Lower bounds on mixing norms for the advection diffusion equation in $$\mathbb {R}^d$$

Abstract

AbstractAn algebraic lower bound on the energy decay for solutions of the advection-diffusion equation in$$\mathbb {R}^d$$Rdwith$$d=2,3$$d=2,3is derived using the Fourier-splitting method. Motivated by a conjecture on mixing of passive scalars in fluids, a lower bound on the$$L^2$$L2- norm of the inverse gradient of the solution is obtained via gradient estimates and interpolation.