This talk is devoted to the study of the equation $u u_x – u_{yy}=f$ in the domain $(x_0,x_1)\times (-1,1)$, in the vicinity of the shear flow profile $u(x,y)=y$. This equation serves as a toy model for more complicated fluid equations such as the Prandtl system. The difficulty lies in the fact that we are interested in changing sign solutions. Hence the equation is forward parabolic in the region where $u>0$, and backward parabolic in the region $u<0$. The line $u=0$ is a free boundary and an unknown of the problem. Unexpectedly, we prove that even when the data (i.e. the source term $f$ or the boundary data) are smooth, existence of strong solutions of the equation fails in general. This phenomenon is already present at the linear level, and linked to the existence of singular profiles for the homogeneous linearized equation. In fact, we prove that strong solutions exist (both for the linearized and for the nonlinear system) if and only if the data satisfy a finite number of orthogonality conditions, whose purpose is to avoid the presence of singular profiles in the solution. A key difficulty of our work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. This is a joint work with Frédéric Marbach and Jean Rax.