Mathematical Theories on the Prandtl System in Sobolev Spaces

As a fundamental equation in the boundary layer theory, the Prandtl system was derived by Prandtl in 1904. It describes the behavior of an incompressible viscous flow near a boundary in a small viscosity limit with non-slip boundary condition. Despite of its importance in physics, the general well-posedness theories and the justification of the Prandtl system remain as challenging unsolved problems in mathematics. So far, there are only a few rigorous mathematical results on the well-posedness theory and the limit is only justified in the framework of analytic functions locally.In fact, since the classical work by Oleinik and her collaborators in 1960s about the local well-posedness theories for two space dimensional (2D) Prandtl system, important progress has been made on the global existence of weak solution, well-posedness framework for analytic data, instability and ill-posedness theories in Sobolev spaces, etc. Since Crocco transformation plays an important role in Oleinik's theory but it can not be applied to the study on the Navier-Stokes equations, recently, we introduced a direct energy method to study the well-posedness of the 2D Prandtl system in Sobolev spaces without using the Crocco transformation. Some local in time results have been obtained. We hope that this approach will shed some light on the three space dimensional (3D) problem and the justification of the limit from the Navier-Stokes equations to the superposition of the Prandtl system and the Euler equations.Hence, in this research proposal, we will investigate whether under some conditions, classical solutions can persist for a long time so that we can study their large time behavior, or some kind of singularity will eventually form so that the solutions will blowup in finite time in the setting of Sobolev spaces. Moreover, we will investigate the 3D Prandtl system. Here, notice that so far the global existence of weak solution in 2D was obtained by Xin-Zhang under a favorable condition on the pressure in addition to the monotonicity condition on the tangential velocity. Moreover, almost all the existing mathematical theories are for 2D problems. Hence, the problems proposed in this project are extremely challenging so that the expected outcome may only be obtained in some restrictive settings. However, we believe that even a small progress in these directions will lead to the enrichment of the mathematical theories for the Prandtl system.