Digital Object Identifier (DOI) 10.1007/s00205-010-0324-8
Singular Limits of the Klein–Gordon Equation
Chi-Kun Lin & Kung-Chien Wu
Communicated by T.-P. Liu
Dedicated to Professor Yuh-Jia Lee on his sixtieth birthday.
Abstract
We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated de-focusing nonlinear Klein–Gordon equation. For the semiclassical limit, → 0, we show that the limit wave function of the modulated defocusing cubic non-linear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c→ ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the de-focusing nonlinear Schrödinger equation. The nonrelativistic-semiclassical limit,
→ 0, c = −α → ∞ for some α > 0, of the modulated defocusing cubic non-linear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.
1. Introduction
One of the fundamental partial differential equations is the nonlinear Klein– Gordon equation 2 2mc2∂ 2 tΨ − 2 2mΔΨ + mc2 2 Ψ + V (|Ψ |2)Ψ = 0, (1.1) where m is mass, c is the speed of light and is the Planck constant. Here Ψ (x, t) is a complex-valued vector field over a spatial domainΩ ⊂ Rn, V is the first derivative of a twice differentiable nonlinear real-valued function overR+. Thus Vis the potential energy and V is the potential energy density of the fields. Since the Planck constant has dimension of action [] = [energy] × [time] = [action],
This work is partially supported by National Science Council of Taiwan under the grant NSC98-2115-M-009-004-MY3.
it is easy to check that (1.1) is dimensionally balanced. When the potential energy V = 0, (1.1) is the typical Klein–Gordon equation for a free particle and its dis-persion relation has the form E2= p2c2+ m2c4, p being the momentum, hence E = ±mc2for a resting particle. In quantum field theory the state of a particle
with negative energy is interpreted as the state of an antiparticle possessing positive energy, but opposite electric charge.
The Klein–Gordon equation for the complex scalar field is the relativistic ver-sion of the Schrödinger equation, which is used to describe spinless particles. It was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. However, this equation was named after the physicists Oskar Klein and Walter Gordon, who in 1927 proposed that it describes relativistic electrons. Although it turned out that the Dirac equation describes the spinning electron, the Klein–Gordon equation correctly describes the spinless pion. The reader is referred to [26] for a general introduction to nonlinear wave equations and [22] for the physical background. It is not straightforward to identify the nonrelativistic limit, that is, c→ ∞, from Equation (1.1). To this end we notice that mc2t and the Planck constant have the same dimension of action, [mc2t] = [] = [action], and we may consider the modulated wave function [18]
ψ(x, t) = Ψ (x, t) exp(imc2
t/), (1.2)
where the factor exp(imc2t/) describes the oscillations of the wave function, then ψ satisfies the modulated nonlinear Klein–Gordon equation
i∂tψ − 2 2mc2∂ 2 tψ + 2 2mΔψ − V (|ψ|2)ψ = 0. (1.3) The second term of Equation (1.3) shows the relativistic effect, which is small when considering the light speed c to be large, while the Planck constant is kept fixed. Thus, in the limit as c→ ∞, Equation (1.3) goes over into the defocusing nonlinear Schrödinger equation i∂tψ + 2 2mΔψ − V (|ψ|2)ψ = 0. (1.4) However, the semiclassical limit, that is → 0, is not clear from (1.3). One way to tackle this problem is the hydrodynamical formulation, as had been done for the Schrödinger equation [7–9]. In fact, we have discussed in detail the hydrodynamical structure of the modulated Klein–Gordon equation and its relation to the nonlinear Schrödinger equation and to the compressible (relativistic) Euler equations [15]. The most important local conservation laws associated with the modulated Klein– Gordon equation (1.3) are the charge and energy, given respectively by
∂ ∂t |ψ|2+ i 2mc2 ψ∂tψ − ψ∂tψ + ∇ · i 2m ψ∇ψ − ψ∇ψ= 0, (1.5) ∂ ∂t 1 c2|∂tψ| 2+ |∇ψ|2 +2m2V − ∇ ·∇ψ∂tψ + ∇ψ∂tψ = 0. (1.6)
Examining the charge equation (1.5) we see that although|ψ|2, the Schrödinger part, is positive-definite, the Klein–Gordon part 2mci2(ψ∂tψ − ψ∂tψ) is not. Here
we face one of the major difficulties with the Klein–Gordon equation. However, the energy density is positive-definite and can be employed to obtain an estimate of the charge of the Schrödinger part. Thus we introduce the charge-energy inequality to establish singular limits. This is consistent with Einstein’s relativity of mass-energy equivalence.
The question of the singular limits of the nonlinear Klein–Gordon equation and related equations has received considerable attention. Quite often, the limiting solu-tion (when it exists) satisfies a completely different nonlinear partial differential equation. The nonrelativistic limit of the modulated nonlinear Klein–Gordon equa-tion is one physical problem involving quantum (dispersion) effects where such a singular limiting process is interesting. In particular, Machihara et al. [18] gave a very complete answer of the Cauchy problem for the modulated Klein–Gordon equation; they proved that any finite energy solution converges to the corresponding solution of the nonlinear Schrödinger equation in the energy space, after infinite oscillation in time is removed. The Strichartz estimate plays a most important role in obtaining the uniform bound in space and time (see also [21,23] and references therein). Furthermore, it was shown by Masmaudi and Nakanishi in [20] that the solutions for the nonlinear Klein–Gordon equation can be described by using a sys-tem of two coupled nonlinear Schrödinger equations as the speed of light c tends to infinity. Thus, we may think of the coupled nonlinear Schrödinger equations as the singular limit (nonrelativistic limit) as c→ ∞ of the nonlinear Klein–Gordon equation.
The semiclassical limit of the defocusing nonlinear Schrödinger equation (1.3) is very well studied theoretically and numerically. In this limit, the Euler equation for an isentropic compressible flow is formally recovered [7–9,16] through the WKB analysis or Madelung transform (see [3,10,11] for the derivative nonlinear Schrödinger equation). On the other hand, if we forget the hydrodynamical formu-lation and return to the original nonlinear Schrödinger equation, then the limit wave function is shown to satisfy the wave map equation by Colin and Soyeur [2] for the case when there are no vortices, and by Lin and Xin [16] when there are vortices in two space dimensions. The vortex dynamics is also studied by Colliander and
Jerrard in [5]. For long wave-short wave equations we have a similar result for the weak coupling case [14]. However, to the best of our knowledge, the semiclassical limit of the nonlinear Klein–Gordon equation has not been studied theoretically. According to the correspondence principle, the classical or relativistic world should emerge from the quantum world whenever the Planck’s constant is negligible. But the semiclassical limit, that is → 0, is mathematically singular and is not clear from Equation (1.3) directly. However, as we will discuss in Section2, the semiclassical limit of the modulated nonlinear Klein–Gordon equation (1.3) can be introduced in the same way as the semiclassical limit of the Schrödinger type equa-tions, and the nonrelativistic-semiclassical limit can be discussed in a similar way. Let us briefly summarize our results. In Section2, we investigate the semiclas-sical limit of the Cauchy problem for the modulated defocusing cubic nonlinear Klein–Gordon Equations (2.1)–(2.2). We prove that any finite charge-energy
solution converges to the corresponding solution of the relativistic wave map, and the scattering sound wave is shown to satisfy a linear relativistic wave equation (see Theorem2.2below). Unlike the Schrödinger equation, the charge is not positive definite for the Klein–Gordon equation, so we have to introduce a charge-energy inequality obtained by combining the conservation laws of charge and the energy of the nonlinear Klein–Gordon equation. Besides the linear momentum W of the Schrödinger part, we have to introduce one more term Z , defined by (2.16), of the relativistic part. By rewriting the conservation of charge in terms of W and Z we can prove convergence to the relativistic wave map by the compactness argument.
Shatah [24] has proved the existence of global weak solutions of the wave map. For completeness we also prove the nonrelativistic limit of the relativistic wave map in Theorem2.4.
In Section3we employ the same idea used in Section2to obtain the nonrela-tivistic limit of the Cauchy problem for the modulated Klein–Gordon equation for general defocusing nonlinearity V(|ψc|2) = |ψc|p, p > 0, and the main result is described in Theorem3.2, which states that any finite charge-energy solution converges to the corresponding solution of the defocusing nonlinear Schrödinger equation in the energy space. For the sharper Strichartz estimate approach and a more complete result, the reader is referred to [18]. The main difference is that we combine the charge and energy conservation laws together to obtain the charge-energy inequality. Let us remark that in the case of the semiclassical limit, we have L∞t L2xas a bound for∂tψ, but for the non-relativistic limit, we only have L∞t L2xas a bound for1c∂tψc. Thus we need an extra argument to obtain strong convergence for the non-relativistic limit.
In Section4, under restrictions similar to those for the semiclassical limit, we study the nonrelativistic-semiclassical limit of the Cauchy problem for the modu-lated defocusing cubic nonlinear Klein–Gordon equation. We prove that any finite charge-energy solution converges to the corresponding solution of the wave map, and the associated phase function is shown to satisfy a linear wave equation. The main result is stated in Theorem4.2. Finally, we give a detail proof of Theorem3.1
in the appendix. The strategy of the proof follows that introduced by Leray in the context of the Navier–Stokes equations, as well as many other existence proofs for weak solutions of other equations.
Notation. In this paper, Lp(), (p 1) denotes the classical Lebesgue space
with norm f p = (
Ω| f |pdx)1/p, the Sobolev space of functions with all its kth partial derivatives in L2() will be denoted by Hk(), and its dual space is H−k(). We use f, g = Ω f gdx to denote the standard inner product on the Hilbert space L2(Ω). Without loss of generality, the units of length maybe chosen so thatΩdx= 1. Given any Banach space X with norm ·Xand p 1, the space of measurable functions u= u(t) from [0, T ] into X such that uX∈ Lp([0, T ]) will be denoted Lp([0, T ]; X). And C([0, T ]; w-Hk()) will denote the space of a continuous function from[0, T ] into w-Hk(). This means that for every ϕ ∈ H−k(), the function ϕ, u(t) is in C([0, T ]). Finally, we abbreviate “C” to “ ”, where C is a positive constant depending only on a fixed parameter.
2. Semiclassical limit
The specific problem we will consider in this section is the semiclassical limit of the modulated nonlinear Klein–Gordon equation (1.3) with the potential func-tion given by V(|ψ|2) = |ψ|2− 1. For the formal analysis of the semiclassical limit from the point of view of hydrodynamics, the reader is referred to [15]. For convenience let us call it the modulated defocusing cubic nonlinear Klein–Gordon equation. After dividing by, we relabel it as
i∂tψ− 2c2∂ 2 tψ+ 2Δψ − |ψ|2− 1 ψ= 0. (2.1)
The initial conditions are supplemented by
ψ(x, 0) = ψ0(x), ∂tψ(x, 0) = ψ1(x), x ∈ Ω. (2.2)
The superscript in the wave function ψindicates the-dependence and the light speed c is assumed to be a fixed number in this section. To avoid complications at the boundary, we concentrate below on the case where x∈ Ω = Tn, the n-dimen-sional torus. Notice that the fourth term |ψ|2−1of (2.1) can serve as the density fluctuation of the sound wave, which is similar to the acoustic wave as discussed in the low Mach number limit of the compressible fluid [1,12,13,17,19]. For this model (2.1)–(2.2) we have the following existence result.
Theorem 2.1. Let c, T > 0 and 0 < 1. Given initial data (ψ0, ψ1) ∈ H1(Tn) ⊕ L2(Tn) and |ψ0|2−1
∈ L2(Tn), there exists a function ψsuch that ψ∈ L∞[0, T ]; H1(Tn)∩ C[0, T ]; L2(Tn), (2.3) ∂tψ∈ L∞ [0, T ]; L2(Tn)∩ C[0, T ]; H−1(Tn), (2.4) |ψ|2− 1 ∈ L∞ [0, T ]; L2(Tn), (2.5) and satisfies the weak formulation of (2.1) given by
0= i ψ(·, t2) − ψ(·, t1), ϕ − 2c2 ∂tψ(·, t2) − ∂tψ(·, t1), ϕ − 2 t2 t1 ∇ψ(·, τ), ∇ϕdτ − t2 t1 |ψ|2− 1 ψ(·, τ), ϕ dτ, (2.6) for every[t1, t2] ⊂ [0, T ] and for all ϕ ∈ C0∞(Tn). Moreover, for all t ∈ [0, T ], it satisfies the charge-energy inequality
Tn|ψ |2+ 1 c2|∂tψ| 2+ |∇ψ|2+1 2 |ψ|2− 1 2 dx 2C1+ 1+2 2 c2 C2, (2.7)
where C1= Tn|ψ 0|2+ c2 i 2 ψ1ψ0− ψ1ψ0 dx, C2= Tn 1 c2|ψ1|2+ |∇ψ0|2+ 1 2 |ψ0| 2− 1 2 dx, (2.8)
are the initial charge and energy respectively.
The charge consists of the Schrödinger part (positive definite) and the Klein– Gordon part (not positive definite). However, it can be bounded by the energy. We denote by “∩” the intersection of topological spaces equipped with the relative topology induced by the inclusion maps. Since we are concerned with the semi-classical limit in this section, so the proof of this theorem, Theorems3.1and4.1
of the following two sections will be given in the appendix.
We assume that the initial datum satisfy|ψ0| = 1 almost everywhere and (ψ0, ψ1) → (ψ0, 0) in H
1(Tn) ⊕ L2(Tn) as → 0, thus |ψ0| = 1 almost everywhere. First we deduce from the charge-energy inequality (2.7) that
{ψ} is bounded in L∞[0, T ]; H1(Tn), (2.9) {∂tψ} is bounded in L∞ [0, T ]; L2(Tn), (2.10) |ψ|2− 1 is bounded in L∞[0, T ]; L2(Tn), (2.11)
then the classical compactness argument shows that there exists a subsequence still denoted by{ψ}and a functionψ satisfying
ψ ∈ L∞[0, T ]; H1(Tn), ∂ tψ ∈ L∞ [0, T ]; L2(Tn) such that ψ ψ weakly ∗ in L∞[0, T ]; H1(Tn), (2.12) ∂tψ ∂tψ weakly ∗ in L∞ [0, T ]; L2(Tn). (2.13) Next, from (2.11), we have
|ψ|2→ 1 almost everywhere and strongly in L2(Tn). (2.14) Note that (2.11) shows only that{|ψ|2−1}is a weakly relative compact set in L∞([0, T ]; L2(Tn)). Thus, to overcome the difficulty caused by nonlinearity, that is, the fourth term on the right-hand side of (2.6), we have to proveψ → ψ strongly in C([0, T ]; L2(Tn)).
Lemma 2.1. For all 0 < 1, the sequence {ψ} is a relatively compact set in C([0, T ]; L2(Tn)) endowed with its strong topology, that is, there exists ψ ∈ C([0, T ]; L2(Tn)) such that
ψ→ ψ strongly in C[0, T ]; L2(Tn). (2.15) Proof. In this case, compactness requires more than just boundedness because of
the strong topology over the time variable t. We appeal to the Arzela–Ascoli the-orem, which asserts that{ψ}is a relatively compact set in C([0, T ]; L2(Tn)) if and only if
(1) {ψ(t)}is a relatively compact set in L2(Tn) for all t 0; (2) {ψ}is equicontinuous in C[0, T ]; L2(Tn).
From (2.7) or (2.9) we know that{ψ(t)}is a bounded set in H1(Tn) and hence is a relatively compact set in L2(Tn) by the Rellich lemma, which states that
H1(Tn) → L2(Tn) is a compact embedding.
In order to establish condition (2), we apply the fundamental theorem of calculus and the uniform bound of{∂tψ}to obtain
ψ(t2) − ψ(t1)L2(Tn) |t2− t1|∂tψ(s)L2(Tn) |t2− t1|
for some s ∈ (t1, t2). This completes the proof of Lemma2.1.
The quantity|ψ(x,t)| 2−1is bounded in L∞([0, T ]; L2(Tn)), and hence it con-verges weakly∗ to some function w ∈ L∞([0, T ]; L2(Tn)). To find the explicit form ofw, we define two functions W(ψ) and Z(ψ), respectively, by
W(ψ) = i 2 ψ∇ψ− ψ∇ψ, Z(ψ) = i 2c2 ψ∂tψ− ψ∂ tψ . (2.16) We rewrite the conservation of charge (1.5) as
∂ ∂t |ψ|2− 1 + Z(ψ) + div W(ψ) = 0, (2.17) then, integrating (2.17) with respect to t and using the initial condition|ψ0|2= 1, we have |ψ(x, t)|2− 1 = −Z(ψ) + Z(ψ(x, 0)) − t 0 div W(ψ)dτ. (2.18)
Thus, to obtain the compactness of the sequence{|ψ(x,t)| 2−1}, we have to treat the compactness of{Z(ψ)}and{W(ψ)}separately. First we have the following lemma.
Lemma 2.2. Assume the hypothesis of Theorem2.1, then ψ∂tψ ψ∂tψ (2.19) t 0 div ψ∇ψdτ t 0 div ψ∇ψdτ (2.20) inD((0, T ) × Tn).
Proof. We observe thatψ∈ C([0, T ]; L2(Tn)) implies ψ∈ L2([0, T ] × Tn)
and∂tψ∈ L∞([0, T ]; L2(Tn)) implies ∂tψ∈ L2([0, T ] × Tn). Also ψ con-verges strongly to ψ in L2([0, T ] × Tn) and ∂
tψconverges weakly to∂tψ in L2([0, T ] × Tn). Thus for all ϕ ∈ C0∞(Tn) we have
lim →0 t2 t1 Tnψ (x, t)∂ tψ(x, t)ϕ(x)dx dt = t2 t1 Tnψ(x, t)∂tψ(x, t)ϕ(x)dx dt.
Similarly∇ψ∈ L∞([0, T ]; L2(Tn)) implies ∇ψ∈ L2([0, T ] × Tn) and ∇ψ converges weakly to∇ψ in L2([0, T ] × Tn), then integration by parts, then by the Fubini theorem and the Lebesgue dominated convergence theorem, we conclude that − t2 t1 Tn t 0 div ψ(x, τ)∇ψ(x, τ) − ψ(x, τ)∇ψ(x, τ)dτϕ(x)dx dt = t2 t1 t 0 Tn ψ(x, τ) − ψ(x, τ)∇ψ(x, τ) · ∇ϕ(x)dx dτ dt + t2 t1 t 0 Tn ∇ψ(x, τ) − ∇ψ(x, τ)ψ(x, τ) · ∇ϕ(x)dx dτ dt → 0 as → 0. This completes the proof of Lemma2.2.
It follows from Lemma2.2that Z(ψ) Z(ψ), Z(ψ(x, 0)) 0 and
t 0 div W(ψ)dτ t 0 div W(ψ)dτ inD((0, T ) × Tn), thus |ψ(x, t)|2− 1 −Z(ψ) − t 0 div W(ψ)dτ (2.21)
inD((0, T ) × Tn), and the limit function w is given explicitly by w = −Z(ψ) −
t 0
div W(ψ)dτ.
Passage to the limit( → 0). The uniform boundness of the sequences {ψ}in L∞[0, T ]; H1(Tn)and{∂tψ}in L∞ [0, T ]; L2(Tn)imply 2c2 ∂tψ(·, t2), ϕ → 0, 2c2 ∂tψ(·, t1), ϕ → 0, (2.22) 2 t2 t1 ∇ψ(·, τ), ∇ϕdτ → 0 (2.23)
as → 0. The strong convergence of ψin C([0, T ]; L2(Tn)) implies ψ(·, t2), ϕ → ψ(·, t2), ϕ , ψ(·, t1), ϕ → ψ(·, t1), ϕ . (2.24) The convergence of the nonlinear term follows by combining (2.15) and (2.21), so that for all t > 0
|ψ|2− 1 ψ − Z(ψ) + t 0 div W(ψ)dτ ψ (2.25) inD((0, T ) × Tn) and hence t2 t1 |ψ|2− 1 ψ(·, τ), ϕ dτ → − t2 t1 Z(ψ) + t 0 div W(ψ)dτ ψ(·, τ), ϕ dτ. (2.26)
Putting all the above convergent results into the weak formulation (2.6), the limit wave functionψ satisfies
i∂tψ + Z(ψ) + t 0 div W(ψ)dτ ψ = 0 (2.27)
in the sense of distribution. Note |ψ|2 = 1, we have ψ∂tψ + ψ∂tψ = 0 and ψ∇ψ + ψ∇ψ = 0, hence 1 2 ψ∂tψ − ψ∂tψ = ψ∂tψ = −ψ∂tψ, 1 2 ψ∇ψ − ψ∇ψ= ψ∇ψ.
Differentiating (2.27) with respect to t, we have ∂2 tψ + 1 c2∂t(ψ∂tψ) − div (ψ∇ψ) ψ −∂tψ ψ ∂tψ = 0, (2.28) or ∂2 tψ + 1 c2 ψ∂2 tψ + ∂tψ∂tψ −ψΔψ + ∇ψ · ∇ψψ + |∂tψ|2ψ = 0. (2.29) Thereforeψ satisfies the relativistic wave map equation
1+ 1 c2 ∂2 tψ − Δψ = |∇ψ|2− 1+ 1 c2 |∂tψ|2 ψ, |ψ| = 1 almost everywhere (2.30) supplemented with the initial conditions
ψ(x, 0) = ψ0(x), ∂tψ(x, 0) = 0, x ∈ Tn, |ψ0| = 1 almost everywhere (2.31)
Using the fact|ψ| = 1 and writing ψ = eiθ shows 1+ 1 c2 ∂2 tθ = Δθ , θ(x, 0) = arg ψ0, ∂tθ(x, 0) = 0, (2.32)
that is,θ is a distribution solution of the linear relativistic wave equation. Moreover, θ(x, t) ∈ H1(Tn) and ∂
tθ(x, t) ∈ L2(Tn) for all t ∈ [0, T ] implies that θ is the unique weak solution of (2.32) with finite energy. Thec12 terms in (2.30) and (2.32)
show the relativistic effect and formally, letting c→ ∞, they reduce to the standard wave map and wave equation, respectively (see [2,6,14,16]).
Theorem 2.2. Let(ψ0, ψ1) ∈ H1(Tn) ⊕ L2(Tn), |ψ0| = 1 almost everywhere and(ψ0, ψ1) → (ψ0, 0) in H1(Tn)⊕L2(Tn), |ψ0| = 1 almost everywhere, and let
ψbe the corresponding weak solution of the modulated defocusing cubic nonlin-ear Klein–Gordon equation (2.1)–(2.2). Then the weak limitψ, satisfying |ψ| = 1 almost everywhere, solves the relativistic wave map (2.30)–(2.31). Moreover, let ψ = eiθ; then the phase functionθ satisfies the relativistic wave equation (2.32).
For completeness we also discuss the non-relativistic limit of the relativistic wave map equation (2.30)–(2.31). To indicate the c-dependence of the wave func-tion, we replaceψ by φcand rewrite (2.30)–(2.31) as
1+ 1 c2 ∂2 tφc− Δφc= |∇φc|2− 1+ 1 c2 |∂tφc|2 φc, (2.33) φc(x, 0) = φc 0(x), ∂tφc(x, 0) = 0, x ∈ Tn, (2.34) |φc| = |φc
0| = 1 almost everywhere. Let Re φc andIm φc denote the real and
imaginary parts of φc,φc = Re φc + iIm φc, and uc = (Re φc, Im φc)t then (2.33)–(2.34) can be rewritten as 1+ 1 c2 ∂2 tuc− Δuc= |∇uc|2− 1+ 1 c2 |∂tuc|2 uc, (2.35) uc(x, 0) = uc0(x), ∂tuc(x, 0) = 0, x ∈ Tn, (2.36) where uc0(x) = (Re φ0c, Im φc0)t and|uc| = |uc0| = 1 almost everywhere. When c= ∞, the necessary and sufficient condition for the existence of weak solutions to (2.35)–(2.36) was proved by Shatah [24] (see also [25]). His result is easily ex-tended to general c by replacing the Riemann metricη = diag(1, −1, −1, . . . , −1) byηc= diag(1 + 1/c2, −1, −1, . . . , −1) and ∂α = ηαβ∂β by ∂α = ηαβc ∂β. Lemma 2.3. [24] If|uc| = 1 almost everywhere and satisfies ∇uc ∈ L∞([0, T ]; L2(Tn)), ∂tuc∈ L∞([0, T ]; L2(Tn)), then ucis a weak solution of (2.35)–(2.36) if and only if∂α( ∂αuc∧ uc) = 0, where ∧ denotes the wedge product.
By Lemma2.3, we have the existence of weak solutions of the wave map equation.
Theorem 2.3. [24] Given initial data uc0 ∈ H1(Tn) and |uc0| = 1, there exists a function uc,|uc| = 1 almost everywhere, such that
∇uc∈ L∞[0, T ]; L2(Tn), ∂
tuc∈ L∞
[0, T ]; L2(Tn)
(2.37) and satisfies the wave map equation
1+ 1 c2 ∂2 tuc− Δuc = |∇uc|2− 1+ 1 c2 |∂tuc|2 uc (2.38)
inD((0, T ) × Tn). Moreover, for all t ∈ [0, T ], it satisfies the energy relation Tn 1+ 1 c2 |∂tuc|2+ |∇uc|2dx Tn|∇u c 0| 2dx. (2.39) As before, we assumeφ0c → φ0 strongly in H1(Tn) and |φ0| = 1 almost everywhere, equivalently if u0 = (Re φ0, Im φ0)t,|u0| = 1 almost everywhere,
then uc0→ u0in H1(Tn). We deduce from the energy relation (2.39) and|uc| = 1 almost everywhere that
{uc}
c is bounded in L∞([0, T ]; H1(Tn)) , (2.40) {∂tuc}c is bounded in L∞([0, T ]; L2(Tn)). (2.41) By the classical compactness argument and the diagonalization process, there exists a subsequence, still denoted by {uc}c, satisfying u ∈ L∞([0, T ]; H1(Tn)) and ∂tu ∈ L∞([0, T ]; L2(Tn)) such that
uc u weakly ∗ in L∞[0, T ]; H1(Tn), (2.42) ∂tuc ∂tu weakly∗ in L∞
[0, T ]; L2(Tn). (2.43) Using the same argument as Lemma2.1, we deduce from (2.40)–(2.41) that
uc→ u strongly in C[0, T ]; L2(Tn). (2.44) Combining (2.44) and |uc| = 1 almost everywhere, we derive |u| = 1 almost everywhere. Moreover, using (2.42)–(2.44), we have
∂αuc∧ uc→ ∂αu∧ u in D((0, T ) × Tn). (2.45) Note that ucsatisfies∂α( ∂αuc∧ uc) = 0 in the sense of distribution;
1+ 1 c2 ∂tuc∧ uc(t2, ·) − ∂tuc∧ uc(t1, ·), ϕ + n i=1 t2 t1 ∂iuc∧ uc(·, τ), ∂iϕ dτ = 0 (2.46)
for every [t1, t2] ⊂ [0, T ] and for all ϕ ∈ C0∞(Tn). Letting c → ∞ in (2.46) and using (2.45), we have shown that u satisfies∂α(∂αu∧ u) = 0 in the sense of distribution, and by Lemma2.3it solves the wave map equation
∂2 tu− Δu = |∇u|2− |∂ tu|2 u inD((0, T ) × Tn). (2.47) Denote u = (α, β)t andφ = α + iβ, then we have ∇φc → ∇φ weakly ∗ in L∞([0, T ]; L2(Tn)), φc → φ strongly in L∞([0, T ]; L2(Tn)) and ∂tφc → ∂tφ weakly∗ in L∞([0, T ]; L2(Tn)). Moreover, φ satisfies the wave map equation
∂2 tφ − Δφ = |∇φ|2− |∂ tφ|2 φ, (t, x) ∈ [0, T ] × Tn, (2.48) φ(x, 0) = φ0(x), ∂tφ(x, 0) = 0, x ∈ Tn, (2.49) in the sense of distribution and|φ| = |φ0| = 1 almost everywhere.
Theorem 2.4. Letφ0c, φ0 ∈ H1(Tn), |φ0c| = |φ0| = 1 almost everywhere and
φc
0→ φ0in H1(Tn). Let φcbe the corresponding weak solution of the relativistic wave map (2.33)–(2.34). Then the weak limitφ of {φc}c satisfies|φ| = 1 almost everywhere and solves the wave map (2.48)–(2.49).
3. Nonrelativistic limit
This section is devoted to the non-relativistic limit of the modulated nonlinear Klein–Gordon equation with the potential function given by V(|ψc|2) = |ψc|p,
p> 0, i∂tψc− 2 2c2∂ 2 tψc+ 2 2Δψ c− |ψc|pψc= 0. (3.1) As usual, we supplement the system (3.1) with initial conditions
ψc(x, 0) = ψc
0(x), ∂tψc(x, 0) = ψ1c(x), x ∈ T n.
(3.2) Here the Planck’s constant is a fixed positive number and the superscript c in the wave functionψcindicates c-dependence. Similarly to the semiclassical limit discussed in the previous section, we discuss only the periodic domainTnand state the existence theorem of (3.1)–(3.2) first, leaving the proof to the appendix.
Theorem 3.1. Let p, , T > 0 and c 1. Given initial data (ψ0c, ψ1c) in H1∩ Lp+2(Tn) ⊕ L2(Tn), there exists a function ψcsuch that
ψc∈ L∞[0, T ]; H1(Tn)∩ C[0, T ]; L2(Tn), (3.3) ∂tψc∈ L∞ [0, T ]; L2(Tn)∩ C[0, T ]; H−1(Tn), (3.4) ψc∈ L∞[0, T ]; Lp+2(Tn), (3.5)
and satisfies the weak formulation of (3.1) given by 0= − 2 2c2 ∂tψc(·, t2) − ∂tψc(·, t1), ϕ + i ψc(·, t 2) − ψc(·, t1), ϕ −2 2 t2 t1 ∇ψc(·, τ), ∇ϕ dτ − t2 t1 |ψc|pψc(·, τ), ϕ dτ, (3.6) for every[t1, t2] ⊂ [0, T ] and for all ϕ ∈ C0∞(Tn). Moreover, ψc satisfies the charge-energy inequality Tn|ψ c|2+ 2 2c2|∂tψ c|2+2 2|∇ψ c|2+|ψc|p+2 p+ 2 dx 2C1+ 1+ 2 c2 C2, (3.7) where C1and C2are the initial charge and energy given respectively by
C1= Tn|ψ c 0|2+ c2 i 2 ψc 1ψ0c− ψ c 1ψ c 0 dx, C2= Tn 2 2c2|ψ c 1|2+ 2 2 |∇ψ c 0|2+ 1 p+ 2|ψ c 0|p+2dx. (3.8)
To study the nonrelativistic limit, we will assume that the initial condition (ψc 0, ψ c 1) converges strongly in H 1(Tn) ∩ Lp+2(Tn) ⊕ L2(Tn) to (ψ 0, 0) when the
light speed c tends to∞. We deduce from the charge-energy inequality (3.7) that
{ψc} c is bounded in L∞ [0, T ]; H1(Tn), (3.9) 1 c∂tψ c ! c is bounded in L∞[0, T ]; L2(Tn), (3.10) {ψc} c is bounded in L∞ [0, T ]; Lp+2(Tn). (3.11) In the case of the semiclassical limit, we have L∞t L2x bound for∂tψ, but for the non-relativistic limit, we have only L∞t L2xas a bound for1c∂tψ
c, so we need further argument to showψc → ψ in C([0, T ]; L2(Tn)).
Lemma 3.1. For all c 1, the sequence {ψc}c is a relatively compact set in C([0, T ]; w-H1(Tn)), thus there exists ψ ∈ C([0, T ]; w-H1(Tn)) such that
ψc→ ψ in C[0, T ]; w-H1(Tn)
as c→ ∞.
Furthermore,{ψc}cis a relatively compact set in C([0, T ]; L2(Tn)) endowed with its strong topology and
ψc→ ψ in C[0, T ]; L2(Tn) as c→ ∞.
Proof. As discussed in the previous section, we appeal to the Arzela–Ascoli
theorem, which states that the sequence {ψc}
c is a relatively compact set in C([0, T ]; w-H1(Tn)) if and only if
(1) {ψc(t)} is a relatively compact set in w-H1(Tn) for all t 0;
(2) {ψc} is equicontinuous in C[0, T ]; w-H1(Tn), that is, for every ϕ ∈ H−1(Tn) the sequence {ψc, ϕ }cis equicontinuous in the space C([0, T ]). Since{ψc(t)}cis uniformly bounded in H1(Tn), thus {ψc(t)}cis a relatively com-pact set in w-H1(Tn) for every t > 0. In order to establish condition (2), let A⊂ Cc∞(Tn) be an enumerable set which is dense in H−1; then for anyρ ∈ A, we have i ψc(·, t2) − ψc(·, t1), ρ = 2 2c2 ∂tψc(·, t2) − ∂tψc(·, t1), ρ +2 2 t2 t1 ∇ψc(·, τ), ∇ρ dτ + t2 t1 |ψc|pψc(·, τ), ρ dτ, hence |ψc(·, t 2)−ψc(·, t1), ρ | c−1ρL2(Tn)+|t2− t1| ρH1(Tn)+ρL∞(Tn).
Thus for any > 0, we can choose δ = such that if |t2− t1| < δ and c−1< ,
then
|ψc(·, t
2) − ψc(·, t1), ρ | .
Moreover, by the density argument we can prove
|ψc(·, t
2) − ψc(·, t1), ϕ | , (3.12)
for allϕ ∈ H−1(Tn). Thus {ψc}c is equicontinuous in C
[0, T ]; w-H1(Tn)for c larger. The second statement follows immediately by the Rellich lemma, which states that H1(Tn) → L2(Tn) compactly, that is, w-H1(Tn) → L2(Tn) continu-ously. This completes the proof of Lemma3.1.
In order to overcome difficulties caused by nonlinearity, we need the following lemma.
Lemma 3.2. Assume the hypothesis of Theorem3.1. Letψcbe a sequence of weak solution to (3.1)–(3.2) then there existsψ ∈ L∞([0, T ]; Lp+1(Tn)) such that
ψc→ ψ in L∞[0, T ]; Lp+1(Tn).
Proof. The proof is divided into two cases. First, for 0 < p 1, since L2(Tn) ⊂ Lp+1(Tn) for bounded measure |Tn| < ∞, the strong conver-gence in the space L∞([0, T ]; L2(Tn)) also implies the strong convergence in L∞([0, T ]; Lp+1(Tn)). Second, p > 1, the strong convergence in the space L∞([0, T ]; L2(Tn)) and the weakly ∗ convergence in L∞([0, T ]; Lp+2(Tn)) com-bined with an interpolation argument yields the result. Indeed,ψc ψ weakly
∗ in L∞([0, T ]; Lp+2(Tn)), the sequence {ψc − ψ}
c is a norm bounded set in L∞([0, T ]; Lp+2(Tn)), there exists a constant K > 0 such that
lim sup c→∞ ψ
c− ψp+2
L∞([0,T ];Lp+2(Tn))= K < ∞. (3.13)
Next, letη > 0 be arbitrary, and choose δ < η/K , the Young inequality gives
|ψc− ψ|p+1= |ψc− ψ|p+1−2/p|ψc− ψ|2/p
δ|ψc− ψ|p+2+ C|ψc− ψ|2. (3.14)
Integrating this inequality overTn, we have
ψc− ψp+1 Lp+1(Tn) δψ c− ψp+2 Lp+2(Tn)+ Cψ c− ψ2 L2(Tn). (3.15) Thus lim sup c→∞ ψ c− ψp+1 L∞([0,T ];Lp+1(Tn)) K δ η. (3.16)
Becauseη > 0 is arbitrary, we have ψc → ψ in L∞([0, T ]; Lp+1(Tn)). Passage to limit(c → ∞). The uniform boundedness of the sequence {1c∂tψc}cin L∞([0, T ]; L2(Tn)) yields 2 2c2 ∂tψc(·, t2) − ∂tψc(·, t1), ϕ → 0. (3.17)
The weak∗ convergence of ψcin L∞([0, T ]; H1(Tn)) and the strong convergence in C([0, T ]; L2(Tn)) imply t2 t1 ∇ψc(·, τ), ∇ϕ dτ → t2 t1 ∇ψ(·, τ), ∇ϕ dτ, (3.18) ψc(·, t 2) − ψc(·, t1), ϕ → ψ(·, t2) − ψ(·, t1), ϕ . (3.19)
For the nonlinear term, we rewrite it as
t2 t1 Tn |ψc|pψc(x, τ) − |ψ|pψ(x, τ) ϕ(x)dx dτ = t2 t1 Tn ψc(x, τ) − ψ(x, τ) |ψc|p(x, τ)ϕ(x)dx dτ + t2 t1 Tn |ψc|p(x, τ) − |ψ|p(x, τ) ψ(x, τ)ϕ(x)dx dτ ≡ I + II (3.20) for allϕ ∈ C0∞(Tn). We will estimate the integrals I and II separately. First, by Hölder inequality, we have
I ψc− ψLp+1([t1,t2]×Tn)ϕL∞(Tn)ψcp
Lp+1([t
1,t2]×Tn)→ 0, (3.21)
thus I tends to 0 as c → ∞ by Lemma3.2. The estimate of II requires higher integrability. Since|ψc|p |ψ|pweakly in Lp+2p ([0, T ] × Tn) for T < ∞ and
ψ is bounded in Lq([0, T ] × Tn), 1 q p + 2, hence we can choose q = p+2 2 such that II = t2 t1 Tn |ψc|p(x, τ) − |ψ|p(x, τ) ψ(x, τ)ϕ(x)dx dτ → 0. (3.22) Combining the above convergent results into the weak formulation (3.6), as c→ ∞, we deduce thatψ is a distribution solution of the defocusing nonlinear Schrödinger equation; i∂tψ + 2 2 Δψ − |ψ| pψ = 0, (x, t) ∈ Tn× (0, T ), (3.23) ψ(x, 0) = ψ0(x), x ∈ Tn. (3.24) Theorem 3.2. Let(ψ0c, ψ1c) ∈ H1∩ Lp+2(Tn) ⊕ L2(Tn), (ψ0c, ψ1c) → (ψ0, 0) in H1∩ Lp+2(Tn)⊕ L2(Tn), and ψcbe the corresponding weak solution of the modu-lated defocusing nonlinear Klein–Gordon equation (3.1)–(3.2). Then the weak limit ψ of {ψc}
csolves the defocusing nonlinear Schrödinger equation (3.23)–(3.24).
4. Nonrelativistic-semiclassical limit
In this section we will consider the nonrelativistic-semiclassical limit of the modulated nonlinear Klein–Gordon equation with the potential function given by V(|ψ|2) = |ψ|2− 1. In order to avoid carrying out a double limit, the parameters c and must be related. For simplicity, we take = ε, 1c = εα for someα > 0, 0 < ε 1 and rewrite the modulated defocusing cubic nonlinear Klein–Gordon equation as i∂tψε− 1 2ε 1+2α∂2 tψε+ ε 2Δψ ε−|ψε|2− 1 ε ψε = 0, (4.1)
supplemented with initial conditions ψε(x, 0) = ψε
0(x), ∂tψε(x, 0) = ψ1ε(x), x ∈ T n.
(4.2) Here the superscriptε in the wave function ψε indicates theε-dependence. As discussed in Sections2and3, we discuss only the periodic domainTnand state the following existence theorem.
Theorem 4.1. Given(ψ0ε, ψ1ε) ∈ H1(Tn) ⊕ L2(Tn) and|ψ0ε|2−1
ε ∈ L2(Tn), there exists a functionψεsuch that
ψε ∈ L∞([0, T ]; H1(Tn)) ∩ C([0, T ]; L2(Tn)), (4.3) ∂tψε∈ L∞([0, T ]; L2(Tn)) ∩ C([0, T ]; H−1(Tn)), (4.4) |ψε|2− 1 ε ∈ L∞([0, T ]; L 2(Tn)), (4.5)
and satisfies the weak formulation of (4.1) given by 0= −1 2ε 1+2α ∂ tψε(·, t2) − ∂tψε(·, t1), ϕ + i ψε(·, t2) − ψε(·, t1), ϕ −ε 2 t2 t1 ∇ψε(·, τ), ∇ϕ dτ − t2 t1 |ψε|2− 1 ε ψε(·, τ), ϕ dτ, (4.6) for every [t1, t2] ⊂ [0, T ] and for all ϕ ∈ C∞0 (Tn). Moreover, it satisfies the charge-energy inequality Tn|ψ ε|2+ ε2α|∂ tψε|2+ |∇ψε|2+ 1 2 |ψε|2− 1 ε 2 dx 2C1+ (1 + 2ε2+2α)C 2 (4.7)
where C1and C2denote the initial charge and energy given respectively by C1= Tn|ψ ε 0| 2+ ε1+2αi 2 ψ1εψ0ε− ψ1εψ0ε dx, C2= Tnε 2α|ψε 1|2+ |∇ψ0ε|2+ 1 2 |ψε 0|2− 1 ε 2 dx. (4.8)
To study the nonrelativistic-semiclassical limit, we still assume|ψ0ε| = |ψ0| = 1 and(ψ0ε, ψ1ε) → (ψ0, 0) in H1(Tn) ⊕ L2(Tn) as ε → 0. It follows immediately
from the charge-energy inequality (4.7) that
{ψε}ε is bounded in L∞[0, T ]; H1(Tn), (4.9) {εα∂tψε}ε is bounded in L∞ [0, T ]; L2(Tn), (4.10) |ψε|2− 1 ε ! ε is bounded in L∞[0, T ]; L2(Tn). (4.11) We deduce from (4.11) that
|ψε|2→ 1 almost everywhere and strongly in L2(Tn)
asε tends to 0. As discussed above (4.11) shows only that the quantity{|ψε|ε2−1}ε is a weakly relative compact set in L∞([0, T ]; L2(Tn)), then (up to a subse-quence) the sequence {|ψε|ε2−1}ε converges weakly ∗ to some function w in L∞([0, T ]; L2(Tn)). In order to find w explicitly, we rewrite the conservation of charge as |ψε|2− 1 ε = −Z(ψε) + Z(ψε(x, 0)) − t 0 div W(ψε)dτ, (4.12) where Z(ψε) and W(ψε) are defined similarly to (2.16). We deduce from (4.9) and (4.10) that Z(ψε) 0 in D((0, T )×Tn), and the same discussion as Lemma2.2, we can prove
t 0 div W(ψ)dτ t 0 div W(ψ)dτ inD((0, T ) × Tn), hence |ψε|2− 1 ε − t 0 div W(ψ)dτ (4.13) inD((0, T ) × Tn). Therefore |ψε|2− 1 ε − t 0 div W(ψ)dτ (4.14)
weakly∗ in L∞([0, T ]; L2(Tn)), and thus
|ψε|2− 1 ε ψε −ψ t 0 div W(ψ)dτ in D((0, T ) × Tn). (4.15) By combining the above convergent results, one can pass to the limit in each term of (4.6) and conclude that the limitψ satisfies |ψ| = 1 almost everywhere and
i∂tψ + t 0 div W(ψ)dτ ψ = 0 (4.16)
inD((0, T ) × Tn). Similar to the discussion in the case of the semiclassical limit, using|ψ| = |ψ0| = 1 almost everywhere, we can prove that ψ satisfies the wave map equation ∂2 tψ − Δψ = |∇ψ|2− |∂ tψ|2 ψ , |ψ| = 1 almost everywhere (4.17) ψ(x, 0) = ψ0(x), ∂tψ(x, 0) = 0, x ∈ Tn. (4.18) Using the fact|ψ| = |ψ0| = 1 again and writing ψ = eiθ shows
∂2
tθ = Δθ, θ(x, 0) = arg ψ0, ∂tθ(x, 0) = 0. (4.19) Theorem 4.2. Let(ψ0ε, ψ1ε) ∈ H1(Tn) ⊕ L2(Tn), |ψ0ε| = 1, and (ψ0ε, ψ1ε) →
(ψ0, 0) in H1(Tn)⊕ L2(Tn), |ψ0| = 1, and let ψεbe the corresponding weak solu-tion of the modulated cubic nonlinear Klein–Gordon equasolu-tion (4.1)–(4.2). Then the weak limitψ satisfies |ψ| = 1 almost everywhere and solves the wave map (4.17)–(4.18). Moreover, letψ = eiθ; then the phase functionθ satisfies the wave equation (4.19).
Acknowledgments. The final version of this paper was carried out while C.-K. Lin was visit-ing the Department of Mathematical and Statistical Science, University of Alberta, Canada. He thanks Prof. Yau-Shu Wong for the support, hospitality and friendship during his visit.
Appendix: Proof of Theorem3.1
The goal of this appendix is a short and direct proof of Theorem3.1. (The proof of Theorem2.1or Theorem4.1proceeds along the same lines with modification.) We employ the Fourier–Galerkin method to construct a sequence of approxima-tion soluapproxima-tions, and use the compactness argument to prove the existence of weak solutions; this technique was applied to complex Ginzburg–Landau equation by
Doering et al. in [4]. The light speed c and Planck’s constant are assumed to be fixed numbers (or both equal to 1 after proper rescaling) and the proof is decomposed into four steps.
Step 1. Construction of approximation solutions ψδ by the Fourier–Galerkin method. Let Pδ denote the L2orthogonal projection onto the span of all Fourier modes of wave vectorξ with |ξ| 1/δ. Define ψ0δ = Pδψ0,ψ1δ = Pδψ1and let
ψδ= ψδ(t) be the unique solution of the ODE
− 2 2c2∂ 2 tψδ+ i∂tψδ+ 2 2 Δψ δ− Pδ(|ψδ|pψδ) = 0, (5.1) with initial conditions
ψδ(x, 0) = ψδ
0(x), ∂tψδ(x, 0) = ψ1δ(x), x ∈ Tn. (5.2)
The regularized initial data are chosen such that(ψ0δ, ψ1δ) → (ψ0, ψ1) in H1∩ Lp+2(Tn) ⊕ L2(Tn) as δ tends to zero. These solutions will satisfy the regularized version of the weak formulation
0= − 2 2c2 ∂tψδ(·, t2) − ∂tψδ(·, t1), ϕ + i ψδ(·, t2) − ψδ(·, t1), ϕ −2 2 t2 t1 ∇ψδ(·, τ), ∇ϕ dτ − t2 t1 |ψδ|pψδ(·, τ), ϕ dτ
for every[t1, t2] ⊂ [0, ∞) and for all ϕ ∈ C0∞(Tn). Furthermore, the approximate solutionψδ≡ Pδψ will converge to ψ in C∞asδ tends to zero and satisfies the conservation laws of charge and energy given, respectively, by
Tn|ψ δ|2+ c2 i 2 ψδ∂tψδ− ψδ∂ tψδ dx= C1δ, (5.3) Tn 2 2c2|∂tψ δ|2+2 2 |∇ψ δ|2+ 1 p+ 2|ψ δ|p+2 dx = C2δ. (5.4)
Here C1δand C2δdenote the initial charge and initial energy, respectively. By Young’s inequality and uniform boundedness of the charge and energy, we derive
Tn|ψ δ|2 dx c2 Tn|∂tψ δ||ψδ|dx + Cδ 1 1 2 Tn|ψ δ|2+2 c4|∂tψδ| 2 dx+ C1δ 1 2 Tn|ψ δ|2dx+ 1 c2C2δ+ Cδ1,
that is, Tn|ψ δ|2dx 2Cδ 1+ 2 c2C2δ. (5.5)
Adding (5.4) and (5.5) together, we have shown that the approximate solutionψδ satisfies the charge-energy inequality
Tn|ψ δ|2+ 2 2c2|∂tψδ| 2+2 2 |∇ψ δ|2+|ψδ|p+2 p+ 2 dx 2C δ 1+ 1+ 2 c2 C2δ. (5.6)
Step 2. Show that {ψδ} is a relatively compact set in C([0, T ]; L2(Tn)) ∩ L∞([0, T ]; Lp+1(Tn)) and {∂tψδ} is relatively compact in C([0, T ]; H−1(Tn)). We deduce from the charge-energy bound (5.6) that
{ψδ}δ is bounded in L∞[0, T ]; H1(Tn), (5.7) {∂tψδ}δ is bounded in L∞ [0, T ]; L2(Tn), (5.8) {ψδ}δ is bounded in L∞[0, T ]; Lp+2(Tn). (5.9) It follows from (5.7)–(5.9) and the classical compactness argument that there exists a subsequence of {ψδ}δ, which we still denote by {ψδ}δ, and ψ ∈ L∞([0, T ]; H1(Tn)), ∂tψ ∈ L∞([0, T ]; L2(Tn)) such that ψδ ψ weakly ∗ in L∞[0, T ]; H1(Tn), (5.10) ∂tψδ ∂tψ weakly ∗ in L∞ [0, T ]; L2(Tn), (5.11) ψδ ψ weakly ∗ in L∞[0, T ]; Lp+2(Tn). (5.12) Using the same technique discussed in Lemma2.1and Lemma3.2, we can apply the Arzela–Ascoli theorem and interpolation theorem to conclude
ψδ→ ψ in C[0, T ]; L2(Tn)∩ L∞[0, T ]; Lp+1(Tn).
The convergence of∂tψδ → ∂tψ in C([0, T ]; w − L2(Tn)) also follows by the Arzela–Ascoli theorem. First, it is obvious that{∂tψδ(t)}δis a relatively compact set inw-L2(Tn) for all t 0 by energy bound. To show {∂tψδ} is equicontinuous in C([0, T ]; w-L2(Tn)), let A ⊂ C∞0 (Tn) be an enumerable set which is dense in L2(Tn), then for any ρ ∈ A, we have
2 2c2 ∂tψδ(·, t2) − ∂tψδ(·, t1), ρ = i t2 t1 ∂tψδ(·, τ), ρ dτ −2 2 t2 t1 ∇ψδ(·, τ), ∇ρ dτ − t2 t1 |ψδ|pψδ(·, τ), ρ dτ,
so we derive the estimate
|∂tψδ(·, t2) − ∂tψδ(·, t1), ρ | |t2− t1|
ρH1(Tn)+ ρL∞(Tn).
The rest follows by density argument, and this proves the equicontinuity of{∂tψδ} in C([0, T ]; w-L2(Tn)), so ∂tψδ → ∂tψ in C([0, T ]; w-L2(Tn)). Indeed, we have the strong convergence∂tψδ→ ∂tψ in C([0, T ]; H−1(Tn)) by the Rellich lemma, which states that L2 → H−1is a compact embedding.
Step 3. Passage to the limit (δ → 0). The weak ∗ convergence of ψδ in L∞([0, T ]; H1(Tn)), the strong convergence of ψδin C([0, T ]; L2(Tn)) and the strong convergence of∂tψδin C([0, T ]; H−1(Tn)) give the following convergent results: t2 t1 ∇ψδ(·, τ), ∇ϕ dτ → t2 t1 ∇ψ(·, τ), ∇ϕ dτ, (5.13) ψδ(·, t2) − ψδ(·, t1), ϕ → ψ(·, t2) − ψ(·, t1), ϕ , (5.14) ∂tψδ(·, t2) − ∂tψδ(·, t1), ϕ → ∂tψ(·, t2) − ∂tψ(·, t1), ϕ . (5.15)
Moreover, the same argument we applied to the non-relativistic limit shows
|ψδ|pψδ→ |ψ|pψ in the sense of distribution, that is, t2 t1 |ψδ|pψδ(·, τ), ϕ dτ → t2 t1 |ψ|pψ(·, τ), ϕ dτ. (5.16) Thereforeψ satisfies the weak formulation of (3.1).
Step 4. Proof of the charge-energy inequality. The strong convergence ofψδ in C([0, T ]; L2(Tn)) implies Tn|ψ δ|2dx→ Tn|ψ| 2dx. (5.17)
The weak convergence ofψδ in L∞([0, T ]; H1(Tn)) ∩ L∞([0, T ]; Lp+2(Tn)), together with the fact that the norm of the weak limit of a sequence is a lower bound for the inferior limit of the norms, yields
Tn|∇ψ| 2 dx lim inf δ→0 Tn|∇ψ δ|2dx, (5.18) Tn|ψ| p+2 dx lim inf δ→0 Tn|ψ δ|p+2 dx. (5.19)
Similarly, the weak convergence of∂tψδin L∞([0, T ]; L2(Tn)) implies Tn|∂tψ| 2 dx lim inf δ→0 Tn|∂tψ δ|2 dx. (5.20)
By combining (5.6) and the above inequalities, we obtain the charge-energy inequality Tn|ψ| 2+ 2 2c2|∂tψ| 2+2 2|∇ψ| 2+|ψ|p+2 p+ 2 dx 2C1+ 1+ 2 c2 C2, (5.21)
where the two constants C1= Tn|ψ0| 2+ c2 i 2 ψ1ψ0− ψ1ψ0 dx, C2= Tn 2 2c2|ψ1| 2+2 2|∇ψ0| 2+ 1 p+ 2|ψ0| p+2dx, (5.22)
represent the initial charge and energy, respectively. This completes the proof of Theorem3.1.
References
1. Bresch, D., Desjardins, B., Grenier, E., Lin, C.K.: Low Mach number limit of vis-cous polytropic flows: formal asymptotics in the periodic case. Stud. Appl. Math. 109, 125–149 (2002)
2. Colin, T., Soyeur, A.: Some singular limits for evolutionary Ginzburg Landau equa-tions. Asymptot. Anal. 13, 361–372 (1996)
3. Desjardins, B., Lin, C.K., Tso, T.C.: Semiclassical limit of the derivative nonlinear Schrödinger equation. Math. Models Methods Appl. Sci. 10, 261–285 (2000)
4. Doering, C.R., Gibbon, J.D., Levermore, C.D.: Weak and strong solutions of the complex Ginzburg–Landau equation. Phys. D 71, 285–318 (1994)
5. Colliander, J.E., Jerrard, R.L.: Vortex dynamics for the Ginzburg–Landau– Schrödinger equation. Int. Math. Res. Notices 7, 333–358 (1998)
6. Desjardins, B., Lin, C.K.: On the semiclassical limit of the general modified NLS equation. J. Math. Anal. Appl. 260, 546–571 (2001)
7. Gasser, I., Lin, C.K., Markowich, P.: A review of dispersive limit of the (non)linear Schrödinger-type equation. Taiwanese J. Math. 4, 501–529 (2000)
8. Grenier, E.: Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Am. Math. Soc. 126, 523–530 (1998)
9. Jin, S., Levermore, C.D., McLaughlin, D.W.: The semiclassical limit of the defo-cusing NLS hierarchy. Commun. Pure Appl. Math. 52, 613–654 (1999)
10. Lee, J.H., Lin, C.K.: The behavior of solutions of NLS equation of derivative type in the semiclassical limit. Chaos Solitons Fractals 13, 1475–1492 (2002)
11. Lee, J.H., Lin, C.K., Pashaev, O.K.: Shock waves, chiral solitons and semioclassical limit of one-dimensional anyons. Chaos Solitons Fractals 19, 109–128 (2004) 12. Lin, C.K.: On the incompressible limit of the compressible Navier-Stokes equations.
Commun. Partial Differ. Equ. 20, 677–707 (1995)
13. Lin, C.K.: Potential flow for the compressible viscous fluid and its incompressible limit. Indiana Univ. Math. J. 49, 1539–1561 (2000)
14. Lin, C.K., Wong, Y.S.: Zero-dispersion limit of the short-wave-long-wave interaction equation. J. Differ. Equ. 228, 87–110 (2006)
15. Lin, C.K., Wu, K.C.: On the fluid dynamical approximation to the nonlinear Klein– Gordon equation (2009, submitted)
16. Lin, F.H., Xin, J.X.: On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation. Commun. Math. Phys. 200, 249–274 (1999)
17. Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)
18. Machihara, S., Nakanishi, K., Ozawa, T.: Nonrelativistic limit in the energy space for nonlinear Klein–Gordon equations. Math. Ann. 322, 603–621 (2002)
19. Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables Appl. Math. Sci., vol. 53. Springer, Berlin, 1984
20. Masmoudi, N., Nakanishi, K.: From nonlinear Klein–Gordon equation to a system of coupled nonlinear Schrödinger equations. Math. Ann. 324, 359–389 (2002)
21. Masmoudi, N., Nakanishi, K.: Nonrelativistic limit from Maxwell-Klein–Gordon and Maxwell–Dirac to Poisson–Schrödinger. Int. Math. Res. Notice 13, 697–734 (2003) 22. Messiah, A.: Quantum Mechanics, vol. 1 & 2 (two volumes bound as one). Dover
Publications, New York, 1999
23. Nakanishi, K.: Nonrelativistic limit of scattering theory for nonlinear Klein–Gordon equations. J. Differ. Equ. 180, 453–470 (2002)
24. Shatah, J.: Weak solutions and development of singularities of the SU(2)σ -Model. Comm. Pure Appl. Math. 41, 459–469 (1998)
25. Shatah, J., Struwe, M.: Geometric Wave Equations, Courant Lecture Notes in Math-ematics, vol. 2. American Mathematical Society, Providence, 1998
26. Strauss, W.: Nonlinear Wave Equations, CBMS, vol. 73. American Mathematical Soci-ety, Providence, 1989
Department of Applied Mathematics and Center of Mathematical Modeling and Scientific Computing,
National Chiao Tung University, Hsinchu 30010, Taiwan. e-mail: [email protected] e-mail: [email protected] (Received January 11, 2009 / Accepted October 15, 2009) Published online April 22, 2010 – © Springer-Verlag (2010)
