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 (−∆) α . First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with 5/6 < α ≤ 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 R 3 × (0, +∞). In particular, when α = 1, we prove that the solution constructed by Korobkov-Tsai [23, 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 [17, Jia andŠverák, Invent. Math. 196 (2014), 233-265]. n 2 , γ(r) = +∞ 0 s r−1 e −s ds, r > 0.

Global Regularity of weak solutions to the generalized Leray equations and its applications

“…Via the differences characterization of Besov spaces and the bootatrap argument, we improve the regularity for weak solution from H α ω (R 3 ) to H 1+α ω (R 3 ). This regularity result, together linear theory for the non-local stokes system, deduces the pointwise estimates of V which makes us to obtain the natural pointwise property of the self-similar solution constructed in [12]. In particular, we obtain the optimal decay estimate to the self-similar solution of the classical Naiver-Stokes equations by means of the special structure of Oseen tensor under the low regularity condition on initial data.…”

“…The advantage of this method can be used to deal with the case of half-space, but this method can not provide the pointwise decay estimates. Recently, Lai-Miao-Zheng [12] used the L 2 weighted estimate to prove the solution constructed in [15] satisfying the pointwise decay. Besides, for the hypo-dissipative case, i.e., 5 8 < α < 1, the authors proved there exists a weak solution to (1.5) in Sobolev space H α (R 3 ), which is stated as following: Theorem 1.1 (Theorem 3.6, [12]).…”

“…The keypoint of constructing a solution in [12] by making use of the topological degree theory consists of the blowup argument and viscous approximation method. In contrary to the framework of weighted Hölder space used in [23], Theorem 1.1 only provide uniformly H α (R 3 ) bound of solution V , and don't provide any information on pointwise behavior for the case α < 1.…”

“…In contrary to the framework of weighted Hölder space used in [23], Theorem 1.1 only provide uniformly H α (R 3 ) bound of solution V , and don't provide any information on pointwise behavior for the case α < 1. By the way, the authors also established the more high local regularity of V via classical the elliptic regularity theory and bootstrap technique in [12].…”

Global existence of a uniformly local energy solution for the incompressible fractional Navier–Stokes equations

“…Lately, Lai, Miao and Zheng in [12] proved the existence of forward self-similar solutions to (FNS)-(FNSI) with 5/6 < s ≤ 1 for arbitrary large self-similar initial data via the blow-up argument. Since E 2 contains non-trivial scale-invariant functions, for example: σ(x) |x| 2s−1 , inspired by [9], we think that, with the help of the global-in-time existence of local Leray solutions in Theorem 1.5, the result given in [12] may be proved in another way.…”