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Volume 38, Issue 4
Gevrey Well-Posedness of the Hyperbolic Prandtl Equations

Wei-Xi Li & Rui Xu

DOI: 10.4208/cmr.2021-0104

Commun. Math. Res., 38 (2022), pp. 605-624.

Published online: 2022-10

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  • 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.

  • Keywords

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

  • AMS Subject Headings

35Q35, 35Q30, 76D10, 76D03

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-38-605, author = {Li , Wei-Xi and Xu , Rui}, title = {Gevrey Well-Posedness of the Hyperbolic Prandtl Equations}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {4}, pages = {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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0104}, url = {http://global-sci.org/intro/article_detail/cmr/21074.html} }
TY - JOUR T1 - Gevrey Well-Posedness of the Hyperbolic Prandtl Equations AU - Li , Wei-Xi AU - Xu , Rui JO - Communications in Mathematical Research VL - 4 SP - 605 EP - 624 PY - 2022 DA - 2022/10 SN - 38 DO - http://doi.org/10.4208/cmr.2021-0104 UR - https://global-sci.org/intro/article_detail/cmr/21074.html KW - Hyperbolic Prandtl boundary layer, well-posedness, Gervey space, abstract Cauchy-Kowalewski theorem. AB -

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.

Li , Wei-Xi and Xu , Rui. (2022). Gevrey Well-Posedness of the Hyperbolic Prandtl Equations. Communications in Mathematical Research . 38 (4). 605-624. doi:10.4208/cmr.2021-0104
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