Connections between the convective diffusion equation and theforced Burgers equation

[摘要] The convective diffusion equation withdriftb(x)and indefinite weightr(x),∂ϕ∂t=∂∂x[a∂ϕ∂x−b(x)ϕ]+λr(x)ϕ,   (1)is introduced as a model for population dispersal. Strong connections betweenEquation (1) and the forced Burgers equation with positive frequency(m≥0),∂u∂t=∂2u∂x2−u∂u∂x+mu+k(x),   (2)are established through the Hopf-Cole transformation. Equation (2) is a primeprototype of the large class of quasilinear parabolic equations given by∂u∂t=∂2u∂x2+∂(f(v))∂x+g(v)+h(x).  (3)A compact attractor and an inertial manifold for the forced Burgers equation areshown to exist via the Kwak transformation. Consequently, existence of aninertial manifold for the convective diffusion equation is guaranteed. Equation (2)can be interpreted as the velocity field precursed by Equation (1). Therefore, thedynamics exhibited by the population density in Equation (1) show their effects onthe velocity expressed in Equation (2). Numerical results illustrating some aspectsof the previous connections are obtained by using a pseudospectral method.

[学科分类] 应用数学