Local existence of solutions to 3D Prandtl equations with a special structure

Yuming Qin; Xiuqing Wang

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arXiv · 2307.08030

Local existence of solutions to 3D Prandtl equations with a special structure

Yuming Qin·Xiuqing Wang

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Abstract

In this paper, we consider the 3D Prandtl equation in a periodic domain and prove the local existence and uniqueness of solutions by the energy method in a polynomial weighted Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the Crocco transform has always been used with the general outer flow $U\neq\text{constant}$, this Crocco transform is not needed here for 3D Prandtl equations. We use the skill of cancellation mechanism and construct a new unknown function to show that the existence and uniqueness of solutions to 3D Prandtl equations (cf. Masmoudi and Wong, Comm. Pure Appl. Math., 68(10)(2015), 1683-1741) which extends from the two dimensional case in \cite {12} to the present three dimensional case with a special structure.

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Yuming Qin, Xiuqing Wang. 2023-07-16. Local existence of solutions to 3D Prandtl equations with a special structure. https://arxiv.org/abs/2307.08030

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