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 viscosity limit

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 is done in [Pierre Lissy, A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transportdiffusion equation, C. R. Math. Acad. Sci. Paris, Volume 352, 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.