Well-posedness for the Chern-Simons-Schrödinger equations

Jishan Fan; Tohru Ozawa; Jishan Fan; Tohru Ozawa

doi:10.3934/math.2022955

AIMS Mathematics

2022, Volume 7, Issue 9: 17349-17356. doi: 10.3934/math.2022955

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Well-posedness for the Chern-Simons-Schrödinger equations

Jishan Fan ¹,
Tohru Ozawa ^{2
,
,}

1.
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2.
Department of Applied Physics, Waseda University, Tokyo, 169-8555, Japan

First, we prove uniform-in- $\epsilon$ regularity estimates of local strong solutions to the Chern-Simons-Schrödinger equations in $\mathbb{R}^2$ . Here $\epsilon$ is the dispersion coefficient. Then we prove the global well-posedness of strong solutions to the limit problem $(\epsilon = 0)$ .

Keywords:

Citation: Jishan Fan, Tohru Ozawa. Well-posedness for the Chern-Simons-Schrödinger equations[J]. AIMS Mathematics, 2022, 7(9): 17349-17356. doi: 10.3934/math.2022955

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Abstract

1. Introduction

In this paper, we consider the following Chern-Simons-Schrödinger equations ^[1,2]:

$\begin{eqnarray} &&i\partial_t\phi+(\epsilon\nabla+iA)^2\phi = A_0\phi-\lambda|\phi|^2\phi+if(x,t)\phi, \end{eqnarray}$

(1.1)

$\begin{eqnarray} &&\partial_tA-\nabla A_0 = \mbox{Im}(\epsilon\overline\phi\nabla^\bot\phi)+A^\bot|\phi|^2, \end{eqnarray}$

(1.2)

$\begin{eqnarray} && \mathrm{rot}\, A = -\frac12|\phi|^2\ \ \mbox{in}\ \ \mathbb{R}^2\times(0,\infty), \end{eqnarray}$

(1.3)

$\begin{eqnarray} &&\phi(\cdot,0) = \phi_0(\cdot)\ \ \mbox{in}\ \ \mathbb{R}^2, \end{eqnarray}$

(1.4)

where $\phi$ is the complex scalar field, $A_0$ and $A: = \left(\begin{array}{c} A_1\\ A_2 \end{array} \right)$ are the real gauge fields. $\lambda > 0$ is a coupling constant representing the strength of interaction potential. $\epsilon \geq 0$ is the dispersion coefficient. $i: = \sqrt{-1}, \nabla^\bot: = \left(\begin{array}{r} -\partial_2\\ \partial_1 \end{array} \right), A^\bot: = \left(\begin{array}{r} -A_2\\ A_1 \end{array} \right), \mathrm{rot}\, \phi: = \left(\begin{array}{r} \partial_2\phi\\ -\partial_1\phi \end{array} \right) = -\nabla^\bot\phi$ and $\mathrm{rot}\, A: = \partial_1A_2-\partial_2A_1$ . $f$ is a complex smooth function.

The system (1.1)–(1.3) 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.

The system (1.1)–(1.3) is invariant under the following gauge transformations:

$\begin{equation} \phi\to\phi e^{i\chi}, A_0\to A_0-\partial_t\chi, A\to A-\nabla\chi, \end{equation}$

(1.5)

where $\chi:\mathbb{R}^{2+1}\to\mathbb{R}$ is a smooth function. In this work, we fix the Coulomb gauge:

$\begin{equation} \mathrm{div}\, A = 0\ \ \mbox{in}\ \ \mathbb{R}^2\times(0,\infty). \end{equation}$

(1.6)

Taking $\mathrm{div}\,$ to (1.2), $\mathrm{rot}\,$ to (1.3), using $\mathrm{rot}\, ^2A = -\Delta A+\nabla \mathrm{div}\, A$ and (1.6), we can reformulate (1.1)–(1.3) as follows.

$\begin{eqnarray} &&i\partial_t\phi+\epsilon^2\Delta\phi+2i\epsilon A\cdot\nabla\phi-|A|^2\phi = A_0\phi-\lambda|\phi|^2\phi+if\phi, \end{eqnarray}$

(1.7)

$\begin{eqnarray} &&-\Delta A_0 = \epsilon\mbox{Im}(\partial_1\phi\partial_2\overline\phi-\partial_2\phi\partial_1\overline\phi)- \mathrm{rot}\,(|\phi|^2A), \end{eqnarray}$

(1.8)

$\begin{eqnarray} &&\Delta A = \frac12 \mathrm{rot}\,|\phi|^2\ \ \mbox{in}\ \ \mathbb{R}^2\times(0,\infty), \end{eqnarray}$

(1.9)

$\begin{eqnarray} &&\phi(\cdot,0) = \phi_0(\cdot)\ \ \mbox{in}\ \ \mathbb{R}^2. \end{eqnarray}$

(1.10)

Bergé-de Bouard-Saut ^[3] proved that the Cauchy problem is locally well-posed when $\phi_0\in H^2$ . Huh ^[4] improved it to the case $\phi_0\in H^1$ . Lim ^[5] refined it to the case $\phi_0\in H^s$ for $s\geq1$ . In ^[3], the authors also showed, by deriving a virial identity, that solutions blow up in finite time under certain conditions. The existence of a standing wave solution has also been proved in ^[6,7]. Liu-Smith-Tataru ^[8] proved the local well-posedness of (1.1)–(1.3) for small data $\phi_0\in H^\sigma$ with any $\sigma > 0$ under the Lorentz (heat) gauge:

$\begin{equation} A_0 = \mathrm{div}\, A. \end{equation}$

(1.11)

Please see ^{[9,10,11,12,13]} for other studies of the problem (1.1)–(1.3).

All the above results dealt with the case $\epsilon = 1$ . The aim of this paper is to prove the uniform regularity estimates independent of $\epsilon$ and prove the global well-posedness of strong solutions to the limit problem $(\epsilon = 0)$ . We will prove

Theorem 1.1. Let $\phi_0\in H^s$ with $s > 1$ . Then there exists $T_0 > 0$ such that the problem $(1.7)$ – $(1.10)$ has a unique local strong solution $(\phi, A_0, A)$ on $[0, T_0]$ satisfying

$\begin{equation} \underset{\epsilon \in [0,1]}{\sup} \ \underset{t \in [0,T_0]}{\sup}\|(\phi,\nabla A_0,\nabla A)(\cdot,t)\|_{H^s}\leq C. \end{equation}$

(1.12)

Remark 1.1. Our approach is very much technically simpler than that in ^[5] when $s > 1$ .

Remark 1.2. We are unable to prove a similar result for the Maxwell-Schrödinger system.

When $\epsilon = 0$ , the problem (1.7)–(1.10) reads as follows.

$\begin{eqnarray} &&\partial_t\phi = -iA_0\phi-i|A|^2\phi+\lambda i|\phi|^2\phi+f\phi, \end{eqnarray}$

(1.13)

$\begin{eqnarray} &&\Delta A_0 = \mathrm{rot}\,(|\phi|^2A), \end{eqnarray}$

(1.14)

$\begin{eqnarray} &&\Delta A = \frac12 \mathrm{rot}\,|\phi|^2, \end{eqnarray}$

(1.15)

$\begin{eqnarray} && \mathrm{div}\, A = 0\ \ \mbox{in}\ \ \mathbb{R}^2\times(0,\infty), \end{eqnarray}$

(1.16)

$\begin{eqnarray} &&\phi(\cdot,0) = \phi_0(\cdot)\ \ \mbox{in}\ \ \mathbb{R}^2. \end{eqnarray}$

(1.17)

We have

Theorem 1.2. Let $\phi_0\in H^s\cap L^\infty$ with $s\geq1$ . Then the problem $(1.13)$ – $(1.17)$ has a unique global strong solution $(\phi, A_0, A)$ satisfying $(1.12)$ .

Remark 1.3. For recent literature on the concept of "well-posedness, " we refer the reader to ^[14,15,16]. Regarding vanishing dispersion limit for related equations, see ^[17,18,19].

In the following proofs, we will use the bilinear commutator and product estimates due to Kato-Ponce ^[20]:

$\begin{eqnarray} &&\|\Lambda^s(fg)-f\Lambda^sg\|_{L^p}\leq C(\|\nabla f\|_{L^{p_1}}\|\Lambda^{s-1}g\|_{L^{q_1}}+\|g\|_{L^{p_2}}\|\Lambda^sf\|_{L^{q_2}}), \end{eqnarray}$

(1.18)

$\begin{eqnarray} &&\|\Lambda^s(fg)\|_{L^p}\leq C(\|f\|_{L^{p_1}}\|\Lambda^sg\|_{L^{q_1}}+\|\Lambda^sf\|_{L^{p_2}}\|g\|_{L^{q_2}}) \end{eqnarray}$

(1.19)

with $s > 0, \Lambda: = (-\Delta)^\frac12$ and $\frac1p = \frac{1}{p_1}+\frac{1}{q_1} = \frac{1}{p_2}+\frac{1}{q_2}$ .

2. Proof of Theorem 1.1

Since the local well-posedness of smooth solution is well-known ^[5], we only need to show (1.12).

(1.7) can be written as

$\begin{equation} \partial_t\phi = i\epsilon^2\Delta\phi-2\epsilon A\cdot\nabla\phi-iA_0\phi-i|A|^2\phi+i\lambda|\phi|^2\phi+f\phi. \end{equation}$

(2.1)

Testing (2.1) by $\overline\phi$ , taking the real parts and using (1.5), we have

$\frac12\frac{\mathrm{d}}{\mathrm{d}t}\int|\phi|^2\mathrm{d}x = \mbox{Re}\int f|\phi|^2\mathrm{d}x\leq\|f\|_{L^\infty}\|\phi\|_{L^2}^2,$

which gives

$\begin{equation} \|\phi\|_{L^2}\leq C. \end{equation}$

(2.2)

Applying $\Lambda^s$ to (2.1), testing by $\Lambda^s\overline\phi$ , taking the real parts, and using (1.5), (1.17) and (1.18), we get

$\begin{eqnarray} &&\frac12\frac{\mathrm{d}}{\mathrm{d}t}\int|\Lambda^s\phi|^2\mathrm{d}x = -2\epsilon\mbox{Re}\int(\Lambda^s(A\cdot\nabla\phi)-A\cdot\nabla\Lambda^s\phi)\Lambda^s\overline\phi\mathrm{d}x\\ &&-\mbox{Re}\ i\int\Lambda^s(A_0\phi)\Lambda^s\overline\phi\mathrm{d}x-\mbox{Re}\ i\int\Lambda^s(|A|^2\phi)\Lambda^s\overline\phi\mathrm{d}x\\ &&+\mbox{Re}\lambda i\int\Lambda^s(|\phi|^2\phi)\Lambda^s\overline\phi\mathrm{d}x+\mbox{Re}\int\Lambda^s(f\phi)\Lambda^s\overline\phi\mathrm{d}x\\ &\leq&C\|\nabla A\|_{L^\infty}\|\Lambda^s\phi\|_{L^2}^2+C\|\nabla\phi\|_{L^p}\|\Lambda^sA\|_{L^\frac{2p}{p-2}}\|\Lambda^s\phi\|_{L^2}\\ &&\qquad\qquad\left(p: = \frac{2}{(2-s)^+}\ \ \mbox{if}\ \ s < 2\ \ \mbox{and}\ \ p = 4\ \ \mbox{if}\ \ s\geq2\right)\\ &&+C\|A_0\|_{L^\infty}\|\Lambda^s\phi\|_{L^2}^2+C\|\phi\|_{L^\infty}\|\Lambda^sA_0\|_{L^2}\|\Lambda^s\phi\|_{L^2}\\ &&+C\|A\|_{L^\infty}^2\|\Lambda^s\phi\|_{L^2}^2+C\|\phi\|_{L^\infty}\|A\|_{L^\infty}\|\Lambda^sA\|_{L^2}\|\Lambda^s\phi\|_{L^2}\\ &&+C\|\phi\|_{L^\infty}^2\|\Lambda^s\phi\|_{L^2}^2+C\|f\|_{L^\infty}\|\Lambda^s\phi\|_{L^2}^2 +C\|\phi\|_{L^\infty}\|\Lambda^sf\|_{L^2}\|\Lambda^s\phi\|_{L^2}. \end{eqnarray}$

(2.3)

Noting

$\begin{eqnarray} \|\nabla A\|_{L^\infty}+\|\Lambda^sA\|_{L^\frac{2p}{p-2}}+\|\Lambda^sA\|_{L^2}&\leq&C\|\nabla A\|_{L^2}+C\|\Lambda^{s+1}A\|_{L^2}\\ &\leq&C\|\phi\|_{L^4}^2+C\|\Lambda^s(|\phi|^2)\|_{L^2}\\ &\leq&C\|\phi\|_{L^4}^2+C\|\phi\|_{L^\infty}\|\Lambda^s\phi\|_{L^2}\leq\|\phi\|_{H^s}^2 \end{eqnarray}$

(2.4)

and

$\begin{eqnarray} \|\Lambda^sA_0\|_{L^2}&\leq&C\|\Lambda^{s-2} \mathrm{div}\,(\mbox{Im}(\epsilon\overline\phi\nabla^\bot\phi)+A^\bot|\phi|^2)\|_{L^2}\\ &\leq&C\|\phi\|_{H^s}^2+C\|\phi\|_{L^\infty}^2\|\Lambda^{s-1}A\|_{L^2}+C\|A\|_{L^\infty}\|\phi\|_{L^\infty}\|\Lambda^{s-1}\phi\|_{L^2}\\ &\leq&C\|\phi\|_{H^s}^4+C, \end{eqnarray}$

(2.5)

we obtain

$\begin{eqnarray} \|A_0\|_{L^\infty}&\leq&C\|A_0\|_{L^4}+C\|\Lambda^sA_0\|_{L^2}\\ &\leq&C\|\nabla A_0\|_{L^\frac43}+C\|\Lambda^sA_0\|_{L^2}\\ &\leq&C\|\overline\phi\nabla^\bot\phi\|_{L^\frac43}+C\|A^\bot|\phi|^2\|_{L^\frac43}+C\|\Lambda^sA_0\|_{L^2}\\ &\leq&C\|\phi\|_{L^4}\|\nabla\phi\|_{L^2}+C\|A\|_{L^\infty}\|\phi\|_{L^\frac83}^2+C\|\Lambda^sA_0\|_{L^2}\\ &\leq&C\|\phi\|_{H^s}^4+C \end{eqnarray}$

(2.6)

due to (1.2), and

$\begin{eqnarray} \|A\|_{L^\infty}&\leq&C\|A\|_{L^4}+C\|\Lambda^{s+1}A\|_{L^2}\\ &\leq&C\|\nabla A\|_{L^\frac43}+C\|\Lambda^{s+1}A\|_{L^2}\\ &\leq&C\|\phi\|_{L^\frac83}^2+C\|\Lambda^s(|\phi|^2)\|_{L^2}\\ &\leq&C\|\phi\|_{L^\frac83}^2+C\|\phi\|_{L^\infty}\|\Lambda^s\phi\|_{L^2}\\ &\leq&C\|\phi\|_{H^s}^2. \end{eqnarray}$

(2.7)

Inserting (2.4), (2.5), (2.6), and (2.7) into (2.3), we arrive at

$\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\int|\Lambda^s\phi|^2\mathrm{d}x\leq C\|\phi\|_{H^s}^6+C, \end{equation}$

(2.8)

which gives (1.12).

The proof is complete.

3. Proof of Theorem 1.2

First, we still have (2.2).

It is clear that

$\frac{\mathrm{d}}{\mathrm{d}t}|\phi|^2 = (f+\overline f)|\phi|^2\leq 2\|f\|_{L^\infty}\|\phi\|_{L^\infty}^2,$

and thus

$\|\phi\|_{L^\infty}^2\leq\|\phi_0\|_{L^\infty}^2+2\int_0^t\|f\|_{L^\infty}\|\phi\|_{L^\infty}^2\mathrm{d}\tau,$

which yields

$\begin{equation} \|\phi\|_{L^\infty}\leq C. \end{equation}$

(3.1)

Applying $\nabla$ to (1.12), testing by $\nabla\overline\phi$ , taking the real parts and using (1.15), (2.2) and (3.1), we see that

$\begin{eqnarray*} \frac12\frac{\mathrm{d}}{\mathrm{d}t}\int|\nabla\phi|^2\mathrm{d}x& = &-\mbox{Re}\ i\int\phi\nabla A_0\nabla\overline\phi\mathrm{d}x-\mbox{Re}\ i\int\phi\nabla|A|^2\cdot\nabla\overline\phi\mathrm{d}x\\ &&+\lambda\mbox{Re}\ i\int\phi\nabla\overline\phi\nabla|\phi|^2\mathrm{d}x+\mbox{Re}\int\nabla(f\phi)\nabla\overline\phi\mathrm{d}x\\ &\leq&\|\phi\|_{L^\infty}\|\nabla A_0\|_{L^2}\|\nabla\phi\|_{L^2}+2\|\phi\|_{L^\infty}\|A\|_{L^4}\|\nabla A\|_{L^4}\|\nabla\phi\|_{L^2}\\ &&+2\lambda\|\phi\|_{L^\infty}^2\|\nabla\phi\|_{L^2}^2+\|f\|_{L^\infty}\|\nabla\phi\|_{L^2}^2+\|\phi\|_{L^\infty}\|\nabla f\|_{L^2}\|\nabla\phi\|_{L^2}\\ &\leq&C\||\phi|^2A\|_{L^2}\|\nabla\phi\|_{L^2}+C\|A\|_{L^4}\|\nabla A\|_{L^4}\|\nabla\phi\|_{L^2}+C\|\nabla\phi\|_{L^2}^2+C\\ &\leq&C\|\phi\|_{L^8}^2\|A\|_{L^4}\|\nabla\phi\|_{L^2}+C\|\nabla A\|_{L^\frac43}\|\nabla A\|_{L^4}\|\nabla\phi\|_{L^2}+C\|\nabla\phi\|_{L^2}^2+C\\ &\leq&C\|\nabla A\|_{L^\frac43}\|\nabla\phi\|_{L^2}+C\|\nabla A\|_{L^\frac43}\|\nabla A\|_{L^4}\|\nabla\phi\|_{L^2}+C\|\nabla\phi\|_{L^2}^2+C\\ &\leq&C\||\phi|^2\|_{L^\frac43}\|\nabla\phi\|_{L^2}+C\||\phi|^2\|_{L^\frac43}\||\phi|^2\|_{L^4}\|\nabla\phi\|_{L^2}+C\|\nabla\phi\|_{L^2}^2+C\\ &\leq&C\|\nabla\phi\|_{L^2}^2+C, \end{eqnarray*}$

which implies

$\begin{equation} \|\phi(\cdot,t)\|_{H^1}\leq C. \end{equation}$

(3.2)

Here we have used the estimates

$\begin{equation} \|\nabla A\|_{L^q}\leq C\||\phi|^2\|_{L^q}\leq C\ \ \mbox{for}\ \ 1 < q < \infty, \end{equation}$

(3.3)

and

$\begin{equation} \|\nabla A_0\|_{L^2}\leq C\|A^\bot|\phi|^2\|_{L^2}\leq C\|A\|_{L^4}\leq C\|\nabla A\|_{L^\frac43}\leq C, \end{equation}$

(3.4)

On the other hand, noting that

$\begin{eqnarray} \|A_0\|_{L^\infty}&\leq&C\|A_0\|_{L^4}+C\|\nabla A_0\|_{L^4}\\ &\leq&C\|\nabla A_0\|_{L^\frac43}+C\|\nabla A_0\|_{L^4}\\ &\leq&C\||\phi|^2A\|_{L^\frac43}+C\||\phi|^2A\|_{L^4}\\ &\leq&C\|A\|_{L^4}\leq C, \end{eqnarray}$

(3.5)

and

$\begin{equation} \|A\|_{L^\infty}\leq C\|A\|_{L^4}+C\|\nabla A\|_{L^4}\leq C. \end{equation}$

(3.6)

Then applying $\Lambda^s$ to (1.12), testing by $\Lambda^s\overline\phi$ , taking the real parts, using (1.18), (3.1), (3.5) and (3.6), we obtain

$\begin{eqnarray*} &&\frac12\frac{\mathrm{d}}{\mathrm{d}t}\int|\Lambda^s\phi|^2\mathrm{d}x = -\mbox{Re}\ i\int\Lambda^s(A_0\phi)\Lambda^s\overline\phi\mathrm{d}x-\mbox{Re}\ i\int\Lambda^s(|A|^2\phi)\Lambda^s\overline\phi\mathrm{d}x\\ &&+\lambda\mbox{Re}\ i\int\Lambda^s(|\phi|^2\phi)\Lambda^s\overline\phi\mathrm{d}x+\mbox{Re}\int\Lambda^s(f\phi)\Lambda^s\overline\phi\mathrm{d}x\\ &\leq&C\|A_0\|_{L^\infty}\|\Lambda^s\phi\|_{L^2}^2+C\|\phi\|_{L^\infty}\|\Lambda^sA_0\|_{L^2}\|\Lambda^s\phi\|_{L^2}\\ &&+C\|A\|_{L^\infty}^2\|\Lambda^s\phi\|_{L^2}^2+C\|\phi\|_{L^\infty}\|A\|_{L^\infty}\|\Lambda^sA\|_{L^2}\|\Lambda^s\phi\|_{L^2}\\ &&+C\|\phi\|_{L^\infty}^2\|\Lambda^s\phi\|_{L^2}^2+C\|f\|_{L^\infty}\|\Lambda^s\phi\|_{L^2}^2 +C\|\phi\|_{L^\infty}\|\Lambda^sf\|_{L^2}\|\Lambda^s\phi\|_{L^2}\\ &\leq&C\|\Lambda^s\phi\|_{L^2}^2+C\|\Lambda^sA_0\|_{L^2}\|\Lambda^s\phi\|_{L^2}+C\|\Lambda^sA\|_{L^2}\|\Lambda^s\phi\|_{L^2}+C\\ &\leq&C\|\Lambda^s\phi\|_{L^2}^2+C\|\Lambda^{s-1}(|\phi|^2A)\|_{L^2}\|\Lambda^s\phi\|_{L^2} +C\|\Lambda^{s-1}(|\phi|^2)\|_{L^2}\|\Lambda^s\phi\|_{L^2}+C\\ &\leq&C\|\Lambda^s\phi\|_{L^2}^2+C\|\Lambda^{s-1}A\|_{L^2}\|\Lambda^s\phi\|_{L^2}+C\|\Lambda^{s-1}\phi\|_{L^2}\|\Lambda^s\phi\|_{L^2}+C\\ &\leq&C\|\Lambda^s\phi\|_{L^2}^2+C, \end{eqnarray*}$

which leads to (1.12).

Here we have used the estimates

$\begin{eqnarray} \|\Lambda^sA\|_{L^2}&\leq&C\|\Lambda^{s-1}(|\phi|^2)\|_{L^2}\\ &\leq&C\|\Lambda^{s-1}\phi\|_{L^2}\leq C\|\Lambda^s\phi\|_{L^2}+C, \end{eqnarray}$

(3.7)

and

$\begin{eqnarray} \|\Lambda^sA_0\|_{L^2}&\leq&C\|\Lambda^{s-1}(|\phi|^2A)\|_{L^2}\\ &\leq&C\|\Lambda^{s-1}\phi\|_{L^2}+C\|\Lambda^{s-1}A\|_{L^2}\\ &\leq&C\|\Lambda^s\phi\|_{L^2}+C. \end{eqnarray}$

(3.8)

The proof is complete.

4. Conclusions

We have obtained the Sobolev estimates on local time interval uniformly in the dispersion coefficient $\epsilon \in (0, 1]$ . Moreover, we have proved the existence and uniqueness of global solutions to the limit problem $\epsilon = 0$ .

Acknowledgments

Fan is partially supported by NSFC (No. 11971234).

Conflict of interest

The authors declare no conflict of interest.

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AIMS Mathematics

Well-posedness for the Chern-Simons-Schrödinger equations

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Abstract

1. Introduction

2. Proof of Theorem 1.1

3. Proof of Theorem 1.2

4. Conclusions

Acknowledgments

Conflict of interest

References

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通讯作者: 陈斌, bchen63@163.com

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AIMS Mathematics

Well-posedness for the Chern-Simons-Schrödinger equations

Related Papers:

Abstract

1. Introduction

2. Proof of Theorem 1.1

3. Proof of Theorem 1.2

4. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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