The forward self-similar solution of fractional incompressible Navier-Stokes system: the critical case. (English) Zbl 07890708
Summary: In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli, Kohn and Nirenberg-type iteration, \(L^{\infty}\) estimates of the self-similar solution’s profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [B. Lai et al., Trans. Am. Math. Soc. 374, No. 10, 7449–7497 (2021; Zbl 1479.35622)] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution’s profile, a new technique is required due to lack of the related regularity theory.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
self-similar solution; weighted estimate; pointwise estimate; fractional Navier-Stokes equationsCitations:
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