LOW REGULARITY GLOBAL WELL-POSEDNESS FOR THE ZAKHAROV AND KLEIN-GORDON-SCHRÖDINGER SYSTEMS

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1 LOW REGULARITY GLOBAL WELL-POSEDNESS FOR THE ZAKHAROV AND KLEIN-GORDON-SCHRÖDINGER SYSTEMS JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS Abstract We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schrödinger system, which are systems in two variables u : R d x R t C and n : R d x R t R The Zakharov system is known to be locally well-posed in (u, n) L H 1/ and the Klein-Gordon-Schrödinger system is known to be locally well-posed in (u, n) L L Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the L norm of u and controlling the growth of n via the estimates in the local theory 1 Introduction The initial-value problem for the one-dimensional Zakharov system is i t u + xu = nu (11) t n x n = x u u(x, 0) = u 0 (x), n(x, 0) = n 0 (x), t n(x, 0) = n 1 (x) Here u : [0, T ) R R, n : [0, T ) R R This problem arises in plasma physics Sufficiently regular solutions of (11) satisfy conservation of mass (1) M[u](t) = u(t) dx = u 0 dx = M[u 0 ] and conservation of the Hamiltonian ( x (13) H[u, n, ν](t) = u(t) + 1 n(t) + n(t) u(t) + 1ν(t)) dx = H[u 0, n 0, ν 0 ] where ν(t) is such that t n = x ν and t ν = x (n + u ) The local-in-time theory in X s,b spaces has been established in [4], [11], the latter paper obtaining local well-posedness (LWP) for the one-dimensional equation (11) with (u 0, n 0, n 1 ) L H 1/ H 3/ and for some more regular spaces H k H s H s Mathematics Subject Classification 35Q55 Key words and phrases Zakharov system, Klein-Gordon-Schrödinger system, global wellposedness JC is partially supported by NSERC Grant RGPIN and the Sloan Foundation JH is supported by an NSF postdoctoral fellowship 1

2 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS with various 1 k, s As an immediate consequence of the local theory and (1),(13), one has global well-posedness (GWP) for k = 1, s = 0 Pecher [16], using the lowhigh frequency decomposition method of Bourgain [6], proved GWP for 9 < k < 1, 10 s = 0 This result was improved in [18] using the I-method of [8] to obtain GWP for 5 < k < 1, s = k 1 The preceding GWP results are all based on the conservation 6 of the Hamiltonian (13) or certain variants of the Hamiltonian In this paper, we prove GWP for k = 0, s = 1, using a scheme based on mass conservation (1) and subcritical slack in certain multilinear estimates at this regularity threshold In [1], it is shown that the one-dimensional LWP theory of [11] is effectively sharp by adapting techniques of [5] and [7] Thus, we establish GWP in the largest space for which LWP holds Theorem 11 The Zakharov system (11) is globally well-posed for (u 0, n 0, n 1 ) L H 1/ H 3/ and the solution (u, n) satisfies (1) and n(t) H 1 x + t n(t) H 3 x exp(c t u 0 L ) max( n 0 H 1 + n 1 H 3, u 0 L ) Remark 1 Since Theorem 11 is based on the mass conservation property (1) and the local theory, the same result applies to certain Hamiltonian generalizations of (11) for which global well-posedness was previously unknown Indeed, if we write H[u, n, ν](t) = x u(t) + α n(t) + β ν(t) + γn(t) u(t) dx and calculate t u i 0 0 H u n = 0 0 H n ν 0 0 H ν we find the evolution system { i t u + x u = γnu t n αβ xn = βγ x u If we then choose α = β = 1 and γ = 1, we encounter a Hamiltonian evolution problem similar to (11) but with +nu replaced by nu The local theory for these problems coincides but the appearance of α = β = 1 in the Hamiltonian H precludes its use in obtaining a globalizing estimate 1 The paper [11] actually gives a systematic treatment of LWP for higher dimensional versions of (11) as well Their result in dimension one uses the calculus techniques for obtaining X s,b bilinear estimates developed by Kenig, Ponce and Vega [14], [15] The only LWP result in [11] when k = 0 is for dimension one, s = 1, and thus we have restricted exclusively to this case Although the additional assumption n 1 Ḣ 1 appears in the papers [16] and [18], it can likely be removed by introducing suitable low frequency modifications to the energy identity; see [10] Lemma A1 p 358

3 GWP FOR ZAKHAROV AND KGS 3 The initial-value problem for the d-dimensional Klein-Gordon-Schrödinger system with Yukawa coupling is i t u + u = γnu x R d, t R (14) t n + αβ(1 )n = βγ u u(x, 0) = u 0 (x), n(x, 0) = n 0 (x), t n(x, 0) = n 1 (x) Here α, β, γ are real constants The solution satisfies conservation of mass (15) M[u](t) = u(t) dx = u 0 dx = M[u 0 ] and conservation of the Hamiltonian ( H[u, n, ν](t) = u(t) + 1 β tn(t) + α ) + 1 n(t) + γn(t) u(t) dx (16) = H[u 0, n 0, ν 0 ] Pecher [17] proved that (14) is LWP for d = 3 in L L H 1 and some more regular spaces H k H s H s 1 for various k, s by following the scheme developed for the Zakharov system in [11] Provided that α > 0 and β > 0, energy conservation (16) yields GWP in the setting 1 d 3 and k = 1, s = 1 In the case when α > 0 and β > 0 where the energy gives control on the H 1 norm, the low-high frequency separation method of Bourgain [6] has been applied to (14) in [17] and the method of almost conservation laws of [8] has been applied to (14) in [], to obtain GWP under the following assumptions: for d = 1, k = s, s > 1 ; for d =, k = s, s > 17 3 ; for d = 3, k = s, s > 7 ; for d = 3, k, s > 7, k + s > 3 Moreover, in each of these cases, a polynomial in time bound is obtained for the growth of the norms The method of almost conservation laws has been also applied in [1], to obtain global solutions for a wave-schrödinger system in dimensions 3 and 4 that in particular, for d = 3, implies GWP for (14) with k = s, s > 57 5 In this paper, we prove 4 GWP for d = 3, k = s = 0, by a scheme involving (15) and direct application of the Strichartz estimates for the Schrödinger operator and Minkowski s integral inequality applied to the Klein-Gordon Duhamel term 3 Theorem 13 If αβ > 0, the Klein-Gordon-Schrödinger system (14) in dimension d = 3 is globally well-posed for (u 0, n 0, n 1 ) L L H 1 Moreover, the solution (u, n) satisfies (15) and (17) n(t) L + t n(t) H 1 exp(c t u 0 L ) max(( n 0 L + n 1 H 1), u 0 L ) Remark 14 In the case where α < 0 and β < 0, global well-posedness of (14) for large smooth data was previously unknown Since our proof of Theorem 13 is based 3 Similar results hold for d = 1, d =, although for expositional convenience, we have restricted to the most delicate case d = 3

4 4 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS on the conservation of u(t) L, we do not require any Sobolev norm control obtained from the Hamiltonian and obtain global well-posedness for this case as well The proof of both Theorem 11 and 13 apply essentially the same scheme, although invoke a different space-time norm in the local theory estimates 11 Outline of method We describe the globalization scheme for the Zakharov system and the Klein-Gordon-Schrödinger system using the abstract initial value problem posed at some time t = T j Ku = F (u, n) (18) Ln = G(u) (u, n)(t j ) = (u j, n j ) Here K and L are linear differential operators of evolution type, F is a nonlinear term coupling the two equations together and G is a nonlinear term depending only upon u The fact that G does not depend upon n is used in our scheme Let W (t)n 0 denote the linear group W (t) applied to initial data n 0 solving the initial value problem Ln = 0, n(0) = n 0 Similarly, let S(t)u 0 denote the solution of Ku = 0, u(0) = u 0 We denote with W n 0 + L 1 g the solution of the linear initial value problem Lu = g, n(0) = n 0 Similarly, Su 0 + K 1 g denotes the solution of Ku = g, u(0) = u 0 We solve the second equation in our system to define n in terms of the initial data n j and u (19) n = W n j + L 1 G(u) and insert the result into the solution formula for u to obtain an integrodifferential equation for u (110) u = Su j + K 1 F (u, W n j + L 1 G(u)) Local well-posedness for problems of the form (18) often follows from a fixed point argument applied to (110) The fixed point analysis is carried out in a Banach space X [Tj,T j 1 ] of functions defined on the spacetime slab [T j, T j+1 ] R d The initial data are considered in function spaces having the unitarity property with respect to the linear solution maps (111) W (t)n 0 W = n 0 W, S(t)u 0 S = u 0 S, t, and X [Tj,T j+1 ] C([T j, T j+1 ]; S) For the applications we have in mind, the length of the time interval j := [T j, T j+1 ] is chosen to be small enough to prove a contraction estimate and the smallness condition is of the form for certain γ, β > 0 j min( u j γ S, n j β W )

5 GWP FOR ZAKHAROV AND KGS 5 Suppose that u(t) S = u 0 S for all times t where solutions of (18) are well defined If we iterate the local well-posedness argument, we will have successive time intervals [T j, T j+1 ] with uniformly lower bounded lengths unless n j W grows without bound as we increase j Suppose then at some time T j we have n j W u j γ/β S so that j = n j β W Since we have that (19) and (111) hold, any growth in n(t) W as t moves through the time interval [T j, T j+1 ] is due to the nonlinear influence of u upon n through the term L 1 G(u) Therefore, an estimate of the form (11) L 1 G(u) L [Tj,T j+1 ] W δ j G( u L [Tj,T j+1 ] S ) n j W permits an iteration of the local theory Observe that the appearance of the conserved S norm of u in this step suggests that we should retain this smallness property of the W increment of n over [T j, T j+1 ] uniformly with respect to j We then iterate the local well-posedness argument ( ) n j W m = O δ G( u 0 S ) times with time steps of uniform size = ( n j W ) β This extends the solution to the time interval [T j, T j + m ] with m = C( u 0 S ) n j 1 β+δβ W If 1 β + δβ 0, the scheme progresses to give global well-posedness for (18) Implementing this abstract scheme for specific systems requires a quantification of the parameters β and δ using the local-in-time theory for the system Notice that one way to force 1 β + δβ 0 is by demanding β 1 But β is always bigger than 1 Still 1 β + δβ 0 can be greater or equal to zero because of the contribution of δβ term for certain β > 1 and δ > 0 Calculations are required to obtain the parameters β, δ in any particular system Unfortunately, we will often find that δ is very close to zero Thus the condition that 1 β δβ 0 fails to hold for many physical systems Nevertheless for the Klein-Gordon-Schrödinger system the local well-posedness theory that we develop using the Strichartz s norms is sufficient for the above condition to hold In particular we have that β = 4 and δ = 3 (see the 4 proof of Theorem 13) and thus 1 β + δβ = 0 This approach cannot be used for the Zakharov system The main reason is that the nonlinearity G(u) has two derivatives (see equation (11)) and the local estimates are not as generous The idea now is to perform the contraction argument for (110) in a ball } B X[T {u = : u,t j j+1 ] X[T,T ( j j+1 ] j )α u 0 L where j = [T j, T j+1] This idea is implemented here through the use of the X s,b spaces with b < 1/ An easy consequence of this new iteration is that the local time

6 6 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS interval j is larger or in other words (since < 1) β is smaller In addition when we calculate the growth of the n(t) W norm, it takes more time for this norm to double in size In other words the new δ is bigger Thus since m > m and > we have a better chance to meet the requirement of m 1 1 β + δβ 0 The details are explained in the proof of Theorem 11 Remark 15 The abstract scheme described above can be applied to other evolution systems that have common features with systems (11) and (14) In particular, the scheme requires a satisfactory local well-posedness theory in a Banach space that embeds in C([T j, T j+1 ]; S), with u(t) S = u 0 S holding true, and that the nonlinear term of the second equation is independent of n As examples we mention the following systems The initial-value problem for the coupled Schrödinger-Airy equation i t u + xu = αun + β u u x R, t R (113) t n + x 3 n = γ x u u(x, 0) = u 0 (x), n(x, 0) = n 0 (x) This system arises in the theory of capillary-gravity waves The local well-posedness theory has been successively sharpened in [], [3], [9], the last paper establishing local well-posedness for (u 0, n 0 ) L H s for 3 < s 1, and in some more regular 4 spaces In [0], Pecher proved global well-posedness using I-method techniques for the harder Schrödinger-KdV system where the left hand side of the second equation of (113) includes n x n, with (u 0, n 0 ) H s H s and s > 3 when β = 0, and also for 5 s > when β 0 by dropping down from the s = 1 setting in which conservation of 3 energy yields global well-posedness Our scheme also applies to the Schrödinger-Benjamin-Ono system i t u + x u = αun x R, t R (114) t n + ν x x n = β x u u(x, 0) = u 0 (x), n(x, 0) = n 0 (x) with α, β, ν R This system has been studied in [3] where local well-posedness for u 0 H s and n 0 H s 1, with s 0 and ν 1 is established In particular it is locally well-posed for (u 0, n 0 ) L H 1/ Pecher proved [19] global well-posedness for s > 1/3 under the parameter constraints ν > 0, α < 0 and also proved local β well-posedness without the restriction ν 1 but only for s > 0 In a forthcoming paper, we establish global well-posedness results for (113) and (114) with ν 1 for (u 0, n 0 ) L H 1/

7 GWP FOR ZAKHAROV AND KGS 7 Basic estimates for the group and Duhamel terms Let U(t) = e it denote the free linear Schrödinger group For the 1d wave equation, it is convenient to factor the wave operator t x = ( t x )( t + x ), and work with reduced components, as was done in [11] Low frequencies in the time-derivative initial data create some minor difficulties, which we address in a manner slightly different than was done in [11] Consider an initial data pair (n 0, n 1 ), and we look to solve ( t x )n = 0 such that n(0) = n 0, t n(0) = n 1 Split n 1 = n 1L + n 1H into low and high frequencies, and set ˆν(ξ) = ˆn 1H(ξ), so that iξ x ν = n 1H Let (1) so that W + (n 0, n 1 )(t, x) = 1n 0(x t) 1ν(x t) + 1 W (n 0, n 1 )(t, x) = 1 n 0(x + t) + 1 ν(x + t) + 1 x x t x+t x n 1L (y) dy n 1L (y) dy ( t ± x ) (n 0, n 1 )(t, x) = 1n 1L(x) (n 0, n 1 )(x, 0) = 1n 0(x) 1ν(x) t (n 0, n 1 )(x, 0) = 1 xn 0 (x) + 1 xν(x) + 1n 1L(x) and thus n = W + (n 0, n 1 ) + W (n 0, n 1 ) has the desired properties We shall also use the notation W (n 0, n 1 ) = W + (n 0, n 1 ) + W (n 0, n 1 ) Let G(t)(n 0, n 1 ) = cos[t(i ) 1/ ]n 0 + sin[t(i )1/ ] (I ) 1/ n 1 be the free linear Klein-Gordon group, so that ( t + (1 ))G(t)(n 0, n 1 ) = 0, G(0)(n 0, n 1 ) = n 0, t G(0)(n 0, n 1 ) = n 1 Since our analysis involves tracking quantities whose size increments, rather than doubles, from one step to the next, it is imperative that we be precise about the definition of the following Sobolev norms When we write the norm H s, we shall mean exactly ( 1/ f H s = (1 + ξ ) s dξ) ˆf(ξ) ξ Define the norm ( f A s = ˆf(ξ) dξ + ξ 1 Of course, f A s f H s Let () (n 0, n 1 ) W = ( n 0 A 1/ x ξ 1 ) 1/ ξ s ˆf(ξ) dξ + n 1 A 3/ x ) 1/

8 8 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS When working with a function of t, we use the shorthand n(t) W = (n(t), t n(t)) W In our treatment of the Zakharov system, we shall track the size of the wave component n(t) in the above norm Let ( (3) (n 0, n 1 ) G = n 0 L x + n 1 H 1 x ) 1/ Again, for functions of t, we use the shorthand n(t) G = (n(t), t n(t)) G In our treatment of the Klein-Gordon-Schrödinger system, we shall track the size of the wave component n(t) in the above norm In our treatment of the Zakharov system, we shall need to work in the Bourgain spaces We define the Schrödinger-Bourgain space X0,α S, α R, by the norm ( 1/ z X S 0,α = τ + ξ α ẑ(ξ, τ) dξ dτ), ξ,τ and the one-dimensional reduced-wave-bourgain spaces X 1,α, for α R, as ( 1/ z W X ± = ξ 1 τ ± ξ α ẑ(ξ, τ) dξ dτ) 1,α ξ,τ Let ψ C0 (R) satisfy ψ(t) = 1 on [ 1, 1] and ψ(t) = 0 outside of [, ] Let ψ T (t) = ψ(t/t ), which will serve as a time cutoff for the Bourgain space estimates For clarity, we write ψ 1 (t) = ψ(t) The following two lemmas are standard in the subject, although we are focusing attention particularly on the exponent of T in these estimates Lemma 1 (Group estimates) Suppose T 1 (a) Schrödinger U(t)u 0 C(Rt;L x ) = u 0 L x If 0 b 1 1, then ψ T (t)u(t)u 0 X S T 1 b 1 u 0 L (Strichartz Estimates) If q, r, + d = d, excluding the q r case d =, q =, r =, then U(t)u 0 L q u t Lr x 0 L (b) 1-d Wave W (t)(n 0, n 1 ) C([0,T ];Wx) (1 + T ) (n 0, n 1 ) W If 0 b 1, ψ T (t) (t)(n 0, n 1 ) W X ± T 1 b (n 0, n 1 ) W 1,b (c) Klein-Gordon G(t)(n 0, n 1 ) C(Rt;G x) = (n 0, n 1 ) G Remark It is important that the first estimate in (b) and the identity in (c) do not have implicit constant multiples on the right-hand side, as these estimates will be used to deduce almost conservation laws The (1+T ) prefactor in the first estimate of (b) arises from the low frequency terms Had we made the assumption that n 1 Ḣ 1, this term could be removed and the norm W redefined so that equality is obtained The (1 + T ) prefactor will not cause trouble in our iteration since T will be selected so that T (n 0, n 1 ) W functions as an increment whose size is on par with the increment arising from the Duhamel terms (see the proof of Theorem 11 for details)

9 GWP FOR ZAKHAROV AND KGS 9 Proof The Strichartz estimates quoted in (a) were established in [1] (for a more recent reference, see [13]) The first assertion in (a) is immediate by Plancherel s theorem For the second assertion in (a), we note that [ψ T (t)u(t)u 0 ] (ξ, τ) = (ψ T ) (τ + ξ )û 0 (ξ), and consequently ψ T (t)u(t)u 0 X S c ψ T H b 1 u 0 L To complete the proof of the estimate, we note that ψ T H b 1 ψ T L + ψ T Ḣb 1 = T 1 ψ1 L + T 1 b 1 ψ 1 Ḣb 1 by scaling For the first assertion in (b), let f(x, t) solve the linear wave equation (4) t f xf = 0 with initial data f(x, 0) = n 0 (x), t f(x, 0) = n 1 (x) Let P H be the projection onto frequencies ξ 1, and P L be the projection onto frequencies ξ 1 Let D 3/ be the multiplier operator with symbol ξ 3/ By applying D 3/ P H to (4), multiplying by D 3/ P H t f and integrating in x, we obtain the conservation identity (5) P H f(t ) A 1/ x + t P H f(t ) A 3/ x = P H n 0 A 1/ + P H n 1 A 3/ To obtain low frequency estimates, we work directly from the explicit formula (6) f(x, t) = 1 n 0(x + t) + 1 n 0(x t) + 1 By applying P L and then directly estimating, we obtain x+t (7) P L f(t ) L x P L n 0 L + T P L n 1 L After applying t to (6), it can be rewritten as t f(x, t) = 1 x+t x t x t n 1 (y) dy n 0 (y) dy + 1 n 1(x + t) + 1 n 1(x t) Applying P L and then directly estimating, we obtain (8) P L t f(t ) L x T P L n 0 L + P L n 1 L Combining (5), (7), and (8), we obtain the claim The second part of (b) is proved similarly to the second part of (a) For (c), let f(t, x) solve (9) t f f + f = 0 with initial data (f(0), t f(0)) = (n 0, n 1 ) Let E be the multiplier operator with symbol (1 + ξ ) 1/ Apply E to (9), then multiply by t Ef, and finally integrate in x to obtain the asserted conservation law

10 10 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS Let U R z(t, x) = t 0 U(t t )z(t, x) dt denote the Duhamel operator corresponding to the Schrödinger operator, so that (i t + )U R z(t, x) = iz(t, x), U R z(0, x) = 0 Let so that R z(t, x) = 1 t 0 z(t s, x s)ds ( t ± x ) R z(t, x) = 1 z(t, x) R z(0, x) = 0, t R z(0, x) = 1 z(0, x) It follows that if we set n = W + R z W R z, then ( t x)n = x z and n(0, x) = 0, t n(0, x) = 0, so we define (10) W R z = W + R z W R z For the Klein-Gordon equation, let G R z(t, x) = t 0 sin[(t t )(I ) 1/ ] (I ) 1/ z(t, x) dt so that ( t + (I ))G R z = z, G R z(0, x) = 0, t G R z(0, x) = 0 Lemma 3 (Duhamel estimates) Suppose T 1 (a) Schrödinger If 0 c 1 < 1, then U R z C([0,T ];L x ) T 1 c 1 z X S 0, c1 If 0 c 1 < 1, 0 b 1, b 1 + c 1 1, then ψ T U R z X S T 1 b 1 c 1 z X S 0, c1 (Strichartz Estimates) If q, r, + d = d excluding the q r case d =, q =, r =, and similarly for q, r, then U R z C([0,T ];L x )+ U R z L q z, where indicates the Hölder dual exponent ( = 1) [0,T ] Lr x L q p p [0,T ] L r x (b) 1-d Wave If 0 c < 1, then W R z C([0,T ];Wx) T 1 c z X 1, c If 0 c < 1, 0 b, b + c 1, then ψ T R z X 1,b T 1 b c z X 1, c (c) Klein-Gordon G R z C([0,T ];Gx) z L 1 [0,T ] H 1 x Proof The second assertion in each of (a) and (b) is [11] Lemma 1(ii) For the Stricharz estimates quoted in (a), see [1][13] We next establish the first part of (a) We begin by establishing the bound (11) ψ T (t)u R z(x, t) L t L x ct 1 c 1 z X S 0, c1

11 GWP FOR ZAKHAROV AND KGS 11 Let f ξ (t) = e itξ ẑ(ξ, t), where ˆ denotes the Fourier transform in the x-variable only We have (1) ψ T (t)u R z(x, t) L t L x t L = ψ T (t) f ξ (t )dt 0 t L ξ t L ψ T (t) f ξ (t )dt 0 ξ L t Below we shall show that for a function f(t) of the t-variable alone, we have the estimate (13) ψ T (t) t 0 f(t )dt L t Assuming this, then it follows from (1) that ct 1 c 1 f H c 1 ψ T (t)u R z(x, t) L t L ct 1 c 1 f x ξ c H 1 L t ξ = ct 1 c 1 z X S 0, c1 completing the proof of (11) Now we show (13) Break f(t) = f + (t) + f (t) where ˆf (τ) = χ τ < 1 ˆf(τ) and ˆf + (τ) = χ T τ > 1 ˆf(τ) Then for f, we have T We compute ψ T (t) t 0 f (s) ds T 1/ f L T 1 c 1 f H c 1 L t and hence t 0 f + (s) ds = (χ [ t,0] f + )(0) = 1 (χ [ t,0] f + ) (σ) dσ π = 1 1 e itσ ˆf + (σ) dσ π iσ σ [ t T ψ T (t) f + (s) ds ] (τ) ˆψ(T (τ σ)) T ˆψ(T τ) = ˆf + (σ) dσ 0 σ σ

12 1 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS Thus, ψ T (t) t 0 f + (s) ds L t T ˆψ(T τ) L 1 τ ˆf+ (τ) τ ( τ > 1 T dτ τ c 1 L 1 τ ) 1/ f H c1 T 1 c 1 f H c 1 establishing (13) It remains only to show continuity, ie that for a fixed z X S 0, c 1 and each ɛ > 0, there is δ = δ(ɛ, z) > 0 such that if t t 1 < δ, then ψ T U R z(x, t ) ψ T U R z(x, t 1 ) L x < ɛ By an ɛ/3 argument appealing to (11), it suffices to establish this statement for z belonging to the dense class S(R ) X0, c S 1 However, if z S(R ), we have t (U R z) = z + i (U R z) and the fundamental theorem of calculus and (11) imply that U R z(, t ) U R z(, t 1 ) L x c(t t 1 )( z L t L x + z X S 0, c 1 ) The proof of the first assertion of (b) proceeds in analogy to the above proof, first establishing the bound R z L t H 1/ x T 1 c z X 1, c The continuity statement is deduced by a density argument as in the previous paragraph, and finally the bound as stated on W R z follows by the identity t W R z = x W + R z + x W R z The proof of (c) follows from an application of Minkowskii s integral inequality, with the continuity statement deduced by a density argument as in the previous paragraph 3 1-d Zakharov system In this section, we prove Theorem 11 We shall make use of the conservation law (1) to control the growth of u(t) from one local time step to the next We track the growth of n(t) in the norm W defined in () using the estimates from the local theory We now state the needed estimates from the local theory of [11] Lemma 31 (Multilinear estimates) (a) If 1 4 < b 1, c 1, b < 1 and b + b 1 + c 1 1, then n ± u X S 0, c1 n ± X 1,b u X S

13 GWP FOR ZAKHAROV AND KGS 13 (b) If 1 4 < b 1, c < 1 and b 1 + c 1, then x (u 1 ū ) X 1, c u 1 X S u X S We remark that we can simultaneously achieve both optimal conditions b + b 1 + c 1 = 1 and b 1 + c = 1, for example by taking all four indices b = b 1 = c = c 1 = 1 3 Proof (a) is the case k = 0, l = 1 in [11] Lemma 43, and (b) is the case k = 0, l = 1 in [11] Lemma 44 The assumptions b + c 1 > 3, b + b 4 1 > 3 for Lemma 43 4 and b 1 + c > 3 for Lemma 44 that appear in [11] are not needed and we only have 4 the requirements b + b 1 + c 1 1 for Lemma 43 and b 1 + c 1 for Lemma 44 The reason is that equation (430) in [11] on p 44 is finite even if α 1 < 1 since the range of integration is finite 4 (from 0 to ξ1 /4) Because relaxing this condition is essential to our method, we have included these proofs in the appendix so that they can be examined by the reader Proof of Theorem 11 As discussed above, we can reduce the wave component n = n + + n and recast (11) as (31) { i t u + xu = (n + + n )u x R, t R ( t ± x )n ± = ± 1 x u + 1 n 1L which has the integral equation formulation u(t) = U(t)u 0 iu R [(n + + n )u](t) n ± (t) = (t)(n 0, n 1 ) ± R ( x u )(t) Fix 0 < T < 1, and consider the maps Λ S, Λ (3) (33) Λ S (u, n ± ) = ψ T Uu 0 + ψ T U R [(n + + n )u] Λ (u) = ψ T (n 0, n 1 ) ± ψ T R ( x u ) We seek a fixed point (u(t), n ± (t)) = (Λ S (u, n ± ), Λ (u)) Estimating (3) in X0,b S 1, applying the first estimates in Lemma 1(a), 3(a) and following through with Lemma 31(a); and estimating (33) in X 1,b, applying the first estimates in Lemma 1(b), 3(b) and following through with Lemma 31(b), we obtain Λ S (u, n ± ) X S T 1 b 1 u 0 L + T 1 b 1 c 1 n ± X 1,b u X S Λ (u) X 1,b T 1 b (n 0, n 1 ) W + T 1 b c u X S 0,b 1 4 This comment applies in the k = 0, l = 1 which Lemmas 43,44 are stated setting but perhaps not in the general setting in

14 14 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS and also Λ S (u 1, n 1± ) Λ S (u, n ± ) X S T 1 b 1 c 1 ( n 1± X 1,b u 1 u X S + n 1± n ± X 1,b u X S ) Λ (u 1 ) Λ (u ) X 1,b T 1 b c ( u 1 X S + u X S ) u 1 u X S By taking T such that (34) (35) T 3 b 1 c 1 u 0 L 1, T 3 b b 1 c u 0 L 1 T 3 b b 1 c 1 (n 0, n 1 ) W 1 T 3 b 1 c u 0 L (n 0, n 1 ) W one obtains sufficient conditions for a contraction argument yielding the existence of a fixed point u X0,b S 1, n ± X of (3)-(33) such that 1,b (36) u X S T 1 b 1 u 0 L, n ± X 1,b T 1 b (n 0, n 1 ) W Similarly estimating (3) in C([0, T ]; L x) by applying Lemmas 1(a),3(a) and (36) shows that in fact u C([0, T ]; L x) We may therefore invoke the conservation law (1) to conclude u(t ) L x = u 0 L and thus are concerned only with the possibility of growth in n(t) W from one time step to the next Suppose that after some number of iterations we reach a time where n(t) W u(t) L = u 0 x L Take this x time position as the initial time t = 0 so that u 0 L (n 0, n 1 ) W Then (35) is automatically satisfied and by (34), we may select a time increment of size (37) T (n 0, n 1 ) 1/( 3 b b 1 c 1 ) W = (n 0, n 1 ) W where the right-hand side follows by selecting the optimal condition b + b 1 + c 1 = 1 in Lemma 31(a) Since n = W (n 0, n 1 ) + W R ( x u ) we can apply Lemma 1(b), 3(b) and follow through with (36) to obtain n(t ) W (1 + T ) (n 0, n 1 ) W + CT 3 (b 1+c) u 0 L (n 0, n 1 ) W + CT 1 ( u0 L + 1) where C is some fixed constant The second line above follows by selecting the optimal condition b 1 +c = 1 in Lemma 31(b), and using (37) to obtain T (n 0, n 1 ) W CT 1 From this we see that we can carry out m iterations on time intervals, each of length (37), where (38) m (n 0, n 1 ) W T 1 ( u 0 L + 1)

15 GWP FOR ZAKHAROV AND KGS 15 before the quantity n(t) W doubles The total time we advance after these m iterations, by (37) and (38), is 1 mt u 0 L + 1 which is independent of n(t) W We can now repeat this entire procedure, each time advancing a time of length ( u 0 L + 1) 1 (independent of the size of n(t) W ) Upon each repetition, the size of n(t) W will at most double, giving the exponential-in-time upper bound stated in Theorem d Klein-Gordon Schrödinger system The goal of this section is to prove Theorem 13 For the Klein-Gordon-Schrödinger system (14), no special multilinear estimates are needed Instead, we will work in standard space-time norms and use Sobolev imbedding and the Hölder inequality We shall use the conservation law (15) to control the growth of u(t) from one time step to the next and track the growth of n(t) G, where the G norm was defined in (3), by direct estimation For expositional convenience, we restrict to dimension d = 3 although similar results do hold for d = 1 and d = Proof of Theorem 13 (14) has the integral equation formulation Define the maps Λ S, Λ G as (41) (4) u(t) = U(t)u 0 + iu R [nu](t) n(t) = G(t)(n 0, n 1 ) + G R ( u )(t) Λ S (u, n) = Uu 0 + iu R [nu] Λ G (u) = G(n 0, n 1 ) + G R ( u ) Let Str = L 10/3 [0,T ] L10/3 x L 8 [0,T ] L1/5 x We seek a fixed point (u(t), n(t)) = (Λ S (u, n), Λ G (u)) in the space [C([0, T ]; L x ) Str] C([0, T ]; G x) Apply Lemma 3(a) with ( q, r) = ( 0, 5) for d = 3 to obtain 9 Λ S (u, n) C([0,T ];L x ) Str u 0 L + T 1/4 n L [0,T ] L u x L 10 3 [0,T ] L 10 3 x Estimate (4) in C([0, T ]; G x ) and apply Lemma 3(c) followed by Sobolev imbedding to obtain (43) Λ G (u) C([0,T ];Gx) (n 0, n 1 ) G + ct 3/4 u L 8 [0,T ] L1/5 x where we estimated as: u L 1 [0,T ] Hx 1 u L 1 T 3/4 u There [0,T ] L 6/5 x L 8 [0,T ] L1/5 x are similar estimates for the differences Λ S (u 1, n 1 ) Λ S (u, n ) and Λ G (u 1 ) Λ G (u )

16 16 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS If T is such that (44) (45) T 1/4 u 0 L 1 T 1/4 (n 0, n 1 ) G 1 T 3/4 u 0 L (n 0, n 1 ) G then a contraction argument implies there is a solution (u, n) to (14) on [0, T ] such that (46) (47) u C([0,T ];L x ) Str u 0 L n C([0,T ];Gx) (n 0, n 1 ) G + ct 3/4 u 0 L By the conservation of mass, we have u(t) L x = u 0 L, and are thus concerned only with the possibility that n(t) G grows excessively from one local increment to the next Suppose that after some number of iterations n(t) G u(t) L = u 0 L Consider this time as the initial time so that (n 0, n 1 ) G u 0 L Then (45) is automatically satisfied, and by (44), we may thus take (48) T (n 0, n 1 ) 4 G We see from (47) that, after m iterations, each of size (48), where m (n 0, n 1 ) G T 3/4 u 0 L the quantity n(t) G at most doubles The total time advanced after these m iterations is 1 mt u 0 L We can now repeat this entire procedure, each time advancing a time of length u 0 L (independent of the size of n(t) G ) Upon each repetition, the size of n(t) G will at most double, giving the exponential-in-time upper bound stated in Theorem 13 Appendix A Proof of the multilinear estimates (expository) In this section, we prove Lemma 31 The material here is taken from [11] Lemma 43, 44 with only a slight modification at one stage This modification was described in a note under the heading proof following the statement of Lemma 31 Given its importance in our scheme, the full proof is included here in detail We need the calculus lemmas: Lemma A1 ([11] Lemma 41) Let f L q (R), g L q (R) for 1 q, q and = 1 Assume that f, g are nonnegative, even, and nonincreasing for positive q q argument Then f g enjoys the same properties

17 GWP FOR ZAKHAROV AND KGS 17 Define λ if λ > 0 [λ] + = ɛ if λ = 0 0 if λ < 0 Lemma A ([11] Lemma 4) Let 0 a a + and a + + a > 1 Then s R, where α = a [1 a + ] + y y s a + y + s a dy c s α Proof of Lemma 31(a) We shall only do the + case The estimate is equivalent to S c v v 1 v where (A1) S = ˆvˆv 1ˆv ξ 1/ σ b σ 1 c 1 σ b 1 with ˆv = ˆv(ξ, τ), ˆv 1 = ˆv 1 (ξ 1, τ 1 ), ˆv = ˆv (ξ, τ ), σ 1 = τ 1 + ξ 1, σ = τ + ξ, σ = τ + ξ, and indicates the restriction ξ 1 = ξ + ξ, τ 1 = τ + τ Indeed, for ˆv 1 L, n + u(ξ 1, τ 1 ) σ 1 c1ˆv 1 (ξ 1, τ 1 ) dξ 1 dτ 1 ξ 1,τ 1 [ ] = ˆn + (ξ, τ)û(ξ, τ ) σ 1 c1ˆv 1 (ξ 1, τ 1 )dξ 1 dτ 1 ξ 1,τ ξ 1 =ξ+ξ 1 τ 1 =τ+τ Let ˆv(ξ, τ) = ˆn + (ξ, τ) ξ 1/ σ b and ˆv (ξ, τ ) = û(ξ, τ ) σ b 1 We note here that to obtain (A1) σ 1 σ σ = τ 1 + ξ 1 (τ + ξ) (τ + ξ ) = ξ 1 ξ ξ = (ξ 1 1 ) (ξ 1 )

18 18 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS In the case when ξ 1, it suffices to estimate ˆvˆv 1ˆv ξ 1,τ ξ 1 =ξ+ξ 1 σ 1/4 σ 1 1/4+ σ 1/4+ τ 1 =τ+τ ( ) ( ) ˆv1 ˆvˆv = ξ 1,τ 1 σ 1 1/4+ ξ 1 =ξ+ξ σ 1/4 σ dξ dτ 1/4+ dξ 1 dτ 1 τ 1 =τ+τ [ ] [ ] ˆv 1 ˆvˆv = x 1,t 1 σ 1 1/4+ ξ=ξ 1 +ξ σ 1/4 σ dξ dτ 1/4+ τ=τ 1 +τ [ ] [ ] [ ] ˆv 1 ˆv ˆv = σ 1 1/4+ σ 1/4 σ 1/4+ x,t [ ˆv 1 ] L σ 1 1/4+ 8/3 t L 4 x [ ] ˆv σ 1/4 L 4 t L x [ ˆv ] L σ 1/4+ 8/3 t L 4 x Now [ ] ˆv = σ 1/4 e ixξ L 4 t L ξ τ x = e itτ ˆv(ξ, τ) dτ τ τ + ξ 1/4 e itτ ˆv(ξ, τ) dτ τ τ + ξ 1/4 itτ ˆv(ξ, τ ξ) e dτ τ τ 1/4 e itτ ˆv(ξ, τ ξ) dτ τ = ˆv L ξ L τ = v L L 4 t L ξ L ξ L 4 t e itτ ˆv(ξ, τ) τ + ξ L ξ L 4 t L ξ L t dτ dξ 1/4 by Minkowskii by Sobolev L 4 t L x For j = 1,, the estimate on [ ] ˆv j L σ j 1/4+ 8/3 t L 4 x is obtained by interpolating halfway between [ σ j dˆv j ] L 4 t L v x j L t L for d > 1 x (Strichartz) and [ˆv j ] L t L = v x j L t L This leaves us to estimate x (A) S = ˆvˆv 1ˆv ξ 1/ σ b σ 1 c 1 σ b 1

19 GWP FOR ZAKHAROV AND KGS 19 with ˆv = ˆv(ξ, τ), ˆv 1 = ˆv 1 (ξ 1, τ 1 ), ˆv = ˆv (ξ, τ ), σ 1 = τ 1 + ξ 1, σ = τ + ξ, σ = τ + ξ, and indicates the restriction ξ 1 = ξ + ξ, τ 1 = τ + τ Region σ dominant, σ max( σ 1, σ ) (A3) ( S ξ,σ ( ( ) ( 1/ ˆv σ b ˆv 1ˆv ξ 1/ ξ,σ ξ,σ σ 1 c 1 σ b 1 ) 1/ ( ( ˆv σ b ˆv 1ˆv ξ,σ ξ,σ ξ,σ sup σ b σ,ξ σ ξ ξ σ 1 c 1 σ b 1 dξ dσ ) ( ) 1/ ξ,σ ξ σ c 1 σ b 1 ) 1/ v v 1 v )) 1/ The inner integral is taken over fixed σ, ξ, σ Since ξ 1 = ξ + ξ, we have dξ 1 = dξ, and since σ 1 σ σ = (ξ 1 1 ) (ξ 1 ), we have dσ 1 = [+(ξ 1 1 ) (ξ 1 )]dξ 1 = ξdξ Thus, the quantity in parentheses in (A3) is bounded by sup σ b dσ 1 dσ σ σ σ 1 σ 1 c 1 σ b 1 sup σ b σ [1 c 1] + σ [1 b 1] + σ sup σ b+[1 c 1] + +[1 b 1 ] + σ since σ is dominant If b, c 1, b 1 < 1, then the exponent is b c 1 b 1, and it suffices to have b + c 1 + b 1 1 Region σ 1 dominant, σ 1 max( σ, σ ) By the Cauchy-Schwarz method, it suffices to show (A4) sup σ 1 c 1 ξ σ b σ b 1 dξ dσ ξ 1,σ 1 ξ σ is finite Subregion ξ 1 1 ξ 1 Then ξ 3 ξ 1 The inner integral over ξ is taken with σ 1, ξ 1, σ fixed Since σ 1 σ σ = (ξ 1 1 ) (ξ 1 ), we have dσ = (ξ 1 )dξ and thus (A4) is bounded by sup σ 1 σ 1 c 1 σ1 σ =0 σ1 σ b 1 dσ σ b dσ σ=0 sup σ 1 σ 1 c 1+[1 b 1 ] + +[1 b] +

20 0 JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS If b, b 1 < 1, then the exponent here is b 1 b c 1, and thus we need b 1 +b+c 1 1 Subregion ξ 1 1 ξ 1 Since ξ = ξ 1 ξ, we have ξ 3 ξ 1 1 Also 3 (ξ ) (ξ 1 1 ) (ξ 1 ) = σ 1 σ σ 3 σ 1, and thus (ξ 1 1 ) 4 σ 1 Thus (A4) is bounded by sup ξ 1 1 4c 1 σ b σ b 1 dξ dσ ξ 1,σ 1 ξ We change variables y = (ξ 1 ) to obtain y= ξ1 (A5) sup ξ 1 1 4c 1 y 1/ σ b σ b 1 dσ dy ξ 1,σ 1 y= ξ 1 σ σ The inner integral over σ is taken with fixed y = (ξ 1 ), ξ 1, σ 1, and thus σ + σ = σ 1 (ξ 1 1 ) (ξ 1 ) = σ 1 (ξ 1 1 ) y is fixed By Lemma A, σ b σ b 1 dσ c σ 1 (ξ 1 1 ) y α if b 1 + b > 1 σ where α = b [1 b 1 ] + if b 1 b or = b 1 [1 b] + if b b 1 By Lemma A1 with f(y) = χ ξ1 y ξ 1 (y) y 1/ and g(y) = y α, sup σ 1 + ξ1 y= ξ 1 y 1/ σ 1 (ξ 1 1 ) y α dy + ξ1 y= ξ 1 y 1/ y α dy ξ 1 [1 α] + and hence (A5) is controlled by sup ξ 1 ξ 1 1 4c 1+[1 α] + We now consider the exponent Suppose b, b 1 < 1 but b 1 + b > 1 Then α = 1 + b + b 1 Case 1 α > 1 b + b 1 > 3 4 Then we need c Case α = 1 b + b 1 = 3 4 Then we need c 1 > 1 4 Case 3 α < 1 b + b 1 < 3 4 Then the exponent is 4 4c 1 4b 4b 1, and we need b + b 1 + c 1 1 Region σ dominant, σ max( σ, σ 1 ) This is analogous to the σ 1 dominant case, but we carry it out anyway By the Cauchy-Schwarz method, need to show (A6) sup σ b 1 ξ σ b σ 1 c 1 dσ 1 dξ 1 ξ,σ σ 1,ξ 1 is finite Subregion ξ 1 ξ 1 1 Then ξ 3 ξ 1 1 The inner integral over ξ 1

21 GWP FOR ZAKHAROV AND KGS 1 is taken with σ 1, ξ, σ fixed Since σ 1 σ σ = (ξ 1 1 ) (ξ 1 ), we have dσ = (ξ 1 1 )dξ 1 Thus (A6) is bounded by sup σ b 1 σ 1 c 1 dσ 1 σ b dσ σ σ 1 σ sup σ σ b 1+[1 c 1 ] + +[1 b] + If c 1, b < 1, then need b 1 + c 1 + b 1 Subregion ξ 1 ξ 1 1 Then ξ 3 ξ 1 Also, 3(ξ 4 1 ) (ξ 1 ) (ξ 1 1 ) = σ 1 + σ + σ 3 σ and hence (ξ 1 ) 4 σ We change variable y = (ξ 1 1 ) to obtain that (A6) is bounded by ξ (A7) sup ξ 1 4b 1 y 1/ σ b σ 1 c 1 dσ 1 dy σ,ξ y= ξ σ 1 Since σ + σ 1 = σ + (ξ 1 1 ) (ξ 1 ) = σ (ξ 1 ) + y is fixed, by Lemma A, σ 1 σ b σ 1 c 1 dσ 1 σ (ξ 1 ) + y α if b + c 1 > 1 with α = b [1 c 1 ] + if c 1 b and α = c [1 b] + if b c 1 By Lemma A1 with f(y) = χ ξ y ξ (y) y 1/ and g(y) = y α, ξ sup y 1/ σ (ξ 1 ) + y α dy σ y= ξ Hence (A7) is bounded by ξ y= ξ y 1/ y α dy ξ [1 α] + sup ξ ξ 1 4b 1+[1 α] + We now consider the exponent If b, c 1 < 1 and b + c 1 > 1, then α = 1 + b + c 1 Case 1 α < 1 b + c 1 < 3 Then the exponent is 4 4b 4 1 4b 4c 1 so we need b + b 1 + c 1 1 Case α = 1 b + c 1 = 3 Here, we need b 4 1 > 1 4 Case 3 α > 1 b + c 1 > 3 Here, we need b Proof of Lemma 31 (b) We show that the proof of Lemma 31 (b) is actually identical to that of Lemma 31 (a) We discuss only the + case The estimate is equivalent to showing W c v v 1 v

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