Well-Posedness of the Boundary Layer Equations

Abstract

Abstract. We consider the mild solutions of the Prandtl equations on the half space. Requiring analyticity only with respect to the tangential variable, we prove the short time existence and the uniqueness of the solution in the proper function space. The proof is achieved applying the abstract Cauchy-Kowalewski theorem to the boundary layer equations once the convection-diffusion operator is explicitly inverted. This improves the result of [M. Sammartino and R. E. Caflisch, Comm. Math. Phys., 192 (1998), pp. 433-461], as we do not require analyticity of the data with respect to the normal variable.

On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane

“…Thus the solvability of the Prandtl equations itself is not surprising in our setting; cf. [2,32,17]. But we note here that the solvability of the Prandtl equations does not necessarily imply the desired asymptotic expansion, as in the counter example by [10].…”

“…The analyticity condition is in fact required only in the tangential direction [17]. But the solvability for general initial data in a Sobolev class is still an open problem.…”

Zero Viscosity Limit for Analytic Solutions of the Primitive Equations

“…[7,[15][16][17][18]28,41]), we use the following version of the Abstract CauchyKowalevski Theorem (ACK) (cf. [1,35,47] and references therein). Consider the equation u + F(u, t) = 0.…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…One brief remark here is that the Prandtl equations are well posed only under strong conditions on the flow, such as when boundary and the data have some degree of analyticity [5,128,95,23,84] or the data is monotonic in the normal direction to the boundary [122,124,83]. The most classical result verifying (3.9) is [128] in the analytic functional framework, after the pioneering work of [5,6].…”