J. Math. Pures Appl. 98 (2012) 328–345
www.elsevier.com/locate/matpur
Hydrodynamic limits of the nonlinear Klein–Gordon equation
Chi-Kun Lin
a,∗, Kung-Chien Wu
baDepartment of Applied Mathematics and Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu 30010, Taiwan
bDepartment of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 OWA, UK
Received 18 August 2011 Available online 25 February 2012
Abstract
We perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein–Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we further rescale the time variable, then in the semiclassical limit (the light speed kept fixed), the incompressible Euler equations are recovered. The proof involves the modulated energy introduced by Brenier (2000)[1].
©2012 Elsevier Masson SAS. All rights reserved.
Résumé
On obtient une dérivation mathématique des équations d’Euler compressibles et incompressibles à partir de l’équation de Klein– Gordon non linéaire modulée. Avant la formation de singularités pour le système limite, on démontre dans la limite non relativiste et semi-classique la convergence vers les équations d’Euler compressibles. Au moyen d’un changement d’échelle supplémentaire en temps, on démontre la limite semi-classique la vitesse de la lumière restant fixée, la limite vers les équations d’Euler incompres-sibles. La démonstration utilise l’énergie modulée introduite par Brenier (2000)[1].
©2012 Elsevier Masson SAS. All rights reserved.
Keywords: Klein–Gordon equation; Hydrodynamic limits; Euler equations
1. Introduction
In this paper, we study the nonlinear Klein–Gordon equation ¯h2 2mc2∂ 2 tΨ − ¯ h2 2mΨ + mc2 2 Ψ + V |Ψ |2Ψ= 0 , (1.1) where m is mass, c is the speed of light, ¯h is the Planck constant and Ψ (x, t) is a complex-valued vector field over a spatial domain Ω ⊂ Rn. The nonlinear function V is the first derivative of a twice differentiable nonlinear real-valued function overR+. Thus, V is the potential energy and V is the potential energy density of the fields.
* Corresponding author.
E-mail addresses:[email protected](C.-K. Lin),[email protected](K.-C. Wu). 0021-7824/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved.
The Klein–Gordon equation for the complex scalar field is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. The reader should refer to[23,25]for the physical background and[27]for a general introduction to nonlinear wave equations. Since the Planck constant ¯h has the dimension of action, [¯h] = [energy] × [time] = [action], it is easy to check that (1.1) is dimensionally balanced. Furthermore, mc2t and the Planck constant ¯h have the same dimensions, [mc2t] = [¯h] = [action], so we may consider the modulated wave
function
ψ (x, t )= Ψ (x, t) expimc2t /¯h,
where the factor exp(imc2t /¯h) describes the oscillations of the wave function, then ψ satisfies the modulated nonlinear
Klein–Gordon equation i¯h∂tψ+ ¯ h2 2mψ− V |ψ|2ψ= ¯h2 2mc2∂ 2 tψ. (1.2)
The relations between different terms in (1.2) are best seen when the equation is written in terms of dimensionless variables, which will be adorned with carets. The dimensionless independent variables are given by
x= L ˆx, t= T ˆt,
where L and T denote the reference length and time respectively. We also define the reference velocity by U= L/T and rescale the potential energy as
V= mU2Vˆ.
Substituting all of these rescaled quantities into the original equation (1.2), and dropping all carets, yields
iε∂tψ+ 1 2ε 2ψ− V|ψ|2ψ=1 2ε 2ν2∂2 tψ. (1.3)
Note that the first important dimensionless parameter ν is given by the ratio of reference velocity and speed of light,
ν= U/c, and the scaled Planck constant ε = ¯h
mU2T is the second important dimensionless parameter. The two
dimen-sionless parameters ν and ε show the relativistic and quantum effects respectively. Formally letting ν→ 0 or c → ∞ (more precisely U c), i.e., the so-called nonrelativistic limit, the modulated nonlinear Klein–Gordon equation (1.3) will reduce to the nonlinear Schrödinger equation
iε∂tψ+
1 2ε
2ψ− V|ψ|2ψ= 0,
(1.4) though one has to be extremely careful with heuristics due to the double time derivative on the right-hand side of (1.3).
Over the last twenty years, there has been a vast amount of research concerning the nonrelativistic limit of the Cauchy problem for the nonlinear Klein–Gordon equation. In particular, in[18]Machihara–Nakanishi–Ozawa proved that any finite energy solution converges to the corresponding solution of the nonlinear Schrödinger equation in the energy space, after infinite oscillations in time are removed. The Strichartz estimate plays the most important role to obtain the uniform bound in space and time (see also[22,24]and references therein). However, to the best of our knowledge, the semiclassical limit ε→ 0 is not well studied and is not clear from (1.3). On the other hand, based on the hydrodynamical structure, the semiclassical limit, ε→ 0, of the defocusing nonlinear Schrödinger equation is quite well understood (see[4] for the review). In[6], Jin–Levermore–McLaughlin applied the inverse scattering to establish the semiclassical limit of the defocusing cubic nonlinear Schrödinger equation; the complete integrability was exploited to obtain the global characterization of the weak limits of the entire cubic NLS hierarchy. Therefore to study the various singular (hydrodynamics) limits of the nonlinear Klein–Gordon equation (1.1), it is better to start from (1.3) because of its analogue to the nonlinear Schrödinger equation (1.4).
For the defocusing nonlinear Schrödinger equation, the semiclassical limit for initial data with Sobolev regularity in short time has been studied by Grenier[5]. In this limit, the Euler equations for an isentropic compressible flow are recovered. The basic idea is introducing the modified Madelung transform and rewrite the hydrodynamical equations as a linear dispersive perturbation of the quasilinear symmetric hyperbolic system. The same method also works well to other Schrödinger type equations[3,8,9,11]. But unfortunately, this method cannot be applied to the modulated
nonlinear Klein–Gordon equation (1.3). The main difficulty is that we are not able to rewrite it as a quasilinear symmetric hyperbolic system and thus the standard energy estimate cannot be applied.
We proved in [12,28]that the semiclassical limit of the modulated cubic nonlinear Klein–Gordon equation is a relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic-semiclassical limit is the classical wave map for the limit wave function and the typical linear wave equation for the associated phase function. In this paper, we consider the singular limits from the point of view of hydrodynamics (see
[13]for the fluid approximation). As has been pointed out by Masmoudi and Nakanishi in[21](see also[12]), the Klein–Gordon equation behaves like the Schrödinger equation in the nonrelativistic region, and like the wave equation in the relativistic region. Thus, we consider the nonrelativistic-semiclassical limit, i.e. the two parameters ε, ν→ 0 simultaneously, of the modulated nonlinear Klein–Gordon equation (1.3). To avoid carrying out a double limit, we restrict the case when the two parameters ν and ε are related. In this situation, the compressible Euler equations are recovered as the nonrelativistic-semiclassical limit. On the other hand, if we rescale the time variable, then the extra degree of the parameter ε enables us to discuss the semiclassical limit no matter when ν is of order O(1) or tends to zero as ε→ 0 and the limit of the current is shown to satisfy the incompressible Euler equations.
The modulated energy method was introduced by Brenier [1] to prove the convergence of the Vlasov–Poisson system to the incompressible Euler equations. It was immediately extended by Masmoudi in[20] to general ini-tial data allowing the presence of high oscillations in time (see also[10] for the quantum hydrodynamic model of semiconductor). The same idea is also applied to study various singular limits of other equations, for example the Schrödinger–Poisson equation[26], the Gross–Pitaevskii equation[14] and the coupled nonlinear Schrödinger equation[7,15]. In fact, we will employ this method to study the hydrodynamic limits of the modulated nonlinear Klein–Gordon equation (1.3). Similar to[1], we limit ourselves in this paper to the case when the initial data is well prepared. For general initial condition, the method used in[10,20]should be applicable and it will be our next research project.
The modulated energy is designed to control the propagation of the charge and current (or momentum) for Vlasov– Poisson, Schrödinger and the related nonrelativistic type equations. Since the charge (current) of the Klein–Gordon equation is constituted by the Schrödinger and relativistic parts, thus, the main idea is to show that the relativistic charge and current are small and the main contribution of the nonrelativistic-semiclassical limit comes from the Schrödinger part. In contrast with the Schrödinger equation and its variants, we have to introduce one correction term of the modulated energy which controls the propagation of the relativistic charge and current. In fact, the relativistic parts vanishes as ε tends to zero. Thus we prove the convergence of the charge and the current defined by the modulated nonlinear Klein–Gordon equation towards the solution of the γ -law compressible Euler equations. The range of the adiabatic exponent γ is different: γ > 1 for even spatial dimension n and γn−1n for odd n.
Turning to the incompressible limit, we have to rescale the time variable and consider the potential energy designed to represent in the form of pressure instead of the charge (or density). In this case, we show that the current converges to the incompressible Euler equations in the semiclassical limit. For the potential energy given in terms of the power of density, we have a similar result. However, the associated adiabatic exponent γ is different because of the convexity of the potential energy discussed. The difference between the compressible and incompressible flow is typified by consideration of the way in which an acoustic wave travels in the fluid. Besides the correction term of the modulated energy as discussed in the compressible Euler limit, we have to introduce one more correction term which describes the propagation of the density fluctuation in order to obtain the incompressible limit. This is similar to the zero Mach number limit of the compressible fluid [2,17,19]. The convergent result can be improved for n= 2 by the standard bootstrap process.
The rest of the paper is organized as follows. In Section2, we derive the hydrodynamical structure of the modulated nonlinear Klein–Gordon equation and discuss their relation to the compressible and incompressible Euler equations. The proof of the convergence of the modulated nonlinear Klein–Gordon equation to the compressible Euler equations is established in Section3. In Section4, we prove the convergence of the time-scaled modulated nonlinear Klein– Gordon equation to the incompressible Euler equations.
Notation. In this paper, Lp(Ω) (p 1) denotes the classical Lebesgue space with norm f p= (
Ω|f |pdx)1/p,
the Sobolev space of functions with all its k-th partial derivatives in L2(Ω)will be denoted by Hk(Ω). We abbreviate “ C ” to “”, where C is a positive constant depending only on a fixed parameter.
2. Hydrodynamical structure
A fluid mechanical interpretation for the linear Schrödinger equation was put forth by Madelung in 1927, and applies to nonlinear Schrödinger equations. Indeed, as shown in[5], the same idea also applied to the modulated non-linear Klein–Gordon equation (1.3). We introduce the complex wave function, the so-called Madelung transformation,
ψ= A exp(iS/ε), (2.1)
in which both A, the amplitude, and S, the action function, are real-valued functions. It is important that the amplitude is assumed to be non-negative at every point: A 0. Plugging (2.1) into modulated nonlinear Klein–Gordon equation (1.3) and separating the real and imagine parts, we obtain
∂tA+ A 2 S− ν2∂t2S+ ∇A · ∇S − ν2∂tA∂tS= 0, (2.2) ∂tS+ 1 2|∇S| 2−1 2ν 2(∂ tS)2+ V A2=ε 2 2 νA A , (2.3)
where the d’Alerbertian ν is defined by ν ≡ − ν2∂t2. Eqs. (2.2) and (2.3) are equivalent to the modulated
nonlinear Klein–Gordon equation (1.3) for smooth functions A and S. Eq. (2.2) turns out to be the continuity equation for the relativistic quantum fluid and Eq. (2.3) is the relativistic quantum Hamilton–Jacobi equation. Introducing the new functions ρ= A2= ψψ = |ψ|2, u= ∇S =iε 2 1 |ψ|2(ψ∇ψ − ψ∇ψ), ρK= ν2A2∂tS= iεν2 2 (ψ ∂tψ− ψ∂tψ ),
we can rewrite (2.2)–(2.3) as the dispersive perturbation of the compressible Euler type equations
∂t(ρ− ρK)+ ∇ · (ρu) = 0, ν2∂tu= ∇ ρK ρ , (2.4) ∂t(ρu− ρKu)+ ∇ · (ρu ⊗ u) + ∇P (ρ) = ε2 4∇ · ρ∇2log ρ−ε 2ν2 4 ∂t(ρ∇∂tlog ρ), (2.5) where P (ρ)= ρV(ρ)− V (ρ) is the pressure and ∇2 denotes the Hessian. Eqs. (2.4)–(2.5) are constituted by the Euler, relativistic and quantum parts. If the “Euler part” of these equations is to be hyperbolic, then the pressure P (ρ) must be a strictly increasing function of ρ; in that case, P(ρ)= ρV(ρ) >0. This means that V must be a strictly convex function of ρ and corresponds to a defocusing nonlinear Klein–Gordon equation. Defining the Schrödinger part energy density ESand relativistic part energy density EK respectively by
ES= 1 2ρ|u| 2+ε2 8 |∇ρ|2 ρ + V (ρ) = ε2 2|∇ψ| 2+ V|ψ|2, EK= 1 2ν2 ρK2 ρ + ε2ν2 8 |∂tρ|2 ρ = ε2ν2 2 |∂tψ| 2,
we obtain from (2.4)–(2.5) the conservation of energy
∂t(ES+ EK)+ ∇ · ES+ P (ρ) u=ε 2 4 ∇ · uρ− ∇ · (ρu)∇ρ ρ .
In the formal nonrelativistic limit ν→ 0, the relativistic part energy EK gives ρK→ 0, and (2.4)–(2.5) reduce to the
quantum hydrodynamical equations
∂tρ+ ∇ · (ρu) = 0, ∂t(ρu)+ ∇ · (ρu ⊗ u) + ∇P (ρ) = ε2 4 ∇ · ρ∇2log ρ,
which are exactly the fluid formulation of the defocusing nonlinear Schrödinger equation (1.4). In this case the rela-tivistic part energy density EK vanishes and the limit energy density E will be given by
E=1 2ρ|u| 2+ε2 8 |∇ρ|2 ρ + V (ρ),
and will satisfy
∂tE+ ∇ · E+ P (ρ)u=ε 2 4∇ · (ρu)ρ ρ − ∇ · (ρu) ∇ρ ρ .
Next letting ν→ 0 and ε → 0 simultaneously, both the relativistic and quantum correction terms in (2.4)–(2.5) vanish and the limit densities ρ, u and P will satisfy the compressible Euler equations
∂tρ+ ∇ · (ρu) = 0,
∂t(ρu)+ ∇ · (ρu ⊗ u) + ∇P (ρ) = 0,
and the limit energy density E will be given by
E=1
2ρ|u|
2+ V (ρ),
and will satisfy
∂tE+ ∇ ·
E+ P (ρ)u= 0
hence playing the role of a Lax entropy for the Euler system.
In order to investigate the incompressible limit, we introduce the scaling t= εαt, x= x, α > 0.
After dropping the tilde, the modulated nonlinear Klein–Gordon equation (1.3) becomes
iε1+α∂tψ− ε2+2αν2 2 ∂ 2 tψ+ ε2 2 ψ− V |ψ|2ψ= 0.
For this model the corresponding fluid dynamics equations (2.4)–(2.5) turn out to be
∂t(ρ− ρK)+ ∇ · (ρu) = 0, (2.6) ∂t ρu− ρKu+ ε2ν2 4 ρ∇∂tlog ρ + ∇ · (ρu ⊗ u) + 1 ε2α∇P (ρ) = ε2−2α 4 ∇ · ρ∇2log ρ, (2.7) and the associated energy equation becomes
∂t(ES+ EK)+ ∇ · ES+ P (ρ) ε2α u =ε2−2α 4 ∇ · (ρu)ρ ρ − ∇ · (ρu) ∇ρ ρ ,
where the Schrödinger part energy density ES and relativistic part energy density EKare given respectively by
ES= 1 2ρ|u| 2+ε2−2α 8 |∇ρ|2 ρ + 1 ε2αV (ρ), EK= 1 2ν2ε2α ρK2 ρ + ε2ν2 8 |∂tρ|2 ρ .
It follows immediately from the energy equation that 1 2ν2ε2α ρK2 ρ + ε2ν2 8 |∂tρ|2 ρ + 1 2ρ|u| 2+ε2−2α 8 |∇ρ|2 ρ + V (ρ) ε2α dx C (2.8)
for all 0 < t <∞ if the initial energy is bounded. Assuming the minimum of the convex function V (ρ) occurs at
ρ= 1 then the energy bound (2.8) implies ρ→ 1 and ρK→ 0 as ε → 0. Since the density ρ goes to 1, we expect that
∂tu+ ∇ · (u ⊗ u) + ∇ P = 0,
where P is the limit of P (ρ)ε−P (1)2α . In other words, we recover the incompressible Euler equations, for which we refer
to[16].
3. Compressible Euler equations
The first result we shall prove rigorously in this paper is the convergence towards the compressible Euler equations. In fact, we consider the so-called nonrelativistic-semiclassical limit, i.e. ν→ 0 and ε → 0 simultaneously. In order to avoid carrying out a double limits the parameters ν and ε must be related. For convenience we set ν= εκ for some κ > 0, 0 < ε 1 and assume the potential energy V(|ψε|2)= |ψε|2(γ−1). Indeed we consider the modulated nonlinear Klein–Gordon equation
iε∂tψε− 1 2ε 2+2κ∂2 tψε+ 1 2ε 2ψε− ψε 2(γ−1) ψε= 0, (3.1) supplemented with the initial conditions:
ψε(x,0)= ψ0ε(x), ∂tψε(x,0)= ψ1ε(x), x∈ Ω, (3.2) satisfying Tn 1 2ε 2+2κ ψε 1 2 +1 2ε 2 ∇ψε 0 2 +1 γ ψ ε 0 2γ dx C. (3.3)
Here and below, C denotes various positive constants independent of ε. To avoid the complications at the boundary, we concentrate below on the case where x∈ Ω = Tn, the n-dimensional torus.
Associated with (3.1) are the local conservation laws corresponding to charge, momentum (current) and energy conservation. In fact, we have the hydrodynamical variables: Schrödinger part charge ρSε, relativistic part charge ρKε, Schrödinger part momentum (current) JSε, relativistic part momentum (current) JKε and energy eεgiven as follows:
ρSε= ψε 2, ρεK= i 2ε 1+2κψε∂ tψε− ψε∂tψε , JSε=JS,ε1, JS,ε2, . . . , JS,nε = i 2ε ψε∇ψε− ψε∇ψε, JKε =JK,ε 1, JK,ε 2, . . . , JK,nε =1 2ε 2+2κ∂ tψε∇ψε+ ∂tψε∇ψε , eε=1 2ε 2+2κ ∂ tψε 2 +1 2ε 2 ∇ψε 2+ 1 γ ψ ε 2γ. (3.4)
The local conservation laws of the modulated Klein–Gordon equation (3.1) are the charge, momentum (current) and energy given below:
(A) Conservation of charge
∂ ∂t
ρSε− ρεK+ ∇ · JSε= 0, (3.5) (B) Conservation of momentum (current)
∂ ∂t JSε− JKε+1 4ε 2∇ · 2∇ψε⊗ ∇ψε+ ∇ψε⊗ ∇ψε− ∇2 ψε 2 +1 4ε 2+2κ∇∂ t ψε∂tψε+ ψε∂tψε +γ− 1 γ ∇ ψ ε 2γ = 0, (3.6) (C) Conservation of energy ∂ ∂te ε− ∇ · 1 2ε 2∇ψε∂ tψε+ ∇ψε∂tψε = 0. (3.7)
Theorem 3.1. Let γ > 1 for even spatial dimension n and γ n−1n for odd n. Let ψεbe the solution of the modulated nonlinear Klein–Gordon equations (3.1)–(3.2) and the initial condition (ψ0ε, ψ1ε)∈ Hs+1(Tn)⊕ Hs(Tn), s >n2+ 1, satisfying (3.3) and (3.11). Then there exists T∗>0 such that
ρSε− ρ(·, t)Lγ(Tn)→ 0, ρKε(·, t) L 2γ γ+1(Tn)→ 0, JSε− ρu(·, t) L 2γ γ+1(Tn)→ 0, JKε(·, t)L1(Tn)→ 0,
for t∈ [0, T∗) as ε↓ 0, where (ρ, u) ∈ C([0, T∗); Hs(Tn)) is the unique local smooth solution of the γ -law compress-ible Euler equations
⎧ ⎨ ⎩ ∂tρ+ ∇ · (ρu) = 0, x∈ Tn, t∈ [0, T∗), ∂t(ρu)+ ∇ · (ρu ⊗ u) + ∇P (ρ) = 0, ρ(x,0)= ρ0(x), u(x,0)= u0(x), x∈ Tn, (3.8)
where 0 < ρ0∈ Hs(Tn), u0∈ Hs(Tn) and the equation of states is given by P (ρ)=γ−1γ ργ.
Motivated by Brenier’s pioneer work[1], we will prove this theorem by modulated energy. It is easy to see that when the parameter ε is small, the wave function ψεand hydrodynamic variables ρ, u are related according to
ψε 2= ρSε≈ ρ, iε 2 1 |ψε|2 ψε∇ψε− ψε∇ψε≈ u.
The symbol “A≈ B” means that A almost equals B. Moreover, as ε tends to zero, the limiting energy will be12ρ|u|2+
1
γργ. Keeping this term in mind and comparing with the energy of the modulated nonlinear Klein–Gordon equation
(3.1), we have: 1 2ε 2 ∇ψε 2 ≈1 2ρ|u| 2, 1 2ε 2+2κ ∂ tψε 2≈ 0, 1 γ ρSεγ≈ 1 γρ γ+ ργ−1ρε S− ρ .
Thus, we have the relation 1 2ε 2 ∇ψε 2−1 2ρ|u| 2≈ε2 2 ∇ψ ε 2− 2ε−2 ψε 2|u|2+ ε−2 ψε 2|u|2 ≈ε2 2 ∇ψ ε 2 − iε−1ψε∇ψε− ψε∇ψε· u + ε−2 ψε 2 |u|2 =1 2 (ε∇ − iu)ψ ε 2 .
Therefore we can define the modulated energy of (3.1) as
Hε(t)=1 2 Tn (ε∇ − iu)ψε 2dx+1 2 Tn ε1+κ∂tψε 2dx+ Tn ΘρSε, ρdx (3.9) where ΘρSε, ρ=1 γ ρSεγ− ργ− ργ−1ρSε− ρ (3.10) is a convex function, minimum occurs at ρSε= ρ and satisfies Θ(ρSε) 0. We also assume
Hε(0)=1 2 Tn (ε∇ − iu0)ψ0ε 2dx+ 1 2 Tn ε1+κψ1ε 2dx + Tn Θ ψ0ε 2, ρ0 dx= Oεβ, for some β > 0, (3.11)
i.e., we consider the well-prepared initial data. We can rewrite the modulated energy (3.9) in terms of hydrodynamical variables only as Hε(t)= Tn eεdx− Tn u· JSεdx+1 2 Tn ρSε|u|2dx+ Tn γ− 1 γ ρ− ρ ε S ργ−1dx. (3.12)
Therefore to obtain the hydrodynamic limit we have to show that the modulated energy Hε(t)tends to zero as ε→ 0. Indeed, we have the following estimate.
Lemma 3.2. Under the hypothesis of Theorem3.1and let λ= min{1, κ, β}, we have Hε(t) Oελ uniformly in t∈ [0, T ].
Proof. We have to check the evolution of the modulated energy Hε(t)given by (3.12). Differentiating the modulate energy Hεwith respect to time variable t and using the conservation of energy (3.7), we obtain
d dtH ε(t)= −d dt Tn u· JSεdx+1 2 d dt Tn ρSε|u|2dx+ d dt Tn γ− 1 γ ρ− ρ ε S ργ−1dx. (3.13)
We discuss the right-hand side of (3.13) separately. Integration by parts and using conservation of momentum (3.6), the first term of the right-hand side of (3.13) becomes
−d dt Tn u· JSεdx= − Tn ∂tu· JSεdx− Tn γ− 1 γ ρSεγ∇ · u dx −ε2 4 Tn 2∇ψε⊗ ∇ψε+ ∇ψε⊗ ∇ψε: ∇u + ∇ ψε 2 · (∇∇ · u) dx −1 4ε 2+2κ d dt Tn ∂t ψε 2 ∇ · u dx − d dt Tn u· JKε dx +1 4ε 2+2κ Tn ∂t ψε 2 ∇ · ∂tu dx+ Tn ∂tu· JKε dx. (3.14)
Next, by conservation of charge (3.5) and integration by parts, we have 1 2 d dt Tn ρεS|u|2dx= Tn ρSεu· ∂tu dx+ 1 2 Tn ∇|u|2· Jε Sdx +1 2 d dt Tn ρKε|u|2dx− Tn ρKεu· ∂tu dx. (3.15)
The third term of the right-hand side of (3.13) becomes
d dt Tn γ− 1 γ ρ− ρ ε S ργ−1dx= Tn (γ − 1)ργ−2ρ− ρSε∂tρ dx− d dt Tn ργ−1ρKε dx + Tn ∂tργ−1ρKε − ∇ρ γ−1· Jε Sdx. (3.16)
From (3.14)–(3.16) we define the correction term of the modulated energy Hεas
Gε(t)= −1 2 Tn |u|2ρε Kdx+ Tn ργ−1ρKε dx+1 4ε 2+2κ Tn ∂t ψε 2 ∇ · u dx + Tn u· JKε dx.
It is designed to control the propagation of the relativistic charge and current and will be proved to be small as ε→ 0. Using crucially the limit compressible Euler equations (3.8), we have
d dt Hε(t)+ Gε(t)= Tn JSε− ρSεu· (u · ∇u) dx +1 2 Tn JSε· ∇|u|2dx − Tn γ− 1 γ ρSεγ− (γ − 1)ργ−1ρSε ∇ · u dx −ε2 4 Tn ∇ ψε 2· (∇∇ · u) dx +1 4ε 2+2κ Tn ∂t ψε 2 ∇ · ∂tu dx − Tn u· ∂tuρKε dx+ Tn ∂tργ−1ρKε dx+ Tn ∂tu· JKε dx+ R1+ R2, (3.17) where R1= − ε2 2 Tn ∇ψε⊗ ∇ψε+ ∇ψε⊗ ∇ψε: ∇u dx, R2= − Tn (γ − 1)ργ−1∇ · (ρu) dx.
To deal with R1, we can rewrite R1as
R1= − 1 2 Tn (ε∇ − iu)ψε⊗ (ε∇ − iu)ψε
+ (ε∇ − iu)ψε⊗ (ε∇ − iu)ψε: ∇u dx + K
1+ K2+ K3, where K1= ε 2 Tn
(−iu)ψε⊗ ∇ψε+ (−iu)ψε⊗ ∇ψε: ∇u dx, K2= 1 2 Tn
(−iu)ψε⊗ (−iu)ψε+ (−iu)ψε⊗ (−iu)ψε: ∇u dx, K3= ε 2 Tn
∇ψε⊗ (−iu)ψε+ ∇ψε⊗ (−iu)ψε: ∇u dx.
Using integration by part and go back to the hydrodynamic variables (3.4), one can calculate that
K1= − Tn u⊗ JSε: ∇u dx = Tn 1 2|u| 2∇ · Jε Sdx, and K2+ K3= Tn ρSεu⊗ u: ∇u dx − Tn JSε⊗ u: ∇u dx = Tn (u· ∇)u·ρSεu− JSεdx, i.e. R1= − 1 2 Tn
(ε∇ − iu)ψε⊗ (ε∇ − iu)ψε+ (ε∇ − iu)ψε⊗ (ε∇ − iu)ψε: ∇u dx
+ Tn 1 2|u| 2∇ · Jε Sdx+ Tn (u· ∇)u·ρSεu− JSεdx. (3.18)
Also using the identity (γ − 1)ργ−1∇ · (ρu) =γ− 1 γ ∇ργ· u + (γ − 1)ργ∇ · u, we have R2= − Tn (γ− 1)ργ−1∇ · (ρu) dx = −(γ− 1) 2 γ Tn ργ∇ · u dx. (3.19) Employing (3.18) and (3.19), we can rewrite (3.17) as
d dt Hε(t)+ Gε(t)= −1 2 Tn
(ε∇ − iu)ψε⊗ (ε∇ − iu)ψε+ (ε∇ − iu)ψε⊗ (ε∇ − iu)ψε: ∇u dx
− (γ − 1) Tn 1 γ ρSεγ− ργ− ργ−1ρSε− ρ∇ · u dx −ε2 4 Tn ∇ ψε 2· (∇∇ · u) dx − Tn u· ∂tuρKε dx+ Tn ∂tργ−1ρKε dx +1 4ε 2+2κ Tn ∂t ψε 2 ∇ · ∂tu dx+ Tn ∂tu· JKε dx. (3.20)
One can estimate the first term of the right-hand side of (3.20) as follows ∇u: (ε∇ − iu)ψε⊗ (ε∇ − iu)ψε ∇u L∞(Tn) n j,=1 (ε∂j− iuj)ψε(ε∂− iu)ψε n∇uL∞(Tn) (ε∇ − iu)ψε 2.
Furthermore, for t∈ [0, T∗), by (3.3), (3.5) and (3.7) we have ε∇ψεL2(Tn)=ε
1+κ∂
tψεL2(Tn)= O(1) (3.21)
and
ψεLq(Tn)= O(1), 2 q 2γ. (3.22)
Then by Hölder inequality we have the following estimates
ε2 Tn ∇ ψε 2· (∇∇ · u) dx εε∇ψε L2(Tn)ψεL2γ(Tn)∇∇ · u L 2γ γ−1(Tn) εuH s(Tn), (3.23) and Tn ρεKu· ∂tu dx εκu · ∂tuL∞(Tn)ψε L2(Tn)ε1+κ∂tψεL2(Tn) εκu · ∂ tuL∞(Tn) εκu2Hs(Tn). (3.24)
Similar to (3.23)–(3.24), we also have Tn ∂tργ−1ρKε dx ε κργ−1 Hs(Tn), (3.25) ε2+2κ Tn ∂t ψε 2 ∇ · ∂tu dx ε1+κuHs(Tn), (3.26) Tn ∂tu· JKεdx ε κu Hs(Tn). (3.27)
Combing the above estimates we obtain the inequality d dt Hε(t)+ Gε(t) ∇uL∞(Tn)Hε(t)+ εδ uHs(Tn)+ u2Hs(Tn)+ ρ γ−1 Hs(Tn) (3.28) for t∈ [0, T∗)and δ= min{1, κ}. Integrating (3.28) with respect to time variable t yields
Hε(t) Hε(0)+ Gε(0)− Gε(t)+ C1
t
0
Hε(τ ) dτ+ C2εδt.
Similar to (3.23)–(3.27) one can show that Gε(0)− Gε(t)= O(εκ); and hence Hε(t) C1
t
0
Hε(τ ) dτ+ Hε(0)+ C2εδt+ C3εκ.
Employing the initial condition Hε(0) and the Gronwall inequality we derive
Hε(t)C4εβ+ C2εδt+ C3εκ
1+ C1t eC1t
.
This shows Hε(t) O(ελ)for t∈ [0, T∗), where λ= min{1, κ, β}. 2 It is easy to check that the modulated energy can be rewritten as
Hε(t)=ε 2 2 Tn ∇ρSε 2dx+1 2 Tn 1 ρSε JSε− ρSεu 2 dx+1 2 Tn ε1+κ∂tψε 2 dx+ Tn ΘρεS, ρdx. (3.29) Using (3.29) and Lemma 3.2, we have
Tn 1 ρεS JSε− ρεSu 2 dx→ 0, Tn ΘρSε, ρdx→ 0 (3.30) as ε→ 0. Also the elementary computation shows that ([17])
1 γ ρ ε S− ρ γ ΘρSε, ρ if γ 2; (3.31) ρSε− ρ 2 ρ2−γΘρSε, ρ if 1 < γ < 2 and ρεS 2ρ; (3.32) ρSε− ρ γ ΘρSε, ρ if 1 < γ < 2 and ρSε 2ρ; (3.33) and henceρSε− ρLγ(Tn)→ 0 as ε → 0. On the other hand, applying the triangle and Hölder inequalities we have
JSε− ρu L 2γ γ+1(Tn) JSε− ρSεu L 2γ γ+1(Tn)+ ρSε− ρu L 2γ γ+1(Tn) ρSεL2γ(Tn) 1 ρSε JSε− ρSεu L2(Tn) +ρSε− ρLγ(Tn)u L 2γ γ−1(Tn)
which converges to zero as ε→ 0 by (3.30)–(3.33). Combing (3.21) and (3.22) we have ρKε(·, t) L 2γ γ+1(Tn) ε κε1+κ∂ tψεL2(Tn)ψεL2γ(Tn)→ 0, and JKε(·, t)L1(Tn) εκε1+κ∂tψεL2(Tn)ε∇ψεL2(Tn)→ 0
Remark. The main reason why the adiabatic exponent γ depending on either spacial dimension is even or odd comes
from the estimates (3.23) and (3.26) given respectively by, ∇∇ · u L 2γ γ−1(Tn) uH s(Tn), ∇ · ∂tu L 2γ γ−1(Tn) uH s(Tn).
Since∇∇ · u ∈ Hs−2(Tn), s− 2 >n2− 1, we have ∇∇ · u ∈ Hn/2(Tn)for even spatial dimension n, thus∇∇ · u ∈
Lp(Tn)for any 1 p < ∞ by the Sobolev inequality, so we conclude γ > 1 for even spatial dimensions. On the other hand, for odd n,∇∇ · u ∈ H(n−1)/2(Tn), the Sobolev inequality implies∇∇ · u ∈ Lp(Tn), 1 p 2n, so we need γ2γ−1 2n, and hence γ n−1n for odd spatial dimensions. The argument for the estimate of∇ · ∂tuis similar.
4. Incompressible Euler equations
The second result we want to address in this paper concerns the convergence towards the incompressible Euler equations. We still consider only the n-dimensional torus Tn as discussed in the previous section. To obtain the incompressible limit, the time variable need to be rescaled, t → εαt, α > 0, and potential energy is given by
V(|ψε|2)= (|ψε|γ − 1)|ψε|γ−2, γ 2. More precisely, we will investigate the time-scaled modulated nonlinear
Klein–Gordon equation iε1+α∂tψε− ε2+2αν2 2 ∂ 2 tψε+ ε2 2ψ ε− ψε γ − 1 ψε γ−2ψε= 0, (4.1) supplemented with initial conditions
ψε(x,0)= ψ0ε(x), ∂tψε(x,0)= ψ1ε(x), x∈ Tn,
satisfying the uniform bound Tn 1 2ν 2ε2 ψε 1 2 +1 2ε 2−2α ∇ψε 0 2 + 1 γ ε2α ψ ε 0 γ − 12 dx < C. (4.2)
We will consider the limit as the scaled Planck constant ε→ 0 and the parameter ν is kept fixed. To prove the incompressible limit of (4.1) we have to define the hydrodynamical variables; Schrödinger part charge ρSε, relativistic part charge ρKε, Schrödinger part momentum (current) JSε, relativistic part momentum JKε and energy eεas follows:
ρSε= ψε 2, ρεK= i 2ν 2ε1+αψε∂ tψε− ψε∂tψε , JSε=JS,ε1, JS,ε2, . . . , JS,nε = i 2ε 1−αψε∇ψε− ψε∇ψε, JKε =JK,ε 1, JK,ε 2, . . . , JK,nε =ν 2ε2 2 ∂tψε∇ψε+ ∂tψε∇ψε , eε=1 2ν 2ε2 ∂ tψε 2 +1 2ε 2−2α ∇ψε 2 + 1 γ ε2α ψ ε γ − 12 .
The local conservation laws associated with the rescaled modulated nonlinear Klein–Gordon equation (4.1) are the charge, momentum and energy given respectively by:
(A) Conservation of charge
∂ ∂t ρSε− ρεK+ ∇ · JSε= 0, (4.3) (B) Conservation of momentum ∂ ∂t JSε− JKε+1 4ε 2−2α∇ · 2∇ψε⊗ ∇ψε+ ∇ψε⊗ ∇ψε− ∇2 ψε 2 +1 4ν 2ε2∇∂ t ψε∂tψε+ ψε∂tψε + 1 γ ε2α∇ (γ − 1) ψε 2γ − (γ − 2) ψε γ= 0, (4.4)
(C) Conservation of energy ∂ ∂te ε− ∇ · 1 2ε 2−2α∇ψε∂ tψε+ ∇ψε∂tψε = 0. (4.5)
We now state the main theorem of this section.
Theorem 4.1. Let α > 0, γ 2 and ψεbe the solution of the time scale modulated nonlinear Klein–Gordon equation (4.1) with initial condition (ψ0ε, ψ1ε)∈ Hs+1(Tn)⊕ Hs(Tn), s >n2+ 1, satisfying (4.2) and (4.8). Then there exists T∗>0 such that ρSε− 1(·, t) Lγ(Tn)→ 0, ρεK(·, t) L 2γ γ+1(Tn)→ 0, JSε− ρSεu(·, t) L 2γ γ+1(Tn)→ 0, JKε(·, t)L1(Tn)→ 0,
for t∈ [0, T∗) as ε↓ 0, where u ∈ C([0, T∗); Hs(Tn)) is the unique local smooth solution of the incompressible Euler equations
∂
tu+ (u · ∇)u + ∇π = 0, ∇ · u = 0, u(x,0)= u0(x), ∇ · u0= 0.
(4.6)
Similar to the previous section, we define the modulated energy
Hε(t)=1 2 Tn ε1−α∇ − iuψε 2dx+ν 2ε2 2 Tn ∂tψε 2dx 1 γ ε2α Tn ρSε γ 2 − 12dx (4.7)
which satisfies the well-prepared initial condition
Hε(0)=1 2 Tn ε1−α∇ − iu0 ψ0ε 2dx+ν 2ε2 2 Tn ψ1ε 2dx+ 1 γ ε2α Tn ψ0ε γ− 12dx= Oεβ, (4.8) for some β > 0. The modulated energy can be further rewritten in terms of the hydrodynamic variables as
Hε(t)= Tn eεdx− Tn u· JSεdx+1 2 Tn ρSε|u|2dx. (4.9)
Lemma 4.2. Under the hypothesis of Theorem4.1and let λ= min{β, δ}, where δ = 2α/γ , we have Hε(t) Oελ, t∈ [0, T ].
Proof. Differentiating the modulated energy (4.9) with respect to t and using conservation of energy (4.5), we obtain
d dtH ε(t)= −d dt Tn u· JSεdx+ d dt Tn 1 2ρ ε S|u|2dx≡ I1+ I2.
By conservation of momentum (4.4), integration by part and using the fact that u is divergence free, we obtain
I1= − Tn ∂tu· JSε− JKεdx− d dt Tn u· JKεdx−ε 2−2α 4 Tn 2∇ψε⊗ ∇ψε+ ∇ψε⊗ ∇ψε: ∇u dx.
Next employing conservation of charge (4.3) and integration by part, we have
I2= Tn ρSεu· ∂tu dx+ Tn 1 2∇|u| 2· Jε Sdx+ d dt Tn 1 2ρ ε K|u|2dx− Tn ρKεu· ∂tu dx.
Gε(t)= −1 2 Tn ρKε|u|2dx+ Tn u· JKε dx,
then using crucially the incompressible Euler system (4.6), we have
d dt Hε(t)+ Gε(t)= −ε 2−2α 2 Tn ∇ψε⊗ ∇ψε+ ∇ψε⊗ ∇ψε: ∇u dx + Tn JSε− ρSεu· (u · ∇u) dx + Tn 1 2J ε S· ∇|u| 2dx + Tn JSε− ρSεu· ∇π dx − Tn ρKεu· ∂tu dx+ Tn ∂tu· JKεdx. (4.10)
To deal with the first integral of the right-hand side of (4.10), we need the following equality
−ε2−2α 2 Tn ∇ψε⊗ ∇ψε+ ∇ψε⊗ ∇ψε: ∇u dx = −1 2 Tn ε1−α∇ − iuψε⊗ε1−α∇ − iuψε +ε1−α∇ − iuψε⊗ε1−α∇ − iuψε: ∇u dx + Tn 1 2|u| 2∇ · Jε Sdx+ Tn (u· ∇)u·ρεSu− JSεdx. (4.11) This equality similar to (3.18) as discussed in previous section. Combining (4.10) and (4.11), we have the equality
d dt Hε(t)+ Gε(t)= −1 2 Tn ε1−α∇ − iuψε⊗ε1−α∇ − iuψε +ε1−α∇ − iuψε⊗ε1−α∇ − iuψε: ∇u dx + Tn JSε− ρεSu· ∇πdx − Tn ρKεu· ∂tu dx+ Tn ∂tu· JKεdx. (4.12)
Now we will estimate the second, third and fourth integral of right side of (4.12) separately. By (4.2) and (4.5), we have for t∈ [0, T∗) ε1−α∇ψεL2(Tn)=ε∂tψεL2(Tn)= O(1) (4.13) and ρSε γ 2 − 1 L2(Tn)= O εα. (4.14)
We deduce from the inequality
ρSε− 1 γ 2 ρε S γ 2 − 1 , and (4.14) that ρSε− 1Lγ(Tn)= O ε2αγ. (4.15)
Tn ρεS(u· ∇π) dx = Tn ρSε− 1(u· ∇π) + u · ∇πdx = Tn ρSε− 1(u· ∇π) dx ε2αγ u · ∇π L γ γ−1(Tn).
To go further, we need the relation Tn JSε· ∇πdx = Tn π ∂t ρSε− 1− π∂tρεKdx = d dt Tn πρSε− 1− ρKεπ dx− Tn ∂tπ ρSε− 1− ρKε∂tπ dx. (4.16)
The last integral of (4.16) can be estimated by Hölder inequality Tn ∂tπ ρSε− 1− ρKεdx ε2αγ∂ tπ L γ γ−1(Tn)+ ε α∂ tπL∞(Tn),
and the estimates of the third and fourth integrals of the right-hand side of (4.12) are given respectively by Tn ρKεu· ∂tu dx εαu · ∂tuL∞(Tn), and Tn ∂tu· JKεdx εα∂tuL∞(Tn).
To obtain the incompressible limit we have to introduce one more correction term of the modulated energy defined by
Wε(t)=
Tn
ρKε −ρSε− 1π dx.
The correction term Wε(t)can be served as the acoustic part (density fluctuation) of the modulated energy Hε(t). It is designed to control the propagation of the acoustic wave. Hence for t∈ [0, T∗)we have
d dt Hε(t)+ Gε(t)+ Wε(t) ∇uL∞(Tn)Hε(t) + εδu · ∇π L γ γ−1(Tn)+ ∂tπL γ γ−1(Tn)+ ∂tπL∞(T n) + u · ∂tuL∞(Tn)+ ∂tuL∞(Tn) (4.17)
where δ= 2α/γ . Integrating (4.17) yields
Hε(t) Hε(0)+ Gε(0)+ Wε(0)− Gε(t)− Wε(t)+ C1
t
0
Hε(τ ) dτ+ C2εδt.
One can show that Gε(0)+ Wε(0)− Gε(t)− Wε(t)= O(εδ), and hence
Hε(t) C1
t
0
Hε(τ ) dτ+ Hε(0)+ C2εδt+ C3εδ.
Applying the Gronwall inequality and the decay rate of Hε(0) we derive the inequality
Hε(t)C4εβ+ C2εδt+ C3εδ
1+ C1t eC1t
.
It is easy to rewrite the modulated energy (4.7) as Hε(t)=ε 2−2α 2 Tn ∇ρSε 2dx+1 2 Tn 1 ρSε JSε− ρSεu 2 dx +ν2ε2 2 Tn ∂tψε 2dx+ Tn 1 γ ε2α ρSε γ 2 − 12dx, (4.18)
then from Lemma4.2and (4.18) we have Tn 1 ρSε JSε− ρSεu 2 dx→ 0 (4.19)
as ε→ 0. We deduce from (4.19) and Hölder inequality that JSε− ρSεu L 2γ γ+1(Tn) ρSεL2γ(Tn) 1 ρSε JSε− ρSεu L2(Tn)
which converges to zero as ε→ 0. Finally, combing (4.13) and (4.15), we have ρKε(·, t) L 2γ γ+1(Tn) ε αε∂ tψεL2(Tn)ψ ε L2γ(Tn)→ 0 and JKε(·, t) L1(Tn) εαε∂tψεL2(Tn)ε 1−α∇ψε L2(Tn)→ 0
as ε→ 0. This completes the proof of Theorem4.1. 2 When α > 1−λ2, we deduce from (4.18) that
Tn ∇ρSε 2dx=1 2 Tn ∇ρSε ρSε 2dx→ 0 as ε→ 0, and ∇ρεS− 1 L 2γ γ+1(Tn) ∇ρ Sε ρSε L2(Tn) ρSεL2γ(Tn), (4.20)
by Hölder inequality. Thus, ρSε→ 1 strongly in W1,
2γ
γ+1(Tn). Furthermore, by Sobolev inequality we can show that ρSε→ 1 strongly in L
2nγ
n(γ+1)−2γ(Tn)for n 2. In particular n = 2, iterating the estimate (4.20) by the so-called “boot-strap process”, we have ρSε → 1 in Lp(T2)for any 1 p < ∞, and hence we have the following improvement of Theorem4.1.
Theorem 4.3. Assume the same hypothesis of Theorem4.1and Lemma4.2. Let α > 1−λ2and n= 2 then there exists T∗>0 such that for any η > 0,
ρSε− 1(·, t) L 1 η(T2)→ 0, ρ ε K(·, t)L2−η(T2)→ 0, JSε− ρεSu(·, t)L2−η(T2)→ 0, J ε K(·, t)L1(T2)→ 0,
for t∈ [0, T∗) as ε→ 0, where u is the unique local smooth solution of the incompressible Euler equations (4.6).
If we replace the potential energy (|ψε|γ− 1)|ψε|γ−2by (|ψε|2(γ−1)− 1) then all the previous analysis still works
for this model with small modification. Indeed, we investigate the time-scaled modulated nonlinear Klein–Gordon equation
iε1+α∂tψε− ε2+2αν2 2 ∂ 2 tψε+ ε2 2ψ ε− ψε 2(γ−1) − 1ψε= 0, (4.21) supplemented with initial conditions
ψε(x,0)= ψ0ε(x), ∂tψε(x,0)= ψ1ε(x), x∈ Tn.
It follows from the energy conservation law that we may assume the initial condition satisfies the uniform bound Tn 1 2ν 2ε2 ψε 1 2 +1 2ε 2−2α ∇ψε 0 2 + 1 γ ε2α ψ ε 0 2γ + (γ − 1) − γ ψ0ε 2dx < C. (4.22) In this case the associated modulated energy is defined by
Hε(t)=1 2 Tn ε1−α∇ − iuψε 2dx+ν 2ε2 2 Tn ∂tψε 2 dx + 1 γ ε2α Tn ρSεγ+ (γ − 1) − γρSεdx
and is well-prepared, i.e.,
Hε(0)=1 2 Tn ε1−α∇ − iu0 ψ0ε 2dx+ν 2ε2 2 Tn ψ1ε 2dx + 1 γ ε2α Tn ψ0ε 2γ + (γ − 1) − γ ψ0ε 2dx= Oεβ, (4.23) for some β > 0. Then employing the same argument as previous discussion, we can show that
Hε(t) Oελ, t∈ [0, T ],
where δ= 2α/γ and λ = min{β, δ}.
Theorem 4.4. Let γ > 1 for even spatial dimension n, γ n−1n for odd spacial dimension n. Let α > 0 and ψε be the solution of the time scale modulated nonlinear Klein–Gordon equation (4.21) with initial condition (ψ0ε, ψ1ε)∈ Hs+1(Tn)⊕ Hs(Tn), s >n
2+ 1, satisfying (4.22) and (4.23). Then there exists T∗>0 such that
ρSε− 1(·, t)Lγ(Tn)→ 0, ρ ε K(·, t) L 2γ γ+1(Tn)→ 0, JSε− ρSεu(·, t) L 2γ γ+1(Tn)→ 0, JKε(·, t)L1(Tn)→ 0,
for t∈ [0, T∗) as ε↓ 0, where u ∈ C([0, T∗); Hs(Tn)) is the unique local smooth solution of the incompressible Euler equations (4.6). In particular, when n= 2 and α > 1 −λ2, the convergent results can be improved as follows:
ρSε− 1(·, t) L1η(T2)→ 0, ρ ε K(·, t)L2−η(T2)→ 0, JSε− ρSεu(·, t)L2−η(T2)→ 0, J ε K(·, t)L1(T2)→ 0, for any η > 0 and t∈ [0, T∗) as ε→ 0.
Acknowledgements
C.K. Lin is supported by National Science Council of Taiwan under the grant NSC98-2115-M-009-004-MY3. K.C. Wu is supported by the Tsz-Tza Foundation in Institute of Mathematics, Academia Sinica, Taipei, Taiwan. K.C. Wu would like to thank Dr. Clément Mouhot for his kind invitation to visit Cambridge during 2011–2012 academic year.
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