On Wellposedness of the Zakharov System. J. Bourgain and J. Colliander. Summary. Consider first the 2D Zakharov system ( )

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1 IMRN International Mathematics Research Notices 1996, No. 11 On Wellposedness of the Zakharov System J. Bourgain and J. Colliander Summary Consider first the 2D Zakharov system { iut + u = nu ( ) n = tt n n = ( u 2 ) with space variable x R 2.Itisshown,in particular, that ( ) is locally well posed in the energy norm ( u(),n(), ṅ() ) H 1 L 2 Ĥ 1. 1 If the L 2 and energy conservation laws yield a priori bounds on H 1 L 2 Ĥ 1 (which is the case for u 2 sufficiently small), there is uniqueness of the weak solutions obtained in [SS]. This result improves on the work of [OT] where local wellposedness is shown in H 2 H 1 L 2. The method used is very similar to [B2]. We construct local solutions applying the contraction principle in spaces introduced in [B1] for nonlinear Schrödinger equations. Estimates on the nonlinear term are then obtained by direct analysis of the Fourier transform, closely related to [B2], and Strichartz s inequality for the linear Schrödinger equation [Str]. The result on wellposedness in the energy norm is in fact already known in the 1D case (see [C] for more details). Existence of global classical solutions for the Zakharov system in 1D is shown in SS using a priori conserved quantities, and, in 2D, global existence is established in AA, provided u 2 is sufficiently small. The methods used here permit us, however, to deal with a larger class of equations (for instance, considering Zakharov systems with more general nonlinearity, in both 1D and 2D) and to prove global existence of classical solutions provided there is no blowup in the energy norm (in particular, for small data). Details on this issue will appear in [C]. Received 18 March Revision received 19 April Denote by Ĥ 1 the space of distributions φ such that φ = div V, V : R 2 R 2 in L 2 and φ Ĥ 1 = V 2 (in 2D).

2 516 Bourgain and Colliander The preceding is equally valid for the system ( ) in 3D(considering this specific nonlinearity). Smooth initial data yield smooth solutions until there is a blowup in the energy norm. In particular, for u() H 1 small, they exist globally in time. This result solves the 3D Cauchy problem for classical solutions, left open so far. The method is similar as in 2D, except that here Strichartz s inequality for both the Schrödinger and wave equations is used, as well as the Tomas-Stein restriction theorem for the 2-sphere and some refinements of this obtained in [B3]. Sections 1 1 in the paper deal with the 2D case. The modifications of the analysis for 3D appears in Section 11. The results and techniques developed here may be useful in the pursuit of further various investigations, for instance, limiting behaviour when ion speed goes to infinity (see [SW]), and problems related to blowup phenomena (see [GM1], [GM2]). Section 1 Recall the Zakharov system { iut + u = nu n = ( u 2 ) where the space variable x ranges in R D (D = 2, 3). We follow the fixed-point approach from [KPV], [B2] in constructing local solutions. This analysis, which we recall in Section 8, depends especially on certain estimates related to the linear Schrödinger equation. The facts used are standard and are summarized below. Section 2 We have Inequalities related to linear Schrödinger equations in 2D a(k, λ) e i(k.x+λt) dk dλ L p x,t [ ] 1/2 C (1 + λ k 2 ) 4(1/2 1/p)+ a(k, λ) 2 dk dλ (2 p 4) (2.1) (p = 4: Strichartz s inequality; p = 2 : Parseval; interpolate to get (2.1)); a(k, λ) e i(k.x+λt) dk dλ L t L 2 x ( 1/2 C (1 + λ k 2 ) 1+ a(k, λ) 2 dk dλ) (2.2)

3 On Wellposedness of the Zakharov System 517 (trivial); a(k, λ) e i(k.x+λt) dk dλ L q t L2 x (interpolate between (2.2) and Parseval); ( ) 1/2 C (1+ λ k 2 ) 1 2/q+ a(k, λ) 2 (2 q ) (2.3) Proof of (2.4). a(k, λ) e i(k.x+λt) dk dλ L 2 t L q x ( (1 + k ) 2(1 2/q)+ a(k, λ) 2 ) 1/2 (2 q ). (2.4) From Hausdorff-Young and 2-concavity, we get the estimate a(k, λ) e iλt dλ L q (k) L 2 t and (2.4) follows from Hölder s inequality. Section 3 a(k, λ) e iλt dλ ( 1/2 = a(k, λ) dλ) 2 L 2 t L q (k) L q (k) Let S(t) = e it. We first estimate the contribution of (cf. (8.9) below) t S(t τ)[ 1 ( u 2 )]udτ (s 1 > 1 γ, b = 1/2+) (3.1) Xs1,b where the spaces X s,b are defined as in [B1]. 2 Here 1 F = e ik.x { dλ e iλt 1 ( 1 + λ ) e i k t 1 ( 1 λ ) 2 k 2 k The contributions to (3.2) are (cf. [B2] for more details) 2 Thus where k=k 1 +k 2 +k 3 λ=λ 1 +λ 2 +λ 3 (2.5) F(λ, k) e i k t λ 2 k 2. (3.2) 1 + k s1 (3.3) (1 + λ k 2 ) 1 b u Xs,b = u Xs,b [,δ] = { (1 + k 2 ) s (1 + λ k 2 2 ) b û(k, λ) 2 dkdλ 1/2 u(x, t) = û(k, λ)e i(k.x+λt) dkdλ on R 2 [,δ] and must be understood as the Fourier restriction norm.

4 518 Bourgain and Colliander k 1 + k λ 1 + λ 2 k 1 +k 2 û(k 1,λ 1 ) û( k 2, λ 2 ) û(k 3,λ 3 ) (3.4) + k=k 1 +k 2 +k 3 λ=± k 1 +k 2 +λ 3 idem., (3.5) where d = ( ) satisfies d 2 1 (and appears from the duality estimate). thus Write û(k, λ) =(1 + k s 1 ) 1 (1 + λ k 2 b ) 1 c(k, λ); (3.6) c 2 = u Xs1,b. (3.7) Section 4 Estimation of (3.3) From (3.3), (3.5) we get k=k 1 +k 2 +k 3 λ=λ 1 +λ 2 +λ 3 (1 + k s 1) k 1 +k 2 (1 + k 1 s 1) 1 c(k 1,λ 1 ) 1 + λ k 2 1/2 1+ λ 1 +λ 2 k 1 +k 2 1+ λ 1 k 2 1 1/2+ (1 + k 2 s 1) 1 c(k 2,λ 2 ) (1 + k 3 s1) 1 c(k 3,λ 3 ) 1+ λ 2 +k 2 2 1/2+ 1+ λ 3 k 2. (4.1) 3 1/2+ Assume k 1 k 2. We use the notation for 1 + in the denominators. Case 1: k 1 k 3 Case 1.1: k 1 k 2. One has for k 1 1 ( max λ 1 + λ 2 k 1 +k 2 1/2, λ 1 k 2 1 1/2, λ 2 +k 2 2 1/2) > k 1 2 k 2 2 ± k 1 +k 2 1/2 k 1. (4.2) At this stage we will not make use of the λ 1 + λ 2 k 1 +k 2 denominator. Hence the following cases need to be distinguished.

5 On Wellposedness of the Zakharov System 519 Case 1.1.1: λ 1 k 2 1 1/2 > k 1. (4.1) k=k 1 +k 2 +k 3 λ=λ 1 +λ 2 +λ 3 λ k 2 {{ 1/2 Case 1.1.2: λ 2 + k 2 2 1/2 > k 1. (4.1) < λ k 2 {{ 1/2 Case 1.2: k 1 k 2. (1 + k 2 s 1) 1 c(k 2,λ 2 ) (1 + k 3 s 1) 1 c(k 3,λ 3 ) c(k 1,λ 1 ) λ 2 +k 2 2 {{ 1/2+ λ 3 +k 2 3 {{ 1/2+ L 4 xt L 2 xt L xt L xt by (2.1) (s 1 close to 1) (s 1 close to 1). c(k 1,λ 1 ) λ 1 k 2 1 {{ 1/2+ (1 + k 2 s 1 ) 1 c(k 2,λ 2 ) {{ (1 + k 3 s 1) 1 c(k 3,λ 3 ) λ 3 k 2 3 {{ 1/2+ L 8 t L2 x L 4 t L4 x L 2 t L8 x L 8 t L8 x by (2.3) by (2.1) by (2.4) (s 1 close to 1). Write max( λ k 2, λ 1 k 2 1, λ 2 + k 2 2, λ 3 k 2 3 ) > k2 k k2 2 k2 3 (4.3) (4.4) = 2 k 2 + k 1,k 2 +k 3 (4.5) since k = k 1 + k 2 + k 3. Recall also (4.2) Hence max( λ 1 + λ 2 k 1 +k 2, λ 1 k 2 1, λ 2+k 2 2 ) > k 2 1 k2 2 ± k 1+k 2 > k 1 k 2,k 1 +k 2 k 1 +k 2. (4.6) (4.5) + (4.6) > k 1 + k 2,k 2 + k 1 +k 2,k 1 k 2 (1 + k 3 ) k 1 +k 2 Consequently, for k 1 + k 2 > 1, > 1 2 k 1+k 2 2 (1 + k 3 ) k 1 +k 2. (4.7) max( λ k 2 1/2, λ 1 k 2 1 1/2, λ 2 + k 2 2 1/2, λ 3 k 2 3 1/2, λ 1 + λ 2 k 1 +k 2 1/2, k 3 ) > k 1 +k 2. (4.8)

6 52 Bourgain and Colliander The cases max( λ 1 k 2 1 1/2, λ 2 + k 2 2 1/2, λ 1 + λ 2 k 1 +k 2 1/2 )> k 1 +k 2 (4.9) are taken care of by the discussion of Case 1.1. Next we consider the remaining cases. Case 1.2.1: λ k 2 1/2 k 1 + k 2 > 1. Since k 2 k 1 1/2 k 1 +k 2, (4.1) < d(k, {{ λ) c(k 1,λ 1 ) λ 1 k 2 1 {{ 1/2+ (1 + k 2 s1 ) 1 c(k 2,λ 2 ) λ 2 +k 2 2 {{ 1/2+ L 2 xt L 4 xt L xt Case 1.2.2: λ 3 k 2 3 1/2 k 1 + k 2. (4.1) λ k 2 {{ 1/2 (1 + k 3 s 1) 1 c(k 3,λ 3 ) λ 3 k 2 3 {{ 1/2+ L xt by (2.1) (s 1 1) (s 1 1). c(k 1,λ 1 ) λ 1 k 2 1 {{ 1/2+ (1 + k 2 s 1) 1 c(k 2,λ 2 ) λ 2 +k 2 2 {{ 1/2+ (1 + k 3 s 1 ) 1 c(k 3,λ 3 ) {{ L t L 2 x L 4 t L4 x L 8 t L8 x L 2 t L8 x by (2.3) by (2.1) (s 1 1) by (2.4). Case 1.2.3: k 3 > k 1 + k 2. One obtains, since k 2 k 1 >1/2 k 1 +k 2, (4.1) λ k 2 {{ 1/2 Case 2: k 1 < k 3. c(k 1,λ 1 ) λ 1 k 2 1 {{ 1/2+ (1 + k 2 2s1 1 ) 1 c(k 2,λ 2 ) λ 2 +k 2 2 {{ 1/2+ c(k 3,λ 3 ) λ 3 k 2 3 {{ 1/2+ L 4 xt L 4 xt L xt L 4 xt by (2.1) by (2.1) (s 1 1) by (2.1). Thus k 1 + k 2 1 s1 (4.1) < λ k 2 1/2 λ 1 + λ 2 k 1 +k 2 (1 + k 2 s 1) 1 c(k 2,λ 2 ) c(k 3,λ 3 ) λ 2 +k 2 2 1/2+ λ 3 k 2 3 c(k 1,λ 1 ) λ 1 k 2 1 1/2+ (4.1) (4.11) (4.12) 1/2+. (4.13)

7 On Wellposedness of the Zakharov System 521 Case 2.1: k 1 k 2. From (4.2), (4.13) is clearly bounded by λ k 2 {{ 1/2 Case 2.2: k 1 k 2. (4.1) c(k 1,λ 1 ) λ 1 k 2 1 s 1/2+ {{ L 4 xt L 4/(2 s 1) xt (1 + k 2 s 1) 1 c(k 2,λ 2 ) λ 2 +k 2 2 s 1/2+ {{ L xt c(k 3,λ 3 ) λ 3 k 2 3 {{ 1/2+ by (2.1) by (2.1) (s 1 1) (2.1). λ k 2 {{ 1/2 Estimate c(k 1,λ 1 ) λ 1 k 2 1 {{ 1/2+ (1 + k 2 2s1 1 ) 1 c(k 2,λ 2 ) λ 2 +k 2 2 {{ 1/2+ L 4 xt c(k 3,λ 3 ) λ 3 k 2 3 {{ 1/2+ L 4 xt L 4 xt L xt L 4 xt by (2.1) by (2.1) (s 1 1) by (2.1). This completes the discussion of (3.3). Section 5 Remarks 5.1 Discussion of the role of small time interval (4.14) (4.15) From the preceding estimates, it is clear that at least one of the three factors c 2 (cf. (3.5), (3.6)), obtained in the bound for (3.3) may be weakened to (1 + λ k 2 ) γ c(k, λ) 2 (5.1) for some γ>. This follows easily from the fact that some of the denominators were not fully used. Observe that γ>is independent of the choice of b>1/2and s 1. Estimate then by interpolation (5.1) c 1 θ 2 (1 + k s 1 ) û θ 2 for some θ>. (5.2) If u is supported by a time interval [ 2δ, 2δ], we get by Hölder and Hausdorff-Young inequalities (1 + k s 1 ) û 2 2 = (1 + k 2s 1 ) û(k)(t) 2 L 2 [ 2δ,2δ] dk

8 522 Bourgain and Colliander δ 1/2 (1 + k 2s 1 ) û(k)(t) 2 L 4 dk t ( 3/2 <δ (1 1/2 + k 2s 1 ) û(k, λ) dλ) 4/3 dk <δ (1 1/2 + k 2s 1 ) (1 + λ k 2 ) 1/2+ û(k, λ) 2 dλ dk. (5.3) Hence, in particular, (1 + k s 1 ) û 2 δ 1/4 c 2, (5.4) which by (5.2) yields a saving of some factor δ θ,θ > in estimating (3.3). A similar remark will apply later, in the estimates of Sections 6 and Lipschitz estimates One obtains similarly the estimate δ θ u 1 s1,b u 2 s1,b u 3 s1,b (5.5) if in (3.3) the three u-factors are replaced by u 1,u 2,u 3. Thus one may consider (3.1) as a trilinear expression and hence also get Lipschitz estimates, at least for the contribution of (3.3). Section 6 Estimating (3.4) Estimate (3.4) by (ν =±1) 2 r r Z + k = k 1 + k 2 + k 3 λ ν(λ 1 + λ 2 ) λ 3 < 2 r λ 1 + λ 2 k 1 +k 2 <2 r (1 + k s 1) λ k 2 1/2 k 1 +k 2 (1 + k 1 s 1) 1 c(k 1,λ 1 ) λ 1 k 2 1 1/2+ (1 + k 2 s1) 1 c(k 2,λ 2 ) λ 2 +k 2 2 1/2+ (1 + k 3 s1) 1 c(k 3,λ 3 ) λ 3 k 3 1/2+ (6.1)

9 On Wellposedness of the Zakharov System 523 = r Z + 2 r k = k 1 + k 2 + k 3 (1 + k s 1) λ k 2 1/2 k 1 +k 2 λ = ν(λ 1 + λ 2 ) + λ 3 + λ 4 λ 1 + λ 2 k 1 +k 2 <2 r (1 + k 1 s 1) 1 c(k 1,λ 1 ) λ 1 k 2 1 1/2+ (1 + k 2 s1) 1 c(k 2,λ 2 ) λ 2 +k 2 2 1/2+ (1 + k 3 s1) 1 c(k 3,λ 3 ) λ 3 k 2 3 1/2+ χ r (λ 4 ) (6.2) where χ r = χ [ 2 r,2 r ]. We estimate (6.2) for individual r Z +.Fixr Z +. In Cases and 1.1.2, the denominator λ 1 + λ 2 k 1 +k 2 was not used. Compared with (4.3), (4.4), the expression (6.2) contains the additional factor 2 r χ r (λ 4 ), and in the estimate we get (2 r χ r ) q < L 2 r/q, for q sufficiently large, as an extra factor. t We also need to consider the following case. Case 1.1.3: λ 1 + λ 2 k 1 +k 2 1/2 > k 1. k=k 1 +k 2 +k 3 λ=ν(λ 1 +λ 2 )+λ 3 +λ 4 Hence The case Hence k 1 + k 2 < 2 r/2 and (6.2) is bounded by c(k 1,λ 1 ) (1 + k 2 s 1) 1 c(k 2,λ 2 ) (1 + k 3 s 1) 1 c(k 3,λ 3 ) 2 r/2 χ λ k 2 {{ 1/2 λ 1 k 2 1 {{ 1/2+ λ 2 +k 2 2 {{ 1/2+ λ 3 k 2 r (λ 4 ) 3 {{ 1/2+ {{ L 4 t L 4 x L t L2 x L t L x L t L x L 4/3+ t by (2.1) (2.2) 2 r(1/4 ). Next consider Case 1.2. Since λ = ν(λ 1 + λ 2 ) + λ 3 + λ 4, one has (6.3) max( λ k 2, λ 1 k 2 1, λ 2 + k 2 2, λ 3 k 2 3, λ 4 ) > k 2 ν(k 2 1 k2 2 ) k2 3 { (6.4) 2 k1 + k 2,k 2 +k 3 (ν = 1) = 2 k 1 + k 2,k 1 +k 3 (ν = 1). max( λ k 2 1/2, λ 1 k 2 1 1/2, λ 2 + k 2 2 1/2, λ 3 k 2 3 1/2, 2 r/2, k 3 ) > k 1 + k 2. (6.5) max( λ 1 k 2 1 1/2, λ 2 + k 2 2 1/2, 2 r/2 ) > k 1 + k 2 (6.6) was considered above.

10 524 Bourgain and Colliander In Cases 1.2.1, 1.2.2, and 1.2.3, there is an additional factor 2 r χ r (λ 4 ). Clearly, one may estimate with an extra factor (2 r χ r ) q < L 2 r/q,qlarge enough. t In Case 2.1, we get the estimate k=k 1 +k 2 +k 3 λ=ν(λ 1 +λ 2 )+λ 3 +λ 4 c(k 1,λ 1 ) (1 + k 2 s 1) 1 c(k 2,λ 2 ) c(k 3,λ 3 ) λ k 2 {{ 1/2 λ 1 k 2 1 s 1/2+ λ 2 +k {{ 2 2 s 1/2+ λ 3 k {{ 2 2 r(1+s1)/2 χ r (λ 4 ) 3 1/2+ {{{{ L 4 xt L 4/(2 s 1) xt L xt L 4 xt L q t (q >4) (2.1) (2.1) (s 1 1) (2.1) 2 r(1/q (1 s 1)/2). (6.7) In Case 2.2, there is again an extra factor 2 r χ r (λ 4 ) in (4.15) contributing for (2 r χ r ) q < 2 r/q,q>4. In conclusion, the rth term in (6.2) may be bounded by 2 γr, for some γ>. This completes the discussion of (3.4). Summarizing the preceding, we get thus, invoking the remarks from Section 5, t S(t τ)[ 1 ( u 2 ]u(τ) dτ δ θ u 3 X s1,b[,δ] Xs1,b[,δ] for some fixed θ>,and a similar estimate considering the left member as a trilinear expression in u 1,u 2,u 3. Section 7 In this section we estimate t S(t τ)(n.u)(τ) dτ Xs1,b where N is of the form (cf. Section 8) N(x, t) = n(k) e i(k.x± k t) dk, R 2 (6.8) (7.1) ( ) 1/2 n(k) 2 = n 2 < (7.2) and u is as above.

11 On Wellposedness of the Zakharov System 525 and hence Estimate (7.1) by (1 + k s 1) (1 + λ k 2 ) 1 b n(k ) û(k 3,λ 3 ) where d 2 1 λ=± k +λ 3 λ=± k +λ 3 Case 1: k 3 k. λ=± k +λ 3 (1 + k s 1) (1 + λ k 2 ) 1 b n(k ) (1 + k 3 s1) 1 c(k 3,λ 3 ) (1 + λ 3 k 2. (7.3) 3 )b Hence k k 3 and (7.3) is bounded by λ k 2 1 b n(k ) We distinguish the following cases: c(k 3,λ 3 ) λ 3 k 2 (7.4) 3 b. λ k 2 λ 3 k 2 3 (7.5) λ k 2 < λ 3 k 2 3 (7.6) which will be treated similarly. In the case (7.5), estimate (7.4) by λ=± k +λ 3 n(k ) c(k 3,λ 3 ) 1+ λ 3 k 2 3 (7.7) 2 dk dλ n(k c(k k,λ 3 ) ) 1+ λ 3 k k 2 dk dλ 3 λ=± k +λ 3 { [ ] 1 dk dλ 1 + λ ± k k k 2 2 dk 1/2 (7.8) 1/2 n(k ) 2 c(k k,λ 3 ) 2 dk dλ 3 λ=± k +λ 3 (7.9) { 1/2 sup dk λ R,k R λ ± k k k 2 2. n 2. c 2. (7.1) The first factor from (7.1) will be estimated using the following.

12 526 Bourgain and Colliander Lemma R 2 For a R,x R 2,α>1, dx 1 + x 2 ± x x +a α <C α. (7.12) Lemma 7.11 yields, in particular, the bound n 2 u Xs1 for (7.1).,b Proof of (7.12). Rewrite (7.12) in polar coordinates x = re iθ rdrdθ 1+ r 2 ± re iθ x +a α (7.13) and estimate for fixed θ, letting u = r 2, 1 du 1 + u± ue iθ x +a = du (7.14) α ϕ(u) α where ϕ(u) = u + a ± ue iθ x, and hence 1 ϕ (u) 1/1 for u 1. Thus (7.14) (1 + v α ) 1 dv<c α by change of variable, implying (7.12). In the case (7.6), replace (7.4) by λ k 2 n(k ) c(k 3,λ 3 ), (7.15) λ=± k +λ 3 which we treat similarly to (7.7). Remark. From the use of the denominators in the preceding and the comments made in Section 5.1, it follows that for a small time interval [,δ], there is once more a gain of a factor δ θ, for some fixed constant θ>. Case 2: k k 3. One has, since k = k + k 3,λ=± k +λ 3, max( λ k 2, λ 3 k 2 3 ) > ± k k +k k 3 2 k 2 k 2, (7.16) assuming k > 1. Assume 2(1 b) s 1. (7.17)

13 On Wellposedness of the Zakharov System 527 If λ k 2 > λ 3 k 2 3, we write for (7.3) the estimate λ=± k +λ 3 n(k ) (1 + k 3 s 1) 1 c(k 3,λ 3 ) 1+ λ 3 k 2 3 b. (7.18) If λ k 2 λ 3 k 2 3,bound (7.3) by λ=± k +λ 3 Estimation of (7.18). dk dλ 1 + λ k 2 1 b n(k ) (1 + k 3 s1) 1 c(k 3,λ 3 ) 1+ λ 3 k 2. (7.19) 3 b s 1/2 λ=± k +λ 3 As in Case 1, we get Distinguishing the respective cases 2 n(k c(k 3,λ 3 ) ) (1 + k 2 3 λ dk dk 3 b )(1 + k 3 s 3 dλ 3 1 ) 1/2. (7.2) k 2 3 λ 3 > k 3 2 (7.21) k 2 3 λ 3 k 3 2, (7.22) estimate (7.2), respectively, by and { [ ] c(k 3,λ 3 ) 2 1/2 dk dλ n(k k 3 ) λ=± k k 3 +λ 3 1+ k 3 dk s 3 dλ 3 (7.23) 1+2b [ ] 2 c(k 3,λ 3 ) dk dλ n(k k 3 ) λ=± k k 3 +λ 3 1+ k 2 3 λ 3 s 1 2 +b 1/2 (7.24) [ ] 1 1/2 (7.23) < R k 3 dk 2(s 3 n 2 c 2 <C n 2 u Xs1,b (7.25) 1+2b)

14 528 Bourgain and Colliander by Hölder, and (7.24) < [ ] 1/2 1 R k 2 3 λ ± k k 3 s 1+2b dk 3 n 2 c 2 <C n 2 u Xs1,b (7.26) by (7.11). Estimation of (7.19). Distinguish again the cases λ k 2 < k 3 2 (7.27) λ k 2 k 3 2. (7.28) If (7.27), estimate (7.19) by λ=± k +λ λ k 2 1 b+s 1/2 n(k ) c(k 3,λ 3 ), (7.29) which is estimated as (7.15), since 1 b + s 1 /2 > 1/2. If (7.28), estimate (7.19) by λ=± k +λ 3 n(k ) since 2(1 b) + s 1 > 1. c(k 3,λ 3 ) k 3 2(1 b)+s 1 <c n 2 u Xs1,b, (7.3) Remark. The same comment regarding small time intervals applies, except that now θ = θ(s 1 ) s 1 1 <, because of the λ 3 k 2 3 b s1/2 denominator in (7.19) and (7.17). In conclusion, assuming t S(t τ)(nu)(τ) dτ δ c(1 s1) N 2 u Xs1,b[,δ] (7.31) Xs1,b[,δ] s 1 1, 2(1 b) s 1. (7.32)

15 On Wellposedness of the Zakharov System 529 Section 8 Recall that the Zakharov system in D = 2 { iut + u = nu n tt n = ( u 2 ) (8.1) may be rewritten in a Hamiltonian form as iu t + u = nu n t = div V V t = (n+ u 2 ) where V : R 2 R 2. There is conservation of the L 2 -norm (8.2) ( 1/2 u(t) 2 = u(x, t) dx) 2 (8.3) and the energy H(u, n, V) = R 2 { u (n2 + V 2 )+n u 2 dx. (8.4) Recall that u(t) H 1, n(t) 2, V(t) 2 are a priori controlled from (8.3), (8.4), provided u 2 is small, similarly as for the focusing 2D cubic nonlinear Schrödinger equation. Rewrite (8.1) as an integral equation u(t) = S(t)φ + i where n is given by t S(t τ)(nu)(τ) dτ (8.5) n = 1 ( u ) 2 ) + W(t)(a, b) (8.6) and where 1 was defined in (3.2) and W(t)(a, b) = 1 2 [( â(k) + ˆb(k) ) ( e i(k.x+ k t) + â(k) b(k) ) ] e i(k.x k t) dk. (8.7) i k i k Let a = n() L 2 (R 2 ) and b = n t () = div V t= by (8.2), V L 2.

16 53 Bourgain and Colliander Hence W(t)(a, b) is given by expressions of the form (7.2) N(x, t) = n(k)e i(k.x± k t) dk (8.8) as considered in Section 7. From (8.5), (8.6), u(t) = S(t)φ + i t S(t τ)[ 1 u 2 ]u(τ) dτ + i which we solve as a fixed-point problem. t S(t τ)[w(τ)(a, b)u(τ)] dτ, (8.9) Let (φ, a, b) H 1 L 2 Ĥ 1 be given. Fix s 1 < 1 (sufficiently close to 1), b>1/2 with 2(1 b) >s 1. Fix a time interval [,δ] and consider the map F : u (8.9) (8.1) acting on the space X s1,b[,δ]. We have, from the preceding estimate, F(u) Xs1,b[,δ] c φ H 1 + δ θ u 3 X s1,b[,δ] + δθ ( a 2 + b Ĥ 1) u Xs1,b[,δ] (8.11) where θ> is a fixed constant and θ 1 s 1. The estimate on the second term in (8.9) follows from (6.8), and on the third term from (7.31). Choosing δ small enough (depending on (φ, a, b) H 1 L 2 Ĥ 1 and s 1 < 1), it follows from (8.11) that F maps a sufficiently large ball in the space X s1,b[,δ] to itself. One has similarly a contractive estimate F(u) F(v) Xs1,b[,δ] δ θ ( u 2 s 1,b + v 2 s 1,b ) u v s 1,b (8.12) +δ θ ( a 2 + b Ĥ 1) u v s1,b, and hence Picard s theorem yields a unique fixed point. The preceding yields local wellposedness in the space X s1,b(s 1 )[,δ] C H s 1[,δ] ( s1 <1 close enough to 1; δ = δ(s 1 ) ) for given data in H 1 L 2 Ĥ 1. In order to pursue the discussion, we will first consider data in H s H s 1 Ĥs 2 for some s>1 and next the case s = 1. Some of the complications below might be removed by some extra work in the preceding analysis. Section 9 Assume initial values satisfying ( ) u(),n(), ṅ() s<s 2 (H s H s 1 Ĥs 2 ),s 2 >1 (9.1)

17 On Wellposedness of the Zakharov System 531 and ( u(t),n(t),ṅ(t) ) bounded in H 1 H 2 Ĥ 1 on some time interval [,T] (in particular, T = for u 2 small). Then the solution (u, n, ṅ) obtained earlier satisfies ( ) u(t),n(t),ṅ(t) (H s H s 1 Ĥs 2 ) on [,T]. (9.2) s<s 2 We first establish an estimate local in time. Assume (9.1). Fix s 1 < 1, sufficiently close to 1, b 1 =b(s 1 )>1/2,and δ = δ(s 1 ) > such that 2(1 b 1 ) >s 1. Let u X s1,b 1 [,δ]bethe solution obtained above applying the contraction principle in X s1,b 1 [,δ]. Hence, by (8.11), u Xs1,b 1 [,δ] C ( ( u(),n(), ṅ() ) ) H 1 L 2 Ĥ = 1 C1. (9.3) Fix next 1 <s<s 2 and b 1 >b>1/2where b = b(s) will be specified. We estimate u Xs,b [,δ] from (8.9). Thus t u Xs,b [,δ] u() H s + S(t τ)[ 1 u 2 ]u(τ) dτ t + S(t τ)[w(τ)(a, b)u(τ)] dτ, s,b Xs,b [,δ] and we bound the second and third term in (9.4). These bounds are obtained by reviewing the estimates worked out in Sections 3, 4, 6, and 7. From the analysis in Sections 3, 4, and 6, one clearly gets the inequality t (9.4) S(t τ)( 1 u 2 )u(τ) dτ δ θ u 2 X s1,b[,δ] u X s,b [,δ] (9.5) Xs,b [,δ] where θ>is a fixed constant; thus the estimate depends on u s,b only in a linear way. Consider next the analysis of (7.1) t S(t τ)(nu)(τ) dτ Xs,b [,δ]. (9.6) In Case 1, one gets again the estimate (9.6) <δ θ N 2 u Xs,b [,δ] (9.7) for some fixed θ>. The analysis in Case 2 is a bit more delicate. One obtains, clearly, the estimate k s 2(1 b) n(k) 2 u Xs1,b N H s 2(1 b) u X s1,b. (9.8)

18 532 Bourgain and Colliander Since s<s 2,we may choose b>1/2 and s<s <s 2 satisfying s 2(1 b) = s 1. (9.9) From the preceding, this leads thus to the estimate t S(t τ)[w(τ) ( n(), ṅ() ) u(τ)] dτ Xs,b [,δ] δ θ ( n(), ṅ() ) L 2 Ĥ 1 u Xs,b [,δ] + C ( n(), ṅ() ) H s 1 Ĥ s 2 u Xs1,b[,δ]. Hence, from (9.4), (9.5), (9.1), (9.3), it follows that (9.1) u Xs,b [,δ] u() H s + δ θ ( u 2 X s1,b 1 [,δ] + C ( n(), ṅ() ) L 2 Ĥ 1 ) u Xs,b [,δ] (9.11) +C ( n(), ṅ() ) H s 1 Ĥ s 2 u Xs1,b 1 [,δ] u() H s + C 2 δ θ u Xs,b [,δ] + C 1 ( n(), ṅ() ) H s 1 Ĥ s 2. (9.12) Choosing δ> small enough, an estimate on u Xs,b [,δ],b = b(s) > 1/2 is obtained. It is important to notice that the size of the time interval [,δ]isindependent of s<s 2. Consequently, u We verify also that Recall (8.6) s<s 2 C H s[,δ]. (9.13) (n, ṅ) s<s 2 C H s 1 Ĥ s 2 [,δ]. (9.14) n = 1 ( u 2 ) + W(t) ( n(), ṅ() ). (9.15) Clearly, the second term of (9.15) belongs to s<s 2 (H s 1 Ĥs 2 ). From (3.2), Consider the first term of (9.15). Thus, since we are in 2D and s 2 > 1, 1 ( u 2 ) = 1 F, 1 F(t) 2 H s 1 = F (1 + k s 1 ) 2 s<s 2 CĤs 2[,δ]. (9.16)

19 On Wellposedness of the Zakharov System 533 { dλ e iλt 1 ( 1 + λ ) e i k t 1 ( 1 λ ) F(λ, k) e i k t 2 k 2 k λ 2 k dλ 2 [ (1 + k s 1 ) 2 F(k, λ) dλ (1 + λ k )(1 + λ + k ) (1 + k s 2 ) 2 F(k, λ) 2 dk dλ ] 2 2 dk and similarly Hence = F 2 L 2 H s 2 (9.17) t ( 1 F)(t) H s 2 F L 2 H s 2. (9.18) ( 1 ( u 2 ), t 1 ( u 2 ) ) and (9.14) follows. From (9.13), (9.14) ( u(t),n(t),ṅ(t) ) s<s 2 CĤs 1 Ĥ s 2 [,δ] (9.19) s<s 2 (H s H s 1 Ĥs 2 ) for t [,δ], (9.2) and since δ is controlled from the (H 1 L 2 Ĥ 1 )-norm, (9.2) remains by assumption valid on [,T]. It follows in particular that the blowup in H s H s 1 Ĥs 2 for some s>1 (assuming data in that space) may only happen if there is a blowup in the energy norm. Section 1 Consider next data ( u(),n(), ṅ() ) in H 1 L 2 Ĥ 1, and assume, for instance, u() 2 sufficiently small to ensure that ( u(t),n(t),ṅ(t) ) is a priori bounded in H 1 L 2 Ĥ 1. Our aim is to show the uniqueness of the weak solution obtained in [SS]. Let (u α,n α )bea sequence of global classical solutions of the equation so that ( uα (),n α (), ṅ α () ) ( u(),n(), ṅ() ) in H 1 L 2 Ĥ 1. (1.1) It follows from the analysis in Section 8 that (u α,n α,ṅ α ) converges in C Hs1 [,δ],denoting H s = H s H s 1 Ĥs 2. Since there is a uniform bound in H 1, the sequence converges

20 534 Bourgain and Colliander in C Hs [,δ] for all s<1. In order to iterate the procedure, we need thus to improve the preceding local argument in the sense that the assumption (1.1) is weakened to ( uα (),n α (), ṅ α () ) ( u(),n(), ṅ() ) in H s for all s<1. (1.2) Let (u, n) (resp. (u,n )) be solutions corresponding to data (φ, a, b) (resp. (φ,a,b )). From (8.9), we get that u u Xs1,b[,δ] φ φ H s 1 +δ θ( u 2 X s1 b[,δ] + u 2 X s1 u u,b[,δ] Xs1,b[,δ] t + S(t τ) [ W(τ)(a, b)u(τ) ] t dτ S(t τ) [ W(τ)(a,b )u (τ)dτ. (1.3) ) Xs1,b[,δ] The second term in (1.3) is bounded by Cδ θ u u Xs1,b[,δ] and the last term by δ θ ( a 2 + b Ĥ 1 + a 2 + b Ĥ 1) u u Xs1,b[,δ] (1.4) t + τ)) (S(t [ W(τ)(a a,b b )u (τ) ]. (1.5) Xs1,b [,δ] Thus (1.4) Cδ θ u u Xs1,b[,δ]. The only additional point to be made concerns (1.5). Going back to the estimates from Section 7 on (7.1), easy modifications yield the bound t S(t τ)(n.u )(τ) dτ Xs1,b choosing σ> sufficiently small to ensure 2(1 b) >s 1 +σ. and hence It follows that (1.5) C( a a H σ + b b H σ 1) u Xs1 +σ,b C N Ĥ σ u Xs1 +σ,b, (1.6) C (a a,b b ) H σ H σ 1 (1.7) u u Xs1,b[,δ] (φ φ,a a,b b ) H1 σ. (1.8) Thus, under the assumption (1.2), one may still conclude convergence of (u α,n α,ṅ α )in C Hs1 [,δ]. From the arguments presented at the end of next section (dealing with the 3D case), it follows that in fact the sequence (n α, ṅ α ) converges in L 2 Ĥ 1.

21 On Wellposedness of the Zakharov System 535 Section 11 Wellposedness of the Zakharov system in 3D One may try to repeat the preceding analysis in 3D and estimate (3.1) and (7.1), to get the result local in time. An attempt to estimate (3.1) following the same procedure and replacing the 2D inequalities by the corresponding inequalities in 3D just fails. We describe here a slightly different scheme, using moreover the Strichartz inequality for the wave equation. In this case, we are specifically dealing with equation ( ) and its nonlinearity. Let the spaces X s,b be defined as before, and thus u Xs,b = ( (1 + k 2 ) s (1 + λ k 2 2 ) b û(k, λ) 2 dk dλ) 1/2. (11.1) Strichartz s inequality for the linear Schrödinger equation involves in 3D the exponent 1 3 = 2(D + 2) D D=3 instead of 4, and the inequality (2.1) is replaced by u L p x,t C u X,(5/2)(1/2 1/p)+ for 2 p 1/3, (11.2) interpolating between p = 1/3 (Strichartz s inequality) and p = 2 (Parseval). We also need the corresponding function spaces Y s,b associated to the linear wave equation u Ys,b = ( (1 + k 2 ) s (1 + λ k 2 ) b û(k, λ) 2 dk dλ) 1/2 (11.3) and Strichartz s inequality for the linear wave equation u L 4 x,t C u Y1/2,1/2+. (11.4) Again, interpolating (11.4) and Parseval s identity implies that u L p x,t C u Y 2(1/2 1/p),2(1/2 1/p)+ (2 p 4). (11.5) Interpolating (11.2) and the obvious inequality gives also u L q x,t C u X s,(5/2)(1/2 1/q s/5)+ for 1 3 q,3 2 3 q s q. (11.6) We will mainly rely on the following estimate.

22 536 Bourgain and Colliander Lemma There is the inequality u 1.u 2 Y1/2, ρ C u 1 X2σ,ρ u 2 X,ρ (11.7) provided σ<1/2,ρ<1/2 are chosen sufficiently close to 1/2. It is easy to make the statement more precise from the argument given next. Proof of Lemma k=k 1 +k 2 λ=λ 1 +λ 2 By duality, we have to estimate k 1/2 c 1 (k 1,λ 1 ) c 2 (k 2,λ 2 ) λ k ρ k 1 2σ λ 1 k 2 1 ρ λ 2 +k 2 (11.8) 2 ρ where c 1 2 u 1 X2σ,ρ, c 2 2 u 2 X,ρ, d 2 1, and stands for 1 + as before. We distinguish a number of cases. Case 1: k 1 k 2. k=k 1 +k 2 λ=λ 1 +λ 2 Then (11.8) is bounded by k 2σ 1/2 λ k {{ ρ c 1 (k 1,λ 1 ) λ 1 k 2 1 {{ ρ c 2 (k 2,λ 2 ) λ 2 +k 2 2 {{ ρ L 4 x,t L 1/3 x,t L 1/3 x,t (11.5) (11.2) (11.2) where the sign succeeding the exponents depends on how close σ and ρ are to 1/2. Case 2: k 1 k 2. Clearly, (11.9) max ( λ k, λ 1 k 2 1, λ 2 +k 2 2 ) k 2 2. (11.1) Case 2.1: λ k > k 2 2. {{ k=k 1 +k 2 λ=λ 1 +λ 2 L 2 x,t For 2ρ 1/2, (11.8) may be bounded by c 1 (k 1,λ 1 ) k 1 2σ λ 1 k 2 1 {{ ρ c 2 (k 2,λ 2 ) λ 2 +k 2 2 {{ ρ L 1 x,t L 1/3 x,t (11.6) (11.2). (11.11)

23 On Wellposedness of the Zakharov System 537 Case 2.2: λ 1 k 2 1 >k2 2. Write, for (11.8), k=k 1 +k 2 λ=λ 1 +λ 2 k 2ρ 1/2 λ k {{ ρ L 4 x,t (11.5) Case 2.3: λ 2 + k 2 2 >k2 2. Write, for (11.8), k=k 1 +k 2 λ=λ 1 +λ 2 k 2ρ 1/2 λ k {{ ρ which is again conclusive. This proves (11.7). L 4 x,t c 1 (k 1,λ 1 ) k 1 {{ 2σ L 2 t L6 x c 1 (k 1,λ 1 ) k 1 2σ λ 1 k 2 1 {{ ρ L 1 xt (11.5) (11.6) c 2 (k 2,λ 2 ) λ 2 +k 2 2 {{ ρ L t L 2 x. We next come back to expressions (3.3) and (3.4). Estimation of (3.3). Assume k 1 k 2. Case 1: k 1 k 3. (3.3) = Write k=k 1 +k 2 +k 3 λ=λ 1 +λ 2 +λ 3 c 2 (k 2,λ 2 ) {{ L 2 xt k s 1 λ k 2 k 1 + k 2 1 b λ 1 +λ 2 k 1 +k 2 û(k 1,λ 1 ) û( k 2, λ 2 ) û(k 3,λ 3 ) { λ k 2 û(k 3,λ 1 b 3 ) k k 3 =k 1 +k 2 λ λ 3 =λ 1 +λ 2 { ( k 1 s 1 û(k 1,λ 1 ) ) û( k 2, λ 2 ) k k 3 1/2 λ λ 3 k k 3 1/2 k 1 +k 2 1/2 λ 1 +λ 2 k 1 +k 2 1/2 (11.12) (11.13) V Y1/2, 1/2 W Y1/2, 1/2 (11.14)

24 538 Bourgain and Colliander where V(k,λ )= k =k k 3 λ =λ λ 3 λ k 2 1 b û(k 3,λ 3 ) (11.15) and Ŵ(k,λ )= k =k 1 +k 2 λ =λ 1 +λ 2 k 1 s 1 û(k 1,λ 1 ). û( k 2, λ 2 ). (11.16) Apply Lemma 11.7, assuming b>1/2 sufficiently close to 1/2 and s 1 < 1 sufficiently close to 1. Thus and V Y1/2, 1/2 C u X2σ,ρ. d 2 C u Xs1,b (11.17) W Y1/2, 1/2 C u X2σ,ρ ( ) s 1/2 u X,ρ C u 2 X s1,b. (11.18) Case 2: k 1 k 3. where k k 3 =k 1 +k 2 λ λ 3 =λ 1 +λ 2 Estimate (3.3) by { λ k 2 ( k 3 s 1 k k 3 1/2 s 1 û(k 1 b 3,λ 3 ) ) λ λ 3 k k 3 1/2 { ( k1 s1 û(k 1,λ 1 ) ) û( k 2, λ 2 ) k 1 +k 2 1/2 λ 1 +λ 2 k 1 +k 2 1/2 V 1 Y1/2 s1, 1/2 W Y 1/2, 1/2 (11.19) V1 (k,λ )= k =k k 3 λ =λ λ 3 λ k 2 {{ 1 b ( k3 s 1 û(k 3,λ 3 ) ) {{ L 1/3 x,t L 1/3 x,t (11.2) (11.2) (11.2) and W is as before. Hence V 1 L 5/3 x,t C u Xs1,b. (11.21)

25 On Wellposedness of the Zakharov System 539 On the other hand, dualizing (11.5) yields that V 1 Y1/2 s1, 1/2 C V 1 L 4/3+ C u Xs1,b (11.22) by (11.21). The second factor in (11.19) was estimated in (11.18). Estimation of (3.4). Assume again k 1 k 2. Case 1: k 1 k 3. dk k =k 1 +k 2 λ 1,λ 2 k =k k 3 λ =λ λ 3 One may clearly estimate (3.4) by k =k k 3 λ =λ λ 3 λ k 1 k =k 1 +k 2 λ =λ 1 +λ 2 k k 3 1/2 λ k 2 û(k 3,λ 1 b 3 ) ( k 1 s 1 k 1/2 û(k 1,λ 1 ) ) û( k 2, λ 2 ) k λ 1 +λ 2 λ k 2 1 b û(k 3,λ 3 ) k k 3 1/2 λ k 1/2 L 2 k,λ ( k 1 s 1 k 1/2 û(k 1,λ 1 ) ) û( k 2, λ 2 ) k λ 1/2 L 2 k,λ (11.23) (11.24) V Y1/2, 1/2 W Y1/2, 1/2+ (11.25) with V, W defined by (11.15), (11.16). Lemma 11.7 applies also to estimating the second factor of (11.25), and hence we get the bound u 3 X. s1,b Case 2: k 1 k 3. adjustment as in Case 1 above. Proceed now as in Case 2 when estimating (3.3), with the same

26 54 Bourgain and Colliander The conclusion of the preceding is that also in the 3D case t S(t τ)[ 1 ( u 2 )]u(τ) dτ Xs1,b u 3 X s1,b (11.26) and also (6.8), since the denominator exponents were clearly not fully used. We now pass to the analysis and the estimation of (7.3) = λ=± k +λ 3 (1 + k s 1) (1 + λ k 2 ) 1 b n(k ) (1 + k 3 s1) 1 c(k 3,λ 3 ) (1 + λ 3 k 2. (11.27) 3 )b Case 1: max ( λ k 2, λ 3 k 2 3 ) k 2. Assume λ k 2 >k 2.Fors 1 2(1 b) as in (7.17), (11.27) yields by Cauchy-Schwartz (11.27) k,λ λ=± k +λ 3 n(k ) c(k 3,λ 3 ) (1 + k 3 s 1 )(1 + λ3 k 2 3 )b (11.28) 2 c(k 3,λ 3 ) dk dλ n(k k 3 ) (1 + k 3 s 1 )(1 + λ3 k 2 dk 3 )b 3 dλ 3 λ=± k k 3 +λ 3 [ ] dk 1/2 3 sup (1 + k 3 2s 1 )(1 + λ± k k3 k 2 3 2b ) dk dλ n(k k 3 ) 2 c(k 3,λ 3 ) 2 dk 3 dλ 3 λ=± k k 3 +λ 3 The second factor in (2) yields n 2 c 2. To bound the first factor, distinguish the cases k 3 2 < λ ± k k 3 k 2 3 (11.29) k 3 2 λ± k k 3 k 2 3. (11.3) If (11.29), the bound is clear since 2s 1 + 4b >3. If (11.3), estimate by 1/2. 1/2 dk 3 (1 + k 3 )(1 + λ± k k 3 k 2 3 2b+s 1 1/2 )

27 = R + S 2 r 2 drdζ (1 + r)(1 + λ± rζ k r 2 α ) On Wellposedness of the Zakharov System 541 (α>1) dudζ 1 + λ± <C (11.31) uζ k u α as in Lemma Next assume (11.29) and write, for (11.27), λ=± k +λ λ k 2 1 b {{ Case 2: max ( λ k 2, λ 3 k 2 3 ) k k 3. Replace (11.27) by λ=± k +λ 3 n(k ) {{ c(k 3,λ 3 ) 1+ k 3 s 1 {{ L 1/3 xt L t L2 x L 2 t L6 x. by (11.2) (1 + λ k 2 ) 1 b n(k ) (11.32) Hence k 2 + k 2 3 ± k k 3 k 2 and thus 1 + k c(k 3,λ 3 ) (1 + λ 3 k 2 (11.33) 3 )b. Fix R Z + and consider the contribution to (11.33) of the region 1 + k R 1+ k 3. Thus λ k 2 <R 2, λ 3 k 2 3 <R2,and these regions are broken up by defining c a (k) = Thus and λ k 2 a 1 c(k, λ) dλ and d a (k) = ( 1/2 c a 2) 2 c 2, a Z (11.33) a,a Z a, a <R 2 1 a 1 b a b cr b 1/2. c 2.max a,a λ k 2 a 1 dλ for a, a Z. (11.34) ( 1/2 d a 2) 2 d 2 (11.35) a Z k 2 k 2 3 ± k +a a 1 k 2 k 2 3 ± k +a a 1 d a (k) n(k ) c a (k 3 ) d a (k) n(k ) c a (k 3 ), (11.36)

28 542 Bourgain and Colliander denoting by c a = c a c a 2, d a = d a d a 2 (11.37) the L 2 -normalizations. Thus we need to estimate an integral of the form, k, k 3 R k 2 k 2 3 ± k +a 1 d(k) n(k ) c(k 3 ); d 2 = 1, c 2 = 1. (11.38) For j Z +,j Rand s, s Z +,s,s R 2,define and n j = n j 1 k <j (restriction to annuli of width 1) (11.39) c s = c k 2 s 1; d s = d k 2 s 1 With these notations, (11.38) = j,s,s s s± j+a 1 (restriction to annuli of width 1/R). (11.4) d s (k)n j (k )c s (k 3 ). (11.41) Next, we will use some facts about convolution of measures on spheres. Let µ 1,µ 2 M(S 2 ),S 2 =unit sphere in R 3 with invariant measure σ, and dµ i /dσ L 2 (σ)(i=1,2). Then µ 1 µ 2 2 c dµ dσ. dµ 2 2 dσ. (11.42) 2 Also, there is some p < 2 such that if ρ is a localizing function vanishing on an ε- neighborhood of and on the sphere with radius 2, then [(µ 1 µ 2 )ρ] p ε C dµ 1 dσ dµ 2 2 dσ. (11.43) 2 The first estimate (11.42) is the Tomas-Stein restriction theorem for spheres in R 3, since µ 1 µ 2 2 = µ 1 µ 2 2 µ 1 4. µ 2 4 c dµ 2 dµ 2 dσ 2 dσ. (11.44) 2

29 On Wellposedness of the Zakharov System 543 The estimate (11.43) is a refinement of this fact, for which the reader is referred to [B3, p. 55] (see also the remarks). Observe that (11.43) and the Hausdorff-Young inequality imply (µ 1 µ 2 )ρ ε p ε C dµ 1 dσ dµ 2 2 dσ. (11.45) 2 From (11.42), (11.45) we get that and c s d s 2 cr 1/2 c s 2 d s 2 (11.46) ( ) (c. s d s )ρ ε Cε C R 3/p 2 c s 2 d s 2 (11.47) s + s p with ρ = ρ ε vanishing on an ε-neighborhood of and on the unit sphere. We deduce (11.47) from (11.45). We have ( ) ( (c. s d s ) ρ ε = R 3+3/p s + s (cr s dr s )ρ R ) ε (11.48) p s + s p where c R s (x) = c s(rx) is supported by the annulus x s/r < 1/R 2, and similarly for d R s. Define, for t s/r < 1/R 2, the measure µ t by Hence dµ t dσ = cr s ts2. (11.49) c R s = t s/r <1/R 2 µ t dt and c R s 2 = ( Similarly, we define µ t corresponding to d R s. From (11.45), (11.5), it follows that (c R s dr s (R/ )ρ s + s ) p t s/r <1/R 2, t s /R <1/R 2 ε C t s/r <1/R 2 t s /R <1/R 2 dµ t dσ dµ t dσ ( (µ t µ t )ρ R ) dt dt s + s p dµ t 2 dσ dt dt ε C 1 2 R 2 cr s 2 d R s dt) 1/2. (11.5) ε C R 5 c s 2 d s 2 (11.51)

30 544 Bourgain and Colliander and (11.48), (11.51) yield (11.47). Similarly, (11.46) is derived from (11.42). Coming back to (11.41), choose ε> and write (11.41) = s s±j+a 1 + j,s,s s s± j+a 1 d s c s,n j (d s c s )ρ ε ( s s±j+a 1 j <εr or j s s <εr ).,n j s + s Estimate, using (11.39), (11.47), the first sum of (11.52) as ε C R 3/p 2 ε C R 3/p 2 s s± j+a 1 d s c s,n j. (11.52) c s 2 d s 2 n j p s s± j+a 1 1/2 c s 2 d s 2 n j 2 (mes [ j 1 k j]) 1/p 1/2 2 1/2 ε C R 1/p n j 2 2 c s 2 d s 2 j R j R s s± j+a 1 ε C R 1/2 1/p n 2. (11.53) Consider the second sum in (11.52). Observe that since s R 2 and j <R, s = s a+ O(1). From (11.46) we get, thus, R 1/2 s s± j+a 1 j <εr or j s s a <εr = R 1/2 s c s 2 d s 2 c s 2 n j 2 j <εr or j s s a <εr d s± j a 2 n j 2 <R 1/2 (εr) 1/2 c 2 d 2 n 2 =ε 1/2 n 2. (11.54) An appropriate choice of ε permits us to derive, from (11.53), (11.54), the bound R κ n 2 on (11.41), (11.38). Here κ> is some fixed constant. Substitution in (11.36) yields the estimate cr b 1/2 κ c 2 n 2 on the contribution of (11.33) corresponding to 1 + k R 1+ k 3.Forκ>b 1/2,the estimate on (11.33) results from summing over dyadic values of R>1. This completes the analysis of (11.27). Thus, again in the 3D case, an estimate of the form (7.31) on (7.1) is derived.

31 On Wellposedness of the Zakharov System 545 The remainder of the discussion in Sections 8, 9, 1 is similar, except for the estimate of n given by (9.15), requiring in 3D a more specific argument. Estimate, using (11.7), 1 ( u 2 ) L 1 ( u 2 ) Ys H s,1/2+ c ( u 2 ) Ys 1, 1/2+ =c uu Ys +1, 1/2+ c u Xs +1/2,1/2 u X2σ,1/2. (11.55) Requiring s + 1/2 <s 2 yields s <s 2 1/2. Thus, the first term in (9.15) appears in fact smoother than the initial data n(). In particular, (9.14) holds. Observe also from the preceding that in (1.2) one gets convergence in H s L 2 Ĥ 1 for s<1. Finally, in order to derive a priori bounds on u(t) H 1, n(t) 2, V(t) 2 from the Hamiltonian (8.4), one needs to assume u() H 1 small, as in the case of the 3D focusing cubic nonlinear Schrödinger equation. One has, indeed, the inequality in 3D φ 4 dx C φ 3 H 1 φ 2. (11.56) The final result is the following. Theorem Consider the Zakharov system ( ) in3d. (1) There is local wellposedness in the energy norm and, in particular, uniqueness of the weak solutions of [SS]. (2) Smooth initial data yield smooth solutions as long as the energy norm does not blow up. Hence, if u() H 1 is sufficiently small, they are global in time. References [AA] [B1] [B2] [B3] [C] [GM1] H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math. 299 (1894), J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, Geom. Funct. Anal. 3 (1993), , On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J. 76 (1994), , Estimates for cone multipliers in Geometric Aspects of Functional Analysis, Oper. Theory Adv. Appl. 77, Birkhäuser, Basel, 1995, J. Colliander, Thesis, University of Illinois, to appear. L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two, Comm. Math. Phys. 16 (1994),

32 546 Bourgain and Colliander [GM2], Existence of self-similar blow-up solutions for Zakharov equation in dimension two, Comm. Math. Phys. 16 (1994), [KPV] C. Kenig, G. Ponce, and Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal. 127 (1995), [OT] T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equation, Differential Integral Equations 5 (1992), [SW] S. H. Schochet and M. I. Weinstein, The nonlinear limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 16 (1986), [Str] R. Strichartz, Restriction of Fourier transform to quasi surfaces and decay of solutions to the wave equation, Duke Math. J. 44 (1977), [SS] C. Sulem and P. L. Sulem, Quelques résultats de regularité pour les equations de la turbulence de Langmuir, C. R. Acad. Sci. Paris. Sér. I. Math. 289 (1979), Bourgain: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 854, USA; and Department of Mathematics, University of Illinois, Urbana, Illinois 6181, USA Colliander: Department of Mathematics, University of Illinois, Urbana, Illinois 6181, USA