Erratum to: Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations

Abstract

We wish to correct some notations and results contained in the original article. The main results (Theorems 1.2-1.3) remain unchanged but Lemma 2.1 should be slightly corrected. 1 Let us start from the corrections for some notations and typos.(I). On Page 3 of original article, we defined the cylindersThe center of these cylinders are at (x 0 , t 0 ). In fact, we need (x 0 , t 0 ) to be located on the top of these cylinders. Therefore the definitions of Q r (x 0 , t 0 ), Q * r (x 0 , t 0 ), cQ r (x 0 , t 0 ) and cQ * r (x 0 , t 0 ) on page 3 of original article should be replaced by what is shown below:Moreover, the time interval I r in Lemma 2.4 should be I r = [−r 2s , 0]. (II). In part (3) of Definition 1

Partial regularity of solutions to the fractional Navier-Stokes equations

“…Using the Littlewood-Paley theory, Katz and 5310 YUKANG CHEN AND CHANGHUA WEI Pavlović in [9] studied the partial regularity of solutions to the fractional Navier-Stokes equations at the first blow-up time when 2 < s < 5 2 . Recently, Tang and Yu [20,21] studied the partial regularity of solutions to the fractional Navier-Stokes equations in the case of 3 2 < s < 2. The method of Tang and Yu depends on the characterizations for fractional Laplacian established by Caffarelli and Silvestre [3].…”

“…The method of Tang and Yu depends on the characterizations for fractional Laplacian established by Caffarelli and Silvestre [3]. However, the case when s = 3 2 is left open in [20,21]. To take a glance of the criticality of s = 3 2 , let us first recall the following imbedding property…”

“…The definition of suitable weak solutions to fractional Navier-Stokes equations (see [20,21]) is based on the characterizations for fractional powers of Laplacian operator obtained by Caffarelli and Silvestre [3]. These characterizations shown in [3] are proved by the well-known harmonic extensional idea.…”

“…To prove Theorem 1.2, we follow the strategy in [20,21]. At first, we need to get the smallness of more dimensionless quantities in some smaller domains under the assumption (2) (see Lemma 3.4).…”

“…Our approach mainly depends on the selection of the appropriate norm on the pressure p, which is very important for the final inductive arguments. Finally, we emphasize that in Section 3 we also generalize the restriction of the time integrable index of pressure p in [20,21].…”