Ergodic theorems for reaction-diffusion processes

New sufficient conditions are given for the ergodicity of reaction-diffusion processes which improve both Neuhauser's recent result and the present author's previous result. In the main criterion; contrary to the previous ones, the pure birth rate of the reaction plays a critical role. To do this, a new but natural coupling is introduced. It is proved that this coupling is the best one in some sense. One of the main results says that the reaction-diffusion processes are ergodic for all large enough pure birth rates.     

Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

     

Weak solutions and supersolutions in $ L^1 $ for reaction-diffusion systems

We prove here that limits of nonnegative solutions to reaction-diffusion systems whose nonlinearities are bounded in always converge to supersolutions of the system. The motivation comes from the question of global existence in time of solutions for the wide class of systems preserving positivity and for which the total mass of the solution is uniformly bounded. We prove that, for a large subclass of these systems, weak solutions exist globally. RID="h1" ID="h1"A Philippe, mon ma?tre et ami     

Convergence of a splitting method of high order for reaction-diffusion systems

In this article, we prove the convergence of a splitting scheme of high order for a reaction-diffusion system of the form where is an matrix whose spectrum is included in 0 \}$">. This scheme is obtained by applying a Richardson extrapolation to a Strang formula.

     

Polymers on disordered trees,spin glasses,and traveling waves

We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions that move at all possible speeds above a certain minimal speed. The speed of the wavefront is the free energy of the polymer problem and the minimal speed corresponds to a phase transition to a glassy phase similar to the spin-glass phase. Several properties of the polymer problem can be extracted from the correspondence with the traveling wave: probability distribution of the free energy, overlaps, etc.     

Existence of standing pulse solutions for an excitable activator-inhibitory system

Invariant manifold techniques are used to establish the existence of a standing wave solution for a system of reaction-diffusion equations modeled after excitable media. As a by-product of the method of proof, we obtain a local uniqueness result.     

非线性反应扩散方程的数值解

本文采用Petrov-Galerkin有限元法构造了非线性反应扩散方程的数值格式,既适用于全场性的初值,也适用于局部性的初值,利用全场性初值求得的进波数值解与精确解高度吻合,证明本方法与其它数值方法比较,有更高的精度和稳定性,利用各种局部性初值所算出的数值解表明,任何局部的扰动都会得到充分的发展,且当时间充分长后,演变为向左、右传播的行波,其波前的形状及传播速度完全由系统本身所决定,而与初值的类型无关.     

Stability of the numerical solution of the periodic reaction- diffusion systems

In this paper we study the behavior of the numerical solution of nonlinear reaction-diffusion systems, with periodic in time nonlinear term, obtained via the known θ-method. In particular we are interested to the existence and asymptotic stability of the numerical periodic solutions in order to simulate the behaviour of the theoretical solution. To this end, by imposing the positivity of the numerical scheme, we can use some results about M-matrices.So, by means of particular over and upper solutions, we study some conditions for the stability and instability of the trivial solution.Finally we show when a positive numerical periodic solution exists and when it is unique and asymptotically stable.     

Weak solutions to quasilinear wave equations of Klein-Gordon or Sine-Gordon type and relaxation to reaction-diffusion equations

This paper regards the existence of weak solutions for a quasilinear wave equation of Klein-Gordon and Sine-Gordon type with the presence of a linear damping term and the relaxation to the reaction-diffusion equation when the momentum relaxation time tends to zero. In the limit process is fundamental the celebrated Div-curl Lemma of Tartar and Murat. Received February 5, 1996     

Stochastic analysis of a chemical reaction with spatial and temporal structures

A stochastic analysis of the spatial and temporal structures in the Prigogine-Lefever-Nicolis model (the Brusselator) is presented. The analysis is carried out through a Langevin equation derived from a multivariate master equation using the Poisson representation method, which is used to calculate the spatial correlation functions and the fluctuation spectra in the Gaussian approximation. The case of an infinite three-dimensional system is considered in detail. The calculations for the spatial correlation functions and the fluctuation spectra for a finite system subject to different kinds of boundary conditions are also given.