ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS

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1 ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS Abdelhafid Younsi To cite this version: Abdelhafid Younsi. ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS. 4 pages <hal v5> HAL Id: hal Submitted on 4 Dec 212 (v5), last revised 26 Dec 212 (v6) HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS ABDELHAFID YOUNSI Abstract. In this paper, we give a new regularity criterion on the uniqueness results of weak solutions for the 3D Navier-Stokes equations. Moreover, we improve the known condition on the uniqueness of weak solutions for the 3D Navier-Stokes equations obtained by Beirao da Viega [1]. 1. Introduction Two of the profound open problems in the theory of three dimensional viscous flow are the unique solvability theorem for all time and the regularity of solutions. For the three-dimensional Navier-Stokes system weak solutions of problem are known to exist by a basic result by J. Leray from 1934 [5], it is not known if the weak solution is unique or what further assumption could make it unique only the uniqueness of weak solutions remains as an open problem. There are many results that give sufficient conditions for regularity of a weak solution [1, 2, 3, 4, 8, 1, 13]. In this paper, we are interested in the problem of finding sufficient conditions for weak solutions of 3D Navier-Stokes equations such that they become regular and unique. That u L 4 (,T;V 1 ) gives uniqueness in the class of weak solutions satisfying the energy inequality is well known (Beiraoda Viega 1995) [1]. In our main result, we show that u L q (,T;V 1 ) gives uniqueness in the class of weak solutions satisfying the energy inequality for all q This particular result is a significant improvement of those in [1]. 2. Preliminary We denote by Hper m (Ω), the Sobolev space of L-periodic functions endowed with the inner product (u,v) = (D β u,d β v) L 2 (Ω) and the norm u m = ( D β u 2 L 2 (Ω) )1 2. β m β m We define the spaces V m as completions of smooth, divergence-free, periodic, zeroaverage functions with respect to the Hper m norms. V m denotes the dual space of V m and V denotes the space V. We denote by A the Stokes operator Au = u for u D(A). We recall that the operator A is a closed positive self-adjoint unbounded operator, with D(A) = {u V, Au V }. We have in fact, D(A) = V 2. Now define the trilinear 2 Mathematics Subject Classification. 35Q3, 35A2, 35D3. Key words and phrases. Navier-Stokes equations - weak solutions - uniqueness. 1

3 2 ABDELHAFID YOUNSI form b(.,.,.) associated with the inertia terms b(u,v,w) = 3 v j u i w j dx. (2.1) x i i,j=1 The continuity property of the trilinear form enables us to define (using Riesz representation theorem) a bilinear continuous operator B(u,v); V V V will be defined by Recall that for u satisfying.u = we have Ω B(u,v),w = b(u,v,w), w V. (2.2) b(u,u,u) = and b(u,v,w) = b(u,w,v). (2.3) Hereafter, c i N, will denote a dimensionless scale invariant positive constant which might depend on the shape of the domain. We recall some inequalities that we will be using in what follows. Young s inequality Poincaré s inequality ab σ p ap + 1 b q,a,b,σ >,p > 1,q = p qσ q p p 1. (2.4) λ u 2 1 for all u V, (2.5) where λ 1 is the smallest eigenvalue of the Stokes operator A. 3. Navier-Stokes equations The conventional Navier-Stokes system can be written in the evolution form u +νau+b(u,u) = f, t >, t div u =, in Ω (, ) and u(x,) = u, in Ω. (3.1) We recall that a Leray weak solution of the Navier-Stokes equations is a solution which is bounded and weakly continuous in the space of periodic divergence-free L 2 functions, whose gradient is square-integrable in space and time and which satisfies the energy inequality. The proof of the following theorem is given in [6, 9]. Theorem 3.1. Let Ω R n, n = 2,3 and f L 2 (,T;V 1 ), u V be given. Then there exists a weak solution u of (3.1) wich satisfies u L 2 (,T;V 1 ) L (,T;V ), T >. Furthermore if n = 2, u is unique. In this paper we will be especially interested in the case where n = 3. We will prove the weak solution is unique in L 4 (,T;V1 ) for all θ 1. Then our main 4 theorem is as follows Theorem 3.2. Let u and v two suitable weak solutions of the 3D Navier-Stokes equations (3.1 ) with u = v, if u and v are in L 4 (,T;V1 ) then u = v for all θ 1 4.

4 UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS 3 Proof. Let w denote their difference. We obtain the equation for w = u v as t w+aw+b(v,v) B(u,u) =, with divw =. (3.2) Taking the scalar product of (3.2 ) with w, it follows that 1 d 2dt w 2 +ν A 1 2 w 2 = b(w,w,u). (3.3) Using the Holder s inequality, we have b(w,w,u) c 1 w 2 L 4 A 1 2 u L 2. (3.4) A straightforward application of Peetr s theorem [6, 7] H (1 θ)m (Ω) L q θ (Ω), 1 = 1 q θ 2 (1 θ)m, (3.5) n if we consider m = 1 then q θ 4 for θ 1, inequality (3.4 ) means that 4 b(w,w,u) c 2 w 2(1 θ) 1 w 2θ L 2 u 1. (3.6) Hence, applying the Poincaré inequality gives b(w,w,u) c 3 w (1 θ) 1 w 1 w θ L 2 u 1. (3.7) Using Young s inequality on the right-hand side, we obtain d dt w 2 +ν w 2 1 ν ( 2 w 2 1 +c 4 w 1 w θ L 2 u 1. (3.8) Consequently, d dt w 2 c 4 w 2θ 2 w 1. (3.9) Applying Gronwall s inequality on (3.14 ) and using the fact that yields w(t) 2 c 6 w() 2 ) 2 2θ 2, (3.1) θ +1 w 2 Using the Cauchy Schwarz inequality, we obtain ( ) 1 ( w 2 1 ds w 4 2 t 1 ds Since we get 1 ds. (3.11) ) 1 u ds. (3.12) w 1 v 1 + u 1, (3.13) w 4 1 ds Therefore, for 1 < p <, we have Then, because ( v 1 + u 1 ) 4 ds. (3.14) (a+b) p 2 p (a p +b p ), for a, b. (3.15) in (3.14 ) it follows that ( w 4 1 ds v u 4 1 ) ds. (3.16)

5 4 ABDELHAFID YOUNSI By combining all of the above estimates in (3.11 ), we get ( ( ) ) 1 ( w(t) 2 c 6 w() 2 v u 4 2 t 1 ds ) 1 u ds. (3.17) The assumption on u and v guarantees that the right hand side is finite for all t T. On the other hand w() =, so we get w(t) = on [,T], which completes the proof. Since our condition can be verified for suitable weak solutions to the Navier- Stokes system it improves the known results substantially. As a continuation of the previous work [11, 12] on the Navier-Stokes system, this paper introduces a new direction for the study of the uniqueness of weak solutions. References [1] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in R n, Chinese Ann. Math. Ser. B 16 (1995), [2] C. Cao, E. S. Titi, Regularity criteria for the three dimensional NavierStokes equations. Indiana Univ. Math. J. 57, 6, (28), [3] C. R. Doering, The 3D Navier-Stokes problem. Annu. Rev. Fluid Mech. 41(29), [4] I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction. J. Math. Phys. 48, 6523, (27). [5] J. Leray, Sur le mouvement d un liquide visqueux emplissant l espace., Acta Mathematica, 63 (1934), [6] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod Gauthier-Villars, Paris, [7] J. Peetre, Espaces d interpolation et théorème de Sobolev. Ann. Inst. Fourier, 16 (1966), [8] J. Serrin, On the interior regulariy of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, (1962), [9] R. Temam, Navier-Stokes Equations. North-Holland Pub. Company, Amsterdam, [1] J. Wolf, A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, Springer-Verlag, New York, USA (21) [11] A. Younsi, Effect of hyperviscosity on the Navier-Stokes turbulence, Electron. J. Diff. Equ. 11 (21), [12] A. Younsi, Effect of hyperviscosity on the geometry of the vorticity, NoDEA Nonlinear Differential Equations and Appl. 212, DOI 1.17/s (On line first). [13] Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component.j. Math. Phys. 5, (29). Department of Mathematics and Computer Science, University of Djelfa, Algeria. address: younsihafid@gmail.com