Forward self-similar solutions of the fractional Navier-Stokes equations. (English) Zbl 1420.35192
Summary: We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion \((- {\Delta})^\alpha\). First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with \(5 / 6 < \alpha \leq 1\) for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in \(\mathbb{R}^3 \times(0, + \infty)\). In particular, when \(\alpha = 1\), we prove that the solution constructed by M. Korobkov and T.-P. Tsai [Anal. PDE 9, No. 8, 1811–1827 (2016; Zbl 1358.35094)] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [H. Jia and V. Šverák, Invent. Math. 196, No. 1, 233–265 (2014; Zbl 1301.35089)].
MSC:
| 35Q30 | Navier-Stokes equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35R11 | Fractional partial differential equations |
| 35B44 | Blow-up in context of PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
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