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The local well-posedness for nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative
Boundary Value Problems volume 2014, Article number: 90 (2014)
Abstract
We study the Cauchy problem of the nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative: , , , where a, b, and c are real numbers. We obtain the local well-posedness for the Cauchy problem with low regularity initial value data by the Fourier restriction norm method.
1 Introduction
In [1], Fibich et al. discussed the following fourth-order Schrödinger equation:
They gave the sufficient conditions of the existence of solutions in the space . For the case , the mass is still invariant under the scaling . We call this case mass-critical.
Meanwhile, in [1], Fibich et al. mentioned the physical motivation of (1.1). With nonparaxial effects, one obtained the perturbed nonlinear Schrödinger equation:
Furthermore, with vectorial effects, one obtained the following equation:
Evidently, the nonlinearities with derivatives appear. It is well known that nonlinearities with derivatives bring about more difficulties to solve the problem for us. Especially, there are so many nonlinearities with derivatives in (1.2) and (1.3).
So, in this paper we will study the following nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and two-order derivative in one dimension:
where are complex-valued function, is the complex conjugate quantity of . a, b, and c are real numbers. We are interested in obtaining the well-posedness for the Cauchy problem of (1.1) with initial value data under low regularity (which means , ). Tao et al. obtained the global well-posedness for the Schrödinger equations with derivative () by the I-method (see [2, 3]). Bourgain obtained the well-posedness for the nonlinear Schrödinger equation () by the Fourier restriction norm method (see [4]). The character of (1.4) lies in the coexistence of the mass-critical nonlinearity and the two-order derivative. We will discuss the local well-posedness for the fourth-order Schrödinger equation by the Fourier restriction norm method.
For the complicated case, we will discuss it in another paper.
First, we introduce the following notations. We define the Sobolev norms by
where , and denotes the Fourier transformation of .
We also define the spaces (see [5]) on by
where denotes the Fourier transformation of .
We denote by () the fundamental solution operator of the fourth-order Schrödinger equation, i.e.,
where denotes the Fourier transformation of φ, and represents the inverse Fourier transformation.
We use C to denote various constants which may be different from in particular cases of use throughout.
The main result of this paper is the following theorem.
Theorem 1.1 Let , . Then the system (1.4) is locally solvable in , i.e., for any , there exists a corresponding such that the system (1.4) has a unique solution in the class
Moreover, the mapping is Lipschitz continuous from to .
In [6], Cui et al. obtained the local well-posedness with the initial condition satisfying , for in (1.4).
Thus from the above theorem, we can see the following result.
Remark 1.1 When mass-critical nonlinearity and nonlinearity with derivative appear at the same time, nonlinearity with second-order derivative plays more important role. This property is consistent with the classical Schrödinger equation which has both mass-critical nonlinearity and first-order derivative nonlinearity.
2 The preliminary estimates
Definition 2.1 For two integers and , we say that is an admissible pair if the following condition is satisfied:
We have the following Strichartz estimate (see [6]): For any admissible pair
Lemma 2.1 Assume that is an admissible pair. Let . We have
For any , we have
where .
Proof Firstly, we prove the inequality (2.2).
For any , we have
where .
Noting that , using the Strichartz estimate, we obtain
Next we prove (2.3), we only need to prove that for we have
Changing the variable to , we obtain
From (2.6) and (2.7), we can obtain
so that (2.5) holds.
Similarly, (2.4) follows from the following inequality:
This inequality has been proved in Theorem 4.1 of [7]. We omit its detailed proof here. □
Lemma 2.2 If for . Then we have
where is as same as in Lemma 2.1.
Proof Noting that is an admissible pair, so we have
Therefore, using (2.7), Minkowski’s inequality, (2.9), and taking , we can obtain
By interpolation [8] between (2.10) and the following relation:
we immediately obtain for all
□
We take a function with on and . We denote .
Similar to the proof of Lemma 3.2 in [9] (or see [[10], Lemmas 3.1-3.3], [[11], Lemma 2.3], [[12], Lemma 2.7]), we have the following estimates.
Lemma 2.3 For any real s, , , and . We have
Lemma 2.4 [12]
If f, , , and belong to a Schwartz space on , then we have
where
Lemma 2.5 Let , , . Then we have
Proof By the definition of and duality, the inequality (2.11) is reduced to the following estimate:
for all , where
Let , , .
Without loss of generality, we can assume that , for .
We split the domain of integration into two cases and .
Case I. Assume that .
Noting that , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
Case II. Assume that .
Subcase 1. Assume that . Then we have .
Noting that , by Lemma 2.4, the Hölder inequality, and Lemma 2.2, we obtain
Subcase 2. Assume that . We split this domain of integration in several pieces.
Assume that .
Similar to the proof of subcase 1, noting that , , by Lemma 2.4, the Hölder inequality, and Lemma 2.2, we obtain
Assume that .
In this case, the proof is similar to that of the above case, so here we omit the detailed proof.
Assume that .
In this case, we easily get
By a straightforward calculation, we can obtain
3.0: Assume that , so holds.
Noting that , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
3.1: Assume that . By (2.10) and (2.11), we immediately obtain .
Noting that , , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
3.2: Assume that .
The proof is similar to that of the case 3.1, so here omit the detailed proof.
3.3: Assume that . By (2.10) and (2.11), we immediately obtain .
Noting that , , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
□
Lemma 2.6 Let , . Then we have the following inequality:
Proof Firstly, we prove (2.14) holds for .
By the definition, the Hölder inequality, and the Sobolev inequality, we have
That means
Again using the Hölder inequality, the Sobolev inequality in the variable x, and Lemma 2.1, we obtain
where , , , , is admissible pair.
Combining (2.16) and (2.17), we obtain
Secondly, using the Leibniz rule for fractional power, in a similar way, we obtain
where , , , , is admissible pair for .
Taking , , ; and for some , , , ; , , , for in (2.19), noting and , we obtain
which completes the proof. □
3 Proof of main result
Proof of Theorem 1.1 We will prove Theorem 1.1 by using the Banach fixed point theorem. Let . We rewrite (1.4) in integral form:
We denote
and define a mapping S as follows: For ,
In the sequel we will prove that S is well defined and it is a contraction map on Z.
By Lemma 2.3 and Lemmas 2.5-2.6, for , we obtain
If we take δ such that , then .
Similarly, for , in an analogous way to the above, we can obtain
Furthermore, if we take δ such that then S is a contraction mapping of Z into itself. The desired result immediately follows from Banach’s fixed point theorem. That means that there is a unique solution which solves the Cauchy problem (1.4) for . The Lipschitz continuousness from to is easily obtained from the above proof process. □
References
Fibich G, Ilan B, Papanicolaou G: Self-focusing with fourth-order dispersion. SIAM J. Appl. Math. 2002, 62: 1437-1462. 10.1137/S0036139901387241
Colliander J, Keel M, Staffilani G, Takaoka H, Tao T: Global well-posedness for Schrödinger equations with derivative. SIAM J. Math. Anal. 2001, 33: 649-669. 10.1137/S0036141001384387
Colliander J, Keel M, Staffilani G, Takaoka H, Tao T: A refined global well-posedness result for Schrödinger equations with derivative. SIAM J. Math. Anal. 2002, 34: 64-86. 10.1137/S0036141001394541
Bourgain J: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 1993, 3: 107-156. 10.1007/BF01896020
Bourgain J: Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Int. Math. Res. Not. 1998, 5: 253-283.
Cui SB, Guo CH:Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces and applications. Nonlinear Anal. 2007, 67: 687-707. 10.1016/j.na.2006.06.020
Kenig CE, Ponce G, Vega L: Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 1991, 40: 33-69. 10.1512/iumj.1991.40.40003
Bergh J, Löfstöm T: Interpolation Spaces. Springer, New York; 1976.
Kenig CE, Ponce G, Vega L: A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 1996, 9: 573-603. 10.1090/S0894-0347-96-00200-7
Kenig CE, Ponce G, Vega L: The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 1993, 71: 1-21. 10.1215/S0012-7094-93-07101-3
Guo CH, Cui SB: Global existence for 2D nonlinear Schrödinger equations via high-low frequency decomposition. J. Math. Anal. Appl. 2006, 324: 882-907. 10.1016/j.jmaa.2005.12.056
Huo ZH, Jia YL: Well-posedness for the Cauchy problem of coupled Hirota equation with low regularity data. J. Math. Anal. Appl. 2006, 322: 566-579. 10.1016/j.jmaa.2005.09.033
Acknowledgements
This work is supported by Natural Science of Shanxi province (No. 2013011003-2, No. 2013011002-2 and No. 2010011001-1), Natural Science Foundation of China (No. 11071149), Research Project Supported by Shanxi Scholarship Council of China (No. 2011-011).
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Guo, C., Sun, S. & Ren, H. The local well-posedness for nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative. Bound Value Probl 2014, 90 (2014). https://doi.org/10.1186/1687-2770-2014-90
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DOI: https://doi.org/10.1186/1687-2770-2014-90