An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection–diffusion equation in the vanishing

The aim of this short paper is to explore a new connection between a conjecture concerning sharp boundary observability estimates for the 1 -D heat equation in small time and a conjecture concerning the cost of null-controllability for a 1 -D convection–diffusion equation with constant coefficients controlled on the boundary in the vanishing viscosity limit, in the spirit of what was done in Pierre Lissy (2012). We notably establish that the first conjecture implies the second one as soon as the speed of the transport part is non-negative in the transport-diffusion equation.