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The well-posedness of the three space dimensional Prandtl equations isstudied under some constraint on its flow structure. It reveals that theclassical Burgers equation plays an important role in determining this type offlow with special structure, that avoids the appearance of the complicatedsecondary flow in the three-dimensional Prandtl boundary layers. And thesufficiency of the monotonicity condition on the tangential velocity field forthe existence of solutions to the Prandtl boundary layer equations isillustrated in the three dimensional setting. Moreover, it is shown that thisstructured flow is linearly stable for any three-dimensional perturbation.

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