From the Klein-Gordon Zakharov system to a singular ... - NYU (Math)
From the Klein-Gordon Zakharov system to a singular nonlinear Schr¨odinger system Nader Masmoudi ∗and Kenji Nakanishi
†
June 18, 2008
Abstract In this paper, we continue our investigation of the high-frequency and subsonic limits of the Klein-Gordon-Zakharov system. Formally, the limit system is the nonlinear Schr¨odinger equation. However, for some special case of the parameters going to the limits, some new models arise. The main object of this paper is the derivation of those new models, together with convergence of the solutions along the limits.
R´ esum´ e Du syst`eme de Klein-Gordon Zakharov vers un syst`eme de Schr¨odinger nonlin´eaire sigulier. Dans cet article, on continue l’investigation des limites haute fr´equence et subsonique du syst`eme de Klein-Gordon-Zakharov. Formellement, le syst`eme limite est le syst`eme de Schr¨odinger nonlin´eaire. Cependant, pour un cas particulier des param`etres, on trouve un nouveau mod`ele qui contient un terme sigulier. L’objet de ce papier est de donner une d`erivation rigoureuse de ce mo`ele et de montrer la convergence dans l’espace d’´energie. ∗
Courant Institute, New York University, 251 Mercer street, New York, NY 10012, USA. Partially supported by an NSF grant Grant No DMS-0703145. † Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
1
1
Introduction
The Klein-Gordon-Zakharov system in nondimensional variables reads: ( −2 c E¨ − ∆E + c2 E = −nE, α−2 n ¨ − ∆n = ∆|E|2 ,
(1.1)
where E : R1+3 → R3 is the electric field∗ and n : R1+3 → R, is the density fluctuation of ions, c2 is the plasma frequency and α the ion sound speed. This system describes the interaction between Langmuir waves and ion sound waves in a plasma (see Dendy [9] and Bellan [3]). It can be derived from the two-fluid EulerMaxwell system (see Sulem and Sulem [23], Colin and Colin [7] and Texier [24, 25] for some rigorous derivations). We also refer to [16, Introduction] for the rescaling with physical constants. The system (1.1) has the following conserved energy Z 1 ˙ 2 + 1 ||α∇|−1 n| ˙ 2 + |n|2 + n|E|2 dx. (1.2) c2 |E|2 + |∇E|2 + c−2 |E| 2 2 Notice that this energy is at least O(c2 ) due to the first term when c goes to infinity, so it is not useful by itself to get uniform bounds when c goes to infinity and does not give a conserved quantity for the limit system. To explain the main contribution of this paper, we start by some formal consid2 erations. Taking F = eic t E, system (1.1) becomes ( −2 c F¨ + 2iF˙ − ∆F = −nF, (1.3) α−2 n ¨ − ∆n = ∆|F |2 . Its formal limit as c, α → ∞ is given by the nonlinear Schr¨odinger equation: 2iF˙ − ∆F = |F |2 F,
n = −|F |2 .
If we take the limit c → ∞ first, we get the usual Zakharov system: ( 2iF˙ − ∆F = −nF, α−2 n ¨ − ∆n = ∆|F |2 .
(1.4)
(1.5)
If we take the limit α → ∞ first in (1.1), we get the nonlinear Klein-Gordon system : c−2 E¨ − ∆E + c2 E = −|E|2 E. ∗
In our results the range of E may be Rd or Cd with arbitrary d.
2
(1.6)
It is classically known that the limit when α goes to infinity in the Zakharov system (1.5) leads to the cubic nonlinear Schr¨odinger equation (1.4) and that the limit when c goes to infinity in the cubic nonlinear Klein-Gordon system (1.6) also leads to the cubic nonlinear Schr¨odinger equation. However a more precise analysis involving the two different modes of oscillations 2 2 of (1.1), namely writing E = E1 e−ic t + E2 eic t shows that these two limits do not commute. Indeed, the non-relativistic limit of the nonlinear Klein-Gordon was studied in [13, 14]. In [14] we proved that the limit system is a coupled nonlinear Schr¨odinger system ( 2iE˙ 1 − ∆E1 − (|E1 |2 + 2|E2 |2 )E1 = 0, (1.7) 2iE˙ 2 − ∆E2 − (|E2 |2 + 2|E1 |2 )E2 = 0 which differs from the one we can derive from the Zakharov system or the one derived in [16] where we took a simultaneous limit requiring that α < c where the limit system was ( 2iE˙ 1 − ∆E1 − (|E1 |2 + |E2 |2 )E1 = 0, (1.8) 2iE˙ 2 − ∆E2 − (|E2 |2 + |E1 |2 )E2 = 0. In this paper, we will study the case where γα = 2c2 for some fixed constant γ. At the limit we will get a singular Schr¨odinger system (1.11). Formally, we see that when γ goes to infinity we recover the nonlinear Schr¨odinger system (1.8) derived in [16], and when γ goes to zero we recover the nonlinear Schr¨odinger system (1.7) derived in [14]. To write our limit system, we need the following operators, defined as functions √ of |∇| = −∆ (by using the Fourier transform) |∇| |∇| + Aγ := = PV − iγπδ(|∇| − γ), |∇| − γ + 0i |∇| − γ |∇| A− , (1.9) γ := |∇| + γ |∇|2 1 + iγπ − Aγ := (Aγ + Aγ ) = P V − δ(|∇| − γ), 2 2 2 |∇| − γ 2 where P V denotes the principal value. Our main result is as follows. Theorem 1.1. Let (E c , nc ) be a sequence of solutions for (1.1) such that c → ∞ with γ = 2c2 /α > 0 fixed, and initial data satisfying (E c (0), c−2 Ic E˙ c (0)) → (ϕ, ψ) in H 1 , (nc (0), |α∇|−1 n˙ c (0)) bounded in L2 . 3
(1.10)
∞ Let T c be its maximal existence time. Let E∞ := (E∞ 1 , E2 ) be the solution of the following nonlinear Schr¨odinger type system ⊥
2iE˙ − ∆E = |E|2 E + Aγ (E1 · E2 )E , 1 E(0) = (ϕ − iψ, ϕ − iψ), 2
(1.11)
where E⊥ = (E2 , E1 ). Let T ∞ be the maximal existence time for E∞ . Then in the limit c → ∞ with γα = 2c2 , we have lim inf T c ≥ T ∞ , and for any T < T ∞ , 2
2
−ic t ∞ E2 ) → 0 in C([0, T ]; H 1 ). E c − (eic t E∞ 1 +e
(1.12)
We have asymptotic formula also for E˙ c , nc and n˙ c , which we will give in a more precise and general version of the above theorem (see Theorem 3.1). Here we just remark that the singular part in the equation for E∞ actually comes from singular behavior of nc and n˙ c . Remark that in the limit system (1.11), the L2 norm of the solution decreases in t by the nonlinear interaction of E1 · E2 at the frequency of size γ, because of the + dissipative part of A+ γ , i.e. =Aγ = −γπδ(|∇| − γ): ∞
∂t kE
(t)k2L2x
γ =− (2π)2
Z |ξ|=γ
Z 2 −ixξ (E1 · E2 )(t, ξ)e dξ. dx 3
(1.13)
R
This property is used in a forthcoming paper [18] to study the limit when γ goes to infinity in (1.11). A similar phenomenon is known in the context of stability of nonlinear bound states, to cause the radiation damping [22] in the nonlinear Klein-Gordon equation (the linear ground state decays by the nonlinear resonance), and the relaxation of excited states [26] in the nonlinear Schr¨odinger equation (the excited states decay by the nonlinear resonance). In those cases, the operator A+ γ involving a potential gives decay in the ODE governing the amplitude of the bound states. But as far as the authors know, the above theorem seems the first observation with a rigorous proof for a nonlinear resonance leading to decrease of energy for the limit wave functions. The rest of the paper is organized as follows: First in the next section, we will rewrite our equation into a first order system such that we can formally derive the limit system. Then we restate our main result in Section 3 in the new variables, allowing more general initial data, which can introduce some additional singular terms into the limit system. After preparing some notations and tools in Section 4, we prove first a set of uniform estimates in Section 5, and then prove the convergence in Section 6.
4
We conclude the introduction with some notations used throughout the paper. hai := (1 + |a|)1/2 , ha, bi := 3 and k ≥ 1. Let T c be the maximal existence time of (Ec , N c ). Then there is 6
T > 0, depending only on the size of the above initial norm, such that T c ≥ T and kEc (t)kH 1 + kN c (t)kL2 +W k,p is uniformly bounded on [0, T ] for large c. Moreover, assume that the initial data satisfies as c → ∞ Ec (0) → ∃Φ∞ in H 1 , e−σiαγt ei|α∇|t N c (0) → ∃µσ∞ in D0 ((0, ∞) × R3 ), for σ = 0, +. Let E∞ be the solution of the following limit system ∞⊥ ∞ +∞ 2iE˙ ∞ − ∆E∞ = [|E∞ |2 + 0 be the maximal existence time of E∞ . Then we have a lower bound lim inf T c ≥ T ∞ , and for all T < T ∞ we have uniform convergence Ec − E∞ → 0 in C([0, T ]; H 1 ),
(3.4)
and also, by decomposing N c = N f c + N 0c + N +c + N −c according to (2.6), N 0c + |E∞ |2 − N 0Ic → 0 in C([0, T ]; L2 ), −Ic ∞ ∞ → 0 in C([0, T ]; L2 ), N −c + eiαγt A− γ (E1 · E2 ) − N
(3.5)
∞ ∞ +Ic N +c + e−iαγt A+ → 0 in C([0, T ]; L2 + W k,p ), γ (E1 · E2 ) − N
for all p > 3, where N f c = ei|α∇|t N c (0),
N 0Ic = ei|α∇|t |Φ∞ |2 ,
∞ ∞ N +Ic = ei|α∇|t A+ γ (Φ1 · Φ2 ),
∞ ∞ N −Ic = ei|α∇|t A− γ (Φ1 · Φ2 ).
(3.6)
Moreover, we have lim keis|∇| (N c − N f c )(t)kL2 +W k,p = 0
s→∞
(3.7)
uniformly for t ∈ [0, T ] and for large c. Remark 3.2. The uniform bound of N c (0) implies that the convergence to µσ∞ in (3.2) actually holds *-weakly in L∞ (0, ∞; L2 + W k,p ), so that we can make sense of the products with µσ∞ in the limit system. Remark 3.3. (3.7) implies that the singular parts µσ∞ are preserved for later time, namely e−iσαγt eiα(t−t0 )|∇| N c (t0 ) → µσ∞ .
(3.8)
In other words, the singular initial layer N +Ic does not affect these terms (neither do the regular ones N 0Ic and N −Ic ). This follows from the decay property of eit|∇| A+ γ, see Lemma 4.3. 7
In particular, if we start with initial data N c (0) bounded in H σ for some σ ∈ R, then we will never encounter µ∗∞ , because for any χ ∈ C0∞ (R3 ) we have p (3.9) kχ(x/R)ei|α∇|t N c (0)kL2 H σ . R/αkN c (0)kH σ , see [16, Lemma 8.1]. Hence nontrivial µ∗∞ can be created only from singular (in the Fourier space) initial data. For example, if µ00 and µ+ 0 are bounded complex-valued measures on 2 [0, b) and (a, b) × S respectively, then Z b c −1 |ξ|−2 δ(|ξ| − τ /α)µ00 (τ )dτ N (0) := F3 0 Z b (3.10) −1 + F3 δ(|ξ| − γ − τ /α)µ+ 0 (τ, ξ/|ξ|)dτ =
a −1 F3 α|ξ|−2 µ00 (α|ξ|)
+ F3−1 αµ+ 0 (α(|ξ| − γ), ξ/|ξ|)
is bounded in W k,∞ for any k ∈ N, and the limit profiles are given by Z b 0∞ −3 µ (t) = (2π) eiτ t µ00 (τ )dτ = (2π)−2 F1−1 µ00 (τ ), 0 (3.11) Z b +∞ −1 iτ t + −1 + µ (t) = F3 δ(|ξ| − γ) e µ0 (τ, ξ/|ξ|)dτ = F4 δ(|ξ| − γ)µ0 (τ, ξ/|ξ|). a
Note that µ and µ do not see each other because of the rapid oscillation e−iαγt . If we choose N c (0) = N (0) independent of c, then the convergence (3.2) implies that µ0∞ is a constant, and µ+∞ is time-independent with Fourier support on {|ξ| = γ}. We remark that σ = −1 in (3.2) would give always 0 in the limit because of the oscillation eiα(|∇|+γ)t , which is uniformly rapid for all frequency ξ. 0∞
+∞
Remark 3.4. For the uniform bounds, we can sharpen the W k,p norm by replacing Lp with the Lorentz space L3,∞ . Remark 3.5. Theorem 1.1 easily follows from the above theorem by transforming the variables back to the original (E, n), in the case N c (0) is bounded in L2 and hence µ0∞ = µ+∞ = 0. However the singular part W k,p is needed even for the proof in this case. Indeed, to prove the above result, we will work on some small time interval (0, T1 ) on which we can prove some uniform estimates, then we will pass to the limit. Then, to extend the convergence to the maximal existence interval (0, T ∞ ), we need to iterate the same argument on some interval (T1 , T2 ). We notice that at the time T1 , N +c (T1 ) contains the singular part ∞ ∞ N +c (T1 ) → −A+ γ (E1 (T1 ) · E2 (T1 )),
(3.12)
which is bounded in W k,p + L2 for all p > 3 and k ∈ N by Lemma 4.3, but does not belong to L2 in general. 8
Our first order system (2.1) is not exactly invariant for time shift, because of the 2 oscillation factors e±ic t , but for the modulated translation 2
(E, N ) 7→ (eic t0 E(t + t0 ), N (t + t0 )),
(3.13)
for any t0 ∈ R. Correspondingly, we have an immediate Corollary 3.6. In the above theorem, assume instead of (3.2) 2
e−ic t0 Ec (0) → ∃Φ∞ in H 1 , e−σiαγ(t+t0 ) ei|α∇|t N c (0) → ∃µσ∞ in D0 ((0, ∞) × R3 ),
(3.14)
for some t0 ∈ R. Then we have the convergence 2
e−ic t0 Ec − E∞ → 0,
N 0c + |E∞ |2 − N 0Ic → 0,
−Ic ∞ ∞ N −c + eiαγ(t+t0 ) A− → 0, γ (E1 · E2 ) − N
(3.15)
∞ ∞ +Ic N +c + e−iαγ(t+t0 ) A+ → 0, γ (E1 · E2 ) − N
in the same topologies and with the same E∞ and N ∗Ic as above. Proof. Assume by contradiction that one of the convergences fails. Extracting a 2 subsequence of c, we may assume in addition that eic t → eiθ for some θ ∈ R. Then we can apply the above theorem replacing Φ∞ with eiθ Φ∞ and µσ∞ with e2σiθ µσ∞ . Since the limit system is invariant with respect to the “gauge transform” (E∞ , µσ ) 7→ (eiθ E∞ , e2σiθ µσ ),
(3.16)
the theorem gives all the desired convergences for this subsequence, a contradiction.
Strictly speaking, we will be using the above logic implicitly in the proof of the above theorem when extending the convergence from the first time step T1 to the maximal existence time T ∞ . Namely, we should apply the above argument to the 2 modulated translation(eic T1 E(t + T1 ), N (t + T1 )) to get the convergence in the next time step (cf. (3.8) for the persistence of (3.14)). We will not repeat this in the proof given below.
4
Preliminaries and notations
Before starting the proof, we prepare basic settings and estimates together with some notations.
9
4.1
Frequency decomposition
Let χ ∈ C0∞ (R3 ) satisfy 0 ≤ χ ≤ 1, χ(ξ) = 1 for |ξ| ≤ 4/3 and χ(ξ) = 0 for |ξ| ≥ 5/3. For any a > 0 and any function ϕ, we denote ( f≤a − f≤a/2 , (a > 1) f≤a := χ(|∇/a|)f, f>a := f − f≤a , fa := . (4.1) f≤a , (a ≤ 1) Hence we have the inhomogeneous Littlewood-Paley decomposition X f= fj , D := {1, 2, 22 , 23 , 24 . . . }
(4.2)
j∈D
In addition, we denote the non-resonant frequency part by NX := N − Nγ .
(4.3)
We note that the singularity of A+ γ is only around |ξ| = γ in the Fourier space, and so it is regular in the physical space. For bilinear interactions, we denote frequency trichotomy by f g = (f g)LH + (f g)HL + (f g)HH , X X X fh gl + fi gj , := f l gh + l4l
(4.4)
4i≥j≥i/4
where i, j, k, l, h run over the dyadic numbers D, and LH, HL and HH respectively indicate low-high, high-low and high-high frequency interactions. If no ambiguity can occur, we often abbreviate such as (f g)Y l := ((f g)Y )l and (f g)Y +Z := (f g)Y + (f g)Z where Y, Z = HH, HL or LH and l = a, > a, ≤ a. For example, (EF )HLX = ((EF )HL )X , (EF )HH>a = ((EF )HH )>a , etc.
4.2
Strichartz norms
We briefly recall the Strichartz estimate for e−it∆c /2 and eit|α∇| on R3 (see [10, 13]). For the Klein-Gordon equation, we have ke−it∆c /2 ϕkIc−µ Lp (R;Bqσ (R3 )) . kϕkH 1 ,
(4.5)
s where Bps := Bp,2 denotes the inhomogeneous Besov space (cf. [5]), provided that
θ ∈ [0, 1], p ∈ [2, ∞], (p, θ) 6= (2, 0), 1 1 1 θ θ−1 1 2θ = − + , σ =1+ , µ= 1+ , q 2 p 3p p p 3
10
(4.6)
where θ = 0 corresponds to the Strichartz estimate for the wave equation, and θ = 1 without Ic is for the Schr¨odinger equation. Moreover, for n, we have keit|α∇| ϕkα−1/p Lp (R;B˙ qσ (R3 )) . kϕkL2 ,
(4.7)
s denotes the homogeneous Besov space, provided that where B˙ qs := B˙ q,2
2 < p ≤ ∞,
1 1 1 + = , p q 2
2 σ=− . p
(4.8)
kSn f kST (N )3 . kf kST (N )∗4 ,
(4.9)
For the Duhamel terms we have similarly kSE f kST (E)1 . kf kST (E)∗2 ,
where for each ST (E)j (resp. ST (N )j ) we could choose any space in (4.5) (resp. in (4.7)), but for the sake of concreteness we choose the following specific exponents: 2/3
1 ST (E)1 = L∞ H 1 ∩ Ic−5/9 L3 B18/5 ∩ Ic−1/3 L3 B6 , 1 1 ST (E)∗2 = L1 H 1 + Ic25/36 L12/7 B9/7 + Ic1/2 L10/7 B10/7 , −2/3
ST (N )3 = L∞ L2 ∩ α−1/3 L3 B6
(4.10)
,
2/3
1/3
ST (N )∗4 = L1 L2 + α1/3 L3/2 B6/5 + α1/6 L6/5 B3/2 , where θ = 1 for the second, the fifth, and the sixth spaces, and θ = 0 for the third one. In applying the Strichartz estimates, we will write these exponents explicitly.
4.3
Fourier restriction norms
For any s ∈ R and any interval I ⊂ R, we define X s,1 := {e−i∆c t/2 u(t) | u ∈ Ht1 (I; Hxs )}, Y s,1 := {ei|α∇|t u(t) | u ∈ Ht1 (I; Hxs )}.
(4.11)
with the norms kukX s,1 (I) = kei∆c t/2 u(t)kHt1 (I;Hxs ) ,
kvkY s,1 (R) = ke−i|α∇|t v(t)kHt1 (I;Hxs ) .
(4.12)
Those norms on the whole line t ∈ R can be represented by the Fourier transform kukX s,1 (R) = khτ − ω(ξ)ihξis uˆkL2τ,ξ ,
kvkY s,1 (R) = khτ − α|ξ|ihξis vˆkL2τ,ξ .
(4.13)
The distance from the characteristic surface, such as |τ − ω(ξ)| for X s,1 , plays an essential role in using those norms. So, we consider an explicit extension from (0, T ) to R. We define an extension operator ρT for any T ∈ (0, 1) by ρT u(t) = χ(t)u(µT (t)), 11
(4.14)
where µT (t) := max(t, 2T − t, 0) and χ ∈ C0∞ (R) satisfies χ(t) = 1 for |t| ≤ 2 and χ(t) = 0 for |t| ≥ 3. It is clear that ρT u(t) = u(t) for t ∈ (0, T ), and ρT is bounded on Ht1 (0, T ; H s ) → Ht1 H s (R1+3 ) uniformly for s ∈ R and 0 < T ≤ 1. For the bilinear estimates using those norms, we introduce decomposition with respect to the distance from characteristic surface. For any β : R3 → R and δ > 1 and any function u(t, x) on R × R3 , we define P|τ −β(ξ)|≤δ u := F4−1 χ((τ − β(ξ))/δ)F4 u, P|τ −β(ξ)|>δ u := u − P|τ −β(ξ)|≤δ u.
(4.15)
Estimating in the Fourier space, we easily obtain kP|τ −ω(ξ)|>δ ukL2 H s . δ −1 kukX s,1 ,
(4.16)
kP|τ −α|ξ||>δ ukL2 H s . δ −1 kukY s,1 ,
(4.17)
s,b spaces: We can derive similar estimates in L∞ t setting without bypassing X
Lemma 4.1. We have
Z t
ρT (P f )(s)ds |τ |>δ
0
L∞ t (R;X)
. min(δ −1 , T )kf kL∞ , t (R;X)
(4.18)
uniformly for any δ > 1, any T > 0 and any Banach space X. Proof. The left hand side is bounded by (
Z t
kP|τ |>δ f kL1 (0,T ;X) . T kf kL∞ (X) ,
(P|τ |>δ f )(s)ds ≤
∞ k[ψ(δt) ∗ f ]t0 kL∞ (0,T ;X) . δ −1 kf kL∞ (X) , 0 Lt (0,T ;X)
(4.19)
where we denoted ψ(t) := F −1 τ −1 (1 − χ(τ )) ∈ L1 (R). We note here that the proof is simpler than that of lemma 2.3 of [16] due to the different order of the integration and the extension ρT .
4.4
Singular decay estimate
Here we derive some estimates on the singular operator A+ γ together with the wave propagator. First, we have a pointwise decay estimate: Lemma 4.2. For any ϕ ∈ S(R3 ) with symmetry ϕ(x) = ϕ(|x|), we have ( hxi−1 ht − |x|i−1 (|x| < t), |eit|∇| A+ ϕ(x)| . γ hxi−1 (|x| > t), uniformly for t > 0 and x ∈ R3 . 12
(4.20)
Proof. By the Laplace transform, we have it|∇|
e
A+ γ ϕ(x)
= −ie = −ie
Z
it|∇|
lim Z lim ε→+0
itγ
ε→+0
∞
|∇|eis(|∇|−γ+iε) ϕds
0 ∞
(4.21) |∇|e
i(|∇|−γ+iε)(t+s)
ϕds.
0
Let |x| = r. By the Fourier transform, the expression before the limit is equal to Z ∞Z ∞ sin(ρr) 2 −2 (4.22) ρei(ρ−γ+iε)(t+s) F3 ϕ(ρ) (2π) ρ dρds. ρr 0 0 Define f (t) by F1 f (ρ) = ρ2 F3 ϕ(ρ) for ρ > 0, and F1 f (ρ) = 0 for ρ < 0. Then Z ∞ f (t + r + s) − f (t − r + s) −(iγ+ε)(t+s) e ds (4.22) = 2πir 0 Z Z (4.23) 1 1 ∞ 0 −(iγ+ε)(t+s) = f (t + θr + s)e dsdθ. iπ −1 0 Since F1 f ∈ W 2,1 and F1 f 0 = iρF1 f ∈ W 3,1 , we have |f (t)| . hti−2 ,
|f 0 (t)| . hti−3 .
(4.24)
Hence we have for r < t, Z
−1
|(4.22)| . hri
∞
ht − r + si−2 ds . hri−1 ht − ri−1 ,
(4.25)
0
and for any r > 0, −1
|(4.22)| . hri
Z
ht + θr + si−2 ds . hri−1 ,
sup |θ|≤1
(4.26)
R
both uniformly in ε > 0. Thus we get the desired bound by ε → +0. Applied to the Littlewood-Paley decomposition, the above estimate immediately implies the following Lp decay. Lemma 4.3. If q ≥ 1, p ≤ ∞, 1/q − 1/p ≥ 2/3 and (p, q) 6= (3, 1), (∞, 3/2), then we have −3(1/q−1/p−2/3) keit|∇| A+ kϕγ kLq (R3 ) . γ ϕγ kLp (R3 ) . t
(4.27)
In addition, we have keit|∇| A+ γ ϕγ kL3,∞ (R3 ) . kϕγ kL1 (R3 ) ,
keit|∇| A+ γ ϕγ kL∞ (R3 ) . kϕγ kL3/2,1 (R3 ) ,
where Lp,q denotes the Lorentz space. 13
(4.28)
Proof. Let ψ ∈ S(R3 ) be radially symmetric and F3 ψ(ξ) = 1 for |ξ| . γ +1 including supp F3 ϕγ , so that we have it|∇| + eit|∇| A+ Aγ ψ. γ ϕγ = ϕγ ∗ e
(4.29)
Hence by the Young inequality for the Lorentz space, we have for the first case, it|∇| + keit|∇| A+ Aγ ψkLr,∞ , γ ϕγ kLp . kϕγ kLq ke
(4.30)
where 1/r = 1/p − 1/q + 1 ∈ [3, ∞], and applying the above lemma to ψ, −3(1/q−1/p−2/3) keit|∇| A+ . γ ψkLr,∞ . t
(4.31)
The second case is just the critical case for the Young inequality. We will mainly use the above Lp decay with q = 1. From (4.23), it is clear that the pointwise estimate for r > t can not be improved, and hence eit|∇| A+ γ ϕ does not 3 3 belong to L (R ) in general.
5
Uniform estimates
In this section and the next one, we prove the main theorem 3.1. The main part of the proof consists in estimating the following norms uniformly in c (and α) and for small T > 0. k Ekk := kEkStrE (0,T ) + kEkX (0,T ) ,
(5.1)
k Nkk := kN k[Strn (0,T )+L∞ (0,T ;B 1/2 )]∩Y(0,T )+L∞ (0,T ;W k,p ) , t
2,∞
t
for arbitrarily fixed k ≥ 1 and p > 3, where StrE , Strn , X and Y are defined by 1/3
1 StrE := {u ∈ L∞ (H 1 ) | u≤c ∈ L3 (B18/5 ), u>c ∈ c−1/3 L3 (B6 )}, −2/3
Strn := L∞ (L2 ) ∩ α−1/3 L3 (B6 X = Ic5/6 X 0,1 ,
),
(5.2)
Y = Ic−1/6 αY 0,1 .
Note that in the StrE norm, the frequencies lower than c are estimated in the second space in (4.10) and the higher part in the third space. The uniform estimate will be done in this section, while Section 6 will be devoted to the convergence proof. Let us outline the proof for the uniform bounds. First in Section 5.1, we derive the estimates in the space-time Fourier spaces X and Y by simple product estimates, from the Strichartz and energy bounds.
14
To estimate the Strichartz norm of E, we decompose X E =e−i∆c t/2 E(0) − SE Ic [nγ F + (nX F≤c )HH+LH + (nX(≤˜γ k/c) Fk )] k>c
X − SE Ic [ (nX(>˜γ k/c) Fk )LH + (nX F )HL ]
(5.3)
k>c
where F = E + E∗ and γ˜ = γ/ε with ε > 0 given in Lemma 5.4. The terms appearing on the first line of (5.3) will be treated in Proposition 5.2 using only Strichartz bounds. The terms on the second line of (5.3) require the use of the nonresonant property and are treated in Proposition 5.5. 1/2 To estimate NX in Strn + L∞ B2,∞ , we write N = ei|α∇|t N (0) − Sn |α∇|
X X
− Sn |α∇|
X X
k
k
hEk , E + E∗ ij
j≤˜ γ k/c
hEk , E + E∗ ij .
(5.4)
j>˜ γ k/c
For the part where j ≤ γ˜ k/c, we cannot use the nonresonant property but we can gain powers of c because j is much smaller than k. This part can be treated only by Strichartz in Proposition 5.3. The part j > γ˜ k/c, is treated in Proposition 5.6 using the nonresonant property of the interaction. Finally, the estimate on Nγ ∈ L∞ (W k,p ) is done in Section 5.4 by integrating by parts in time.
5.1
X × Y bounds from Strichartz bounds
Now we start the actual proof of theorem 1.1, or the general version 3.1. Here we derive the X and Y type estimate from the Strichartz type bounds. We have the following Proposition. Proposition 5.1. For any functions n, E and F on (0, T ) × R3 , we have kSE Ic nEkX (0,T ) . T 1/6 kN kL∞ (0,T ;L2 +W 1,p ) kEkStrE (0,T ) , kSn |α∇|hE, F ikY(0,T ) . T 1/6 kEkL∞ (0,T ;L2 ) kF kStrE (0,T ) .
(5.5)
Proof. Decompose N = N1 + N2 such that N1 ∈ L∞ L2 and N2 ∈ L∞ W k,p . We use by Sobolev that StrE ⊂ L3 L∞ + c−1/3 L3 L18 , W 1,p ⊂ L∞ , and that L18 × L2 ⊂ H −1/6 to deduce that kn1 EkL3 L2 +c−1/3 L3 H −1/6 . kN1 kL∞ L2 kEkStrE , kn2 EkL∞ L2 . kN2 kL∞ L∞ kEkL∞ L2 . kN2 kL∞ W 1,p kEkL∞ L2 .
15
(5.6)
−1/6
Since c−1/6 H −1/6 ⊂ Ic
L2 , we obtain the first estimate.
kSE Ic nEkX (0,T ) . T 1/6 kN kL∞ (L2 +W 1,p ) kEkStrE .
(5.7)
For N , we get exactly in the same way k∇hE, F ikL3 L2 +c−1/3 L3 H −1/6 . k∇EkL∞ L2 kF kStrE + k∇F kL∞ L2 kEkStrE ,
(5.8)
by putting the low frequency in the Strichartz space and hence we obtain the second estimate.
5.2
Strichartz estimate for regular interactions
To derive H 1 × L2 and Strichartz bounds for E and NX , we decompose the bilinear terms into frequencies as in (4.4). Those interactions where the less regular function has lower or similar frequency are relatively more regular. In [16], these term were treated only by the Strichartz estimate. Here, due to the low regularity, we have to treat some of those terms using their nonresonant property. We have the following estimates, which will be used with E, F = E or E∗ Proposition 5.2. For any functions E and n defined on (0, T ) × R3 , we have kSE Ic (nE≤c )HH+LH kStrE (0,T ) . T 1/4 knkL∞ (0,T ;L2 ) kEkStrE (0,T ) , kSE Ic (nE)kStrE (0,T ) . T knkL∞ (0,T ;W 1,p ) kEkL∞ (0,T ;H 1 ) , X kSE Ic (n≤˜γ k/c Ek )kStrE (0,T )
(5.9)
k>c
. (T + T 1/2 c−1/2 )knkStrn (0,T )+L∞ (0,T ;B 1/2 ) kEkStrE . 2,∞
Proof. For the first estimate, we use the Sobolev and the Strichartz estimate, hence it is bounded by 1 k(nE≤c )HH+LH kL12/7 B9/7 . T 1/4 knkL∞ L2 kEkStrE .
(5.10)
The second estimate easily follows from the energy inequality and 1,p ) kEkL∞ H 1 . knEkL∞ H 1 . knkL∞ t (W
(5.11)
For the third estimate, we decompose n = w + u such that w ∈ Strn and 1/2 u ∈ L∞ B2,∞ . For the part in Strn , we have
X
Ic (w≤˜γ k/c Ek )kH 1 . (c/k)1−2/3 kwkB −2/3 kEk kB 2/3 k 1/3 2 k 6 3 `k>c k>c (5.12) −1/2 .c kwkα−1/3 B −2/3 kE>c kc−1/6 B 2/3 . 6
16
3
Integrating in time, we get X kSE Ic (w≤˜γ k/c Ek )kStrE . T 1/2 c−1/2 kwkα−1/3 L3 B −2/3 kE>c kc−1/6 L6 B 2/3 . 6
3
(5.13)
k>c 1/2
For the part in L∞ B2,∞ , we have k
X
Ic (u≤˜γ k/c Ek )kH 1 . k(c/k)u≤˜γ k/c kL∞ kEk kH 1 `2
k>c
k>c
. kukB 1/2 kEkH 1 . 2,∞
Integrating in time, we get X kSE Ic (u≤˜γ k/c Ek )kStrE . T kukL∞ B 1/2 kEkL∞ H 1 .
(5.14)
(5.15)
2,∞
k>c
1/2
For the estimate of Sn α|∇|hE, F iX in Strn + L∞ B2,∞ where E, F = E, E∗ , we have to use the nonresonant property for almost all the interactions. However, there is a resonant case where we can only use the Strichartz estimate. The resonance we have here is actually less severe than the one at the frequency γ. This is the case when cj ∼ k ∼ l and E = E and F = E∗ . For this case, we use the following proposition Proposition 5.3. For any functions E, F on (0, T ) × R3 , we have X kSn α|∇|(F Ek )≤˜γ k/c kStrn (0,T ) . T 1/2 c−1/2 kF kStrE (0,T ) kEkStrE (0,T ) .
(5.16)
k&c
Proof. Here we use that they are HH interactions. Hence, kSn α|∇|(Fl Ek )j kStrn . αjk(Fl Ek )j kα1/3 L3/2 B 2/3 6/5
.α
2/3 5/3
j
k(Fl Ek )j kL3/2 L6/5
. α2/3 j 5/3 c−1/6 l−1 k −2/3 kFl kL2 H 1 kEk kc−1/6 L6 B 2/3
(5.17)
3
.c
−1/2
5/3
(cj/l)
kFl k
L2 H 1
kE>c kc−1/6 L6 B 2/3 , 3
which can be summed in `2j `1k `1l (cj . k ∼ l), using the Young inequality for convolution in Z (cf. [16, Lemma 2.6]), and yields a factor c−1/2 .
5.3
Bilinear estimate for nonresonant interactions
The remaining terms can not be estimated simply by using the Strichartz estimates. We need to take into account the nonresonance property and use the X s,b norms. Here nonresonance means the following simple trichotomy: one of three interacting 17
functions (including the output) must be away from the characteristic surface in the space-time Fourier space. The X s,b spaces give a gain for functions away from the characteristics as in (4.16), (4.17) and (4.18). Now we make the above statement into precise estimates. We estimate interactions of the form hδ N, E F = P|τ −ω(ξ)|>δ E, E ∗F = P|τ +ω(ξ)+2c2 |>δ E ∗ ,
(5.18)
where δ > 0 will be determined according to Lemma 5.4. We denote nF := 0 (one can take ε = 1/80), such that we have the following (i) and (ii) for large c (say c > 2(γ + 1)). Let j, k, l ∈ D be dyadic numbers. C C (i) If δ ≤ εαj and j > 1, then we have hnC j Ek | Fl it,x = 0. ∗C C (ii) If δ ≤ εαj and min(k, l) < ε γc j, then we have hnC Xj Ek | Fl it,x = 0.
Proof. By the Plancherel identity in space-time, we have Z C C C C C hnj Ek | Fl it,x = nC j Ek F l (t, x)dtdx ZZ ˆC ˆC =C nˆC j (τ0 , ξ0 )Ek (τ1 , ξ1 )Fl (τ, ξ)dξ1 dτ1 dξdτ.
(5.19)
(τ0 ,ξ0 )+(τ1 ,ξ1 )=(τ,ξ)
For the proof of the first point, we want to show that the set ˆC ˆC A = Supp(nˆC j (τ0 , ξ0 )Ek (τ1 , ξ1 )Fl (τ, ξ))∩{(τ0 , ξ0 )+(τ1 , ξ1 ) = (τ, ξ)} = ∅. We denote the distance from each characteristic surface in the integrand on the right hand side by d0 = |τ0 ∓ α|ξ0 || , d1 = |τ1 − ω(ξ1 )|, d = |τ − ω(ξ)|,
(5.20)
where we denote ω(ξ) = c2 (h|ξ|/ci−1). Assume that A 6= ∅ and let (τ0 , ξ0 , τ1 , ξ1 , τ, ξ) ∈ A. By the constraint (τ0 , ξ0 ) + (τ1 , ξ1 ) = (τ, ξ), we have 6δ > d0 + d1 + d ≥ α|ξ0 | − |ω(ξ) − ω(ξ1 )| 1 ≥ α|ξ0 | − c|ξ0 | ≥ α|ξ0 |, 2 18
(5.21)
since α = 2c2 /γ c when c is large. Hence, by choosing ε small enough, we have 6δ > d0 + d1 + d ≥ 21 α|ξ0 | ≥ 31 αj since j > 1, and we get a contradiction. Hence, A = ∅ and (i) is proved. For the proof of the second point, we argue in a similar manner. We use that the characteristic surface for E ∗ is τ + c2 (hξ/ci + 1) = 0, so the distance from the characteristic is given by d1 = |τ1 + c2 (hξ1 /ci + 1)|.
(5.22)
d0 + d1 + d ≥ α|ξ0 | − 2c2 − ω(ξ1 ) − ω(ξ) .
(5.23)
Hence we have
Since |ξ0 | 6∼ γ, we have |α|ξ0 | − 2c2 | ∼ α|ξ0 | + 2c2 & αj,
(5.24)
where we used the fact that if j = 1 > |ξ0 | then γ > 1 by the support of F3 nX and hence 2c2 = αγ > α = αj. The condition on k and l implies that ω(ξ1 ) + ω(ξ) < c(|ξ1 | + |ξ|) ≤ c(2 min(|ξ1 |, |ξ|) + |ξ0 |) ≤ (εα + 2c)j.
(5.25)
Hence we get a contradiction if ε is small enough and α, c are large. This ends the proof of (ii). Now we proceed to bilinear estimates. We start by looking at SE Ic (nE). Proposition 5.5. For any functions N and E on (0, T ) × R3 , we have kSE Ic (nX E)HL kStrE (0,T ) . (T 1/6 + c−1/2 )kNX kL∞ L2 (0,T )∩Y(0,T ) kEkStrE (0,T )∩X (0,T ) ,
(5.26)
X kSE Ic [ (nX(>˜γ k/c) Ek )LH kStrE (0,T ) k>c 1/5
. (T
(5.27) +c
−1/2
)kNX kL∞ L2 (0,T )∩Y(0,T ) kEkStrE (0,T )∩X (0,T )
where n := γ˜ k/c, the condition of the lemma holds for both cases with E = E or E = E∗ , for sufficiently large c. Hence applying to nE the same decomposition as for E, we have F C C F (nj Ek )l = (nFj Ek )l + (nC j Ek )l + (nj Ek )l .
(5.30)
Each term is estimated as follows, where we regard ε just as a constant. 1 + First we prove (5.26), hence k . j ∼ l. Using the Sobolev embedding B18/5 1/3
c−1/3 B6
1/3
1/6
⊂ Ic B∞ , we have
kSE Ic (nFj Ek )l kStrE . hl/ci−1 k(nFj Ek )l kL1 H 1 . hl/ci−1 lknFj kL2 L2 kEk kL2 L∞ α (5.31) . hl/ci−1 lhj/ci1/6 kNj kY T 1/6 hk/ci1/3 k −1/6 kEkStrE (0,T ) αj . hl/ci−1/2 T 1/6 k −1/6 kNj kY kEkStrE , which can be summed in `2l `1j `1k (k . j ∼ l) and gives kSE Ic (nF E)HL kStrE . T 1/6 kN kY kEkStrE . −3/2
Similarly, by using L2 ⊂ B∞
(5.32)
we have
−1 F C 3/2 kSE Ic (nC hk/ci−5/6 (αj)−1 kEk kX j Ek )l kStrE . hl/ci lkNj kL2 L2 k
. hl/ci−1 hk/ci−5/6 (k/c)3/2 c−1/2 kN kL2 L2 kEkX ,
(5.33)
This can be summed in `1l `1j `1k (k . j ∼ l) and gives −1/2 1/2 kSE Ic (nC E F )C T kN kL∞ L2 (0,T ) kEkX . HL kStrE . c
(5.34)
Now, using Lemma 4.1 and the Sobolev embedding, we have −1 3/2 C F −1 kNjC kL∞ L2 k 1/2 kEkC kL∞ H 1 kSE Ic (nC j Ek )l kc−1 L∞ H 3/2 . cδ hl/ci l
. (cl)−1 hl/ci−1 l3/2 k 1/2 kN kL∞ L2 kEkL∞ H 1 ,
(5.35) 3/2
which can be summed in `1j `1k (k . j ∼ l). We then use the fact that c−1 L∞ B2,∞ ⊂ c−1/2 StrE , deducing kSE Ic (nC E C )FHL kStrE . c−1/2 kN kL∞ L2 kEkL∞ H 1 . 20
(5.36)
Next we concentrate on (5.27), hence we have γ˜ k/c ≤ j ≤ k ∼ l and k > c. Using the Strichartz estimate, we have kSE Ic (nFj Ek )l kStrE . hl/ci−1 k(nFj Ek )l kIc1/2 L10/7 B 1
10/7
(0,T )
. T 1/5 hl/ci−1/2 j 9/10 knFj kL2 L2 kEk kL∞ H 1 α . T 1/5 hl/ci−1/2 j 9/10 hj/ci1/6 kN kY kEkStrE , αj
(5.37)
which can be summed in `1l `1j `1k (j ≤ k ∼ l & c), and yields a factor T 1/5 . In the same way as (5.33), we have −1 3/2 F kSE Ic (nC kNjC kL2 L2 hk/ci−5/6 (αj)−1 kEk kX j Ek )l kStrE . hl/ci lj
(5.38)
. hl/ci−11/6 lj 1/2 c−2 kN kL2 L2 kEkX , which can be summed in `1l `1k `1j (j . k ∼ l) and we get a factor c−1/2 T 1/2 . Finally, in the same way as (5.35), we have C F kSE Ic (nC j Ek )l kc−1 L∞ H 3/2 .
1 hl/ci−1 l1/2 j 3/2 kN kL∞ L2 kEkL∞ H 1 cj
(5.39)
3/2
which can be summed `1k `1j (j . k ∼ l). We then use that c−1 L∞ B2,∞ ⊂ c−1/2 StrE , getting a factor c−1/2 . Next we consider the nonresonant term in the equation for n. Proposition 5.6. For any functions E and F on (0, T ) × R3 , we have X X kSn α|∇|hEk , F + F∗ iXj kStrn (0,T )+L∞ (0,T ;B 1/2 ) 2,∞
k
(5.40)
j>˜ γ k/c
. kEkStrE (0,T )∩X (0,T ) kFkStrE (0,T )∩X (0,T ) Proof. We will denote E = E and F = F or F∗ . Decomposing into dyadic pieces, we consider interactions of the form hEk , Fl ij for N with j > γ˜ k/c. Hence, F
hEk , Fl ij = hEkF , Fl ij + hEk , FlF ij − hEkF , FlF ij + hEkC , FlC ij .
(5.41)
By using the Strichartz estimate, we have kSn |α∇|hEkF , Fl ij kStrn . αjk(EkF Fl )j kα1/6 L6/5 B 1/3 3/2
. αjα .
−1/6 1/3
j
kEkF kL2 L2 hl/ci1/3 l−2/3 kFl kIc−1/3 L3 B 2/3 6
αj (j/c)1/3 hk/ci−5/6 hl/ci1/3 l−2/3 kEkX kF kStrE . δ
21
(5.42)
This is summable in `1j `1l `1k for l . j ∼ k and for j . k ∼ l. The above term can be bounded also by αjk(EkF Fl )j kL1 L2 . αjT 1/6 kEkF kL2 L3 hl/ci1/3 l−2/3 kFl kIc−1/3 L3 B 2/3 6
αj . T 1/6 k 1/2 hk/ci−5/6 hl/ci1/3 l−2/3 kEkX kF kStrE , δ
(5.43)
which is summable in `1j `1l `1k for k . l ∼ j, yielding a factor T 1/6 . For hEk , FlF ij , we have just to switch the roles of k and l. For hEkF , FlF ij , we have a better bound kSn |α∇|hEkF , FlF ij kStrn . αjk(EkF Fl F )j kL1 L2 . (αj)−1 (kl)3/4 hk/ci−5/6 hl/ci−5/6 kEk kX kFk kX ,
(5.44)
which is summable for all j, k, l and gives a factor c−1/2 . Finally, for the last term of (5.41), we have F
kSn |α∇|hEkC , FlC ij kL∞ H 1/2 .
αj 3/2 1/2 s j (kl)−1 kEk kL∞ H 1 kFl kL∞ H 1 δ
(5.45)
where s = min(j, k, l). This can be summed in `1l `1k for k . l ∼ j, l . k ∼ j and 1/2 k ∼ l & j and gives a result in L∞ B2,∞ .
5.4
The resonant part of N
To estimate the resonant frequency part Nγ , we integrate by parts as in (2.6). Then the estimate on the boundary terms follows from Lemma 4.3. The estimate on the integral terms use the following proposition Proposition 5.7. For any functions E and F, we have ˙ Fi kL∞ L2 . T kEk ˙ L∞ H −1 kFkL∞ H 1 , kSn hE, γ ˙ F∗ i k ∞ 2 . T kEk ˙ ∞ −1 kFk ∞ kA− S hE, γ
n
γ L L
L H
L H1 ,
(5.46)
˙ ˙ ∗ kA+ γ Sn hE, F iγ kL∞ W k,p . T kEkL∞ H −1 kFkL∞ H 1 Proof. For the proof of the first two estimates, we have just to use that ˙ γ kL2 . k(FE) ˙ γ kL1 k(FE) X ˙ H −1 , . kFk kL2 kE˙ l kL2 . kFkH 1 kEk
(5.47)
k∼l
and then apply the energy estimate. For the last term, we use the above L1x bound together with Lemma 4.3 to deal with A+ γ. ˙ L∞ H −1 ≤ kEkL∞ H 1 + kEkL∞ H 1 kN kL∞ (L2 +W 1,p ) . We then use that kEk 22
5.5
Concluding the estimates
Applying the propositions of the previous subsections, we can estimate all the terms appearing in (5.3) and (5.4). Recall k Ekk and k Nkk defined in (5.1). Proposition 5.8. If (E, N ) is a solution of (2.1) on (0, T ), then we have the following a priori bound k Ekk . kE(0)kH 1 + (T 1/6 + c−1/2 )kkNkkk Ekk, k Nkk . kN (0)kL2 +W k,p + k Ekk2 (1 + TkkNkk).
(5.48)
Hence, it is clear that there exists a c0 big enough and there exists a uniform time T such that the equation can be solved for c > c0 on the time interval (0, T ).
6
Passage to the limit
In this section, we prove the convergence towards the limit system. We denote N 0∞ = −|E∞ |2 , ∞ ∞ N +∞ = −eiαγt A+ γ (E1 · E2 ),
Eω := e−i(∆c −∆)t/2 E∞ ,
∞ ∞ N −∞ = −e−iαγt A− γ (E1 · E2 ),
N σω := N σ∞ + N σI
for σ = 0, ±,
(6.1) (6.2)
where N σI = N σIc were defined in Theorem 3.1. We also denote N ∞ = N 0∞ + N +∞ + N −∞ , N I = N f + N 0I + N +I + N −I ,
(6.3)
N ω = N ∞ + N I = N 0ω + N +ω + N −ω . Taking the real value, we define also nI = K /2)kStrE ∩X . ε.
(6.16)
Hence we may replace Eω by Eω≤K in E 4 . Integrating on ei|α∇|t , we get t I I 2SE Ic N>2γ Eω≤K = ei∆c (s−t)/2 Ic (|α∇|−1 N>2γ )Eω≤K 0 I − 2iSE Ic (|α∇|−1 N>2γ )∂t Eω≤K I )Eω≤K . + SE ∆c Ic (|α∇|−1 N>2γ
(6.17)
I Since |α∇|−1 N>2γ is bounded in α−1 L∞ (H 1 +W 2,p ), the first two lines are bounded in Ic X 1,1 . For the last term, we need to integrate once more, which yields similar terms but with one more |α∇|−1 and ∆c . Since ∆2c Ic ≤ O(α3/2 |∇|) and ∆2c ≤ O(α|∇|2 ) on I H s , and |α∇|−2 N>2γ is bounded in α−2 L∞ (H 2 + W 3,p ), those terms after the second integration are bounded in c−1 StrE ∩ α−1 X . I The other terms including N >2γ or Eω∗ ≤K are treated in the same way, integrating ±i|α∇|t iα(±|∇|−γ) on the phase e or e . The denominators are never singular thanks to
25
the frequency restriction > 2γ. Actually it is not needed for the term N E∗ . Also, the same argument applies to E 5 without any low frequency cut-off, just by integrating on the phase e±iαγt or e−2iαγt . Thus we obtain kE 5 kStrE = o(1).
(6.18)
I and Eω≤K , which we further It remains to estimate the part in E 4 with N≤2γ cut-off in the physical space. Fix χ ∈ C0∞ (R3 ) satisfying χ(x) = 1 for |x| ≤ 1 and χ(x) = 0 for |x| ≥ 2. There exists R > 1 such that k(1 − χ(x/R))Eω≤K kL∞ H 1 ≤ ε and so ω⊥
knI≤2γ [(1 − χ(x/R))Eω≤K ]kL∞ H 1 + kµ∗∞ (1 − χ(x/R))E≤K kL∞ H 1 . ε.
(6.19)
Hence its contribution to E 4 is O(ε) in Ic X 1,1 . Thus we may replace E≤K further by χ(x/R)E≤K . For the singular part, we may replace N I by N f because (N I − N f )≤2γ → 0 in L∞ W k,p by Lp decay for eit|∇| and Lemma 4.3. Then the Fourier and physical cut-offs together with the time integration provide compactness for the convergence in (3.2) such that kSE Ic χ(x/R)(nI≤2γ − c1/3 kStrE . κkE − Eω kStrE + o(1), by using Proposition 5.1. For the time derivative term, we use ˙ L2 . ckEkH 1 + cknEkH −1 . c, kEk
(6.26)
and also StrE ⊂ L3 (L∞ + c−1/3 L18 ) to deduce ˙ L∞ L2 . cκ. k∂t (EE)kL2 L2 . κkEkStrE kEk
(6.27)
By the same product estimate, we have kEEkL2 H 1 . κkEk2StrE .
(6.28)
Using these bounds, we get kNX2 kY . α−1 k(i∂t + |α∇|)NX2 kL2 L2 . o(1) + κkE − Eω kStrE , kNX4 kY . α−1 k∂t (EE)kL2 L2 . c−1 .
(6.29)
Thus by adding them up, k(N + − N +ω )X kY . κkE − Eω kStrE + o(1).
(6.30)
For the Strichartz-type norms, by using Propositions 5.3 and 5.6, we have kN 1 kStrn +L∞ B 1/2 . kE>c1/3 kStrE ∩X 2,∞
. kE − Eω kStrE + kE − Eω − E 4 − E 5 kX + o(1),
(6.31)
where Eω , E 4 and E 5 are negligible for the higher frequency > c1/3 . Decomposing into the resonant frequency and the rest, we have ∞ kN 2 + N 3 kL∞ (W k,p +H 1/2 ) . k(EE)ll − E∞ 1 · E2 kL∞ (L1 ∩H 1/2 )
. kE − Eω kStrE + o(1),
(6.32)
where W k,p and L1 were used for Nγj , and H 1/2 for NXj . Using the α gain in the wave Strichartz, we have kNX4 kStrn . k∂t (EE)ll kα1/3 L3/2 B 2/3 . α−1/3 c5/9 T k∂t (EE)ll kL∞ B −1
6/5
6/5
.c
−1/9
T kE˙ ≤c1/3 kL∞ H −1 kE≤c1/3 kL∞ H 1 . c
−1/9
(6.33)
κ.
Also, by Lemma 4.3 with 1/q = 2/3 + 1/p, we have 1/3+2/p
i|α∇|t kA+ ϕγ kW k,p . kϕγ kL1 γe
27
2/3−2/p
kϕγ kL2
.
(6.34)
Hence combining it with the same Strichartz estimate as above, we get 1/3+2/p
kNγ4 kL∞ W k,p . k∂t (EE)llR kL1 L1 . Tα
−1/3(2/3−2/p)
k∂t (EE)llR k
2/3−2/p 2/3
α1/3 L3/2 B6/5
(6.35)
= o(1),
Gathering the above estimates, we get kN + − N +ω kStrn +L∞ (W k,p +B 1/2 ) 2,∞
ω
ω
. kE − E kStrE + kE − E − E 4 − E 5 kX + o(1).
(6.36)
Using the estimates (6.22), (6.30) and (6.36), as well as their counterparts for N 0 and N − , we deduce that (6.6) holds. This ends the proof of convergence. (3.7) is a direct consequence of Lemma 4.3 and (3.6).
References [1] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schr¨odinger equation: smoothness and approximation.. Funct. Anal. 79 (1988), no. 1, 183–210. [2] P. Bechouche, N. Mauser and S. Selberg, Nonrelativistic limit of Klein-GordonMaxwell to Schr¨odinger-Poisson. Amer. J. Math. 126 (2004), no. 1, 31–64. [3] P. M. Bellan, Fundamentals of plasmas physics, Cambridge University Press, Cambridge, 2006. [4] L. Berg´e, B. Bid´egaray and T. Colin, A perturbative analysis of the timeenvelope approximation in strong Langmuir turbulence. Phys. D 95 (1996), no. 3-4, 351–379. [5] J. Bergh and J. L¨ofstr¨om, Interpolation spaces, An introduction. Grundlehren Math. Wiss. 223, Springer, Berlin–Heiderberg–New York, 1976. [6] T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schr¨odinger equation in H s . Nonlinear Anal. 14 (1990), no. 10, 807–836. [7] M. Colin and T. Colin, On a quasilinear Zakharov system describing laserplasma interactions, Differential Integral Equations 17 (2004), no. 3-4, 297–330. [8] T. Colin, G. Ebrard, G. Gallice and B. Texier, Justification of the Zakharov model from Klein-Gordon–wave systems. Comm. Partial Differential Equations 29 (2004), no. 9-10, 1365–1401. 28
[9] R. O. Dendy, Plasma Dynamics, Oxford University Press, 1990. [10] J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schr¨odinger equations, Ann. Inst. H. Poincar´e Phys. Th´eor. 43 (1985), no. 4, 399–442. [11] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151 (1997), no. 2, 384–436. [12] C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems. J. Funct. Anal. 127 (1995), no. 1, 204–234. [13] S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. Math. Ann. 322 (2002), no. 3, 603–621. [14] N. Masmoudi and K. Nakanishi From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrdinger equations, Math. Ann. 324 (2002), no. 2, 359–389. [15] N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-KleinGordon and Maxwell-Dirac to Poisson-Schr¨odinger, Int. Math. Res. Not. 2003, no. 13, 697–734. [16] N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to the nonlinear Schr¨odinger equation, J. Hyperbolic Differ. Equ. 2 (2005), no. 4, 975–1008. [17] N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math. 172, (2008), no. 3, 535–583. [18] N. Masmoudi and K. Nakanishi, Two asymptotic problems for a singular nonlinear Schr¨odinger system, preprint (2008). Available at http://www.math.nyu.edu/faculty/masmoudi/index.html [19] T. Ozawa, K. Tsutaya, and Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions. Math. Ann. 313 (1999), no. 1, 127–140. [20] T. Ozawa and Y. Tsutsumi, The nonlinear Schr¨odinger limit and the initial layer of the Zakharov equations. Differential Integral Equations 5 (1992), no. 4, 721–745.
29
[21] S. Schochet and M. Weinstein, The nonlinear Schr¨odinger limit of the Zakharov equations governing Langmuir turbulence. Comm. Math. Phys. 106 (1986), no. 4, 569–580. [22] A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136 (1999), no. 1, 9–74. [23] Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schr¨odinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999, Self-focusing and wave collapse. [24] Benjamin Texier, WKB asymptotics for the Euler-Maxwell equations, Asymptot. Anal. 42 (2005), no. 3-4, 211–250. [25] Benjamin Texier, Derivation of the Zakharov equations, preprint (2006). [26] T.-P. Tsai and H.-T. Yau, Relaxation of excited states in nonlinear Schr¨odinger equations. Int. Math. Res. Not. (2002), no. 31, 1629–1673. [27] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35, 908–914.
30
†
June 18, 2008
Abstract In this paper, we continue our investigation of the high-frequency and subsonic limits of the Klein-Gordon-Zakharov system. Formally, the limit system is the nonlinear Schr¨odinger equation. However, for some special case of the parameters going to the limits, some new models arise. The main object of this paper is the derivation of those new models, together with convergence of the solutions along the limits.
R´ esum´ e Du syst`eme de Klein-Gordon Zakharov vers un syst`eme de Schr¨odinger nonlin´eaire sigulier. Dans cet article, on continue l’investigation des limites haute fr´equence et subsonique du syst`eme de Klein-Gordon-Zakharov. Formellement, le syst`eme limite est le syst`eme de Schr¨odinger nonlin´eaire. Cependant, pour un cas particulier des param`etres, on trouve un nouveau mod`ele qui contient un terme sigulier. L’objet de ce papier est de donner une d`erivation rigoureuse de ce mo`ele et de montrer la convergence dans l’espace d’´energie. ∗
Courant Institute, New York University, 251 Mercer street, New York, NY 10012, USA. Partially supported by an NSF grant Grant No DMS-0703145. † Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
1
1
Introduction
The Klein-Gordon-Zakharov system in nondimensional variables reads: ( −2 c E¨ − ∆E + c2 E = −nE, α−2 n ¨ − ∆n = ∆|E|2 ,
(1.1)
where E : R1+3 → R3 is the electric field∗ and n : R1+3 → R, is the density fluctuation of ions, c2 is the plasma frequency and α the ion sound speed. This system describes the interaction between Langmuir waves and ion sound waves in a plasma (see Dendy [9] and Bellan [3]). It can be derived from the two-fluid EulerMaxwell system (see Sulem and Sulem [23], Colin and Colin [7] and Texier [24, 25] for some rigorous derivations). We also refer to [16, Introduction] for the rescaling with physical constants. The system (1.1) has the following conserved energy Z 1 ˙ 2 + 1 ||α∇|−1 n| ˙ 2 + |n|2 + n|E|2 dx. (1.2) c2 |E|2 + |∇E|2 + c−2 |E| 2 2 Notice that this energy is at least O(c2 ) due to the first term when c goes to infinity, so it is not useful by itself to get uniform bounds when c goes to infinity and does not give a conserved quantity for the limit system. To explain the main contribution of this paper, we start by some formal consid2 erations. Taking F = eic t E, system (1.1) becomes ( −2 c F¨ + 2iF˙ − ∆F = −nF, (1.3) α−2 n ¨ − ∆n = ∆|F |2 . Its formal limit as c, α → ∞ is given by the nonlinear Schr¨odinger equation: 2iF˙ − ∆F = |F |2 F,
n = −|F |2 .
If we take the limit c → ∞ first, we get the usual Zakharov system: ( 2iF˙ − ∆F = −nF, α−2 n ¨ − ∆n = ∆|F |2 .
(1.4)
(1.5)
If we take the limit α → ∞ first in (1.1), we get the nonlinear Klein-Gordon system : c−2 E¨ − ∆E + c2 E = −|E|2 E. ∗
In our results the range of E may be Rd or Cd with arbitrary d.
2
(1.6)
It is classically known that the limit when α goes to infinity in the Zakharov system (1.5) leads to the cubic nonlinear Schr¨odinger equation (1.4) and that the limit when c goes to infinity in the cubic nonlinear Klein-Gordon system (1.6) also leads to the cubic nonlinear Schr¨odinger equation. However a more precise analysis involving the two different modes of oscillations 2 2 of (1.1), namely writing E = E1 e−ic t + E2 eic t shows that these two limits do not commute. Indeed, the non-relativistic limit of the nonlinear Klein-Gordon was studied in [13, 14]. In [14] we proved that the limit system is a coupled nonlinear Schr¨odinger system ( 2iE˙ 1 − ∆E1 − (|E1 |2 + 2|E2 |2 )E1 = 0, (1.7) 2iE˙ 2 − ∆E2 − (|E2 |2 + 2|E1 |2 )E2 = 0 which differs from the one we can derive from the Zakharov system or the one derived in [16] where we took a simultaneous limit requiring that α < c where the limit system was ( 2iE˙ 1 − ∆E1 − (|E1 |2 + |E2 |2 )E1 = 0, (1.8) 2iE˙ 2 − ∆E2 − (|E2 |2 + |E1 |2 )E2 = 0. In this paper, we will study the case where γα = 2c2 for some fixed constant γ. At the limit we will get a singular Schr¨odinger system (1.11). Formally, we see that when γ goes to infinity we recover the nonlinear Schr¨odinger system (1.8) derived in [16], and when γ goes to zero we recover the nonlinear Schr¨odinger system (1.7) derived in [14]. To write our limit system, we need the following operators, defined as functions √ of |∇| = −∆ (by using the Fourier transform) |∇| |∇| + Aγ := = PV − iγπδ(|∇| − γ), |∇| − γ + 0i |∇| − γ |∇| A− , (1.9) γ := |∇| + γ |∇|2 1 + iγπ − Aγ := (Aγ + Aγ ) = P V − δ(|∇| − γ), 2 2 2 |∇| − γ 2 where P V denotes the principal value. Our main result is as follows. Theorem 1.1. Let (E c , nc ) be a sequence of solutions for (1.1) such that c → ∞ with γ = 2c2 /α > 0 fixed, and initial data satisfying (E c (0), c−2 Ic E˙ c (0)) → (ϕ, ψ) in H 1 , (nc (0), |α∇|−1 n˙ c (0)) bounded in L2 . 3
(1.10)
∞ Let T c be its maximal existence time. Let E∞ := (E∞ 1 , E2 ) be the solution of the following nonlinear Schr¨odinger type system ⊥
2iE˙ − ∆E = |E|2 E + Aγ (E1 · E2 )E , 1 E(0) = (ϕ − iψ, ϕ − iψ), 2
(1.11)
where E⊥ = (E2 , E1 ). Let T ∞ be the maximal existence time for E∞ . Then in the limit c → ∞ with γα = 2c2 , we have lim inf T c ≥ T ∞ , and for any T < T ∞ , 2
2
−ic t ∞ E2 ) → 0 in C([0, T ]; H 1 ). E c − (eic t E∞ 1 +e
(1.12)
We have asymptotic formula also for E˙ c , nc and n˙ c , which we will give in a more precise and general version of the above theorem (see Theorem 3.1). Here we just remark that the singular part in the equation for E∞ actually comes from singular behavior of nc and n˙ c . Remark that in the limit system (1.11), the L2 norm of the solution decreases in t by the nonlinear interaction of E1 · E2 at the frequency of size γ, because of the + dissipative part of A+ γ , i.e. =Aγ = −γπδ(|∇| − γ): ∞
∂t kE
(t)k2L2x
γ =− (2π)2
Z |ξ|=γ
Z 2 −ixξ (E1 · E2 )(t, ξ)e dξ. dx 3
(1.13)
R
This property is used in a forthcoming paper [18] to study the limit when γ goes to infinity in (1.11). A similar phenomenon is known in the context of stability of nonlinear bound states, to cause the radiation damping [22] in the nonlinear Klein-Gordon equation (the linear ground state decays by the nonlinear resonance), and the relaxation of excited states [26] in the nonlinear Schr¨odinger equation (the excited states decay by the nonlinear resonance). In those cases, the operator A+ γ involving a potential gives decay in the ODE governing the amplitude of the bound states. But as far as the authors know, the above theorem seems the first observation with a rigorous proof for a nonlinear resonance leading to decrease of energy for the limit wave functions. The rest of the paper is organized as follows: First in the next section, we will rewrite our equation into a first order system such that we can formally derive the limit system. Then we restate our main result in Section 3 in the new variables, allowing more general initial data, which can introduce some additional singular terms into the limit system. After preparing some notations and tools in Section 4, we prove first a set of uniform estimates in Section 5, and then prove the convergence in Section 6.
4
We conclude the introduction with some notations used throughout the paper. hai := (1 + |a|)1/2 , ha, bi := 3 and k ≥ 1. Let T c be the maximal existence time of (Ec , N c ). Then there is 6
T > 0, depending only on the size of the above initial norm, such that T c ≥ T and kEc (t)kH 1 + kN c (t)kL2 +W k,p is uniformly bounded on [0, T ] for large c. Moreover, assume that the initial data satisfies as c → ∞ Ec (0) → ∃Φ∞ in H 1 , e−σiαγt ei|α∇|t N c (0) → ∃µσ∞ in D0 ((0, ∞) × R3 ), for σ = 0, +. Let E∞ be the solution of the following limit system ∞⊥ ∞ +∞ 2iE˙ ∞ − ∆E∞ = [|E∞ |2 + 0 be the maximal existence time of E∞ . Then we have a lower bound lim inf T c ≥ T ∞ , and for all T < T ∞ we have uniform convergence Ec − E∞ → 0 in C([0, T ]; H 1 ),
(3.4)
and also, by decomposing N c = N f c + N 0c + N +c + N −c according to (2.6), N 0c + |E∞ |2 − N 0Ic → 0 in C([0, T ]; L2 ), −Ic ∞ ∞ → 0 in C([0, T ]; L2 ), N −c + eiαγt A− γ (E1 · E2 ) − N
(3.5)
∞ ∞ +Ic N +c + e−iαγt A+ → 0 in C([0, T ]; L2 + W k,p ), γ (E1 · E2 ) − N
for all p > 3, where N f c = ei|α∇|t N c (0),
N 0Ic = ei|α∇|t |Φ∞ |2 ,
∞ ∞ N +Ic = ei|α∇|t A+ γ (Φ1 · Φ2 ),
∞ ∞ N −Ic = ei|α∇|t A− γ (Φ1 · Φ2 ).
(3.6)
Moreover, we have lim keis|∇| (N c − N f c )(t)kL2 +W k,p = 0
s→∞
(3.7)
uniformly for t ∈ [0, T ] and for large c. Remark 3.2. The uniform bound of N c (0) implies that the convergence to µσ∞ in (3.2) actually holds *-weakly in L∞ (0, ∞; L2 + W k,p ), so that we can make sense of the products with µσ∞ in the limit system. Remark 3.3. (3.7) implies that the singular parts µσ∞ are preserved for later time, namely e−iσαγt eiα(t−t0 )|∇| N c (t0 ) → µσ∞ .
(3.8)
In other words, the singular initial layer N +Ic does not affect these terms (neither do the regular ones N 0Ic and N −Ic ). This follows from the decay property of eit|∇| A+ γ, see Lemma 4.3. 7
In particular, if we start with initial data N c (0) bounded in H σ for some σ ∈ R, then we will never encounter µ∗∞ , because for any χ ∈ C0∞ (R3 ) we have p (3.9) kχ(x/R)ei|α∇|t N c (0)kL2 H σ . R/αkN c (0)kH σ , see [16, Lemma 8.1]. Hence nontrivial µ∗∞ can be created only from singular (in the Fourier space) initial data. For example, if µ00 and µ+ 0 are bounded complex-valued measures on 2 [0, b) and (a, b) × S respectively, then Z b c −1 |ξ|−2 δ(|ξ| − τ /α)µ00 (τ )dτ N (0) := F3 0 Z b (3.10) −1 + F3 δ(|ξ| − γ − τ /α)µ+ 0 (τ, ξ/|ξ|)dτ =
a −1 F3 α|ξ|−2 µ00 (α|ξ|)
+ F3−1 αµ+ 0 (α(|ξ| − γ), ξ/|ξ|)
is bounded in W k,∞ for any k ∈ N, and the limit profiles are given by Z b 0∞ −3 µ (t) = (2π) eiτ t µ00 (τ )dτ = (2π)−2 F1−1 µ00 (τ ), 0 (3.11) Z b +∞ −1 iτ t + −1 + µ (t) = F3 δ(|ξ| − γ) e µ0 (τ, ξ/|ξ|)dτ = F4 δ(|ξ| − γ)µ0 (τ, ξ/|ξ|). a
Note that µ and µ do not see each other because of the rapid oscillation e−iαγt . If we choose N c (0) = N (0) independent of c, then the convergence (3.2) implies that µ0∞ is a constant, and µ+∞ is time-independent with Fourier support on {|ξ| = γ}. We remark that σ = −1 in (3.2) would give always 0 in the limit because of the oscillation eiα(|∇|+γ)t , which is uniformly rapid for all frequency ξ. 0∞
+∞
Remark 3.4. For the uniform bounds, we can sharpen the W k,p norm by replacing Lp with the Lorentz space L3,∞ . Remark 3.5. Theorem 1.1 easily follows from the above theorem by transforming the variables back to the original (E, n), in the case N c (0) is bounded in L2 and hence µ0∞ = µ+∞ = 0. However the singular part W k,p is needed even for the proof in this case. Indeed, to prove the above result, we will work on some small time interval (0, T1 ) on which we can prove some uniform estimates, then we will pass to the limit. Then, to extend the convergence to the maximal existence interval (0, T ∞ ), we need to iterate the same argument on some interval (T1 , T2 ). We notice that at the time T1 , N +c (T1 ) contains the singular part ∞ ∞ N +c (T1 ) → −A+ γ (E1 (T1 ) · E2 (T1 )),
(3.12)
which is bounded in W k,p + L2 for all p > 3 and k ∈ N by Lemma 4.3, but does not belong to L2 in general. 8
Our first order system (2.1) is not exactly invariant for time shift, because of the 2 oscillation factors e±ic t , but for the modulated translation 2
(E, N ) 7→ (eic t0 E(t + t0 ), N (t + t0 )),
(3.13)
for any t0 ∈ R. Correspondingly, we have an immediate Corollary 3.6. In the above theorem, assume instead of (3.2) 2
e−ic t0 Ec (0) → ∃Φ∞ in H 1 , e−σiαγ(t+t0 ) ei|α∇|t N c (0) → ∃µσ∞ in D0 ((0, ∞) × R3 ),
(3.14)
for some t0 ∈ R. Then we have the convergence 2
e−ic t0 Ec − E∞ → 0,
N 0c + |E∞ |2 − N 0Ic → 0,
−Ic ∞ ∞ N −c + eiαγ(t+t0 ) A− → 0, γ (E1 · E2 ) − N
(3.15)
∞ ∞ +Ic N +c + e−iαγ(t+t0 ) A+ → 0, γ (E1 · E2 ) − N
in the same topologies and with the same E∞ and N ∗Ic as above. Proof. Assume by contradiction that one of the convergences fails. Extracting a 2 subsequence of c, we may assume in addition that eic t → eiθ for some θ ∈ R. Then we can apply the above theorem replacing Φ∞ with eiθ Φ∞ and µσ∞ with e2σiθ µσ∞ . Since the limit system is invariant with respect to the “gauge transform” (E∞ , µσ ) 7→ (eiθ E∞ , e2σiθ µσ ),
(3.16)
the theorem gives all the desired convergences for this subsequence, a contradiction.
Strictly speaking, we will be using the above logic implicitly in the proof of the above theorem when extending the convergence from the first time step T1 to the maximal existence time T ∞ . Namely, we should apply the above argument to the 2 modulated translation(eic T1 E(t + T1 ), N (t + T1 )) to get the convergence in the next time step (cf. (3.8) for the persistence of (3.14)). We will not repeat this in the proof given below.
4
Preliminaries and notations
Before starting the proof, we prepare basic settings and estimates together with some notations.
9
4.1
Frequency decomposition
Let χ ∈ C0∞ (R3 ) satisfy 0 ≤ χ ≤ 1, χ(ξ) = 1 for |ξ| ≤ 4/3 and χ(ξ) = 0 for |ξ| ≥ 5/3. For any a > 0 and any function ϕ, we denote ( f≤a − f≤a/2 , (a > 1) f≤a := χ(|∇/a|)f, f>a := f − f≤a , fa := . (4.1) f≤a , (a ≤ 1) Hence we have the inhomogeneous Littlewood-Paley decomposition X f= fj , D := {1, 2, 22 , 23 , 24 . . . }
(4.2)
j∈D
In addition, we denote the non-resonant frequency part by NX := N − Nγ .
(4.3)
We note that the singularity of A+ γ is only around |ξ| = γ in the Fourier space, and so it is regular in the physical space. For bilinear interactions, we denote frequency trichotomy by f g = (f g)LH + (f g)HL + (f g)HH , X X X fh gl + fi gj , := f l gh + l4l
(4.4)
4i≥j≥i/4
where i, j, k, l, h run over the dyadic numbers D, and LH, HL and HH respectively indicate low-high, high-low and high-high frequency interactions. If no ambiguity can occur, we often abbreviate such as (f g)Y l := ((f g)Y )l and (f g)Y +Z := (f g)Y + (f g)Z where Y, Z = HH, HL or LH and l = a, > a, ≤ a. For example, (EF )HLX = ((EF )HL )X , (EF )HH>a = ((EF )HH )>a , etc.
4.2
Strichartz norms
We briefly recall the Strichartz estimate for e−it∆c /2 and eit|α∇| on R3 (see [10, 13]). For the Klein-Gordon equation, we have ke−it∆c /2 ϕkIc−µ Lp (R;Bqσ (R3 )) . kϕkH 1 ,
(4.5)
s where Bps := Bp,2 denotes the inhomogeneous Besov space (cf. [5]), provided that
θ ∈ [0, 1], p ∈ [2, ∞], (p, θ) 6= (2, 0), 1 1 1 θ θ−1 1 2θ = − + , σ =1+ , µ= 1+ , q 2 p 3p p p 3
10
(4.6)
where θ = 0 corresponds to the Strichartz estimate for the wave equation, and θ = 1 without Ic is for the Schr¨odinger equation. Moreover, for n, we have keit|α∇| ϕkα−1/p Lp (R;B˙ qσ (R3 )) . kϕkL2 ,
(4.7)
s denotes the homogeneous Besov space, provided that where B˙ qs := B˙ q,2
2 < p ≤ ∞,
1 1 1 + = , p q 2
2 σ=− . p
(4.8)
kSn f kST (N )3 . kf kST (N )∗4 ,
(4.9)
For the Duhamel terms we have similarly kSE f kST (E)1 . kf kST (E)∗2 ,
where for each ST (E)j (resp. ST (N )j ) we could choose any space in (4.5) (resp. in (4.7)), but for the sake of concreteness we choose the following specific exponents: 2/3
1 ST (E)1 = L∞ H 1 ∩ Ic−5/9 L3 B18/5 ∩ Ic−1/3 L3 B6 , 1 1 ST (E)∗2 = L1 H 1 + Ic25/36 L12/7 B9/7 + Ic1/2 L10/7 B10/7 , −2/3
ST (N )3 = L∞ L2 ∩ α−1/3 L3 B6
(4.10)
,
2/3
1/3
ST (N )∗4 = L1 L2 + α1/3 L3/2 B6/5 + α1/6 L6/5 B3/2 , where θ = 1 for the second, the fifth, and the sixth spaces, and θ = 0 for the third one. In applying the Strichartz estimates, we will write these exponents explicitly.
4.3
Fourier restriction norms
For any s ∈ R and any interval I ⊂ R, we define X s,1 := {e−i∆c t/2 u(t) | u ∈ Ht1 (I; Hxs )}, Y s,1 := {ei|α∇|t u(t) | u ∈ Ht1 (I; Hxs )}.
(4.11)
with the norms kukX s,1 (I) = kei∆c t/2 u(t)kHt1 (I;Hxs ) ,
kvkY s,1 (R) = ke−i|α∇|t v(t)kHt1 (I;Hxs ) .
(4.12)
Those norms on the whole line t ∈ R can be represented by the Fourier transform kukX s,1 (R) = khτ − ω(ξ)ihξis uˆkL2τ,ξ ,
kvkY s,1 (R) = khτ − α|ξ|ihξis vˆkL2τ,ξ .
(4.13)
The distance from the characteristic surface, such as |τ − ω(ξ)| for X s,1 , plays an essential role in using those norms. So, we consider an explicit extension from (0, T ) to R. We define an extension operator ρT for any T ∈ (0, 1) by ρT u(t) = χ(t)u(µT (t)), 11
(4.14)
where µT (t) := max(t, 2T − t, 0) and χ ∈ C0∞ (R) satisfies χ(t) = 1 for |t| ≤ 2 and χ(t) = 0 for |t| ≥ 3. It is clear that ρT u(t) = u(t) for t ∈ (0, T ), and ρT is bounded on Ht1 (0, T ; H s ) → Ht1 H s (R1+3 ) uniformly for s ∈ R and 0 < T ≤ 1. For the bilinear estimates using those norms, we introduce decomposition with respect to the distance from characteristic surface. For any β : R3 → R and δ > 1 and any function u(t, x) on R × R3 , we define P|τ −β(ξ)|≤δ u := F4−1 χ((τ − β(ξ))/δ)F4 u, P|τ −β(ξ)|>δ u := u − P|τ −β(ξ)|≤δ u.
(4.15)
Estimating in the Fourier space, we easily obtain kP|τ −ω(ξ)|>δ ukL2 H s . δ −1 kukX s,1 ,
(4.16)
kP|τ −α|ξ||>δ ukL2 H s . δ −1 kukY s,1 ,
(4.17)
s,b spaces: We can derive similar estimates in L∞ t setting without bypassing X
Lemma 4.1. We have
Z t
ρT (P f )(s)ds |τ |>δ
0
L∞ t (R;X)
. min(δ −1 , T )kf kL∞ , t (R;X)
(4.18)
uniformly for any δ > 1, any T > 0 and any Banach space X. Proof. The left hand side is bounded by (
Z t
kP|τ |>δ f kL1 (0,T ;X) . T kf kL∞ (X) ,
(P|τ |>δ f )(s)ds ≤
∞ k[ψ(δt) ∗ f ]t0 kL∞ (0,T ;X) . δ −1 kf kL∞ (X) , 0 Lt (0,T ;X)
(4.19)
where we denoted ψ(t) := F −1 τ −1 (1 − χ(τ )) ∈ L1 (R). We note here that the proof is simpler than that of lemma 2.3 of [16] due to the different order of the integration and the extension ρT .
4.4
Singular decay estimate
Here we derive some estimates on the singular operator A+ γ together with the wave propagator. First, we have a pointwise decay estimate: Lemma 4.2. For any ϕ ∈ S(R3 ) with symmetry ϕ(x) = ϕ(|x|), we have ( hxi−1 ht − |x|i−1 (|x| < t), |eit|∇| A+ ϕ(x)| . γ hxi−1 (|x| > t), uniformly for t > 0 and x ∈ R3 . 12
(4.20)
Proof. By the Laplace transform, we have it|∇|
e
A+ γ ϕ(x)
= −ie = −ie
Z
it|∇|
lim Z lim ε→+0
itγ
ε→+0
∞
|∇|eis(|∇|−γ+iε) ϕds
0 ∞
(4.21) |∇|e
i(|∇|−γ+iε)(t+s)
ϕds.
0
Let |x| = r. By the Fourier transform, the expression before the limit is equal to Z ∞Z ∞ sin(ρr) 2 −2 (4.22) ρei(ρ−γ+iε)(t+s) F3 ϕ(ρ) (2π) ρ dρds. ρr 0 0 Define f (t) by F1 f (ρ) = ρ2 F3 ϕ(ρ) for ρ > 0, and F1 f (ρ) = 0 for ρ < 0. Then Z ∞ f (t + r + s) − f (t − r + s) −(iγ+ε)(t+s) e ds (4.22) = 2πir 0 Z Z (4.23) 1 1 ∞ 0 −(iγ+ε)(t+s) = f (t + θr + s)e dsdθ. iπ −1 0 Since F1 f ∈ W 2,1 and F1 f 0 = iρF1 f ∈ W 3,1 , we have |f (t)| . hti−2 ,
|f 0 (t)| . hti−3 .
(4.24)
Hence we have for r < t, Z
−1
|(4.22)| . hri
∞
ht − r + si−2 ds . hri−1 ht − ri−1 ,
(4.25)
0
and for any r > 0, −1
|(4.22)| . hri
Z
ht + θr + si−2 ds . hri−1 ,
sup |θ|≤1
(4.26)
R
both uniformly in ε > 0. Thus we get the desired bound by ε → +0. Applied to the Littlewood-Paley decomposition, the above estimate immediately implies the following Lp decay. Lemma 4.3. If q ≥ 1, p ≤ ∞, 1/q − 1/p ≥ 2/3 and (p, q) 6= (3, 1), (∞, 3/2), then we have −3(1/q−1/p−2/3) keit|∇| A+ kϕγ kLq (R3 ) . γ ϕγ kLp (R3 ) . t
(4.27)
In addition, we have keit|∇| A+ γ ϕγ kL3,∞ (R3 ) . kϕγ kL1 (R3 ) ,
keit|∇| A+ γ ϕγ kL∞ (R3 ) . kϕγ kL3/2,1 (R3 ) ,
where Lp,q denotes the Lorentz space. 13
(4.28)
Proof. Let ψ ∈ S(R3 ) be radially symmetric and F3 ψ(ξ) = 1 for |ξ| . γ +1 including supp F3 ϕγ , so that we have it|∇| + eit|∇| A+ Aγ ψ. γ ϕγ = ϕγ ∗ e
(4.29)
Hence by the Young inequality for the Lorentz space, we have for the first case, it|∇| + keit|∇| A+ Aγ ψkLr,∞ , γ ϕγ kLp . kϕγ kLq ke
(4.30)
where 1/r = 1/p − 1/q + 1 ∈ [3, ∞], and applying the above lemma to ψ, −3(1/q−1/p−2/3) keit|∇| A+ . γ ψkLr,∞ . t
(4.31)
The second case is just the critical case for the Young inequality. We will mainly use the above Lp decay with q = 1. From (4.23), it is clear that the pointwise estimate for r > t can not be improved, and hence eit|∇| A+ γ ϕ does not 3 3 belong to L (R ) in general.
5
Uniform estimates
In this section and the next one, we prove the main theorem 3.1. The main part of the proof consists in estimating the following norms uniformly in c (and α) and for small T > 0. k Ekk := kEkStrE (0,T ) + kEkX (0,T ) ,
(5.1)
k Nkk := kN k[Strn (0,T )+L∞ (0,T ;B 1/2 )]∩Y(0,T )+L∞ (0,T ;W k,p ) , t
2,∞
t
for arbitrarily fixed k ≥ 1 and p > 3, where StrE , Strn , X and Y are defined by 1/3
1 StrE := {u ∈ L∞ (H 1 ) | u≤c ∈ L3 (B18/5 ), u>c ∈ c−1/3 L3 (B6 )}, −2/3
Strn := L∞ (L2 ) ∩ α−1/3 L3 (B6 X = Ic5/6 X 0,1 ,
),
(5.2)
Y = Ic−1/6 αY 0,1 .
Note that in the StrE norm, the frequencies lower than c are estimated in the second space in (4.10) and the higher part in the third space. The uniform estimate will be done in this section, while Section 6 will be devoted to the convergence proof. Let us outline the proof for the uniform bounds. First in Section 5.1, we derive the estimates in the space-time Fourier spaces X and Y by simple product estimates, from the Strichartz and energy bounds.
14
To estimate the Strichartz norm of E, we decompose X E =e−i∆c t/2 E(0) − SE Ic [nγ F + (nX F≤c )HH+LH + (nX(≤˜γ k/c) Fk )] k>c
X − SE Ic [ (nX(>˜γ k/c) Fk )LH + (nX F )HL ]
(5.3)
k>c
where F = E + E∗ and γ˜ = γ/ε with ε > 0 given in Lemma 5.4. The terms appearing on the first line of (5.3) will be treated in Proposition 5.2 using only Strichartz bounds. The terms on the second line of (5.3) require the use of the nonresonant property and are treated in Proposition 5.5. 1/2 To estimate NX in Strn + L∞ B2,∞ , we write N = ei|α∇|t N (0) − Sn |α∇|
X X
− Sn |α∇|
X X
k
k
hEk , E + E∗ ij
j≤˜ γ k/c
hEk , E + E∗ ij .
(5.4)
j>˜ γ k/c
For the part where j ≤ γ˜ k/c, we cannot use the nonresonant property but we can gain powers of c because j is much smaller than k. This part can be treated only by Strichartz in Proposition 5.3. The part j > γ˜ k/c, is treated in Proposition 5.6 using the nonresonant property of the interaction. Finally, the estimate on Nγ ∈ L∞ (W k,p ) is done in Section 5.4 by integrating by parts in time.
5.1
X × Y bounds from Strichartz bounds
Now we start the actual proof of theorem 1.1, or the general version 3.1. Here we derive the X and Y type estimate from the Strichartz type bounds. We have the following Proposition. Proposition 5.1. For any functions n, E and F on (0, T ) × R3 , we have kSE Ic nEkX (0,T ) . T 1/6 kN kL∞ (0,T ;L2 +W 1,p ) kEkStrE (0,T ) , kSn |α∇|hE, F ikY(0,T ) . T 1/6 kEkL∞ (0,T ;L2 ) kF kStrE (0,T ) .
(5.5)
Proof. Decompose N = N1 + N2 such that N1 ∈ L∞ L2 and N2 ∈ L∞ W k,p . We use by Sobolev that StrE ⊂ L3 L∞ + c−1/3 L3 L18 , W 1,p ⊂ L∞ , and that L18 × L2 ⊂ H −1/6 to deduce that kn1 EkL3 L2 +c−1/3 L3 H −1/6 . kN1 kL∞ L2 kEkStrE , kn2 EkL∞ L2 . kN2 kL∞ L∞ kEkL∞ L2 . kN2 kL∞ W 1,p kEkL∞ L2 .
15
(5.6)
−1/6
Since c−1/6 H −1/6 ⊂ Ic
L2 , we obtain the first estimate.
kSE Ic nEkX (0,T ) . T 1/6 kN kL∞ (L2 +W 1,p ) kEkStrE .
(5.7)
For N , we get exactly in the same way k∇hE, F ikL3 L2 +c−1/3 L3 H −1/6 . k∇EkL∞ L2 kF kStrE + k∇F kL∞ L2 kEkStrE ,
(5.8)
by putting the low frequency in the Strichartz space and hence we obtain the second estimate.
5.2
Strichartz estimate for regular interactions
To derive H 1 × L2 and Strichartz bounds for E and NX , we decompose the bilinear terms into frequencies as in (4.4). Those interactions where the less regular function has lower or similar frequency are relatively more regular. In [16], these term were treated only by the Strichartz estimate. Here, due to the low regularity, we have to treat some of those terms using their nonresonant property. We have the following estimates, which will be used with E, F = E or E∗ Proposition 5.2. For any functions E and n defined on (0, T ) × R3 , we have kSE Ic (nE≤c )HH+LH kStrE (0,T ) . T 1/4 knkL∞ (0,T ;L2 ) kEkStrE (0,T ) , kSE Ic (nE)kStrE (0,T ) . T knkL∞ (0,T ;W 1,p ) kEkL∞ (0,T ;H 1 ) , X kSE Ic (n≤˜γ k/c Ek )kStrE (0,T )
(5.9)
k>c
. (T + T 1/2 c−1/2 )knkStrn (0,T )+L∞ (0,T ;B 1/2 ) kEkStrE . 2,∞
Proof. For the first estimate, we use the Sobolev and the Strichartz estimate, hence it is bounded by 1 k(nE≤c )HH+LH kL12/7 B9/7 . T 1/4 knkL∞ L2 kEkStrE .
(5.10)
The second estimate easily follows from the energy inequality and 1,p ) kEkL∞ H 1 . knEkL∞ H 1 . knkL∞ t (W
(5.11)
For the third estimate, we decompose n = w + u such that w ∈ Strn and 1/2 u ∈ L∞ B2,∞ . For the part in Strn , we have
X
Ic (w≤˜γ k/c Ek )kH 1 . (c/k)1−2/3 kwkB −2/3 kEk kB 2/3 k 1/3 2 k 6 3 `k>c k>c (5.12) −1/2 .c kwkα−1/3 B −2/3 kE>c kc−1/6 B 2/3 . 6
16
3
Integrating in time, we get X kSE Ic (w≤˜γ k/c Ek )kStrE . T 1/2 c−1/2 kwkα−1/3 L3 B −2/3 kE>c kc−1/6 L6 B 2/3 . 6
3
(5.13)
k>c 1/2
For the part in L∞ B2,∞ , we have k
X
Ic (u≤˜γ k/c Ek )kH 1 . k(c/k)u≤˜γ k/c kL∞ kEk kH 1 `2
k>c
k>c
. kukB 1/2 kEkH 1 . 2,∞
Integrating in time, we get X kSE Ic (u≤˜γ k/c Ek )kStrE . T kukL∞ B 1/2 kEkL∞ H 1 .
(5.14)
(5.15)
2,∞
k>c
1/2
For the estimate of Sn α|∇|hE, F iX in Strn + L∞ B2,∞ where E, F = E, E∗ , we have to use the nonresonant property for almost all the interactions. However, there is a resonant case where we can only use the Strichartz estimate. The resonance we have here is actually less severe than the one at the frequency γ. This is the case when cj ∼ k ∼ l and E = E and F = E∗ . For this case, we use the following proposition Proposition 5.3. For any functions E, F on (0, T ) × R3 , we have X kSn α|∇|(F Ek )≤˜γ k/c kStrn (0,T ) . T 1/2 c−1/2 kF kStrE (0,T ) kEkStrE (0,T ) .
(5.16)
k&c
Proof. Here we use that they are HH interactions. Hence, kSn α|∇|(Fl Ek )j kStrn . αjk(Fl Ek )j kα1/3 L3/2 B 2/3 6/5
.α
2/3 5/3
j
k(Fl Ek )j kL3/2 L6/5
. α2/3 j 5/3 c−1/6 l−1 k −2/3 kFl kL2 H 1 kEk kc−1/6 L6 B 2/3
(5.17)
3
.c
−1/2
5/3
(cj/l)
kFl k
L2 H 1
kE>c kc−1/6 L6 B 2/3 , 3
which can be summed in `2j `1k `1l (cj . k ∼ l), using the Young inequality for convolution in Z (cf. [16, Lemma 2.6]), and yields a factor c−1/2 .
5.3
Bilinear estimate for nonresonant interactions
The remaining terms can not be estimated simply by using the Strichartz estimates. We need to take into account the nonresonance property and use the X s,b norms. Here nonresonance means the following simple trichotomy: one of three interacting 17
functions (including the output) must be away from the characteristic surface in the space-time Fourier space. The X s,b spaces give a gain for functions away from the characteristics as in (4.16), (4.17) and (4.18). Now we make the above statement into precise estimates. We estimate interactions of the form hδ N, E F = P|τ −ω(ξ)|>δ E, E ∗F = P|τ +ω(ξ)+2c2 |>δ E ∗ ,
(5.18)
where δ > 0 will be determined according to Lemma 5.4. We denote nF := 0 (one can take ε = 1/80), such that we have the following (i) and (ii) for large c (say c > 2(γ + 1)). Let j, k, l ∈ D be dyadic numbers. C C (i) If δ ≤ εαj and j > 1, then we have hnC j Ek | Fl it,x = 0. ∗C C (ii) If δ ≤ εαj and min(k, l) < ε γc j, then we have hnC Xj Ek | Fl it,x = 0.
Proof. By the Plancherel identity in space-time, we have Z C C C C C hnj Ek | Fl it,x = nC j Ek F l (t, x)dtdx ZZ ˆC ˆC =C nˆC j (τ0 , ξ0 )Ek (τ1 , ξ1 )Fl (τ, ξ)dξ1 dτ1 dξdτ.
(5.19)
(τ0 ,ξ0 )+(τ1 ,ξ1 )=(τ,ξ)
For the proof of the first point, we want to show that the set ˆC ˆC A = Supp(nˆC j (τ0 , ξ0 )Ek (τ1 , ξ1 )Fl (τ, ξ))∩{(τ0 , ξ0 )+(τ1 , ξ1 ) = (τ, ξ)} = ∅. We denote the distance from each characteristic surface in the integrand on the right hand side by d0 = |τ0 ∓ α|ξ0 || , d1 = |τ1 − ω(ξ1 )|, d = |τ − ω(ξ)|,
(5.20)
where we denote ω(ξ) = c2 (h|ξ|/ci−1). Assume that A 6= ∅ and let (τ0 , ξ0 , τ1 , ξ1 , τ, ξ) ∈ A. By the constraint (τ0 , ξ0 ) + (τ1 , ξ1 ) = (τ, ξ), we have 6δ > d0 + d1 + d ≥ α|ξ0 | − |ω(ξ) − ω(ξ1 )| 1 ≥ α|ξ0 | − c|ξ0 | ≥ α|ξ0 |, 2 18
(5.21)
since α = 2c2 /γ c when c is large. Hence, by choosing ε small enough, we have 6δ > d0 + d1 + d ≥ 21 α|ξ0 | ≥ 31 αj since j > 1, and we get a contradiction. Hence, A = ∅ and (i) is proved. For the proof of the second point, we argue in a similar manner. We use that the characteristic surface for E ∗ is τ + c2 (hξ/ci + 1) = 0, so the distance from the characteristic is given by d1 = |τ1 + c2 (hξ1 /ci + 1)|.
(5.22)
d0 + d1 + d ≥ α|ξ0 | − 2c2 − ω(ξ1 ) − ω(ξ) .
(5.23)
Hence we have
Since |ξ0 | 6∼ γ, we have |α|ξ0 | − 2c2 | ∼ α|ξ0 | + 2c2 & αj,
(5.24)
where we used the fact that if j = 1 > |ξ0 | then γ > 1 by the support of F3 nX and hence 2c2 = αγ > α = αj. The condition on k and l implies that ω(ξ1 ) + ω(ξ) < c(|ξ1 | + |ξ|) ≤ c(2 min(|ξ1 |, |ξ|) + |ξ0 |) ≤ (εα + 2c)j.
(5.25)
Hence we get a contradiction if ε is small enough and α, c are large. This ends the proof of (ii). Now we proceed to bilinear estimates. We start by looking at SE Ic (nE). Proposition 5.5. For any functions N and E on (0, T ) × R3 , we have kSE Ic (nX E)HL kStrE (0,T ) . (T 1/6 + c−1/2 )kNX kL∞ L2 (0,T )∩Y(0,T ) kEkStrE (0,T )∩X (0,T ) ,
(5.26)
X kSE Ic [ (nX(>˜γ k/c) Ek )LH kStrE (0,T ) k>c 1/5
. (T
(5.27) +c
−1/2
)kNX kL∞ L2 (0,T )∩Y(0,T ) kEkStrE (0,T )∩X (0,T )
where n := γ˜ k/c, the condition of the lemma holds for both cases with E = E or E = E∗ , for sufficiently large c. Hence applying to nE the same decomposition as for E, we have F C C F (nj Ek )l = (nFj Ek )l + (nC j Ek )l + (nj Ek )l .
(5.30)
Each term is estimated as follows, where we regard ε just as a constant. 1 + First we prove (5.26), hence k . j ∼ l. Using the Sobolev embedding B18/5 1/3
c−1/3 B6
1/3
1/6
⊂ Ic B∞ , we have
kSE Ic (nFj Ek )l kStrE . hl/ci−1 k(nFj Ek )l kL1 H 1 . hl/ci−1 lknFj kL2 L2 kEk kL2 L∞ α (5.31) . hl/ci−1 lhj/ci1/6 kNj kY T 1/6 hk/ci1/3 k −1/6 kEkStrE (0,T ) αj . hl/ci−1/2 T 1/6 k −1/6 kNj kY kEkStrE , which can be summed in `2l `1j `1k (k . j ∼ l) and gives kSE Ic (nF E)HL kStrE . T 1/6 kN kY kEkStrE . −3/2
Similarly, by using L2 ⊂ B∞
(5.32)
we have
−1 F C 3/2 kSE Ic (nC hk/ci−5/6 (αj)−1 kEk kX j Ek )l kStrE . hl/ci lkNj kL2 L2 k
. hl/ci−1 hk/ci−5/6 (k/c)3/2 c−1/2 kN kL2 L2 kEkX ,
(5.33)
This can be summed in `1l `1j `1k (k . j ∼ l) and gives −1/2 1/2 kSE Ic (nC E F )C T kN kL∞ L2 (0,T ) kEkX . HL kStrE . c
(5.34)
Now, using Lemma 4.1 and the Sobolev embedding, we have −1 3/2 C F −1 kNjC kL∞ L2 k 1/2 kEkC kL∞ H 1 kSE Ic (nC j Ek )l kc−1 L∞ H 3/2 . cδ hl/ci l
. (cl)−1 hl/ci−1 l3/2 k 1/2 kN kL∞ L2 kEkL∞ H 1 ,
(5.35) 3/2
which can be summed in `1j `1k (k . j ∼ l). We then use the fact that c−1 L∞ B2,∞ ⊂ c−1/2 StrE , deducing kSE Ic (nC E C )FHL kStrE . c−1/2 kN kL∞ L2 kEkL∞ H 1 . 20
(5.36)
Next we concentrate on (5.27), hence we have γ˜ k/c ≤ j ≤ k ∼ l and k > c. Using the Strichartz estimate, we have kSE Ic (nFj Ek )l kStrE . hl/ci−1 k(nFj Ek )l kIc1/2 L10/7 B 1
10/7
(0,T )
. T 1/5 hl/ci−1/2 j 9/10 knFj kL2 L2 kEk kL∞ H 1 α . T 1/5 hl/ci−1/2 j 9/10 hj/ci1/6 kN kY kEkStrE , αj
(5.37)
which can be summed in `1l `1j `1k (j ≤ k ∼ l & c), and yields a factor T 1/5 . In the same way as (5.33), we have −1 3/2 F kSE Ic (nC kNjC kL2 L2 hk/ci−5/6 (αj)−1 kEk kX j Ek )l kStrE . hl/ci lj
(5.38)
. hl/ci−11/6 lj 1/2 c−2 kN kL2 L2 kEkX , which can be summed in `1l `1k `1j (j . k ∼ l) and we get a factor c−1/2 T 1/2 . Finally, in the same way as (5.35), we have C F kSE Ic (nC j Ek )l kc−1 L∞ H 3/2 .
1 hl/ci−1 l1/2 j 3/2 kN kL∞ L2 kEkL∞ H 1 cj
(5.39)
3/2
which can be summed `1k `1j (j . k ∼ l). We then use that c−1 L∞ B2,∞ ⊂ c−1/2 StrE , getting a factor c−1/2 . Next we consider the nonresonant term in the equation for n. Proposition 5.6. For any functions E and F on (0, T ) × R3 , we have X X kSn α|∇|hEk , F + F∗ iXj kStrn (0,T )+L∞ (0,T ;B 1/2 ) 2,∞
k
(5.40)
j>˜ γ k/c
. kEkStrE (0,T )∩X (0,T ) kFkStrE (0,T )∩X (0,T ) Proof. We will denote E = E and F = F or F∗ . Decomposing into dyadic pieces, we consider interactions of the form hEk , Fl ij for N with j > γ˜ k/c. Hence, F
hEk , Fl ij = hEkF , Fl ij + hEk , FlF ij − hEkF , FlF ij + hEkC , FlC ij .
(5.41)
By using the Strichartz estimate, we have kSn |α∇|hEkF , Fl ij kStrn . αjk(EkF Fl )j kα1/6 L6/5 B 1/3 3/2
. αjα .
−1/6 1/3
j
kEkF kL2 L2 hl/ci1/3 l−2/3 kFl kIc−1/3 L3 B 2/3 6
αj (j/c)1/3 hk/ci−5/6 hl/ci1/3 l−2/3 kEkX kF kStrE . δ
21
(5.42)
This is summable in `1j `1l `1k for l . j ∼ k and for j . k ∼ l. The above term can be bounded also by αjk(EkF Fl )j kL1 L2 . αjT 1/6 kEkF kL2 L3 hl/ci1/3 l−2/3 kFl kIc−1/3 L3 B 2/3 6
αj . T 1/6 k 1/2 hk/ci−5/6 hl/ci1/3 l−2/3 kEkX kF kStrE , δ
(5.43)
which is summable in `1j `1l `1k for k . l ∼ j, yielding a factor T 1/6 . For hEk , FlF ij , we have just to switch the roles of k and l. For hEkF , FlF ij , we have a better bound kSn |α∇|hEkF , FlF ij kStrn . αjk(EkF Fl F )j kL1 L2 . (αj)−1 (kl)3/4 hk/ci−5/6 hl/ci−5/6 kEk kX kFk kX ,
(5.44)
which is summable for all j, k, l and gives a factor c−1/2 . Finally, for the last term of (5.41), we have F
kSn |α∇|hEkC , FlC ij kL∞ H 1/2 .
αj 3/2 1/2 s j (kl)−1 kEk kL∞ H 1 kFl kL∞ H 1 δ
(5.45)
where s = min(j, k, l). This can be summed in `1l `1k for k . l ∼ j, l . k ∼ j and 1/2 k ∼ l & j and gives a result in L∞ B2,∞ .
5.4
The resonant part of N
To estimate the resonant frequency part Nγ , we integrate by parts as in (2.6). Then the estimate on the boundary terms follows from Lemma 4.3. The estimate on the integral terms use the following proposition Proposition 5.7. For any functions E and F, we have ˙ Fi kL∞ L2 . T kEk ˙ L∞ H −1 kFkL∞ H 1 , kSn hE, γ ˙ F∗ i k ∞ 2 . T kEk ˙ ∞ −1 kFk ∞ kA− S hE, γ
n
γ L L
L H
L H1 ,
(5.46)
˙ ˙ ∗ kA+ γ Sn hE, F iγ kL∞ W k,p . T kEkL∞ H −1 kFkL∞ H 1 Proof. For the proof of the first two estimates, we have just to use that ˙ γ kL2 . k(FE) ˙ γ kL1 k(FE) X ˙ H −1 , . kFk kL2 kE˙ l kL2 . kFkH 1 kEk
(5.47)
k∼l
and then apply the energy estimate. For the last term, we use the above L1x bound together with Lemma 4.3 to deal with A+ γ. ˙ L∞ H −1 ≤ kEkL∞ H 1 + kEkL∞ H 1 kN kL∞ (L2 +W 1,p ) . We then use that kEk 22
5.5
Concluding the estimates
Applying the propositions of the previous subsections, we can estimate all the terms appearing in (5.3) and (5.4). Recall k Ekk and k Nkk defined in (5.1). Proposition 5.8. If (E, N ) is a solution of (2.1) on (0, T ), then we have the following a priori bound k Ekk . kE(0)kH 1 + (T 1/6 + c−1/2 )kkNkkk Ekk, k Nkk . kN (0)kL2 +W k,p + k Ekk2 (1 + TkkNkk).
(5.48)
Hence, it is clear that there exists a c0 big enough and there exists a uniform time T such that the equation can be solved for c > c0 on the time interval (0, T ).
6
Passage to the limit
In this section, we prove the convergence towards the limit system. We denote N 0∞ = −|E∞ |2 , ∞ ∞ N +∞ = −eiαγt A+ γ (E1 · E2 ),
Eω := e−i(∆c −∆)t/2 E∞ ,
∞ ∞ N −∞ = −e−iαγt A− γ (E1 · E2 ),
N σω := N σ∞ + N σI
for σ = 0, ±,
(6.1) (6.2)
where N σI = N σIc were defined in Theorem 3.1. We also denote N ∞ = N 0∞ + N +∞ + N −∞ , N I = N f + N 0I + N +I + N −I ,
(6.3)
N ω = N ∞ + N I = N 0ω + N +ω + N −ω . Taking the real value, we define also nI = K /2)kStrE ∩X . ε.
(6.16)
Hence we may replace Eω by Eω≤K in E 4 . Integrating on ei|α∇|t , we get t I I 2SE Ic N>2γ Eω≤K = ei∆c (s−t)/2 Ic (|α∇|−1 N>2γ )Eω≤K 0 I − 2iSE Ic (|α∇|−1 N>2γ )∂t Eω≤K I )Eω≤K . + SE ∆c Ic (|α∇|−1 N>2γ
(6.17)
I Since |α∇|−1 N>2γ is bounded in α−1 L∞ (H 1 +W 2,p ), the first two lines are bounded in Ic X 1,1 . For the last term, we need to integrate once more, which yields similar terms but with one more |α∇|−1 and ∆c . Since ∆2c Ic ≤ O(α3/2 |∇|) and ∆2c ≤ O(α|∇|2 ) on I H s , and |α∇|−2 N>2γ is bounded in α−2 L∞ (H 2 + W 3,p ), those terms after the second integration are bounded in c−1 StrE ∩ α−1 X . I The other terms including N >2γ or Eω∗ ≤K are treated in the same way, integrating ±i|α∇|t iα(±|∇|−γ) on the phase e or e . The denominators are never singular thanks to
25
the frequency restriction > 2γ. Actually it is not needed for the term N E∗ . Also, the same argument applies to E 5 without any low frequency cut-off, just by integrating on the phase e±iαγt or e−2iαγt . Thus we obtain kE 5 kStrE = o(1).
(6.18)
I and Eω≤K , which we further It remains to estimate the part in E 4 with N≤2γ cut-off in the physical space. Fix χ ∈ C0∞ (R3 ) satisfying χ(x) = 1 for |x| ≤ 1 and χ(x) = 0 for |x| ≥ 2. There exists R > 1 such that k(1 − χ(x/R))Eω≤K kL∞ H 1 ≤ ε and so ω⊥
knI≤2γ [(1 − χ(x/R))Eω≤K ]kL∞ H 1 + kµ∗∞ (1 − χ(x/R))E≤K kL∞ H 1 . ε.
(6.19)
Hence its contribution to E 4 is O(ε) in Ic X 1,1 . Thus we may replace E≤K further by χ(x/R)E≤K . For the singular part, we may replace N I by N f because (N I − N f )≤2γ → 0 in L∞ W k,p by Lp decay for eit|∇| and Lemma 4.3. Then the Fourier and physical cut-offs together with the time integration provide compactness for the convergence in (3.2) such that kSE Ic χ(x/R)(nI≤2γ − c1/3 kStrE . κkE − Eω kStrE + o(1), by using Proposition 5.1. For the time derivative term, we use ˙ L2 . ckEkH 1 + cknEkH −1 . c, kEk
(6.26)
and also StrE ⊂ L3 (L∞ + c−1/3 L18 ) to deduce ˙ L∞ L2 . cκ. k∂t (EE)kL2 L2 . κkEkStrE kEk
(6.27)
By the same product estimate, we have kEEkL2 H 1 . κkEk2StrE .
(6.28)
Using these bounds, we get kNX2 kY . α−1 k(i∂t + |α∇|)NX2 kL2 L2 . o(1) + κkE − Eω kStrE , kNX4 kY . α−1 k∂t (EE)kL2 L2 . c−1 .
(6.29)
Thus by adding them up, k(N + − N +ω )X kY . κkE − Eω kStrE + o(1).
(6.30)
For the Strichartz-type norms, by using Propositions 5.3 and 5.6, we have kN 1 kStrn +L∞ B 1/2 . kE>c1/3 kStrE ∩X 2,∞
. kE − Eω kStrE + kE − Eω − E 4 − E 5 kX + o(1),
(6.31)
where Eω , E 4 and E 5 are negligible for the higher frequency > c1/3 . Decomposing into the resonant frequency and the rest, we have ∞ kN 2 + N 3 kL∞ (W k,p +H 1/2 ) . k(EE)ll − E∞ 1 · E2 kL∞ (L1 ∩H 1/2 )
. kE − Eω kStrE + o(1),
(6.32)
where W k,p and L1 were used for Nγj , and H 1/2 for NXj . Using the α gain in the wave Strichartz, we have kNX4 kStrn . k∂t (EE)ll kα1/3 L3/2 B 2/3 . α−1/3 c5/9 T k∂t (EE)ll kL∞ B −1
6/5
6/5
.c
−1/9
T kE˙ ≤c1/3 kL∞ H −1 kE≤c1/3 kL∞ H 1 . c
−1/9
(6.33)
κ.
Also, by Lemma 4.3 with 1/q = 2/3 + 1/p, we have 1/3+2/p
i|α∇|t kA+ ϕγ kW k,p . kϕγ kL1 γe
27
2/3−2/p
kϕγ kL2
.
(6.34)
Hence combining it with the same Strichartz estimate as above, we get 1/3+2/p
kNγ4 kL∞ W k,p . k∂t (EE)llR kL1 L1 . Tα
−1/3(2/3−2/p)
k∂t (EE)llR k
2/3−2/p 2/3
α1/3 L3/2 B6/5
(6.35)
= o(1),
Gathering the above estimates, we get kN + − N +ω kStrn +L∞ (W k,p +B 1/2 ) 2,∞
ω
ω
. kE − E kStrE + kE − E − E 4 − E 5 kX + o(1).
(6.36)
Using the estimates (6.22), (6.30) and (6.36), as well as their counterparts for N 0 and N − , we deduce that (6.6) holds. This ends the proof of convergence. (3.7) is a direct consequence of Lemma 4.3 and (3.6).
References [1] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schr¨odinger equation: smoothness and approximation.. Funct. Anal. 79 (1988), no. 1, 183–210. [2] P. Bechouche, N. Mauser and S. Selberg, Nonrelativistic limit of Klein-GordonMaxwell to Schr¨odinger-Poisson. Amer. J. Math. 126 (2004), no. 1, 31–64. [3] P. M. Bellan, Fundamentals of plasmas physics, Cambridge University Press, Cambridge, 2006. [4] L. Berg´e, B. Bid´egaray and T. Colin, A perturbative analysis of the timeenvelope approximation in strong Langmuir turbulence. Phys. D 95 (1996), no. 3-4, 351–379. [5] J. Bergh and J. L¨ofstr¨om, Interpolation spaces, An introduction. Grundlehren Math. Wiss. 223, Springer, Berlin–Heiderberg–New York, 1976. [6] T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schr¨odinger equation in H s . Nonlinear Anal. 14 (1990), no. 10, 807–836. [7] M. Colin and T. Colin, On a quasilinear Zakharov system describing laserplasma interactions, Differential Integral Equations 17 (2004), no. 3-4, 297–330. [8] T. Colin, G. Ebrard, G. Gallice and B. Texier, Justification of the Zakharov model from Klein-Gordon–wave systems. Comm. Partial Differential Equations 29 (2004), no. 9-10, 1365–1401. 28
[9] R. O. Dendy, Plasma Dynamics, Oxford University Press, 1990. [10] J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schr¨odinger equations, Ann. Inst. H. Poincar´e Phys. Th´eor. 43 (1985), no. 4, 399–442. [11] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151 (1997), no. 2, 384–436. [12] C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems. J. Funct. Anal. 127 (1995), no. 1, 204–234. [13] S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. Math. Ann. 322 (2002), no. 3, 603–621. [14] N. Masmoudi and K. Nakanishi From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrdinger equations, Math. Ann. 324 (2002), no. 2, 359–389. [15] N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-KleinGordon and Maxwell-Dirac to Poisson-Schr¨odinger, Int. Math. Res. Not. 2003, no. 13, 697–734. [16] N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to the nonlinear Schr¨odinger equation, J. Hyperbolic Differ. Equ. 2 (2005), no. 4, 975–1008. [17] N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math. 172, (2008), no. 3, 535–583. [18] N. Masmoudi and K. Nakanishi, Two asymptotic problems for a singular nonlinear Schr¨odinger system, preprint (2008). Available at http://www.math.nyu.edu/faculty/masmoudi/index.html [19] T. Ozawa, K. Tsutaya, and Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions. Math. Ann. 313 (1999), no. 1, 127–140. [20] T. Ozawa and Y. Tsutsumi, The nonlinear Schr¨odinger limit and the initial layer of the Zakharov equations. Differential Integral Equations 5 (1992), no. 4, 721–745.
29
[21] S. Schochet and M. Weinstein, The nonlinear Schr¨odinger limit of the Zakharov equations governing Langmuir turbulence. Comm. Math. Phys. 106 (1986), no. 4, 569–580. [22] A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136 (1999), no. 1, 9–74. [23] Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schr¨odinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999, Self-focusing and wave collapse. [24] Benjamin Texier, WKB asymptotics for the Euler-Maxwell equations, Asymptot. Anal. 42 (2005), no. 3-4, 211–250. [25] Benjamin Texier, Derivation of the Zakharov equations, preprint (2006). [26] T.-P. Tsai and H.-T. Yau, Relaxation of excited states in nonlinear Schr¨odinger equations. Int. Math. Res. Not. (2002), no. 31, 1629–1673. [27] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35, 908–914.
30
