Nonuniqueness of Generalised Weak Solutions to the Primitive and Prandtl Equations

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Nonlinear Science Pub Date : 2024-05-27 DOI:10.1007/s00332-024-10032-8

Daniel W. Boutros, Simon Markfelder, Edriss S. Titi

引用次数: 0

Abstract

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the z-coordinate, respectively) that are constructed have different regularities.

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原始方程和普朗特方程广义弱解的非唯一性

我们开发了一种凸积分方案，用于构建二维和三维静水欧拉方程（也称为海洋和大气动力学的无粘性原始方程）的非唯一弱解。我们还为构建三维粘性原始方程和二维普朗特方程的非唯一弱解开发了这样一种方案。在布特罗斯等人（Calc Var Partial Differ Equ 62(8):219, 2023）的文章中，对静水欧拉方程的经典弱解概念进行了概括，而我们在此引入了进一步的概括。对于这种广义弱解，我们证明了一大类初始数据的存在性和非唯一性。此外，我们还构建了无限多的广义弱解实例，这些广义弱解不保存能量。所构建的静力学欧拉方程的气压和气压线性解（分别是水平速度在 z 坐标上的平均值和波动值）具有不同的规律性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Nonlinear Science 数学-力学

CiteScore

5.00

自引率

3.30%

发文量

审稿时长

4.5 months

期刊介绍： The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be. All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.