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

Abstract

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 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 globalin-time Cauchy-Kowalevsakya result for Prandtl system with small analytical data.

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

“…Finally, we mention that there have been increasing activities in recent years on the studies of the unsteady Prandtl's system in both two and three dimensions with substantial results for general initial data, ill-posedness, instability and different methods, we refer to [22][23][24][25][26][27][28][29][30][31][32] and references therein for these further developments. §2 Existence…”

Global tangentially analytical solutions of the 3D axially symmetric Prandtl equations

“…This result was extended to the three case in [20]. Most recently, Global existence of tangentially analytical solutions with small data was proved in [26] for the two dimensional Prandtl equations. This result was improved to the optimal Gevrey class 2 in [28].…”

“…The inequality (3.32) is a special case of Treves inequality that can be found in [11]. Proof of Lemma 3.5 can be found in [26,Lemma 3.1]. See also [12,Lemma 3.3].…”

“…This quantity has a sufficiently fast decay-in-time rate for our constructed weighted analytical energy, which ensures the positive lower bound of the analytical radius for any time. Its two dimensional originality can be traced to Paicu-Zhang [26].…”

“…Proof of the main result relies on the construction of a tangentially weighted analytic energy functional, which acts on a specially designed good unknown. The constructed energy functional can find its two dimensional parallel in Ignatova-Vicol [12] where no tangential weight is introduced and the specially good unknown is set to control the lower bound of the analytical radius, whose two dimensional similarity can be traced to Paicu-Zhang [26]. Our result is an improvement of that in Ignatova-Vicol [12] from the almost global existence to the global existence and an extension of that in Paicu-Zhang [26] from the two dimensional case to the three dimensional axially symmetric case.…”