Asymptotic behavior of null controllability cost for parabolic equations with vanishing diffusivity under Robin and Neumann boundary conditions

In this paper we study the null-controllability cost of a transport-diffusion system under Robin conditions with distributed control and in which the transport coefficient is a gradient field. First, we provide some conditions on transport coefficient and boundary potential to show that the control cost decays exponentially when the viscosity vanishes and the control time is sufficiently large. On the other hand, if the range of the control region by the transport flow does not cover that of Ω, we prove that the control cost explodes exponentially for the Neumann conditions case with vanishing viscosity and all control time.