R E S E A R C H
Open Access
Cauchy problem of the generalized
Zakharov type system in
R
Shujun You
*and Xiaoqi Ning
*Correspondence: [email protected] Department of Mathematics, Huaihua University, Huaihua, 418008, China
Abstract
In this paper, we consider the initial value problem for a two-dimensional generalized Zakharov system with quantum effects. We prove the existence and uniqueness of global smooth solutions to the initial value problem in the Sobolev space through makinga prioriintegral estimates and the Galerkin method.
Keywords: Zakharov system; Cauchy problem; existence; uniqueness
1 Introduction
In the recent years, special interest has been devoted to quantum corrections to the Za-kharov equations for Langmuir waves in a plasma []. By use of a quantum fluid approach, the following modified Zakharov equations are obtained:
iEt+Exx–HExxxx=nE, ()
ntt–nxx+Hnxxxx=|E|xx, ()
whereHis the dimensionless quantum parameter given by the ratio of the ion plasmon and electron thermal energies. ForH= , this system was derived by Zakharov in [] to model a Langmuir wave in plasma. The Zakharov system attracted many scientists’ wide interest and attention [–].
In this paper, we deal with the following generalized Zakharov system:
iEt+E–HE–nE= , ()
ntt–n+Hn–|E|= , ()
where (E,n) : (x,t)∈R×Rand the initial data are taken to be
E|t==E(x), n|t==n(x), nt|t==n(x). ()
To study a smooth solution of the generalized Zakharov system, we transform it into the following form:
iEt+E–HE–nE= , ()
nt–ϕ= , ()
ϕt–n+Hn–|E|= , ()
with initial data
E|t==E(x), n|t==n(x), ϕ|t==ϕ(x). ()
Now we state the main results of the paper.
Theorem . Suppose that E(x)∈Hl+(R),n(x)∈Hl+(R),n(x)∈Hl(R),l≥.Then there exists a unique global smooth solution of the initial value problem()-().
E(x,t)∈L∞,T;Hl+R, Et(x,t)∈L∞
,T;HlR
n(x,t)∈L∞,T;Hl+R, nt(x,t)∈L∞
,T;HlR
ntt(x,t)∈L∞
,T;Hl–R.
The obtained results may be useful for better understanding the nonlinear coupling be-tween the ion-acoustic waves and the Langmuir waves in a two-dimensional space.
2 A priori estimates
Lemma . Suppose that E(x)∈L(R).Then,for the solution of problem()∼(),we have
EL(R)=E(x)
L(R).
Proof Taking the inner product of () andE, then taking the imaginary part, we have
Im(iEt,E) =Re(Et,E) =
d dtE
L, ImE–HE–nE,E= .
Hence, we get
d dtE
L= .
We thus get Lemma ..
Lemma . (Sobolev’s estimations) Assume that u∈Lq(Rn), Dmu∈Lr(Rn), ≤q,r≤
∞, ≤j≤m,we have the estimations DjuLp(Rn)≤CDmu
α
Lr(Rn)u–
α Lq(Rn),
where C is a positive constant, ≤mj ≤α≤,
p= j n+α
r– m
n
+ ( –α)
Lemma . Suppose that E(x)∈H(R),n(x)∈H(R),ϕ(x)∈H(R).Then we have
F(t) =F(),
where
F(t) =∇EL+HEL+
Rn|E| dx+
∇ϕ
L+
n
L+
H
∇n
L.
Proof Take the inner products of () and –Et. Since
Re(iEt, –Et) = , Re(E, –Et) =
d dt∇E
L, Re–HE, –Et
=H
d dtE
L, Re(–nE, –Et) =
R
n|E|tdx
=
d dt
Rn|E| dx–
Rnt|E| dx
=
d dt
Rn|E| dx–
Rnt
ϕt–n+Hn
dx = d dt
Rn|E| dx–
Rnt
ϕtdx+
d dtn
L+
H
d dt∇n
L = d dt
Rn|E| dx–
R
ϕϕtdx+
d dtn
L+
H
d dt∇n
L = d dt
Rn|E| dx+
d dt∇ϕ
L+
d dtn
L+
H
d dt∇n
L,
thus it follows that
d dt
∇EL+HEL+
R
n|E|dx+ ∇ϕ
L+
n
L+
H
∇n
L
= . ()
Letting
F(t) =∇EL+HEL+
R
n|E|dx+ ∇ϕ
L+
n
L+
H
∇n
L,
and noticing (), we obtain
F(t) =F().
Lemma . Suppose that E(x)∈H(R),n(x)∈H(R),ϕ(x)∈H(R).Then we have
sup ≤t≤T
EH+nH+ϕH
Proof By Holder’s inequality, Young’s inequality and Lemma ., it follows that¨
R
n|E|dx≤ nLE
L ≤
n
L+EL ≤
n
L+CELE
L ≤
n
L+
H
E
L+C.
From Lemma . we get
∇EL+
H
E
L+
∇ϕ
L+
n
L+
H
∇n
L≤F() +C.
Take the inner products of Eq. () andϕ. It follows that
ϕt–n+Hn–|E|,ϕ
= ()
since
(ϕt,ϕ) =
d dtϕ
L,
n+|E|,ϕ≤nL+E
L
ϕL≤
ϕ
L+C,
where
EL≤C∇ELEL≤C.
Hn,ϕ=H(n,ϕ) =H(n,nt) =
H
d dtn
L.
Hence, from Eq. () we get
d dt
ϕL+HnL
≤ ϕL+C.
Using Gronwall’s inequality, we obtain that
sup ≤t≤T
ϕL+HnL
≤C.
We thus get Lemma ..
Lemma . Suppose that E(x)∈H(R),n(x)∈H(R),ϕ(x)∈H(R).Then we have
sup ≤t≤T
EH+nH+ϕH+EtL+ntL+ϕtL
Proof Differentiating () with respect tot, then taking the inner products of the resulting equation andEt, we have
iEtt+Et–HEt– (nE)t,Et
= ()
since
Im(iEtt,Et) =
d dtEt
L, Im
Et–HEt–nEt,Et
= ,
Im(–ntE,Et)≤CEL∞ntLEtL≤CEt
L+ntL
.
By Lemma ., it follows that
EL∞≤CE
LE
L;
thus from Eq. () we get
d dtEt
L≤C
EtL+ntL
. ()
Differentiating Eq. () with respect tot, then taking the inner products of the resulting equation andnt, we have
(ntt–ϕt,nt) = ()
since
(ntt,nt) =
d dtnt
L,
(–ϕt,nt) =
–n+Hn–|E|,nt
=
d dt∇n
L+
H
d dtn
L–
|E|,nt
.
Noting that
|E|,nt≤CEL∞ELntL≤Cnt
L+
,
from Eq. () we get
d dt
ntL+∇nL+HnL
≤CntL+
. ()
From Eq. () and () we get
d dt
EtL+ntL+∇nL+HnL
≤CEtL+ntL+
.
By Gronwall’s inequality, it follows that
sup ≤t≤T
EtL+ntL+∇nL+HnL
Take the inner products of Eq. () andϕt. It follows that
ϕt–n+Hn–|E|,ϕt
= ()
since
(ϕt,ϕt) =ϕtL,
–n+Hn–|E|,ϕt
≤nL+HnL+E
L
ϕtL ≤C+
ϕt
L.
From Eq. () we get
ϕtL≤C.
Take the inner products of Eq. () andE. It follows that
iEt+E–HE–nE,E
= ()
since
(iEt–nE,E)≤
EtL+EL∞nL
EL≤C,
E–HE,E=EL+H∇E
L.
From Eq. () we get
∇E
L ≤C.
From () we obtain
HEL≤ EtL+EL+nEL ≤C,
where
nEL≤ nLEL≤C∇n
Ln
L∇E
LE
L≤C.
From () we obtain
ϕL=ntL≤C.
We thus get Lemma ..
Lemma . Suppose that f,f∈Hs(),⊆Rn.Then we have Ds(f·f)L≤Cs
fLDsf
L+DsfLfL
,
Lemma . Suppose that E(x)∈Hm+(R),n(x)∈Hm+(R),ϕ(x)∈Hm+(R),m≥. Then we have
sup ≤t≤T
E(x,t)Hm++n(x,t)Hm++ϕ(x,t)Hm+
≤C
sup ≤t≤T
Et(x,t)Hm+nt(x,t)Hm+ϕtHm≤C.
Proof Lemma . is true whenm= (Lemma .). Suppose that Lemma . is true when
m=k, (k≥). Take the inner products of () and (–)k+k+ϕ. It follows that
ϕt–n+Hn–|E|, (–)k+k+ϕ
= ()
since
ϕt, (–)k+k+ϕ
=
d dt∇
k+ϕ
L,
–n, (–)k+k+ϕ=–n, (–)k+k+nt
=
d dt∇
k+n
L,
Hn, (–)k+k+ϕ=Hn, (–)k+k+nt
=H
d dt∇
k+n
L, –|E|, (–)k+k+ϕ=∇k+|E|,∇k+ϕ≤∇k+|E|L∇k+ϕL
≤CEL∇k+E
L∇k+ϕL ≤C∇k+ϕ
L+
,
thus from Eq. () it follows that
d dt∇
k+ϕ
L+∇k+n
L+H∇k+n
L
≤C∇k+ϕ
L+
. ()
By using Gronwall’s inequality, we have
sup ≤t≤T
∇k+ϕ
L+∇k+n
L+∇k+n
L
≤C.
From () and () we get
∇k+n
tL=∇k+ϕL≤C, ∇k+ϕ
tL≤C∇k+nL+∇k+nL+EL∇k+E
L
≤C.
Differentiating () with respect tot, then taking the inner products of the resulting equa-tion and (–)k+k+E
t, we obtain
iEtt+Et–HEt– (nE)t, (–)k+k+Et
= . ()
Since
ImiEtt, (–)k+k+Et
=
d dt∇
k+E
t
ImEt–HEt, (–)k+k+Et
= ,
Im
–ntE, (–)k+k+Et≤∇k+(ntE),∇k+Et
≤C∇k+ntLEL+∇k+E
LntL∇k+Et
L ≤C∇k+Et
L+
,
Im
–nEt, (–)k+k+Et≤∇k+(nEt),∇k+Et
≤C∇k+nLEtL+∇k+Et
LnL∇k+Et
L ≤C∇k+Et
L+
,
thus from Eq. () we get
d dt∇
k+E
t
L≤C∇k+Et
L+
. ()
By using Gronwall’s inequality, we get
sup ≤t≤T
∇k+E
t
L≤C.
From () we obtain
∇k+E
L≤C∇k+EtL+∇k+EL+∇k+nLEL+nL∇k+E
L
≤C.
Hence
sup ≤t≤T
E(x,t)Hk++n(x,t)Hk++ϕ(x,t)Hk+
≤C,
sup ≤t≤T
Et(x,t)Hk++nt(x,t)Hk++ϕtHk+
≤C.
This means Lemma . is true whenm=k+ . Thus Lemma . is proved completely.
3 Existence and uniqueness of solution
Now, with these lemmas, we are able to prove Theorem .. First we obtain the existence and uniqueness of the global generalized solution of problem ()-().
Definition . The functions E ∈L∞(,T;H)∩ W,∞(,T;L), n ∈L∞(,T;H)∩ W,∞(,T;L) andϕ∈L∞(,T;H)∩W,∞(,T;L) are called a generalized solution of
problem ()-() if for anyω∈Lthey satisfy the integral equality
(iEjt,ω) + (Ej,ω) =H
Ej,ω
+ (nEj,ω), j= , , . . . ,N,
(nt,ω) = (ϕ,ω),
(ϕt,ω) +H(n,ω) = (n,ω) +
|E|,ω
with initial data
Now, one can estimate the following theorem.
Theorem . Suppose that E(x)∈Hl+(R),n(x)∈Hl+(R),ϕ(x)∈Hl+(R), l≥. Then there exists a global smooth solution of the initial value problem()-().
E(x,t)∈L∞,T;Hl+R, Et(x,t)∈L∞
,T;HlR,
n(x,t)∈L∞,T;Hl+R, nt(x,t)∈L∞
,T;HlR,
ϕ(x,t)∈L∞,T;Hl+R, ϕt(x,t)∈L∞
,T;HlR.
Proof By using the Galerkin method, choose the basic periodic functions{ωκ(x)}as
fol-lows:
–ωκ(x) =λκωκ(x), ωκ(x)∈H(),κ= , . . . ,l.
The approximate solution of problem ()-() can be written as
El(x,t) =
l
κ=
ακl(t)ωκ(x), nl(x,t) = l
κ=
βκl(t)ωκ(x),
ϕl(x,t) =
l
κ=
γκl(t)ωκ(x),
where
El(x,t) =El,El, . . . ,ENl, ακl(t) =αlκ,ακl, . . . ,αlκN.
is a two-dimensional cube with Din each direction, that is,={x= (x,x)||xj| ≤
D,j= , }. According to Galerkin’s method, these undetermined coefficientsαl
κ(t),βκl(t)
andγκl(t) need to satisfy the following initial value problem of the system of ordinary dif-ferential equations:
iEljt,ωκ
+Elj,ωκ
=HEjl,ωκ
+nlEjl,ωκ
, j= , , . . . ,N, ()
nlt,ωκ
=ϕl,ωκ
, ()
ϕtl,ωκ
+Hnl,ωκ
=nl,ωκ
+El,ωκ
()
with initial data
El(x, ) =El(x), nl(x, ) =nl(x), ϕl(x, ) =ϕl(x), () where
El(x) H
−→E(x), nl(x)
H
−→n(x), ϕl(x)
H
−→ϕ(x), l→ ∞.
Similarly to the proof of Lemmas .-., for the solutionEl(x,t),nl(x,t),ϕl(x,t) of
prob-lem ()-(), we can establish the following estimates:
sup ≤t≤T
El
H+nlH+ϕlH+EtlL+nltL+ϕtlL
where the constantsCare independent oflandD. By compact argument, some subse-quence of (El,nl,ϕl), also labeled byl, has a weak limit (E,n,ϕ). More precisely
El(x,t)→E(x,t) inL∞,T;Hweakly star,
nl(x,t)→n(x,t) inL∞,T;Hweakly star,
ϕl(x,t)→ϕ(x,t) inL∞,T;Hweakly star and
Elt→Et inL∞
,T;Lweakly star,
nlt→nt inL∞
,T;Lweakly star,
ϕtl→ϕt inL∞
,T;Lweakly star.
By using Guo and Shen’s method [], one can prove the existence of a local solution for the periodic initial problem ()-(). Similarly to Zhou and Guo’s proof [], lettingD→ ∞, the existence of a local solution for the initial value problem ()-() can be obtained. By the continuation extension principle, from the conditions of the theorem anda priori
estimates in Section , we can get the existence of a global generalized solution for problem ()-(). By Lemma . and the Sobolev imbedding theorem, Theorem . is proved.
Next, we prove the uniqueness of a solution for problem ()-().
Theorem . Suppose that E(x)∈Hl+(R),n(x)∈Hl+(R),ϕ(x)∈Hl+(R), l≥. Then the global solution of the initial value problem()-()is unique.
Proof Suppose that there are two solutionsE,n,ϕandE,n,ϕ. Let
E=E–E, n=n–n, ϕ=ϕ–ϕ.
From ()-() we get
iEt+E–HE–nE+nE= , ()
nt–ϕ= , ()
ϕt–n+Hn–|E|+|E|= , ()
with initial data
E|t== , n|t== , ϕ|t== , x∈R. ()
Take the inner product of () andE. Since
Im(iEt,E) =
d dtE
Im(nE–nE,E)≤(nE+nE,E)
≤CEL∞nL+nL∞EL
EL ≤CnL+EL
,
thus we obtain
d dtE
L≤C
nL+EL
. ()
Take the inner product of () andϕ. Since
(ϕt,ϕ) =
d dtϕ
L, (–n,ϕ)≤C
nL+ϕL
,
Hn,ϕ=Hn,ϕ=Hn,nt
=H
d dtn
L, –|E|+|E|,ϕ=(E–E)E+E(E–E),ϕ
≤C(EL∞EL+EL∞EL)ϕL ≤CEL+ϕL
,
thus we get
d dt
ϕL+HnL
≤CEL+nL+ϕL
. ()
Hence from () and () we get
d dt
EL+nL+ϕL
≤CEL+nL+ϕL
. ()
By using Gronwall’s inequality and noticing (), we arrive at
E≡, n≡, ϕ≡.
Theorem . is proved. This completes the proof of Theorem ..
4 Results and discussion
One can regard ()-() as the Langmuir turbulence parameterized by H and study the asymptotic behavior of systems ()-() whenHgoes to zero.
5 Conclusions
Bya prioriintegral estimates and the Galerkin method, we have the following conclusion. Suppose thatE(x)∈Hl+(R),n(x)∈Hl+(R),n(x)∈Hl(R),l≥. Then there exists
a unique global smooth solution of the initial value problem ()-().
E(x,t)∈L∞,T;Hl+R, Et(x,t)∈L∞
,T;HlR
n(x,t)∈L∞,T;Hl+R, nt(x,t)∈L∞
,T;HlR
ntt(x,t)∈L∞
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
SY carried out the existence studies and drafted the manuscript. XN carried out the uniqueness of the solution and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the National Natural Science Foundation of China (Grant No. 11501232), Research Foundation of Education Bureau of Hunan Province (Grant Nos. 15B185 and 16C1272) and Scientific Research Fund of Huaihua University (Grant No. HHUY2015-05) for the support.
Received: 29 September 2016 Accepted: 25 January 2017 References
1. Garcia, LG, Haas, F, de Oliveira, LPL, Goedert, J: Modified Zakharov equations for plasmas with a quantum correction. Phys. Plasmas12, 012302 (2005)
2. Zakharov, VE: Collapse of Langmuir waves. Sov. Phys. JETP35, 908-914 (1972)
3. Holmer, J: Local ill-posedness of the 1D Zakharov system. Electron. J. Differ. Equ.2007, 24 (2007)
4. Pecher, H: An improved local well-posedness result for the one-dimensional Zakharov system. J. Math. Anal. Appl. 342, 1440-1454 (2008)
5. Guo, B, Zhang, J, Pu, X: On the existence and uniqueness of smooth solution for a generalized Zakharov equation. J. Math. Anal. Appl.365, 238-253 (2010)
6. Linares, F, Matheus, C: Well posedness for the 1D Zakharov-Rubenchik system. Adv. Differ. Equ.2009, 14 (2009) 7. Linares, F, Saut, JC: The Cauchy problem for the 3D Zakharov-Kuznetsov equation. Discrete Contin. Dyn. Syst.24,
547-565 (2009)
8. You, S: The posedness of the periodic initial value problem for generalized Zakharov equations. Nonlinear Anal. TMA 71, 3571-3584 (2009)
9. You, S, Guo, B, Ning, X: Initial boundary value problem for a generalized Zakharov equations. Appl. Math.57, 581-599 (2012)
10. Masmoudi, N, Nakanishi, K: Energy convergence for singular limits of Zakharov type systems. Invent. Math.172, 535-583 (2008)
11. Seadawy, AR: Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl.67, 172-180 (2014)
12. Ismail, A: The first integral method for constructing exact and explicit solutions to nonlinear evolution equations. Math. Methods Appl. Sci.35, 716-722 (2012)
13. Morris, R, Kara, AH, Biswas, A: Soliton solution and conservation laws of the Zakharov equation in plasmas with power law nonlinearity. Nonlinear Anal., Model. Control18, 153-159 (2013)
14. You, S, Ning, X: On global smooth solution for generalized Zakharov equations. Comput. Math. Appl.72, 64-75 (2016) 15. Guo, B, Shen, L: The existence and uniqueness of the classical solution and the periodic initial value problem for
Zakharov equation. Acta Math. Appl. Sin.3, 310-324 (1982)
