In this paper, we consider large time behaviour of solutions to the 3D Navier–Stokes equations with damping term \(|u|^{\beta -1}u\)(\(\beta \ge 1\)). For \(\beta >\frac{7}{3}\), we derive decay rates of the \(L^2\)-norm of the solutions. By using Fourier splitting method, we also prove that the solutions are asymptotically equivalent to the solutions of the classic 3D Navier–Stokes equations.

中文翻译：

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带阻尼的3D Navier–Stokes方程解的长时间行为

在本文中，我们考虑了带有阻尼项\（| u | ^ {\ beta -1} u \）（\（\ beta \ ge 1 \））的3D Navier–Stokes方程解的较大时间行为。对于\（\ beta> \ frac {7} {3} \），我们得出解的\（L ^ 2 \）-范数的衰减率。通过使用傅立叶分裂法，我们还证明了该解在渐近上等同于经典3D Navier–Stokes方程的解。