The Zakharov system in dimension $d \geqslant 4$

The sharp range of Sobolev spaces is determined in which the Cauchy problem for the classical Zakharov system is well-posed, which includes existence of solutions, uniqueness, persistence of initial regularity, and real-analytic dependence on the initial data. In addition, under a condition on the data for the Schr\"odinger equation at the lowest admissible regularity, global well-posedness and scattering is proved. The results cover energy-critical and energy-supercritical dimensions $d \geqslant 4$.


Introduction
Consider an at most weakly magnetized plasma with ion density fluctuation v : R 1+d → R and complex envelope u : R 1+d → C of the electric field.In [37] Zakharov derived the equations for the dynamics of Langmuir waves, which are rapid oscillations of the electric field in a conducting plasma.A scalar version of his model, called the Zakharov system, is given by i∂ t u + ∆u = vu □v = ∆|u| 2  (1.1) with the d'Alembertian □ = ∂ 2 t − ∆.We refer to [37,8,36] and the books [17,35] for more details of the model and its derivation.
The Zakharov system is Lagrangian, and formally the L 2 -norm of u and the energy are constant in time.
(1.2) In recent years, this initial value problem has attracted considerable attention, partly driven by the close connection to the focusing cubic nonlinear Schrödinger equation (NLS) which arises as a subsonic limit of the Zakharov system (1.1) [34,1,32,26,29].In addition, bound states for the focusing cubic NLS are closely intertwined with the global dynamics of (1.1).More precisely, if Q ω : R d → R is a bound state for the focusing cubic NLS, in other words if Q ω solves ) is a global (non-dispersive) solution of (1.1).This connection has been used to analyze the blow-up behaviour [15,16,30] in dimension d = 2, and also in the periodic case [28].Furthermore, we can write the Zakharov energy as is the energy for the focusing cubic NLS.As the cubic NLS is energy-critical in d = 4, the Zakharov system is also frequently referred to as energy-critical in dimension d = 4 although, in contrast to the cubic NLS, the Zakharov system lacks scale-invariance, see [20] for further discussion.
In the Zakharov system, the interplay between the different dispersive effects of solutions to Schrödinger and wave equations leads to a rich local and global well-posedness theory [1,31,26,5,13,9,12,2,4,27,3].In particular, it turned out that the required regularity of the Schrödinger component can go below the scaling critical one (s = d/2−1) for the cubic nonlinear Schrödinger equation.Concerning the asymptotic behaviour of global solutions, scattering results have been proven in certain cases [33,14,22,3,21,19,18,24,20].
The aim of this paper is twofold.First, we give a complete answer to the question of local well-posedness in dimension d ⩾ 4, i.e. the energy-critical and super-critical dimensions.Second, we prove that these local solutions are global in time and scatter, provided that the Schrödinger part is small enough.To be more precise, consider the case d ⩾ 4, and (s, ℓ) satisfying Our first main result is Theorem 1.1.The Zakharov system (1.1) with initial condition (1.2) is locally well-posed with a real-analytic flow map, if and only if (s, ℓ) ∈ R 2 satisfies (1.3).
To be more precise, we consider mild solutions to an equivalent first order system (2.1), as usual.For this we show local well-posedness results, Theorem 7.6, which applies to the non-endpoint case, and Theorem 7.7, for the endpoint case.Finally, we provide two examples in Subsection 9.1, which show that if the flow map exists for (s, ℓ) in the exterior of the region defined by (1.3), it does not have bounded directional derivatives of second order at the origin.Partial ill-posedness results have been obtained earlier in [13,23,2,11].In the specific point (s, ℓ) = (2, 3) in d = 4 a stronger form of ill-posedness was proved in [3, Section 7], namely that there is no distributional solution at this regularity.and depends real-analytically on the initial data.This solution scatters as t → ±∞.
Theorem 1.2 is a consequence of Theorems 7.6, 7.7 and 8.1, which again apply to the first order system (2.1) in the mild formulation, see Subsection 9.2.In fact, we prove something stronger, and show that the smallness condition in Theorem 1.2 can be replaced with the weaker condition ∥f ∥ 1 2 We remark that Theorems 7.6, 7.7 (setting g * = 0) also imply that the smallness condition on f does not depend on (g 0 , g 1 ) provided that ∥(g 0 , g 1 )∥ ≪ 1 is also sufficiently small.For readers primarily interested in this important and much easier case, we provide a simplified approach and results in Section 5.
In general, ϵ > 0 in Theorem 1.2 must depend on the wave initial data (g 0 , g 1 ), and it is not even uniform with respect to its norm, at least when (s, ℓ) is on a segment of the lowest regularity (ℓ = d/2 − 2 and (d − 3)/2 ≤ s < d/2 − 1): Take any non-negative f 0 ∈ C ∞ 0 (R d ) \ {0}.Multiplying it with a large number a ≫ 1, we can make the NLS energy negative E S (af 0 ) < 0. Imposing g 0 = −|af 0 | 2 and g 1 = 0 makes the Zakharov energy the same: E Z (af 0 , g 0 , g 1 ) = E S (af 0 ).When the energy is negative, scattering is impossible, because the global dispersion would send the negative nonlinear part to zero as t → ∞.Finally, to make the Schrödinger data small, we can use the scaling-invariance of the NLS: Let f (x) = λaf 0 (λx) with λ → ∞.Since this is the Ḣd/2−1 -invariant scaling, all Ḣs norms with s < d/2−1 tend to zero as the data concentrate, including the L 2 norm (s = 0).For the wave component, the scaling leaves Ḣd/2−2 invariant, which is the lowest (critical) regularity.In other words, we can make the Schrödinger data as small in H s as we like for s < d/2 − 1, while keeping the wave norm in Ḣd/2−2 .
Further, in the energy-critical case (d = 4), we observe that there exist non-scattering solutions as soon as ∥g 0 ∥ L 2 > ∥W 2 ∥ L 2 , where W (x) = (|x| 2 /(d(d − 2)) + 1) −1 is the ground state of the NLS.To see this, start with f (x) = aW χ(x/R) with a smooth cut-off function χ (which is needed since W barely fails to be in L 2 (R 4 )).Choosing a > 1, and then R > 1 large enough depending on a, we obtain E S (f ) < E S (W ) and ∥|f | 2 ∥ L 2 > ∥W 2 ∥ L 2 , so that we can apply the grow-up result (with g 0 = −|f | 2 and g 1 = 0 as above) in the radial case obtained in [20].The large data case in the energy-critical dimension d = 4 is addressed in a follow-up paper [7].
The key contributions of Theorems 1.1 and 1.2 are firstly that we give a complete characterisation of the region of well-posedness in arbitrary space dimension d ⩾ 4, and secondly that we obtain global wellposedness and scattering for wave data of arbitrary size, only requiring the Schrödinger data to be small enough.In particular, in the energy-critical dimension d = 4 this extends [3] to the subregion where (s, ℓ) = (1, 0) or s ≥ 4ℓ + 1 or s > 2ℓ + 11  8 and the scattering to wave data of arbitrary size.Note that [3] covers the energy space (s, ℓ) = (1, 0) but by a compactness argument, from which it is not immediately clear whether the solution map is analytic.Further, if d = 4, the large data threshold result in [20] is restricted to radial data.In higher dimensions, this is an extension of the local well-posedness results in [13], which apply in the subregion where ℓ ≤ s ≤ ℓ + 1 and 2s > ℓ + d−2 2 , and the global well-posedness and scattering result in [24], which applies if (s, ℓ) = ( d−3 2 , d−4 2 ) and both the wave and the Schrödinger data are small.The recent well-posedness results cited above rely on a partial normal form transformation.This strategy introduces certain boundary terms which are non-dispersive and difficult to deal with in the low regularity setup.In this paper, we introduce a new perturbative approach which is based on Strichartz and maximal L 2 t,x norms with additional temporal derivatives allowing us to exploit the different dispersive properties of the wave and the Schrödinger equation.Further, the global well-posedness result allows for wave data of arbitrary size, which is achieved by treating the free wave evolution as a potential term in the Schrödinger equation.
One of the main challenges in proving the global well-posedness results in Theorem 1.1 and Theorem 1.2 in the range where s > ℓ + 1 lies in the fact that it seems impossible to control the endpoint Strichartz norm, i.e. to prove that . To some extent, this is explained by considering as a toy model for (1.1), where ϕ λ = e it|∇| f λ is a free wave, ψ µ = e it∆ g µ is a free solution to the Schrödinger equation, the wave data f λ has spatial frequencies |ξ| ≈ λ, and the Schrödinger data g µ has spatial frequencies |ξ| ≈ µ with µ ≪ λ.Note that this is essentially the first Picard iterate for (1.1).A computation shows that the product ϕ λ ψ µ has spacetime Fourier support in the set {|τ | ≪ λ 2 , |ξ| ≈ λ} and hence (modulo a free Schrödinger wave) we can write In particular, we expect that (in the case d = 4 for ease of notation) x (R 1+4 ) .If we assume the wave endpoint regularity, in d = 4 we can only place ϕ λ ∈ L ∞ t L 2 x .Thus applying Hölder's inequality together with the sharp Sobolev embedding and the endpoint Strichartz estimate for the free Schrödinger equation we see that Note that the above chain of inequalities is essentially forced if we may only assume the regularity ϕ λ ∈ L ∞ t L 2 x .Consequently, we obtain Again, as we can only place f λ ∈ L 2 x , this imposes the restriction s ⩽ 1.It is very difficult to see a way to improve the above computation, and in fact this high-low interaction is essentially what led to the restriction s < 1 in [3,24].Note however that this obstruction only leads to ⟨∇⟩ s u ̸ ∈ L 2 t L 4 x (R 1+4 ), and is not an obstruction to well-posedness.In other words, provided only that s ⩽ 2 we still have u ∈ L ∞ t H s x since similar to the above computation In summary, the above example strongly suggests that it is not possible to construct solutions to the Zakharov system by iterating in the endpoint Strichartz norms L 2 t W s,4 (R 1+4 ), or even any space which contains the endpoint Strichartz space.Thus an alternative space is required, and this is what we construct in this paper.
A partial solution to the above problem of obtaining well-posedness in the regularity region s ≥ ℓ + 1 was given in [3].The approach taken there was to replace the endpoint Strichartz space L 2 t W s,4 x with the intermediate Strichartz spaces L q t W s,r x for appropriate (non-endpoint, i.e. q > 2) Schrödinger admissible (q, r).However, the argument given in [3] requires additional regularity for the wave component v as it exploits Strichartz estimates for the wave equation to compensate for the loss in decay in the intermediate Schrödinger Strichartz spaces, and thus misses a neighbourhood of the corner (s, l) = ( d 2 , d 2 − 2).The key observation that gives well-posedness in the full region (1.3) is that the output of the above high-low interaction has small temporal frequencies.Consequently, the endpoint Strichartz space only loses regularity at small temporal frequencies.This observation can be exploited by using norms of the form (1.4) Note that if u = e it∆ f is a free solution to the Schrödinger evolution, then u has temporal Fourier support in {|τ | ≈ |ξ| 2 } and hence x .Thus the norm (1.4) is equivalent to the standard endpoint Strichartz space for free Schrödinger waves.On the other hand, if u has Fourier support in {|τ | ≲ |ξ|}, i.e. u has only small temporal frequencies, then In other words, we only have ) and thus we allow for a loss of regularity in the small temporal frequency region of the Strichartz norm.Moreover, again considering the above high-low interaction, we can control the output (i∂ t + ∆) −1 (ϕ λ ψ µ ) in the temporal derivative Strichartz space (1.4) provided that a ⩾ s − 1.In particular choosing a ∼ 1 gives the full range s < 2. Thus roughly speaking, the norm (1.4) matches the standard endpoint Strichartz space for the Schrödinger like portion of the evolution of u (i.e. when |τ | ≈ |ξ| 2 ), but allows for a loss of regularity in the small temporal frequency regions |τ | ≪ |ξ| 2 of u which are strongly influenced by nonlinear wave-Schrödinger interactions.We refer to estimate (2.5) and Remark 7.3 below for further related comments.
1.1.Outline of the paper.In Section 2, notation is introduced, the crucial function spaces are defined, and their key properties are discussed.Further, a product estimate for fractional time-derivatives is proved.Bilinear estimates for the Schrödinger and the wave nonlinearities are proved in Section 3 and 4, respectively.In Section 5 we provide a shortcut to simplified local and small data global well-posedness and scattering results which do not use the refined results of the following Sections.Local versions of the bilinear estimates in the endpoint case are proved in Section 6.In Section 7 the technical well-posedness results are established, most notably Theorems 7.6 and Theorem 7.7.Persistence of regularity is established in Section 8. Finally, the proofs of Theorem 1.1 and Theorem 1.2 are completed in Section 9.

Notation and Preliminaries
The Zakharov system has an equivalent first order formulation which is slightly more convenient to work with.Suppose that (u, v) is a solution to (1.1) and let V = v − i|∇| −1 ∂ t v. Then (u, V ) solves the first order problem (2.1) Conversely, given a solution (u, V ) to (2.1), the pair (u, ℜ(V )) solves the original Zakharov equation (1.1).
To restrict the Fourier support to larger sets, we use the notation and define C >µ = I − C ⩽µ .For ease of notation, for λ ∈ 2 N we often use the shorthand P λ f = f λ .In particular, note that u 1 = P 1 u has Fourier support in {|ξ| < 2}, and we have the identity For brevity, let us denote the frequently used decomposition into high and low modulation by so that u λ = P N λ u + P F λ u.Similarly, we take Note that u = P N u + P F u, and these multipliers all obey the Schrödinger scaling, for instance where P N 2 is a space-time convolution with a Schwartz function, so that we can easily deduce that P N λ and P F λ are bounded on any L p t L q x uniformly in λ ∈ 2 N , and that P N and P F are bounded on any L 2 t B s q,2 .
2.2.Function spaces.In the sequel, by default we consider tempered distributions.We define the inhomogeneous Besov spaces B s q,r and Sobolev spaces W s,p via the norms We use the notation 2 * = 2d d−2 and 2 * = (2 * ) ′ = 2d d+2 to denote the endpoint Strichartz exponents for the Schrödinger equation.Thus for d ⩾ 3 we have x by the (double) endpoint Strichartz estimate [25].To control the frequency localised Schrödinger component of the Zakharov evolution, we take parameters s, a, b ∈ R, λ ∈ 2 N and define .
The parameters a, b ∈ R are required to prove the bilinear estimates in the full admissible region (1.3).Roughly speaking a measures a loss of regularity in the small temporal frequency regime |τ | ≪ ⟨ξ⟩, for instance (when b = 0) if supp u ⊂ {τ ≲ ⟨ξ⟩ ≈ λ} we have Thus, when the temporal frequencies are small, the non-L ∞ t H s x component of the norm S s,a,b λ loses λ −a derivatives when compared to the standard scaling for the Schrödinger equation.On the other hand the b parameter simply gives a gain in regularity in the high-modulation regime, for instance we have The choice of a and b will depend on (s, ℓ), there is some flexibility here, but one option is to choose Thus in the region ℓ + 1 ⩽ s ⩽ ℓ + 2, when the Schrödinger component of the evolution is more regular, we require a > 0 positive (depending on the size of s − ℓ) and can take b = 0. On the other hand, in the "balanced region" ℓ < s < ℓ + 1 we can simply take a = b = 0.In the final region ℓ − 1 ⩽ s ⩽ ℓ, when the wave is more regular, we can take a = 0 and require b > 0 positive.
Remark 2.1.It is worth noting that due to the factor (λ only controls the endpoint Strichartz estimate without loss when a = 0.In particular, if 0 ⩽ a ⩽ 1, we only have (2.5) In view of the choice (2.4), this means that in the region s − ℓ ⩾ 1 we no longer have control over the endpoint Strichartz space L 2 t W s,2 * x .On the other hand, in the small modulation regime, we retain control of the endpoint Strichartz space.More precisely, provided that 0 ⩽ a ⩽ 1, an application of Bernstein's inequality gives the characterisation To control the Schrödinger nonlinearity we take Remark 2.2.In the special case 0 ⩽ a < 1 2 we have To see this, let 1 r = 1 2 − a and apply Bernstein's inequality together with the Sobolev embedding to obtain which implies the claim, since b ⩾ 0.
We also require a suitable space in which to control the evolution of the wave component.To this end, for ℓ, α, β ∈ R, we let Thus for small temporal frequencies we essentially take ( λ+|∂t| λ ) α V ∈ L ∞ t H ℓ x , while for large temporal frequencies (in the Schrödinger like regime) the wave component V has roughly β derivatives.Eventually we will take α = a and β = s − 1 2 .Consequently, in the high temporal frequency regime, the wave component V essentially inherits the regularity of the Schrödinger evolution u.To bound the right-hand side of the half-wave equation at frequency λ, we define Similarly, if ℓ, α, α ′ , β, β ′ ∈ R with α ′ ⩽ α and β ′ ⩽ β we have Proof.The first claim follows from the characterisation (2.6).The remaining inequalities are clear from the definitions.□ To control the evolution of the full solution, we sum the dyadic terms in ℓ 2 , and define the norms and Then, we define the corresponding spaces as the collection of all tempered distributions with finite norm.
For a general potential V ∈ L ∞ t L 2 x , we let where U V (t, s)f denotes the homogeneous solution operator for the Cauchy problem We show later that the operators U V and I V are well-defined on suitable function spaces, provided only that x solution to the wave equation.Similarly, we define the solution operator for the inhomogeneous half-wave equation by We record here two straightforward energy inequalities which we exploit in the sequel.
Proof.The estimate for the free solutions follows from the fact that the temporal frequency is of size λ 2 and the endpoint Strichartz estimate.
In order to prove the estimate for the Duhamel term, in view of the characterisation (2.6) it suffices to bound the high-modulation contribution λ s ∥I 0 x due to the (double) endpoint Strichartz estimate.To this end, we first claim that for any µ > 0 and (2.8) Assuming (2.8) for the moment, we conclude that x . (2.9) To improve this, we again use (2.8) and observe that Hence the claimed inequality follows.
To complete the proof of the norm bounds, it only remains to verify the claimed bound (2.8).Define Therefore the bound (2.8) follows by writing We now turn to the proof of continuity.In view of the definition of the time restricted space N s,a,b (I), it suffices to consider the case I = R.Moreover, the norm bound proved implies that it is enough to prove x and the continuity follows from the dominated convergence theorem.□ The energy inequality has the following useful consequence.
Proof.After writing , the energy inequality in Lemma 2.4 implies that it suffices to prove that for every λ ∈ 2 N we have lim We decompose into low and high modulation contributions F λ = P N λ F + P F λ F .For the former term, we observe that the endpoint Strichartz estimate gives x .For the remaining high modulation contribution and therefore an application of Sobolev embedding gives, uniformly for M ⩾ 1, Since x , for any ϵ > 0, by choosing M sufficiently large, and letting t, t ′ → ∞ the Riemann-Lebesgue lemma implies that lim sup As this holds for every ϵ > 0, result follows.□ We also require an energy type inequality for the wave equation.
and for λ > 2 16 , Proof.The estimate for free solutions follows from the fact that their temporal frequencies are of size λ.
For the Duhamel integral we have x .Similarly to (2.8) above we also obtain, for λ > 2 16 , and deduce (2.10) Since the bound for the L 2 t,x component of the norm ∥ • ∥ W ℓ,α,β follows directly from the definition, it only remains to bound The first term on the righthand side of (2.11) can be bounded directly from (2.10).We turn to the second contribution in (2.11) and write where the identity is due to the fact that d λ = P (t) Again, since the temporal frequencies of e it|∇| z are ≈ λ, we conclude that x and the continuity follows from the dominated convergence theorem as in Lemma 2.4.□

2.4.
A product estimate for fractional derivatives.The definition of the norms ∥ • ∥ S s,a,b λ involves three distinct regions of temporal frequencies, the low modulation case |τ + |ξ| 2 | ≪ λ 2 , the medium modulation case |τ | ≪ λ 2 , and the high modulation case |τ | ≫ λ 2 .When estimating bilinear quantities, this leads to a large number of possible frequency interactions.To help alleviate the number of possible cases we have to consider, we prove the following bilinear estimate which we later exploit as a black box.
Lemma 2.7.Let a ∈ R, µ > 0, and 1 ⩽ p, q, r, p, q, r ⩽ ∞ with 1 p = 1 q + 1 r and 1 p = 1 q + 1 r .Then x .Proof.The proof is essentially well-known, and thus we shall be somewhat brief.The main obstruction is that we allow the endpoint case r = ∞, and, as we are working with fractional derivatives in time, this causes the usual difficulties due to the failure of the Littlewood-Paley theory.In particular, to avoid summation issues, we closely follow the proof of the endpoint Kato-Ponce type inequality contained in [6].
To simplify notation, and in contrast to the rest of the paper, we temporarily adopt the convention that the temporal frequency multipliers P (t) ν give an inhomogeneous decomposition over ν ∈ 2 N , thus where φ is as in Subsection 2.1.We first consider the case a > 0 and prove the stronger estimate (2.12) Clearly, after rescaling, this implies the required estimate in the case a > 0. The proof of the estimate (2.12) is a straightforward adaption of the argument given in [6].In more detail, we decompose By symmetry, it is enough to consider the first term.To deal with the problem of summation over frequencies, we introduce a commutator term and write The bound for the second term in (2.13) follows directly from Hölder's inequality.To bound the third term in (2.13), we note that for any M ∈ 2 N we have Optimising in M then gives and hence (2.12) follows for the third term in (2.13).Finally, to bound the first term in (2.13), we first claim that for any 0 < θ < 1 a we have the commutator estimates and Assuming these bounds for the moment, we then have for any Optimising in M , we conclude that and hence (2.12) follows.It only remains to prove the standard commutator bounds (2.14) and (2.15).We begin by noting that for any a ∈ R, we have the related estimate which follows by writing ≪ν u (t − ss ′ )ds ′ ds for some ψ 1 ∈ S(R) (i.e.some smooth rapidly decreasing kernel independent of ν, u, and v), ψ 2 (s) = sψ 1 (s), and so applying Hölder's inequality and using translation invariance, we obtain (2.16).To conclude the proof of (2.14), we note that if a > 0, then (2.16) also holds with ≪ν u replaced with ⩽ν u (this is simply another application of Hölder and Bernstein), and hence (2.14) follows from the interpolation type bound which holds for any 0 ⩽ θ < 1/a.Finally, the second commutator bound (2.15) follows by simply discarding the commutator structure and applying Hölder and Bernstein's inequalities.This completes the proof of (2.12) and hence the required estimate holds in the case a > 0.
It only remains to consider the case a < 0, but this follows by arguing via duality.Namely, the estimate (2.12) gives After a shift, we may assume that (−ϵ, ϵ) ⊂ I 1 ∩ I 2 for some ϵ > 0, and that I 1 lies to the left of I 2 (i.e.inf I 1 ⩽ inf I 2 ).Define ρ 1 (t) = ρ(ϵ −1 t) and ρ 2 (t) = ρ(−ϵ −1 t) and let u j be an extension of u| Ij to R such that ∥u∥ S s,a,b (Ij ) ∼ ∥u j ∥ S s,a,b .By construction we have u = ρ 1 u 1 + ρ 2 u 2 on I 1 ∪ I 2 , and hence by definition of the restriction norm provided that S s,a,b enjoys a localisability estimate of the form Taking ϵ > 0 as large as possible (namely ϵ ≈ |I 1 ∩ I 2 |) leads to the desired estimate.
It remains to prove the above localisability, which follows from the product estimate Lemma 2.7.Indeed, for every frequency λ ∈ 2 N , we have x , where the norm of ρ 1 is bounded uniformly in λ by .
To bound the first term, we decompose u into high and low temporal frequencies and observe that another application of Lemma 2.7 gives On the other hand, for the second term, we have which implies the required bound, since b ≤ 1. □

Bilinear Estimates for Schrödinger nonlinearity
In this section we prove that we can bound the Schrödinger nonlinearity in the space N s,a,b .
Proof.In view of the definition of N s,a,b and W ℓ,a,β , a short computation shows that it suffices to prove the bounds and, under the additional assumption that supp v ⊂ {|τ | ≪ ⟨ξ⟩ 2 }, that we have More precisely, assuming that the bounds (3.1) -(3.4) hold, we decompose An application of (3.1), (3.3), and (3.4) (together with the invariance of the righthand side with respect to complex conjugation) gives On the other hand, for the V 2 contribution, we note that since ≲ ∥V ∥ W ℓ,a,β an application of (3.2) and (3.3) implies that and consider the high-low interactions λ 0 ≫ λ 1 , low-high interactions λ 0 ≪ λ 1 , and the balanced interactions case λ 0 ≈ λ 1 .
Case 2: λ 0 ≪ λ 1 .We begin by observing that an application of the Sobolev embedding On the other hand, again applying the product estimate Lemma 2.7 gives Hence, provided that Case 3: λ 0 ≈ λ 1 .Similar to above, we have This completes the proof of (3.1).
We now turn to the proof of the second estimate (3.2).As previously, we apply the frequency decomposition (3.5) and consider each frequency interaction separately.
Case 1: λ 0 ≫ λ 1 .We start by noting that an application of Sobolev embedding gives sup provided that Hence via Hölder's inequality we obtain Case 2: λ 0 ≪ λ 1 .An application of Bernstein's inequality together with the square function characterisation of L p x gives ≲ λ s+b x .
Therefore applying Bernstein's inequality and Hölder's inequality we conclude that Similar to the above, an application of Sobolev embedding gives ).Consequently, via Bernstein's inequality we have This completes the proof of (3.2). The ).The proof is standard, and follows by adapting the proof of the product estimate ∥f g∥ We now turn to the proof of the final estimate (3.4).As before, we decompose the inner sum into highlow interactions λ 0 ≫ λ 1 , low-high interactions λ 0 ≪ λ 1 , and the balanced interactions case λ 0 ≈ λ 1 , and consider each case separately.
Case 1: The assumption on the Fourier support of v implies the non-resonant identity Hence the disposability of the multiplier P N λ0 , and Bernstein's inequality, gives Consequently, we conclude that Case 2: λ 0 ≪ λ 1 .We first observe that the Fourier support assumption on v implies that ). Bernstein's inequality and the temporal product estimate in Lemma 2.7 implies which is certainly summable under the assumption that 2 .Case 3: λ 0 ≈ λ 1 .We now consider the remaining high-high interactions.Via the product estimate in Lemma 2.7 we obtain where we have used that ℓ ⩾ d−4 2 for the Sobolev embedding, and the summation is trivial in this case.□ We require a local version of the bilinear estimate, with the advantage that we can place v in dispersive norms of the form 2 ).There exists C > 0 such that for any interval 0 ∈ I ⊂ R we have x (I×R d ) ∥u∥ S s,a,0 (I) .Proof.In view of Lemma 2.4 and Theorem 3.1, it suffices to prove that for any s ⩾ 0 and 0 ⩽ a ⩽ 1 we have x (I×R d ) ∥u∥ S s,a,0 (I) . (3.8) An application of Bernstein's inequality together with (2.6) gives x and hence the endpoint Strichartz estimate implies that after extending F from I to R by zero, that The inequality (3.8) then follows from the elementary product estimate which holds for any s ⩾ 0. □

Bilinear estimates for the wave nonlinearity
Here we give the bilinear estimates required to control solutions to . The main estimate we prove is the following.
Proof.An application of the energy inequality in Lemma 2.6 implies that it suffices to prove the bounds ≲ ∥φ∥ S s,a,b ∥ψ∥ S s,a,b , (4.3) We start with the proof of (4.1) and decompose the product φψ into the standard frequency trichotomy In view of the fact that the left hand side of (4.1) is invariant with respect to complex conjugation, it suffices to consider the first two terms in (4.5), i.e. the high-low and high-high frequency interactions.
Proof of (4.1) case 1: high-low interactions.Note that in this case we must have µ ≫ 1.A computation then gives the non-resonant identity To bound the A 1 term, we observe that we can sum up over µ ≫ 1 to obtain (4.1) for the A 1 contribution.To bound A 2 , we apply the temporal product estimate in Lemma 2.7 which gives This can be summed up over µ ≫ 1 to give (4.1) for the A 2 contribution provided that Proof of (4.1) case 2: high-high interactions.An application of the product estimate in Lemma 2.7 together with Bernstein's inequality gives On the other hand, since ℓ + 1 − a ⩾ 0, we have x .Therefore summing up gives where we used the assumption 2s − ℓ − d−2 2 ⩾ 2a.This completes the proof of (4.1).
Proof of (4.2).This is slightly easier than the previous estimate (4.1) as we no longer have to deal with the temporal weight (µ + |∂ t |) a .To bound the high-low interactions, we observe that and hence provided that Similarly, to deal with the high-high interactions, we note that for any where we used the assumption 2s − ℓ − d−2 2 ⩾ 0. In view of the frequency decomposition (4.5), together with the invariance of the left hand side of (4.2) under complex conjugation, this completes the proof of the L ∞ t L 2 x bound (4.2).Proof of (4.3).We now turn to the proof of the L 2 t,x bound (4.3), and again decompose the product into the standard frequency trichotomy as in (4.5).For the high-low interaction terms, we note that x and hence, provided that Similarly, to bound the high-high interaction terms, we have for any Therefore, noting that since β ⩾ 0 we have we conclude that This completes the proof of (4.3).Proof of (4.4).To prove the remaining estimate (4.4), we can simply use Bernstein and Hölder inequalities and the endpoint Strichartz estimate with loss (2.5) As in the Schrödinger case, we additionally provide a local version of the bilinear estimate which contains a dispersive norm.
There exists 0 < θ < 1 and C > 0 such that for any interval 0 ∈ I ⊂ R, if φ, ψ ∈ S s,a,0 (I), then Proof.Let λ 1 , λ 2 ∈ 2 N .It suffices to show that there exists δ, N > 0 such that together with an estimate with a derivative loss, but the Strichartz norm on the righthand side x (I×R d ) ∥ψ∥ S s,a,0 (I) ∥φ∥ S s,a,0 (I) We start with the proof of (4.6).Choose s ′ < s such that An application of Theorem 4.1 implies that and hence (4.6) follows.
We now turn to the proof of (4.7).An application of the standard energy inequality for the wave equation together with the convexity of L p t and Bernstein's inequality implies that x Therefore there exists N > 0 such that x (I×R d ) ∥ψ∥ S s,a,0 (I) ∥φ∥ S s,a,0 (I) and the proof is complete.□

Simplified small data global and large data local theory
As a warm up to the proof of the main results contained in Theorem 1.1 and Theorem 1.2, we show how the bilinear estimates in the previous two sections can be used to prove a simplified small data global well-posedness and scattering result and a large data local well-posedness result in the non-endpoint case.
Suppose that (s, ℓ) lies in the region (1.3), and define the parameters (a, b) as in (2.4).A computation shows that the energy inequality in Lemma 2.4 together with the bilinear estimates in Theorem 3.1 and Theorem 4.1 implies that we have the bound Therefore Φ : S s,a,b → S s,a,b .Repeating this argument with differences, shows that provided ∥f ∥ H s + ∥g∥ H ℓ is sufficiently small, there exists a fixed point and again applying Theorem 4.1, we then obtain a solution (u, V ) ∈ C(R, H s × H ℓ ) to the Zakharov system (2.1).The scattering property follows from Lemma 2.5 and a similar analogue for the wave part.Note that the above argument requires that we have the smallness condition ∥f ∥ H s + ∥g∥ H ℓ ≪ 1.Our later arguments will significantly improve this to just requiring g ∈ H d− 4 2 and ∥f ∥ In other words we only require smallness of f in the endpoint Sobolev space.In addition, we also obtain a stronger uniqueness claim, as well as persistence of regularity.
Let us now sketch a simplified, large data local well-posedness result in the non-endpoint case s > d− 3  2 .Suppose that (s, ℓ) satisfies (1.3) with s > d−3 2 , and take (a, b) as in (2.4).Define l = min{s − 1 2 , ℓ} and take the map Φ as above.The non-endpoint condition s > d− 3  2 is due to the use of Corollary 4.2, while the choice of l is made to ensure that we can construct a fixed point for Φ in S s,a,0 (I) via Corollary 3.2.Once we have a fixed point u ∈ S s,a,0 (I), we use an additional argument to upgrade this to u ∈ S s,a,b (I), which is needed to get the correct regularity for the wave component where ϵ > 0 is fixed later (depending on f , g, and the absolute constants in the above bilinear estimates).Define the subset Ω ⊂ S s,a,0 (I) as x )ϵ, ∥ψ∥ S s,a,0 (I) ≲ ∥f ∥ H s .An application of Corollary 3.2 and Corollary 4.2 gives θ > 0 such that for every ψ ∈ Ω we have ∥ψ∥ S s,a,0 (I) ≲ ∥f ∥ H s + ϵ∥ψ∥ S s,a,0 (I) + ϵ 2θ (1 + ∥g∥ L 2 x ) 2θ ∥ψ∥ 3−2θ S s,a,0 (I) .On the other hand, in view of the endpoint Strichartz estimate we have S s,a,0 (I) .Consequently, choosing ϵ > 0 sufficiently small, we see that Φ : Ω → Ω.A similar argument shows that Φ is a contraction on Ω (with respect to the norm ∥•∥ S s,a,0 (I) ), and hence there exists a fixed point u ∈ Ω ⊂ S s,a,0 (I) for Φ.
We now upgrade the (far paraboloid) regularity to u ∈ S s,a,b (I).Note that this is immediate if s > ℓ since b = 0 in this case.If s ⩽ ℓ, then an application of Theorem 3.1 together with Lemma 2.4 gives ∥u∥ S s,a,0 (I) .
To check conditions of Theorem 3.1, it is helpful to note that a = 0 when s ⩽ ℓ.Theorem 4.1 implies ≲ ∥u∥ 2 S s,a,0 (I) and hence we conclude that ∥u∥ S s,a,b (I) ≲ ∥f ∥ H s + ∥g∥ H ℓ ∥u∥ S s,a,0 (I) + ∥u∥ 3 S s,a,0 (I) .
Consequently, if u ∈ S s,a,0 (I) is a solution to Φ(f, g; u) = u, then we have the improved regularity u ∈ S s,a,b (I).As above, we now define V = e it|∇| g − J 0 (|∇||u| 2 ).
Since u ∈ S s,a,b (I), an application of Theorem 4.1 then gives V ∈ W ℓ,a,s− 1 2 (I).In particular, we have a local solution (u, V ) ∈ C(I, H s × H ℓ ) to the Zakharov system (2.1).

Local bilinear estimates in the endpoint case
In this section, we deal with the endpoint case (s, ℓ) = ( d−3 2 , d−4 2 ) and establish bilinear estimates which include both dispersive norms and a slightly weaker frequency summation on the right hand side.Define the norms ∥u∥ S s,0,0 The notation here is chosen to match that introduced earlier.In particular we have ∥u∥ S s,0,0 w ≲ ∥u∥ S s,0,0 and ∥v∥ W ℓ,0,β w ≲ ∥v∥ W ℓ,0,β .We start with an improvement of Theorem 3.1 in the endpoint case.
S s,0,0 w (I) .( Proof.Suppose for the moment that we can prove that for any α ≫ 1 we have an application of (6.2) and (6.3), together with the definition of the restricted spaces N s,0,0 (I), S s,0,0 w (I), and W ℓ,0,ℓ w (I), then implies that for any M ≫ 1 we have ∥vu∥ N s,0,0 (I) ⩽ λ P λ (vu ≳λ ) N s,0,0 (I) Optimising in M then gives (6.1).Thus it remains to prove the bounds (6.2) and (6.3).For the former estimate, we observe that since s = ℓ + 1 where the last line followed via Hölder's inequality and Sobolev embedding.The proof of (6.3) is more involved, and exploits the fact that the high-low interactions are non-resonant.In particular, since λ ⩾ α ≫ 1, the non-resonant identity To estimate the first term in (6.4), we observe that since s = ℓ + 1 2 , we have To bound the second term in (6.4), again using the fact that ℓ x .Finally, for the last term in (6.4), since s − ℓ − 1 = − 1 2 , an application of Bernstein's inequality gives

□
We have a related estimate to deal with the wave nonlinearity.
Optimising in M then gives (6.5).It remains to prove the bounds (6.6), (6.7), and (6.8).We begin by noting that an application of Hölder's inequality and the Sobolev embedding together with the assumptions on (s, ℓ) imply that On the other hand, again applying a combination of Hölder's inequality and Sobolev embedding, we have The bound (6.6) now follows from the standard energy inequality as (6.10) We now turn to the proof of (6.7), this requires exploiting the fact that the high-low interactions are non-resonant.More precisely, since λ ⩾ α ≫ 1, the non-resonant identity and the bounds (6.10) and (6.9) implies that λ⩾α |∇|J 0 (P On the other hand, for the high temporal frequency term, we note that the same argument giving (2.8) together with Berstein's inequality gives ∥J 0 (P and hence (6.7) now follows from another application of (6.9).The final estimate is the high-high case (6.8).An application of Sobolev embedding gives and hence (6.8) follows from the energy estimate (6.10) together with the L 2 t,x bound (6.9) in the special case α ≈ 1.
where we assume that f ∈ H s and ∥F ∥ N s,a,0 < ∞.In particular, this shows that the Duhamel operators I V are well-defined as maps from N s,a,0 to S s,a,0 , even for large wave potentials V .
then there exists a unique solution u Moreover, there exists a constant C = C(V L ) > 0 (independent of I, f , V , and F ), such that ∥u∥ S s,a,0 (I) ⩽ C(V L ) ∥f ∥ H s + ∥F ∥ N s,a,0 (I) and, writing I = (T − , T + ) with −∞ ⩽ T − < T + ⩽ ∞, there exists f ± ∈ H s such that Remark 7.2 (Free wave potentials).The potential V in Theorem 7.1 should be thought of as a small perturbation of the free wave V L = e it|∇| g.In particular, in the special case where the potentially is simply a free wave, i.e.V = V L , the smallness condition is trivially satisfied.Consequently, for any f ∈ H s , g ∈ H ℓ , F ∈ N s,a,0 , Theorem 7.1 gives a global solution u ∈ S s,a,0 to the Schrödinger equation Thus no smallness condition is required on the potential V L or the data f .Moreover, for any open interval I ⊂ R and g ∈ H ℓ , the Duhamel integral is a continuous map I V L : N s,a,0 (I) → S s,a,0 (I), and we have the bound ∥I V L [F ]∥ S s,a,0 (I) ≲ ∥F ∥ N s,a,0 (I) .
Remark 7.3 (Strichartz control).When a > 0, the solution space S s,a,0 does not control the Strichartz space x .On the other hand, when 0 ⩽ s < ℓ+1, we have a * (s, ℓ) = 0. Therefore, an application of (2.5) and Theorem 7.1 implies that solutions to the Schrödinger equation (7.1) satisfy the (global) Strichartz estimate In particular, for any 0 The first step in the proof of Theorem 7.1 is to prove a local version with the additional assumption that the potential V is small in some dispersive type norm.
x (I×R d ) < ϵ, then the Cauchy problem x (I × R d ) and we have the bound ∥u∥ S s,a,0 (I) ⩽ C ∥f ∥ H s + ∥F ∥ N s,a,0 (I) .

Moreover, writing I
Proof.This is a direct application of Lemma 2.4, Lemma 2.5, and Corollary 3.2.Define the sequence u j ∈ S s,a,0 (I) for j ⩾ 1 by solving and let u 0 = 0.An application of Corollary 3.2 together with the smallness assumption on V implies that ∥u j ∥ S s,a,0 (I) ≲ ∥f ∥ H s + ϵ∥u j−1 ∥ S s,a,0 (I) + ∥F ∥ N s,a,0 (I) and ∥u j − u j−1 ∥ S s,a,0 (I) ≲ ϵ∥u j−1 − u j−2 ∥ S s,a,0 (I) Thus provided ϵ > 0 is sufficient small (depending only on the constant in Corollary 3.2), the sequence u j is a Cauchy sequence and hence converges to a (unique) solution u ∈ S s,a,0 (I).Uniqueness in the larger space x follows by standard arguments from the Strichartz estimate x (I×R d ) .Finally, to prove the existence of the limits lim t→T± e −it∆ u(t), it suffices to show that e −it∆ u is a Cauchy sequence as t → T + .To this end, we first observe that by Corollary 3.2 we have G = ℜ(V )u + F ∈ N s,a,0 (I).Let G ′ ∈ N s,a,0 be any extension of G from I to R. Then for any t, t ′ ∈ I and therefore, an application of Lemma 2.4 and Lemma 2.5 implies that e −it∆ u(t) is a Cauchy sequence as required. □ To apply the previous proposition, we need to decompose R into intervals on which V L is small.This exploits the dispersive properties of the free wave V L = e it|∇| g.More precisely, we have the following minor variation of [3,Lemma 4.1].On the other hand, the definition of the norm W ℓ,a,β implies that Therefore, for every j = 1, . . ., N , we have The proof of Theorem 7.1 now follows by repeatedly applying Proposition 7.4 together with the decomposability property in Lemma 2.8.
Proof of Theorem 7.1.Let ϵ > 0 and suppose that An application of Lemma 7.5 gives finite number of open intervals (I j ) j=1,...,N and points t j ∈ I j−1 ∩ I j such that I = ∪ N j=1 I j , min |I j ∩ I j+1 | > 0, and sup j=1,...,N Assuming ϵ > 0 is sufficiently small, Proposition 7.4 gives a (unique) solution u ∈ C(I j , H s ) ∩ L 2 t L 2 * x (I j × R d ) on the interval 0 ∈ I j to the Cauchy problem such that ∥u∥ S s,a,0 (Ij ) ≲ ∥f ∥ H s + ∥F ∥ N s,a,0 (Ij ) ≲ ∥f ∥ H s + ∥F ∥ N s,a,0 (I) .Taking new data u(t j ) and u(t j−1 ), and again applying Proposition 7.4, we get a unique solution ∥u∥ S s,a,0 (I k ) ≲ ∥f ∥ H s + ∥F ∥ N s,a,0 (I) .
Continuing in this manner, after at most N steps, we obtain a unique solution u where the first inequality is a consequence of Lemma 2.8.Finally, to show that the claimed limits as t → sup I and t → inf I exist, we simply repeat the argument at the end of the proof of Proposition 7.4.□

7.2.
Local and small data global results for the Zakharov system.We first consider the non-endpoint case s > d−3 2 .Theorem 7.6 (LWP and small data GWP: non-endpoint case).Let d ⩾ 4 and suppose that (s, ℓ) satisfies the conditions (1.3) and s > d−3 2 .Let a = a * (s, ℓ) and b = b * (s, ℓ) as in (2.4).For some 0 < θ < 1 and any then for all (f, g) in there exists a unique solution (u, V ) ∈ S s,a,b (I) × W ℓ,a,s− 1 2 (I) to (2.1).The flow map Proof.Fix (s, ℓ) satisfying the conditions (1.3) and s > d− 3  2 , and define a = a * (s, ℓ) and b = b * (s, ℓ) as in (2.4).Let l = min{ℓ, s − 1 2 } and define V L = e it|∇| g * to the free wave evolution of g * ∈ H ℓ , and similarly u L = e it∆ f * for f * ∈ H s in case of the free Schrödinger evolution.
Let us recall that I V L [F ] denotes the solution to the inhomogeneous Schrödinger equation and similarly, J 0 [G] denotes the solution to the inhomogeneous wave equation We claim there exists 0 < θ < 1 such that ∥ψ∥ S s,a,0 (I) (7.4) The estimate (7.4) follows from Theorem 7.1 and Theorem 3.1.To prove (7.5), we apply Remark 7.3 and observe that via the Littlewood-Paley square function estimate and Bernstein's inequality see also (2.7).Therefore and so (7.5) follows.The final estimate (7.6) is a direct application of Corollary 4.2.Set ρ = V − V L and g * = g − g * .Then the pair (u, ρ) solves After noting that ρ = e it|∇| g * − J 0 (|∇||u| 2 ) it suffices to find a fixed point u ∈ S s,a,0 (I) for the map x (I) + ∥u 2 ∥ S s,a,0 (I) .Note that (1 + Λ −θ ) −1 ∥u∥ S s,a,0 (I) ⩽ ∥u∥ Z ⩽ ∥u∥ S s,a,0 (I) and hence ∥ • ∥ Z is an equivalent norm on S s,a,0 (I).Moreover, since a x (I) ≲ ∥u∥ S s,a,0 , a short computation using the bounds (7.4), (7.5), and (7.6) implies that Moreover, in view of the endpoint Strichartz estimate and the definition of Λ we have Consequently, for any (f, g) ∈ D and u, v ∈ S s,a,0 (I) we see that Let C = C(g * ) denote the largest of the above implicit constants and take K = {u ∈ S s,a,0 | ∥u∥ Z ⩽ 2Cϵ}.
Then provided ϵ = ϵ(g * ) > 0 is chosen sufficiently small, we get unique fixed point u ∈ K ⊂ S s,a,0 .In addition, as a consequence of the above estimates, for (f, g) ∈ D and u ∈ K, we have that for any v ∈ S s,a,0 (I), the linear map T v = v −D v Φ(f, g; u) is a small perturbation of the identity (with respect to the norm ∥ • ∥ Z ), and hence T is a linear homeomorphism onto S s,a,0 (I).Furthermore, the map Φ is real-analytic (as a composition of linear, bi-and trilinear maps over R).If u[f, g] denotes the solution with initial data (f, g), the implicit function theorem [10,Theorem 15.3] I) is a composition of real-analytic maps and therefore real-analytic.In the case s ≥ ℓ + 1 2 we have ℓ = l and b = 0, so this is the claim.
To prove that the solution scatters, we note that writing I = (T 0 , T 1 ), then as in the proof of Theorem 7.1, a computation shows that for any sequence of times t j ↗ T 1 , the sequence (e −itj ∆ u(t j ), e −itj |∇| V (t j )) forms a Cauchy sequence in H s × H ℓ .In particular the limits lim t↗T1 e −it∆ u(t), e −it|∇| V (t) and lim t↘T0 e −it∆ u(t), e −it|∇| V (t) exist in H s × H ℓ .Therefore, if I = R, the solution scatters to free solutions as t → ±∞.

Persistence of Regularity
In this section our goal is show that under suitable assumptions on a solution (u, V ) to (2.1), any additional regularity of the data (u, V )(0) persists in time.
We break the proof of Theorem 8.1 into three main steps.

□ 7 .-posedness results 7 . 1 .
WellGlobal well-posedness for the model problem.The first step in the proof of Theorem 1.2, is to prove the following global result for the model problem