A well-posedness theory for the Prandtl equations in three space variables

Abstract

The well-posedness of the three space dimensional Prandtl equations is studied under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. And the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the Prandtl boundary layer equations is illustrated in the three dimensional setting. Moreover, it is shown that this structured flow is linearly stable for any three-dimensional perturbation.

arXiv comment40 pages

References in corpus (2)

• On the ill-posedness of the Prandtl equation
• A note on the Prandtl boundary layers

Cited by in corpus (4)

• Almost global existence for the Prandtl boundary layer equations
• Long time well-posdness of the Prandtl equations in Sobolev space
• Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin Boundary Condition in Weighted Sobolev Spaces
• Full Regularity and Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin or Dirichlet Boundary Condition