Global Well-posedness and Regularity of Weak Solutions to the Prandtl's System

Abstract

We continue our study on the global solution to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik provided that the pressure is favorable. First, by using a different method from [13], we gave a direct proof of existence of a global weak solution by a direct BV estimate. Then we prove the uniqueness and continuous dependence on data of such a weak solution to the initialboundary value problem. Finally, we show the smoothness of the weak solutions and then the global existence of smooth solutions.

Global Regularity of the 2d Steady Compressible Prandtl Equations

“…For the unsteady compressible Prandtl equations, similar results are obtained in [3]. Recently, Xin, Zhang and Zhao [35] put forward a direct proof of the existence of global weak solutions to the Prandtl equations by a direct BV estimate. The key ingredients of this paper are that they studied the uniqueness and the regularity of a weak solution.…”

Hydrostatic approximation and optimal convergence rate for the Second-Grade fluid system

“…Noting that if the region Ω in (1.5) is the upper-half space, then it is converted to the wellknown Prandtl system which was widely studied in the past decades. Readers can refer to [1,5,6,8,10,12,16,18,19,21,26,30,31]. Compared with the usual Prandtl system, the pressure in hydrostatic Navier-Stokes system (1.5) is an unknown function, thus the study for (1.5) is more difficult.…”