We are interested in controllability problems of equations coming from a boundary layer model. We simplify the problem by considering only equations with constant coefficients. The problem is described by a degenerate parabolic equation (a linearized Crocco-type equation) where phenomena of diffusion and transport are coupled. First we give a geometric characterization of the influence domain of a locally distributed control. Then we prove regional null controllability results on this domain. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by decomposition of the space-time domain and Carleman-type estimates along characteristics. In the second part of this paper, we treat the case of a boundary control.