Nonlinear Schrodinger-Helmholtz Equation as Numerical Regularization of the Nonlinear Schrodinger Equation
aa r X i v : . [ m a t h . A P ] J un NONLINEAR SCHR ¨ODINGER-HELMHOLTZ EQUATION AS NUMERICALREGULARIZATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION
YANPING CAO, ZIAD H. MUSSLIMANI, AND EDRISS S. TITI
Abstract.
A regularized α − system of the Nonlinear Schr¨odinger Equation (NLS) with 2 σ nonlinearpower in dimension N is studied. We prove existence and uniqueness of local solution in the case1 ≤ σ < N − and existence and uniqueness of global solution in the case 1 ≤ σ < N . When α → + ,this regularized system will converge to the classical NLS in the appropriate range. In particular, thepurpose of this numerical regularization is to shed light on the profile of the blow up solutions of theoriginal Nonlinear Schr¨odinger Equation in the range N ≤ σ < N , and in particular for the criticalcase σ = N . MSC Classification : 35Q40, 35Q55
Keywords : Schr¨odinger-Newton equation, Hamiltonian regularization of the nonlinear Schr¨odingerequation, Schr¨odinger-Helmholtz equation.1.
Introduction
The Nonlinear Schr¨odinger equation (NLS): iv t + △ v + | v | σ v = 0 , x ∈ R N , t ∈ R , (1) v (0) = v , where v is a complex-valued function in R N × R , arises in various physical contexts describing wavepropagation in nonlinear media (see, e.g., [14], [21], [22] and [26]). For example, when σ = 1, equation(1) describes propagation of a laser beam in a nonlinear optical medium whose index of refraction isproportional to the wave intensity. Also, the Nonlinear Schr¨odinger Equation successfully models otherwave phenomena such as water waves at the free surface of an ideal fluid as well as plasma waves. In allcases, it is interesting to note that Eq. (1) describes wave propagation in nonhomogeneous linear mediawith self-induced potential given by | v | σ .As it is mentioned above, the σ = 1 case is particularly interesting for laser beam propagation inoptical Kerr media. Depending on the dimensionality of the space upon which the beam is propagatingin, the wave dynamics can be either “simple” or “intricate”. In one space dimension, the NLS equationis known to be integrable and possesses soliton solution that preserves their structure upon collision [1].The picture in two-dimensional (2D) space is totally different. The 2D NLS equation is not integrable,hence no exact soliton solutions are known. Instead, the 2D NLS equation admits the waveguide solution(also known as Townes soliton) v ( x, y, t ) = R ( r ) exp( it ) with r = p x + y where R > d Rdr + 1 r dRdr − R + R = 0 , dRdr (0) = 0 , lim r → + ∞ R ( r ) = 0 . (2) Date : June 27, 2007.
Importantly, the L norm (or power in optics) of the Townes soliton defines a critical value for blow up.If initially the beam’s power is larger than that of the Townes soliton, || v || L > || R || L , then the beamundergoes a finite time blow up. If on the other hand || v || L < || R || L , then the wave will diffract.Various mechanisms to arrest collapse have been suggested such as nonparaxiality [6], or higher orderdispersion [7].As a result, an important issue that arises in the mathematical study of the NLS is the questionof local and global existence of solutions, their uniqueness, as well as the profile of blow up solutions.Knowing answers to such questions may have some consequences on possible physical observations ofphenomenon governed by the NLS and in validating its derivation. In the works of Ginibre and Velo[10] and Weinstein [24], it is proved that equation (1) has a unique global solution when 0 < σ < N ,and that it has a unique global solution for “small” initial data for the critical case σ = N . Theproof of global existence uses the fact that the energy N ( v ) = R R N | v ( x, t ) | dx and the Hamiltonian H ( v ) = R R N (cid:16) |∇ v ( x, t ) | − | v ( x,t ) | σ +2 σ +1 (cid:17) dx are conserved quantities of the dynamics of (1). In the case of σ ≥ N , Glassey [11] proved that there exist solutions that develop singularities in finite time. In recentyears there was an intensive remarkable computational work concerning the blow up for the critical case σ = N . For instance, Merle and Raphael have obtained a sharp lower bound on the blow up rate forthe L norm of the NLS in R N (see [15] and references therein). Moreover, Fibich and Merle [8] studiedself-focusing in bounded domains using a combination of rigorous, asymptotic and numerical results.Instead of the potential | v | σ , physicists consider self-gravational potential (see, e.g., [18], [20]) andcome to a new system: Schr¨odinger-Newton equation (SN): iv t + △ v + ψv = 0 , x ∈ R N , t ∈ R , (3) − α △ ψ = | v | ,v (0) = v , where α > H ( v ) = R R N (cid:16) |∇ v ( x, t ) | − ψ ( x,t ) | v ( x,t ) | (cid:17) dx and can be obtained formally by the variational princi-ple i ∂v∂t = δ H ( v ) δv ∗ , where v ∗ denotes the complex conjugate of v . System (3), or at least its stationarystate, has been studied [17], [23], [19]. This coupled system of equations consists of the Schr¨odingerequation for a wave function v moving in a potential ψ , where ψ is obtained by solving the Poissonequation with source ρ = | v | . It can be thought of as the Schr¨odinger equation for a particle movingin its own gravatational field. As in the NLS, the energy N ( v ) = R R N | v ( x, t ) | dx and the Hamiltonian H ( v ) = R R N (cid:16) |∇ v ( x, t ) | − ψ ( x,t ) | v ( x,t ) | (cid:17) dx are also conserved in this system. The question of existenceand uniqueness of local and global solutions for system (3) has not been answered completely yet.Inspired by the α − models of turbulence (see, e.g., [3], [4], [5], [9], [12], [13] and references therein),we introduce a generalization of (3), the Schr¨odinger-Helmholtz (SH) regularization of the classical NLS: iv t + △ v + u | v | σ − v = 0 , x ∈ R N , t ∈ R , (4) u − α △ u = | v | σ +1 ,v (0) = v , where α > σ ≥
1. System (4) is a Hamiltonian system with the corresponding Hamiltonian H ( v ) = R R N (cid:16) |∇ v ( x, t ) | − u ( x,t ) | v ( x,t ) | σ +1 σ +1 (cid:17) dx and can be obtained formally by the variational principle i ∂v∂t = δ H ( v ) δv ∗ , where again v ∗ denotes the complex conjugate of v . In this system, we can regard thewave function v moves in a potential u | v | σ − , where u is obtained by solving the Helmholtz elliptic AMILTONIAN REGULARIZATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION 3 problem u − α △ u = | v | σ +1 . Observe that the energy N ( v ) = R R N | v ( x, t ) | dx and the Hamiltonian H ( v ) = R R N (cid:16) |∇ v ( x, t ) | − u ( x,t ) | v ( x,t ) | σ +1 σ +1 (cid:17) dx are conserved in this system. When σ = 1, we have thepotential u as in the SN with the only difference that the Possion equation is modified as a Helmholtzequation. So we consider this system as a generalized system of SN. A more important fact is that when α = 0, one recovers the classical NLS, therefore we regard this system as a regularization of the classicalNLS. In this paper we focus on the case α > α → + . In particular, we will investigate the case σ = N , which is not completelyunderstood.In this paper, we will study the question of local and global existence of unique solution for system(4). Specifically, we will prove the short time existence of unique solution, when 1 ≤ σ < N − (we defineonce and for all N − = ∞ when N ≤ ≤ σ < N . The proof will follow the ideas of [10] and [24] and use the important fact of theconservation of the corresponding energy and the Hamiltonian of (4). All the proofs presented here willapply directly to system (3) as well. So simultaneously we have the same results for system (3): we haveshort time existence of unique solution when 1 ≤ σ < N − , and global existence of unique solution when1 ≤ σ < N . Comparing to the results of the classical NLS (1) ( σ < N − for local existence and σ < N for global existence), one expects these “better” results for (3) and (4) since the nonlinear terms in (3)and (4) are milder than that of the classical NLS (1). The parametre α plays an important role in ourproofs. In a subsequential paper, we will investigate numerically the blow up profiles of the NLS, in therelevant range of σ , when α → + .In section 2, we will introduce some essential notations and definitions, and some preliminary resultsthat will be used throughout the paper. Following the work of Ginibre and Velo [10], we prove in section3 local (in time) existence and uniqueness of solution for system (4) using the contraction mapping prin-ciple. In section 4, we will extend the local solution to global existence, for 1 ≤ σ < N , after establishingthe required a priori estimates for the H norm of the solution, which remains finite for every finiteinterval of time. 2. Notations and Preliminaries
In this section we introduce some preliminary results and the basic notations and definitions that willbe used throughout this paper.We denote by k · k p the norm in the space L p = L p ( R N ) (1 ≤ p ≤ ∞ ), except for p = 2 where thesubscript 2 will be omitted. We will denote by h· , ·i the scalar product in L . The conjugate pair p, p ′ satisfies the relation p + p ′ = 1. For any real number l , we denote by H l = H l ( R N ), the usual Sobolevspace. Of special interest is the H Sobolev space with the norm defined by k v k H = Z R N (cid:0) | ξ | (cid:1) | ˆ v ( ξ ) | dξ, (5)or equivelently, k v k H = k v k + k∇ v k . (6)We denote by k u k W k,p = (cid:0) Σ | α |≤ k R R N | D α u | p dx (cid:1) /p , ≤ p < ∞ , for u belongs to the Sobolev space W k,p ( R N ). For any interval I of the real line R , and for any Banach space B , we denote by C ( I, B )(respectively C b ( I, B )) the space of continuous (repectively bounded continuous) functions from I into B .In this paper C and C α will denote constants which might depend on various parameters of the problem. Y.CAO, Z. H. MUSSLIMANI, AND E.S. TITI
They might vary in value from one time to another, but they are independent of the solution. Whenit is relevant we will comment on the asymptotic behavior of these constants as they depend on thecorresponding parameters.First, we recall some classical Gagliardo-Nirenberg and Sobolev inequalities (see, e.g., [2]).
Proposition 1. (1) For any N ≥ , we have k v k q ≤ C k v k − q − q N k∇ v k q − q N for every v ∈ H , < q − q N ≤ k v k q ≤ C k v k W ,m for every v ∈ W ,m , q ≥ m, m > N (8) k v k q ≤ C k v k W ,m for every v ∈ W ,m , q ≥ m − N ≥ , q < ∞ (9) In particular , (10) k v k q ≤ C k v k W , = C k v k H for every v ∈ H , ≤ q ≤ ∞ , N ≤ . (11) (2) For N ≤ , k v k q ≤ C k v k H for every v ∈ H , ≤ q < ∞ . (12)With these inequalities at hand, we can process the nonlinear term. Let us rewrite the term f ( v ) = u | v | σ − v = B ( | v | σ +1 ) | v | σ − v, (13)where B = ( I − α △ ) − , the inverse of the Helmholtz operator. Then f is a locally Lipschitz mappingfrom H into L r ′ , for some r ∈ (2 , NN − ], where r + r ′ = 1. Proposition 2.
Let N ≥ and ≤ σ < N − . For every v , v ∈ H ⊂ L r , where r depends onthe given σ and belongs to the range r ∈ (2 , NN − ] (we consider NN − as ∞ when N ≤ ), we have k f ( v ) − f ( v ) k r ′ ≤ k k v − v k r , where k = C α ( k v k H + k v k H ) σ and r + r ′ = 1 , for some constant C α . Before we prove this proposition, we will state the following Lemmas:
Lemma 3.
Let N ≥ and ≤ σ < N − . For every v , v , v ∈ H ⊂ L r , where r depends on the given σ and belongs to the range r ∈ (2 , NN − ] , we have k B ( | v | σ ) | v | σ − v ( v − v ) k r ′ ≤ C α k v k σH k v − v k r , (14) k B ( | v | σ − v ) | v | σ − v ( v − v ) k r ′ ≤ C α k v k σH k v − v k r . (15) Proof.
First, denote I = k B ( | v | σ ) | v | σ − v ( v − v ) k r ′ . Case 1. N ≤ : By H¨older’s inequality, we have I ≤ k B ( | v | σ ) | v | σ − v k r ′ β k v − v k r ′ γ = k B ( | v | σ ) | v | σ − v k rr − k v − v k r , (16)where in the last equality, we choose γ = r − > β + γ = 1 such that r ′ γ = r and r ′ β = rr − .By Cauchy-Schwarz inequality, we have k B ( | v | σ ) | v | σ − v k rr − ≤ k B ( | v | σ ) k rr − k| v | σ k rr − . (17) AMILTONIAN REGULARIZATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION 5
Now, for the elliptic equation u − α △ u = f in R N , we have the regularity property [16], [25] k u k W ,p ≤ C α k f k p for any 1 < p < ∞ , (18)where C α depends on N, p and α , and C α ∼ α as α → + . Moreover, for α fixed, C α ∼ p as p → ∞ .Since rr − >
2, by (11) and (18), we have k B ( | v | σ ) k rr − ≤ C k B ( | v | σ ) k W , ≤ C α k| v | σ k = C α k v k σ σ , and k| v | σ k rr − = k v k σ rr − σ . Since σ ≥
1, combining the above two terms and applying (12), we have I ≤ C α k v k σH k v − v k r . Case 2. N ≥ : Applying H¨older’s inequality, we have I ≤ k B ( | v | σ ) k r ′ θ k| v | σ − v k r ′ β k v − v k r ′ γ = k B ( | v | σ ) k r ′ θ k v k σσr ′ β k v − v k r ′ γ , where θ + β + γ = 1.Now by (8), (9) and (18), we have k B ( | v | σ ) k r ′ θ ≤ C k B ( | v | σ ) k W ,m ≤ C α k| v | σ k m = C α k v k σσm , where we require r ′ θ ≥ m − N when m − N ≥
0, or r ′ θ ≥ m when m − N <
0, for m > I ≤ C α k v k σσm k v k σσr ′ β k v − v k r ′ γ . (19)Now, by requiring σm = σr ′ β = r ′ γ = r , we have θ = r − r − σ − > ⇒ σ < r − β = r − σ > ⇒ σ < r − γ = r − > m = rσ > σ < N + 22 N r − σ > r − . Since 2 < r ≤ NN − , we conclude that σ < N − , i.e., I ≤ C α k v k σH k v − v k r . By exactly the same steps, inequality (15) follows readily. (cid:3)
Y.CAO, Z. H. MUSSLIMANI, AND E.S. TITI
Lemma 4.
Let N ≥ and ≤ σ < N − . For every v , v , v ∈ H ⊂ L r , where r depends on the given σ and belongs to the range r ∈ (2 , NN − ] , we have k B ( | v | σ +1 ) | v | σ − ( v − v ) k r ′ ≤ C α k v k σH k v − v k r , (20) k B ( | v | σ +1 ) | v | σ − v ( v − v ) k r ′ ≤ C α k v k σH k v − v k r . (21) Proof.
Denote I = k B ( | v | σ +1 ) | v | σ − ( v − v ) k r ′ . Case 1. N ≤ : By H¨older’s inequality, we obtain I ≤ k B ( | v | σ +1 ) | v | σ − k r ′ β k v − v k r ′ γ = k B ( | v | σ +1 ) | v | σ − k rr − k v − v k r , where we choose the same β , γ as in (16).Now, when σ >
1, again by H¨older’s inequality, we have k B ( | v | σ +1 ) | v | σ − k rr − ≤ k B ( | v | σ +1 ) k rr − β k| v | σ − k rr − γ (22)= k B ( | v | σ +1 ) k rr − β k v k σ − σ − rr − γ . (23)By choosing 1 < β < σ and 2 < r ≤
4, one can easily verify that rr − β ≥ σ − rr − γ > k B ( | v | σ +1 ) k rr − β ≤ C k B ( | v | σ +1 ) k W , ≤ C α k| v | σ +1 k = C α k v k σ +12( σ +1) . (24)Since 2( σ + 1) > σ − rr − γ > σ >
1, by (12) we conclude I ≤ C α k v k σH k v − v k r . (25)Now, when σ = 1, by choosing 2 < r ≤ β = 1, we have k B ( | v | σ +1 ) | v | σ − k rr − = k B ( | v | σ +1 ) k rr − ≤ C α k v k σ +12( σ +1) . By (12), we conclude that I ≤ C α k v k σH k v − v k r . (26) Case 2. N ≥ : By H¨older’s inequality, we have I ≤ k B ( | v | σ +1 ) k r ′ θ k| v | σ − k r ′ β k v − v k r ′ γ = k B ( | v | σ +1 ) k r ′ θ k v k σ − σ − r ′ β k v − v k r ′ γ , (27)where θ + β + γ = 1. AMILTONIAN REGULARIZATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION 7
Now, by (8), (9) and (18), we have k B ( | v | σ +1 ) k r ′ θ ≤ C k B ( | v | σ +1 ) k W ,m ≤ C α k| v | σ +1 k m = C α k v k σ +1( σ +1) m , where r ′ θ ≥ m − N when m − N ≥
0, and r ′ θ ≥ m when m − N <
0, and m > I ≤ C α k v k σ +1( σ +1) m k v k σ − σ − r ′ β k v − v k r ′ γ . (28)Now, by requiring ( σ + 1) m = ( σ − r ′ β = r ′ γ = r (choose β = ∞ when σ = 1), we have θ = r − r − − σ > ⇒ σ < r − β = r − σ − > ⇒ σ < rγ = r − > m = rσ + 1 > σ < N + 22 N r − σ > r − . Since 2 < r ≤ NN − , we obtain σ < N − , i.e., I ≤ C α k v k σH k v − v k r . The same can be shown for (21). (cid:3)
Now, we are ready to prove Proposition 2.
Proof.
Recall that f ( v ) = B ( | v | σ +1 ) | v | σ − v , by direct calculation, we have ∂f ( v ) ∂v = σ + 12 [ B ( | v | σ ) | v | σ − v + B ( | v | σ +1 ) | v | σ − ] (29) ∂f ( v ) ∂v ∗ = σ + 12 B ( | v | σ − v ) | v | σ − v + σ − B ( | v | σ +1 ) | v | σ − v (30)when v = 0, where v ∗ is the conjugate of the complex-valued function v .For v = 0, we have ∂f ( v ) ∂v = 0 ,∂f ( v ) ∂v ∗ = 0 . Now, by Lemma 3 and Lemma 4 we obtain k ∂f ( v ) ∂v ( v − v ) k r ′ ≤ C α k v k σH k v − v k r , k ∂f ( v ) ∂v ∗ ( v − v ) k r ′ ≤ C α k v k σH k v − v k r . When v v = 0, by Mean-Value Theorem, we have | f ( v ) − f ( v ) | ≤ max {| ∂f ( v ) ∂v | , | ∂f ( v ) ∂v ∗ |}| v − v | . Y.CAO, Z. H. MUSSLIMANI, AND E.S. TITI
That is, for some intermediate point v between v and v k f ( v ) − f ( v ) k r ≤ ˜ C α k v k σH k v − v k r ≤ C α ( k v k H + k v k H ) σ k v − v k r . When v v = 0, without loss of generality, we assume that v = 0, then k f ( v ) − f ( v ) k r ′ = k f ( v ) − k r ′ = k B ( | v | σ +1 ) | v | σ − v k r ′ ≤ C α ( k v k H + k v k H ) σ k v k r , where the last inequality follows from direct application of Lemma 4.Therefore, we conclude that, for any v , v ∈ H ⊂ L r , k f ( v ) − f ( v ) k r ′ ≤ C α ( k v k H + k v k H ) σ k v − v k r . (cid:3) Next, we will give some elementary properties of the free evolution (linear Schr¨odinger equation)formally defined by the group of operators U ( t ) = exp( it △ ) , (31)where t ∈ R . In the following, we will state some well-known results about the operator U ( t ) withoutproving them (see, e.g., [10],[21]). Lemma 5.
For any r ≥ , and for any t = 0 , U ( t ) is a bounded linear operator from L r ′ to L r , and themap t → U ( t ) is strongly continous. Moreover, for all t ∈ R \ { } , one has k U ( t ) v k r ≤ (4 π | t | ) Nr − N k v k r ′ (32) for all v ∈ L r ′ . Corollary 6.
Let I be an interval of R , and let v ∈ C ( I, L r ′ ) . Then for all t ∈ R the map τ → U ( t − τ ) v ( τ ) is continous from I \ { t } into L r . Existence and Uniqueness of Local Solutions
In this section we will prove a local existence and uniquenss theorem of solutions to system (4) by afixed point technique.The integral equation v ( t ) = U ( t − t ) v + i Z tt U ( t − τ ) f ( v ( τ )) dτ (33)may be considered as the integral version of the initial value problem for equation (4).Defining the subspace Y ( I ) ⊂ C ( I, X ) and Y b ( I ) ⊂ C b ( I, X ) by Y ( I ) = { v : v ∈ C ( I, X ) and v ( t ) = U ( t − s ) v ( s ) for all s and t ∈ I } Y b ( I ) = Y ( I ) ∩ C b ( I, X ) . Here for special interest we choose the Banach space X = L r ( R N ), for some r >
2, which is specified inthe proof of Lemma 3 and Lemma 4, and ¯ X = L r ′ ( R N ).If v ∈ C b ( I, X ), we shall denote its norm by | v | I , and for v ∈ C b ( I, H ), we denote its norm by | v | H ,I .The ball of radius R in C b ( I, X ) will be denoted by B ( I, R ). AMILTONIAN REGULARIZATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION 9
Let t , t ∈ R and let v ( t ) be a family of complex-valued functions defined on R N , depending on aparameter t ∈ R . We formally define the operators[ G ( t , t ) v ]( t ) = i Z t t U ( t − τ ) f ( v ( τ )) dτ, (34)where f is the nonlinear term defined in (13). The first lemma below gives a meaning to the expressiondefined by (34) and contains some of its properties. Lemma 7.
For any interval I ⊂ R (possibly unbounded), the maps ( t , t , v ) → G ( t , t ) v are continuousfrom I × I × C ( I, X ) to Y b ( R ) . Moreover, for any t , t ∈ I, ( t < t ) , for any compact sub-interval J such that [ t , t ] ⊂ J ⊂ I , and for any t ∈ [ t , t ] , for any v , v ∈ C ( I, X ) the G operator satisfies theestimates k G ( t , t ) v ( t ) − G ( t , t ) v ( t ) k r ≤ k ′ k v − v k J | t − t | Nr − N +1 where k ′ = k (4 π ) Nr − N , k = C α ( k v k H + k v k H ) , which is derived in the proof of Proposition 2.Proof. For any v ∈ C ( I, X ) the function τ → f ( v ( τ )) belongs to C ( I, ¯ X ) as consequence of Proposition2. Therefore, by Lemma 3, for any t ∈ R \ { t } the function τ → U ( t − τ ) f ( v ( τ )) (35)is continous from I to X . To check the integrability of the function (35) it will be enough to show theintegrability of its norm. More generally one is interested in the integrability of k U ( t − τ )[ f ( v ( τ )) − f ( v ( τ ))] k r , (36)for any v , v ∈ C ( I, H ) ⊂ C ( I, X ).This is a direct consequence of Proposition 2 and Lemma 3: For t ∈ R , for every compact sub-interval J ⊂ I and τ ∈ J , we have k U ( t − τ )[ f ( v ( τ )) − f ( v ( τ ))] k r ′ ≤ (4 π | t − τ | ) Nr − N k k v − v k J . Finally, we come to the conclusion that k G ( t , t ) v ( t ) − G ( t , t ) v ( t ) k ≤ k ′ k v − v k J | t − t | Nr − N +1 . (cid:3) Now, in order to study the equation (33) one needs the operators[ F ( t ) v ]( t ) = [ G ( t , t ) v ]( t ) . (37)The existence and properties of F follow immediately from Lemma 5.For every v ∈ C ( I, H ) ⊂ C ( I, X ),[ A ( t , v ) v ]( t ) = [ F ( t ) v ]( t ) + U ( t − t ) v (38)is a continuous map from C ( I, H ) ⊂ C ( I, X ) into C ( I, X ).With these notations equation (33) may be rewritten as A ( t , v ) v = v. (39)The next lemma gives some elementary properties of the solutions of equation (33). In particular, itexpresses the consistency of the change of the initial time t . Lemma 8.
Let I and J be two intervals of R , J ⊂ I , let t ∈ J , let v ∈ H be such that the function t → U ( t − t ) v belongs to Y ( I ) , and let v ∈ C ( J, X ) be a solution of the equation (39) ( i ) The function φ ( v ) : s → U ( · − s ) v ( s ) = [ φ ( v )]( s ) (40) belongs to C ( J, Y ( I )) and satisfies for all s, s ′ ∈ J the equality [ φ ( v )]( s ) − [ φ ( v )]( s ′ ) = G ( s ′ , s ) v. (41) Furthermore, if for some s ∈ J, [ φ ( v )]( s ) ∈ Y b ( I ) , then φ ( v ) ∈ C ( J, Y b ( I )) . If in addition J is bounded,then φ ( v ) ∈ C b ( J, Y b ( I )) . ( ii ) For any s ∈ J , u satisfies the equation A ( s, v ( s )) v = v. (42) Proof.
Apply the operator U ( t − s ) to equation [ A ( t , v ) v ]( s ) = v ( s ) and use the fact that U ( t − s )[ G ( t , t ) v ]( s ) = [ G ( t , t ) v ]( t ) (for the proof of this identity, we refer to Ginibre and Velo [10]) yields[ φ ( v )]( s ) = U ( · − t ) v + G ( t , s ) v. (43)¿From which (41) follows immediately. The continuity properties of the left-hand side of (43) are then aconsequence of the assumptions made on v and of Lemma 7. Finally, putting s ′ = t in (41) and takingthe values of both members at t one obtains equation (42) at time t . (cid:3) We are now ready to discuss the problem of the existence and uniqueness of solutions of equation (39).
Theorem 9.
For any ρ > , there exists a T ( ρ ) > , depending only on ρ , such that for any t ∈ R and for any v ∈ H , for which k v k H ≤ ρ , equation (39) has a unique solution on C ( I, X ) , where I = [ t − T ( ρ ) , t + T ( ρ )] and X = L r .Proof. Let ρ be a fixed postive number, let t and T ∈ R , T >
0, and let I = [ t − T, t + T ]. Then forevery v , v ∈ H and k v − v k H ≤ ρ , Lemma 5 and (37) yield the inequality | F ( t ) v − F ( t ) v | I ≤ k ′ | t − t | Nr − N +1 | v − v | I . (44)In particular, if we take T = T ( ρ ) with T ( ρ ) defined by4 k ′ | T ( ρ ) − t | Nr − N +1 = 1 (45)in equality (44) it gives | F ( t ) v − F ( t ) v | I ≤ | v − v | I . (46)Let now v ∈ X be such that U ( · − t ) v ∈ B ( I, ρ ). Definition (38) and estimate (46) imply | A ( t , v ) v | I ≤ ρ, (47)and | A ( t , v ) v − A ( t , v ) v | I ≤ | v − v | I , (48)for all v, v , v ∈ B ( I, ρ ), from which it follows that A ( t , v ) is a contraction from the ball B ( I, ρ ) intoitself. The result is now a consequence of the contraction mapping theorem. (cid:3) AMILTONIAN REGULARIZATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION 11 Global Existence of Solutions
In this section we will study global existence of solutions to system (4) under the condition of σ ≥ ≤ σ < N . Comparing this to the results ofthe classical NLS ( σ < N ), we “gain” global regularity for larger range of values of σ . As we stated inthe introduction, system (4) will recover the classical NLS as the parameter α → + , in a subsequencialpaper, we will study numerically the blow up profile of the classical NLS by focusing on system (4) with N ≤ σ < N when N ≤
3. To be more specific, the profile of blow-up in the critical case σ = N in theclassical NLS has not been known completely, in a subsequential work, we will compute SH system (4)and try to find out the blow up profile by forcing the parameter σ to approach zero. Theorem 10.
Let v ∈ H ( R N ) . If ≤ σ < N , then there exists a unique solution v ∈ C (( −∞ , ∞ ); H ( R N )) of the initial-value problem (4), in the sense of the equivalent integral equation.Furthermore, as long as v ( x, t ) remains in H ( R N ) , the energy N ( v ) = Z R N | v ( x, t ) | dx (49) and Hamiltonian H ( v ) = Z R N (cid:18) |∇ v ( x, t ) | − u ( x, t ) | v ( x, t ) | σ +1 σ + 1 (cid:19) dx (50) remain constant in time. In the local existence theorem in section 3, we have shown that the length T , of the interval ofexistence [ t , t + T ], can be taken to depend only on k v k H . It follows that if v ( x, t ) is a maximallydefined solution on [ t , T max), then either T max = + ∞ or lim t → T − max k v ( t ) k H = + ∞ . The heart of the global existence proof lies in the use of the invariants (49) and (50), which enable us toobtain an a priori bound of the following type: k v ( x, t ) k H ≤ C ( N , H ) . (51) Proof.
We proceed as follows:¿From (50) we have k∇ v ( x, t ) k ≤ H + 1 σ + 1 Z R N u | v | σ +1 dx. (52)Observe that (cid:12)(cid:12)(cid:12)(cid:12)Z R N u | v | σ +1 dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k u k p k| v | σ +1 k p ′ = k u k p k v k σ +1 p ′ ( σ +1) (53)where 1 p + 1 p ′ = 1 , < p, p ′ < ∞ . Case 2. N ≤ : By (8) and (18), we have k u k p ≤ C k u k W ,p = C k B ( | v | σ +1 ) k W ,p ≤ C α k| v | σ +1 k p = C α k v k σ +1( σ +1) p for any p > p = p ′ = 2, we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z R N u | v | σ +1 dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C α k v k σ +1)2( σ +1) . By (7), taking q = 2( σ + 1), we obtain k v k σ +1) ≤ C k v k − σ +1) − · σ +1) N k∇ v k σ +1) − · σ +1) N , with 0 < σ + 1) − · σ + 1) N < , which is always satisfied when N ≤ k∇ v ( t ) k ≤ H + C α k v k σ +1) − ( σN − k∇ v k σN − . (54) Case 2. N ≥ : By (9) and (18), we obtain k u k p ≤ C k u k W ,m = C k B ( | v | σ +1 ) k W ,m ≤ C α k| v | σ +1 k m = C α k v k σ +1( σ +1) m , where 1 p = 1 m − N > ⇒ m < N . (55)Pluging into (53) and requiring m = p ′ , i.e., m = NN +2 , we get (cid:12)(cid:12)(cid:12)(cid:12)Z R N u | v | σ +1 dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C α k v k σ +1) m ( σ +1) . By (7), taking q = m ( σ + 1), we obtain k v k m ( σ +1) ≤ C k v k − m ( σ +1) − m ( σ +1) N k∇ v k m ( σ +1) − m ( σ +1) N , with 0 < m ( σ + 1) − m ( σ + 1) N < ⇒ σ < N − . (56)Then (52) yields k∇ v ( t ) k ≤ H + C α k v k σ +1) − ( σN − k∇ v k σN − . (57) AMILTONIAN REGULARIZATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION 13
For (54) and (57), k∇ v k is bounded when σN − <
2, i.e., σ < N . Therefore, the H norm ofthe solution v is bounded uniformly independent of time t , so we can conclude that we have globalsolution for any 1 ≤ σ < N . (cid:3) In conclusion, we have shown that the Schr¨odinger-Newton system (3) and the Schr¨odinger-Helmholtzsystem (4) admit short time unique solution when 1 ≤ σ < N − (by definition, N − = ∞ when N ≤ ≤ σ < N . Comparing to the result of classical NLS (1)( σ < N − for local existence and σ < N for global existence), one expects this “better” result since thenonlinear terms in system (3) and system (4) are milder than that of the classical nonlinear Schr¨odingerequation (1). Acknowledgements
This work was supported in part by the NSF, grant no. DMS-0504619, the BSF grant no. 2004271and the ISF grant no. 120/06.
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E-mail address : [email protected] (Z. H. Musslimani) Department of Mathematics, Florida State University, Tallahassee FL 32306, USA
E-mail address : [email protected] (E.S. Titi) Department of Mathematics, and Department of Mechanical and Aerospace Engineering, Uni-versity of California, Irvine, CA 92697-3875, USA,
ALSO , Department of Computer Science and AppliedMathematics, Weizmann Institute of Science, Rehovot 76100, Israel
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