Strong Ill-Posedness in [math] of the 2D Boussinesq Equations in Vorticity Form and Application to the 3D Axisymmetric Euler Equations

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-09-03 DOI:10.1137/23m159384x

Roberta Bianchini, Lars Eric Hientzsch, Felice Iandoli

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引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5915-5968, October 2024.
Abstract. We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorticity form in [math] without boundary, building upon the method that Elgindi and Shikh Khalil [Strong Ill-Posedness in [math] for the Riesz Transform Problem, arXiv:2207.04556v1, 2022] developed for scalar equations. We provide examples of initial data with vorticity and density gradient of small [math] size, for which the horizontal density gradient [math] has a strong [math]-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi’s decomposition of the Biot–Savart law, we apply our method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with vorticity uniformly bounded and small in [math] provides a solution whose gradient of the swirl has a strong [math]-norm inflation in infinitesimal time. The norm inflation is quantified from below by an explicit lower bound which depends on time and the size of the data and is valid for small times.

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涡度形式的二维布森斯克方程[math]中的强拙问题及其在三维轴对称欧拉方程中的应用

SIAM 数学分析期刊》，第 56 卷第 5 期，第 5915-5968 页，2024 年 10 月。摘要。在 Elgindi 和 Shikh Khalil [Strong Ill-Posedness in [math] for the Riesz Transform Problem, arXiv:2207.04556v1, 2022] 针对标量方程提出的方法基础上，我们证明了无边界涡度形式的二维 Boussinesq 系统的强假设不成立性。我们举例说明了具有小[math]涡度和密度梯度的初始数据，对于这些初始数据，水平密度梯度[math]在无穷小时间内具有强烈的[math]-norm膨胀，而涡度和垂直密度梯度仍然是有界的。此外，我们利用 Elgindi 对 Biot-Savart 定律分解的三维版本，将我们的方法应用于有漩涡且远离垂直轴的三维轴对称欧拉方程，结果表明，一大类涡度均匀有界且[math]较小的初始数据提供了一个解，其漩涡梯度在无穷小时间内具有强烈的[math]-norm 膨胀。通过一个取决于时间和数据大小的显式下限，从下往上量化了这种正态膨胀，它对小时间有效。

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来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.