Research Article | Open Access
On the Weak Solution to a Fractional Nonlinear Schrödinger Equation
We obtain the existence of a global weak solution to a fractional nonlinear Schrödinger equation by the Galerkin method. Its uniqueness is also discussed. In our proof, we use harmonic analysis techniques and compactness arguments.
This paper is concerned with the following fractional partial differential equations in the dimensional torus : where and are real numbers and . is given and is a complex-valued function. Here, , the fractional Laplacian with respect to .
The fractional Laplacian operator appears in a wide class of physical systems and engineering problems, including Lévy flights, viscoelasticity, electrochemistry, control, porous media, electromagnetic, stochastic interfaces, and anomalous diffusion problems, and attracts the interests of many mathematicians; see [1–5], for example. The quasigeostrophic equation with fractional dissipation has been also extensively studied; see Constantin et al. [6–10], for example. In mathematical physics, the fractional Laplacian is often applied to describe many complicated phenomena via partial differential equations.
The Schrödinger equation is a fundamental equation in physics, which describes nonrelativistic quantum mechanical behavior. It is well known that Feynman and Hibbs  used path integrals over Brownian paths to derive the standard Schrödinger equation ( in (1)). Recently, Laskin [12, 13] showed that the path integral over the Lévy-like quantum mechanical paths allows us to generalize the classical quantum mechanics. Namely, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation. The fractional Schrödinger equation includes the space derivative of order instead of second-order space derivative in the standard Schrödinger equation. Laskin  showed the hermiticity of the fractional Hamilton operator and established the parity conservation law. Guo et al.  obtained the existence of a unique global smooth solution to the periodical boundary value problem for the fractional nonlinear Schrödinger equation.
In studying (1), there exist some essential difficulties. First, since the fractional differential operator is defined by Fourier series and is nonlocal except when , which means that depends not only on for near , but also on for all . Moreover, integration by part for nonlinear term is not valid. These bring new difficulties when doing energy estimate. And thus new harmonic analysis methods must be introduced to overcome these difficulties. Second, there are some difficulties in the convergence of the approximate solutions because of the nonlinear term. A compactness device should be given to treat this case.
We now collect the notations in this paper. The fractional Laplacian , for , can be defined as where is the Fourier coefficients of :
We shall also invoke the notion of inhomogeneous Sobolev space , which comprises all tempered distributions on such that
We now end this introduction by outlining the rest of this paper. In Section 2, we prove the existence of a weak solution to (1); see Theorem 8. The uniqueness of such weak solutions is discussed in Section 3; see Theorem 9.
2. Existence of a Weak Solution
In this section, we prove the existence of a weak solution to the following system:
Let us first recall and prove some fundamental Lemmas.
Lemma 1 (see ). Let be a Banach space; consider for some ; then is continuous map from to .
Lemma 2 (see ). Let be a bounded domain in , and , are in with then weakly in .
Lemma 3 (see ). Let , , , and , , be three Banach spaces satisfying , where , are reflexive and the embedding is compact. Endow the space with the norm Then the embedding is compact.
In the following developments, we modify the methods in .
Proposition 4. Let , , and . Then with .
Proof. First, we have
Let , where with ,
and is a standard approximation of identity, , supp , and . Now we can write
Poisson’s summation then yields
Due to the fact that
for some .
Lemma 5. Let , and let be a complex-valued function satisfying with and . Then
Proof. For , by invoking Proposition 4 and changing of variables, we have
In the following Lemma, we give a characterization of the eigenvalues and eigenvectors of the pseudodifferential operators .
Lemma 6. Let . Then there exist a sequence of real numbers and a sequence of periodical functions , such that Moreover, is a basis of satisfying for , .
Notice that the lemma is a direct consequence of elliptic regularity and functional analysis; see , for example.
Definition 7. Let , , , and . A measurable complex-valued function is said to be a weak solution to (5)-(6) on , provided the following:(1) and ;(2) satisfies (5) in the sense of distributions;(3) a.e. .
Here and hereafter, .
Now, we state our existence results in the following theorem.
Proof. We use Galerkin method.
Step 1. Construction of approximate solution.
Let be given as in Lemma 6. We consider the approximate solution which has the form where satisfy the following ordinary differential system: Here and hereafter, is the inner product in .
The system (26)-(27) is nonsingular because are linear independent. Thus we may apply standard theory of ordinary differential equations to obtain the existence of a local solution to equations (26)-(27) on , for some . We shall then, in the next step, establish some a priori estimates of the obtained solutions. This will ensure that .
Step 2. A priori estimates.
By multiplying (26) by and summing with , we have Taking real part of (28) and invoking Hölder inequality then yields By Gronwall’s inequality, we have
By multiplying (26) by , summing with , and noticing (23), we get Thanks to Lemma 5, we may consider the real part of (31) to obtain By Gronwall’s inequality again, we have
By multiplying (26) by , summing with , and taking , we have Looking into the real part of (34), we see Thus,
Now we can obtain the estimate for . By differentiating (26) with respect to , we get By multiplying (37) by and summing with , we have Notice that Taking the real part of (38) then yields
By Gronwall’s inequality, we deduce
Step 3. Convergence process.
By (30), (33), and (41), we have, up to a subsequence, still denoted by , that Thus by Lemma 3, we find By (44), there exists a function such that with . By Lemma 2 and the fact that we see Fix ; we now pass to limit in (26) to deduce A simple density argument then shows for all .
The proof of Theorem 8 is completed.
3. Uniqueness of Weak Solutions
Proof. Let and be two weak solutions of (5)-(6), given in Theorem 8, with the same datum. Then satisfies
Also, we have
By taking the inner product of (51) with in , we obtain
By taking the real part of (54) and noticing we have Thus and a.e. in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Zujin Zhang was partially supported by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007 and 20122BAB201014) and the National Natural Science Foundation of China (11326138 and 11361004). Xiaofeng Wang was partially supported by the Guangzhou higher Education Science and Technology Projection (2012A018). Zheng-an Yao was partially supported by the National Natural Science Foundation of China (11271381).
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