The Navier-Stokes Equations with Time Delay Werner Varnhorn Faculty of Mathematics University of Kassel, Germany AMS: 35 (A 35, D 5, K 55, Q 1), 65 M 1, 76 D 5 Abstract In the present paper we use a time delay ε > for an energy conserving approximation of the nonlinear term of the non-stationary Navier-Stokes equations. We prove that the corresponding initial value problem (N ε ) in smoothly bounded domains G R 3 is well-posed. Passing to the limit ε we show that the sequence of stabilized solutions has an accumulation point such that it solves the Navier-Stokes problem (N ) in a weak sense (Hopf). 1 Introduction Let T > be given and G R 3 be a bounded domain with a smooth compact boundary G. In G we consider a non-stationary viscous incompressible fluid flow and assume that it can be described by the Navier-Stokes equations t v ν v + p + (v )v = F, v G =, v t= = v. v =, (N ) These equations represent a system of nonlinear partial differential equations concerning four unknown functions: the velocity vector v = (v 1 (t, x), v 2 (t, x), v 3 (t, x)) and the (scalar) kinematic pressure function p = p(t, x) of the fluid at time t (, T ) in position x G. The constant ν > (kinematic viscosity), the external force density F, and the initial velocity v are given data. In (N ) t v means the partial derivative with respect to the time t, is the Laplace operator in R 3, and = ( 1, 2, 3 ) the
gradient, where j = x j denotes the partial derivative with respect to x j (j = 1, 2, 3). From the physical point of view, the nonlinear convective term (v )v is a result of the total derivative of the velocity field. Here the operator (v ) has to be applied to each component v j of v. In the fourth equation v = 1 v 1 + 2 v 2 + 3 v 3 defines the divergence of v, which vanishes due to the incompressibility of the fluid. Finally, the no-slip boundary condition v G = expresses that the fluid adheres to the boundary G. Let us assume that smooth data are given without any smallness assumptions. Then the problem to construct a solution v, p of (N ), which is uniquely determined and exists globally in time, has not been solved in the 3-d case considered here (see for example [3], [4]). Consequently, there is no globally stable approximation scheme for (N ) up to now. In the present paper we use a smoothing procedure for the Navier-Stokes equations based on a time delay in the nonlinear term: Setting ε = T/N > (N N) and we replace (N ) by v ε ( t ) = v(t ε) t v ν v + p + (v ε )v = F, v =, (N ε ) v G =, v t = v. Here the initial value v from (N ) has to be extended to the time interval [ ε, ] in a suitable way. We show that these equations have strongly H 2 (G) - continuous global unique solutions (see the next section for the notations): Theorem 1: Let f H 1 (, T, H(G)), ε >, and v C([ ε, ], H 2 (G) V (G)) with t v C([ ε, ], H(G)) L 2 (G)( ε,, V (G)) be given. Then there is a unique solution v, p of the equations (N ε ) such that v C([, T ], H 2 (G) V (G)), t v C([, T ], H(G)) L 2 (, T, V (G)), p C([, T ], L 2 (G)). Moreover, v satisfies for all t [, T ] the energy equation and the estimates v(t) 2 + 2ν v(s) 2 ds = v 2 + 2 (F (s), v(s))ds (1.1) t v(t) 2 dt K, v(t) 2 K, t v(t) K (t [, T ]) (1.2) with some constant K = K(G, T, ε, ν, F, v ). 2
We investigate the behavior of the solution of (N ε ) if the delay ε tends to zero. Up to now, without any additional smallness assumptions on the data, the only solutions of the Navier-Stokes system (N ), for which global existence (but not uniqueness) for all time t [, T ] has been proved, are weak solutions. Let us recall the definition of a weak solution of the Navier-Stokes equations (N ) (compare also [5], p. 22, [6], p. 173, [7], p. 72, [9], p. 28): Definition: Let v H(G) and f L 2 (, T, H(G)) be given. function v L 2 (, T, V (G)) L (, T, H(G)) Then we call a a weak solution of the Navier-Stokes equations (N ), if v : [, T ] H(G) is weakly continuous, if v(t) v as t (strong H(G) continuity at t = ), and if for all test function Φ C ((, T ) G) with Φ(t) C,σ (G) the following identity holds: { (v(t), t Φ(t)) + ν( v(t), φ(t)) ((v(t) )Φ(t), v(t)) (F (t), Φ(t))}dt =. We show that weak solutions can be constructed by the solution of (N ε ), if the delay parameter ε = T/N tends to zero (T remains fixed): Theorem 2: Let T > be fixed. For ε N = T/N > (N N) let v N denote the solution constructed in Theorem 1. Then the sequence (v N ) N N has an accumulation point v, which solves the Navier-Stokes equations (N ) weakly and satisfies for all t [, T ] the energy inequality v(t) 2 + 2ν v(s) 2 ds v 2 + 2 (F (s), v(s))ds. (1.3) 2 Notations and Auxiliaries Throughout this paper, G R 3 is a bounded domain having a compact boundary G of class C 2. In the following, all functions are real valued. As usual, C (G) denotes the spaces of smooth functions defined in G with compact support, and L p (G) is the Lebesgue space with the norm f,p (1 p ; if p = we use f instead of f, ). The space L 2 (G) is a Hilbert space with scalar product and norm defined by (f, g) = f(x) g(x)dx, f = f,2 = (f, f) 1 2, respectively. G 3
The Sobolev space H m (G) (m N = {, 1, 2,...}) is the space of functions f such that α f L 2 (G) for all α = (α 1, α 2, α 3 ) N 3 with α = α 1 + α 2 + α 3 m. Its norm is denoted by where k = f m = f H m (G) = x k α m 1 2 α f 2 (k = 1, 2, 3) is the distributional derivative., α = α 1 1 α 2 2 α 3 3, The completion of C (G) with respect to the norm m is denoted by H m (G). In particular, we have H (G) = H (G) = L 2 (G). For the corresponding spaces of vector functions u = (u 1, u 2, u 3 ) and their norms we use the same symbols as in the scalar case, hence, for example, (u, v) = 3 (u k, v k ), u = (u, u) 1 2 = k=1 G u(x) 2 dx 1 2, where u(x) = (u 1 (x) 2 + u 2 (x) 2 + u 3 (x) 2 ) 1 2 is the Euclidian norm of u(x) R 3. We shall often use the Poincare inequality u 2 c u 2. (2.4) Here the constant λ 1 = c 1 is the smallest eigenvalue of the Laplace operator in G with vanishing boundary condition, and it depends on the size of the domain G ([6], p. 11). We will also use the estimates (see [1], p. 97) u c 1 u 2, v,p = c 2 v (1 p 6), (2.5) valid for functions u H 2 (G) and v H 1 (G), respectively. The completion of the set of vector functions C,σ(G) = {u C (G) div u = } with respect to the norm and 1 are basic spaces for the treatment of the Stokes and the Navier-Stokes equations. We denote theses spaces by H(G) and V (G), respectively. In H 1 (G) and V (G) we also use ( u, v) = 3 ( k u j, k v j ), u = ( u, u) 1 2 k,j=1 as scalar product and norm ([9], p. 5). Let P : L 2 (G) H(G) denote Weyl s orthogonal projection ([9], p. 15) such that L 2 (G) = H(G) {v L 2 (G) v = p for some p H 1 (G)}. 4
Then the stationary linear Stokes system can be written as u + p = f in G, u = in G, u G =, (2.6) Au = P f in G, (2.7) where A = P denotes the Stokes operator with domain of definition D(A) = H 2 (G) V (G) (see [3], p. 27). It is well known (compare [2]) that for f = P f H(G) this system has a unique solution u H 2 (G) V (G) satisfying the estimate u 2 c P u = c Au = c f. (2.8) Finally, we use the B-valued spaces C m (J, B) and H m (a, b, B), m N. Here J R is a compact interval, a, b R (a < b), and B is any of the spaces above. For C (, ) we simply write C(, ), and we sometimes use H, V (G), H m,... instead of H(G), V (G), H m (G),..., if the domain of definition is clear from the context. All constants appearing in this note are generic, i. e. its value may be different in different estimates. Throughout the paper we deal with two kinds of constants. Constants, which may depend on the domain G (on its size or on the regularity of its boundary G) and on the viscosity ν, but not on the data v and F, are always denoted by C, C 1, C 2,... Constants, which, in addition, may also depend on the data v and F, on T, and on the delay ε, are always denoted by K, K 1, K 2,... For the derivation of energy estimates we need the following version of Gronwall s Lemma: Lemma 1: Let α, β, γ R (β, γ > ) and let Φ, f, g C([, T ]) be real functions (f, g in [, T ]) satisfying for all t [, T ] the inequality Φ(t) + Then for all t [, T ] Proof: Let φ(t) + f(s)ds α + βt + G(t) = f(s)ds = α + βt + α + βt + g(s)ds + Φ(s)ds. (2.9) g(s)ds e γt. (2.1) g(s)ds, δ >, and define Ψ(t) = (δ + G(t))e γt. Then Ψ (t) = γψ(t) + G (t)e γt, which implies Ψ(t) = α + δ + γ Ψ(s)ds + βγ 1 (e γt 1) + 5 g(s)e γs ds.
We show that Φ(t) < Ψ(t) for all t [, T ]: Obviously, Φ() α < α + δ = Ψ(). Now let t > be the first position with Φ(t ) = Ψ(t ). Then Φ(t) Ψ(t) for all t [, t ], hence Φ(t ) α + βt + g(s)ds + γ < α + β + γ 1 (e γt 1) + Thus Φ < Ψ for every δ >, which implies Φ(t) α + βt + Φ(s)ds g(s)e γs ds + γ g(s)ds e γt for all t [, T ], and the last integral in (2.9) can be estimated by Ψ(s)ds = Ψ(t ). = = s (α + βs + α + βt + α + βt + α + βt + g(r)dr)γ eγs ds g(s)ds e γt α βγ 1 (e γt 1) g(s)ds e γt α βt g(s)ds (e γt 1). g(s)ds g(s)e γs ds Addition of the other terms of the right hand side of (2.9) proves (2.1). 3 Proof of the Theorems As a first result we prove that the stabilized equations (N ε ) for ε > are well-posed: Proof of Theorem 1: Let us first assume t [, ε]. Here the system (N ε ) is linear, and a spectral Galerkin method based on the eigenfunctions of the Stokes operator as in [4] shows the existence and uniqueness of the solution v 1, p 1 such 6
that v 1 C([, ε], H 2 (G) V (G)) with t v 1 C([, ε], H(G)) L 2 (, ε, V (G)) and p 1 C([, ε], L 2 (G)). Due to the orthogonality relation (see for example [9], p. 163) ((u )w, w) = (u V (G), w H 1 (G)), v 1 satisfies for t [, ε] the energy equation (1.1). Thus from the given initial function v, defined for t [ ε, ], all regularity properties carry over to the solution v = v 1, defined for t [, ε]. Now using v 1 as new initial function, we consider the system (N ε ) for t [ε, 2ε] only. Again, we obtain a unique solution v 2, p 2 having the same regularity properties on [ε, 2ε] as above v 1, p 1 on [, ε]. Because v 1 and v 2 are strongly H 2 (G)-continuous on [, ε] and [ε, 2ε], respectively, and because v 1 (ε) = v 2 (ε), we have t v 2 (ε) = t v 1 (ε), and the resulting solution v (defined by v = v 1 on [, ε], v = v 2 on [ε, 2ε]) has on [, 2ε] the asserted continuity properties, too (it should be carefully noted that the jump of the time derivative at t =, which is due to the initial construction, leads to a jump of the second order time derivative at ε, 2ε,..., only). Repeating this procedure N times on all subintervals of length ε, the global existence, uniqueness, and conservation of energy of the solution v, p is proved. Concerning the estimates, it is sufficient to show the existence of some constant K 1 = K 1 (G, ε, ν, F, v ) such that (1.2) holds for the case T = ε. Because then, by the same procedure, we find some constant K 2 = K 2 (G, ε, ν, F, v ) such that (1.2) holds for the case T = 2ε, too. Thus, repeating this technique a finite number (= N) of times (ε remains fixed), the asserted estimates (1.2) are proved. To do so, we start with the energy equation (1.1). Here we find v(t) 2 + 2ν v(s) 2 ds v 2 + F (s) 2 ds + v(s) 2 ds, and Lemma 1 leads to v(t) 2 + 2ν v(s) 2 ds v 2 + ( v 2 + εk 1 )e ε, F (s) 2 ds e t (3.1) where here K 1 depends on F only (note that (3.1) holds even for all t [, T ] without any additional considerations). Hence, in particular, v(t) K for all t [, ε]. Now multiplying the equations (N ε ) scalar in L 2 (G) with 2P v(t), we obtain after integration with respect to t v(t) 2 +2ν P v(s) 2 ds v 2 + 2( F (s) + (v ε (s) )v(s) ) P v(s) ds, 7
where the integrand on the right hand side can be estimated as follows: Thus Lemma 1 yields 2( F (s) + (v ε (s) )v(s) ) P v(s) ν 1 ( F (s) + (v ε (s) )v(s) ) 2 P v(s) 2 2ν 1 ( F (s) 2 + (v ε (s) )v(s) 2 ) + ν P v(s) 2 2ν 1 ( F (s) 2 + v ε (s) 2 v(s) 2 ) + ν P v(s) 2. v(t) 2 + ν P v(s) 2 ds ( v 2 + K 2 ε)e K 2ε with some constant K 2 = K 2 (ν, F, v ). Hence, in particular, v(t) 1 K for all t [, ε]. Next, differentiating the equations (N ε ) with respect to t, multiplying scalar in L 2 (G) by 2 t v, and integrating with respect to t, implies t v(t) 2 + 2ν t v() 2 + t v(s) 2 ds 2( t F (s) t v(s) (( t v ε (s) )v(s), t v(s)))ds, because ((v ε (t) ) t v(t), t v(t)) =. The last summand can be estimated using the generalized Hölder inequality uv u,3 v,6, and the imbeddings (2.5) as follows: 2(( t v ε (s) )v(s), t v(s)) = +2(( t v ε (s) ) t v(s), v(s)) 2 t v ε (s),6 t v(s) v(s),3 2(c 1 t v ε (s) v(s) )( t v(s) ) c 2 t v ε (s) 2 v(s) 2 + ν t v(s) 2 K 3 t v ε (s) 2 + ν t v(s) 2, with some constant K 3 = K 3 (G, ε, ν, F, v ). This implies t v(t) 2 + ν t v() 2 + t v(s) 2 ds t F (s) 2 ds + t v(s) 2 ds + K 3 t v(s) 2 ds + νp v 2 + K 4 (G, ε, ν, F, v ), 8 t v ε (s) 2 ds
because t F L 2 (, T, H(G)), t v L 2 ( ε,, V (G)), and because Hence, in particular, t v() 2 νp v 2 + K 5 (G, ε, ν, F, v ). ε t v(t) 2 dt K and t v(t) 2 K for all t [, ε]. Finally, because νp v(t) = t v(t) + P ((v ε (t) )v(t)) F (t), by means of Cattabriga s estimate [2], we find v(t) 2 C( t v(t) + v ε (t) v(t) + F (t) ), hence v(t) 2 K for all t [, ε]. This proves (1.2) and thus the theorem. Next we show that the time delay ε in the nonlinear term can be removed again in a certain sense. In this case the resulting sequence of stabilized solutions has an accumulation point v such that v is a weak solution of the Navier-Stokes equations (N ): Proof of Theorem 2: Let T be given. For every N = 1, 2,... let v N denote the solution of the stabilized equations (N ε ), ε = T/N >, according to Theorem 1. By (3.1) and Lemma 1, for all t [, T ] we find v N (t) 2 + 2ν v N (s) 2 ds v 2 + F (s) 2 ds e t K (3.2) with some constant K = K(F, v ) independent of N and t [, T ]. Because H(G), as a Hilbert space, is separable, it has a complete orthonormal system E = {e 1, e 2,...} in C,σ, where we use that C,σ is dense in H(G). For N, i N consider now on [, T ] the function t g i,n (t) = (v N (t), e i ). From (3.2) we find g i,n (t) K i for all N N and t [, T ] with some constant K i = K i (e i, F, v ), i. e. for every i N the function sequence (g i,n ) N is uniformly bounded on [, T ]. Moreover, for every i N this sequence is also equi-continuous on [, T ]. This follows from the fact that the functions g i,n (N N) satisfy on [, T ] a Lipschitz condition having the same Lipschitz constant K i for all N N: ( t v N (t), e i ) = ν(v N (t), P e i ) + ((v N ε (t) )e i, v N (t)) + (F (t), e i ) ν v N (t) P e i + v N ε (t) e i v N (t) + F (t) e i (3.3) K i. Here the Lipschitz constant K i = K i (ν, e i, F, v ) is independent of N and t [, T ] because of (3.2). Now applying the Arzela-Ascoli Theorem we obtain a subsequence 9
of (g i,n ) N being uniformly convergent on [, T ]. Moreover, by a suitable diagonal procedure we can find a subsequence of (v N ) N which we again denote by (v N ) N such that the resulting subsequences (g i,n ) are uniformly convergent on [, T ] für all i N, i. e. there exists a weakly continuous function v : [, T ] H(G) such that v N (t) v(t) weakly for all t [, T ]. Due to (3.2) we even may assume that v L 2 (, T, V (G)) L (O, T, H(G)), and that v N v weakly in L 2 (, T, V (G)) and strongly in L 2 (, T, H(G)). Next we show that v satisfies the weak Navier-Stokes equations according to Definition 1.3, where we can restrict ourselves to functions (t, x) Φ(t, x) = s(t)e i (x) with some scalar function s C ((, T )). Because the stabilized solution vn certainly fulfills the integral identity and because { s (v N, e i ) + νs( v N, e i ) s((v N ε )e i, v N ) s(f, e i )}dt =, (v N, e i )dt (v, e i )dt, ( v N, e i )dt ( v, e i )dt as N due to the weak convergence in L 2 (, T, V (G)), it remains to prove {((v )e i, v) ((v N ε )e i, v N )}dt (N ). (3.4) To do so, we use the following decomposition of the integral above: {(((v v N ) )e i, v)+((v N )e i, (v v N ))+((v N v N ε )e i, v N )}dt = A N +B N +C N. The integrals A N, B N, C N can be estimated as follows: A N B N v v N e i v dt v v n e i v N dt 1 2 v v N 2 dt 1 2 v v N 2 dt 1 e 2 v 2 dt, e 2 v N 2 dt. Here we used the strong convergence of (v N ) N in L 2 (, T, H(G)) and the estimate (3.2). Finally, again by (3.2), ε C N v N vε N e i v N dt K i v N vε N dt + v N vε N dt 1 ε 2 1 2
with some constant K i = K i (e i, F, v ). For the first integral we have ε v N v N ε dt ε ( v N + v N ε )dt Kε = KT/N (N ). The second integral can be treated with Friedrichs inequality (see [8], p. 147): Let δ > be given. Then there is a number M δ N such that v N v N ε = v N (t) v N (t ε) M δ m=1 (v N (t) v N (t ε), e m rangle + δ{ v N (t) + v N (t ε) }. Using (3.3) we find (v N (t) v N (t ε), e m ) K m ε = K m T/N, hence the finite sum above can be estimated by ( T M δ m=1 K m ) /N (N ). Finally, the second integral remains bounded independent of ε and N: { v N (t) + v N (t ε) }dt 2 v N (t) dt 2 T ε 1 v N (t) 2 dt 2 K. This proves that v satisfies the weak Navier-Stokes equations. The validity of the energy inequality follows from the energy equation (1.1) for the stabilized solution by means of (see [8], p. 148) v(t) 2 + 2ν v(s) 2 ds lim inf n v 2 + 2 v N (t) 2 + 2ν (F (s), v(s))ds, v N (s) 2 ds (3.5) for all t [, T ]. In particular, lim v(t) 2 v 2 as t +, which, together with the weak continuity of v : [, T ] H(G), implies the strong continuity at initial time t = as asserted: v(t) v as t. This proves the theorem. Remark: As already mentioned in the proof of Theorem 1, there is a jump of the time derivative at t =, i. e. t v C([, T ], H(G)) and t v C([ ε, ], H(G)), but not necessarily t v(t ) = t v (t ) (in a sense of a one-sided derivative). 11
This jump, which leads to a jump of the second order time derivatives at the points ε, 2ε,..., can be avoided choosing a special initial function v as follows: Because t v() = νp v() P ((v( ε) )v()) + F (), we can determine α = t v(), if β = v(), γ = v( ε), and F () are given. Now, for example, choose v to be a second order polynomial with v ( ε) = γ, v () = β and v () = α, i. e. t v (t) = (γ + εα β)(t/ε) 2 + αt + β. References [1] Adams, R. A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, Oxford 23. [2] Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Sem. Mat. Univ. Padova 31 (1964), 38 34. [3] Fujita, H., T. Kato: On the Navier-Stokes initial value problem. I. Arch. Rat. Mech. Anal. 16 (1964), 269 315. [4] Heywood, J. G.: The Navier-Stokes equations: On the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (198), 639 681. [5] Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213 231. [6] Ladyzhenskaya, O. A.: The mathematical theory of viscous incompressible flow. Gordon & Breach, New York et al., 1969. [7] Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear problems. Proc. Symp. Madison (1963), 69 98. [8] Shinbrot, M.: Lectures on fluid mechanics. Gordon & Breach, New York 1973. [9] Temam, R.: Navier-Stokes Equations. AMS Chelsea, New York, 21. 12