Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number

This is the third in a series of our study of Rayleigh‐Bénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh‐Bénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection. © 2007 Wiley Periodicals, Inc.     

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.     

Reaction-diffusion front speed enhancement by flows

We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction-diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in three and more dimensions. We also prove conjectures from Audoly, Berestycki and Pomeau (2000) [1], Berestycki (2003) [3], Fannjiang, Kiselev and Ryzhik (2006) [11] for cellular flows, concerning the rate of speed-up of fronts and the minimal flow amplitude necessary to quench solutions with initial data of a fixed (large) size.     

Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number

We study asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the global attractors to the Boussinesq system for Rayleigh‐Bénard convection converge to that of the infinite‐Prandtl‐number model for convection as the Prandtl number approaches infinity. This offers partial justification of the infinite‐Prandtl‐number model for convection as a valid simplified model for convection at large Prandtl number even in the long‐time regime. © 2006 Wiley Periodicals, Inc.     

On the Response of an Elastic Solid to Droplet Impact

Expressions are obtained in closed form for the transient stressin a homogeneous isotropic elastic half-space whose surfaceis subjected to a constant uniform pressure over a circle whoseradius increases in proportion to the square root of time. Thepositions of the wave fronts are derived, together with themagnitudes of the discontinuities of the stress components acrossthem. It is shown that a Rayleigh surface wave appears at themoment when the radius of the area under load is increasingwith the Rayleigh wave-speed. The behaviour of the stress andstrain is described in the region of the stress singularityat its leading edge.     

Pinning and mode-locking of reaction fronts by vortices

We present results of experiments on the behavior of reaction fronts in the presence of vortex-dominated flows. The flow is either a single vortex or a chain of vortices in an annular configuration, and the reaction is the excitable Belousov–Zhabotinsky chemical reaction. If the vortex chain oscillates periodically in the lateral direction, the reaction front often mode-locks to the oscillations, propagating an integer number of wavelengths of the flow (two vortices) in an integer number of drive periods. In the presence of a uniform “wind”, the front often freezes, remaining pinned to the leading vortex and neither propagating forward against the wind nor being blown backward by it. Studies with an individual vortex verify the ability of a moving vortex to pin and drag a reaction front. We use this pinning behavior to explain the mode-locking for the oscillating case.     

Traveling wave solutions for an autocatalytic reaction–diffusion model

In this paper we consider an autocatalytic reaction–diffusion model which has many applications. We extend previous results using qualitative analysis and show the existence of an exponentially decaying traveling wave front for a minimum speed and algebraically decaying wave fronts for large speeds. Further, the wave front profiles are calculated and the minimum speed is accurately determined using different numerical methods.     

Stability of Generalized Transition Fronts

We study the qualitative properties of the generalized transition fronts for the reaction–diffusion equations with the spatially inhomogeneous nonlinearity of the ignition type. We show that transition fronts are unique up to translation in time and are globally exponentially stable for the solutions of the Cauchy problem. The results hold for reaction rates that have arbitrary spatial variations provided that the rate is uniformly positive and bounded from above.     

Speed-up of combustion fronts in shear flows

This paper is concerned with the analysis of speed-up of reaction-diffusion-advection traveling fronts in infinite cylinders with periodic boundary conditions. The advection is a shear flow with a large amplitude and the reaction is nonnegative, with either positive or zero ignition temperature. The unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity, solving an open problem from Berestycki (Nonlinear PDEs in condensed matter and reactive flows, Kluwer, Doordrecht, 2003). The asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem, and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional Hörmander-type condition on the flow.     

Evolutes of fronts in the Minkowski plane

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.     

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction–diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.     

Travelling Wave Solutions in Delayed Reaction Diffusion Systems with Partial Monotonicity

This paper deals with the existence of travelling wave fronts of delayed reaction diffusion systems with partial quasi-monotonicity. We propose a concept of ＂desirable pair of upper-lower solutions＂, through which a subset can be constructed. We then apply the Schauder＇s fixed point theorem to some appropriate operator in this subset to obtain the existence of the travelling wave fronts.     

Global existence and blowup of a localized problem with free boundary

This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large.     

Global existence and blowup of a nonlocal problem in space with free boundary

This paper concerns a double fronts free boundary problem for the reaction–diffusion equation with a nonlocal nonlinear reaction term in space. For such a problem, we mainly study the blowup property and global existence of the solutions. Our results show that if the initial value is sufficiently large, then the blowup occurs, while the global fast solution exists for a sufficiently small initial data, and the intermediate case with a suitably large initial data gives the existence of the global slow solution.     

Two-dimensional curved fronts in a periodic shear flow

This paper is devoted to the study of traveling fronts of reaction-diffusion equations with periodic advection in the whole plane R². We are interested in curved fronts satisfying some “conical” conditions at infinity. We prove that there is a minimal speed c^∗ such that curved fronts with speed c exist if and only if c≥c^∗. Moreover, we show that such curved fronts are decreasing in the direction of propagation, that is, they are increasing in time. We also give some results about the asymptotic behaviors of the speed with respect to the advection, diffusion and reaction coefficients.     

Determination of the equilibrium boundaries of a two-layer system convection stability : PMM vol. 43, no. 6, 1979, pp. 1008–1013

The method which permits the determination of a large number of critical Rayleigh numbers and respective critical motions is applied to the determination of convection stability boundaries of a two-layer system. The method consists essentially of reducing the problem to the algebraic problem of eigenvalues by discretizing the equations by the method of finite elements or finite differences. Only the scheme of the method based on discretization by finite elements is then presented.     

Haar wavelet method for solving Fisher’s equation

In this paper, we develop an accurate and efficient Haar wavelet solution of Fisher’s equation, a prototypical reaction-diffusion equation. The solutions of Fisher’s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt Haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.     

Existence,uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model

This paper is concerned with the existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model. We first establish the existence of traveling wave fronts by using geometric singular perturbation theory. Then the asymptotic behavior and uniqueness of traveling wave fronts are obtained by using the standard asymptotic theory and sliding method. In addition, our method is also suitable to establish the uniqueness and asymptotic behavior of traveling wave fronts for a cooperative system.     

Multiplicity of supercritical fronts for reaction–diffusion equations in cylinders

We study multiplicity of the supercritical traveling front solutions for scalar reaction–diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions.