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The Ginzburg-Landau equation with nonzero Neumann boundary data
Nonlinear Anrrlysis, Theory, Methods & Applrcutronr, Vol. 23, No. 3, PP. 399-404, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .oO
Pergamon
THE GINZBURG-LANDAU EQUATION WITH NONZERO NEUMANN BOUNDARY DATA CHARLES Department
of Mathematics,
(Received Key words and phrases:
Wellesley
Bu
College,
Wellesley,
11 May 1993; received for publication
Ginzburg-Landau
equation,
nonzero
MA 02181, U.S.A. 14 July 1993)
Neumann
data,
Galerkin’s
method.
I. PRELIMINARIES
THE Ginzburg-Landau equation (GL) U, = (v + ior) AU - (K + i/l) 1I/ [*U + y Uis occasionally claimed, on the basis of heuristic arguments, to describe the amplitude evolution of instability waves in fluid dynamics [l, 21. The Cauchy problem for the 1D GL with x E Q = [0, L], U(x, 0) E H’(a) and either zero Dirichlet or zero Neumann boundary data is shown to be well-posed via classical techniques of nonlinear parabolic equations [3]. When the Dirichlet boundary data are nonzero, a weak solution is available via the Galerkin-Vishik method [4]. For the GL with D spatial dimensions posed on a bounded domain 52 where D = 1,2, it is shown in [5] that a finite-dimensional attractor captures all the solutions by studying the longtime behavior of solutions. When fz = R*, several results on the Cauchy problem of the GL equation are available [6]. For fi = [0, co) one imposes initial and boundary conditions such as U(x, 0) = U&x), U(0, t) = Q(t) or r/,(0, t) = Q(t) for x 1 0, t 2 0. It is called the forced partial differential equation (see [7]). For example, the forced nonlinear Schodinger equation (NLS) can be treated via inverse scattering (e.g. [8, 91) and semigroup theory [lo, I 11. It is shown that, under certain conditions on initial and boundary data, the forced Kortewegde Vries equation (KdV) is well-posed and there exists a unique smooth solution 112, 131. Existence of a unique local classic solution to the forced Ginzburg-Landau equation in u, = (v + icX)U,, - (K + @)I Ul*U + yU on the semiline with U(0, t) given is investigated [14] while global existence is established if I/31 5 6~ or CY~> 0. In this paper we shall study the GL equation when x E n = [0, L] with the Neumann boundary data U,(O, t) = P(t), U,(L, t) = R(t) and initial data U(x, 0) = U,(x). Under the assumption that IpI < V%K and certain conditions on the initial-boundary data, we show that there exists a unique weak solution. The forced GL is posed as follows (a, a, y real, v, K > 0)
u, = XEc2
= [O,L],
t E LO,Tl,
(v + icY)U,, - (K +
WC 0) = U,(x),
ij?)lul*u +
(1.1)
)Ju
U,(O,f) = P(t),
U,(L,
t)
= R(t). (1.2)
Here P(t), R(I), U,(x) are complex functions. We shall, throughout P(f), R(t) E C2[0, m), U,(x) E H’(Q), u;(o) = P(O), u;(L) = R(0). 399
this paper,
assume
that
C.
400
Bu
First we write
- o2 $os~R(t)- (x 2L
U(x, t) = u(x, t) + g(x, t) = u(x, t) Then
P(t)
.
(1.3)
(1.1) becomes f4, = (v + i(Y)& + (v + iol)g,, - g, + yg + yu
=
and (1.2) is equivalent
-
(K
+
(v
+
icY)u,
i~)(li#U -
(K
+
gU2
+
+
iB)lU12U
2glU12 +
+
c,,
+
21g12ti
+
1g12g)
c,(U)
+
c2(U2)
(1.4)
to
x E Q = [O, L],
U(X, 0) = u&x) = U,(x) - g(x, 0) E H’(Q)
t E [O, Tl,
(1.5) U,(O, t) = u,(L, t) = 0. Next we use the appropriate function
following satisfying
transformation (here w(*, t) E C”(H2(sZ)) fl C’(L2(i2)) ~(0, I) = w(L, t) = 0)
is an
x
u(x, t> = u(x, t) + f(x, t> = u(x, t> + and this substitution
0
w(x, 0 dx
(l-6)
in (1.4) yields u, = (v + icY)u,, -
(K
+
i/3)lv12u
G2(u2) + G,(v) + Go
+
(1.7)
X v,(O,
t) =
u,(L,t)
=
0,
u(x,O)
=
l&(x) -
w(x, O)dx
(1.8)
s0 where G2(v2) = G,(u) Go
=
-g,
-
f,
+
(v
Now it is time to determine
-(K
=
-(K
+
W(&x
+ +
@)(2(g
+_f)juj2
$)((g
+ f)2i?
+ f,>
-
the function
(K
+
+
+
(g
+f)u2)
)g + f12u)
$)k
+ fi2k
+
(1.9)
(1.10)
yu
+ f)
W, which plays an important
+
Yk
+ fh
(1.11)
role in our scheme.
LEMMA 1.1. Assume that IpI < d?K. Then there exists w E C’(L2(sZ)) fI C”(H2(sZ)), ~(0, t) = w(L, t) = Osuch that Go E L’(H’(sZ)), 13,G,(0, t) = d,G,(L, t) = Oand a,G, E L2(Q). Proof.
Differentiate
(1.11) with respect
to x and set a,G,
= 0
(1.12)
Ginzburg-Landau
This is a problem
of a differential-integral
equation
W, = (v + iol)W,, - (k + @)(21f12w + c6fw + c5 Refw
401
equation
with the initial-boundary
values
+ f2W)
+ c, Re w + c3 Im w + c2 Refg
+ c,f + co
(1.13) (1.14)
w(0, t) = w(L, t) = w(x, 0) = 0.
Here all cis are C2 functions with values in w2(0). Define A = (v + i(y)@, D(A) = H2(a) tl H,‘(a), then D(A) and H,‘(Q) are dense in L2(sZ) and A is closed. For w E H;(Q) and A > 0 one has - A)w, w)l 1 Ia4l;
I(0
+ vIIw%
+ 4w’ll;l
2 dlwl?f~(n,.
Thus, A - A maps D(A) l-l onto L2(sZ), 111 - AI] 5 l/A for A > 0 and A generates a continuous contraction semigroup. To show that c6fw + c5 Refw + c, Re w + cj Im w + c2 Refg + c,f is locally Lipschitz continuous in the D(A) norm, we notice that (1.15) Ilf’llz
= IlWII2>
Ilf “112= Ilwxll2.
Clearly c6fw + c5 Re fw + c, Re w + c3 Im w E D(A) (1.13) one finds = -(K
c2 Refg
+
ip)4g, Refg,
C,f=
since -(K
(1.16) w E D(A).
From
(1.12)
$)2g&f.
+
and (1.17)
If we go back to (1.3), we can verify that g(0, t)f(O, t) = g(L, t)f(L, t) = 0 since g(L, t) = 0 and f(0, t) = 0 from (1.6). Therefore, c2 Refi + c,f E D(A). Finally it comes to our attention that atcO E C1(H2(sZ)). Now by standard arguments as described in [ll, 15, 161, there is a unique local solution w E C ‘(0, TM; L2(sZ)) fl C’(O, TM; D(A)) to (1.13) along with the initialboundary condition (1.14). But we still need TM = m, i.e. the solution is global. To accomplish that, note
d,llwll: =
n
W,Wdx 12
(wt% + WWJ dx = 2 Re
= -2vllw,112
-
2 Re(K
+
ip)
(21f121 w12 + f2ii12) dx - 2 Re(K + @)
50 X
(c6fw + c,Refw
+ c,Rew
+ c,Imw
+ c,Ref
+ c,f + c,)Wdx.
(1.18)
.i a
Since IpI < OK, -2Re(K
there exists m > 0 such that (here fW = a + ib)
(21f121w12 + f2ti2)dx
+ ip)
= -2
n
(2KG2
+ 2Kb2
+ Ka2 -
Kb2 + 2pab) ti
(3Ka2
+ Kb2 + 2pab) dx
n = -2 5
sn
-m
(a2 + b2)dx
n
= -m
If 121w12d_x. (1.19) sD
C. Bu
402
From
(l.lS),
(1.19)
&Ilwll: 5 -2vlIwXll;
5 4lw,ll:
+
~ll~,Il,Il4,
- df4:
+ dw,llt + ;
+ NI-~211;+ 4l.f4:
+ m’ll.flG+ fillwll;
IIWII:- f4lf4f
+ ; Il.f4: + g Ilwll; + ma-1121142 + ~‘llfll: + fi’lI4 I M, + Mllwll;.
(1.20)
By Gronwall’s lemma it is evident from (1.20) that IIwI(~ is bounded on [0, T] for all on [0, T] for all T > 0. Hence, T > 0. Similarly we can prove that IIw,ll$ is bounded estimate [17] indicates that by (1.15) and (1.16) so is IlfI]$~~c,. The Gagliardo-Nirenberg IIwIL 5 ~llwxl11’2114Y2 so II4L andllfllmare bounded on [0, T] for all T > 0. Now use the fact that the semigroup is continuously contractive in L’(Q) and by a standard estimate on w, (cf. [16]) one finds I(w~~(~< m for any t > 0. Thus, IIwIIDcAj < co by (1.13). Consequently w E C’(L2(Q)) n C”(H2(Q)) is the unique global solution to (1.13) with initial-boundary data (1.14). This effectively shows that a,G, = 0. From (1.1 l), Go E L2(Q), a,G, E L*(n) is obvious. n 2. THE
To show the existence (cf. [3, 181). Let
WEAK
and uniqueness
(a,~,,
EXISTENCE
AND
of the weak solution,
UNIQUENESS
we use the Galerkin-Vishik
-AWN = ~jWj,
a,mj(0)= d,mj(L)= 0,
Wj E H2(n),
then clearly approximate
SOLUTION:
method
j = 1,2,...
(2.1)
Wj E I/ = (v: u E H’(Q), v,(O) = u,(L)= 0). Suppose U, = Cygj,(t)wj solution where (gj,]y’ , are determined by the conditions Wj) +
(V +
i~)(ax~,,d,Wj)
+
(K
+
$)(~Iu,~~u,,
is an
Wj) = (Go + G, + G,, Wj) (2.2)
Vom + ug = U(X, 0) E v.
&z(O) = Uom E [WI, ..*, WA, Multiply
(2.2) by gjm (t) and add them for j = 1,2, . . . , m. Now one has (a turn, u,) + (v + iol)]l%v,ll:
From
+ (K + $)(21~,1~~,,
u,) = (Go + G, +
G,,
U,).
(2.3)
(1.9)-( 1.11) it is clear that IG,(QJ/
IGo1 5 aoY
5 a,lurnl,
1’32&A
5
(2.4)
d4n12.
Take the real part of (2.3)
+a,Il~,,$ I -ulla,u,,$ - Re(K + $)(21u,12u,, u,) + WG, + G, + G,, u,) I
-Vlla,U,II;
-
2K
b,i4~ I .n
+
/(Go s 61
+
G,
+
G2hn1
b.
(2.5)
equation
403
-41&hnll~- 34bmll:+ 4&nll: + d47zll: + 4hnl12 5 c’.
(2.6)
Ginzburg-Landau
Putting
(2.4) in (2.5) one has + U%X
Thus, AWj
5
by Gronwall’s in (2.2) to get
(a f~,,,, Au,)
lemma
one has IIv,ll:
+ (V + iol)(Au,,
I M on [0, T]. By (2.1), one could replace
Au,)
AU,) + (K + @)(21~,1$,z,
= (G, + G, + G,, Au,).
Wj by
(2.7)
If we take the real part of (2.7) then
+%unlll; +vllAu,llt 5
6 + IPI)Ibmllilbuml12 + lb%+ cl + G21i21buml12
I
;
IlAu,ll;
+
(Ic;F’)’IIu,,II:+ ;
IlAu,,ll: +
&IlGo+ G, + G&. (2.8)
Now use IIu,,ll~ I A4 and the following
Gagliardo-Nirenberg
estimates
(cf. [17]) (2.9)
Il~,llZ 5 ~~lla,~,llfII~,11”2 5 ~‘~2lla,& IIu,ll: Then
5 ~llax~,llzll~,ll:
(2.10)
5 ~~3%0,112.
(2.8) becomes
a,Ila,U,1l: 5 m,U,llt
+
;
II&
+
G,
+
4;
5
~kN,ll:
+
h%~,,~~2
+
c.
(2.11)
This certainly implies that IIa, u, II2 is bounded on any [0, T]. Together with the fact that llum]]2 is also bounded we conclude that u, is bounded in L-(0, T, V) for any T. Now differentiate (2.2) with respect to t Wj)
(afu,, =
One could replace
-(K
(v + h)(a, a, u, , a,
+ +
i~)(a,(al~,l2~,),
Wj)
+
(atGo + dtGl + d,G,,
Wj).
(2.12)
wj by a,~,,, in (2.12) and take the real part
+a,lla,~,ll$+ ~ll+w& This together
Wj)
with the boundedness
5 c~(IIu~II; + lb + G, + ~2112~lla~hA12. (2.13) of IIu,II~~(~) and IIG, + G, + G2]12 implies
a,Ila,u,ll:
that
5 G, + Glla,~,II:.
By [3], a,~,,, is bounded in L”(0, T; L2(sZ)) and we can extract u, from u, so that u, + u weakly in L*(O, T, V); a,u@ + x weakly in L*(O, T, L*(n)), a,u = x and u solves
(&u, h) + (v + ia)(a,u, r3,h) + (K + $)(lu12u, h) = (G, + G, + G,, h) for all h E I’. We deduce (a+,
from (1.5), (1.6) and (2.14) that u = u +
h) + (V + icu)(a,24, a,h)
(2.14)
f satisfies
+ (K + ij3)(1241224,h) = (co + c,
+ c,,
U)
(2.15)
404
C. Bu
for all h E Vwith u E L-(0, T; H’(Q), &u E L”(0, Z L’(Q)), u(x, 0) = uO(x) = UO(x) - g(x, 0), ~~(0, t) = u,(L, t) = 1. And this implies that U satisfies (&U,h)
+ (v + i(Y)(a,u,
&h) + (K + @)(Iu~*u,h)
= y(U,h)
(2.16)
for all h E V with U E H’(a), U(x, 0) = U,(x), U,(O, t) = P(t), U,(L, t) = R(t). The uniqueness proof is similar to that shown in [4]. We have, therefore, completed the proof of the following theorem. THEOREM 2.1 (weak solution). Let Ipi < a~. For the forced GL equation Neumann boundary data and initial data (1.2) there exists a unique solution Remark 2.2 (regularity). strong solution.
We indicate
here that if uO(x) E Hz then the solution
(1.1) with nonzero U satisfying (2.16). to (2.1) is a global
Remark 2.3 (mixed boundary data). There is a more general boundary condition as far as forcing is concerned. One could assume that u,(O, t) + au(0, t) and uX(L, 0) + bu(L, 0) be given (in C2, of course): here a, b are real numbers. For the nonlinear Schr6dinger equation on the semiline with u,(O, f) + au(0, t) given, a global solution is available provided that the initial-boundary data are smooth [ 151. We expect that with the mixed boundary data, we will be able, through a slightly more complicated scheme, to obtain the unique weak solution via Galerkin’s method as described here. REFERENCES 1. LANGE C. & NEWELL A., A stability
criterion
for the envelope
equation,
SIAM J. uppl. Math. 27, 441-456
(1974).
2. NEWELL A. & WHITEHEAD J., Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279-304 (1969). 3. LIONS J., Quelques mPthodes de r&solution des probldmes aux limites non lineaires. Dunod, Paris (1969). 4. Bu C., An Initial-boundary Value Problem for the Ginzburg-Landau Equation. Applied Mathematics Letters (to appear).
5. GHIDAGLIA J. & HERON R., Dimension of the attractors
associated
to the Ginzburg-Landau
equation,
Physica 28D,
282-304 (1987).
6. I. 8. 9.
Bu C., On the Cauchy problem for the 1 + 2 Ginzburg-Landau equation (preprint). KAUP D., Wave Phenomena, pp. 163-174. North-Holland, Amsterdam (1984). CARROLL R., On the forced nonlinear Schriidinger equation, Japan J. appl. Math. 7, 321-344 (1990). FOKAS A., An initial-boundary value problem for the nonlinear Schriidinger equation, Physica 35D, 167-185 (1989). of the forced nonlinear SchrGdinger equation, Applicable Analysis 46, 219-239 (1992). 10. Bu C., On well-posedness Schr6dinger equation (NLS) using PDE techniques, 11. CARROLL R. & Bu C., Solutions of the forced nonlinear Applicable Analysis 41, 33-51 (1991). Vries equation in a quarter plane, continuous dependence results, DifJ 12. BONA J. & WINTHER R., The Korteweg-de
Integral Eqns 2, 228-250 (1989). 13. BONA _I. & WINTHER R., The Korteweg-de
Vries equation:
posed in a quarter
plane,
SIAM J. math. Analysis 6,
1056-1106 (1983). 14. 15. 16. 17. 18.
equation: posed in a quarter plane, J. math. Analysis Applic. (to appear). value problem for the nonlinear SchrGdinger equation, Nonlinear Analysis (to appear). PAZY A., Semigroups of Linear operators and Applications to PDE. Springer, New York (1983). NIRENBURG L., On elliptic partial differential equations, Anna/i Scu. norm. sup. Pisa 13, 115-162 (1959). CARROLL R., Mathematical Physics. North-Holland, Amsterdam (1988). Bu C., The G&burg-Landau Bu C., An initial-boundary
Pergamon
THE GINZBURG-LANDAU EQUATION WITH NONZERO NEUMANN BOUNDARY DATA CHARLES Department
of Mathematics,
(Received Key words and phrases:
Wellesley
Bu
College,
Wellesley,
11 May 1993; received for publication
Ginzburg-Landau
equation,
nonzero
MA 02181, U.S.A. 14 July 1993)
Neumann
data,
Galerkin’s
method.
I. PRELIMINARIES
THE Ginzburg-Landau equation (GL) U, = (v + ior) AU - (K + i/l) 1I/ [*U + y Uis occasionally claimed, on the basis of heuristic arguments, to describe the amplitude evolution of instability waves in fluid dynamics [l, 21. The Cauchy problem for the 1D GL with x E Q = [0, L], U(x, 0) E H’(a) and either zero Dirichlet or zero Neumann boundary data is shown to be well-posed via classical techniques of nonlinear parabolic equations [3]. When the Dirichlet boundary data are nonzero, a weak solution is available via the Galerkin-Vishik method [4]. For the GL with D spatial dimensions posed on a bounded domain 52 where D = 1,2, it is shown in [5] that a finite-dimensional attractor captures all the solutions by studying the longtime behavior of solutions. When fz = R*, several results on the Cauchy problem of the GL equation are available [6]. For fi = [0, co) one imposes initial and boundary conditions such as U(x, 0) = U&x), U(0, t) = Q(t) or r/,(0, t) = Q(t) for x 1 0, t 2 0. It is called the forced partial differential equation (see [7]). For example, the forced nonlinear Schodinger equation (NLS) can be treated via inverse scattering (e.g. [8, 91) and semigroup theory [lo, I 11. It is shown that, under certain conditions on initial and boundary data, the forced Kortewegde Vries equation (KdV) is well-posed and there exists a unique smooth solution 112, 131. Existence of a unique local classic solution to the forced Ginzburg-Landau equation in u, = (v + icX)U,, - (K + @)I Ul*U + yU on the semiline with U(0, t) given is investigated [14] while global existence is established if I/31 5 6~ or CY~> 0. In this paper we shall study the GL equation when x E n = [0, L] with the Neumann boundary data U,(O, t) = P(t), U,(L, t) = R(t) and initial data U(x, 0) = U,(x). Under the assumption that IpI < V%K and certain conditions on the initial-boundary data, we show that there exists a unique weak solution. The forced GL is posed as follows (a, a, y real, v, K > 0)
u, = XEc2
= [O,L],
t E LO,Tl,
(v + icY)U,, - (K +
WC 0) = U,(x),
ij?)lul*u +
(1.1)
)Ju
U,(O,f) = P(t),
U,(L,
t)
= R(t). (1.2)
Here P(t), R(I), U,(x) are complex functions. We shall, throughout P(f), R(t) E C2[0, m), U,(x) E H’(Q), u;(o) = P(O), u;(L) = R(0). 399
this paper,
assume
that
C.
400
Bu
First we write
- o2 $os~R(t)- (x 2L
U(x, t) = u(x, t) + g(x, t) = u(x, t) Then
P(t)
.
(1.3)
(1.1) becomes f4, = (v + i(Y)& + (v + iol)g,, - g, + yg + yu
=
and (1.2) is equivalent
-
(K
+
(v
+
icY)u,
i~)(li#U -
(K
+
gU2
+
+
iB)lU12U
2glU12 +
+
c,,
+
21g12ti
+
1g12g)
c,(U)
+
c2(U2)
(1.4)
to
x E Q = [O, L],
U(X, 0) = u&x) = U,(x) - g(x, 0) E H’(Q)
t E [O, Tl,
(1.5) U,(O, t) = u,(L, t) = 0. Next we use the appropriate function
following satisfying
transformation (here w(*, t) E C”(H2(sZ)) fl C’(L2(i2)) ~(0, I) = w(L, t) = 0)
is an
x
u(x, t> = u(x, t) + f(x, t> = u(x, t> + and this substitution
0
w(x, 0 dx
(l-6)
in (1.4) yields u, = (v + icY)u,, -
(K
+
i/3)lv12u
G2(u2) + G,(v) + Go
+
(1.7)
X v,(O,
t) =
u,(L,t)
=
0,
u(x,O)
=
l&(x) -
w(x, O)dx
(1.8)
s0 where G2(v2) = G,(u) Go
=
-g,
-
f,
+
(v
Now it is time to determine
-(K
=
-(K
+
W(&x
+ +
@)(2(g
+_f)juj2
$)((g
+ f)2i?
+ f,>
-
the function
(K
+
+
+
(g
+f)u2)
)g + f12u)
$)k
+ fi2k
+
(1.9)
(1.10)
yu
+ f)
W, which plays an important
+
Yk
+ fh
(1.11)
role in our scheme.
LEMMA 1.1. Assume that IpI < d?K. Then there exists w E C’(L2(sZ)) fI C”(H2(sZ)), ~(0, t) = w(L, t) = Osuch that Go E L’(H’(sZ)), 13,G,(0, t) = d,G,(L, t) = Oand a,G, E L2(Q). Proof.
Differentiate
(1.11) with respect
to x and set a,G,
= 0
(1.12)
Ginzburg-Landau
This is a problem
of a differential-integral
equation
W, = (v + iol)W,, - (k + @)(21f12w + c6fw + c5 Refw
401
equation
with the initial-boundary
values
+ f2W)
+ c, Re w + c3 Im w + c2 Refg
+ c,f + co
(1.13) (1.14)
w(0, t) = w(L, t) = w(x, 0) = 0.
Here all cis are C2 functions with values in w2(0). Define A = (v + i(y)@, D(A) = H2(a) tl H,‘(a), then D(A) and H,‘(Q) are dense in L2(sZ) and A is closed. For w E H;(Q) and A > 0 one has - A)w, w)l 1 Ia4l;
I(0
+ vIIw%
+ 4w’ll;l
2 dlwl?f~(n,.
Thus, A - A maps D(A) l-l onto L2(sZ), 111 - AI] 5 l/A for A > 0 and A generates a continuous contraction semigroup. To show that c6fw + c5 Refw + c, Re w + cj Im w + c2 Refg + c,f is locally Lipschitz continuous in the D(A) norm, we notice that (1.15) Ilf’llz
= IlWII2>
Ilf “112= Ilwxll2.
Clearly c6fw + c5 Re fw + c, Re w + c3 Im w E D(A) (1.13) one finds = -(K
c2 Refg
+
ip)4g, Refg,
C,f=
since -(K
(1.16) w E D(A).
From
(1.12)
$)2g&f.
+
and (1.17)
If we go back to (1.3), we can verify that g(0, t)f(O, t) = g(L, t)f(L, t) = 0 since g(L, t) = 0 and f(0, t) = 0 from (1.6). Therefore, c2 Refi + c,f E D(A). Finally it comes to our attention that atcO E C1(H2(sZ)). Now by standard arguments as described in [ll, 15, 161, there is a unique local solution w E C ‘(0, TM; L2(sZ)) fl C’(O, TM; D(A)) to (1.13) along with the initialboundary condition (1.14). But we still need TM = m, i.e. the solution is global. To accomplish that, note
d,llwll: =
n
W,Wdx 12
(wt% + WWJ dx = 2 Re
= -2vllw,112
-
2 Re(K
+
ip)
(21f121 w12 + f2ii12) dx - 2 Re(K + @)
50 X
(c6fw + c,Refw
+ c,Rew
+ c,Imw
+ c,Ref
+ c,f + c,)Wdx.
(1.18)
.i a
Since IpI < OK, -2Re(K
there exists m > 0 such that (here fW = a + ib)
(21f121w12 + f2ti2)dx
+ ip)
= -2
n
(2KG2
+ 2Kb2
+ Ka2 -
Kb2 + 2pab) ti
(3Ka2
+ Kb2 + 2pab) dx
n = -2 5
sn
-m
(a2 + b2)dx
n
= -m
If 121w12d_x. (1.19) sD
C. Bu
402
From
(l.lS),
(1.19)
&Ilwll: 5 -2vlIwXll;
5 4lw,ll:
+
~ll~,Il,Il4,
- df4:
+ dw,llt + ;
+ NI-~211;+ 4l.f4:
+ m’ll.flG+ fillwll;
IIWII:- f4lf4f
+ ; Il.f4: + g Ilwll; + ma-1121142 + ~‘llfll: + fi’lI4 I M, + Mllwll;.
(1.20)
By Gronwall’s lemma it is evident from (1.20) that IIwI(~ is bounded on [0, T] for all on [0, T] for all T > 0. Hence, T > 0. Similarly we can prove that IIw,ll$ is bounded estimate [17] indicates that by (1.15) and (1.16) so is IlfI]$~~c,. The Gagliardo-Nirenberg IIwIL 5 ~llwxl11’2114Y2 so II4L andllfllmare bounded on [0, T] for all T > 0. Now use the fact that the semigroup is continuously contractive in L’(Q) and by a standard estimate on w, (cf. [16]) one finds I(w~~(~< m for any t > 0. Thus, IIwIIDcAj < co by (1.13). Consequently w E C’(L2(Q)) n C”(H2(Q)) is the unique global solution to (1.13) with initial-boundary data (1.14). This effectively shows that a,G, = 0. From (1.1 l), Go E L2(Q), a,G, E L*(n) is obvious. n 2. THE
To show the existence (cf. [3, 181). Let
WEAK
and uniqueness
(a,~,,
EXISTENCE
AND
of the weak solution,
UNIQUENESS
we use the Galerkin-Vishik
-AWN = ~jWj,
a,mj(0)= d,mj(L)= 0,
Wj E H2(n),
then clearly approximate
SOLUTION:
method
j = 1,2,...
(2.1)
Wj E I/ = (v: u E H’(Q), v,(O) = u,(L)= 0). Suppose U, = Cygj,(t)wj solution where (gj,]y’ , are determined by the conditions Wj) +
(V +
i~)(ax~,,d,Wj)
+
(K
+
$)(~Iu,~~u,,
is an
Wj) = (Go + G, + G,, Wj) (2.2)
Vom + ug = U(X, 0) E v.
&z(O) = Uom E [WI, ..*, WA, Multiply
(2.2) by gjm (t) and add them for j = 1,2, . . . , m. Now one has (a turn, u,) + (v + iol)]l%v,ll:
From
+ (K + $)(21~,1~~,,
u,) = (Go + G, +
G,,
U,).
(2.3)
(1.9)-( 1.11) it is clear that IG,(QJ/
IGo1 5 aoY
5 a,lurnl,
1’32&A
5
(2.4)
d4n12.
Take the real part of (2.3)
+a,Il~,,$ I -ulla,u,,$ - Re(K + $)(21u,12u,, u,) + WG, + G, + G,, u,) I
-Vlla,U,II;
-
2K
b,i4~ I .n
+
/(Go s 61
+
G,
+
G2hn1
b.
(2.5)
equation
403
-41&hnll~- 34bmll:+ 4&nll: + d47zll: + 4hnl12 5 c’.
(2.6)
Ginzburg-Landau
Putting
(2.4) in (2.5) one has + U%X
Thus, AWj
5
by Gronwall’s in (2.2) to get
(a f~,,,, Au,)
lemma
one has IIv,ll:
+ (V + iol)(Au,,
I M on [0, T]. By (2.1), one could replace
Au,)
AU,) + (K + @)(21~,1$,z,
= (G, + G, + G,, Au,).
Wj by
(2.7)
If we take the real part of (2.7) then
+%unlll; +vllAu,llt 5
6 + IPI)Ibmllilbuml12 + lb%+ cl + G21i21buml12
I
;
IlAu,ll;
+
(Ic;F’)’IIu,,II:+ ;
IlAu,,ll: +
&IlGo+ G, + G&. (2.8)
Now use IIu,,ll~ I A4 and the following
Gagliardo-Nirenberg
estimates
(cf. [17]) (2.9)
Il~,llZ 5 ~~lla,~,llfII~,11”2 5 ~‘~2lla,& IIu,ll: Then
5 ~llax~,llzll~,ll:
(2.10)
5 ~~3%0,112.
(2.8) becomes
a,Ila,U,1l: 5 m,U,llt
+
;
II&
+
G,
+
4;
5
~kN,ll:
+
h%~,,~~2
+
c.
(2.11)
This certainly implies that IIa, u, II2 is bounded on any [0, T]. Together with the fact that llum]]2 is also bounded we conclude that u, is bounded in L-(0, T, V) for any T. Now differentiate (2.2) with respect to t Wj)
(afu,, =
One could replace
-(K
(v + h)(a, a, u, , a,
+ +
i~)(a,(al~,l2~,),
Wj)
+
(atGo + dtGl + d,G,,
Wj).
(2.12)
wj by a,~,,, in (2.12) and take the real part
+a,lla,~,ll$+ ~ll+w& This together
Wj)
with the boundedness
5 c~(IIu~II; + lb + G, + ~2112~lla~hA12. (2.13) of IIu,II~~(~) and IIG, + G, + G2]12 implies
a,Ila,u,ll:
that
5 G, + Glla,~,II:.
By [3], a,~,,, is bounded in L”(0, T; L2(sZ)) and we can extract u, from u, so that u, + u weakly in L*(O, T, V); a,u@ + x weakly in L*(O, T, L*(n)), a,u = x and u solves
(&u, h) + (v + ia)(a,u, r3,h) + (K + $)(lu12u, h) = (G, + G, + G,, h) for all h E I’. We deduce (a+,
from (1.5), (1.6) and (2.14) that u = u +
h) + (V + icu)(a,24, a,h)
(2.14)
f satisfies
+ (K + ij3)(1241224,h) = (co + c,
+ c,,
U)
(2.15)
404
C. Bu
for all h E Vwith u E L-(0, T; H’(Q), &u E L”(0, Z L’(Q)), u(x, 0) = uO(x) = UO(x) - g(x, 0), ~~(0, t) = u,(L, t) = 1. And this implies that U satisfies (&U,h)
+ (v + i(Y)(a,u,
&h) + (K + @)(Iu~*u,h)
= y(U,h)
(2.16)
for all h E V with U E H’(a), U(x, 0) = U,(x), U,(O, t) = P(t), U,(L, t) = R(t). The uniqueness proof is similar to that shown in [4]. We have, therefore, completed the proof of the following theorem. THEOREM 2.1 (weak solution). Let Ipi < a~. For the forced GL equation Neumann boundary data and initial data (1.2) there exists a unique solution Remark 2.2 (regularity). strong solution.
We indicate
here that if uO(x) E Hz then the solution
(1.1) with nonzero U satisfying (2.16). to (2.1) is a global
Remark 2.3 (mixed boundary data). There is a more general boundary condition as far as forcing is concerned. One could assume that u,(O, t) + au(0, t) and uX(L, 0) + bu(L, 0) be given (in C2, of course): here a, b are real numbers. For the nonlinear Schr6dinger equation on the semiline with u,(O, f) + au(0, t) given, a global solution is available provided that the initial-boundary data are smooth [ 151. We expect that with the mixed boundary data, we will be able, through a slightly more complicated scheme, to obtain the unique weak solution via Galerkin’s method as described here. REFERENCES 1. LANGE C. & NEWELL A., A stability
criterion
for the envelope
equation,
SIAM J. uppl. Math. 27, 441-456
(1974).
2. NEWELL A. & WHITEHEAD J., Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279-304 (1969). 3. LIONS J., Quelques mPthodes de r&solution des probldmes aux limites non lineaires. Dunod, Paris (1969). 4. Bu C., An Initial-boundary Value Problem for the Ginzburg-Landau Equation. Applied Mathematics Letters (to appear).
5. GHIDAGLIA J. & HERON R., Dimension of the attractors
associated
to the Ginzburg-Landau
equation,
Physica 28D,
282-304 (1987).
6. I. 8. 9.
Bu C., On the Cauchy problem for the 1 + 2 Ginzburg-Landau equation (preprint). KAUP D., Wave Phenomena, pp. 163-174. North-Holland, Amsterdam (1984). CARROLL R., On the forced nonlinear Schriidinger equation, Japan J. appl. Math. 7, 321-344 (1990). FOKAS A., An initial-boundary value problem for the nonlinear Schriidinger equation, Physica 35D, 167-185 (1989). of the forced nonlinear SchrGdinger equation, Applicable Analysis 46, 219-239 (1992). 10. Bu C., On well-posedness Schr6dinger equation (NLS) using PDE techniques, 11. CARROLL R. & Bu C., Solutions of the forced nonlinear Applicable Analysis 41, 33-51 (1991). Vries equation in a quarter plane, continuous dependence results, DifJ 12. BONA J. & WINTHER R., The Korteweg-de
Integral Eqns 2, 228-250 (1989). 13. BONA _I. & WINTHER R., The Korteweg-de
Vries equation:
posed in a quarter
plane,
SIAM J. math. Analysis 6,
1056-1106 (1983). 14. 15. 16. 17. 18.
equation: posed in a quarter plane, J. math. Analysis Applic. (to appear). value problem for the nonlinear SchrGdinger equation, Nonlinear Analysis (to appear). PAZY A., Semigroups of Linear operators and Applications to PDE. Springer, New York (1983). NIRENBURG L., On elliptic partial differential equations, Anna/i Scu. norm. sup. Pisa 13, 115-162 (1959). CARROLL R., Mathematical Physics. North-Holland, Amsterdam (1988). Bu C., The G&burg-Landau Bu C., An initial-boundary