On the Time Analyticity Radius of the Solutions of the Two-Dimensional Navier-Stokes Equations

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1 uj-~xi+ Journal of Dynamics and Differential Equations, Vol. 3, No. 4, 1991 On the Time Analyticity Radius of the Solutions of the Two-Dimensional Navier-Stokes Equations Igor Kukavica ~ Received June 1, 1990 In the case of solutions of the two-diensional Navier-Stokes equations, the following analyticity property is established. If the initial datum lies on the global attractor and is close enough to a stationary solution, then the analyticity radius at t = 0 of the solution can be made arbitrarily large. KEY WORDS: Dissipation; stationary solution. 0. INTRODUCTION We consider the Navier-Stokes equations on s [0, 1] x [0, 1] periodic boundary conditions. They read as follows: Ou k Ou -=- v Au + _ Ot := 1 VP= f div u = 0 u(0, x2, t) =_u(1, x2, t), u(xl, 0, t) = u(xt, 1, t) u_(x, o) = ~_o(x) with (0.1) (0.2) (0.3) (0.4) where v > 0 is the viscosity coefficient, _u = (Ua, u2) is the velocity, p is the pressure, f is a given function which does not depend on time, and _u o is a given periodic function. In this note we study the behavior of the time analyticity radius of solutions of (0.1)-(0.4) starting near stationary solutions. In order to Department of Mathematics, Indiana University, Bloomington, Indiana /3/ /91/ /0 9 i991 Plenum Publishing Corporation

2 612 Kukavica motivate the results stated below we first consider a simple autonomous ODE case, )~ =f(x) (0.5) x(0) = Xo (0.6) where x:r ~ R is unknown, with a real analytic function f and x0 ~ R being given. Assume that 0 is a fixed point of (0.5), i.e., f(0)= 0, f is analytic in a neighborhood of 0; in this case f(x)= ~ n=l c,x" is the Taylor development of f around 0, and there exist constants M, p>0, such that [c,] <~M/p", for n= 1, 2... Let [Xo[ <p. By the method of majorants, the analyticity radius of the solution to (0.5), (0.6) is greater or equal to the one of the solution of Yc = g(x) = M = M - - n=a p--x X(0)----IXol The analyticity radius of the solution x can be computed explicitly and is equal to P(ln p -In IXol)_1 (p _ Ix01) From this formula we see that if the initial value of the solution is close enough to the fixed point, the analyticity radius can be made arbitrarily large. Let us remark that the same type of argument applies also when x, Xo, and f are assumed to be vector valued. In the case of the Navier-Stokes equations certain modifications in the result and the proof are necessary.,first, the solution is known to be analytic only for t > 0 and does not need to exist at all for negative t. This is why we restrict our attention to the case when the solution starts (hence lies) on the global attractor. In this situation, as was proven by Foias and Temam (1979), we get the analyticity at t=0 (hence for every t~r), and the analyticity radius is larger then a positive constant depending only on f and v, and not on a particular choice of the initial data. As far as the proof of our result below is concerned, we use the method developed by Foias and Temam (1979), that is, we introduce the complexified time, since

3 Time Analyticity Radius of Solutions of Navier-Stokes 613 the real method (as in the ODE case above) would involve direct estimates on higher order derivatives. The result which is proved below wilt hopefully lead to better approximations of the global attractor around stationary solutions of Navier-Stokes equations. We emphasize that the same result and the same method (with obvious minor changes) apply also to other dissipative equations with a polynomial nonlinearity of mathematical physics for which an existence of the global attractor is known, like 2D Navier-Stokes equations on a general domain with Dirichlet boundary conditions, 1D or 2D Ginzburg-Landau equations, the Kuramoto-Sivashinsky equation, etc. (see Temam, 1988). 1. NOTATIONS In addition to (0.1)-(0.4) we assume that the average flow vanishes f u_(x)dx=o (1.1) In order to rewrite (0.1)-(0.4) in the usual functional form, we introduce spaces H and V as closures of ~U = {vecg~(g?)2 c~ cg(~)2: vloe periodic, div v=o in f2, fav(x) dx=o } in the Hilbert spaces L2(Q) 2 and Hi(f2) 2, respectively. Then H, V are also Hilbert spaces with scalar products 2 (u, v)= y, f ujvjdx j=i 2 fa 0u~ ~vk ((.,<= E onog& j,k=l and norms ]ul = (u, U) 1/2, lib/h = ((U, U)) 1/2. Let P: L2(~r 2 ~ H be the orthogonal projection and let A = -PA be the closed unbounded operator with domain = H2(Q) 2 c~ V and values in H. Then (0.1)-(0.4), (1.1) read as follows: du ~ + vau + B(u, u)= f (1.2) u(o)=uo (1.3)

4 614 Kukavica where B(u, v)= P((u.V)v) (for u, v where this makes sense), and f, UoE H are given. In order to complexify time, we introduce Vc, Hc as complexifications of V and H, respectively. We recall the following estimates for the nonlinear term B: IB(u, v)l ~< Cllull 1/2 IAul 1/2 Ilvl] (1.4) IB(u, v)l ~< Cllull Ilvll 1/2 iav[1/2 (1.5) The constant C can be chosen in such a way that the inequalities are valid for real as well as for complex cases. The estiates are not the best possible, but they will suffice for our purposes. For more detailed explanation of the notation, the reader is referred to Constantin and Foias (1988) or Foias and Temam (1979). 2. THE STATEMENT AND THE PROOF OF THE MAIN RESULT Let d be the global attractor of (1.2), (1.3), and let u = t~ be a stationary solution of (f.2) i.e., The following is the main result of this note. va~t+b(fi, fi)=f (2.1) Theorem. For every M1, M2 > 0 there exists 6 > 0 such that the following is true. If uo E d and IlUo- ~ll < 6, then the solution of (1.2), (1.3) is analytic in the rectangle D= {t~c: ]~tlt[ <ml, l~;t] <m2} Moreover, 6 can be chosen in such a way that it depends on M1, M2, v, Ifl but not on the, particular choice of the stationary solution ~t. The proof of the theorem is based on the two lemmas below. Lemma 1. For every M > 0 there exists fi > 0 such that the following is true. If uo~h and Iluo-~ll <O, then the solution of (1.2), (1.3) is analytic in the region if)= {t=sei~ -r~/4 < 0 < re/4, 0<s<M} Moreover, 6 can be chosen in such a way that it does not depend on the particular choice of the stationary solution ~.

5 Time Analyticity Radius of Solutions of Navier-Stokes 615 Lemma 2 (see Ladyzhenskaya, 1972). For every M, ~ > 0 there exists 6>0 such that the following is true. If UoS ~ and IIuo-~ll <& then the solution of (1.2), (1.3) satisfies Ilu(t)- ~l[ <~, for te(-m,m) Again, 6 can be chosen in such a way that it does not depend on the particular choice of the stationary solution ~. Note that Uo e d implies the existence of the solution u= u(t) for all real t and u(t) e d, for t e R. Proof of Lemma 1. The proof is based on some a priori estimates on the solution u of (1.2), (1.3). First, v = u-~ satisfies the following: dv dt + ray + B(v, v) + B(v, ~) + B(~, v) = 0 (2.2) v(0) = u o - ~ (2.3) and this is obtained by subtracting (2.1) from (1.2). Let 0e (-7r/4, re/4) and t = se i~ for s > 0. We multiply (2.2) by e i~ then scalarly by Av(sei~ and take real parts ld --- Ilv(sei~ + v cos 0 IA(v(ee~ 2ds = -9t{ei~ v, Av) + b(v, (t, Av) + b(fi, v, Av)] } (2.4) where b(u, v, w)= (B(u, v), w). Now, Ib(v, v, Av)l <<. CIEvll 3/2 [Avl3/Z<~ 89 cos 0 IAvl ) tlvll 6 (2.5) where we denoted Similarly, E(v' O)= 2C4 (-~v 9cos 0)3 Ib(v, 5, Av)l <~ Ctlvl] v2 IAvl 1/2 II~l! lay[ where <~ 89 cos 0 IAvl O, II~ll) tlvll 2 (9)3 F(v, 0, l[~[[)=c4l]~]l 4 4vc-osO (2.6)

6 616 Kukavica The estimate for Ib(f, v, Av)l is the same as (2.6), just' instead of (1.4) we use (1.5). Now, (2.4), (2.5), and (2.6) together give d ds IIv(sei~ 2 <. Eo Ilvll 6.3!_ No ilvll 2 (2.7) where E o = E(v, ~/4) Fo=F(v, re/4, Ilfl[) Suppose IIv(0)ll -- Iluo- fill < 1. Then as long as Ilvll < 1, inequality (2.7) is dominated by the following differential inequality for y(s)= Ilv(sei~ 12: dy <~ (Eo+ Fo)y (2.8) ds y(0) = Hv(0)]l 2 (2.9) Since (2.8), (2.9) give we have for all y(s) ~ Ilv(O)ll2 e (E~ f~ y(s) = IIv(se~~ = < 1 s (o, - 21neo+Fo[, (0)H / (2.10) Using the Galerkin method and the above a priori estimates, we can prove (as did Constantin and Foias, 1988; Foias and Temam, 1979) that the solution of (2.2), (2.3) is bounded and analytic in the region { -2 In ]Iv(0)ll; if)= t=sei~ - Eo+F ~ J provided Ilv(0)ll = Iluo- fill < 1. Now note that //4, 3 (V COS 72/4) 3 In Iluo- ~,l S O = -- [X~) C4(1 + ]1f114/2) diverges to oc as Iluo-fll converges to 0, hence the existence of an appropriate fi follows. Moreover, note that sup I1~[I ~< sup Ilfll ~<2rclfl ff stationary 5 ~,~ Y

7 Time Analyticity Radius of Solutions of Navier-Stokes 617 therefore 6 can be chosen in such a way that it depends only on M, v, and f For completeness we include also an easy proof of Lemma 2. Proof of Lemma 2. Let = {u solution of (1.2), (1.3): u(0)= Uo~ d} We use the following result proved by Foias and Temam (1979). There exist constants K, r0 > 0, dependent on [fl and v such that every u e o~ is analytic in D= {t~c" I~t[ <ro} and IAu(t)l <<. K, for t ~ D. Note that the last fact implies that ff is a normal family of analytic functions On D with values in Vc. Now fix M > 0, take an arbitrary u e Y, and let v--u-~, as in the proof of Lemma 1. Then (2.10) applied for the case 0 = 0 yields ]lv(t)ll ~< Ilv(O)l[e 1/2(F"~176 for te (0, to) (2.11) where t o = ( -2 In llv(0)ll )/(Eo + Fo), provided to > 0, i.e., Ilv(0)ll = Iluo- ~711 < 1. If to > M, then IIv(t)[I ~< IIv(0)ll e 1/2(E~176 for t e [0, M] Hence IlUo- ult --" 0 implies Ilu(t)- fill convergence is uniform on the interval and prove the lemma, the same has to be valid true, we could choose e > 0 and a sequence lim sup llu,(t)- n te [-0, M] sup tlun(t) - t~ [ M,O] 0, for t e[0, M] and the uniform in u e o~. In order to on I-M, M]. If this.was not {un}~=l ~-ff such that ill[ =0 (2.12) ill] 7> e (2.13) Since ~- is a normal family we can, by passing to a subsequence, assume that un converge to a Vc-valued analytic function, uniformly on compact subsets of D. By (2.12) the limit function is ~ and this contradicts (2.13). Now we are in a position to conclude the proof of the theorem. Let 6o denote the 6 > 0 provided by Lemma 1 for M= m 2 %/~. Apply now Lemma 2 with ~=6o and M=M I+M 2 obtaining a new 6>0. This satisfies the requirements. Indeed, let ze(-mi-m2, M1-M2) be arbitrary and Ilu(0)-fi[l<6. By Lemma 2, [lu(z)-~[[<6o, hence by Lemma 1 (take Uo = u(z)), the solution u is analytic in be= {t=sei~ z~c: -n/4<o<n/4,0<s<mzx/2}

8 618 Kukavica But as one can readily check, hence the theorem follows. z~(--m 1 U M2, MI--M2) b,~_d ACKNOWLEDGMENTS I would like to thank Professor Ciprian Foias for encouragement, interest, and valuable discussions. This work was partially supported by DOE Grant DE-FG02-86ER25020 and NSF Grant DMS I would also like to thank the Institute of Mathematics and Its Applications of the University of Minnesota for their hospitality during completion of the work. REFERENCES Constantin, P., and Foias, C. (1988). Navier-Stokes Equations, Chicago Lectures in Mathematics, Chicago/London. Foias, C., and Temam, R. (1979). Some analytic and geometric properties of the solutions of the Navier-Stokes equations. J. Math. Pures AppL 58, Ladyzhenskaya, O. A. (1972). On the dynamical system generated by the Navier-Stokes equations. Zap. Nauch. Sem. LOMI 27, [English translation, J. Soviet Math. 28, (1975).] Temam, R. (1988). Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. No. 68, Springer, New York. Printed in Belgium