Well-Posedness and Adiabatic Limit for Quantum Zakharov System

Well-Posedness and Adiabatic Limit for Quantum Zakharov System Yung-Fu Fang (joint work with Tsai-Jung Chen, Jun-Ichi Segata, Hsi-Wei Shih, Kuan-Hsiang Wang, Tsung-fang Wu) Department of Mathematics National Cheng Kung University Tainan, 701 Taiwan Talk at Institute of Mathematics, Academia Sinica International Conference on Nonlinear Analysis Boundary Phenomena for Evolutionary PDE DEC. 20 24, 2014

Abstract : We consider a quantum Zakharov system in 1-D and investigate the LWP, GWP, adiabatic limit, and least energy solution. In the derivation of multilinear estimates of (QZ), we characterize the dependence of the quantum parameter within the constants, which leads us to improve the result of [GTV] in 1D. We obtain the adiabatic limit for (QZ) system to a quantum modified nonlinear Schrödinger equation. We also prove the existence of homoclinic solutions with the least energy. Future study for (QZ) system: LWP in 2D and 3D, Ill-posedness, classical limit, subsonic limit, ground state, soliton train, and asymptotic behavior. Finally we investigate the scattering problem.

1. Introduction : Zakharov System describes the propagation of Langmuir waves in an ionized plasma. Langmuir waves are rapid oscillations of the electron density in conducting media such as plasmas or metals. ie t + 2 xe = ne, x R, n tt 2 xn = 2 x E 2, E(x, 0) = E 0 (x), n(x, 0) = n 0 (x), n t (x, 0) = n 1 (x). E = the rapidly oscillating electric field, n = the deviation of the ion density from its mean value. For the discussion of derivation of (Z) system, see [Z] and [OT]. (Z)

Conservation of Mass: E(x, t) 2 dx = constant Conservation of Hamiltonian: x E(t) 2 + 1 2 n(t)2 + n(t) E(t) 2 + 1 2 ν(t)2 dx = constant where t n = x ν and t ν = x ( n + E 2 ).

(Ginibre-Tsutsumi-Velo, 1997): Local Well-Posedness: (Z) system with initial data (E 0, n 0, n 1 ) H k H l H l 1 provided 1 2 < k l 1, 0 l + 1 2 2k. Figure 1:

(Colliander-Holmer-Tzirakis, 2008): Global Well-Posedness: (Z) system with initial data (E 0, n 0, n 1 ) L 2 H 1 2 H 3 2. Figure 2:

Taking quantum effects into account, we consider ie t + 2 xe ε 2 4 xe = ne, x R; n tt 2 xn + ε 2 4 xn = 2 x E 2 E(x, 0) = E 0 (x), n(x, 0) = n 0 (x), n t (x, 0) = n 1 (x). (QZ) ε = ω i κ B T e Proton (6.63 10 27 )(8 10 9 ) 2π(1.38 10 16 )(10 5 ) 6.12 10 5 = Planck s constant /2π, κ B = Boltzmann constant, ω i = ion plasma frequency T e = electron fluid temperature

Conservation of Mass: E(x, t) 2 dx = constant Conservation of Hamiltonian: x E 2 + ε 2 2 xe 2 + 1 2 n2 + n E 2 + 1 2 ν2 + ε2 2 xn 2 dx = constant where t n = x ν and t ν = x ( n + E 2 ε 2 2 xn ).

(Jiang-Lin-Shao, 2013) Local Well-Posedness: (QZ) system with initial data (E 0, n 0, n 1 ) H k H l H l 2 provided 3 4 < k, k 3 2 < l < 2k + 3 2, 3 2 < k l < 3 2. Figure 3:

Due to the effect of the terms of 4th order derivatives, the limit of (QZ) as ε 0 heuristically suggests: 1. change the norms for the solution 2. characterize the quantum effect depicted by ε 3. change k and l into 2k and 2l Figure 4:

(Chen-Fang-Wang, 2014) Global Well-Posedness: (QZ) for initial data (E 0, n 0, n 1 ) L 2 H l ε H l 1 ε with 3 4 l 3 4. Figure 5:

(Fang-Shih-Wang, 2014) Local Well-Posedness: (QZ) for initial data (E 0, n 0, n 1 ) H k H l ε H l 1 ε with max{ k 2 2 k 5 8, 3 4 l 2k + 3 4, and, k 2, k 11 2 8 } < l < k + 2. Figure 6:

(Fang-Shih-Wang, 2014) Local Well-Posedness: (Z) for initial data (E 0, n 0, n 1 ) H k H l ε H l 1 ε with 5 4 < l k 1 and 1 2 l 2k 1 2. Figure 7:

2. Notations and Solution formulae Denote D ε := 1 ε 2 2 x and ξ ε := ξ 1 + ε 2 ξ 2. Decompose (QZ) we get ie t + xd 2 εe 2 = (n + + n )E, x R t n ± ± x D ε n ± = 1 1 x E 2 + 1 2D ε 2 n 1L, (QZ ± ) Schrödinger Propagator: U ε (t) = e it 2 xd 2 ε Modified Wave Propagators: (via Fourier transform) 2F ( W ε (t)(n 0, n 1 ) ) = e ±itξε n 0 ± e±itξ ε iξ ε n 1H ± e±itξε 1 iξ ε n 1L

Duhamel Operators: U ε R z(t, x) = and W ε± R z(t, x) = t The integral formulae for solutions: 0 t 0 e i(t s) xd ε z(s, x)ds. U ε (t s)z(s, x)ds E(t, x) = U ε (t)e 0 (x) iu ε R ( (n+ + n )E ) (t, x) (2.1) n ± (t, x) = W ε± (t)(n 0, n 1 ) W ε± R ( D 1 ε x E 2) (t, x) (2.2) Denote ξ := (1 + ξ 2 ) 1 2 and f 2 H l := ξ 2l f(ξ) 2 dξ, f 2 H l ε := ξ ε 2l f(ξ) 2 dξ, (2.3)

f 2 A := f(ξ) 2 dξ + ξ l ε ε 2l f(ξ) 2 dξ and the following ξ 1 1 ξ equivalent norm. f(ξ) 2 dξ + ξ 2l f(ξ) 2 dξ + ε 2l ξ 4l f(ξ) 2 dξ ξ 1 1 ξ ε 1 ε 1 ξ Notice that the H 2l, Hε, l and A l ε norms are equivalent: ε max{0, l} f H 2l f H l ε f A l ε ε min{l, 0} f H 2l. (2.4) Under suitable conditions, we have f H l ε f H l as ε 0. (2.5) Denote the Sobolev norm for acoustic wave n(t) Wε := ( n(t), t n(t) ) Wε := n(t) A l ε + t n(t) A l 1 ε

Bourgain norm for Schrödinger part: E X Sε k,b 1 := ( ξ ε 2k τ + ξ 2 ε 2b 1 Ê(τ, ξ) 2 dτdξ)1 2 (2.6) Bourgain norm for Wave part: n ± X Wε± l,b Y norm for Schrödinger part: E Y Sε k := ( )1 := ξ ε 2l τ ± ξ ε 2b n(τ, ξ) 2 2 dτdξ ( ( ξ ε k τ + ξ 2 ε 1 Ê(τ, ξ) dτ ) 2dξ )1 2 (2.7) (2.8) Y norm for Wave part: n ± Y Wε± l := ( ( ξ ε l τ ± ξ ε 1 n ± (τ, ξ) dτ ) 2dξ )1 2 (2.9)

3. Estimates for Iteration Argument Lemma 1. (Homogeneous Estimates) T 1. For Schrödinger solution, we have (a1) U ε (t)e 0 C([0,T ];H k ε ) = E 0 H k ε. (a2) ψ T (t)u ε (t)e 0 X Sε k,b 1 T 1 2 b 1 E 0 H k ε. (a3) (Strichartz estimates) For Wave solution, we have D θ 2 ε U ε (t)e 0 L q t Lr x E 0 L 2. (b1) W ε (t)(n 0, n 1 ) C([0,T ];Wε ) (1 + T ) (n 0, n 1 ) Wε. (b2) ψ T (t)w ε± (t)(n 0, n 1 ) X Wε± l,b Proof. See [GTV], [CHT], and [KPV]. T 1 2 b (n 0, n 1 ) Wε.

Lemma 2. (Duhamel Estimates) T 1. For Schrödinger solution, we have (a1) U ε R F C([0,T ];H k ε ) T 1 2 c 1 F X Sε. k, c 1 (a2) If F Y S ε k, then U ε R F C(R; Hε k ). (a3) ψ T (t)u ε R F X Sε k,b 1 T 1 b 1 c 1 F X Sε + T 1 2 b 1 ( χ F ) { τ+ξ 2 k, c ε T 1 } Y Sε. 1 (a4) ψ T (t)u ε R F X Sε k,b 1 T 1 b 1 c 1 F X Sε k, c 1. (a5) (Strichartz estimates) D θ 2 ε E L q E t Lr x X S ε. 0,d k

For Wave solution, we have (b1) W ε R G C([0,T ];Wε ) T 1 c( 2 G X Wε+ l, c (b2) If G Y W ε± l, then W ε± R G C(R; Hε). l (b3) ψ T (t)w ε± R G X Wε± l,b T 1 b c G X Wε± l, c (b4) ψ T (t)w ε± R G X Wε± l,b Proof. See [GTV], [CHT], and [KPV]. + G X Wε l, c ). + T 1 2 b ( χ { τ±ξε T 1 }Ĝ) Y Wε±. l T 1 b c G X Wε±. l, c

Lemma 3. (Multilinear Estimates) Let 0 < ε 1. n ± E X Sε C(ε) n ± k, c 1 X Wε± E X Sε. (3.1) k,b 1 l,b Dε 1 x (E 1 Ē 2 ) X Wε± l, c C(ε) E 1 X Sε k,b 1 E 2 X Sε k,b 1. (3.2) n ± E Y Sε k C(ε) n ± X Wε± E l,b X Sε. (3.3) k,b 1 Dε 1 ( ) x E1 Ē 2 Y Wε± l C(ε) E 1 X Sε k,b 1 E 2 X Sε k,b 1. (3.4)

Corollary 1. ([CHT] Lemma 3.1) Let k = 0 and l = 1 2. (a) If 1 4 b < 1 2, 1 4 < b 1, c 1 < 1 2 and b + b 1 + c 1 1, then n ± E X Sε 0, c 1 n ± X Wε± 1 2,b E X Sε 0,b 1. (b) If 1 4 < b 1, c < 1 2 and 2b 1 + c 1, then Dε 1 x (E 1 Ē 2 ) X Wε± E 1 X Sε E 2 1 2, c 0,b X Sε. 1 0,b 1

Theorem 1. (Chen, Fang, & Wang 2014) Let 0 < ε 1. (QZ) is globally well-posed for (E 0, n 0, n 1 ) L 2 H l ε H l 1 ε with 3 4 l 3 4, the solution (E; n) satisfies conservation of mass and n(t) H 2l + t n(t) H 2l 2 max( n 0 H 2l + n 1 H 2l 2; E 0 2 L 2 ) exp c t ( E 0 2 L 2 + 1). Proof. (Outline) Define the maps Λ Sε (E, n ± ) = ψ T U ε E 0 + ψ T U ε R [(n + + n )E], Λ Wε± (E) = ψ T W ε± (n 0, n 1 ) ± ψ T W ε± R ( x u 2 ). For some 0 < T < 1, we seek a fixed point (E(t), n ± (t)) = (Λ Sε (E, n ± ), Λ Wε± (E)).

Using homogeneous estimates, Duhamel estimates, multilinear estimates, and Strichartz estimates, we obtain the existence of a fixed point E X S ε 0,b 1 and n ± X W ε± l,b ψ T E X Sε 0,b 1 T 1 2 b 1 E 0 L 2 and ψ T n ± X Wε± l,b such that T 1 2 b (n 0, n 1 ) Wε. For global existence when 3 4 l 1 2, we interpolate between (3.1) with l = 3 4 and (3.1) with l = 1 2. For global existence when 1 2 l 3 4, we inerpolate between (3.2) with l = 3 4 and (3.2) with l = 1 2. We then invoke the conservation law of mass (1.1) and keep tracking the growth of n(t) H l ε + t n(t) H l 1 ε an exponential bound for the growth, also see [CHT]. in time. We can obtain

Remark 1.. (1) For k 0, we establish GWP in the largest space for LWP. (2) Characterize the dependence of ε and the constants in the multilinear estimates (3) For l = 1 2, we recover the result of [CHT] as ε 0. Figure 8:

Theorem 2. (Fang, Shih, & Wang 2014) Let 0 < ε 1. (QZ) is locally well-posed for (E 0, n 0, n 1 ) Hε k Hε l Hε l 1 provided k 5 8, 3 4 l 2k + 3 4, and max{ k 2 2, k 2 2, k 11 8 } < l < k + 2. Figure 9:

Theorem 3. (Fang, Shih, & Wang 2014) The Zakharov system (Z) is locally well-posed for initial data (E 0, n 0, n 1 ) H k H l H l 1, provided 5 4 < l k 1 and 1 2 l 2k 1 2. Figure 10:

4. Proof of Multilinear Estimates Proof. We only outline the proof for Dε 1 x (E 1 E 2 ) X Wε± l, c C 2 (ε) E 1 X Sε k,b 1 E 2 X Sε k,b 1. First we set ξ = ξ 1 ξ 2 and decompose the ξ 1 -ξ 2 plane into Ω 1 Ω 2 Ω 3 Ω 4, where Ω 1 = {(ξ 1, ξ 2 ) : ξ 2 ξ 1 ξ }, Ω 2 = {(ξ 1, ξ 2 ) : ξ 1 ξ 2 ξ }, Ω 3 = {(ξ 1, ξ 2 ) : ξ 1 ξ 2 ξ }, Ω 4 = {(ξ 1, ξ 2 ) : ξ ξ 1 ξ 2 }.

By duality argument, the estimate Dε 1 x (E 1 E 2 ) X Wε+ l, c is equivalent to The norms E 1 X Sε k,b 1 E 2 X Sε k,b 1 D 1 ε x (E 1 E 2 ), g E 1 X Sε k,b 1 E 2 X Sε g k,b 1 X Wε+. l,c and g X Wε+ l,c = ξ ε l τ + ξ ε c ĝ L 2 E j X Sε k,b 1 = ξ ε k τ + ξ 2 ε b 1 Ê j L 2 suggest that we set v = ξ ε l τ + ξ ε c ĝ, v j = ξ ε k τ + ξ 2 ε b 1 Ê j, j = 1, 2.

Thus we can rewrite the left hand side of the formula as iξ ξ ε l v(τ, ξ) v 2 (τ 2, ξ 2 ) v 1 (τ 1, ξ 1 ) 1 + ε2 ξ 2 ξ 1ε k ξ 2ε k σ c σ 2 b 1 σ 1 b dτ 2 dξ 2 dτ 1 dξ 1, 1 where τ = τ 1 τ 2, ξ = ξ 1 ξ 2, σ = τ + ξ ε, σ 2 = τ 2 + ξ 2 2ε, σ 1 = τ 1 + ξ 2 1ε. We want to show that it is bounded by v L 2 v 1 L 2 v 2 L 2. ξ ε l Denote ξ 1ε k ξ 2ε k = A and analyze A in Ω 1, Ω 2, Ω 3, and Ω 4. The case of k 0. On Ω 1, A ξ ε m 1 ξ 2ε k, where m 1 = l k. On Ω 2, A ξ ε m 1 ξ 1ε k, where m 1 = l k. On Ω 3, A ξ ε m 0, where m 0 = l 2k. On Ω 4, A ξ ε m 0, where m 0 = l 2k.

The case of k < 0 and l 0. On Ω 1, A ξ ε m 0, where m 0 = l 2k. On Ω 2, A ξ ε m 0, where m 0 = l 2k. On Ω 3, A ξ ε m 0, where m 0 = l 2k. On Ω 4, A ξ 2ε m 0, where m 0 = l 2k. The case of k < 0 and l < 0. On Ω 1, A ξ ε m 0, where m 0 = l 2k. On Ω 2, A ξ ε m 0, where m 0 = l 2k. On Ω 3, A ξ ε m 0, where m 0 = l 2k. On Ω 4, A ξ ε l ξ 2ε m 2, where m 2 = 2k.

Thus it is sufficient to show the following estimates. ξ ξ ε m v v 0 2 v 1 D ε σ c σ 2 b 1 σ 1 b dτ 2 dξ 2 dτ 1 dξ 1 v 1 L 2 v 1 L 2 v 2 L 2. For k 0, i + j = 3, and j = 1, 2, we prove ξ ξ ε m 1 k v v 2 v 1 ξ jε Ω i D ε σ c σ 2 b 1 σ 1 b dτ 2 dξ 2 dτ 1 dξ 1 v 1 L 2 v 1 L 2 v 2 L 2 For k < 0 and l 0, we prove ξ ξ 2ε m v v 0 2 v 1 Ω 4 D ε σ c σ 2 b 1 σ 1 b dτ 2 dξ 2 dτ 1 dξ 1 v 1 L 2 v 1 L 2 v 2 L 2. For k < 0 and l < 0, we prove ξ ξ ε l ξ 2ε m v v 2 2 v 1 Ω 4 D ε σ c σ 2 b 1 σ 1 b 1 dτ 2 dξ 2 dτ 1 dξ 1 v L 2 v 1 L 2 v 2 L 2.

We first split the frequency ξ into 3 parts, { ξ 6}, {6 < ξ 12ε 1 }, and {12ε 1 < ξ }. For the low frequency part, { ξ 6}, we use the Strichartz estimates. For the rest two parts, {6 < ξ 12ε 1 } and {12ε 1 < ξ }, we split the integrals into regions depending on the sizes of σ, σ 1, σ 2. { σ σ 1, σ 2 }, { σ 1 σ, σ 2 }, and { σ 2 σ, σ 1 }. The condition m 0 = l 2k 3 4 is required. Remark 2. The condition m 0 3 4 is optimal. Crossing the corresponding boundary of the region, we have counterexamples. Question: Local Well-Posedness for (QZ) in 2D and 3D?

We extend the region of (k, l) of LWP for (QZ) as follows and recover the result of Ginibre-Tsutsumi-Velo. We also extend the region of LWP for (Z). Figure 11:

5. Existence of a Least Energy Solution Consider the stationary solution and static solution to (QZ) in 1D by setting E(x, t) = e iωt Q(x) and n(x, t) = n(x). Then (QZ) { ωq + Q ε 2 Q (4) = nq, n + ε 2 n = Q 2. Question 1. Does solution Q ε,ω (x) and n ε,ω (x) exist? Question 2. Does lim ε 0 (Q ε,ω, n ε,ω ) = (Q 0,ω, Q 2 0,ω)? Question 3. Numerical Results? Question 4. Stability of such solutions? Soliton? Soliton train?

Lemma 4. Let n = 1. n ε,q (x) = R 1 2ε e x y ε Q 2 (y) dy Q 2 (x) as ε 0. Plug n ε,q back into the system to get an ODE with a nonlocal term. For 2D and 3D, ωq + Q ε 2 Q (4) = Q R 1 2ε e x y ε Q 2 (y) dy. (E ε ) ωq + Q ε 2 Q (4) = Q( 1 + ε 2 ) 1 Q 2. (E ε ) Theorem 4. (Fang & Wu, 2014) Let n = 1, 2, 3. For each 0 < ε 1, Problem (E ε ) has a least energy homoclinic solution Q ε.

6. Adiabatic Limit of (QZ) We consider the quantum Zakharov system: i t E + E ε 2 2 E = ne t R, x R d, λ 2 t 2 n n + ε 2 2 n = E 2 t R, x R d, E(0) = E 0, n(0) = n 0, t n(0) = n 1 x R d, Zakharov system: i t E + E = ne t R, x R d, λ 2 t 2 n n = E 2 t R, x R d, E(0) = E 0, n(0) = n 0, t n(0) = n 1 x R d, (λqz) (λz) Let us formally take λ for the second equation in (λqz). Under suitable conditions, we obtain the relation n ε, = (1 ε 2 ) 1 E ε, 2.

Substituting this relation into (λqz), E ε, satisfies the quantum modified NLS i t E ( + ε 2 2 )E = {(1 ε 2 ) 1 E 2 }E. (4NLS) Theorem 5. (Fang & Segata) [d = 1] Let 3M m + 6. For (E 0, n 0, n 1 ) H 3M H 3M 1 H 3M 3,! solution to (λqz) satisfying (E ε,λ, n ε,λ, t n ε,λ ) C([0, ); H 3M H 3M 1 (H 3M 3 Ḣ 1 )) and! solution to (4NLS) satisfying E ε, C([0, ); H 3M ). Further assumptions imply sup E ε,λ (t) E ε, (t) H m Cλ 1. 0 t T

Theorem 6. (Fang & Segata) [d = 2, 3] Let 3M m + 6. For (E 0, n 0, n 1 ) H 3M H 3M 1 H 3M 3,! solution to (λqz) satisfying (E ε,λ, n ε,λ, t n ε,λ ) C([0, ); H 3M H 3M 1 (H 3M 3 Ḣ 1 )) and T (0, ) and a unique solution to (4NLS) satisfying E ε, C([0, T ); H 3M ). Further assumptions imply that Question: sup 0 t T E ε,λ (t) E ε, (t) H m Cλ 1. lim ε,λ λ = E ε,, lim E ε, ε 0 = E 0,?? lim ε,λ ε 0 = E 0,λ, lim E 0, λ = E 0,??

7. ILL-Posedness Problem of (Z) in 1D Holmer proved ill-posedness for (Z) in 1D such as norm inflation, phase decoherence, or data-to-solution map C 2. We expect to improve Holmer s result on (Z) and get some results for (QZ). Figure 12:

References 1. Colliander-Holmer-Tzirakis-2008-Low Rgularity GWP for Zakharov and Klein-Gordon-Schrödinger Systems 2. Fang-Segata-2014-Adiabatic Limit for Quantum Zakharov 3. Fang-Wu-2014-Ground State for Quantum Zakharov 4. Ginibre-Tsutsumi-Velo-1997-On the Cauchy Problem for the Zakharov System 5. Jiang-Lin-Shao-2014-On 1D Quantum Zakharov System 6. Kenig-Ponce-Vega-1991-Oscillatory Integrals and Regularity of Dispersive Equations 7. Haas-Shukla-2009-Quantum and Classical Dynamics of Langmuir Wave Packets 8. Schochet-Weinstein-1986-The NLS of the Zakharov Equations

Governing Langmuir Turbulence 9. Added-Added-1988-Equations of Langmuir Turbulence and NLS-Smoothness and approximation 10. Ozawa-Tsutsumi-1992-The NLS Limit and the Initial Layer of the Zakharov Equations 11. Masmoudi-Nakanishi-2008-Energy Convergence for Singular Limits of Zakharov Type Systems 12. Guo-Zhang-Guo-2013-GWP and the Classical Limit of the Solution for the Quantum Zakharov System 13. Guo-Nakanishi-2013-Small Energy Scattering for the Zakharov System with Radial Symmetry 14. Hani-Pusateri-Shatah-2013-Scattering for Zakharov System 15. Zakharov-1972-Collapse of Langmuir Waves, Sov.Phys.JETP

Acknowledgement Yung-fu Fang wants to express his gratitude toward Manoussos Grillakis, Yoshio Tsutsumi, and Kenji Nakanishi for the inspiring conversations and suggestions. In 2009, Tai-Ping said Let PDE workshops and conferences be held at Tainan. 2009, PDE workshop at Tainan. 2011, Annual DE Workshop at Tainan. 2013, PDE workshop at Tainan 2014, Annual Meeting of Taiwanese Math. Soc. 2015,???

Figure 13:

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