Jonathan Bennett University of Birmingham Mathematics We describe certain elementary, yet striking, monotonicity phenomena pertaining to well-known euclidean Strichartz inequalities as the data flows under suitable diffusion equations. We contrast the Schr\"odinger-like and wave settings with that of the closely-related kinetic transport equation. Finally we discuss certain conjectured multilinear Strichartz inequalities and their relation with the classical restriction conjecture for the Fourier transform.