Alexander Kiselev

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

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.

Bulk Burning Rate in¶Passive–Reactive Diffusion

AbstractWe 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 physical sense in general situations and extends the often ill-defined notion of front speed. We establish rigorous lower bounds for the bulk burning rate that are linear in the amplitude of the advecting velocity for a large class of flows. These “percolating” flows are characterized by the presence of tubes of streamlines connecting distant regions of burned and unburned material and generalize shear flows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are finer than the width of the laminar burning region. We give also examples of very different kind of flows, cellular flows with closed streamlines, and rigorously prove that these can produce only sub-linea enhancement of the bulk burning rate.

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

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. Percolating flows, which are characterized by the presence of long tubes of streamlines mixing hot and cold material, lead to strong speed-up of burning which is linear in the amplitude of the flow, U. On the other hand the cellular flows, which have closed streamlines, are shown to produce weaker increase in reaction. For such flows we get a lower bound which grows as U1/5 for a large amplitude of the flow.

DYNAMICAL UPPER BOUNDS ON WAVEPACKET SPREADING

We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger 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 the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time-independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian—the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.

Flame enhancement and quenching in fluid flows

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 interested in comparing the numerical results with recently predicted analytical upper and lower bounds. We focus on the reaction enhancement and quenching phenomena for two classes of imposed model flows with different geometries: periodic shear flow and cellular flow. We are primarily interested in the fast advection regime. We find that the bulk burning rate v in a shear flow satisfies ν ∼ aU + b where U is the typical flow velocity and a is a constant depending on the relationship between the oscillation length scale of the flow and laminar front thickness. For cellular flow, we obtain ν ∼ U 1/4. We also study the flame extinction (quenching) for an ignition-type reaction law and compactly supported initial data for the scalar field. We find that in a shear flow the flame of size W can be typically quenched by a flow with amplitude U ∼ αW. The constant α depends on the geometry of the flow and tends to infinity if the flow profile has a plateau larger than a critical size. In a cellular flow, we find that the advection strength required for quenching is U ∼ W 4 if the cell size is smaller than a critical value.

Biomixing by Chemotaxis and Enhancement of Biological Reactions

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 reaction enhancement by chemotaxis has not yet received extensive treatment either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and diffusion. On the other hand, when chemotaxis is present, the fertilization rate can be arbitrarily close to being complete provided that the chemotactic attraction is sufficiently strong. Moreover, an interesting feature of the estimates on fertilization rate and timescales in the chemotactic case is that they do not depend on the amplitude of the reaction term.

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

Abstract: 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 essential support of the the absolutely continuous spectrum equals ℝ+. Almost every generalized eigenfunction is bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to Lp with respect to a weight |x|γ with γ >0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one.

Global well-posedness of slightly supercritical active scalar equations

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 logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.

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

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 model of the 2D Boussinesq system motivated by a particular finite time blow up scenario. We prove that finite time blow up is possible for the solutions to the model system.

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

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 involving conservation of a certain family of moduli of continuity.