Journals
Resources
About Us
Open Access

Gevrey Well-Posedness of the Hyperbolic Prandtl Equations

Gevrey Well-Posedness of the Hyperbolic Prandtl Equations

Year:    2022

Author:    Wei-Xi Li, Rui Xu

Communications in Mathematical Research, Vol. 38 (2022), Iss. 4 : pp. 605–624

Abstract

We study 2D and 3D Prandtl equations of degenerate hyperbolic type, and establish without any structural assumption the Gevrey well-posedness with Gevrey index ≤ 2. Compared with the classical parabolic Prandtl equations, the loss of the derivatives, caused by the hyperbolic feature coupled with the degeneracy, cannot be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart. Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature, we give in this text a straightforward proof, basing on an elementary $L^2$ energy estimate. In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmr.2021-0104

Communications in Mathematical Research, Vol. 38 (2022), Iss. 4 : pp. 605–624

Published online:    2022-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    20

Keywords:    Hyperbolic Prandtl boundary layer well-posedness Gervey space abstract Cauchy-Kowalewski theorem.

Author Details

Wei-Xi Li

Rui Xu

  1. Prandtl Equations and Related Boundary Layer Equations

    Local Existence of Solutions to 3D Prandtl Equations with a Special Structure

    Qin, Yuming | Dong, Xiaolei | Wang, Xiuqing

    2024

    https://doi.org/10.1007/978-981-97-4565-4_6 [Citations: 0]
  2. Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations

    De Anna, Francesco | Kortum, Joshua | Scrobogna, Stefano

    Zeitschrift für angewandte Mathematik und Physik, Vol. 75 (2024), Iss. 2

    https://doi.org/10.1007/s00033-023-02179-3 [Citations: 2]
  3. Global‐in‐time well‐posedness of solutions for the 2D hyperbolic Prandtl equations in an analytic framework

    Dong, Xiaolei

    Mathematical Methods in the Applied Sciences, Vol. 48 (2025), Iss. 3 P.3895

    https://doi.org/10.1002/mma.10523 [Citations: 0]
  4. Gevrey Solutions of Quasi-Linear Hyperbolic Hydrostatic Navier–Stokes System

    Li, Wei-Xi | Paicu, Marius | Zhang, Ping

    SIAM Journal on Mathematical Analysis, Vol. 55 (2023), Iss. 6 P.6194

    https://doi.org/10.1137/22M1526290 [Citations: 2]
  5. Local existence of solutions to 3D Prandtl equations with a special structure

    Qin, Yuming | Wang, Xiuqing

    Journal de Mathématiques Pures et Appliquées, Vol. 194 (2025), Iss. P.103670

    https://doi.org/10.1016/j.matpur.2025.103670 [Citations: 0]