ... Full text Cite

Quenching of reaction by cellular flows

Journal Article Geometric and Functional Analysis · February 1, 2006 We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenarios for the initial data that is compactly supported and the size of the support is large enough. If the flow cells are large ... Full text Cite

Enhancement of combustion by drift in a coupled reaction-diffusion model

Journal Article Communications in Mathematical Sciences · 2006 Full text Cite

Quenching and propagation in KPP reaction-diffusion equations with a heat loss

Journal Article Archive for Rational Mechanics and Analysis · October 1, 2005 We consider a reaction-diffusion system of KPP type in a shear flow and with a non-zero heat-loss parameter. We establish criteria for the flame blow-off and propagation, and identify the propagation speed in terms of the exponential decay of the initial d ... Full text Cite

On discrete models of the Euler equation

Journal Article International Mathematics Research Notices · August 28, 2005 Full text Cite

Imbedded singular continuous spectrum for Schrödinger operators

Journal Article Journal of the American Mathematical Society · July 1, 2005 Full text Cite

Transfer matrices and transport for Schrödinger operators

Journal Article Annales de l’institut Fourier · 2004 Full text Cite

Fronts in Reactive Convection: Bounds, Stability, and Instability

Journal Article Communications on Pure and Applied Mathematics · December 1, 2003 This paper examines a simplified active combustion model in which the reaction influences the flow. We consider front propagation in a reactive Boussinesq system in an infinite vertical strip. Nonlinear stability of planar fronts is established for narrow ... Full text Cite

Flame enhancement and quenching in fluid flows

Journal Article Combustion Theory and Modelling · September 1, 2003 We perform direct numerical simulations of an advected scalar field which diffuses and reacts according to a nonlinear reaction law. The objective is to study how the bulk burning rate of the reaction is affected by an imposed flow. In particular, we are i ... Full text Cite

Stability of singular spectral types under decaying pertubations

Journal Article Journal of Functional Analysis · February 20, 2003 We look at invariance of a.e. boundary condition spectral behavior under perturbations, W , of half-line, continuum or discrete Schrödinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported W's to suitable short-range W. We ... Full text Cite

Absolutely continuous spectrum of Stark operators

Journal Article Arkiv for Matematik · January 1, 2003 We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum ... Full text Cite

Dynamical upper bounds on wavepacket spreading

Journal Article American Journal of Mathematics · January 1, 2003 We derive a general upper bound on the spreading rate of wavepackets in the framework of Sohrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where th ... Full text Cite

Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-type operators

Journal Article Mathematische Nachrichten · August 23, 2002 Let g(z,x) denote the diagonal Green's matrix of a self-adjoint m × m matrix-valued Schrödinger operator H = -d2/dx2Im + Q in L2(ℝ)m, m ∈ ℕ. One of the principal results proven in this paper states that for a fixed x0 ∈ ℝ and z ∈ ℂ+, g(z,x0) and g′(z,x0) u ... Full text Cite

Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials

Journal Article Geometric and Functional Analysis · January 1, 2002 We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in LP(ℝ). p < 2. If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completen ... Full text Cite

Quenching of flames by fluid advection

Journal Article Communications on Pure and Applied Mathematics · November 1, 2001 We consider a simple scalar reaction-advection-diffusion equation with ignition-type nonlinearity and discuss the following question: What kinds of velocity profiles are capable of quenching any given flame, provided the velocity's amplitude is adequately ... Full text Cite

An upper bound for the bulk burning rate for systems

Journal Article Nonlinearity · September 1, 2001 We consider a system of reaction-diffusion equations with passive advection term and Lewis number Le not equal to one. Such systems are used to describe chemical reactions in a flow in a situation where temperature and material diffusivities are not equal. ... Full text Cite

WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials

Journal Article Journal of Functional Analysis · February 1, 2001 We prove the WKB asymptotic behavior of solutions of the differential equation -d2u/dx2+V(x)u=Eu for a.e. E>A where V=V1+V2, V1∈Lp(R), and V2 is bounded from above with A=limsupx→∞V(x), while V′2(x)∈Lp(R), 1≤p<2. These results imply that Schrödinger operat ... Full text Cite

Maximal functions associated to filtrations

Journal Article Journal of Functional Analysis · February 1, 2001 Let T be a bounded linear, or sublinear, operator from Lp(Y) to Lq(X). A maximal operator T*f(x)=supjT(f·χYj)(x) is associated to any sequence of subsets Yj of Y. Under the hypotheses that q>p and the sets Yj are nested, we prove that T* is also bounded. C ... Full text Cite

Enhancement of the traveling front speeds in reaction-diffusion equations with advection

Journal Article Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire · January 1, 2001 We establish rigorous lower bounds on the speed of traveling fronts and on the bulk burning rate in reaction-diffusion equation with passive advection. The non-linearity is assumed to be of either KPP or ignition type. We consider two main classes of flows ... Full text Cite

WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials

Journal Article Communications in Mathematical Physics · January 1, 2001 Consider a Schrödinger operator on L2 of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a derivative of some order in L1 + Lp for some exponent p < 2, then an ... Full text Cite

Absolutely continuous spectrum of perturbed stark operators

Journal Article Transactions of the American Mathematical Society · January 1, 2000 We prove new results on the stability of the absolutely continuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stabl ... Full text Cite

Solutions, spectrum, and dynamics for schrödinger operators on infinite domains

Journal Article Duke Mathematical Journal · January 1, 2000 Full text Cite

Bulk burning rate in passive-reactive diffusion

Journal Article Archive for Rational Mechanics and Analysis · January 1, 2000 We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction. We introduce a quantity, the bulk burning rate, that makes both mathematical and physic ... Full text Cite

Effective perturbation methods for one-dimensional Schrödinger operators

Journal Article Journal of Differential Equations · January 20, 1999 Full text Cite

An interpolation theorem related to the A.E. convergence of integral operators

Journal Article Proceedings of the American Mathematical Society · January 1, 1999 We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on Lorentz spaces follows from the appropriate norm ... Full text Cite

Some examples in one-dimensional "geometric" scattering on manifolds

Journal Article Journal of Mathematical Analysis and Applications · August 1, 1997 We consider "geometric" scattering for a Laplace-Beltrami operator on a compact Riemannian manifold inserted between "wires," that is, two half-lines. We discuss applicability and correctness of this model. With an example, we show that such a scattering p ... Full text Cite

Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials

Journal Article Communications in Mathematical Physics · January 1, 1996 We prove that for any one-dimensional Schrödinger operator with potential V(x) satisfying decay condition |V(x)| ≦ Cx-3/4-ε, the absolutely continuous spectrum fills the whole positive semi-axis. The description of the set in ℝ+ on which the singular part ... Full text Cite

Rank one perturbations with infinitesimal coupling

Journal Article Journal of Functional Analysis · January 1, 1995 We consider a positive self-adjoint operator A and formal rank one perturbations B = A + α(φ, ·)φ, where φ ∈ H-2(A) but φ ∉ H-1 (A), with Hs(A) the usual scale of spaces. We show that B can be defined for such φ and what are essentially negative infinitesi ... Full text Cite

Indefinite metric and scattering by a domain with a small hole

Journal Article Mathematical Notes · January 1, 1995 For the problem of plane waves scattered by a domain with a small hole, we suggest a model based on the theory of self-adjoint extensions of symmetric operators in a space with indefinite metric. For two-dimensional problems of scattering on a line with a ... Full text Cite

Eigenfrequencies and eigenfunctions of the Laplacian for Neumann boundary conditions in a system of two coupled cavities

Journal Article Theoretical and Mathematical Physics · September 1, 1994 A model Laplacian with Neumann boundary conditions (Neumann problem) in a system of two cavities joined by a thin channel is investigated. An expression is obtained for the resolvent and also the first terms in the asymptotic expansions of the eigenvalues ... Full text Cite

Essential spectrum of the Laplacian for the Neumann problem in a model region of complicated structure

Journal Article Theoretical and Mathematical Physics · April 1, 1994 A class of regions in which the Laplacian for the Neumann problem has an essential spectrum is considered. The connection between the geometrical characteristics of the region and spectral properties of the Laplacian for the Neumann problem is studied in s ... Full text Cite

A simple model for asset price bubble formation and collapse

Journal Article We consider a simple stochastic differential equation for modeling bubbles in social context. A prime example is bubbles in asset pricing, but similar mechanisms may control a range of social phenomena driven by psychological factors (for example, po ... Cite

Analysis of a Singular Boussinesq Model

Journal Article Research in the Mathematical Sciences Recently, a new singularity formation scenario for the 3D axi-symmetric Euler equation and the 2D inviscid Boussinesq system has been proposed by Hu and Luo based on extensive numerical simulations [15, 16]. As the firrst step to understand the scenario, m ... Link to item Cite

Stirring Speeds Up Chemical Reaction

Internet Publication We consider absorbing chemical reactions in a fluid current modeled by the coupled advection-reaction-diffusion equations. In these systems, the interplay between chemical diffusion and fluid transportation causes the enhanced dissipation phenomenon. We sh ... Link to item Cite