# Existence and Uniqueness of Solutions for a Type of Generalized Zakharov System

Authors: Rui Li; Xing Lin; Zongwei Ma; Jingjun Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study the Cauchy problem for a type of generalized Zakharov system. With the help of energy conservation and approximate argument, we obtain global existence and uniqueness in Sobolev spaces for this system. Particularly, this result implies the existence of classical solution for this generalized Zakharov system.


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## Body

## 1. Introduction

In this paper, we study a type of generalized Zakharov system which is given by(1)iEt-Λ2αE-nE=0,(2)ntt-nxx+Λ2α|E|2=0,
with initial data
(3)E(0,x)=E0(x),n(0,x)=n0(x),nt(0,x)=n1(x),
where x∈ℝ, t>0, and α∈(1/2,1) is a fixed constant. In the above system, Λ:=(-∂x2)1/2 is a fractional differential operator. With this definition, Λ2α maps u to Λ2αu:=ℱx-1(|ξ|2αℱxu) with ℱx the Fourier transform of u(t,x) with respect to the variable x. In particular, Λ2=-∂x2.Whenα=1, system (1) and (2) reduces to the usual Zakharov system
(4)iEt+Exx=nE,ntt-nxx=|E|xx2,
which was first obtained by Zakharov [1]; here, E:ℝ+×ℝ→ℂ is the slowly varying amplitude of high-frequency electric field and n:ℝ+  ×  ℝ→ℝ is the disturbing quantity of ion from its equilibrium. This model turned out to be very useful in laser plasmas, and many contributions have been made both in the physical and mathematical literature. For the local or global existence and uniqueness of smooth solutions for system (4), we refer to [2–6]. Well-posedness of (4) in lower regularity spaces was obtained in [7]. Existence of global attractors for dissipative Zakharov system was studied in [8–11]. For related Zakharov system including magnetic effects, one can see [12–15].On the other hand, Laskin [16, 17] discovered that the path integral over the Lévy-like quantum mechanical paths allows developing the generalization of the quantum mechanics. That is, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation. So fractional Schrödinger equation is fundamental in the fractional quantum mechanics, and its global well-posedness is studied in [18, 19]. Inspired by this, we then replace the Laplacian in the Schrödinger equation of (4) by the fractional differential operator Λ2α, and this is the main motivation of the paper.In this work, we study global existence and uniqueness of smooth solutions for system (1) and (2). The main result is stated in the following theorem.Theorem 1.
Letα∈(1/2,1), let m≥2 be an integer, E0∈Hmα(ℝ),n0∈H(m-2)α+1(ℝ), and n1∈H(m-2)α(ℝ)∩H˙-α(ℝ). Then system (1)~(3) has a unique solution (E,n,nt) satisfying
(5)(E,n,nt)∈C((H(m-2)α(ℝ)∩H˙-α(ℝ))ℝ+;Hmα(ℝ)×H(m-2)α+1(ℝ)×(H(m-2)α(ℝ)∩H˙-α(ℝ))).Theorem1 will be proved by using energy conservation and approximate argument. To this end, in the next section, we present some notations and useful lemmas which will be used throughout the paper. In Section 3, we study a regularized system of (1) and (2). Finally, the proof of Theorem 1 is given in Section 4.


## 2. Preliminaries

Firstly, we set some notations. Fors∈ℝ, we use H˙s to denote the fractional homogeneous Sobolev space, consisting of all tempered distribution u such that ∥u∥H˙s is finite, where ∥u∥H˙s is defined via the Fourier transform
(6)∥u∥H˙s:=∥Λsu∥L2=∥|ξ|sℱxu(ξ)∥L2,Λ:=(-∂x2)1/2.
Similarly, one can define the inhomogeneous Sobolev space Hs equipped with the norm
(7)∥u∥Hs:=∥(1+|ξ|2)s/2ℱxu(ξ)∥L2.
In particular, we have ∥u∥Hs~∥u∥L2+∥Λsu∥L2 for s≥0. Throughout the paper, the initial data (3) is given in the product space Ym defined by
(8)Ym:=Hmα(ℝ)×H(m-2)α+1(ℝ)×(H(m-2)α(ℝ)∩H˙-α(ℝ)).
We endow Ym with the natural norm
(9)∥(u,v,w)∥Ym:=∥u∥Hmα+∥v∥H(m-2)α+1(ℝ)+∥w∥H(m-2)α(ℝ)∩H˙-α(ℝ).Next, we introduce the following calculus inequality, the proof of which can be found, for example, in [20–22].Lemma 2.
Lets>0 and f,g∈𝒮(ℝ) (the class of Schwarz functions); then
(10)∥Λs(fg)∥Lp≤C(∥f∥Lp1∥Λsg∥Lp2+∥Λsf∥Lp3∥g∥Lp4),
where p, p2, p3∈(1,+∞), (1/p1)+(1/p2)=(1/p3)+(1/p4)=1/p.We end this section with the following lemma, which states two conserved quantities for the smooth solutions of (1)~(3). Here, we say a solution (E,n,nt) is a smooth solution of system (1)~(3) provided that (E,n,nt)∈Yk with k sufficiently large and (1)~(3) hold in the classical sense.Lemma 3.
Suppose that(E,n,nt) is a smooth solution of system (1)~(3); then there hold
(11)∥E(t)∥L22=∥E0∥L22,Ψ(t):=∥ΛαE(t)∥L22+12∥Λ1-αn(t)∥L22+12∥Λ-αnt(t)∥L22+∫ℝ‍n(t)·|E(t)|2dx=Ψ(0).Proof.
MultiplyingE¯ on both sides of (1) and then choosing the imaginary part after integration in ℝ, it is easy to obtain
(12)∥E(t)∥L22=∥E0∥L22.
Now, we give the proof of the second conserved quantity. On one hand, multiplyingEt¯ on both sides of (1) and choosing the real part after integration in ℝ, we then get
(13)ddt∥ΛαE(t)∥L22+∫ℝn(t)|E(t)|t2dx=0.
On the other hand, taking inner product of (2) with Λ-2αnt, we then obtain
(14)12ddt∥Λ-αnt(t)∥L22+12ddt∥Λ1-αn(t)∥L22+∫ℝ|E(t)|2nt(t)dx=0.
Combining the above two equalities gives Ψ(t)=Ψ(0).


## 3. Global Existence and Uniqueness for a Regularized System

In order to prove Theorem1, we firstly study a regularized system for (1)~(3) in this section. For ϵ∈(0,1), let us consider the following regularized system:
(15)iEtϵ=Λ2αEϵ+(ℬϵnϵ)Eϵ,Eϵ(x,0)=E0ϵ,
where the operator ℬϵ:=(I+ϵΛ4α)-1 and nϵ=nϵ(Eϵ) is the solution of the equation
(16)nttϵ-Δnϵ=-ℬϵΛ2α|Eϵ|2,
with initial data nϵ(x,0)=n0ϵ, ntϵ(x,0)=n1ϵ. It is easy to see that the operator ℬϵ satisfies the following properties:(1)
∥ℬϵf∥Hs≤∥f∥Hs, ∥ℬϵf∥H˙s≤∥f∥H˙s, ∀s∈ℝ;(2)
(ℬϵf,f)=∫ℝ(Bϵf)·f¯dx≥0;(3)
(ℬϵf,g)=(f,ℬϵg);(4)
∫ℝ(ℬϵΛf)gdx=∫ℝ(Λℬϵf)gdx=∫ℝ(ℬϵf)(Λg)dx=∫ℝ(Λf)(ℬϵg)dx.Roughly speaking, the fourth property says that the operator ℬϵ commutes with the operator Λ; of course, the operator Λ can be replaced by other differential operators such as Λs.From the semigroup theory we know that the linear equationEt=-iΛ2αE generates a unitary group U(t)=exp(-itΛ2α) in Hs(ℝ), so the solution of (15) can be expressed by the following integral form:
(17)Eϵ(t)=U(t)E0ϵ+∫0tU(t-τ)[(ℬϵnϵ)Eϵ](τ)dτ.A few words about the regularized system (15) or (17). If we study directly the integral equation of the original system (1)~(3), that is,
(18)E(t)=U(t)E0+∫0tU(t-τ)(nE)(τ)dτ,
where n=n(E) solves (2), we will find that it is difficult to apply fixed point theorem for this integral equation because the regularity of E and n is not the same (note that (E,n)∈C(ℝ+;Hmα(ℝ)×H(m-2)α+1(ℝ))). In fact, when estimating the L∞Hmα norm of E, we have
(19)∥E∥L∞Hmα≤∥E0∥Hmα+T∥nE∥L∞Hmα,
where we need nE∈C(ℝ+;Hmα). However, this is not correct since n only belongs to C(ℝ+;H(m-2)α+1). For this reason, we first study the regularized system (15) by introducing the operator ℬϵ, and we can see that ℬϵnϵ∈C(ℝ+;H(m+2)α+1)⊂C(ℝ+;Hmα) if nϵ∈C(ℝ+;H(m-2)α+1). Then the well-posedness result of the regularized system can be easily proved through the integral equation (17) (see Theorem 6). Based on the solution of (15) and (16), we have to take ϵ→0 in the regularized system to obtain the desired result as stated in Theorem 1. This step requires some uniform estimates for the solution of the regularized system, and these a priori estimates will be given in Section 4.The main aim in this section is to obtain the existence and uniqueness of global solution for the regularized system (15) and (16). Due to the “good” operator ℬϵ, the global well-posedness result for the regularized system can be proved more easily. Before stating Theorem 6, we need the following two lemmas.Lemma 4 (conserved quantities).
Suppose that(Eϵ,nϵ,ntϵ)∈Yk is a smooth solution of the regularized system (15) and (16); then there hold
(20)Φϵ(t):=∥Eϵ(t)∥L22=Φϵ(0),Ψϵ(t):=∥ΛαEϵ(t)∥L22+12∥Λ1-αnϵ(t)∥L22+12∥Λ-αntϵ(t)∥L22+∫ℝ(ℬϵnϵ)(t)|Eϵ(t)|2dx=Ψϵ(0).The proof of Lemma4 is similar to Lemma 3; thus, it is omitted here.Lemma 5.
Assume that(Eϵ,nϵ,ntϵ)∈Yk is a smooth solution of the regularized system (15) and (16); then there holds
(21)∥Eϵ∥Hα+∥Λ1-αnϵ∥L2+∥Λ-αntϵ∥L2≤C,∀t>0,
where the constant C depends on ∥E0ϵ∥Hα, ∥n0ϵ∥H˙1-α and ∥n1ϵ∥H˙-α. In particular, the above estimate implies that
(22)∥Eϵ∥Lq≤C,∀q∈[2,+∞].Proof.
From Lemma4, we know that
(23)∥ΛαEϵ∥L22+12∥Λ1-αnϵ∥L22+12∥Λ-αntϵ∥L22≤|Ψϵ(0)|+|∫ℝ(ℬϵnϵ)|Eϵ|2dx|.
By Hölder’s inequality and the embedding H˙1-α(ℝ)↪L2/(2α-1)(ℝ), the last term in the above inequality can be estimated as follows:
(24)|∫ℝ(ℬϵnϵ)|Eϵ|2dx|≤∥ℬϵnϵ∥L2/(2α-1)∥Eϵ∥L4/(3-2α)2≤η∥ℬϵnϵ∥L2/(2α-1)2+14η∥Eϵ∥L4/(3-2α)4≤Cη∥Λ1-αnϵ∥L22+14η∥Eϵ∥L4/(3-2α)4.
Applying the Gagliardo-Nirenberg inequality
(25)∥Eϵ∥L4/(3-2α)≤C∥Eϵ∥L2(2α+1)/4α∥ΛαEϵ∥L2(2α-1)/4α,
we have
(26)∥Eϵ∥L4/(3-2α)4≤C∥Eϵ∥L2(2α+1)/α∥ΛαEϵ∥L2(2α-1)/α≤C∥ΛαEϵ∥L2(2α-1)/α.
Using this inequality and Young’s inequality, there holds
(27)|∫ℝ(ℬϵnϵ)|Eϵ|2dx|≤Cη∥Λ1-αnϵ∥L22+14η∥Eϵ∥L4/(3-2α)4≤Cη∥Λ1-αnϵ∥L22+C4η∥ΛαEϵ∥L2(2α-1)/α≤Cη∥Λ1-αnϵ∥L22+Cδ4η∥ΛαEϵ∥L22+C(δ,η),
where C(δ,η) is a constant depending on δ, η. Combining the above arguments, one can see that
(28)∥ΛαEϵ∥L22+12∥Λ1-αnϵ∥L22+12∥Λ-αntϵ∥L22≤|Ψ(0)|+Cη∥Λ1-αnϵ∥L22+Cδ4η∥ΛαEϵ∥L22+C(δ,η).
We firstly choose η small enough to make sure that Cη∥Λ1-αnϵ∥L22 is absorbed by the term (1/2)∥Λ1-αnϵ∥L22; for such fixed η, we then choose δ small enough to make sure that (Cδ/4η)∥ΛαEϵ∥L22 is absorbed by the term ∥ΛαEϵ∥L22. Thus, we get
(29)∥Eϵ∥Hα+∥Λ1-αnϵ∥L2+∥Λ-αntϵ∥L2≤C.
Since 1/2<α≤1, the estimate (22) follows easily from the embedding Hα(ℝ)↪Lq(ℝ) (q∈[2,+∞]). The proof of Lemma 5 is complete.Now, we state the main result of this section.Theorem 6.
Letk≥2013 be an integer, and assume (E0ϵ,n0ϵ,n1ϵ)∈Yk; then for arbitrary ϵ∈(0,1), the regularized system (15) and (16) has a unique solution (Eϵ,nϵ,ntϵ)∈C(ℝ+;Yk).Proof.
The proof consists of two parts: the first part is to prove local existence of the solution for the regularized system by using the standard Banach’s fixed point theorem, and the second part is to extend this local solution to be a global one with the help of somea priori estimates.
Step
1. Firstly, we get the local existence by using the contracting mapping principle. In order to achieve this aim we define 𝒯 by
(30)𝒯Eϵ:=U(t)E0ϵ+∫0tU(t-τ)[(ℬϵnϵ)Eϵ](τ)dτ.
As nϵ satisfies (16), there holds
(31)nϵ=cos(tΛ)n0ϵ+sin(tΛ)Λn1ϵ-∫0tsin((t-τ)Λ)Λ2α-1ℬϵ|Eϵ(τ)|2dτ.
ForT∈(0,1), we now define the space
(32)X={E(t,x)∈C([0,T];Hkα):∥E∥X≤2∥E0ϵ∥Hkα},
where ∥E∥X:=supt∈[0,T]∥E(t)∥Hkα. From (31), it is easy to see that
(33)nϵ(Eϵ)L∞H(k-2)α+1≤C(∥Eϵ∥X2+1),∀Eϵ∈X,∥nϵ(Eϵ)-nϵ(Fϵ)∥L∞H(k-2)α+1≤C∥Eϵ-Fϵ∥X,∀Eϵ,Fϵ∈X,
where C depends on ∥E0ϵ∥Hkα. By the definition of ℬϵ, one also has
(34)∥(ℬϵnϵ)Eϵ∥X≤C∥ℬϵnϵ∥L∞Hkα∥Eϵ∥L∞Hkα≤C(ϵ,∥E0ϵ∥Hkα),∀Eϵ∈X,∥[ℬϵnϵ(Eϵ)]Eϵ-[ℬϵnϵ(Fϵ)]Fϵ∥X≤C(ϵ)[∥(ℬϵnϵ(Eϵ))(Eϵ-Fϵ)∥L∞Hkα+∥[ℬϵ(nϵ(Eϵ)-nϵ(Fϵ))]Eϵ∥L∞Hkα]≤C(ϵ,∥E0ϵ∥Hkα)∥Eϵ-Fϵ∥X,∀Eϵ,Fϵ∈X.
Combining the above estimates, we have
(35)∥𝒯Eϵ∥X≤∥E0ϵ∥Hkα+TC(ϵ,∥E0ϵ∥Hkα),∥𝒯Eϵ-𝒯Fϵ∥X≤TC(ϵ,∥E0ϵ∥Hkα)∥Eϵ-Fϵ∥X.
Hence, if we choose T=T(ϵ,∥E0ϵ∥Hkα) sufficiently small, then 𝒯 maps X into itself and 𝒯 is contractive. From the contraction mapping principle, (15) admits a unique solution Eϵ∈C([0,T]; Hkα(ℝ)), which, by (31), gives (nϵ,ntϵ)∈C([0,T];H(k-2)α+1(ℝ)×H(k-2)α(ℝ)). Moreover, from the above procedure, we know that if T* is the largest existence time of the solution, then T*=∞ or ∥Eϵ(t)∥Hkα→∞ as t→T*.
Step  2. In order to get the global existence result, it suffices to prove that ∥Eϵ(t)∥Hkα≤C(T*) for all t∈[0,T*). To this end, applying the operator Λkα to (15), then multiplying the resulted equation by ΛkαEϵ¯, and integrating the imaginary part, one can obtain
(36)ddt∥ΛkαEϵ∥L22=2Im∫ℝΛkα[(ℬϵnϵ)Eϵ]ΛkαEϵ¯dx≤C(∥ℬϵnϵ∥L∞∥ΛkαEϵ∥L2+∥Λkα(ℬϵnϵ)∥L2∥Eϵ∥L∞)∥ΛkαEϵ¯∥L2.
By (31) and Lemma 5, one can see that
(37)∥nϵ∥H1≤∥n0ϵ∥H1+T*∥n1ϵ∥H1+T*supt∈[0,T*)∥Λ2α-1ℬϵ|Eϵ|2∥H1≤∥n0ϵ∥H1+T*∥n1ϵ∥H1+T*C(ϵ)supt∈[0,T*)∥|Eϵ|2∥H-2α≤C+Csupt∈[0,T*)∥|Eϵ|2∥Hα≤C,
which implies
(38)∥ℬϵnϵ∥L∞≤C∥ℬϵnϵ∥H1≤C∥nϵ∥H1≤C.
Using this estimate and the fact that (k-4)α<(k-2)α+1, one gets from (36) and Lemma 5 that
(39)ddt∥ΛkαEϵ∥L22≤C(∥ΛkαEϵ∥L2+∥nϵ∥H(k-4)α)∥ΛkαEϵ¯∥L2≤C(∥ΛkαEϵ∥L22+∥Λ(k-2)α+1nϵ∥L22+1).
With similar arguments as above, one can deduce from (16) and Lemma 2 that
(40)ddt(∥Λ(k-2)αntϵ∥L22+∥Λ(k-2)α+1nϵ∥L22)=-2∫ℝΛ(k-2)αntϵΛkαℬϵ|Eϵ|2dx≤C∥Λ(k-2)αntϵ∥L2∥ΛkαEϵ∥L2≤C(∥Λ(k-2)αntϵ∥L22+∥ΛkαEϵ∥L22).
Finally, collecting the above two estimates and using Gronwall’s inequality, there holds(41)∥ΛkαEϵ∥L2+∥Λ(k-2)α+1nϵ∥L2+∥Λ(k-2)αntϵ∥L2≤C(ϵ,T*).
This inequality together with Lemma 5 gives
(42)∥Eϵ(t)∥Hkα≤C(ϵ,T*),∀t∈[0,T*),
which implies that T*=+∞. The proof of Theorem 6 is complete.


## 4. Proof of Theorem1

In this section, we will present the proof of Theorem1. In this proof, the key step is to obtain uniform estimates for the approximate solution (Eϵ,nϵ,ntϵ) with respect to ϵ. Note that the constant C in (42) depends on ϵ, so this estimate is not useful in proving our global existence result for system (1)~(3).For(E0,n0,n1)∈Ym, we now choose the regularized initial data (E0ϵ,n0ϵ,n1ϵ)∈Yk with k≥2013 sufficiently large satisfying
(43)∥(E0ϵ-E0,n0ϵ-n0,n1ϵ-n1)∥Ym⟶0,asϵ⟶0.
Now, we are going to give the uniform estimates for (Eϵ,nϵ,ntϵ). These uniform estimates are stated in the following propositions.Proposition 7.
Suppose that(Eϵ,nϵ,ntϵ)∈Yk is the solution of the regularized system (15) and (16) with (E0ϵ,n0ϵ,n1ϵ) satisfying (43); then for sufficiently small ϵ>0, there holds
(44)∥Eϵ∥Hα+∥Λ1-αnϵ∥L2+∥Λ-αntϵ∥L2≤C,∀t>0,
where the constant C depends on ∥E0∥Hα, ∥n0∥H˙1-α and ∥n1∥H˙-α, but C is independent of ϵ and t. In particular, the above estimate implies that
(45)∥Eϵ∥Lq≤C,∀q∈[2,+∞].Proposition7 follows easily from Lemma 5 and (43).Proposition 8.
Under the same assumption as Proposition7, there holds
(46)∥Eϵ∥H2α+∥nϵ∥H1+∥ntϵ∥L2≤C,∀t∈[0,T]
for sufficiently small ϵ>0, where the constant C depends on ∥E0∥H2α,∥n0∥H1, ∥n1∥L2∩H˙-α, and T. In particular, this estimate implies that
(47)∥ΛαEϵ∥Lq≤C,∥nϵ∥Lq≤C,∀q∈[2,+∞].Proof.
Taking one derivative with respect tot on both sides of (15), one gets
(48)iEttϵ-Λ2αEtϵ=(ℬϵntϵ)Eϵ+(ℬϵnϵ)Etϵ.
Then multiplying this equation by Etϵ¯ and integrating the imaginary part, one gets
(49)12ddt∥Etϵ∥L22=Im∫ℝ[(ℬϵntϵ)Eϵ]Etϵ¯dx≤∥ntϵ∥L2∥Eϵ∥L∞∥Etϵ∥L2≤C∥ntϵ∥L2∥Etϵ∥L2≤C(∥ntϵ∥L22+∥Etϵ∥L22).
Next, we take the inner product of (16) with ntϵ and obtain
(50)12ddt(∥ntϵ∥L22+∥∇nϵ∥L22)=-∫ℝntϵΛ2αℬϵ|Eϵ|2dx≤∥ntϵ∥L2∥Λ2α|Eϵ|2∥L2≤C(∥ntϵ∥L22+∥Λ2αEϵ∥L22),
where we have used the following estimate:
(51)∥Λ2α(Eϵ·Eϵ¯)∥L2≤C(∥Eϵ∥L∞∥Λ2αEϵ¯∥L2+∥Λ2αEϵ∥L2∥Eϵ¯∥L∞)≤C∥Λ2αEϵ∥L2.
Note that (15) implies that
(52)∥Λ2αEϵ∥L2≤∥iEtϵ∥L2+∥(ℬϵnϵ)Eϵ∥L2≤∥Etϵ∥L2+∥ℬϵnϵ∥L2/(2α-1)∥Eϵ∥L1/(1-α)≤∥Etϵ∥L2+C∥Λ1-αnϵ∥L2≤∥Etϵ∥L2+C.
Now, it follows from (49)~(52) that
(53)ddt(∥Etϵ∥L22+∥ntϵ∥L22+∥∇nϵ∥L22)≤C(∥ntϵ∥L22+∥Etϵ∥L22)+C.
By Gronwall’s inequality, we have
(54)∥Etϵ∥L22+∥ntϵ∥L22+∥∇nϵ∥L22≤C,
which gives, by (52) and Proposition 7, that
(55)∥Eϵ∥H2α+∥nϵ∥H˙1⋂‍H˙1-α+∥ntϵ∥L2≤C,∀t∈[0,T].
Since
(56)ddt∥nϵ∥L22=2∫ℝnϵntϵdx≤2∥nϵ∥L2∥ntϵ∥L2≤C+∥nϵ∥L22,
Gronwall’s inequality gives ∥nϵ∥L2≤C for all t∈[0,T]. This estimate together with (55) yields the desired estimate.Proposition 9.
Under the same assumption as Proposition7, there holds
(57)∥Eϵ∥H3α+∥nϵ∥H1+α+∥ntϵ∥Hα≤C,∀t∈[0,T],
where the constant C depends on ∥E0∥H3α,∥n0∥H1+α, ∥n1∥Hα∩H˙-α, and T.Proof.
Applying the operatorΛα on both sides of (15) and (16), we get
(58)i(ΛαEϵ)t-Λ3αEϵ+Λα[(ℬϵnϵ)Eϵ]=0,(59)(Λαnϵ)tt+Λ2+αnϵ+ℬϵΛ3α|Eϵ|2=0.
Differentiating (58) with respect to t, we can get
(60)i(ΛαEϵ)tt-Λ3αEtϵ+Λα[(ℬϵntϵ)Eϵ]+Λα[(ℬϵnϵ)Etϵ]=0.
Multiplying ΛαEtϵ¯ on both sides of the above equation and integrating the imaginary part, we have
(61)12ddt∥ΛαEtϵ∥L22=-Im∫ℝΛα[(ℬϵntϵ)Eϵ]·ΛαEtϵ¯dx-Im∫ℝΛα[(ℬϵnϵ)Etϵ]·ΛαEtϵ¯dx.
By Cauchy-Schwarz inequality, Lemma 2, Proposition 8, and Sobolev interpolation inequality, it is easy to get
(62)ddt∥ΛαEtϵ∥L22≤C(∥ΛαEtϵ∥L22+∥Λαntϵ∥L22)+C.
Taking inner product of (59) with Λαntϵ and then using (45) and Lemma 2, one can obtain
(63)12ddt(∥Λαntϵ∥L22+∥Λ1+αnϵ∥L22)=-∫ℝℬϵΛ3α|Eϵ|2Λαntϵdx≤C(∥Λαnt∥L22+∥Λ3αEϵ∥L22).
Moreover, from (58), Lemma 2, and Proposition 8, we know that
(64)∥Λ3αEϵ∥L22≤∥ΛαEtϵ∥L22+∥Λα[(ℬϵnϵ)Eϵ]∥L22≤∥ΛαEtϵ∥L22+C.
From (62)~(64), we have
(65)ddt(∥ΛαEtϵ∥L22+∥Λαntϵ∥L22+∥Λ1+αnϵ∥L22)≤C(∥ΛαEtϵ∥L22+∥Λαntϵ∥L22)+C.
Then Gronwall’s inequality gives
(66)∥ΛαEtϵ∥L22+∥Λαntϵ∥L22+∥Λ1+αnϵ∥L22≤C.
This estimate and (64) yield the desired estimate.Applying the above procedure step by step, we finally obtain the following proposition.Proposition 10.
Under the same assumption as Proposition7, there holds
(67)∥Eϵ∥Hmα+∥nϵ∥H(m-2)α+1+∥ntϵ∥H(m-2)α≤C,∀t∈[0,T],
where the constant C depends on ∥E0∥Hmα, ∥n0∥H(m-2)α+1, ∥n1∥H(m-2)α∩H˙-α, and T.Now, we give the proof of Theorem1.Proof of Theorem1.
By Propositions7~10 and (15) and (16), there exists a subsequence of (Eϵ,nϵ) which converges weakly to (E,n)∈L∞(0,T;Hmα×H(m-2)α+1) (for simplicity, we use the same notation for the subsequence); that is,
(68)(Eϵ,nϵ)⟶(E,n)weaklystarinL∞×(0,T;Hmα×H(m-2)α+1),(Etϵ,ntϵ)⟶(Et,nt)weaklystarinL∞×(0,T;H(m-2)α×H(m-2)α).
Moreover, by Sobolev compact embedding theorem, we also have
(69)(Eϵ,nϵ)⟶(E,n)inL∞(0,T;Hlocm~α×Hloc(m~-2)α+1)
for all m~<m. Note that this convergence result implies that (Eϵ,nϵ) converges to (E,n) in the a.e. sense.
Since(70)∥(ℬϵnϵ)Eϵ∥L∞H(m-2)α+1≤C∥nϵ∥L∞H(m-2)α+1∥Eϵ∥L∞H(m-2)α+1≤C,∥|Eϵ|2∥L∞Hmα≤C∥Eϵ∥L∞Hmα∥Eϵ¯∥L∞Hmα≤C,
one has
(71)((ℬϵnϵ)Eϵ,|Eϵ|2)⟶(χ1,χ2)weaklystarinL∞×(0,T;H(m-2)α+1×Hmα).
Thus, by (69)~(71) and the uniqueness of weak limit, there are
(72)χ1=nE,χ2=|E|2.
Now, letting ϵ→0 in the regularized equations (15) and (16) and using the above convergence results as well as (43), we know that (E,n) is a solution to system (1)~(3). Moreover, with similar arguments as in [12], the solution is a continuous flow with respect to time; namely, (E,n,nt)∈C(ℝ+;Ym). The existence part of Theorem 1 is finished.
It remains for us to prove the uniqueness. Suppose that(E1,n1), (E2,n2)∈L∞(0,T;Hmα×H(m-2)α+1) are both the solutions of system (1)~(3). We set E:=E1-E2, n:=n1-n2 and then (E,n) satisfies the following equations:
(73)iEt-Λ2αE-n1E-nE2=0,(74)ntt-Δn=-Λ2α(|E1|2-|E2|2),(75)E(0)=0,n(0)=0.
From (73) and (74), we can obtain
(76)12ddt∥E∥L22≤∥E2∥L∞∥n∥L2∥E¯∥L2≤C∥n∥L2∥E∥L2,(77)12ddt(∥nt∥L22+∥∇nt∥L22)=-∫ℝΛ2α(E1E¯+EE2¯)ntdx≤C∥E∥H2α∥nt∥L2.
Taking one derivative of (73) with respect to t, we get
(78)iEtt-Λ2αEt-n1tE-n1Et-ntE2-nE2t=0.
Multiplying Et¯ on both sides of the above equation and integrating the imaginary part, we can obtain the following estimate (attention: E2t∈H(m-2)α and n1t∈H(m-2)α):
(79)12ddt∥Et∥L22≤C(∥n∥H1+∥nt∥L2+∥E∥H2α)∥Et∥L2.
Furthermore, it is easy to see that
(80)12ddt∥n∥L22≤∥n∥L2∥nt∥L2,
and by (73),
(81)∥Λ2αE∥L2≤∥Et∥L2+C(∥E∥L2+∥n∥L2).
Finally, it follows from (76)~(81) that
(82)ddt(∥E∥L22+∥Et∥L22+∥nt∥L22+∥n∥H12)≤C(∥E∥L22+∥Et∥L22+∥nt∥L22+∥n∥H12),
where the constant C depends on ∥Ei∥L∞(0,T;Hmα) and ∥ni∥L∞(0,T;H(m-2)α+1), i=1,2. By Gronwall’s inequality and the zero initial condition (75), we get E≡0 and n≡0. Thus the solution of system (1)~(3) is unique. This ends the proof of Theorem 1.



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*Source: 193589-2013-12-09.xml*