Well-Posedness and Adiabatic Limit for Quantum Zakharov System - - PowerPoint PPT Presentation

Well-Posedness and Adiabatic Limit for Quantum Zakharov System Yung-Fu Fang (joint work with Tsai-Jung Chen, Chi-Kun Lin, Jun-Ichi Segata, Hsi-Wei Shih, Kuan-Hsiang Wang, Tsung-fang Wu) Department of Mathematics National Cheng Kung University Tainan, 701 Taiwan Talk at NSYSU The 24th Annual Workshop on Differential Equations

• Feb. 22 ∼ 23, 2016

Abstract : For a Quantum Zakharov system, we study the LWP and GWP, adiabatic limit, and least energy solution. We obtain the adiabatic limit for (QZ) to a quantum modified

• NLS. We also prove the existence of homoclinic solutions with the

least energy. We show an ill-posedness problem. Future study for (QZ) : LWP in 2D and 3D, semi-classical limit and subsonic limit, ground state, soliton waves, asymptotic behavior, and eventually the scattering problem.

• 1. Introduction :

Zakharov System describes the propagation of Langmuir waves in an ionized plasma. ( rapid oscillations of the electron density)        iEt + ∂2

xE = nE,

x ∈ R, ntt − ∂2

xn = ∂2 x|E|2,

E(x, 0) = E0(x), n(x, 0) = n0(x), nt(x, 0) = n1(x). (Z) E = the rapidly oscillating electric field, n = the deviation of the ion density from its mean value. For the derivation of (Z) system, see [Z] and [OT].

Conservation of Mass:

• |E(x, t)|2dx = constant

Conservation of Hamiltonian:

• |∂xE(t)|2 + 1

2n(t)2 + n(t)|E(t)|2 + 1 2ν(t)2dx = constant where ∂tn = ∂xν and ∂tν = ∂x

• n + |E|2

.

Global Well-Posedness: (Z) system with initial data (E0, n0, n1) ∈ Hk ⊕ Hl ⊕ Hl−1(R).

Figure 1:

Taking quantum effects into account, we consider        iEt + ∂2

xE − ε2∂4 xE = nE,

x ∈ R; ntt − ∂2

xn + ε2∂4 xn = ∂2 x|E|2

E(x, 0) = E0(x), n(x, 0) = n0(x), nt(x, 0) = n1(x). (QZ) ε = ωi κBTe

Proton

∼ (6.63 × 10−27)(8 × 109) 2π(1.38 × 10−16)(105) ∼ 6.12 × 10−5 = Planck’s constant /2π, ωi = ion plasma frequency κB = Boltzmann constant, Te = electron fluid temperature

Conservation of Mass:

• |E(x, t)|2dx = constant

Conservation of Hamiltonian:

• |∂xE|2 + ε2|∂2

xE|2 + 1

2n2 + n|E|2 + 1 2ν2 + ε2 2 |∂xn|2dx = constant where ∂tn = ∂xν and ∂tν = ∂x

• n + |E|2 − ε2∂2

xn

• .

Local Well-Posedness: (QZ) with initial data (E0, n0, n1) ∈ Hk ⊕ Hℓ

ε ⊕ Hℓ−1 ε

Figure 2:

Local Well-Posedness: (Z) with initial data (E0, n0, n1) ∈ Hk ⊕ Hℓ

ε ⊕ Hℓ−1 ε

Figure 3:

ILL-Posedness Problem:

Figure 4:

• 2. Notations and Solution formulae

Denote Dε :=

• 1 − ε2∂2

x

and ξε := ξ

• 1 + ε2ξ2.

Decompose (QZ) we get    iEt + ∂2

xD2 εE = (n+ + n−)E,

x ∈ R ∂tn± ± ∂xDεn± = ∓1 2 1 Dε ∂x|E|2 + 1 2n1L, (QZ±) The integral formulae for solutions: E(t, x) = Uε(t)E0(x) − iUε ∗R

• (n+ + n−)E
• (t, x)

(2.1) n±(t, x) = Wε±(t)(n0, n1) ∓ Wε± ∗R

• D−1

ε ∂x|E|2

(t, x) (2.2)

Denote the Sobolev norms f2

Hℓ :=

• ξ2ℓ|

f(ξ)|2dξ, f2

Hℓ

ε :=

• ξε2ℓ|

f(ξ)|2dξ, (2.3) f2

Aℓ

ε :=

• |ξ|≤1

| f(ξ)|2dξ +

• 1≤|ξ|

|ξε|2ℓ| f(ξ)|2dξ. Denote the Sobolev norm for acoustic wave n(t)Wε :=

• n(t), ∂tn(t)
• Wε := n(t)Aℓ

ε + ∂tn(t)Aℓ−1 ε

Bourgain norm for Schr¨

• dinger part:

EXSε

k,b1 :=

ξε2kτ + ξ2

ε2b1|

E(τ, ξ)|2dτdξ 1

2

(2.5) Bourgain norm for Wave part: n±X

Wε± ℓ,b

:= ξε2ℓτ ± ξε2b| n(τ, ξ)|2dτdξ 1

2

(2.6) Y norm for Schr¨

• dinger part:

EY Sε

k :=

ξεkτ + ξ2

ε−1|

E(τ, ξ)|dτ 2 dξ 1

2

(2.7) Y norm for Wave part: n±Y

Wε± ℓ

:= ξεℓτ ± ξε−1| n±(τ, ξ)|dτ 2 dξ 1

2

(2.8)

• 3. Estimates for Iteration Argument

Lemma 1. (Homogeneous Estimates) Lemma 2. (Duhamel Estimates) Lemma 3. (Multilinear Estimates) Let 0 < ε ≤ 1. n±EXSε

k,−c1 C(ε)n±X Wε± ℓ,b EXSε k,b1.

(3.1) D−1

ε ∂x(E1 ¯

E2)X

Wε± ℓ,−c C(ε)E1XSε k,b1E2XSε k,b1.

(3.2) n±EY Sε

k C(ε)n±X Wε± ℓ,b EXSε k,b1.

(3.3) D−1

ε ∂x

• E1 ¯

E2

• Y

Wε± ℓ

C(ε)E1XSε

k,b1E2XSε k,b1.

(3.4)

• 4. Proof of Multilinear Estimates
• Proof. We only outline the proof for

D−1

ε ∂x(E1E2)X

Wε± ℓ,−c C2(ε)E1XSε k,b1E2XSε k,b1.

First we set ξ = ξ1 − ξ2 and decompose the ξ1-ξ2 plane into Ω1 ∪ Ω2 ∪ Ω3 ∪ Ω4, where Ω1 = {(ξ1, ξ2) : |ξ2| ≪ |ξ1| ∼ |ξ|}, Ω2 = {(ξ1, ξ2) : |ξ1| ≪ |ξ2| ∼ |ξ|}, Ω3 = {(ξ1, ξ2) : |ξ1| ∼ |ξ2| ∼ |ξ|}, Ω4 = {(ξ1, ξ2) : |ξ| ≪ |ξ1| ∼ |ξ2|}.

By duality argument, the estimate D−1

ε ∂x(E1E2)XWε+

ℓ,−c E1XSε k,b1E2XSε k,b1

∼

• D−1

ε ∂x(E1E2), g

• E1XSε

k,b1E2XSε k,b1gXWε+ −ℓ,c .

We set gXWε+

−ℓ,c = ξε−ℓτ + ξεc

gL2 ≡ vL2 EjXSε

k,b1 = ξεkτ + ξ2

εb1

EjL2 ≡ vjL2 We can rewrite

• D−1

ε ∂x(E1E2), g

• as
• iξ
• 1 + ε2ξ2

ξεℓ ξ1εkξ2εk

• v(τ, ξ)

σc

• v2(τ2, ξ2)

σ2b1

• v1(τ1, ξ1)

σ1b1 dτ2dξ2dτ1dξ1, τ = τ1 − τ2, ξ = ξ1 − ξ2, σ = τ + ξε, σ2 = τ2 + ξ2

2ε, σ1 = τ1 + ξ2 1ε.

• vL2v1L2v2L2.

• 5. Existence of a Least Energy Solution

Consider the stationary solution and static solution to (QZ) in 1D by setting E(x, t) = eiωtQ(x) and n(x, t) = n(x). Then (QZ) ⇒

• −ωQ + Q′′ − ε2Q(4) = nQ,

−n + ε2n′′ = Q2. Question 1. Does solution Qε,ω(x) and nε,ω(x) exist? Question 2. Does limε→0(Qε,ω, nε,ω) = (Q0,ω, −Q2

0,ω)?

Question 3. Numerical Results? Question 4. Stability of such solutions? Soliton?

Lemma 4. Let n = 1. nε,Q (x) = −

• R

1 2εe

−|x−y| ε Q2 (y) dy → −Q2(x) as ε → 0.

Plug nε,Q back into the system to get an ODE with a nonlocal term. −ωQ + Q′′ − ε2Q(4) = −Q

• R

1 2εe

−|x−y| ε Q2 (y) dy.

(Eε) For 2D and 3D, we also have the solution formula for −ωQ + ∆Q − ε2∆2Q = −Q(1 − ε2∆)−1Q2. (Eε) Theorem 1. (Fang & Wu, 2015) Let n = 1, 2, 3. For each 0 < ε ≤ 1, Problem (Eε) has a least energy homoclinic solution Qε.

Consider the Zakharov system in 1D,        iEt + ∂2

xE = nE,

x ∈ R, λ−2ntt − ∂2

xn = ∂2 x|E|2,

E(x, 0) = E0(x), n(x, 0) = n0(x), nt(x, 0) = n1(x). (Z) When wave speed λ tends to infinity, naturally acoustic wave passed and disappeared, only remained the Schr¨

• dinger part.

Review the ground state for (Z),

• −ωQ + Q′′ = nQ,

−n = Q2. Thus we get −ωQ + Q′′ + Q3 = 0. The solution can be derived via Newton’s method starting with Gaus- sian function. Hence the exact solution is Q(x) = √ 2βsech(βx), where β = √ω. Finally solitary solutions for (Z) can be given as Eλ,0(x, t) = ei(β2−α2)t eiαx 1 − 4λ−2α2√ 2βsech

• β(x − 2αt)
• .

nλ, 0(x, t) = −2β2sech2 β(x − 2αt)

• .

Consider a system that is close related to (QZ),    iEt + ∂2

xE − ε2∂4 xE = nE,

x ∈ R; n = −|E|2 + ε23 2|E|4 + 3(∂xE)2 ¯ E E + 2|∂xE|2 + E∂2

x ¯

E + 4 ¯ E∂2

xE

• .

It possesses solitary waves E,ε(x, t) = ei(ω−αc)t eiαx√ 2βsech

• β(x − ct)
• ,

where ω − αc = −ε2(α4 + β4 − 6α2β2) − α2 + β2 and c = ε2α(4α2 − 4β2) + 2α. Question: can we estimate Eλ,ε − E∞,ε? nλ,ε − n∞,ε?

• 6. Adiabatic Limit of (QZ)

Consider the quantum Zakharov system for d = 1, 2, 3,        i∂tE + ∆E − ε2∆2E = nE t ∈ R, x ∈ Rd, λ−2∂2

tn − ∆n + ε2∆2n = ∆|E|2

t ∈ R, x ∈ Rd, E(0) = E0, n(0) = n0, ∂tn(0) = n1 x ∈ Rd, (λQZ) with the solution (Eλ, nλ). Let us formally take λ → ∞ for the second equation in (λQZ). Under suitable conditions, we obtain the relation n∞ = −(1 − ε2∆)−1|E∞|2.

Substituting this relation into (λQZ), ⇒ E∞ satisfies the quan- tum modified NLS i∂tE − (−∆ + ε2∆2)E = −{(1 − ε2∆)−1|E|2}E. (4NLS) The difference Eλ − E∞ satisfies          i∂t(Eλ − E∞) + ∆ε(Eλ − E∞) = −{I−1

ε |Eλ|2}Eλ + {I−1 ε |E∞|2}E∞ + QλEλ,

(Eλ − E∞)(0, x) = 0, (EE) where Qλ = nλ + I−1

ε |Eλ|2 and Iε = 1 − ε2∆.

To evaluate Eλ − E∞, we rewrite (EE) into the integral equation Eλ(t) − E∞(t) = t Uε(t − s)

• {I−1

ε |Eλ|2}Eλ − {I−1 ε |E∞|2}E∞

• − (QλEλ)
• ds,

where Uε(t) = exp(it∆ε). Qλ satisfies the equation        λ−2∂2

tQλ − ∆εQλ = λ−2∂2 tI−1 ε |Eλ|2,

Qλ(0, x) = n0(x) + I−1

ε |E0|2(x),

∂tQλ(0, x) = n1(x) + 2Im{E0∆εE0}, (Q) which is equivalent to the integral equation

Qλ(t) = cos(λt

• −∆ε){n0 + I−1

ε |E0|2}

+sin(λt√−∆ε) λ√−∆ε {n1 + 2Im(E0∆εE0)} + t sin(λ(t − s)√−∆ε) λ√−∆ε ∂2

tI−1 ε |Eλ|2(s)ds.

(INT) Theorem 2 (Case d = 1). (i) Assume n0+I−1

ε |E0|2 = 0. We have

sup

0≤t≤T ∗ Eλ(t) − E∞(t)Hm ≤ Cλ−1.

(ii) Assume n0 + I−1

ε |E0|2 ≡ 0. Then

sup

0≤t≤T ∗ Eλ(t) − E∞(t)Hm ≤ Cλ−2

The quantity n0 + I−1

ε |E0|2 is called the initial layer.

Theorem 3 (Case d = 2). (i) Assume n0 + I−1

ε |E0|2 = 0.

sup

0≤t≤T ∗ Eλ(t) − E∞(t)Hm ≤ Cλ−1.

(ii) Assume n0 + I−1

ε |E0|2 ≡ 0. Then

sup

0≤t≤T ∗ Eλ(t) − E∞(t)Hm ≤ Cλ−2 log λ.

Theorem 4 (Case d = 3). (i) Assume n0 + I−1

ε |E0|2 = 0.

sup

0≤t≤T ∗ Eλ(t) − E∞(t)Hm ≤ Cλ−1.

(ii) Assume n0 + I−1

ε |E0|2 ≡ 0. Then

sup

0≤t≤T ∗ Eλ(t) − E∞(t)Hm ≤ Cλ−2

• iEt + ∂2

xE − ε2∂4 xE = nE

λ−2ntt − ∂2

xn + ε2∂4 xn = ∂2 x|E|2 ε→0 − →

• iEt + ∂2

xE = nE

λ−2ntt − ∂2

xn = ∂2 x|E|2

Figure 5:

.

Figure 6:

.

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