Journal ArticleCommunications on Pure and Applied Mathematics · April 1, 2025
We consider the patch problem for the (Formula presented.) -(surface quasi-geostrophic) SQG system with the values (Formula presented.) and (Formula presented.) being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches ...
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Journal ArticleJournal of Functional Analysis · October 1, 2024
In this paper, we show that the Keller-Segel equation equipped with zero Dirichlet Boundary condition and actively coupled to a Stokes-Boussinesq flow is globally well-posed provided that the coupling is sufficiently large. We will in fact show that the dy ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2024
Chemotaxis, which involves the directed movement of cells in response to a chemical gradient, plays a crucial role in a broad variety of biological processes. Examples include bacterial motion, the development of single-cell or multicellular organisms, and ...
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Journal ArticleJournal 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 ...
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Journal ArticleArchive 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. ...
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Journal ArticleJournal 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 ...
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Journal ArticleArchive 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleAnnals 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 ...
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Journal ArticleBulletin 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, ...
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Journal ArticleInvolve · 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 ...
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Journal ArticleSIAM 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 ...
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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 ...
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Journal ArticleJournal 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 ...
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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 AccessLink to itemCite
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 ...
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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 ...
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Journal ArticleDuke 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. ...
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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 ...
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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 ...
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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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleArchive 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 ...
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Journal ArticleAdvances 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 ...
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Journal ArticleSIAM 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleArchive 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 ...
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Journal ArticleAnnals 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleCommunications 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 ...
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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 ...
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Journal ArticleAnnals 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 ...
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Journal ArticleAnalysis 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 ...
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Journal ArticleNonlinearity · 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 ...
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Journal ArticleNonlinear 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 ...
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Journal ArticleJournal 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, ...
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Journal ArticleNonlinearity · 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleAdvances 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleNonlinearity · 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 ...
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Journal ArticleMathematical 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleMathematische 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 ...
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Journal ArticleIndiana 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 ...
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Journal ArticleDynamics 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 ...
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Journal ArticleAnnals 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 ...
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Journal ArticleDuke 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 ...
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Journal ArticleGeometric 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 ...
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Journal ArticleArchive 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleCombustion 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleArkiv 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 ...
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Journal ArticleAmerican 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 ...
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Journal ArticleMathematische 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 ...
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Journal ArticleGeometric 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleNonlinearity · 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. ...
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Journal ArticleJournal 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleAnnales 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleTransactions 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 ...
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Journal ArticleArchive 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 ...
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Journal ArticleProceedings 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleMathematical 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 ...
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Journal ArticleTheoretical 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 ...
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Journal ArticleTheoretical 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 ...
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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 ...
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Journal ArticleResearch 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 ...
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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 ...
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