Abstract
We study the Cauchy problem of the nonlinear fourthorder Schrödinger equation with masscritical nonlinearity and derivative: , , , where a, b, and c are real numbers. We obtain the local wellposedness for the Cauchy problem with low regularity initial value data by the Fourier restriction norm method.
Keywords:
nonlinear fourthorder Schrödinger equation with derivative; Fourier restriction norm method; Cauchy problem1 Introduction
In [1], Fibich et al. discussed the following fourthorder 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 masscritical.
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 fourthorder Schrödinger equation with masscritical nonlinearity and twoorder derivative in one dimension:
where are complexvalued function, is the complex conjugate quantity of . a, b, and c are real numbers. We are interested in obtaining the wellposedness for the Cauchy problem of (1.1) with initial value data under low regularity (which means , ). Tao et al. obtained the global wellposedness for the Schrödinger equations with derivative () by the Imethod (see [2,3]). Bourgain obtained the wellposedness for the nonlinear Schrödinger equation () by the Fourier restriction norm method (see [4]). The character of (1.4) lies in the coexistence of the masscritical nonlinearity and the twoorder derivative. We will discuss the local wellposedness for the fourthorder 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 fourthorder 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 1.1Let, . Then the system (1.4) is locally solvable in, i.e., for any, there exists a correspondingsuch that the system (1.4) has a unique solution in the class
Moreover, the mappingis Lipschitz continuous fromto.
In [6], Cui et al. obtained the local wellposedness with the initial condition satisfying , for in (1.4).
Thus from the above theorem, we can see the following result.
Remark 1.1 When masscritical nonlinearity and nonlinearity with derivative appear at the same time, nonlinearity with secondorder derivative plays more important role. This property is consistent with the classical Schrödinger equation which has both masscritical nonlinearity and firstorder 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 2.1Assume thatis an admissible pair. Let. We have
Proof Firstly, we prove the inequality (2.2).
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. □
whereis 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 take a function with on and . We denote .
Similar to the proof of Lemma 3.2 in [9] (or see [[10], Lemmas 3.13.3], [[11], Lemma 2.3], [[12], Lemma 2.7]), we have the following estimates.
Lemma 2.3For any reals, , , and. We have
Lemma 2.4[12]
Iff, , , andbelong to a Schwartz space on, then we have
where
Lemma 2.5Let, , . Then we have
Proof By the definition of and duality, the inequality (2.11) is reduced to the following estimate:
Without loss of generality, we can assume that , for .
We split the domain of integration into two cases and .
Noting that , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
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.
Similar to the proof of subcase 1, noting that , , by Lemma 2.4, the Hölder inequality, and Lemma 2.2, we obtain
In this case, the proof is similar to that of the above case, so here we omit the detailed proof.
In this case, we easily get
By a straightforward calculation, we can obtain
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
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 2.6Let, . 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 2.52.6, 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. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by Natural Science of Shanxi province (No. 20130110032, No. 20130110022 and No. 20100110011), Natural Science Foundation of China (No. 11071149), Research Project Supported by Shanxi Scholarship Council of China (No. 2011011).
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