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.
Keywords:nonlinear fourth-order Schrödinger equation with derivative; Fourier restriction norm method; Cauchy problem
In , Fibich et al. discussed the following fourth-order Schrödinger equation:
Meanwhile, in , 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 ). 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.
We also define the spaces (see ) on by
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.
In , 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
We have the following Strichartz estimate (see ): For any admissible pair
Proof Firstly, we prove the inequality (2.2).
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 . We omit its detailed proof here. □
By interpolation  between (2.10) and the following relation:
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
The proof is similar to that of the case 3.1, so here omit the detailed proof.
By the definition, the Hölder inequality, and the Sobolev inequality, we have
Again using the Hölder inequality, the Sobolev inequality in the variable x, and Lemma 2.1, we obtain
Combining (2.16) and (2.17), we obtain
Secondly, using the Leibniz rule for fractional power, in a similar way, we obtain
which completes the proof. □
3 Proof of main result
In the sequel we will prove that S is well defined and it is a contraction map on Z.
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. □
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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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