Global existence and decay estimates of solutions for the compressible Prandtl type equations in Sobolev spaces

Abstract

This paper addresses the global existence and large-time decay of solutions to the two-dimensional compressible Prandtl equations in Sobolev spaces. We study small perturbations around a monotone shear flow and establish global weighted Sobolev estimates together with precise time-decay rates for the solutions. The main difficulties are that the normal velocity causes a loss of one tangential derivative, the variable density gives rise to nonstandard boundary and shear-flow coupling terms, and the global-in-time argument requires sufficiently strong decay to close the time-weighted energy estimates. To overcome these difficulties, we combine the cancellation method introduced by Masmoudi and Wong~\cite{MW2015} with the good-quantity approach developed by Paicu and Zhang~\cite{PZ2021}, together with weighted shear-flow estimates and a careful treatment of the natural boundary structure. Under suitable assumptions on the density perturbation, the shear flow, and the function $f(t)$ describing the outer tangential flow, we prove the global-in-time well-posedness of solutions in weighted Sobolev spaces. In this sense, the present result may be viewed as a Sobolev-space counterpart of the recent analytic-framework result obtained by Chen, Huang and Li (J. Funct. Anal. 286 (2024), 110322)~\cite{CHL2024} for the two-dimensional compressible Prandtl-type equations, while requiring a weaker time-decay assumption on the density perturbation.