Forward self-similar solutions of the fractional Navier-Stokes Equations

Abstract

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 $5/6<\alpha \le 1$ 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 Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] 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 [Jia and Šverák, Invent. Math. 196 (2014), 233-265].

46pages

References in corpus (1)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations

Cited by in corpus (5)

• Spatial decay of discretely self-similar solutions to the Navier-Stokes equations
• Global Regularity of weak solutions to the generalized Leray equations and its applications
• Global existence of uniformly locally energy solutions for the incompressible fractional Navier-Stokes equations
• Global well-posedness for 2D fractional inhomogeneous Navier-Stokes equations with rough density
• Global regularity and decay behavior for Leray equations with critical-dissipation and Its Application to Self-similar Solutions