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Alexander Kiselev

Publications and source records attributed to Alexander Kiselev.

At least 37 records · Page 2Linked to original sources
arXiv2021

Chemotactic Reaction Enhancement in One Dimension

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 directs bacteria's movement to find food (e.g., glucose) by swimming toward the highest concentration of food molecules. In multicellular organisms, chemotaxis is critical to early development (e.g., movement of sperm towards the egg during fertilization). Chemotaxis also helps mobilize phagocytic and immune cells at sites of infection, tissue injury, and thus facilitates immune reactions. In this paper, we study a PDE system that describes such biological processes in one dimension, which may correspond to a thin channel, the setting relevant in many applications: for example, spermatozoa progression to the ovum inside a Fallopian tube or immune response in a blood vessel.

math.AP↗
arXiv2021

Small scale formations in the incompressible porous media equation

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 stratified steady states of IPM.

math.AP↗
arXiv2020

The Flow of Polynomial Roots Under Differentiation

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 model dynamics of roots of polynomials under differentiation. Interestingly, the same equation has also been recently obtained formally by Dimitri Shlyakhtenko and Terence Tao as the evolution equation for free fractional convolution of a measure - an object in free probability that is also related to minor processes for random matrices. The partial differential equation bears striking resemblance to hydrodynamic models used to describe the collective behavior of agents (such as birds, fish or robots) in mathematical biology. We consider periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data. In the second part of the paper we connect rigorously solutions of the Steinerberger's PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian $(-Δ)^{1/2}$.

math.AP↗
arXiv2020

Boundary layer models of the Hou-Luo scenario

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 \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the fluid. An important role is played by a small boundary layer where intense growth is observed. Several simplified models of the scenario have been considered, all leading to finite time blow up \cite{CKY15,CHKLVY17,HORY,KT1,HL15,KY1}. In this paper, we propose two models that are designed specifically to gain insight in the evolution of fluid near the hyperbolic stagnation point of the flow located at the boundary. One model focuses on analysis of the depletion of nonlinearity effect present in the problem. Solutions to this model are shown to be globally regular. The second model can be seen as an attempt to capture the velocity field near the boundary to the next order of accuracy compared with the one-dimensional models such as \cite{CKY15,CHKLVY17}. Solutions to this model blow up in finite time.

math.AP↗
arXiv2020

Chemotaxis and Reactions in Biology

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 to fight invading bacteria. Another example is reproduction, where eggs release pheromones that attract sperm. A macro scale example is flower scent appealing to pollinators. In this paper we consider a system of PDE designed to model such processes. Our interest is to quantify the effect of chemotaxis on reaction rates compared to pure reaction-diffusion. We limit consideration to surface chemotaxis, which is well motivated from the point of view of many applications. Our results provide the first insight into situations where chemotaxis can be crucial for reaction success, and where its effect is likely to be limited. The proofs are based on new analytical tools; a significant part of the paper is dedicated to building up the linear machinery that can be useful in more general settings. In particular we establish precise estimates on the rates of convergence to ground state for a class of Fokker-Planck operators with potentials that grow at a logarithmic rate at infinity. These estimates are made possible by a new sharp weak weighted Poincaré inequality improving in particular a result of Bobkov and Ledoux.

math.AP↗
arXiv2019

Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. I

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media.

math.AP↗
arXiv2018

Stellar Imaging Coronagraph and Exoplanet Coronal Spectrometer: Two Additional Instruments for Exoplanet Exploration Onboard The WSO-UV 1.7 Meter Orbital Telescope

The World Space Observatory for Ultraviolet (WSO-UV) is an orbital optical telescope with a 1.7 m-diameter primary mirror currently under development. The WSO-UV is aimed to operate in the 115-310 nm UV spectral range. Its two major science instruments are UV spectrographs and a UV imaging field camera with filter wheels. The WSO-UV project is currently in the implementation phase, with a tentative launch date in 2023. Recently, two additional instruments devoted to exoplanets have been proposed for WSO-UV, which are the focus of this paper. UVSPEX, a UV-Spectrograph for Exoplanets, aims to determine atomic hydrogen and oxygen abundance in the exospheres of terrestrial exoplanets. The spectral range is 115-130 nm which enables simultaneous measurement of hydrogen and oxygen emission intensities during an exoplanet transit. Study of exosphere transit photometric curves can help differentiate among different types of rocky planets. The exospheric temperature of an Earth-like planet is much higher than that of a Venus-like planet, because of the low mixing ratio of the dominant coolant (CO2) in the upper atmosphere of the former, which causes a large difference in transit depth at the oxygen emission line. Thus, whether the terrestrial exoplanet is Earth-like, Venus-like, or other can be determined. SCEDI, a Stellar Coronagraph for Exoplanet Direct Imaging is aimed to directly detect the starlight reflected from exoplanets orbiting their parent stars or from the stellar vicinity including circumstellar discs, dust, and clumps. SCEDI will create an achromatic (optimized to 420-700 nm wavelength range), high-contrast stellocentric coronagraphic image of a circumstellar vicinity. The two instruments: UVSPEX and SCEDI, share common power and control modules. The present communication outlines the science goals of both proposed instruments and explains some of their engineering features.

astro-ph.IM↗
arXiv2018

Analysis of a Singular Boussinesq Model

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, models with simplified sign-definite Biot-Savart law and forcing have recently been studied in [7, 6, 8, 12, 14, 18]. In this paper, we aim to bring back one of the complications encountered in the original equation - the sign changing kernel in the Biot-Savart law. This makes analysis harder, as there are two competing terms in the fluid velocity integral whose balance determines the regularity properties of the solution. The equation we study here is based on the CKY model introduced in [6]. We prove that finite time blow up persists in a certain range of parameters.

math.AP↗
arXiv2018

Global Regularity and Fast Small Scale Formation for Euler Patch Equation in a Smooth Domain

It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this paper, we study Euler vortex patches in a general smooth bounded domain. We prove global in time regularity by providing an upper bound on the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.

math.AP↗
arXiv2017

Global regularity for 1D Eulerian dynamics with singular interaction forces

The Euler-Poisson-Alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in the paper by Do, Kiselev, Ryzhik and Tan. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.

math.AP↗
arXiv2017

Global regularity for the fractional Euler alignment system

We study a pressureless Euler system with a nonlinear 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 natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian $(-\partial_{xx})^{α/2}$, $α\in (0, 1)$. The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all $α\in (0, 1)$. To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.

math.AP↗
arXiv2016

Finite time blow up in the hyperbolic Boussinesq system

In recent work of Luo and Hou, 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 two dimensional model that we call "hyperbolic Boussinesq system". This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot-Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.

math.AP↗
arXiv2016

Stability of Blow Up for a 1D model of Axisymmetric 3D Euler Equation

The question of the global regularity vs finite time blow up 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 hyperbolic boundary blow up scenario for the 3D Euler equation proposed by Hou and Luo based on extensive numerical simulations. These models generalize the 1D Hou-Luo model suggested in Hou & Luo's paper, for which finite time blow up has been established in the paper by Kyudong Choi, Thomas Y. Hou, Alexander Kiselev, Guo Luo, Vladimir Sverak, Yao Yao. The main new aspects of this work are twofold. First, we establish finite time blow up for a model that is a closer approximation of the three dimensional case than the original Hou-Luo model, in the sense that it contains relevant lower order terms in the Biot-Savart law that have been discarded in the paper by Hou & Luo's and the paper by Choi et al.. Secondly, we show that the blow up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou-Luo model. Such blow up stability result may be useful in further work on understanding the 3D hyperbolic blow up scenario.

math.AP↗
arXiv2016

Suppression of chemotactic explosion by mixing

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 Keller-Segel equation, which describes a population density of bacteria or mold which attract chemically to substance they secrete. Solution of Keller-Segel equation can exhibit dramatic collapsing behavior, where the density concentrates positive mass in a measure zero region. A natural question is whether presence of fluid flow can affect singularity formation by mixing the density thus making concentration harder to achieve. In this paper, we consider parabolic-elliptic Keller-Segel equation in two and three dimensions with additional advection term modeling ambient fluid flow. We prove that for any initial data, there exist incompressible fluid flows such that the solution to the equation stays globally regular. On the other hand, it is well known that when the fluid flow is absent, there exist initial data leading to finite time blow up. Thus presence of fluid can prevent the singularity formation. We discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers. The first class are relaxation enhancing (RE) flow. These flows are stationary. The second class of flows are Yao-Zlatos optimal mixing flows, which are time dependent. The proof is based on the nonlinear version of the relaxation enhancement construction of the paper by Peter Constantin, Alexander Kiselev, Lenya Ryzhik, Andrej Zlatos, and on some variations of global regularity estimate for Keller-Segel model.

math.AP↗