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Global dynamics below the ground state energy for the Zakharov system in the 3D radial case

We consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup.

This system (1.1) in d dimensions was introduced by Zakharov [21] as a mathematical model for the Langmuir turbulence in unmagnetized ionized plasma. It has been extensively studied. Local wellposedness (without symmetry) is well known. For example, the well-posedness in the energy space was proved in [4] for d = 2, 3 and in [6] for d = 1, and in weighted Sobolev space in [11]. It has been improved to the critical regularity in [6,3] for d = 1, 2, and to the full subcritical regularity in [6,2] for d ≥ 4, d = 3. The well-posedness for the system on the torus was studied in [20,12]. These results except for [11] follow from the iteration argument using Bourgain space, where the estimates depend on α, while in [11] the well-posedness is obtained uniformly for α. For more results on the subsonic limit to NLS (as α → ∞), see [18,16,13]. Concerning the long-time behavior, Merle [14] obtained blow-up in finite or infinite time for negative energy (which we will call grow-up for brevity), while the scattering theory was studied in [19,5,17], dealing with solutions for given asymptotic free profiles. Recently, in [7] the authors obtained scattering for radial initial data with small energy in the 3D case, by using the normal form reduction and radial-improved Strichartz estimates. The purpose of this paper is to consider the global dynamics for larger data under the radial symmetry.
To simplify the presentation, we rewrite the system into the first order as usual. Let N := n − iD −1ṅ /α. Then (1.1) can be rewritten as where E S (u) denotes the Hamiltonian for the cubic NLS (the limit α → ∞) namely (1.8) Let Q be the ground state for NLS (1.7), that is the unique positive radial solution for the following equation among all nontrivial solutions of (1.9) (see, e.g., [8] for further properties of Q). For λ > 0, let then we have (1.12) Thus the Zakharov system (1.4) has the following family of radial standing waves (u, N) = (e i(θ−λ 2 t) Q λ , Q 2 λ ), (1.13) where λ > 0 and θ ∈ R can be chosen arbitrarily. The goal of this study is to determine global dynamics of all the radial solutions "below" the above family of special solutions, in the spirit of Kenig-Merle [10], namely the variational dichotomy into the scattering solutions and the blowup solutions. Such a result has been obtained for the limit equation (1.7) by Holmer-Roudenko [8] in the radial case, as well as in the nonradial case [9]. For the dichotomy, we need to introduce another functional (for NLS), which is the scaling derivative of the action J: (1.14) We would like to get the same result as in [8] for NLS, but by the virial argument as in [14] we can only prove grow-up, due to the poor control of the wave component N. In fact, existence of any blowup in finite time is still an open question for the 3D Zakharov system. The main result of this paper is Then we have (a) if K(u 0 ) ≥ 0, then (1.4) has a unique global solution (u, N), which scatters both as t → ∞ and as t → −∞ in the energy space. More precisely, there are (u ± , N ± ) ∈ H 1 × L 2 such that Remark 1.1. 1) Assuming K(u 0 ) = 0 and (1.15), one can actually get by variational estimates that u 0 = 0, so u ≡ 0 and N = e itαD N 0 , see Section 2.
2) The condition (1.15) is sharp in view of the standing wave solutions (1.13).
The difficulty for the scattering even for small data can be observed by comparing the time decay with the NLS of general power nonlinearity iu − ∆u = |u| p u, u : R 1+d → C. (1.17) It is well known that the scattering for NLS requires p > 2/d, corresponding to the time integrability of the optimal decay of to the potential while the scattering in H s for any s has been proven only for p ≥ 4/d. The 3D Zakharov system would be on the boarderline in the above sense, since the potential n can decay only by as it is solving the 3D wave equation. This suggests that the decay estimates are far insufficient for the scattering in H 1 , and so it is essential to exploit nonlinear oscillations, e.g. by the normal form. This part for small radial data has been resolved in the previous paper [7]. Hence our main task in this paper is to carry out the Kenig-Merle approach [10] in accordance with the normal form. Since the normal form produces nonlinear terms without time integration, we need to modify Kenig-Merle's formulation, as well as some estimates in [7]. As a crucial ingredient for that approach, we will derive a virial identity, which is slightly different from Merle's one in [14] and more suitable for the scattering.

Hamiltonian and variational structures
2.1. Virial identity. We derive a virial identity on R d , which is slightly different from [14]. Recall that the Zakharov system can be rewritten in the Hamiltonian form where J and E ′ Z denote the symplectic operator and the Frechét derivative given by Let A be the generator for the family of scaling transforms 1 Let w := (u, N), J := J −1 and denote the real part of L 2 inner product by ·|· . Then the virial identity for the Zakharov system is given by where we used for the third equality that Therefore, we have proved N) is a smooth decaying solution to Zakharov system (1.1). Then (2.6) The virial identity by Merle [14] is slightly different from the above one. In our notation, it can be written as (2.7) The left hand side differs from (2.6) since ir∂ r D −1 is not self-adjoint, but ir(∂ r + (d − 1)/2)D −1 is so. Precisely, the difference is (2.8) The advantage of our identity is that it is monotone both in the scattering region (K > 0) and in the blow-up region (K < 0), as we will show in the next section, while (2.7) is not monotone when u(t) and n(t) are very small compared withṅ(t). Although Merle's identity is more convenient in the blow-up region, our identity can also be used there, as we will see in Section 3.

Variational estimates.
In the 3D case d = 3, the cubic nonlinearity is L 2supercritical andḢ 1 subcritical. Hence Q is obtained by the constrained minimization Indeed, Q is the unique minimizer modulo the phase e iθ and spatial tranlation. By scaling, we also have for any λ > 0 and Q λ is the unique minimizer modulo phase and translation. Then for some λ > 0 we have E Z (u, N) + λ 2 M(u) < λJ(Q). Moreover, either u ≡ 0 on I, or K(u(t)) = 0 for all t ∈ I. In other words, K(u(t)) does not change its sign on I.
Proof. From Lemma 2.2 (b) we get that if K(u 0 ) = 0, then u ≡ 0, and hence this case is trivial. Thus we may assume K(u 0 ) > 0, hence K(u(t)) > 0 by Lemma 2.2 (b). From the assumption, we get (2.15) immediately from and the Sobolev inequality u L 4 u H 1 . So (u, N)(t) is a priori bounded in H 1 ×L 2 , and thus by the local wellposedness we have I = (−∞, ∞).

(2.22)
For K(ϕ) > 0, or equivalently µ > 1, f (X, µ) is increasing in X unless which is solved µ > ( √ 33 − 1)/2. In the latter case, we have since µ > 3/2. Otherwise, the minimum is attained at the boundary and equal to For K(ϕ) < 0, or equivalently 0 < µ < 1, −f (X, µ) is increasing in X, so its minimum is attained at the boundary and equals to Therefore, the proof of the lemma is completed.
Remark 2.1. Applying the lemma above by letting we get from Lemma 2.1 that the virial Jv|Av is monotone in our consideration. This fact will play crucial role in our consequent analysis.

Growup at infinity
This section is devoted to prove part (b) of Theorem 1.1. We assume that under the assumption of part (b), the solution exists for all t > 0. We will show that sup We consider the localized virial quantity in the form Then similarly to the non-localized virial identity, we can computė Putting ν := n − |u| 2 , the right hand side can be written componentwisė where A j := x · ∇ + (d + j)/2. The right hand side is decomposed into the NLS part: the quadratic terms in ν: and the cubic cross terms: i.e.,V R = NS +QN +CC. Since the NLS part has been treated by Ogawa-Tsutsumi [15] and Holmer-Roudenko [8], while the cross terms are higher order, the main problem for us is to control QN. Indeed, our way of the localization is motivated by a better cancellation in QN, while some other multipliers such as AXv in (3.2) could make the other terms simpler.
It is further decomposed QN = (QN 1 + QN 2 + QN 3 )/2 with where we used the symmetry of the bilinear form as well as X = X * and A 0 = −A * 0 . Putting η := D −1 ν, the other two terms are computed as follows.
where we used DA −1 = A 1 D and ∇A −1 = A 1 ∇, where we used the radial symmetry of ψ R but not of η. Thus we obtain The first two terms are less than ν 2 2 = ∇η 2 2 since ψ R ≤ 1, while the third term is nonpositive since ψ ′ R ≤ 0. The last term is bounded from above and below by 3 In short, we have ρ R (t) → 0 as R → ∞ for each fixed t, but some uniform decay is needed for the main termV ∞ (t) = 4K + ν 2 2 + (1 − d) ν||u| 2 to absorb the error. For that we use the equation of η: 14) and the corresponding integral equation We use the above equation only for very low frequency. More precisely, with a small parameter 0 < δ < 1 independent of t, decompose η smoothly in the Fourier space then we have η >δ 2 ≤ δ −1 ν 2 . For the low frequency part, we have and by the L ∞ decay of the wave equation, Thus we obtain and so (3.20) Next we estimate the cubic cross terms CC = CC 1 + CC 2 + CC 3 , (3.21) For the last term we use the commuting relations: and so (3.23) and the second term on the right is bounded by since the functions in the brackets [] vanish on |x| R. We used the radial Sobolev inequality For the commutator terms CC ′ 3 , we use the elementary commutator estimate together with the (radial/nonradial) Sobolev (3.28) Finally we estimate the NLS part where the bracket denotes the multiplication with the inside function. Using A * 0 = −A 0 as well, we obtain Since ψ R ≤ 1 and ψ ′ R ≤ 0, the first term is less than 2 ∇u 2 2 and the second is nonpositive. The last term equals The quartic terms equal ) and the last term is bounded by using the radial Sobolev inequality. In short, we have obtained Gathering the above estimates on QN, CC and NS, we obtaiṅ (3.39) Also we have Now suppose for contradiction that sup The variational estimate provides us with an upper boundV where log(RM 2 /(δκ)) may be replaced with (RM 2 /(δκ)) 1/6 for example. Then for , which is contradicting the above bound on |V R |.

Concentration-compactness procedure
It remains to prove the scattering in part (a) of Theorem 1.1. Thanks to the variational estimates in Section 2, we can proceed as Kenig-Merle. For each 0 ≤ a ≤ J(Q) and λ > 0, let where S denotes a norm containing almost all the Strichartz norms for radial free solutions, including L ∞ t (H 1 × L 2 ). See (4.25) for the precise definition. For any time interval I, we will denote by S(I) the restriction of S onto I.
From Corollary 2.3 we already know that all solutions starting from K + λ (a) stays there globally in time. What we want to prove is the uniform scattering below the ground state energy, i.e. S λ (a) < ∞ for all a < J(Q). Let The small data scattering in [7] implies that E * λ > 0, and the existence of the ground state soliton implies that E * λ ≤ J(Q). We will prove E * λ = J(Q) by contradiction, and thus finish the proof of Theorem 1.1 (a). The main result in this section is We will prove this lemma by following the concentration-compactness procedure. The main difference from NLS is that we need to work with the solutions after the normal form transform. In particular, we have some nonlinear terms without time integration (or the Duhamel form). Besides that, we have various different interactions, for which we need to use different norms or exponents.

4.1.
Profiles for the radial Zakharov. First we recall the free profile decomposition of Bahouri-Gérard type [1]. Actually we do not need its full power, as we can freeze scaling and space positions of the profiles thanks to the radial symmetry and the regularity room of our problem. Hence the setting is essentially the same as the NLS case [8].
2) The norms in (4.7) are related to the Sobolev embedding L 2 ⊂Ḃ −3/2 ∞ . Interpolation with the Strichartz estimate extends the smallness to any Strichartz norms as far as the exponents are not sharp either in L p or in regularity (including the low frequency of N).
We call such a sequence of free solutions {(u j n , N j n )} n∈N a free concentrating wave. Now we introduce the nonlinear profile associated to a free concentrating wave (u n (t), N n (t)) = U(t − t n )(f, g), t ∞ = lim n→∞ t n ∈ {0, ±∞}, (4.10) where U(t) = e −it∆ ⊕ e itαD denotes the free propagator. With it, we associate the nonlinear profile (u, N), defined as the solution of the Zakharov system satisfying which is obtained by solving the initial data problem (if t ∞ = 0) or by solving the final data problem (if t ∞ = ±∞). When t ∞ = ±∞, the existence of wave operators will be given at the end of this paper as appendix . We call (u n (t), N n (t)) := (u(t − t n ), N(t − t n )) the nonlinear concentrating wave associated with (u n (t), N n (t)). By the above construction we have Given a sequence of solutions to the Zakharov system with bounded initial data, we can apply the free profile decomposition Lemma 4.2 to the sequence of initial data, and associate a nonlinear profile with each free concentrating wave. If all nonlinear profiles are scattering and the remainder is small enough, then we can conclude that the original sequence of nonlinear solutions is also scattering with a global Strichartz bound. More precisely, we have Lemma 4.3. For each free concentrating wave (u j n , N j n ) in Lemma 4.2, let (u j n , N j n ) be the associated nonlinear concentrating wave. Let (ũ n ,Ñ n ) be the sequence of nonlinear solutions with (ũ n ,Ñ n )(0) = (f n , g n ). If (u j n , N j n ) S(0,∞) < ∞ for all j <J, then lim sup n→∞ (u n , N n ) S(0,∞) < ∞. (4.13) To prove Lemma 4.3, we need some global stability. In the next subsection, we will refine the normal form reduction and the nonlinear estimates that was used in [7], and then prove Lemma 4.3 and Lemma 4.1.

4.2.
Nonlinear estimates with small non-sharp norms. In order to obtain the nonlinear profile decomposition, we need that the non-sharp smallness (4.7) is sufficient to reduce the nonlinear interactions globally. The idea is to use interpolation, thus we need to do some refined estimates than in [7], more precisely, to avoid using the sharp (or endpoint) norms with L 2 t or L ∞ t .

4.2.1.
Modifying the nonresonant part. The first problem in following the Strichartz analysis in [7] is the L 2 t -type norms. In fact, one can observe that the use of L 2 t -type Strichartz norm for N is inevitable for the low-high interactions of nu in very low frequencies, since the regularity exponent becomes bigger than that for the dual Schrödinger admissible exponent as we move the Strichartz norm of N to L 2+ t . However, this problem can be avoided by applying the normal form to those interactions. In fact, there is no resonance in very low frequencies because −|ξ| 2 ± α|ξ − η| + |η| 2 ∼ α|ξ − η| (4.14) when all of |ξ|, |ξ − η|, |η| are small. Hence we include them into the "non-resonant" interactions, which are integrated in time before the Strichartz estimate. The second problem is that our solution is no longer small, so the nonlinear terms without time integration (i.e. the boundary terms from the partial integration) do not contain any small factor for the perturbation argument. To overcome this difficulty, we shrink the "non-resonant" part to either higher or lower frequencies, for which we gain a small factor, depending on the frequencies, from the regularity room. Hence our decomposition into the "resonant" and "non-resonant" interactions depends on the solution size.
Thus we are lead to divide the bilinear interactions nu and |u| 2 as follows. Let u = k∈Z P k u be the standard homogeneous Littlewood-Paley decomposition such that supp F P k u ⊂ {2 k−1 < |ξ| < 2 k+1 }. For a parameter β ≥ 5 + | log 2 α|, let XL := {(j, k) ∈ Z 2 | j ≥ max(k + 5, β)}, RL := {(j, k) ∈ Z 2 | |j| < β and j ≥ k + 5}, where all the unions are disjoint. For any set A ⊂ Z 2 , and any functions f (x), g(x), we denote the bilinear frequency cut-off to A by For the nonlinear term nu, we apply the time integration by parts on XL ∪ LL, where the phase factor ω = −|ξ| 2 ± α|ξ − η| + |η| 2 is estimated |ω| ∼ |ξ − η| ξ − η ∼ |ξ − η| ξ , (4.18) which is gained in the bilinear operator For the nonlinear term uū, we integrate by parts on XL ∪ LX. Then we get a bilinear operator of the form  For the fixed free solution u 0 , the iteration u ′ → u is given by where the bilinear forms B, Q and the trilinear form T are defined by B( u 1 , u 2 ) :=(Ω(N 1 , u 2 ), D Ω(u 1 , u 2 )), For brevity, we denote We can estimate each term in the Duhamel formula using some powers of Strichartz norms with non-sharp exponents. For brevity of Hölder-type estimates, we denote the space-time norms by (4.23) Using the above notation, we introduce nearly full sets of the radial Strichartz norms for the Schrödinger and the wave equations. Fix small numbers 0 < κ ≪ ε ≪ 1, (4.24) and let (4.25) Also we denote the smallness in (4.7) by using (4.26) In the nonlinear terms, we should choose appropriate Strichartz exponents so that all can be controled by interpolation between S and Z. For that purpose, we will choose (b, d, s) for u and N respectively to be H s admissible with 0 < s < 1 and L 2 admissible for radial functions. Moreover, b < 1/2 and (b, d) = (0, 1/2). Besides that, we will use the sum space 4 with small ε > 0 for N and the intersection for u, so that we can dispose of very low or high frequencies, and sum over the dyadic decomposition without any difficulty.

Duhamel bilinear terms.
Next we consider the remaining bilinear terms in the Duhamel form after the normal form transform. Here we have to use the radial improvement of the Strichartz norms. For brevity, we denote the integrals in the Duhamel formula by There exists θ > 0 and C(β) > 1 such that for any u and u ′ , we have (4.45) The left hand sides are L 2 -admissible norms for radial functions. Thus the proof is completed by the radial improved Strichartz.

Duhamel trilinear terms.
Finally we estimate the trilinear terms which appear after the normal transform. These are supposedly the easiest, but there is a small complication due to the fact that we have to use negative Sobolev spaces for N in some of the products: r,2 f Ḃ −s p,2 g Ḃs q,2 0 ≤ s < 3/q, 1/r = 1/p + 1/q − s/3. In the next lemma, the constant may decay as β → ∞, but we do not need it.
Lemma 4.6. (a) There exists θ > 0 such that for any u, v, w, n, n ′ , we have (b) There exists θ > 0 such that for any n, u, u ′ , we have , and by non-sharp Sobolev embedding and interpolation, For Ω(n j , (n ′ u) k ), we have either 2 j ≫ 2 k or 2 j + 2 k ≪ 1. In the first case, we have where we used the product estimate for negative Sobolev spaces for n ′ u. In the second case 2 j + 2 k ≪ 1, we have (4.50) Hence, by non-sharp Sobolev embedding and interpolation, where we used the product estimate twice, but did not use any restriction on j, k.
Hence we have the same estimate on Ω(u j , (n ′ u) k ), and so (4.53) Thus, the proof is completed.
Note that in the above estimates we needed the L ∞ t -type norms only for the bare bilinear terms, but not for the Duhamel terms. Thus we have obtained Lemma 4.7. There exist θ > 0, η > 0 and C(β) > 1 such that for each β ≫ 1 and any u 1 , u 2 , u 3 , we have (4.54) For the Duhamel terms we have also  Noting that Z ⊃ S, by (4.54) we have for each j. Let ς = ς(A, σ, β, J) small such that Then by (4.60) on I 1 , we have u ⊲ S(I 1 ) ≤ 8ς and Using the same analysis as above, we can get u ⊲ S(I 2 ) ≤ 8 3 ς. Iterating this for I 2 , I 3 , . . . , I J , we obtain u a − u 1 S 8 2J ς ≪ σ, the desired result was obtained.
With J close toJ and large n, our approximate solution is given by To prove Lemma 4.3, we only need to prove that u J n is an approximate solution of the Zakharov system. In fact, we have Note that (u j n , N j n ) S does not depend on n.
In fact, For each i = j, we have |t i n − t j n | → ∞. for the subsequence t i n − t j n → ∞, we have by (4.54),  as t i n − t j n → ∞. The case t i n − t j n → −∞ is treated similarly, as well as the other terms Q and T . Thus we obtain from which (4.64) follows immediately. In order to prove (4.65), we need a uniform bound on the approximate solutions u J n for J →J. Note that (4.9) implies that u n j (0) H 1 ×L 2 ≪ 1 except for a bounded number of j. Let A be the set of j in the latter case. Then for all j ∈ A, the small data scattering implies that Then by the orthogonality in H 1 Since the number of the remaining components j ∈ A are bounded, we obtain sup J<J sup n u J n S < ∞. The left hand side of (4.65) is bounded by To see this, first note that since (u n , N n ) are in K + λ (a), they are bounded in H 1 × L 2 by the energy. Hence if (u n , N n ) S → ∞ then the L 2 t part must diverge, and we can translate (u n , N n ) in t so that the norm diverges both on (−∞, 0) and on (0, ∞).
So, the only case left is t n → 0. In this case, (4.92) Applying the above argument to the sequence (u n (t), N n (t)) := (u(t+t n ), N(t+t n )), we see that (u(t + t n ), N(t + t n )) is precompact in H 1 × L 2 . Thus we obtain the desired result.

Rigidity Theorem
The main purpose of this section is to disprove the existence of critical element that was constructed in the previous section under the assumption E * λ < J(Q). The main tool is the spatial localization of the virial identity. We prove Theorem 5.1 (Rigidity Theorem). Let (u, N) be a global solution to (1.4) satisfying K(u) ≥ 0, and E Z (u, N) + λ 2 M(u) < J λ (Q λ ) for some λ > 0. Moreover, assume {(u, N)(t) : t ∈ R} is precompact in H 1 × L 2 . Then u = N ≡ 0.
Proof. By contradiction, we assume (u, N) = (0, 0). Then by the compactness we may assume further u = 0, since otherwise N would be a free wave and dispersive. We divide the proof into the following three steps: Step 1: Energy trapping. We claim that c := inf t∈R K(u) > 0. (5.1) If not, then there exists {t n } with t n → t * ∈ [−∞, ∞], and K(u(t n )) → 0. By the precompactness of {u(t) : t ∈ R}, we get that up to a sequence (u(t n ), N(t n )) converges to some (f, g) in H 1 × L 2 . Then we have K(f ) = 0, J λ (f ) ≤ E Z (f, g) + λ 2 M(f ) = E Z (u, N) + λ 2 M(u) < J λ (Q λ ). By the variational characterization of Q λ , we get f ≡ 0 which contradicts to the M(f ) = M(u) = 0.
Step 3: Contradiction to the local virial estimates. We recall the local virial estimates obtained in Section 3. For any R > 0 V R (t) := Ju|(AX + XA)u . where X = X * be the operator of smooth trancation to |x| < R by multiplication with ψ R (x). From the proof in Section 3 and Corollary 2.3 we have On the other hand, from Step 2, Step 1, and Lemma 2.4, we get V ′ R (t) = ν|Xν /2 + ∇η|X∇η /2 + η r |rψ ′ R η r − Similar estimates imply that u n are scattering as t → ∞ for large n. Also similarly to (4.54), we have for some θ > 0 Then by applying (4.54) to the difference equation, we obtain the convergence u n → u in S(T, ∞). Since u n solves the Zakharov system, so does the limit u. If the former is uniformly bounded in H 1 × L 2 , so is the latter, and the convergence is also extended to arbitrary (T, ∞) by the local wellposedness.