Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 184674, 9 pages

http://dx.doi.org/10.1155/2012/184674

## A Note on the Regularity Criterion of Weak Solutions of Navier-Stokes Equations in Lorentz Space

School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 3 July 2012; Accepted 7 August 2012

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Xunwu Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the regularity of Leray weak solutions to the 3D Navier-Stokes equations in Lorentz space. It is proved that the weak solution is regular if the horizontal velocity denoted by satisfies The result is obvious and improved that of Dong and Chen (2008) on the Lebesgue space.

#### 1. Introduction and Main Results

In this note, we consider the regularity criterion of weak solutions of the Navier-Stokes equations in the whole space Here and denote the unknown velocity field and the unknown scalar pressure field. is a given initial velocity. For simplicity, we assume that the external force is zero, but it is easy to extend our results to the nonzero external force case. Here and in what follows, we use the notations for vector functions ,

For a given initial data , Leary in the pioneer work, [1] constructed a global weak solution
From that time on, although much effort has been made on the uniqueness and regularity of weak solutions, the question of global regularity or finite time singularity for weak solutions in is still open. One important observation is that the regularity can be derived when certain growth conditions are satisfied. This is known as a regularity criterion problem. The investigation of the regularity criterion on the weak solution stems from the celebrated work of Serrin [2]. Namely, Serrin's regularity criterion can be described as follows. * A weak solution ** of Navier-Stokes equations is regular if the growth condition on velocity field **holds true. *

It should be mentioned that the Serrin's condition (1.4) is important from the point of view of the relation between scaling invariance and regularity criteria of weak solutions; indeed, if a pair solves (1.1), then so does defined by Scaling invariance means that holds for all and this happens if and only if and satisfy (1.4).

Actually, the condition described by (1.4) which involves all components of the velocity vector field is known as degree growth condition (see Chen and Xin [3] for details), since The degree growth condition is critical due to the scaling invariance property. That is, solves (1.1) if and only if is a solution of (1.1).

Moreover, this pioneer result [2] has been extended by many authors in terms of velocity , the gradient of velocity or vorticity in Lebesgue spaces or Besov spaces, respectively (refer to [4β7] and reference therein).

Actually, the weak solution remains regular when a part of the velocity components or vorticity is involved in a growth condition. On one hand, regularity of the weak solution was recently obtained by Dong and Chen [8] when two velocity components denoted by satisfy the critical growth condition

It should be mentioned that the weak solution remains regular if the single velocity component satisfies the higher (subcritical) growth conditions (see Zhou [9], Penel and PokornΓ½ [10], Kukavica and Ziane [11], and Cao and Titi [12]). One may also refer to some interesting regularity criteria [13β15] for weak solutions of micropolar fluid flows. It seems difficult to show regularity of weak solutions by imposing Serrin's growth condition on only one component of velocity field for both Navier-Stokes equations and micropolar fluid flows.

However, whether or not the result (1.9) can be improved to the critical weak spaces is an interesting and challenging problem, that is to say, when the weak critical growth condition is imposed to only two velocity components. The main difficulty lies in the lack of *a priori* estimates on two-velocity components due to the special structure of the nonlinear convection term in monument equations.

The aim of the present paper is to improve the two-component regularity criterion (1.9) from Lebesgue space to the critical Lorentz space (see the definitions in Section 2) which satisfies the scaling invariance property.

Before stating the main results, we firstly recall the definition of the Leray weak solutions.

*Definition 1.1 (Temam, [16]). *Let and . A vector field is termed as a Leray weak solution of (1.1) on if satisfies the following properties:(i);
(ii) in the distribution space ;(iii) in the distribution space ;(iv) satisfies the energy inequality

The main results now read as follows.

Theorem 1.2. *Suppose , and in the sense of distributions. Assume that is a Leray weak solution of the Navier-Stokes equations (1.1) in . If the horizontal velocity denoted by satisfies the following growth condition:
**
then is a regular solution on . *

*Remark 1.3. *It is easy to verify that the spaces (1.11) satisfy the degree growth conditions due to the scaling invariance property. Moreover, since the embedding relation , Theorem 1.2 is an important improvement of (1.9).

*Remark 1.4. *Unlike the previous investigations via two components of vorticity (see [17, 18]) in weak space, of which the approaches are mainly based on the vorticity equations and seem not available in our case here due to the special structure of convection term, the present examination is directly based on the momentum equations. In order to make use of the structure of the nonlinear convection term , we study every component of and estimate them one by one with the aid of the identities .

#### 2. Preliminaries and *A Priori* Estimates

To start with, let us introduce the definitions of some functional spaces. with are usual Lebesgue space and Sobolev space.

To define the Lorenz space with , we consider a measurable function and define for the Lebesgue measure of the set .

Then if and only if Actually, Lorentz space may be alternatively defined by real interpolation (see Bergh and LΓΆfstrΓΆm [19] and Triebel [20]) with

Especially, is equivalent to the norm and thus it readily seen that

In order to prove Theorem 1.2, it is sufficient to examine * a priori* estimates for smooth solutions of (1.1) described in the following.

Theorem 2.1. *Let , with . Assume that is a smooth solution of (1.1) on and satisfies the growth conditions (1.11). Then
**
holds true. *

*Proof of Theorem 2.1. *Taking inner product of the momentum equations of (1.1) with and integrating by parts, one shows that
In order to estimate the right-hand side of (2.8), with the aid of the divergence-free condition and integration by parts, observe that

The estimation of the terms is now estimated one by one.

In order to estimate and , employing integration by parts deduces that
For , the divergence-free condition and integration by parts imply

Thus, plugging the above inequalities into (2.8) to produce

We now carry out the estimation of (2.12) based on the assumption described by (1.11).

Applying HΓΆlderβs inequality and Youngβs-inequality, we have for the right-hand side (RHS) of (2.12)
where we have used the following HΓΆlder inequalityβs in Lorentz space in the last line (refer to O'Neil [21, Theorems 3.4 and 3.5])
for
with

We now claim that the term in (2.13) can be estimated by applying the following Gagliardo-Nirenberg inequality in Lorentz space

Indeed, choosing and such that
and then applying Gagliardo-Nirenberg inequality, it follows that
Thus, applying the interpolation inequality (2.3), we have
that is to say,
and (2.17) is derived. Therefore, by employing (2.17) and Youngβs inequality, the inequality (2.13) becomes

Inserting (2.22) into (2.12) to produce

Taking Gronwallβs inequality into account yields the desired estimate,
note that
This completes the proof of Theorem 2.1.

#### 3. Proof of Theorem 1.2

According to* a priori* estimates of smooth solutions described in Theorem 2.1, the proofs of Theorem 1.2 are standard.

Since with , by the local existence theorem of strong solutions to the Navier-Stokes equations (see, e.g., Fujita and Kato [22]), there exist a and a smooth solution of (1.1) satisfying
Note that the Leray weak solution satisfies the energy inequality (1.10). It follows from Serrin's weak-strong uniqueness criterion [2] that
Thus, it is sufficient to show that
Suppose that . Without loss of generality, we may assume that is the maximal existence time for . Since on and by the assumptions (1.11), it follows from *a priori* estimate (2.7) that the existence time of can be extended after which contradicts with the maximality of .

Thus, we complete the proof of Theorem 1.2.

#### Acknowledgments

This work is partially supported by NNSF of China (11071185) and NSF of Tianjin (09JCYBJC01800).

#### References

- J. Leray, βSur le mouvement d'un liquide visqueux emplissant l'espace,β
*Acta Mathematica*, vol. 63, no. 1, pp. 193β248, 1934. View at Publisher Β· View at Google Scholar - J. Serrin, βOn the interior regularity of weak solutions of the Navier-Stokes equations,β
*Archive for Rational Mechanics and Analysis*, vol. 9, pp. 187β195, 1962. View at Google Scholar Β· View at Zentralblatt MATH - Z. M. Chen and Z. Xin, βHomogeneity criterion for the Navier-Stokes equations in the whole spaces,β
*Journal of Mathematical Fluid Mechanics*, vol. 3, no. 2, pp. 152β182, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Q. Chen and Z. Zhang, βSpace-time estimates in the Besov spaces and the Navier-Stokes equations,β
*Methods and Applications of Analysis*, vol. 13, no. 1, pp. 107β122, 2006. View at Google Scholar Β· View at Zentralblatt MATH - Z.-M. Chen and W. G. Price, βBlow-up rate estimates for weak solutions of the Navier-Stokes equations,β
*Proceedings of the Royal Society A*, vol. 457, no. 2015, pp. 2625β2642, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z.-M. Chen and W. G. Price, βMorrey space techniques applied to the interior regularity problem of the Navier-Stokes equations,β
*Nonlinearity*, vol. 14, no. 6, pp. 1453β1472, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. Dong, G. Sadek, and Z. Chen, βOn the regularity criteria of the 3D Navier-Stokes equations in critical spaces,β
*Acta Mathematica Scientia Series B*, vol. 31, no. 2, pp. 591β600, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B.-Q. Dong and Z.-M. Chen, βRegularity criterion for weak solutions to the 3D Navier-Stokes equations via two velocity components,β
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 1, pp. 1β10, 2008. View at Publisher Β· View at Google Scholar - Y. Zhou, βA new regularity criterion for weak solutions to the Navier-Stokes equations,β
*Journal de Mathématiques Pures et Appliquées*, vol. 84, no. 11, pp. 1496β1514, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. Penel and M. Pokorný, βSome new regularity criteria for the Navier-Stokes equations containing gradient of the velocity,β
*Applications of Mathematics*, vol. 49, no. 5, pp. 483β493, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - I. Kukavica and M. Ziane, βOne component regularity for the Navier-Stokes equations,β
*Nonlinearity*, vol. 19, no. 2, pp. 453β469, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - C. Cao and E. S. Titi, βRegularity criteria for the three-dimensional Navier-Stokes equations,β
*Indiana University Mathematics Journal*, vol. 57, no. 6, pp. 2643β2661, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B.-Q. Dong and Z.-M. Chen, βRegularity criteria of weak solutions to the three-dimensional micropolar flows,β
*Journal of Mathematical Physics*, vol. 50, no. 10, Article ID 103525, 2009. View at Publisher Β· View at Google Scholar - B.-Q. Dong, Y. Jia, and Z.-M. Chen, βPressure regularity criteria of the three-dimensional micropolar fluid flows,β
*Mathematical Methods in the Applied Sciences*, vol. 34, no. 5, pp. 595β606, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B.-Q. Dong and W. Zhang, βOn the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 73, no. 7, pp. 2334β2341, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Temam,
*Navier-Stokes Equations, Theory and Numerical Analysis*, North-Holland, Amsterdam, The Netherlands, 1977. - H. Kozono and N. Yatsu, βExtension criterion via two-components of vorticity on strong solutions to the 3D Navier-Stokes equations,β
*Mathematische Zeitschrift*, vol. 246, no. 1-2, pp. 55β68, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Zhifei and C. Qionglei, βRegularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in ${\mathit{R}}^{3}$,β
*Journal of Differential Equations*, vol. 216, no. 2, pp. 470β481, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Bergh and J. Löfström,
*Interpolation Spaces*, Springer, New York, NY, USA, 1976. - H. Triebel,
*Interpolation Theory, Function Spaces, Differential Operators*, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978. - R. O'Neil, βConvolution operators and $L(p,q)$ spaces,β
*Duke Mathematical Journal*, vol. 30, pp. 129β142, 1963. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Fujita and T. Kato, βOn the Navier-Stokes initial value problem. I,β
*Archive for Rational Mechanics and Analysis*, vol. 16, pp. 269β315, 1964. View at Google Scholar