We study the Cauchy problem in n-dimensional space for the system of Navier–Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong, and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper demonstrate the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Young’s inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz–Leray projection, and the boundedness of the Riesz transform are developed in mixed-norm Lebesgue spaces. These analysis results are topics of independent interests, and they are potentially useful in other problems.

中文翻译：

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临界混合范数Lebesgue空间中Navier-Stokes方程的适定性

我们研究n中的柯西问题临界混合范数Lebesgue空间中Navier–Stokes方程组的三维空间。临界混合范数Lebesgue空间的类别确定了解决方案的局部适定性和全局适定性。在混合范数Lebesgue空间中，初始数据和解都可能在某些点处是奇异的，或者在无穷大处以不同的空间变量方向以不同的速率衰减为零。这些奇异速率中的某些可能非常强，而某些衰减速率可能会非常慢。除其他兴趣外，本文的结果证明了演化过程中初始数据的各向异性行为的持久性。为了达到目标，我们采用了诸如杨氏不等式，热方程解的时间衰减等基础分析理论，在混合范数Lebesgue空间中发展了Helmholtz-Leray投影的有界性和Riesz变换的有界性。这些分析结果是具有独立利益的主题，在其他问题中可能很有用。