Connections between the convective diffusion equation and the forced Burgers equation
Summary: The convective diffusion equation with drift \(b(x)\) and indefinite weight \(r(x)\), \[ \frac{\partial\varphi} {\partial t}= \frac{\partial}{\partial x}\left[a\frac{\partial\varphi}{\partial x}-b(x) \varphi\right] +\lambda r(x)\varphi, \tag{1} \] is introduced as a model for population dispersal. Strong connections between equation (1) and the forced Burgers equation with positive frequency \((m\geq 0)\), \[ \frac {\partial u}{\partial t}= \frac{\partial^2u} {\partial x^2}-u \frac {\partial u}{\partial x}+ mu+k(x), \tag{2} \] are established through the Hopf-Cole transformation. Equation (2) is a prime prototype of the large class of quasi-linear parabolic equations given by \[ \frac{\partial v} {\partial t}=\frac {\partial^2v}{\partial x^2}+ \frac{\partial\bigl(f(v) \bigr)} {\partial x}+g(v)+h(x). \tag{3} \] A compact attractor and an inertial manifold for the forced Burgers equation are shown to exist via the Kwak transformation. Consequently, existence of an inertial manifold for the convective diffusion equation is guaranteed.
convective diffusion equation with drift
scientific article; zbMATH DE number 1777334