GENERIC SOLVABILITY FOR THE 3-D NAVIER-STOKES EQUATIONS WITH NONREGULAR FORCE

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1 Electronic Journal of Differential Equations, Vol. 2(2), No. 78, pp ISSN: UR: or ftp ejde.math.txstate.edu (login: ftp) GENERIC SOVABIITY FOR THE 3-D NAVIER-STOKES EQUATIONS WITH NONREGUAR FORCE JIHOON EE Abstract. We show that the existence of global strong solutions for the Navier-Stokes equations with nonregular force is generically true. Similar results for equations without the nonregular force have been obtained by Fursikov [5]. Our main tools are the Galerkin method and estimates on its solutions. 1. Introduction We are interested in the generic solvability for the 3-dimensional Navier-Stokes equations with nonregular force on the periodic domain T 3 [, ). u g + (u )u + p = ν u + f + t t, (1.1) div u =, (1.2) u(x, ) = u (x), (1.3) where u is the fluid velocity vector field, p is the scalar pressure, ν is the positive viscosity constant and f + g t s the external force. u is a given initial data. The nonregular part is denoted by g t. We assume g C([, ); V 2 ) which means g(t) is a continuous function in V 2, where the space V 2 is defined below. Since we only consider the periodic domain T 3 = [, 2π] 3, every function can be regarded as a periodic vector field with period 2π, i.e., u(x 1 + 2π, x 2, x 3 ) = u(x 1, x 2, x 3 ), etc. For this above Navier-Stokes equations with nonregular force, the existence of the weak solution was shown in []. Recently, Flandoli and Romito[3] proved the paths of a martingale suitable weak solution for the Navier-Stokes equations with nonregular force have a set of singular points of one-dimensional Hausdorff measure zero. Also the stochastic Navier- Stokes equations have been intensively studied by many authors (see [1], [2], [6] and references therein). One of the most important problems in nonlinear partial differential equations is to show existence of global strong solution for three-dimensional Navier-Stokes equations or to construct an example of the finite blow-up of the solution for the three-dimensional Navier-Stokes equations. Although, it is still far 2 Mathematics Subject Classification. 35A5, 76N1. Key words and phrases. Stochastic Navier-Stokes equations, generic solvability. c 2 Texas State University - San Marcos. Submitted February 2, 2. Published June, 2. Supported by the post-doctoral Fellowship Program of KOSEF. 1

2 2 JIHOON EE EJDE-2/78 from being proved the global existence, it is known to be generically true for (1.1) (1.3) without the nonregular part g t (see [5] and [7]). In this paper, we show that generic solvability is still true even with the nonregular force. We assume f and g are divergence free vector fields for simplicity. In the following, we consider the Banach spaces p (, T ; B) for any Banch spaces B, i.e. we say f p (, T ; B) if and only if f p (,T ;B) < (we use the same notation for the Banach space of 3-dimensional vector fields with the Banach space for scalar valued function for simplicity). We denote <T < p (, T ; B) is denoted by p loc (, ; B). et Hm be the usual Sobolev space(see [7]). Following the notation in [7], we denote the space of 2 divergence-free vector fields by H, the space of H m divergence free vector fields by V m (V 1 will be denoted by V for simplicity and convention). We define the projection operator P div as the projection to the divergence free vector fields. Note that { e i e ik x i = 1, 2, 3, k Z 3 }, where e i is an i-th standard unit vector, is a complete orthonormal basis for 2. Hence projection operator, P div, is defined as ( P div ( ae ik x ) = a k k ) a e ik x. k et α i (k) = P div ( e i e ik x ) 2. Hence K = { 1 α P i(k) div( e i e ik x ) : k Z 3, i = 1, 2, 3 } is a complete orthonormal basis for H. Define B(u, v) = P div (u )v, Λ 2 u = P div u, where P div is the 2 projection operator as above. By projecting (1.1) (1.2) to the divergence-free vector field, we obtain du + νλ2 u(t) B(u, u) = f(t) + dg(t). (1.) For the three-dimensional Navier-Stokes equations with regular force, Fursikov[5] and Temam [7] proved that for any initial data u V, there exists a set F which is included in 2 (, T ; H) and dense in q (, T ; V ) with 1 q < 3 (V is the dual space of V ), such that for every external force f F, the equations have a unique strong solution u. Using methods similar to those developed in [], [5] and [7], we obtain the following generic existence and uniqueness of the Navier-Stokes equations with nonregular force. Theorem 1.1. Assume that the initial data is u V. We also assume that f 2 loc (, ; H). There exist f m 2 loc (, ; H) satisfying f m f in q loc (, ; 6/5 ) for all q satisfying 1 q < 3 such that (1.) corresponding to u and f m possesses a unique strong solution in loc (, ; V ) 2 loc (, ; V 2 ). We remark that since 6/5 V, Theorem 1.1 can be regarded as a slight generalization of the results in [7]. 2. Proof of main Theorem For the proof of Theorem 1.1, we use the methods developed in [], [5], and [7]. First, we consider the following Galerkin approximation of the system (1.1) (1.2). du m + νλ 2 u m P m B(u m, u m ) = P m f + P m dg, u m () = P m u. (2.1)

3 EJDE-2/78 GENERIC SOVABIITY FOR THE 3-D NAVIER-STOKES EQUATIONS 3 The projection onto the space spanned by { 1 α P i(k) div( e i e ik x ) k m} is denoted by P m. Note that (2.1) is equivalent to the integral equation u m (t) + ν = u m () + Λ 2 u m (s)ds P m f(s)ds + P m g(t). P m B(u m (s), u m (s))ds (2.2) Using contraction mapping argument, we can show, local in time, existence of solution u m for (2.2). Following the argument given in [], we consider the following auxiliary equation for given z V. z m (t) + ν Λ 2 z m (s)ds = P m z + P m g(t), t. (2.3) This equation has a unique solution, which is continuous with values in V and global in time. We show z m converges in C([, ); V ) 2 loc (, ; V 2 ). (2.3) is equivalent to z m (t) = e νtλ2 P m z + P m g(t) ν Note that for c(γ) = max x x 2γ e x, we have Λ 2 e ν(t s)λ2 P m g(s)ds. Λ 2γ e tλ2 (H) c(γ) e t 2 λ1 t γ, (2.) where F (H) = sup f H 1 F (f) H and λ 1 is the smallest eigenvalue of Λ 2. Hence Λz m (t) = e νtλ2 ΛP m z + ΛP m g ν Λ 2 2ɛ e ν(t s)λ2 Λ 1+2ɛ P m g(s)ds is a continuous function in H. Since z m C([, ); V ), it follows that z m converges in C([, ); V ). We let z be the limit of z m. Set ρ m (t) = e νtλ2 P m z Then ρ m satisfies the linear equation νλ 2 e ν(t s)λ2 P m g(s)ds. dρ m + νλ 2 ρ m = νλ 2 P m g, ρ m () = P m z. It follows that d Λρ m ν Λ2 ρ m 2 2 ν Λ2 P m g 2 2. Therefore, integrating we have ν Λ 2 ρ m (s) 2 2ds ΛP mz 2 + ν Λ 2 P 2 m g(s) 2 2ds. By taking subsequence, ρ m converges in 2 loc(, ; V 2 ). et ρ be the limit of ρ m. Since we have g 2 loc (, ; V 2 ), we have z m = ρ m + P m g converges to z := ρ + g in 2 loc(, ; V 2 ). Now define ũ m = u m z m. et ũ m satisfy the equations dũ m + νλ 2 ũ m P m B(ũ m + z m, ũ m + z m ) = P m f, t [, ), (2.5) ũ m () = u m () P m z. (2.6)

4 JIHOON EE EJDE-2/78 In [], the boundedness of ũ m in (, T ; 2 ) 2 (, T ; V ) is shown, i.e., ũ m (t) 2 + ν Λũ 2 m (s) 2 2ds ũ m () 2 + C ũ 2 m 2 2 z m 8 + z m + z m 2 + P mf 2 2 V ds. Thus νλ 2 ũ m 2 loc (, ; V ) /3 loc (, ; V ). Note that ũ m ũ m V C ũ m 1/2 2 ũ m 3/2 V, ũ m z m V C ũ m 2 z m 1/2 V z m 1/2 V 2, z m ũ m V C z m 1/2 2 z m 1/2 V ũ m V, z m z m V C z m 1/2 2 z m 3/2 V. Hence, we have P m B(ũ m + z m, ũ m + z m ) /3 loc (, ; V ). Thus we conclude that ũ m t is bounded in /3 loc (, ; V ). Since {ũ m } is bounded in W 1, 3 loc([, ); V ), we have ũ m converges strongly in 2 loc(, ; H). Thus there exists u such that u m converges to u in 2 (, T ; V 1 ɛ ) and 1 ɛ (, T ; H) for any small ɛ >. Since u m is a finite Galerkin approximation, we have u m V m C(m) u m 2. Hence u m is in (, T ; V ) 2 (, T ; V 2 ). Thus we have showed the following lemma. emma 2.1. If u H, then u m converges in 2 (, T ; V 1 ɛ ) and 1 ɛ (, T ; H) for any small ɛ > as m. The sequence u m is bounded in (, T ; H) 2 (, T ; V ). Furthermore, u m is in loc (, ; V ) 2 loc (, ; V 2 ). To proceed further, we consider the linear equations v m t + νλ 2 v m = (I P m )f + (I P m ) g t, v m () = (I P m )u, (2.7) where I P m is the projection onto the space spanned by { 1 α i(k) P div( e i e ik x ) k > m}. emma 2.2. If u H, then there exist a unique solution v m of (2.7) in the space 2 (, T ; V ) (, T ; H) and v m in 2 (, T ; V ) (, T ; H) as m. Furthermore, if u V, then v m 2 (, T ; V 2 ) (, T ; V ) and v m in 2 (, T ; V 2 ) (, T ; V ) as m. Proof. Since (2.7) is a simple linear dissipative system, the existence and uniqueness are immediate consequence of the standard results. Similarly to the proof of emma 2.1, we can prove v m 2 (, T ; V ) (, T ; H) and v m in 2 (, T ; V ) (, T ; H). We only provide the proof of the second claim of this lemma. Equation (2.7) is equivalent to the integral equation v m (t) =e νtλ2 (I P m )u + (I P m )g(t) + νλ 2 e ν(t s)λ2 (I P m )g(s) ds. e ν(t s)λ2 (I P m )f(s) ds

5 EJDE-2/78 GENERIC SOVABIITY FOR THE 3-D NAVIER-STOKES EQUATIONS 5 Since Λv m (t) =e νtλ2 Λ(I P m )u + Λ(I P m )g(t) + νλ 2(1 ɛ) e ν(t s)λ2 Λ 1+2ɛ (I P m )g(s) ds Λe ν(t s)λ2 (I P m )f(s) ds is a continuous function in H (see (2.)), we have v m C([, T ); V ). Set h m (t) =e νtλ2 (I P m )u + e ν(t s)λ2 (I P m )f(s) ds, νλ 2 e ν(t s)λ2 (I P m )g(s) ds i.e., v m (t) = (I P m )g(t) + h m (t). It follows that h m satisfies the equation dh m + νλ 2 h m = νλ 2 (I P m )g + (I P m )f. Taking inner product with Λ 2 h m in 2 produces 1 d 2 Λh m ν Λ2 h m (t) 2 ν 2 2 Λ2 h m C Λ2 (I P m )g 2 + C (I P m)f Integrating over [, T ), we obtain Λ 2 h m (t) 2 + ν Λ 2 h 2 m (t) 2 2 Λ(I P m )u 2 + C Λ 2 (I P 2 m )g(t) C (I P m )f(t) 2 2. Thus h m 2 (, T ; V 2 ). Since g 2 (, T ; V 2 ), we obtain v m 2 (, T ; V 2 ). ebesgue s dominated convergence Theorem produces v m in 2 (, T ; V 2 ) (, T ; V ) as m. For the uniqueness, if there exists two solutions vm 1 and vm 2 of (2.7), then we denote by ρ(t) = vm(t) 1 vm(t). 2 We have the following deterministic equations. dρ + νλ2 ρ =, ρ() =. Thus we have ρ(t) =, which completes the proof. From emma 2.1, we have u m is in (, T ; V ) 2 (, T ; V 2 ). Now for every m, we consider also the solution v m of the linearized problems (2.7) We then set w m = u m + v m and observe w m satisfies w m 2 (, T ; V 2 ) (, T ; V ) if u V. By adding two equations, we have the following Navier-Stokes equations with nonregular force dw m + νλ 2 w m B(w m, w m ) = f m + dg, (2.8) where f m = f B(v m, v m ) B(v m, u m ) B(u m, v m ) (I P m )B(u m, u m ). et w m be another solution of (2.8). Then by letting ρ m = w m w m, we have d ρ m + νλ 2 ρ m = B( ρ m, w m ) + B( w m, ρ m ).

6 6 JIHOON EE EJDE-2/78 Hence we have d ρ m 2 2 2ν Λ ρ m C ρ m 6 w m 3 ρ m 2 ν Λ ρ m C w m V 2 w m V ρ m 2 2. Using Gronwall s inequality, we have ρ m (t) 2 2 ρ m() 2 2 exp (C ) w m V 2 w m V ds. Since w m 2 (, T ; V 2 ) (, T ; V ), it is clear that w m is the unique solution in 2 loc (, ; V 2 ) loc (, ; V ). Hence for the proof of Theorem 1.1, it is sufficient to show that f m converges to f in q (, T ; 6/5 ) with 1 q < 3. Proof of Theorem 1.1. For the remaining of this proof, we use only the weaker assumption u H instead of u V. Since we have v m in 2 (, T ; V ) (, T ; H) as m, it is clear that B(v m, v m ) in q (, T ; 6/5 ) by the inequalities B(v m, v m ) q C 6/5 C ( C v m v m q 3 v m q V v m q/2 v 2 m 3 2 q V 2q 3q 2 q ) 3q ( CT 3q 3q v m (,T ;H) v m 3q/2 It is well known from the interpolation inequality that Then ) 3q/ v m 2 V 2 (,T ;V ). B(u m, v m ) 6/5 C u m 6 v m 3/2 C u m 2 v m 1/2 2 v m 1/2 V. C B(u m, v m ) q 6/5 u m q v 2 m q/2 v 2 m q/2 V ( ) q/2 ( C u m 2 2 v m Similarly, we have 2q 3q 2 ) 3q ( CT 3q u m q 2 (,T ;V ) v m q/2 (,T ;H) v m q/2 2 (,T ;V ). B(v m, u m ) q 6/5 v m q 3 u m q 2 ) q/ v m 2 V CT 3q u m q 2 (,T ;V ) v m q/2 (,T ;H) v m q/2 2 (,T ;V ). To complete the proof, it is sufficient to show that (I P m )B(u m, u m ) in q (, T ; 6/5 ) as m.

7 EJDE-2/78 GENERIC SOVABIITY FOR THE 3-D NAVIER-STOKES EQUATIONS 7 First we recall that u m converges to its limit u from emma 2.1. We rewrite u as an expansion by the complete orthonormal basis K, i.e., u = k Z 3 i=1,2,3 u i k 1 α i (k) P div( e i e ik x ) =: u k e k (x), k Z 3 where u i k is the corresponding coefficient, and for simplicity of notation we introduced the right-hand-side. Then we have (I P m )B(u, u) = (I P m )P div ((u )u) = (I P m )P div ( u k e k (x) k )u k e k (x) k Z 3 k Z 3 ( ( ) ) = P div u k e k (x) u k e k (x) k [ m 2 ] k ( ( + P div u k e k (x) ) ) u k e k (x), k k [ m 2 ] (2.9) where [a] denotes the largest integer less than or equal to a, and h denotes the summation over all h Z 3 satisfying h+j > m when j [ m 2 ]. Using the identity (2.9), we obtain (I P m )B(u, u) C u k e k (x) 3 u 2 + C u 6 u k e k (x) 3/2 k [ m 2 ] k m 2 C u k e k (x) 1/2 u 3/ k [ m 2 ] We obtain that for any q < /3, (I P m )B(u, u) q 6/5 C u 3q 2 2 u k e k (x) q/2 2 k [ m 2 ] ( ) 3q/ ( C u 2 2 u k e k (x) k [ m 2 ] Since K is a complete orthonormal basis for H, we have 2q 3q 2 (I P m )B(u, u) in q (, T ; 6/5 ) as m. ) 3q Thus it only remains to prove that B(u m, u m ) B(u, u) in q (, T ; 6/5 ). From emma 2.1, we have u m u in 2 (, T ; V 1 ɛ ) 1 ɛ (, T ; H) for any ɛ >. We complete the proof by showing that B(u m u, u m ), B(u, u m u) in q (, T ; 6/5 ) for all q < /3..

8 8 JIHOON EE EJDE-2/78 By the interpolation inequality, we have for ɛ < 1/2, Setting r = 1 2ɛ 1 2(1 ɛ) 2(1 ɛ) B(u m u, u m ) 6/5 C u m u u 2 m u V u 1 ɛ m 2, B(u, u m u) 6/5 C u 6 u m u 1/2 2 u m u 1/2 V. 2q(1 2ɛ) 3q 2ɛ(2 q), we have q 2 + q q(1 2ɛ) + (1 ɛ) 2r(1 ɛ) = 1. By Hölder s inequality and emma 2.1, we obtain B(u m u, u m ) q 6/5 ( C u m u r 2 ) q(1 2ɛ) 2r(1 ɛ) ( u m u 2 V 1 ɛ ) q (1 ɛ) ( u m 2 2 ) q/2 which approaches zero as m approaches. Again using Hölder s inequality(note that q 2 + q + 3q = 1), we have B(u, u m u) q 6/5 ( ) q/2 ( C u 2 2q V u m u 2 This completes the proof of Theorem q ) 3q ( q/ u m u 2 V ). Acknowledgements. The author would like to thank Professor D. Chae for the many helpful suggestions on the generic solvability, and Professor Ya Sinai for his helpful discussions. The author also wants to thank the anonymous referee for his/her suggestions and corrections. References [1] G. Da Prato and J. Zabczyk, Ergodicity for Infinite dimensional systems, ondon Mathematical Society ecture Note Series, 229. Cambridge University Press, Cambridge, [2] W. E, J. C. Mattingly, and Y.G. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys., 22 (21), pp [3] F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. AMS., 35 (22), pp [] F. Flandoli and B. Schmalfuß, Weak solutions and attractors for the three-dimensional Navier-Stokes equations with nonregular force, J. Dynamical and Differential Equations, 11 (1999), pp [5] A. V. Fursikov, On some problems of control, Dokl. Akad. Nauk SSSR, (198), pp [6] M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Statist. Physics, 11(2), pp [7] R. Temam, Navier-Stokes equations: Theory and Numerical analysis, North-Holland, Amsterdam, 198. Max-Planck institute for mathematics in the sciences, Insel str eipzig, Germany address: zhlee@math.snu.ac.kr