Abstract We consider the three-dimensional Navier–Stokes equations whose initial data may have infinite kinetic energy. We establish unique existence of the mild solution to the Navier–Stokes equations for small initial data in the whole space ℝ 3 {\mathbb{R}^{3}} and a vertically periodic product space ℝ 2 × 𝕋 1 {\mathbb{R}^{2}\times\mathbb{T}^{1}} which may be constant in vertical direction so that it includes the Oseen vortex. We further discuss its asymptotic stability under arbitrarily large three-dimensional perturbation in ℝ 2 × 𝕋 1 {\mathbb{R}^{2}\times\mathbb{T}^{1}} .

中文翻译：

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三维大扰动下小Oseen型涡的渐近稳定性

摘要 我们考虑初始数据可能具有无限动能的三维 Navier-Stokes 方程。我们在整个空间 ℝ 3 {\mathbb{R}^{3}} 和一个垂直周期的乘积空间 ℝ 2 × 𝕋 1 {\mathbb{ 中建立了 Navier-Stokes 方程的温和解的唯一存在性R}^{2}\times\mathbb{T}^{1}} 在垂直方向上可能是恒定的，因此它包括 Oseen 涡流。我们进一步讨论了它在 ℝ 2 × 𝕋 1 {\mathbb{R}^{2}\times\mathbb{T}^{1}} 中任意大的三维扰动下的渐近稳定性。