Existence Of Global Solution Of Nonlinear Ginzburg-Landau Equation With External Force Term In Two-Dimensional Space

Ginzburg-Landau equation has very abundant Physical backgroundand connotation. In the recent twenty years, many physicists andmathematicians have deeply studied it and obtained plenteous results.In the hydromechanics systems, plasma transmit andsuperconductor theory, the complex Ginzburg-Landau equation is givenin the formu_t= ρu + (1 + iγ ) Δ u-(1 +i μ)|u|~2u .In the physics, it is dissipative nonlinear Schrodinger equation. Atpresent, there are many papers concerning it .However, many of themwere studied in Low-dimensional space. For example, Ghidaglia andHeron [10],Doering [11]etc. have studied the finite dimensional globalattractor and related dynamics systems of the cubic nonlinearGinzburg-Landau equation in one-dimensional and two-dimensionalspaces . Doering, Gibbon and Levermore[18] posed the existence anduniqueness on weak and strong solutions of the nonlinear equation withany order in all spaces .In addition, R.Temam have proved the solution of the complexGinzburg-Landau equationis bounded in two-dimensional sobolev space H ( ? ), H 1( ? ). In thispaper, we consider the existence and uniqueness of global solution of theinitial-boundary value problem on the following nonlinearGinzburg-Landau equation with external force termu ( i ) u ( k i ) u 2u u f ( x , t) (2.1)?? t ? |? + |á ? + + |? ? |?=with the initial conditionu ( x ,0) = u 0( x ), x?ê? , (2.2)Here ? = [0, L1 ] ?á [0, L2],u is a ?-periodic complex function definedon the ??á R, f ( x , t ) ?ê H1( ? ) is a real function which is uniformbounded in t, parameters |? , |á , k, |? ,|? is real number and satisfy|? > 0, k> 0, |?> 0,? is Laplace operator of the R 2.In order to discuss the nonlinear problem, the second chapter listssome theory and preliminaries concerning this theme, simultaneity givesthe definition of sobolev space{ }W m , p ( ? ) = u u ?ê Lp ( ? ), ? |á u ?ê Lp ( ? ),|á?ü m.The norm of W m , p( ? ) is defined by1/u m , p , ? = ( |á???üm ?|áupp ,?)p, 1 ?ü p< ?T .letW m ,2 ( ? ) = Hm( ? ),H = L2 ( ? )= { u ?ê L2 ( ? ), u is a ? -periodic complex function }H k = Hk( ? )={ u ?ê Hk( ? ),u is a ?-periodic complex function }.Where ? = [0, L1 ] ?á [0, L2] is an boundary set in R 2. i is the norm ofL2 ( ? ),and (u , v ) is the internal product. i p ( i 2=i ) is the norm ofLp ( ? ) for 1 ?ü p?ü ?T .In the third chapter ,by the existence of local solution of PazyA [42]we have:Lemma 3.1 For every u 0?ê H2, there exists a unique solution ofthe initial-boundary value problem(2.1) ? (2.2),such that( ) ( )2 1 2u ?ê C [0, Tm ax );H ?é C (0, Tm ax);H,where t ?ê [0, Tmax),and Tm ax= ?T .Moreover, if Tm ax< ?T , then2maxtl ?úiT m u (t ) H?ú?T.By the lemma,we give the definition of the global solution:Definition A function u = u (t ) is said to be a global solution of(2.1) ? (2.2),if u (t )is the solution in the Lemma 3.1 and Tm ax= ?T .In order to prove the existence and uniqueness of global solutionin the sobolev space , we need to establish the priori estimates for thesolution u (t ) in H ( ? ), H 1( ? ) and H 2( ? ).We consider the problem by three steps. Firstly , we establishthe priori estimate for the solution u (t ) in the H ( ? ) and prove that theu (t ) is bounded in the H ( ? ),then obtain the following lemma:Lemma 3.2 Assume u 0 ?ê H( ? ),then the solution u (t ) of theproblem (2.1) ? (2.2) satisfies2 2u (t ) ?ü K1 ( u 0, T), ?t ?ê[0, T ],where K1 depends on initial value and T , T depends on theconstant R0 , u 0 ?ü R0.Secondly, we establish the priori estimate for the solution u (t ) inthe H 1( ? ) and prove that the u (t ) is bounded in the H 1( ? ),thenobtain the following lemma:Lemma 3.3 Assume u 0 ?ê H 1( ?),then the solution u (t ) of theproblem (2.1) ? (2.2) satisfies12 2? u ?ü K 2 ( u 0H, T ), ?t ?ê[0, T ],where K 2 depends on initial value and T , T depends on theconstant R0 , u 0 ?ü R0.At last ,we establish the priori estimate for the solution u (t ) in theH 2 ( ? ) and prove that the u (t ) is bounded in the H 2 ( ? ),then obtainthe following lemma:Lemma 3.4 Assume u 0 ?ê H2( ? ),then the solution u (t ) of theproblem (2.1) ? (2.2) satisfies22 2?u ?ü K 3 ( u 0H, T ), ?t ?ê[0, T ],where K 3 depends on initial value and T , T depends on theconstant R0 , u 0 ?ü R0.By the above three lemmas and [42], we haveTheorem If u 0 ?ê H2( ? ), the initial value problem20( ) ( ) ( , )( ,0) ( )u i u k i u u u f x ttu x u x??? ??? |? + |á ? + + |? ? |?=?? =exists a unique global solution u (t ),such that( ) ( )u ?ê C [0, ?T );H 2 ?é C 1 (0, ?T);H2.