Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
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DOI：
10.4007/annals.2022.196.1.3
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发表时间：
2022-07-01
影响因子：
4.9
通讯作者：
Colombo, Maria
中科院分区：
文献类型：
--
作者：
Albritton, Dallas;Brue, Elia;Colombo, Maria
在一项开创性的工作中，Leray（1934）证明了三维Navier-Stokes方程的全局弱解的存在性。我们展示了两种不同的Leray解，具有零初速度和相同的体力。我们的方法是构建一个“背景”。相似变量下Navier-Stokes动力学的不稳定解；其相似剖面是一个光滑的、紧支撑的涡环，其横截面是Vishik（2018）构建的不稳定二维涡的改进。第二个解是与背景解相关的不稳定流形上的轨迹，与Jia和S? ver??的预测一致。k(2015)。我们的解正好处于已知的适位性理论的边缘。
In a seminal work, Leray (1934) demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a ???background??? solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik (2018). The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and S??ver??k (2015). Our solutions live precisely on the borderline of the known well-posedness theory.