Research Article | Open Access
Energy Solution to the Chern-Simons-Schrödinger Equations
We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space . The Coulomb gauge provides elliptic features for gauge fields . The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.
We study herein the initial value problem of the Chern-Simons-Schrödinger (CSS) equations where denotes the imaginary unit; , , and for ; is the complex scalar field; is the gauge field; is the covariant derivative for , and is a coupling constant representing the strength of interaction potential. The summation convention used involves summing over repeated indices and Latin indices are used to denote .
The CSS system of equations was proposed in [1, 2] to deal with the electromagnetic phenomena in planar domains, such as the fractional quantum Hall effect or high-temperature superconductivity. We refer the reader to [3, 4] for more information on the physical nature of these phenomena.
To figure out the optimal regularity for the CSS system, we observe that the CSS system is invariant under scaling: Therefore, the scaled critical Sobolev exponent is for . In view of (2) we may say that the initial value problem of the CSS system is mass critical.
The CSS system is invariant under the following gauge transformations: where is a smooth function. Therefore, a solution to the CSS system is formed by a class of gauge equivalent pairs . In this work, we fix the gauge by imposing the Coulomb gauge condition of , under which the Cauchy problem of the CSS system may be reformulated as follows: where the initial data . For the formulation of (6)–(8) we refer the reader to Section 3.
The initial value problem of the CSS system was investigated in [6, 7]. It was shown in  that the Cauchy problem is locally well posed in , and that there exists at least one global solution, , provided that the initial data are made sufficiently small in by finding regularized equations. They also showed, by deriving a virial identity, that solutions blow up in finite time under certain conditions. Explicit blow-up solutions were constructed in  through the use of a pseudo-conformal transformation. The existence of a standing wave solution to the CSS system has also been proved in [9, 10].
The adiabatic approximation of the Chern-Simons-Schrödinger system with a topological boundary condition was studied in , which provides a rigorous description of slow vortex dynamics in the near self-dual limit.
Taking the conservation of energy (3) into account, it seems natural to consider the Cauchy problem of the CSS system with initial data . Our purpose here is to supplement the original result of  by showing that there is a unique local- in-time solution in the energy space . We follow a rather direct means of constructing the solution and prove the uniqueness. We adapt the idea discussed in [12, 13] where a low regularity solution of the modified Schrödinger map (MSM) was studied. In fact, the CSS and MSM systems have several similarities except for the defining equation for . In the MSM, can be written roughly as , where denotes the Riesz transform. The local existence of a solution to the MSM was proved in  for the initial data in with , and similarly, the uniqueness was proved in  for with . To show the existence and uniqueness of the solution to the CSS system, the estimate of the gauge field, , is important for situations in which special structures of nonlinear terms in the defining equation for are used. The following describes are our main results.
We present some preliminaries in Section 2. Theorems 1 and 2 are proved in Sections 3 and 4, respectively. We conclude the current section by providing a few notations. We denote space time derivatives by and is used for spacial derivatives. We use the standard Sobolev spaces , with the norm and with the norm , where and . The space denotes . We define the space time norm as . We use to denote various constants. Because we are interested in local solutions, we may assume that . Thus, we replace the smooth function of with . We also use the convention of writing as shorthand for .
Lemma 3. Let and be two functions in and let be the solution of where is small at infinity. Then, and
Lemma 4. Let u be a solution of where and are real-valued functions. Then, for there exists an absolute constant such that wherein one means the homogeneous Sobolev space when and simply when .
Lemma 5. Let and be a solution to the equation Then, for and , one has where and .
Lemma 6. For , one has
3. The Proof of Theorem 1
Theorem 1 is proved in this section. Because the local well-posedness for smooth data is already known in , we simply present an a priori estimate for the solution to (6)–(8). Let us first explain (8). To derive it, note the following identities: where and . Note that the second-order terms are cancelled out. Combined with the above algebra, the equation for comes from the second and third equations in (1): We then have the formulation (6)–(8) in which is the only dynamical variable and , , and are determined through (7) and (8).
The constraint equation and the Coulomb gauge condition provide an elliptic feature of ; that is, the components can be determined from by solving the elliptic equations Taking into account that the Coulomb gauge condition in Maxwell dynamics deduces a wave equation, the previous observation was used in . Using (20), we have the following representation of :
3.1. Estimates for and
Proposition 7. Let and . One also assumes that if or if . Then, one has
Proof. The above can be checked by applying Calderon-Zygmund and Hardy-Littlewood-Sobolev inequalities. We refer to [2, Section 2] for the details.
To estimate , the special algebraic structure and divergence form of the nonlinear terms in (19) are used.
Proposition 8. Let be the solution of (19). Then, one has
Proof. Decompose as follows: We first estimate the quantity . Applying Lemma 3 to (24), we deduce that To estimate we use the Gagliardo-Nirenberg inequality with small : Applying Hardy-Littlewood-Sobolev's inequality to (25) we deduce where Proposition 7 and Lemma 6 are used. We can also derive the following from (25): The first term can be estimated as follows: where is used. The second term can be estimated as follows: where is used. Therefore, we obtain with , that is, , Therefore, we conclude that On the other hand, Lemma 3 shows that We also have from (25) that Therefore, we have
3.2. The Energy Solution to (CSS)
Proof. From the conservation of mass, we derive the first estimate. We apply Lemma 4 to (6) with and . Combined with Proposition 7, we have where . We are then left to estimate . By Proposition 8, we obtain Combining (40) and (41), we obtain where and .
Proof. Applying Lemma 5 with and , we obtain where , and . Considering Proposition 8, we obtain The other terms can be treated, as mentioned in Section 1, by similar arguments to those in [2, Section 3]. Applying Proposition 7, we have Plugging estimates (45)–(48) into (44) with , we obtain
4. The Proof of Theorem 2
Let and be solutions of (6)–(8) with the same initial data. If we set , then the equation for is We will derive where is a constant in Theorem 2 and . Then we have Considering and , we obtain Letting , for the time interval satisfying , we conclude that for , which thus proves Theorem 2.
In the remainder of this section, we derive inequality (51). Multiplying to both sides of (50) and integrating the imaginary part of , we have The integrals (II)–(V), that is, those not containing , can be controlled by applying similar arguments to those described in [2, Section 4]. Integral (II) can be estimated, considering , by for which we omit the proof.
We simply present how to control integral (I), for which we have where , . Applying Lemma 6, we obtain To control , we consider the equation for Decomposing and as (24) and (25), we have Taking into account we can rewrite the equation for as follows: where should be noted. Using the Hardy-Littlewood-Sobolev inequality, we have where and , from which we deduce . Then, we have
The term can be bounded as follows: Since , we have Since , we may check Then, we have Combining estimates (57) and (69), and denoting , we obtain where . We then obtain (51) by combining (55) and (70).
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2011-0015866), and was also partially supported by the TJ Park Junior Faculty Fellowship.
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