A new proof of a fundamental supnorm estimate for one-dimensional advection-diffusion equations. (English) Zbl 1372.35149

Schütz, L.; Ziebell, J. S.; Zingano, J. P.; Zingano, P. R.

Adv. Differ. Equ. Control Process. 11, No. 1, 41-51 (2013).

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

Keywords:

advection-diffusion equations; initial-value problem; energy method; heterogeneous media; forced advection; supnorm estimates; large time behavior