In the first part of this paper, we prove a new theorem concerning non-smooth solutions of nonlinear Navier–Stokes type PDE as they arise in atmospheric and fluid dynamics, but here in arbitrary dimension. In its simplest form, the theorem states that the velocity field must be tangent to the hypersurface on which it has a jump discontinuity, i.e., its singular support. The theorem is proved using Colombeau algebras of generalized functions, providing yet another example of the fruitfulness of this concept for nonlinear problems with singularities, ill posed in distribution theory. In the second part, we discuss a numerical and analytic study of a two-dimensional model which, in spite of its simplicity, predicts remarkably correctly the “wall of the eye” of a hurricane and allows us to get analytic expressions for the asymptotic behaviour of radial and tangential wind field near this wall. These results are consistent with and confirm the theoretical results of the first part.