zbMATH — the first resource for mathematics

A new proof of a fundamental supnorm estimate for one-dimensional advection-diffusion equations. (English) Zbl 1372.35149
Summary: We give a short derivation of supnorm estimates for solutions of one-dimensional advection-diffusion equations $u_t+f(u)_=u_{xx},\qquad x\in\mathbb R,\;t>0$ and some of their generalizations, assuming initial data $$u(\cdot,0)\in L^p (\mathbb R)\cap L^\infty (\mathbb R)$$ for some $$1\leq p<\infty$$. A few related results and open questions are also given.
MSC:
 35K57 Reaction-diffusion equations 35K59 Quasilinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs
Full Text: