A note on regularity criterion for the Navier–Stokes equations in terms of the pressure

Applied Mathematics and Computation 248 (2014) 1–3

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A logarithmically improved on regularity criterion for the Navier–Stokes equations in terms of the pressure Wenying Chen College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou 404000, Chongqing, China

a r t i c l e

i n f o

a b s t r a c t In this note, a logarithmically improved regularity criterion for the Navier–Stokes equations is established in terms of the pressure in the nonhomogeneous Besov space. Ó 2014 Elsevier Inc. All rights reserved.

Keywords: Navier–Stokes equations Regularity criteria Besov space

1. Introduction Consider the Navier–Stokes equations in R3

@ t u þ u  ru  Du þ rp ¼ 0; ðx; tÞ 2 R3  ð0; 1Þ; div u ¼ 0; ðx; tÞ 2 R3  ð0; 1Þ;

ð1:1Þ

3

uðx; 0Þ ¼ u0 ðxÞ; x 2 R ; where u ¼ uðx; tÞ is the velocity field, p ¼ pðx; tÞ is the scalar pressure and u0 ðxÞ with div u0 ¼ 0 in the sense of distribution is the initial velocity field.   In last century, Leray [10] and Hopf [7] constructed weak solutions u of (1.1) for arbitrary u0 2 L2 R3 with r  u0 ¼ 0. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier–Stokes equations. Different criteria for regularity of the weak solutions have been proposed. The Prodi-Serrin conditions (see Prodi [13], Ohyama [12] and Serrin [14]) were established in terms of the velocity field. Later logarithmically improved regularity criteria were proved in [11,1,20,5,19,3]. While, final Serrin-type regularity criterion for the pressure p was established in [16,17,15] in Lp space version. Some extensions can be found in [18,4]. In this short note, we will prove. Theorem 1.1. Let u0 2 L2 \ L4 with div u0 ¼ 0 in R3 . Assume that u0 is a weak solution to (1.1) on (0,T). If for 1 < r 6 1, the pressure pðx; tÞ satisfies

Z 0

T

kpðtÞkqBr

p;1

þ

1 þ log kpðtÞkBrp;1

dt < 1;

with

then uðx; tÞ is smooth on ð0; T. þ Here f ðxÞ ¼ maxf0; f ðxÞg.

E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.09.032 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

2 3 3 þ ¼ 2 þ r; < p < 1; q p 2þr

ð1:2Þ

2

W. Chen / Applied Mathematics and Computation 248 (2014) 1–3

2. Proof of Theorem 1.1 Due Lp theory for the Navier–Stokes equations established by Kato [8] and Giga [6], it is sufficient to show the L4 norm of the solution is bounded up to time T under (1.2). If (1.2) holds, one can deduce that for any small  > 0, there exists T  < T, such that

Z

kpðtÞkqBr

T

T

p;1

þ

1 þ log kpðtÞkBrp;1

dt 6 :

ð2:1Þ

First, let us start from an inequality which has been proved in [2]:

 Z t q  sup kuðsÞk4L4 6 CkuðT  Þk4L4 exp C 1 þ kpðsÞkBrp;1 ds

T  6s6t

0

T

1  q   r 1 þ k p ðsÞk B p;1 B   ds sup log 1 þ kpðsÞkBr C 6 CkuðT  Þk4L4 exp @C A p;1 T  6s6t T  log 1 þ kpðsÞk r Bp;1 !!    C 6 C exp C  log sup 1 þ kpðsÞkBrp;1 6 C sup 1 þ kpðsÞkBrp;1 ; Z

t

T  6s6t

ð2:2Þ

T  6s6t

where we have used (2.1). Multiplying the first equation of (1.1) by Du, after integration by parts, we see that

1 d kruðtÞk2L2 þ kDuðtÞk2L2 ¼ 2 dt 6

Z R3

6

9

ðu  rÞuðx; tÞ  Duðx; tÞdx 6 kuðtÞkL4 kruðtÞkL4 kDuðtÞkL2 6 CkuðtÞk5L4 kDuðtÞk5L2

1 kDuðtÞk2L2 þ CkuðtÞk12 L4 ; 2

1

4

where we used krf kL4 6 Ckf k5L4 kDf k5L2 . Integrating the above inequality over ðT  ; tÞ, we have

!C  sup kr T  6s6t

uðsÞk2L2

6 C 1 þ sup kpðsÞkBrp;1

!C  6 C 1 þ sup

T  6s6t

T  6s6t

kuðsÞk2H3

!2C  6 C 1 þ sup kuðsÞkH3

;

ð2:3Þ

T  6s6t

where (2.2) and an inequality between p and u (see [5]) were used. So the remaining thing is to control the H3 norm of the solution. In order to get the boundedness of H3 norm, we will use the following commutator estimate due to Kato and Ponce [9]:

   a  a1  K ðfg Þ  f Ka g  p 6 C  K g  L for a > 1, and

1 p

Lq1

   krf kLp1 þ Ka f Lp2 kg kLq2 ;

ð2:4Þ

1 2

¼ p11 þ q11 ¼ p12 þ q12 , where K ¼ ðDÞ .

Now, taking K3 to the first equation of (1.1), multiplying it by K3 u, after integration by parts, we obtain

Z   1 d kK3 uðtÞk2L2 þ kK4 uðtÞk2L2 ¼  K3 ðu  ruÞ  u  rK3 u ðx; tÞ  K3 uðx; tÞdx 2 dt R3 6 CkruðtÞkL3 kK3 uðtÞk2L3 3

1

1

5

6 CkruðtÞk4L2 kK3 uðtÞk4L2  kruðtÞk3L2 kK4 uðtÞk3L2 13 3 1 6 kK4 uðtÞk2L2 þ CkruðtÞkL22 kK3 uðtÞk2L2 ; 2

ð2:5Þ

where (2.4) and the following two Sobolev inequalities were used:

 1 3  4 krukL3 6 Ckruk4L2 K3 u 2 ; L

   5 1  3   4 6 K u 3 6 Ckruk6L2 K u 2 :

Substitute (2.3) into (2.5) and choose

L

L

 to be sufficiently small, then applying Gronwall’s inequality on (2.5) yields

3

sup kK uðsÞkL2 6 C:

T  6s6t

This completes the proof.

h

References [1] C.H. Chan, A. Vasseur, Log Improvement of the Prodi-Serrin Criteria for Navier–Stokes Equations, Methods Appl. Anal. 14 (2007) 197–212. [2] B. Dong, W. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal. 73 (2010) 2334–2341.

W. Chen / Applied Mathematics and Computation 248 (2014) 1–3

3

[3] J. Fan, Y. Fukumoto, Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier–Stokes and related equations, Kinet. Relat. Models 6 (2013) 545–556. [4] J. Fan, S. Jiang, G. Ni, On regularity criteria for the n-dimensional Navier–Stokes equations in terms of the pressure, J. Differ. Eqs. 244 (2008) 2936–2979. [5] J. Fan, S. Jiang, G. Nakamura, Y. Zhou, Logarithmically improved regularity criteria for the Navier–Stokes and MHD equations, J. Math. Fluid Mech. 13 (2011) 557–571. [6] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier–Stokes system, J. Differ. Eqs. 62 (1986) 186–212. [7] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951) 213–231. [8] T. Kato, Strong Lp -solutions of the Navier–Stokes equation in Rm , with applications to weak solutions, Math. Z. 187 (1984) 471–480. [9] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988) 891–907. [10] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta. Math. 63 (1934) 183–248. [11] S. Montgomery-Smith, Conditions implying regularity of the three dimensional Navier–Stokes equation, Appl. Math. 50 (2005) 451–464. [12] T. Ohyama, Interior regularity of weak solutions of the time-dependent Navier–Stokes equation, Proc. Jpn. Acad. 36 (1960) 273–277. [13] G. Prodi, Un teorema di unicità per le equazioni di Navier–Stokes, Ann. Mat. Pura Appl. 48 (1959) 173–182. [14] J. Serrin, The initial value problem for the Navier–Stokes equations, in: Nonlinear Problems Proc. Sympos., Madison, Wis., Univ. of Wisconsin Press, Madison, Wis., 1963, pp. 69–98. [15] M. Struwe, On a Serrin-type regularity criterion for the Navier–Stokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2007) 235–242. [16] Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in R3 , Proc. Amer. Math. Soc. 134 (2005) 149–156. [17] Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier–Stokes equations in RN, Z. Angew. Math. Phys. 57 (2006) 384–392. [18] Y. Zhou, Regularity criteria in terms of pressure for the 3-D Navier–Stokes equations in a generic domain, Math. Ann. 328 (2004) 173–192. [19] Y. Zhou, J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math. 24 (2012) 691–708. [20] Y. Zhou, S. Gala, Logarithmically improved regularity criteria for the Navier–Stokes equations in multiplier spaces, J. Math. Anal. Appl. 356 (2009) 498–501.