The linearized Crocco equation
The authors study a two-dimensional boundary layer over a flat plate in the context of the linearized Crocco equation around the steady state solution. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations. The main aim of this paper is to prove the existence and uniqueness of a weak solutions associated with the adjoint linearized Crocco equation with the homogeneous boundary conditions. This equation is doubly degenerate: the coefficient in front of \(\partial_x\) vanishes on a part of the boundary, and the coefficient of the elliptic operatore vanishes in another part of the boundary, \(x\)-variable plays the role of time variable. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operatore is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument. This is also used to prove a priori estimates, which cannot be obtained in a classical way. Finally, the results obtained are compared with other existing in the literature.
scientific article; zbMATH DE number 5126601