The local well-posedness for nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative
© Guo et al.; licensee Springer. 2014
Received: 9 January 2014
Accepted: 4 April 2014
Published: 6 May 2014
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.
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.
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).
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.
where , and denotes the Fourier transformation of .
where denotes the Fourier transformation of .
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.
Moreover, the mapping is Lipschitz continuous from to .
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
Proof Firstly, we prove the inequality (2.2).
so that (2.5) holds.
This inequality has been proved in Theorem 4.1 of . We omit its detailed proof here. □
where is as same as in Lemma 2.1.
We take a function with on and . We denote .
Lemma 2.4 
Let , , .
Without loss of generality, we can assume that , for .
We split the domain of integration into two cases and .
Case I. Assume that .
Case II. Assume that .
Subcase 1. Assume that . Then we have .
Subcase 2. Assume that . We split this domain of integration in several pieces.
Assume that .
Assume that .
In this case, the proof is similar to that of the above case, so here we omit the detailed proof.
Assume that .
3.0: Assume that , so holds.
3.1: Assume that . By (2.10) and (2.11), we immediately 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 .
Proof Firstly, we prove (2.14) holds for .
where , , , , is admissible pair.
where , , , , is admissible pair for .
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.
If we take δ such that , then .
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. □
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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