On the Liouville type theorems for selfsimilar solutions to the NavierStokes equations
Abstract
We prove Liouville type theorems for the selfsimilar solutions to the NavierStokes equations. One of our results generalizes the previous ones by NečasRu̇žičkaŠverak and Tsai.
Using the Liouville type theorem we also remove a scenario of asymtotically selfsimilar blowup for the NavierStokes equations with the profile belonging to with .
AMS Subject Classification Number: 35Q30, 76D03, 76D05
keywords: NavierStokes equations, selfsimilar solution, Liouville’s theorem
1 Introduction
We consider the NavierStokes equation in the space time cylinder
(1) 
where , . The aim of the present paper is to exclude a possible self similar blow up at the point under more general assumptions than in [8]. More precisely, we assume that and respectively are given by a self similar profiles and such that
(2)  
(3) 
where is a positive constant. Then solves the following system proposed by Leray (cf. [6]).
(4) 
It is already known that if for some , then for , while for . The case is proved in [7], while the case has been proved by Tsai in [8]. In fact Tsai proved a more general result, namely that if satisfies a the local energy bound
(5) 
for some ball and some .
We extend the results mentioned above in different directions. Our first main result is the following
Theorem 1.1.
Below we remove the condition on , and instead we restrict the range of so that . Our second main result is the following
Theorem 1.2.
Remark 1.3.
If , then (7) is obviously satisfied with the choice of . In general, if for implies (7). Indeed, for we have
as . Thus, Theorem 1.2 leads to the following Corollary
Corollary 1.4.
Let be a solution to (4). Suppose that for some
(8) 
The above corollary shows clearly that Theorem 1.2 improves the previous results of [7, 8]. As an application the above result one can remove a scenario of asymptotically selfsimilar blowup with a profile given by (8) as follows, which could viewed as an improvement of the corresponding result in [2].
2 Local estimate for local suitable weak solutions to the NavierStokes equation without pressure
The aim of the present section is to provide a local bound for local suitable weak solutions to the NavierStokes equations without pressure.
First, let us recall the notion of the local pressure projection for a given bounded domain , , introduced in [11]. Appealing to the theory of the steady Stokes system (cf. [3]), for any there exists a unique pair which solves in the weak sense the steady Stokes system
Then we set , where denotes the gradient functional in defined by
Here we have denoted by the space of all with .
Remark 2.1.
1. The operator is bounded from into itself with for all . The norm of depends only on and the geometric properties of , and independent on , if is a ball or an annulus, which is due to the scaling properties of the Stokes equation.
2. In case using the canonical embedding and the elliptic regularity we get together with the estimate
(10) 
where the constant in (10)
depends only on and . In case is a ball or an annulus this constant depends only on
(cf. [3] for more details). Accordingly the restriction of to the Lebesgue space defines
a projection in . This projection will be denoted still by .
Below for a class of vector fields we denote
by the set of such that in the sense of distribution.
By using the projection , we introduce the following notion of local suitable weak solution to the NavierStokes equations
Definition 2.2 (Local suitable weak solution).
Remark 2.3.
1. Note that due to the pressure is harmonic, and thus smooth in . Furthermore, as it has been proved in [11] the pressure gradient is continuous in .
2. The notion of local suitable weak solutions to the NavierStokes equations satisfying the local energy inequality (15) has been introduced in [10]. As it has been shown there such solutions enjoy the same partial regularity as the standard suitable weak solution as proved in the paper by CaffarelliKohnNirenberg[1]. Furthermore, the following regularity criterion has been proved for solution satisfying (15):

There exists and absolute number such that if for any it holds
(cf. also [10]).
Before turning to the statement of this result we will fix the notations used throughout this section For and we define the parabolic cylinders
By we denote the space . Furthermore for we set
Remark 2.4.
According to Lemma 4.1[10] the following Caccioppolitype inequality holds true
(16) 
where denotes an absolute constant.
Our main result of this section is the following regularity criterion
Theorem 2.5.
Let be a local suitable weak solution to (1). Let . There exist two positive constants and , both depending on only, such that if for , , the condition
(17) 
implies , and it holds
(18) 
Before turning to the proof of Theorem 2.5 we provide some lemmas, which will be used in our discussion below. We begin with a Caccioppolitype inequlities similar to (16).
Lemma 2.6.
Let be local suitable weak solution to (1). Then for every
(19) 
where denotes a constant depending only on .
Proof: Let be fixed. Set , and define , where . Let denote a suitable cut off function for . As it has been proved in [10] (cf. estimate (4.4) therein), applying Hölder’s inequality, the following inequality holds
(20)  
(21)  
(22) 
By means of Sobolev’s inequality together with Hölder’s inequality, (10) and (22)
(23)  
(24)  
(25)  
(26)  
(27) 
We recall the following Caccioppoli inequality for a harmonic function
which will be repeatedly used below. The proof is immediate from the formula by multiplying , integrating over , and then using integration by part. Recalling that is harmonic, by using (10) with we get first
(28)  
(29)  
(30) 
from which, integrating it over , we obtain
Using this estimate, we have
(31)  
(32) 
Combining (27) with (32), we arrive at
(33) 
Once more using the fact that is harmonic applying integration by parts, Caccioppoli type inequality together (10), we evaluate for almost all
Integration of both side of the above inequality together with Hölder’s inequality gives
(34) 
Combining (22) with (34) we are led to
(35) 
Thus, adding (33) to (35), we obtain
(36)  
(37) 
Let be chosen so that . Applying Hölder’s inequality together with PoincaréSobolev’s inequality, we see that
Integrating this inequality over , and applying Hölder’s inequality, we are led to
(38) 
We now estimate the righthand side of (37) by the aid of (38), and applying Young’s inequality. This gives
(39)  
(40)  
(41) 
By using a standard iteration argument (e.g. see [4]) we deduce from (41) together with Young’s inequality that
(42)  
(43) 
Multiplying both sides of (43) by , and applying Hölder’s inequality, we obtain the desired inequality (19).
We continue our discussion with some useful iteration lemmas. Let be a bounded domain. By , , we denote the image of under the Laplacian , which is a closed subspace of . By we denote the complementary space, which contains all being harmonic in such that
(44) 
By using the wellknown CalderónZygmund inequality, and the elliptic regularity of the Biharmonic equation we get the following
Lemma 2.7.
1. Let . Then there exists a unique such that
(45) 
in the sense of distributions ^{1)}^{1)}1) Here (45) means for all . . In addition, it holds
(46) 
Lemma 2.8.
Let . Let . Suppose, there exists and , such that for all and
(48) 
where . Let . Then for all and it holds
(49) 
Proof: Let and be arbitrarily chosen, but fixed. Let , specified below. According to (44) there exist unique and such that . Noting that , it follows that