Global Existence and the Decay of Solutions to the Prandtl System with Small Analytic Data

Archive for Rational Mechanics and Analysis - Tập 241 - Trang 403-446 - 2021
Marius Paicu1, Ping Zhang2
1Institut de Mathématiques de Bordeaux, Université Bordeaux, Talence Cedex, France
2Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, The Chinese Academy of Sciences, Beijing, China

Tóm tắt

In this paper, we prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive a sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with its primitive one, and which basically controls the evolution of the analytical radius to the solutions. Our result can be viewed as a global-in-time Cauchy–Kowalevsakya result for the Prandtl system with small analytical data, which in particular improves the previous result in Ignatova and Vicol (Arch Ration Mech Anal 220:809–848, 2016) concerning the almost global well-posedness of a two-dimensional Prandtl system.

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