The expansion for ρ~ and u~ω takes the form
ρ~= 1 + ~ρ(1)+ ~2ρ(2)+ ~3ρ(3)+ · · · ,
u~ω = ~u(1)+ ~2u(2)+ ~3u(3)+ · · · .
(6.1)
Substituting (6.1) into equations (2.12) and (2.14) and considering the O(~) terms, we have
∂tρ(1)+ ∇ · u(1) = 0, (6.2)
∂tu(1)+ g
m∇ρ(1)+ 2ω(u(1))⊥= ω2x − 1
m∇V. (6.3)
We may abbreviate this system by using the matrix
A =
0 ∇·
g
m∇ 2ωJ
, (6.4)
where
J =
0 −1 1 0
. (6.5)
Suppose that X = ρ(1), u(1)T
and X solves the initial problem for the system . Then the system may be written as
∂tX(t, x) + AX(t, x) = G(x), X(0, x) = X0, (6.6) where
G(x) =
0 ω2x − 1
m∇V
. (6.7)
Taking the Fourier transform of both the system and the initial condition with respect to the space variables, we reduce the problem to an ordinary differential equation in the time variable
∂tX(t, ξ) + bb A bX(t, ξ) = bG(ξ), X(0, ξ) = cb X0, (6.8)
49
where
The matrix bA has three distinct eigenvalues 0, ±ir g
m|ξ|2+ 16ω2π2δ(ξ). The space of eigenfunctions associated to 0 coincides with the null-space of bA
Ker( bA) =
Taking the inverse Fourier transform in the ξ variables leads to
X(t, x) = 1
An alternative approach to the semiclassical limit is discussed here. Let us consider
|ψ~|2 − 1
~
which is served as the density fluctuation of the sound wave. In other words, (ρ~, µ~) is near the constant state (1, 0). Multiplying (2.19) by ψ~∗
and (2.20) by ψ~ leads to the conservation law
∂t |ψ~|2− 1
6.1. DISPERSION LIMIT 51
To know the condition of V (x), equation (1.1) can be recast as i~∂tψ~= −~2 wave-function ψ~ satisfies the energy inequality
E(t) = E(ψ~) < C. (6.18)
We substitute ∇ω for the operator ∇ − imωx⊥
~
and refer to Thierry Cazenave [19] for defining a new space and having its properties.
Definition 6.1. The space Hω1 is defined as Hω1(R2) = equipped with the norm
kϕk2H1
ω = k∇ωϕk2L2 + kϕk2L2.
Lemma 6.2 (Lemma 9.1.2, Thierry Cazenave [19]). The following properties hold : (i) Hω1 ,→ L2.
(ii) L2 ,→ (Hω1)∗. (iii) k|ϕ|kH1 ≤ kϕkHω1.
The weak formulation of (6.16) is given by i ψ~(t2, ·) − ψ~(t1, ·), φ = ~
2m Z t2
t1
∇ωψ~, ∇ωφ dt
+ Z t2
t1
g |ψ~|2− 1
~
ψ~, φ
dt +
Z t2
t1
1
~
g + m 1
mV −1 2ω2|x|2
ψ~, φ
dt (6.19) for all φ ∈ Cc∞(R2) ∩ Hω1(R2).
Let us now search for the convergence of ψ~. We use Lemma 6.2 to assist us in proceeding with our work.
Lemma 6.3. Let T > 0. For all 0 < ~ 1, the sequence ψ~
~ is a relatively compact set in C([0, T ]; L2(R2)); that is, there exists ψ ∈ C([0, T ]; L2(R2)) such that
ψ~ → ψ strongly in C([0, T ]; L2(R2)).
Proof. Assume that the initial data ψ0~ satisfies |ψ0~| = 1 almost everywhere and ψ0~ → ψ0 strongly in Hω1(R2) as ~ → 0; hence, |ψ0| = 1 almost everywhere. We deduce from the energy inequality (6.18) that
∇ωψ~
~ is bounded in L∞(R+; L2(R2)), (6.20)
|ψ~|2− 1
~
~
is bounded in L∞(R+; L2(R2)); (6.21) therefore,
ψ~
~ is bounded in L∞(R+; Hω1(R2)). (6.22) It can be observed from (1.1) or (6.16) that
∂tψ~
~ is bounded in L∞(R+; Hω−1(R2)). (6.23) The classical compactness argument shows that there exists a function ψ satisfying
ψ ∈ L∞(R+; Hω1(R2)), ∂tψ ∈ L∞(R+; Hω−1(R2)), (6.24)
6.1. DISPERSION LIMIT 53
such that
ψ~* ψ weakly ∗ in L∞(R+; Hω1(R2)), (6.25)
∂tψ~* ∂tψ weakly ∗ in L∞(R+; Hω−1(R2)). (6.26) The properties (i) and (ii) in Lemma 6.2 tell that we can apply the Lions-Aubin Lemma to both (6.22) and (6.23) so that ψ~
~ is a relatively compact set in C([0, T ]; L2(R2)) for T > 0. It is worth pointing out, in passing, that
|ψ~|2 → 1 a.e. and strongly in L2(R2) (6.27) in the sense of distributions.
Proof. Let ∂τ = ∂t− ω x⊥· ∇; the conservation law (6.15) can be recast as
Integrating (6.28) with respect to τ and using the initial condition |ψ0~| = 1, we have
|ψ~|2− 1
along the characteristic dt
1 = dx
−ωx⊥. The main step in proving Lemma 6.4 is in treating the convergence of −
Z
∇·h ψ~∗
∇ψ~i
dτ in view of the weak topology. Using integration by parts and Fubini theorem yields
−
Moreover, ∇ψ~ converges weakly ∗ to ∇ψ in L∞(R+; L2(R2)). The weak convergence of
Based on the above findings, we present the results of passage to the limit.
Theorem 6.5. Assume that ψ~0 satisfies |ψ0~| = 1 almost everywhere and ψ0~ → ψ0 strongly in Hω1(R2) as ~ → 0. Let ψ~ be the weak solution of (6.16). Then ψ~ converges to the weak limit ψ satisfying the wave map equation
∂ttψ − g
m∆ψ = −ψ
|∂tψ|2− g
m|∇ψ|2
, |ψ| = 1 a.e..
Equivalently, ψ = eiθ with the phase function θ satisfies the wave equation
∂ttθ − g We can recognize from both Lemma 6.3 and Lemma 6.4 that
|ψ~|2 − 1
2ω2|x|2. From the energy inequality (6.18), there is further information to suggest that the quantity 1
~(g + mVω) is uniformly bounded. Hence,
6.1. DISPERSION LIMIT 55
In conclusion, the wave function ψ satisfies i∂tψ = −g
Z
∇ · 1
mIm (ψ∗∇ψ)
dτ
ψ + 1
~
g + m 1
mV − 1 2ω2|x|2
ψ (6.37) in the sense of distributions. A more clear expression of (6.37) could be showed. Differ-entiating (6.37) with respect to t, we have the wave map equation
∂ttψ − g
m∆ψ = −ψ
|∂tψ|2− g
m|∇ψ|2
, |ψ| = 1 a.e.. (6.38) Using the fact that |ψ| = 1, we write ψ = eiθ and insert it into (6.37) or (6.38) to show the linear wave equation
∂ttθ − g
m∆θ = 0. (6.39)
Remark 6.6. In (6.38), the terms inside the parentheses showing up in geometrical optics is the eikonal equation.
The dispersion limit suggests that we treat the right side of (6.2)–(6.3) as a perturba-tion, and we can study the linear wave equation instead of (6.2)–(6.3). Much remains to be done, but we intend to continue pursuing this interesting line of inquiry.
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