Formation of small scales in nonlinear PDEs

Rigorous Justification of Taylor Dispersion via Center Manifolds and Hypocoercivity

Margaret Beck

Boston University


Taylor diffusion (or dispersion) refers to a phenomenon discovered experimentally by Taylor in the 1950s where a solute dropped into a pipe with a background shear flow experiences diffusion at a rate proportional to $1/\nu$, which is much faster than what would be produced by the static fluid if its viscosity is small, $0 < \nu \ll 1$. This phenomenon is analyzed rigorously using the linear PDE governing the evolution of the solute. It is shown that the solution can be split into two pieces: an approximate solution and a remainder term. The approximate solution is governed by an infinite- dimensional system of ODEs that possesses a finite-dimensional center manifold, on which the dynamics correspond to diffusion at a rate proportional to $1/\nu$. The remainder term is shown to decay at a rate that is much faster than the leading order behavior of the approximate solution. This is proven using a spectral decomposition in Fourier space and a hypocoercive estimate to control the intermediate Fourier modes.