Existence of physically meaningful velocity fields that induce enhanced dissipation in the whole space

Determine whether there exists an incompressible velocity field on R^n that belongs to suitable L^p classes in space and time and induces enhanced dissipation for the advection–diffusion equation ∂_t θ + u·∇θ = κΔθ posed on R^n, in the sense that the flow accelerates decay relative to pure diffusion despite the absence of boundary-induced dissipation.

Background

In the whole space, classical mechanisms that aid dissipation via boundary effects (e.g., Poincaré inequality) are absent. Existing enhanced dissipation results in R^n often rely on special unbounded or growing velocity fields (such as circularly symmetric shears with radial growth), or yield decay rates that depend sensitively on the initial data via Fourier-splitting methods.

Green’s function bounds (Aronson-type estimates) provide general upper and lower heat kernel estimates under mild assumptions on the velocity field, but these do not directly produce lower bounds on solutions nor demonstrate enhanced dissipation. This raises the question of whether more physical flows—e.g., divergence-free vector fields controlled in L^p in space and time—can yield enhanced dissipation in the whole space.