Convergence to a Single Wave in the Fisher-KPP Equation

The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation,with an initial condition that is a compact perturbation of a step function.A well-known result of Bramson states that,in the reference frame moving as 2t-(3/2) log t+x∞,the solution of the equation converges as t-→ +o∞ to a translate of the traveling wave corresponding to the minimal speed c* =2.The constant x∞ depends on the initial condition u(0,x).The proof is elaborate,and based on probabilistic arguments.The purpose of this paper is to provide a simple proof based on PDE arguments.     

Boundary Layers and KPP Fronts in a Cellular Flow

We study an eigenvalue problem associated with a reaction-diffusion-advection equation of the KPP type in a cellular flow. We obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude A ≪ 1. It follows that the minimal pulsating traveling front speed c _*(A) satisfies the upper and lower bounds C ₁ A ^1/4≦ c _*(A)≦ C ₂ A ^1/4. Physically, the speed enhancement is related to the boundary layer structure of the associated eigenfunction – accordingly, we establish an “averaging along the streamlines” principle for the unique positive eigenfunction.     

Exponential decay for the fragmentation or cell-division equation

We consider a classical integro-differential equation that arises in various applications as a model for cell-division or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady distribution (first positive eigenvector) under general assumptions in the variable coefficients case. We also prove the exponential convergence, for large times, of solutions toward such a steady state.     

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.     

Precursors for waves in random media

We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty–Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully three-dimensional situation.     

From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A  , in a two-dimensional domain with L²$L^2$ cells. For fixed A  , and L→∞$L\to \infty$, the problem homogenizes, and has been well studied. Also well studied is the limit when L   is fixed, and A→∞$A\to \infty$. In this case the solution equilibrates along stream lines.     

Relaxation in Reactive Flows

We consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. We show that the resulting flows possess relaxation-enhancing properties in the sense of [CoKRZ]. In particular, we show that solutions of the nonlinear problems become small when gravity is sufficiently strong due to the improved interaction with the cold boundary. As an application, we deduce that the explosion threshold for power-like nonlinearities tends to infinity in the large Rayleigh number limit. We also discuss the behavior of the principal eigenvalues of the corresponding advection-diffusion problem and the quenching phenomenon for reaction-diffusion equations. Received: March 2007, Revision: May 2007, Accepted: May 2007     

Limit of Fluctuations of Solutions of Wigner Equation

We consider fluctuations of the solution W _ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W _ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, W _ε (0, x, k) = δ(x)f(k) and \({f\in {\mathcal{S}}(\mathbb{R}^d)}\), then the laws of the rescaled fluctuation \({Z_\varepsilon(t):=\varepsilon^{-1/2}[W_\varepsilon(t,x,k)-\bar{W}(t,x,k)]}\) converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered.