Here are some observations. First, 1p and 1q are linear with slope mn = −n2, for fixed n. The increase of p costs the decrease of q. Second, they all pass through (2, ∞) which also means that (2, ∞) is always admissible for all n.
We portray as in Figure 1.
Finally, we end Part I by going back to the Theorem 1.2. If the initial datum u0 is given in L2x, the the solution u is in Lpx with p > 2. We gain more integrability, that is the so-called smooth effect. This also reflects the dispersive nature of Schr¨odinger equation partially.
n p +2
q = n 2
w, : admissible
g: endpoint, non-adimissible
⊗ : endpoint, admissible
slope mn= −n
Figure 1: exponent pair
Dimension n Tabular 1: admissible pair
Part II
Semiclassical Limit of the Long Wave-Short Wave Interaction Equations
5 Introduction
In the Part II, we consider the existence and uniqueness of solutions of the initial value problem for the three coupled long wave-short wave interaction (LSI) equations
i~∂tψ~+~2
2 ∂xxψ~ = β(|ψ~|2+ w~)ψ~ (5.1) i~∂tφ~ +~2
2 ∂xxφ~ = β(|φ~|2+ w~)φ~ (5.2)
∂tw~ = β∂x |ψ~|2 + |φ~|2
(5.3) with initial values
ψ~(0, x) = ψ0~(x) (5.4)
φ~(0, x) = φ~0(x) (5.5)
w~(0, x) = w0~(x) (5.6)
where β > 0, w~ is real-valued and ψ~, φ~ are complex-valued. w~ char-acterizes the long wave and ψ~, φ~ represent the short waves.This system describes the resonance when the group velocity of the short waves and the phase velocity of the long wave coincide.
In section 2, we employ the Madelung transformation to LSI equations (5.1)–(5.3) and rewrite them as a perturbation of the Euler equations. The conservation laws are also derived.
In section 3, we apply the modified Madelung transformation to LSI equa-tions (5.1)–(5.3) and rewrite them as a perturbation of a quasilinear hyper-bolic system. For suitable assumptions on initial data, there exists local classical solution to the quasilinear hyperbolic system as well as the LSI equations. Furthermore, the solution that we establish is uniformly bounded in ~. This allows us to pass to the limit ~ → 0.
Notations. Hs = Ws,2 represents the Sobolev space with norm kf kHs = kf kWs,2 = P
α6sR |Dαf |2dx12
where Dαf , the αth derivatives of f, exists in the weak sense. C([0, T ]; X) consists of f : [0, T ] → X with kf kC([0,T ];X)= max06t6Tkf kX < ∞.
6 Hydrodynamical Structures and Conserva-tion Laws
In this section, we will derive some conservation laws of the LSI equations (5.1)–(5.3) first. For further references (6.1)–(6.26),(6.46)–(6.51), we ignore the superscript ~.
By Madelung transformation, we introduce the complex-valued wave func-tions
ψ = A1exp
iS1
~
, (6.1)
φ = A2exp
iS2
~
, (6.2)
where A1, A2, S1 and S2 are real-valued functions. A1, A2 are called the amplitudes, and S1, S2 the classical actions. Substituting (6.1) (resp.(6.2)) into (5.1) (resp.(5.2)), (A1, S1, A2, S2) obeys the following equations
∂tA1+ ∂xA1∂xS1+ 1
2A1∂xxS1 = 0, (6.3)
∂tS1+ 1
2(∂xS1)2+ βA21+ βw = ~2 2
∂xxA1
A1 , (6.4)
∂tA2+ ∂xA2∂xS2+ 1
2A2∂xxS2 = 0, (6.5)
∂tS2+ 1
2(∂xS2)2+ βA22+ βw = ~2 2
∂xxA2
A2 . (6.6)
Consider the new variables
ρ1 ≡ A21, u1 ≡ ∂xS1, (6.7) ρ2 ≡ A22, u2 ≡ ∂xS2, (6.8) we have the following two conservation laws
∂tρ1+ ∂x(ρ1u1) = 0, (6.9)
∂tu1+ ∂x 1
2u21+ βw
= ~2
2 ∂x∂xx√ ρ1
√ρ1 , (6.10)
∂tρ2+ ∂x(ρ2u2) = 0, (6.11)
∂tu2+ ∂x 1
2u22+ βw
= ~2
2 ∂x∂xx√ ρ2
√ρ2 . (6.12)
Equations (6.9)–(6.12) have the form of a perturbation of the Euler equations with w satisfying
∂tw = β∂x(ρ1+ ρ2), (6.13) which is equivalent to
w(t, x) = w0(x) + β Z t
0
∂x(ρ1+ ρ2)dτ. (6.14) Here (6.9) and (6.11) are conservation laws of mass. From (6.9), (6.10) (resp.(6.11), (6.12)), we can also derive the equation of the canonical mo-mentum ρ1u1 (resp. ρ2u2) which is not conservative. However, adding (6.15), (6.16) together and em-ploying (6.13), we have the conservation law of momentum as follows
∂t So far, we complete the conservation laws of mass and momentum. Next, we will seek for the conservation laws of energy. Multiply (6.9) by −12u21 and βw respectively, and (6.15) by u1, we have Summing (6.18), (6.19) and (6.20), we obtain
∂t 1
Also, from the symmetry point of view, we have Equations (6.21) and (6.22) are not in the conservative forms yet. Adding (6.21), (6.22) together and employing (6.13), we then have the conservation law of energy then we can rewrite (6.23) as
∂t(Eψ+ Eφ) The total energy of the LSI equations (5.1)–(5.3) is constituted by the classi-cal part, Eψ,1+ Eφ,1 the kinetic energy, Eψ,3+ Eψ,4+ Eφ,3+ Eφ,4 the potential energy, and the quantum part Eψ,2+ Eφ,2 which is of order O(~2).
The general problem of the semiclassical limit is to determine the limiting behavior of any function of the field ψ~, φ~ and w~ as ~ → 0. It is natural to conjecture that the dispersive term O(~2) which appears in (6.15) and (6.16) is negligible as ~ → 0 and the limiting density (ρ1, u1, ρ2, u2) satisfies the limiting Euler system with initial values
∂tρ1+ ∂x(ρ1u1) = 0, (6.27)
∂t(ρ1u1) + ∂x
ρ1u21+β 2ρ21
+ βρ1∂xw = 0, (6.28)
∂tρ2+ ∂x(ρ2u2) = 0, (6.29)
∂t(ρ2u2) + ∂x
ρ2u22+β 2ρ22
+ βρ2∂xw = 0, (6.30) with initial values
ρ1,0(x) = ρ1(0, x) = A21,0(x), (6.31) u1,0(x) = u1(0, x) = ∂xS1,0(x), (6.32) ρ2,0(x) = ρ2(0, x) = A22,0(x), (6.33) u2,0(x) = u2(0, x) = ∂xS2,0(x), (6.34) which w satisfies
∂tw = β∂x(ρ1+ ρ2), (6.35)
w(0, x) = w0(x). (6.36)
This argument is self-consistent only if the limiting Euler system (6.27)–
(6.36) remains classical. Furthermore, the limiting energy densities will be given by
Eψ = Eψ,1+ Eψ,3+ Eψ,4
= 1
2ρ1u21+ β
2ρ21+ βρ1w, (6.37) Eφ= Eφ,1+ Eφ,3 + Eφ,4
= 1
2ρ2u22+ β
2ρ22+ βρ2w, (6.38) and will satisfy
∂t(Eψ + Eφ) + ∂x
(Eψ+ Eψ,3)u1+ (Eφ+ Eφ,3)u2− β2
2 (ρ1+ ρ2)2
= 0. (6.39)
Moreover we introduce the modified Madelung transformation as follows ψ = A1exp
iS1
~
, (6.40)
A1 =√
ρ1exp(iθ1), u1 = ∂xS1, (6.41) φ = A2exp
iS2
~
, (6.42)
A2 =√
ρ2exp(iθ2), u2 = ∂xS2, (6.43) which A1and A2 are complex-valued. Plugging (6.40)–(6.43) into (5.1),(5.2), (ρ1, θ1, u1, ρ2, θ2, u2) satisfies
∂tρ1+ ∂x(ρ1u1 + ~ρ1∂xθ1) = 0, (6.44)
∂tθ1+ u1∂xθ1+~
2(∂xθ1)2 = ~ 2
∂xx
√ρ1
√ρ1 , (6.45)
∂tu1+ u1∂xu1+ β∂x(ρ1+ w) = 0, (6.46)
∂tρ2+ ∂x(ρ2u2 + ~ρ2∂xθ2) = 0, (6.47)
∂tθ2+ u2∂xθ2+~
2(∂xθ2)2 = ~ 2
∂xx√ ρ2
√ρ2 , (6.48)
∂tu2+ u2∂xu2+ β∂x(ρ2+ w) = 0, (6.49) which w is given by
∂tw = β∂x(ρ1+ ρ2), (6.50) or is equivalent to
w(t, x) = w0(x) + β Z t
0
∂x(ρ1+ ρ2)dτ. (6.51) It is remarkable that the quantum effect in this system is of order O(~) different from the perturbation of the Euler equations (6.9)–(6.14) of order O(~2).
7 Semiclassical Limit
In this section, we will derive the existence and uniqueness of local clas-sical solutions for LSI equations (5.1)–(5.3) with initial values (5.4)–(5.6).
Then we will study their semiclassical limit by utilizing the hydrodynamical structures presented in the previous section.
First, we employ the modified Madelung transformation [4] to rewrite (5.1)–(5.3) into a perturbation of a quasilinear hyperbolic system [5, 14]. Let
ψ~ = A~1exp
iS1~
~
, (7.1)
A~1 = a~1+ ib~1, u~1 = ∂xS1~, (7.2) φ~ = A~2exp
iS2~
~
, (7.3)
A~2 = a~2+ ib~2, u~2 = ∂xS2~, (7.4) then substituting (7.1) (resp.(7.3)) into (5.1) (resp.(5.2)), we have
∂tA~1+ ∂xS1~∂xA~1+1
2A~1∂xxS1~ = i~
2∂xxA~1, (7.5)
∂tS1~+1
2(∂xS1~)2+ β|A~1|2+ βw~ = 0, (7.6)
∂tA~2+ ∂xS2~∂xA~2+1
2A~2∂xxS2~ = i~
2∂xxA~2, (7.7)
∂tS2~+1
2(∂xS2~)2+ β|A~2|2+ βw~ = 0. (7.8) Differentiating (7.6) (resp.(7.8)) w.r.t. x and replacing (A~1, S1~) (resp.(A~2, S2~)) by (7.2) (resp.(7.4)), we have
∂ta~1+ u~1∂xa~1+1
2a~1∂xu~1 = −~
2∂xxb~1, (7.9)
∂tb~1+ u~1∂xb~1+1
2b~1∂xu~1 = ~
2∂xxa~1, (7.10)
∂tu~1+ u~1∂xu~1+ 2βa~1∂xa~1+ 2βb~1∂xb~1+ β∂xw~ = 0, (7.11)
∂ta~2+ u~2∂xa~2+1
2a~2∂xu~2 = −~
2∂xxb~2, (7.12)
∂tb~2+ u~2∂xb~2+1
2b~2∂xu~2 = ~
2∂xxa~2, (7.13)
∂tu~2+ u~2∂xu~2+ 2βa~2∂xa~2+ 2βb~2∂xb~2+ β∂xw~ = 0, (7.14) with initial values
a~1(0, x) = a~1,0(x), b~1(0, x) = b~1,0(x), u~1(0, x) = u~1,0x = ∂xS1~(0, x), (7.15) a~2(0, x) = a~2,0(x), b~2(0, x) = b~2,0(x), u~2(0, x) = u~2,0x = ∂xS2~(0, x). (7.16) According to (5.3), w~ is given explicitly by
w~(x, t) = w0~(x) + β Z t
0
∂x(a~1)2+ (b~1)2+ (a~2)2+ (b~2)2 dτ. (7.17)
Hence, (7.9)–(7.17) form a quasilinear hyperbolic system which is equivalent to the LSI equations (5.1)–(5.3) with initial values (5.4)–(5.6). The system can be rewritten in the vector form
∂tU~+ A(U~)∂xU~+ G(w~) = ~
Now, we introduce S,
S =
which is symmetry and positive define for β > 0. Multiplying (7.18) by S, we have the quasilinear symmetry hyperbolic system
S∂tU~+ eA(U~)∂xU~+ eG(w~) = ~
2LUe ~, (7.22)
where eG(w~) = SG(w~), eL = SL and eA~ = SA~ is symmetry. The local existence in time for the initial values (7.19) of the quasilinear symmetry hyperbolic system (7.22) follows the iteration scheme as below. For con-venience, we ignore the superscript ~ in (7.23)–(7.30) and some calculating process. Define U0(t, x) = U0(x), w0(t, x) = w0(x) where U0(x), w0(x) are the given initial values and define Uk+1(t, x), wk+1(t, x) inductively as the solution of the linear initial value problem
S∂tUk+1+ eA(Uk)∂xUk+1+ eG(wk+1) = ~
2LUe k+1, (7.23) wk+1(t, x) = w0(x) + β
Z t 0
∂x(ak1)2+ (bk1)2+ (ak2)2+ (bk2)2 dτ, (7.24) Uk+1(0, x) = U0k+1(x) = U0(x), (7.25) for k = 0, 1, 2, . . .. Assume U0 ∈ Hs and w0 ∈ Hs+1 where s is to be determined. Let U be a solution of (7.18) and belongs to C1([0, T ]; C2(Ω)) which is of compact support for each t. The canonical energy associated with the quasilinear symmetry hyperbolic system (7.18) is defined by
(SU, U ) = Z
UtSU dx. (7.26)
The classical energy estimate follows immediately by the symmetry of S, eA and antisymmetry of eL. Indeed,
( eLU, U ) = Z
UtLU dx =e Z
(UtLU )e tdx
= Z
Ut Let
U dx = − Z
UtLU dxe
= −( eLU, U )
and this implies ( eLU, U ) = 0. So, if eA together with its derivatives of any de-sire order are continuous and bounded uniformly in [0, T ] × Ω, by integration by parts, then
d
dt(SU, U ) = (S∂tU, U ) + (SU, ∂tU )
= 2(S∂tU, U )
= ~( eLU, U ) − 2( eA∂xU, U ) − 2( eG, U )
= 0 + ((∂xA)U, U ) − 2( ee G, U ) 6 c1(t)(SU, U ).
By applying Gronwall inequality, we deduce the energy inequality
(SU, U ) ≤ (SU0, U0)eR0tc1(τ )dτ, (7.27) and hence
06t6TmaxkU~(t)kL2 6 c2kU0~kL2. (7.28) The higher energy estimate can be obtained in the similar way. We differen-tiate (7.18) w.r.t. x, then multiply on both sides by S, we have
S∂x∂tU + eA∂x2U + ∂xA∂e xU + ∂xG =e ~
2L∂e xU, (7.29)
∂xU (0, x) = ∂xU0(x). (7.30) With similar calculation,
d
dt(S∂xU, ∂xU ) = (S∂t∂xU, ∂xU ) + (S∂xU, ∂t∂xU )
= 2(S∂t∂xU, ∂xU )
= ~( eL∂xU, ∂xU ) − 2(∂xA∂e xU, ∂xU ) − 2( eA∂x∂xU, ∂xU ) − 2(∂xG, ∂e xU )
= 0 − 2(∂xA∂e xU, ∂xU ) + (∂xA∂e xU, ∂xU ) − 2(∂xG, ∂e xU )
= −(∂xA∂e xU, ∂xU ) − 2(∂xG, ∂e xU ) 6 c3(t)(S∂xU, ∂xU ).
By Gronwall inequality again, we have max
06t6Tk∂xU~(t)kL2 6 c4k∂xU0~kL2. (7.31) Moreover, the estimate of the time derivative ∂tU is directly derived from the equation (7.18) itself.
max
06t6Tk∂tU~kHs−2 = max
06t6T
~
2LU~− A∂xU~− G(w~) Hs−2
6 c5 max
06t6TkU~kHs+ c6 max
06t6TkG(w~)kHs. (7.32)
∂tU~ only belongs to Hs−2 because of the twice derivative appearing in L.
So far, we have shown that for fixed ~,
U~,k ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2) (7.33)
for all k. Hence U~,k
k∈N is uniformly bounded in k. Moreover, by mean value theorem,
06t6Tmax kU~,k(t + h) − U~,k(t)kHs−2
= max
06t6Tk∂tU~,k(ξ) · hkHs−2, ξ ∈ (t, t + h) ⊂ [0, T ]
= h · max
06t6Tk∂tU~,k(t)kHs−2
tends to 0 as h goes to 0, for all k. Thus the sequence U~,k
k∈N is equicon-tinuous. Following the Arzela-Ascoli theorem, there exists
U~ ∈ L∞([0, T ]; Hs) ∩ Lip([0, T ]; Hs−2), such that as k → ∞
U~,k → U~ in C([0, T ]; Hs−2).
Thus, by interpolation inequality, max
06t6TkU~,k1 − U~,k2kHs−θ 6 c7 max
06t6TkU~,k1− U~,k2kHs−2 max
06t6TkU~,k1 − U~,k2kHs 6 c8 max
06t6TkU~,k1− U~,k2kHs−2 for 0 < θ < 2, we have the convergence
U~,k → U~ in C([0, T ]; Hs−θ).
In addition, we discuss the convergence A(Uk)∂xUk+1 to A(U )∂xU . Indeed, it can be done with the fact that
∂xU~,k → ∂xU~, as k → ∞, since
kA(Uk)∂xUk+1− A(U )∂xU kHs−1
= kA(Uk)∂xUk+1− A(Uk)∂xU + A(Uk)∂xU − A(U )∂xU kHs−1
6 kA(Uk)kHs−1k∂xUk+1− ∂xU kHs−1+ kA(Uk) − A(U )kHs−1k∂xU kHs−1 Consequently, we have
U~ ∈ C([0, T ]; Hs).
Then the original equation (7.18) implies U~ ∈ C1([0, T ]; Hs−2); hence we have the solution
U~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2). (7.34) Also, from the relation between U~ and w~ in (7.17), we have
w~ ∈ C([0, T ]; Hs−1) ∩ C1([0, T ]; Hs−3). (7.35) Furthermore, by Sobolev type inequality, if s > 12 + 4 then
Hs−2 ,→ C2.
This can be easily checked by the dimensions of two function spaces Hs−2 and C2, 12 − (s − 2) < ∞1 − 2. Then we have
U~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2) ,→ C1([0, T ]; C2), (7.36) w~ ∈ C([0, T ]; Hs−1) ∩ C1([0, T ]; Hs−3) ,→ C1([0, T ]; C1), (7.37) and hence the solution (U~, w~) of the quasilinear hyperbolic system (7.18)–
(7.20) is classical.
The uniqueness of the classical solution of (7.18) follows from the energy estimate for the difference of two given solutions. Make U and V two so-lutions with the same initial data. Define U∗ = U − V , and we have the equation
S∂tU∗+ eA(U )∂xU∗+ [ eA(U ) − eA(V )]∂xV = ~
2LUe ∗. (7.38) With previously similar arguments and U , V are of compact support, we have
d
dt(SU∗, U∗) = (S∂tU∗, U∗) + (SU∗, ∂tU∗)
= 2(S∂tU∗, U∗)
= ~( eLU∗, U∗) − 2( eA(U )∂xU∗, U∗) − 2([ eA(U ) − eA(V )]∂xV, U∗)
= 0 + ((∂xA(U ))Ue ∗, U∗) − 2([ eA(U ) − eA(V )]∂xV, U∗) 6 c9(t)(SU∗, U∗).
By Gronwall inequality, we have
(SU∗, U∗) 6 (SU0∗, U0∗)eR0tc9(τ )dτ = 0. (7.39) This implies U∗ = 0 and hence U = V . Therefore the classical solution (U~, w~) is unique.
To summarize all this, we have the following result:
Theorem 7.1. Let s > 12 + 4. Assume the initial values
U0~ = (a~1,0, b~1,0, u~1,0, a~2,0, b~2,0, u~2,0) ∈ Hs× Hs× Hs× Hs× Hs× Hs, (7.40)
w0~ ∈ Hs+1, (7.41)
then there exists T > 0 such that the quasilinear hyperbolic system (7.18) with initial values (7.19),(7.20) has a unique classical solution
U~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2) ,→ C1([0, T ]; C2), (7.42) w~ ∈ C([0, T ]; Hs−1) ∩ C1([0, T ]; Hs−3) ,→ C1([0, T ]; C1), (7.43) for all t ∈ [0, T ].
As an immediate consequence, we have the similar result for the LSI equations (5.1)–(5.6).
Theorem 7.2. Let s > 12 + 4. Assume the initial values
(A~1,0, S1,0~ , A~2,0, S2,0~ , w0~) ∈ Hs× Hs+1× Hs× Hs+1× Hs+1, (7.44) then there exists T > 0 such that the LSI equations (5.1)–(5.3) with initial values (5.4)–(5.6) have a unique classical solution (ψ~, φ~, w~) of the form
ψ~ = A~1exp
iS1~
~
, φ~ = A~2exp
iS2~
~
, w~(t, x) = w~0(x) + β
Z t 0
∂x(A~1)2+ (A~2)2 dτ,
which A~1, S1~, A~2, S2~ (resp. w~) are bounded in L∞([0, T ]; Hs) (resp. L∞([0, T ];
Hs−1)) uniformly in ~.
Proof. Since
ψ~ = A~1exp
iS1~
~
and φ~ = A~2exp
iS2~
~
where A~1 = a~1+ ib~1, u~1 = ∂xS1~, A~2 = a~2 + ib~2 and u~2 = ∂xS2~, by Theorem 7.1, we have
A~1 ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2),
∂xS1~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2),
and hence S1~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2).
Similarly,
A~2 ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2),
∂xS2~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2), and hence S2~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2).
By Moser type calculus inequality, we conclude that
ψ~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2) ,→ C1([0, T ]; C2), φ~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2) ,→ C1([0, T ]; C2).
Moreover,
w~(t, x) = w~0(x) + β Z t
0
∂x(A~1)2+ (A~2)2 dτ
∈ C1([0, T ]; C1), and thus the theorem follows.
Because of the nature of the antisymmetry of eL, the term ~(LU, U) van-ishs in our estimates. The time interval [0, T ] and the boundary for U~ in Hs are independent of ~. These will allow us to pass to the limit ~ → 0 in (7.18).
Proposition 7.3. Let (ρ~1, θ~1, u~1, ρ~2, θ~2, u~2, w~) be in C1([0, T ]; C2) and be the solution of equations (6.44)–(6.51). For i = 1, 2, if ρ~i,0(x) > 0 then ρ~i(t, x) > 0, ∀t > 0. Furthermore, when the ~ varies, ρ~i will not be too small; that is, too closed to zero.
Proof. Since u~i, θ~i ∈ C1([0, T ]; C2), u~i + ~∂xθ~i ∈ C1([0, T ] × R). From (6.44), we have
∂tρ~i + ∂xρ~i(u~i + ~∂xθ~i) = 0, (7.45) or
∂tρ~i + (u~i + ~∂xθ~i)∂xρ~i = −ρ~i∂x(u~i + ~∂xθi~). (6.46) In addition, the ordinary differential equations
dx
dt = u~i + ~∂xθ~i, (7.47)
x(τ ) = ξ, (7.48)
has a unique solution x = Γ(t) which belongs to C1([0, T ] × R). Equation (7.46) implies
d
dtρ~i(t, Γ(t)) = −ρ~i(t, Γ(t))∂x(u~i + ~∂xθi~). (7.49) Integrating over [0, τ ], we have
ρ~i(τ, ξ) = ρ~i(0, Γ(0)) exp
− Z τ
0
∂x(u~i + ~∂xθi~)dt
. (7.50) Hence ρ~i(t, x) > 0 if ρ~i,0(x) > 0. Moreover, the integration in the r.h.s. of (7.50) will not tend to the infinity when the ~ varies, hence ρ~i will not be too closed to zero.
The limiting system of the quasilinear hyperbolic system (7.18) with ini-tial value (7.19) is also a quasilinear hyperbolic system as the following shows:
(formally letting ~ → 0)
Ut+ A(U )Ux+ G(w) = 0 (7.51)
U (0, x) = U0(x) (7.52)
w(0, x) = w0(x) (7.53)
where w is given by
∂tw = β∂x(a21+ b21+ a22+ b22), (7.54) or is equivalent to
w(t, x) = w0(x) + β Z t
0
∂x(a21+ b21+ a22+ b22)dτ. (7.55) This is equivalent to the limiting Euler system (6.27)–(6.36) as long as the solutions are smooth. Next, we will show the existence and uniqueness of the local smooth solution to the system (6.27)–(6.36).
Theorem 7.4. Let s > 12+ 4 and [0, T ] be the fixed time interval determined in Theorem 3.1. Given initial values U0~, U0 ∈ Hs, and U0~ converges to U0 in Hs as ~ → 0. Then, there exists
U ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2) ,→ C1([0, T ]; C2), w ∈ C([0, T ]; Hs−1) ∩ C1([0, T ]; Hs−3) ,→ C1([0, T ]; C1),
which is a classical solution to the IVP for the limiting quasilinear hyperbolic system (7.51)–(7.55), and so is to the IVP for the limiting Euler system (6.27)–(6.36).
Proof. SinceU~
~ is bounded uniformly in ~, by Arzela-Ascoli theorem and interpolation inequality, we have a function U such that, as ~ → 0
U~ → U in C([0, T ]; Hs−θ),
for 0 < θ < 2. Also, from the equation (7.18) itself, we have U~ → U in C1([0, T ]; Hs−2−θ),
for 0 < θ < 2. LU~ is uniformly bounded in Hs−2, so the perturbation term
~
2LU~ tends to 0 as ~ → 0. Hence the sequence converges to a solution of the limiting quasilinear hyperbolic system (7.51)–(7.55). The solution w is then given by (7.55) and belongs to C1([0, T ]; C1).
Theorem 7.5. Let (ρ1, u1, ρ2, u2, w) be a solution of the limiting Euler system (6.27)–(6.36) on [0, T ], which initial value (ρ1,0, u1,0, ρ2,0, u2,0, w0) belongs to Hs× Hs× Hs× Hs× Hs+1. Assume A~1,0 (resp. A~2,0, w~0) converges strongly to A1,0 (resp. A2,0, w0) in Hs (resp. Hs, Hs+1) as ~ → 0. Then, for ~ small enough, there exists a unique classical solution (ψ~, φ~, w~) to the IVP for the LSI equations (5.1)–(5.6).
Proof. Consider the difference of (7.18) and (7.51). Define eU~ = U~ − U , then we have
∂tUe~ + A( eU~+ U )∂xUe~ + [A( eU~+ U ) − A(U )]∂xU +G(w~) − G(w)
= ~
2L( eU~ + U ). (7.56)
We introduce S = S( eU~ + U ) which is symmetry, positive define and can symmetrize A( eU~ + U ). Multiplying (7.56) by S, we have
S∂tUe~+ SA( eU~+ U )∂xUe~ + S[A( eU~+ U ) − A(U )]∂xU + SG(w~) − G(w)
= ~
2SL( eU~+ U ). (7.57)
The energy associated with (7.56) is defined by
(S eU~, eU~) = Z
( eU~)tS eU~dx. (7.58)
We apply the energy estimate again.
d
dt(S eU~, eU~) = (S∂tUe~, eU~) + (S eU~, ∂tUe~)
= 2(S∂tUe~, eU~)
= ~(SL( eU~+ U ), eU~) − 2(SA( eU~+ U )∂xUe~, eU~)
− 2(S[A( eU~ + U ) − A(U )]∂xU, eU~) − 2(S[G(w~) − G(w)], eU~).
By the antisymmetry of L, we have
~(SL eU~, eU~) = 0.
The Cauchy-Schwarz inequality implies
~(SLU, eU~) 6 ~c10kLU kL2k eU~kL2 6 ~c11kU kH2k eU~kL2 6 c12k eU~k2L2 ;
− 2(SA( eU~+ U )∂xUe~, eU~) = (S(∂xA( eU~+ U )) eU~, eU~) 6 c13k eU~k2L2 ;
(S[A( eU~+ U ) − A(U )]∂xU, eU~) 6 c14k[A( eU~+ U ) − A(U )]∂xU kL2k eU~kL2 6 c15k∂xU kL2k eU~kL2 6 c16kU kH1k eU~kL2 6 c17k eU~k2L2 ;
(S[G(w~) − G(w)], eU~) 6 c18k eU~k2L2 . Hence we have the inequality
d
dt(S eU~, eU~) 6 c19(t)(S eU~, eU~).
By Gronwall inequality,
(S eU~, eU~) 6 (S eU0~, eU0~)eR0tc19(τ )dτ, (7.59) which the r.h.s. tends to 0 as ~ → 0 because of eU0~ = U0~ − U0 tends to 0.
Then the theorem follows.
We conclude that the behavior of the quasilinear hyperbolic system (7.18) resembles the limiting system (7.51). That is to say, the ~ appearing in the Euler equations (6.9)–(6.13) is negligible. Hence the quantum equations can be depicted by the classical hydrodynamics equations.
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