Research Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping

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1 Mathematical Problems in Engineering Volume 15, Article ID 194, 5 pages Research Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping Weihua Wang 1 and Guopeng Zhou 1 School of Mathematics and Statistics, Hubei University, Wuhan 46, China Institute of Engineering and echnology, Hubei University of Science and echnology, Xianning 471, China Correspondence should be addressed to Guopeng Zhou; zhgpeng@16.com Received 16 March 15; Accepted 6 May 15 Academic Editor: Zenghui Wang Copyright 15 W. Wang and G. Zhou. his 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. his paper is concerned with the regularity criterion of weak solutions to the three-dimensional Navier-Stokes equations with nonlinear damping in critical weak L q spaces. It is proved that if the weak solution satisfies (( u 1 q/(q ) + u q/(q ) Lq, )/(1 + ln(e + u L)))ds <, q>/, then the weak solution of Navier-Stokes equations with nonlinear damping is regular on (, ]. 1. Introduction In this study we consider the Cauchy problem of the threedimensional Navier-Stokes equations with the nonlinear damping t u+(u ) u+ π+ u r u Δu, together with the initial data u, (1) u (x, ) u, () where u(u 1 (x, t), u (x, t), u (x, t)) and π(x, t) denote the unknownvelocityfieldsandtheunknownpressureofthe fluid. u r u, r> is the nonlinear damping. Moreover, represents the gradient operator, ( 1,, ) () Δ i1 u (x, t) x i (4) denotes the Laplacian operator, and (u ) u u i1 i1 u (x, t) u i, x i u i (x, t) x i. he mathematical model (1) is from the resistance to the motion of the flows. It describes various physical situations such as drag or friction effects, porous media flow, and some dissipative mechanisms [1, ]. When the nonlinear damping term u r u in (1) disappears, the system reduces the classic Navier-Stokes equations [, 4] t u+(u ) u + π Δu, (6) u. In the mathematical viewpoint, therefore, Navier-Stokes equations with the nonlinear damping are a modification of the classic Navier-Stokes equations. here is a large literature on the well-posedness and large time behavior for solutions of Navier-Stokes equations with the nonlinear damping (see [1, 5, 6]). However it is not known whether the weak solution of Navier-Stokes equations with the nonlinear damping (1) is (5)

2 Mathematical Problems in Engineering regularorsmoothforagivensmoothandcompactlysupported initial velocity u. Fortunately, the regularity of weak solutions for Navier-Stokes equations with the nonlinear damping (1) can be derived when certain growth conditions aresatisfied.hisisknownasaregularitycriterionproblem. Recently, Zhou [7] studied the regularity criterion for weak solutions for Navier-Stokes equations with the nonlinear damping (1) in critical Lebesgue spaces. hat is, if a weak solution u of Navier-Stokes equations with the nonlinear damping (1) satisfies u L p (,;L q (R )) for p +, <p, (7) q then the weak solution is smooth on (,]. Since Navier-Stokes equations with the nonlinear damping (1) are a modification of the classic Navier-Stokes equations, it is necessary to mention some regularity criteria of weak solutions for Navier-Stokes equations and related fluid models [8, 9]. As for this direction, the first result of Navier- Stokes equations is studied by He [1]andimprovedbyDong and Zhang [11], Pokorý[1, 1], and Zhou [14]. One may also refer to some interesting regularity criteria on related fluid models (see [15] and the references therein). he main purpose of this paper is to investigate the regularity criteria of weak solutions with the aid of two components of velocity fields in critical weak L q space. o do so,werecallthedefinitionoftheweaksolutionofnavier- Stokes equations with the nonlinear damping (1). Definition 1. A measurable function u(x, t) is called a weak solution of Navier-Stokes equations with the nonlinear damping (1) on (,)if u satisfies the following properties: (i) u L (,;L (R )) and u L (,;L (R )); (ii) for any φ C (R [,))with φ (u t φ u φ+ φ:u u+ u R r uφ) dx dt R u φ () dx; (iii) uinthe distribution space D ((,) R ); (iv) u satisfies the energy inequality for t u (t) t L + u (x, s) R dx ds t + u (x, s) R p dx ds u L. he main result on the regularity criteria of the weak solutions of Navier-Stokes equations with the nonlinear damping (1) is now read. heorem. Suppose u L (R ) H 1 (R ) and u is a weak solution of Navier-Stokes equations with the nonlinear (8) (9) damping (1) in (,).Ifanytwocomponentsofvelocityfields satisfy u 1 q/(q ) L + q, u q/(q ) 1 + ln (e + u L ) ds <, q >, (1) then u is smooth on (,]. his result improves the earlier regularity criterion involving (7). Furthermore,heorem also implies the following regularity criterion for weak solutions of Navier- Stokes equations with the nonlinear damping (1). heorem. Suppose u L (R ) H 1 (R ) and u is a weak solution of Navier-Stokes equations with the nonlinear damping (1) in (,).Ifanytwocomponentsofvelocityfields satisfy u 1 q/(q ) L + q, u q/(q ) L ds <, q, q >, (11) then u is smooth on (,]. Remark 4. he main idea in the proof of heorem is borrowing from the argument of previous results on classic Navier-Stokes equations [16] and together with energy methods.. Preliminaries o start with, let us recall the definitions of some functional spaces. L q (R ) with 1 q is a Lebesgue space under the norm g L { ( R g (x) q 1/q dx), 1 q<, q { ess sup { g (x), q, (1) x R and H m (R ) the Hilbert space {g L (R ); m g L < }. (1) o define the Lorenz space L p,q (R ) with 1 p, q, g L p,q (R ) if and only if g L ( p,q g L p, where t q 1/q q/p dt (m (g, t)) t ) < sup (t (m (g, t)) 1/p )< t for 1 q<, for q, (14) m(f,t):m{x R : g (x) >t} (15) is Lebesgue measure of the set {x R : g(x) >t}.

3 Mathematical Problems in Engineering Actually Lorentz space L p,q (R ) may be alternatively defined by real interpolation (see riebel [17]) with L p,q (R )(L p 1 (R ),L p (R )) θ,q (16) 1 p 1 θ + θ, 1 p p 1 p 1 <p<p, <θ<1. In particular, g is equivalently to the norm (17) sup S 1/q 1 g (x) dx. (18) < S < S Furthermore, the definition implies the continuous relationship L q (R ) (R ), 1 <q<. (19) In fact it is easy to check and thus it is readily seen that x /q L q (R ), but x /q (R ). () then the a priori estimate holds true. sup u (t) L 4 C (5) <t< Proof of heorem 6. Multiplying both sides of the Navier- Stokes equations with the nonlinear damping (1) with Δu and integrating in R,wehave wherewehaveused 1 d dt u dx + Δu dx R R + (r 1) R u r u dx R u uδu dx, (6) R pδu dx R pδ (divu) dx. (7) For the right hand side of (6) we have R u uδu dx R u i i u j kk u j dx We also recall the Hölder inequality in Lorentz space which plays an important role in the next section. Lemma 5 (O Neil [18]). Let f L p 1,q 1 (R ) and g L p,q (R ) with 1 p 1, p, (1) 1 q 1, q. hen fg L p,q (R ) satisfies the Hölder inequality of Lorentz spaces where fg L C p,q f L p 1,q 1 g L p,q, () 1 p 1 + 1, p 1 p 1 q q 1 q () k (u i i u j ) k u j dx R k u i i u j k u j dx R + 1 R u i i ( k u j k u j )dx k u i i u j k u j dx R i1 j,k1 R k u i i u j k u j dx + k u u j k u j dx R j1 k1 (8). A Priori Estimates In this section we will prove a priori estimates for smooth solutionsof (1) described in the following. heorem 6. Let >, letu L (R ) H 1 (R ),andletu be a local smooth solution of the Navier-Stokes equations with the nonlinear damping (1).Ifu also satisfies (11),namely, u 1 q/(q ) L + q, u q/(q ) 1 + ln (e + u L ) ds <, q >, (4) + k1 m1 I m, R k u u k u dx wherewehaveusedthefactthatthedivergence-freecondition k1 k u k. (9)

4 4 Mathematical Problems in Engineering For I 1,wehave I 1 i1 j,k1 For I, similarly we obtain R k u i i u j k u j dx R ( u 1 + u ) u dx. I k u u j k u j dx j1 k1 R R ( u 1 + u ) u dx. Finally for I, applying the fact u u 1 u 1, I k u u k u dx k1 R k u ( u 1 u 1 ) k u dx k1 R R ( u 1 + u ) u dx. () (1) () Applying Hölder inequality and Young inequality, we have for the right hand side of () ( R u 1 + u ) u u dx (4) C( u 1 L + q, u ) u L. q/(q 1), Applying the Gagliardo-Nirenberg inequality in Lorentz spaces, that is, u L q/(q 1), C u (q )/q Δu /q, (5) L L thus we have from () 1 d dt u dx + Δu dx R R + (r 1) u R r u dx C( u 1 + u L 1 Δu L q, ) u (q )/q L +C( u 1 q/(q ) + u q/(q ) ) u L which implies Δu /q L (6) d dt u dx + Δu dx R R (7) C( u 1 q/(q ) L + q, u q/(q ) L ) q, u L. Plugging the estimates I i into the right hand side of (6), it follows that 1 d dt u dx + Δu dx R R + (r 1) u R r u dx ( C R u 1 + u ) u u dx. () In particular, d dt R u dx + R Δu dx C( u 1 q/(q ) L + q, u q/(q ) 1 + ln (e + u L ) ) (1 + ln (e + u L )) u L. Employing the Gronwall inequality, it follows that (8) u R dx R u dx exp { ( u 1 q/(q ) L + q, u q/(q ) 1 + ln (e + u L ) {1 + ln (e + u L)}) dt}. (9) Hencewehave We take the Gronwall inequality into account again to get ln (e + R u dx) ln (e + R u dx) ln {e + R u dx} C (u ) + ( u 1 q/(q ) L + q, u q/(q ) 1 + ln (e + u L ) {1 (4) exp { u 1 q/(q ) L + q, u q/(q ) 1 + ln (e + u L ) ds} (41) + ln (e + u L)}) dt. <.

5 Mathematical Problems in Engineering 5 Henceweobtainaprioriestimatesof u: 4. Proof of heorem ess sup u <t< R dx<c(u ). (4) Under the a priori estimates in heorem 6, wenowarein apositiontocompletetheproofofheorem. Sinceu L (R ) H 1 (R ) with u, by the existence theorem of local strong solutions to the Navier-Stokes equations with nonlinear damping r>, there exist a constant > anda unique smooth solution u of (1) satisfying (refer to [19]) u BC([, );H 1 ), u (x, ) u. (4) Note that the weak solution satisfies the energy inequality (9). It follows from the weak-strong uniqueness criterion that u u on [, ). (44) husitissufficienttoshowthat.supposethat <. Without loss of generality, we may assume that is the maximal existence time for u. Since u uon [, ),bythe assumptions (11), u 1 q/(q ) L + q, u q/(q ) 1 + ln (e + u L ) ds <, q >. (45) herefore it follows from (5) that the existence time of u can be extended after t which contradicts the maximality of t. his completes the proof of heorem. Conflict of Interests he authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments hisworkissupportedbythenationalnaturalsciencefoundation of China (Grant no. 6144), the Natural Science Foundation of Hubei Province (Grant no. 1CFC11), and the Project of the Education Department of Hubei Province (Grant nos. 19 and D181). References [1] D. Bresch and B. Desjardins, Existence of global weak solutions foradviscousshallowwaterequationsandconvergencetothe quasi-geostrophic model, Communications in Mathematical Physics,vol.8,no.1-,pp.11,. [] Z.-Q. Luo, Optimal convergence rates for solutions of the monopolar non-newtonian flows, Abstract and Applied Analysis,vol.14,ArticleID7879,6pages,14. [] O. A. Ladyzhenskaya, he Mathematical heory of Viscous Incompressible Fluids,GordenBrech,NewYork,NY,USA,1969. [4] P. G. Lemarié-Rieusset, Recent Developments in the Navier- Stokes Problem, Chapman & Hall/CRC, Boca Raton, Fla, USA,. [5] X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, Mathematical Analysis and Applications,vol.4,no.,pp , 8. [6]Y.Jia,X.Zhang,andB.-Q.Dong, heasymptoticbehavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping, Nonlinear Analysis. Real World Applications,vol.1,no.,pp ,11. [7] Y. Zhou, Regularity and uniqueness for the D incompressible Navier-Stokes equations with damping, Applied Mathematics Letters, vol. 5, no. 11, pp , 1. [8] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis,vol.9,pp ,196. [9] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, vol. 19, no., pp , 6. [1] C. He, Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electronic Differential Equations,no.9,1pages,. [11] B.-Q. Dong and Z. Zhang, he BKM criterion for the D Navier-Stokes equations via two velocity components, Nonlinear Analysis: Real World Applications, vol.11,no.4,pp , 1. [1] P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Applications of Mathematics,vol.49,no.5,pp.48 49,4. [1] M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electronic Journal of Differential Equations,vol.1,pp.1 8,. [14] Y. Zhou, A new regularity criterion for weak solutions to the Navier-Stokes equations, Journal de Mathématiques Pures et Appliquées, vol. 84, no. 11, pp , 5. [15] B.-Q. Dong and Z.-M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, Mathematical Physics,vol.5,no.1,ArticleID155,1pages, 9. [16] B.-Q. Dong and Z.-M. Chen, Regularity criterion for weak solutions to the D Navier-Stokes equations via two velocity components, JournalofMathematicalAnalysisandApplications,vol.8,no.1,pp.1 1,8. [17] H. riebel, Interpolation heory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, he Netherlands, [18] R. O Neil, Convolution operators and L(p, q) spaces, Duke Mathematical Journal,vol.,pp.19 14,196. [19] Y. Giga, Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system, Differential Equations,vol.6,no.,pp.186 1,1986.

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