About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis

Volume 2014 (2014), Article ID 950132, 7 pages

http://dx.doi.org/10.1155/2014/950132
Research Article

A Time-Oscillating Hartree-Type Schrödinger Equation

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received 14 March 2014; Revised 19 May 2014; Accepted 19 May 2014; Published 9 June 2014

Academic Editor: Claudio Cuevas

Copyright © 2014 Xu Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the time-oscillating Hartree-type Schrödinger equation , where is a periodic function. For the mean value of , we show that the solution converges to the solution of for their local well-posedness and global well-posedness.

1. Introduction

In this paper, we discuss the following Hartree-type Schrödinger equation: where represents the convolution operator, , , and is a periodic function belonging to . People are interested in Hartree equation since it has many applications in the quantum theory of large systems of nonrelativistic bosonic atoms and molecules. The numbers of bosons in such systems are very large, but the interactions between them are weak. Hartree equation arises in the study of the mean-field limit of such systems; see, for example, [13].

Different from the classical Hartree-type Schrödinger equation, the coefficient of nonlinearity of is a function, especially a periodic function, not some constant, although its norm is finite. We assume is the period of ; then we can define the mean value

One can take such mean value as the coefficient of nonlinearity of Hartree-type Schrödinger equation: Then, is a time-oscillating equation and is the corresponding deterministic one. In this paper, our purpose is to discuss the relationship of well-posedness of solutions between and .

The Cauchy problem has been settled by Cazenave and Weissler [4, 5] and Miao et al. [68]. For the sake of conciseness, we only state the results without any detailed proof. The definition of admissible pair is arranged in Section 2, although we use it here.

Proposition 1. For any initial data , there exists a unique solution of (or ) defined on the maximal life interval with , . Moreover, the following properties hold.(1) for any admissible pair ; (2)(blow-up alternative) if (resp., ), then, for , one has and, for , one has .

As mentioned above, we are concerned with the behavior of solution of , when . Precisely, in the maximal life interval of solution of , we attempt to find the relationship of solutions between and as is sufficiently large. Mimicking the approach of Cazenave and Scialom [9] and Fang and Han [10] in the case of the Schrödinger equation with the local nonlinear term, we obtain the following theorems for Hartree-type.

Theorem 2. Assume the initial data and define as the solutions of . Let be the solution of with the maximal life interval . Then, we have(1)for any time satisfying , if is sufficiently large, the solution of exists in ;(2)for any admissible pair and time , in as . In particular, the convergence holds in .

Theorem 2 describes the relationship of local well-posedness of solutions between and . Furthermore, if the solution of is globally existent, that is, , we want to know whether Theorem 2 still holds. The following theorem gives the positive answer if the solution of owns sufficient decay as .

Theorem 3. Under the assumptions of Theorem 2, suppose that and and then it follows that solution of is global; that is, . Moreover, solution of is also global if is sufficiently large, and in as , for all admissible pairs .

The assumption (2) makes sure the solution of owning sufficient decay, by which deduces not only is global but also has scattering state (the details can be referred to in [68]). In fact, (2) shows that is global when immediately, according to the blow-up alternative in Proposition 1. And for , the norm of can be controlled by (2), for any , which shows by the blow-up alternative in Proposition 1. The details can be found in Lemma 9.

Many people show that the condition (2) holds in different cases. Cazenave in [4] shows (2) is true for defocusing case ( ) when . When , Miao et al. in [6] show (2) is true for defocusing case with the radial initial data and for focusing case with the radial initial data and its energy and kinetic energy smaller than the ground state’s.

When solution of is global but (2) does not hold, we are not sure the behavior of solution of even is sufficiently large. In order to have a good understanding of the development of , we think that we should understand the development of firstly, especially the blow-up rate of .

In Section 2, we introduce some notations and some useful lemmas. Theorem 2 is proved in Section 3, and Section 4 is devoted to proving Theorem 3.

2. Notations and Some Tools

In this section, we introduce some notations and useful lemmas. In order to discuss nonlinear Schrödinger equation conveniently, we always consider the equivalence of (or ): where represents the Schrödinger group.

Definition 4 (admissible pair). A pair is called admissible if and , (if , then ; if , then ).

Before stating the useful lemma, we describe the Classical Strichartz estimates. The proofs of Strichartz estimates are referred to in [5, 1114].

Lemma 5 (classical Strichartz estimates). The following properties hold.(i)For any and any admissible pair , the function belongs to In addition, there exists a constant such that (ii)Let be an interval in , , and . If is an admissible pair and , then for any admissible pair , the function belongs to . Moreover, there exists a constant independent of such that

We also need the following maximal estimate, which follows immediately from the sharp Hardy inequality (see [15]).

Lemma 6. Let ; one has

The following lemma is the key to discussing the relationship between and , which shows that when goes to infinity, the nonlinearity of converges to the nonlinearity of . The lemma has been proved by Cazenave and Scialom [9]; therefore, we only state it here without any detailed proof.

Lemma 7. Let be an admissible pair, and fix a time . Given , it follows that in , for any admissible pair .

Lemma 8. Let the initial data . For any , define as the solution of , and is the solution of with the maximal life interval . Fix a time satisfying , and suppose exists in the interval when is sufficiently large. Suppose the following conditions hold: where Then, for any admissible pair , one has

Proof. From the conditions (10), we can choose two constants and such that when , we have Set and then Proposition 1 deduces .

It follows from (3) that , where By Lemma 6, Hardy-Littlewood-Sobolev inequality, Hölder inequality, and Sobolev embedding, we obtain where Therefore, we can obtain from Strichartz estimates and Lemma 7 that It follows from Strichartz estimates, Lemma 6, Hardy-Littlewood-Sobolev inequality, Hölder inequality, and Sobolev embedding that Equations (19) and (18) can deduce that The conclusion (12) can be obtained from the above inequality, if we can show

Divide into subintervals , , with , such that in each subinterval, we have where only depends on and .

In the initial interval , since , (19), (18), and (22) deduce that where we let by the special choice of .

Then by the continuity argument, we have Since and both belong to , we choose and obtain

In the interval , Strichartz estimates and inequalities (18), (22), and (25) deduce that Let and apply the continuity argument; we have Furthermore, let again; we have Therefore, by induction argument, we obtain where .

Finally, put all estimates in each subinterval together; we have which shows (21) is true and finishes the proof of lemma.

At the end of section, we give a blow-up alternative for (or ), which is useful for the proof of Theorem 3.

Lemma 9. For any initial data and , there exists a unique solution of (or ) defined on the maximal life interval with . If one supposes then one has and with any admissible pair .

Proof. We assume ; then according to Proposition 1, we obtain and for any , . Since , we can choose sufficiently close to such that where is sufficiently small.

For any admissible pair , Strichartz estimate deduces that Note that It follows from Hardy-Littlewood-Sobolev inequality and Hölder inequality that From (33) and (35), we obtain By Sobolev embedding and the definition of , we have If we choose , we have , which is uniformly bounded for any . Then let converge to ; we have , which is a contradiction. Now, we know . Then, by (33) and (35), we know . The similar way can show ; thus we finish the proof.

3. The Proof of Theorem 2

In this section, we prove the Theorem 2. In view of Lemma 8, we only need to show that the solution of exists in the interval for sufficiently large and the condition (10) holds.

Proof. For any , let and ; furthermore, we have , where the norm is defined as (11). Divide the interval into subintervals , , and , , such that in each subinterval , we have where only depends on and , and is a sufficiently small constant which is chosen later.

In each subinterval , the integral forms (3), (19) (let , and ), and (38) apply where we choose sufficiently small such that and

On , since , then by Strichartz estimate, (19) (let ), and (39), we obtain Then the continuity argument deduces that if is sufficiently small such that , where the second inequality comes from the Strichartz estimate and the definition of .

Therefore, if is sufficiently large, the solution exists on and (10) holds. By Lemma 8, we have .

On , Strichartz estimate and (19) deduce Applying the continuity argument again, we have

Therefore, when is sufficiently large, the solution still exists in , and (10) holds. Furthermore, by Lemma 8, we have .

By induction, on each subinterval , we always have since the number of subintervals is finite, which only depends on and . So, if is sufficiently large, the solution exists in , and the condition (10) holds. Therefore, Theorem 2 follows from Lemma 8; thus we complete the proof.

4. The Proof of Theorem 3

The last section is devoted to the proof of Theorem 3. By blow-up alternative in Proposition 1, the key point is to show the boundness of as being sufficiently large.

Proof. The global existence of solution of is followed from Lemma 9. For any , Theorem 2 shows that the solution of exists in for sufficiently large and in as with any admissible pair . In particular,

Choose sufficiently large such that (32) holds, where satisfies the smallness in the proof of Lemma 9. According to the proof of Lemma 9, we have For any , triangle inequality deduces that Let go to on both sides; then we obtain from (45) (let ) that The arbitrary value of shows that when is sufficiently large, the solution is global existence by blow-up alternative in Proposition 1. Then Lemma 9 can deduce that

Finally, we show that in as for all admissible pairs . Theorem 2 shows that for any therefore, our attention is focused on

We note that Strichartz estimates and Theorem 2 show that By Lemma 9, we know ; then Lemma 7 deduces that Since then it follows from Strichartz estimates, Hardy-Littlewood-Sobolev inequality, and Hölder inequality that

By Sobolev embedding and interpolation, for any time interval , we have

We set . By (52)–(57), we have where we suppose .

Choose sufficiently large such that Since and both are admissible pairs, then it follows from (58) with that The standard continuity argument shows that if is sufficiently small, that is, large enough, we have Finally, (53), (54), (58), and (61) deduce that

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks the anonymous referees for constructive remarks.

References

  1. J. Fröhlich and E. Lenzmann, “Mean-field limit of quantum Bose gases and nonlinear Hartree equation,” in Séminaire: Équations aux Dérivées Partielles (2003-2004), Exp. No. XIX, p. 26, École Polytechnique, Palaiseau, France, 2004. View at MathSciNet
  2. Y. B. Gaididei, K. Ø. Rasmussen, and P. L. Christiansen, “Nonlinear excitations in two-dimensional molecular structures with impurities,” Physical Review E, vol. 52, no. 3, pp. 2951–2962, 1995. View at Publisher · View at Google Scholar · View at Scopus
  3. L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Soviet Physics—Journal of Experimental and Theoretical Physics, vol. 13, pp. 451–454, 1961.
  4. T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York, NY, USA; American Mathematical Society, Providence, RI, USA, 2003.
  5. T. Cazenave and F. B. Weissler, “The Cauchy problem for the nonlinear Schrödinger equation in H1,” Manuscripta Mathematica, vol. 61, no. 4, pp. 477–494, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. X. Miao, G. X. Xu, and L. F. Zhao, “Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data,” Journal of Functional Analysis, vol. 253, no. 2, pp. 605–627, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. C. X. Miao, G. X. Xu, and L. F. Zhao, “Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case,” Colloquium Mathematicum, vol. 114, no. 2, pp. 213–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. X. Miao, G. X. Xu, and L. F. Zhao, “The Cauchy problem of the Hartree equation,” Journal of Partial Differential Equations, vol. 21, no. 1, pp. 22–44, 2008. View at Zentralblatt MATH · View at MathSciNet
  9. T. Cazenave and M. Scialom, “A Schrödinger equation with time-oscillating nonlinearity,” Revista Matemática Complutense, vol. 23, no. 2, pp. 321–339, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. D. Fang and Z. Han, “A Schrödinger equation with time-oscillating critical nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, vol. 74, no. 14, pp. 4698–4708, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. S. Strichartz, “Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations,” Duke Mathematical Journal, vol. 44, no. 3, pp. 705–714, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Ginibre and G. Velo, “The global Cauchy problem for the nonlinear Schrödinger equation revisited,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 2, no. 4, pp. 309–327, 1985. View at Zentralblatt MATH · View at MathSciNet
  13. K. Yajima, “Existence of solutions for Schrödinger evolution equations,” Communications in Mathematical Physics, vol. 110, no. 3, pp. 415–426, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Keel and T. Tao, “Endpoint Strichartz estimates,” The American Journal of Mathematics, vol. 120, no. 5, pp. 955–980, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 2nd edition, 1980.