Abstract
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 Rn, where n≥1 is arbitrary. Our approach can be used for other parabolic equations as well, including degenerate cases like porous medium type equations and p-Laplacian evolution equations. Some open problems and related results of interest are also given.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
