On some energy inequalities and supnorm estimates for advection–diffusion equations in RnRn

We show how some suitable standard energy inequalities can be used for a simple short derivation of the fundamental supnorm estimate ‖u(⋅,t)‖L∞(Rn)≤K(n,p)‖u(⋅,0)‖Lp(Rn)t−n2p∀t>0 for solutions of advection–diffusion equations of the form ut+divf(u)=Δu (and some generalizations) in the whole space RnRn, where n≥1n≥1 is arbitrary. Our approach can be used for other parabolic equations as well, including degenerate cases like porous medium type equations and pp-Laplacian evolution equations. Some open problems and related results of interest are also given.