Instability and Critical Regularity Indice for Degenerate PDEs of Prandtl-type Systems
DescriptionThe study on the hydrodynamics (in)stability and the inviscid limit of viscous fluid has a very long history both in physics and mathematics. The earliest systematic mathematical works are traced back to Rayleigh, Orr, Sommerfeld and Heisendberg. In early years, the spectral approach leads to famous equations like Raylaigh equation and Orr-Sommerfeld equation.The problem becomes more challenging if physical boundary is taken into account. For this, the most fundamental equations were derived by Prandtl in 1904  that describe the motion of incompressible viscous fluid close a no-slip boundary with high Reynolds number. Due to the degeneracy of the system, the well-posedness theories are still basically limited to two space dimensions(2D). This proposal aims to study the instability phenomena of some degenerate PDE systems of Prandtl-type, so that some critical regularity indice for well-posedness theories will then be investigated. Indeed, related to the Rayleigh criterion for Euler ow, the author in  showed that the unstable Euler shear ow leads to the instability of the Prandtl equations. Since then there has been a lot of investigation on the instability of Prandtl equations. Precisely,  shows that non-degenerate critical points in a shear ow lead to strong linear ill- posedness. These results have been further developed in varies settings. In 2D, this instability result reveals that the critical regularity index for well-posedness without monotonicity condition on the velocity field is Gevrey function space with index 2 for tangential space variable that has been recently proved.Applying Prandtl ansatz to the MHD system when both the Reynolds and magnetic Reynolds numbers tend to infinity in the same rate, a Prandtl-type boundary layer system was derived recently in  with well-posedness theory in 2D established in some weighted Sobolev space, under the condition that the tangential magnetic field is not degenerate. Hence, this leads to a question about whether such non-degeneracy in the tangential magnetic field is necessary for well-posedness in Sobolev space in analog to the monotonicity condition of the tangential velocity field for the classical Prandtl equations. To answer this question, we need to study the instability mechanism of such degenerate PDE system, and then investigate the critical regularity index for well-posedness in Gevrey function space. We believe that the study on these problems will not only enrich the mathematical theories for the Prandtl-type systems, but also shed some light on the study of degenerate PDEs with loss of derivatives.
|Effective start/end date||1/08/18 → 6/07/22|