A Liouville theorem for the Chern--Simons--Schr{\"o}dinger equation

TL;DR

This paper proves a Liouville theorem for the Chern--Simons--Schr{"o}dinger equation, supporting the soliton resolution conjecture for certain initial data, and extends understanding of the equation's solutions.

Contribution

The paper establishes a Liouville theorem for the Chern--Simons--Schr{"o}dinger equation, contributing to the theoretical understanding of its solution behavior.

Findings

Liouville theorem proven for the equation

Supports soliton resolution conjecture in unweighted spaces

Extends previous results in weighted spaces

Abstract

In this paper we prove a Liouville theorem for the Chern--Simons--Schr{\"o}dinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [KKO22] for the soliton resolution result in a weighted space.

Equations

$$i({\partial }_t+i{A}_t[u])u+{\partial }_r^{2}u+\frac{1}{r}{\partial }_{r}u-(\frac{m+A_{\theta }[u]}{r}{)}^{2}u+|u{|}^{2}u=0,$$

$$A_t[u]=-{\int }_r^{\infty }(m+A_{\theta }[u])|u{|}^2\frac{d{r}^{\prime }}{r^{\prime }},\text{and}{A}_{\theta }[u]=-\frac{1}{2}{\int }_0^r|u{|}^2{r}^{\prime }d{r}^{\prime }.$$

$$\varphi (t,x)=u(t,r)e^{im\theta }.$$

$$E[u]$$

$$M[u]$$

$$⟨f,g⟩=Re\int f(r)\overline{g(r)}rdr.$$

$$\frac{d}{dt}E[u]=\int Re({\partial }_r\bar{u})({\partial }_r{u}_t)rdr+\int (\frac{m+A_{\theta }[u]}{r}{)}^{2}Re(\bar{u}{u}_t)rdr$$

$$-\int |u{|}^{2}Re(\bar{u}{u}_t)rdr-\int \frac{m+A_{\theta }[u]}{r}|u{|}^2{\int }_0^{r}Re(\bar{u}{u}_t)r^{\prime }d{r}^{\prime }dr$$

$$=-⟨({\partial }_{rr}+\frac{1}{r}{\partial }_r)\bar{u},u_t⟩+⟨(\frac{m+A_{\theta }[u]}{r}{)}^{2}u,u_t⟩-⟨|u{|}^{2}u,u_t⟩-{\int }_0^{\infty }|u{|}^2{\int }_r^{\infty }\frac{m+A_{\theta }[u]}{r}Re(\bar{u}{u}_t)dr{r}^{\prime }d{r}^{\prime }$$

$$=-⟨({\partial }_{rr}+\frac{1}{r}{\partial }_r)\bar{u},u_t⟩+⟨(\frac{m+A_{\theta }[u]}{r}{)}^{2}u,u_t⟩-⟨|u{|}^{2}u,u_t⟩+⟨A_t[u]\bar{u},u_t⟩=⟨i{u}_t,u_t⟩=0.$$

$${\partial }_{t}u=-i\nabla E[u],$$

$${\partial }_{t}M[u]=Re\int \bar{u}{u}_t=Re\int \bar{u}(i{\partial }_r^{2}u+\frac{i}{r}{\partial }_{r}u)=0.$$

$$E[u]=\int \frac{1}{2}|D_{u}u{|}^2,$$

$$D_{u}f={\partial }_{r}f-\frac{m+A_{\theta }[u]}{r}f.$$

$$-Re\int ({\partial }_r\bar{f})\cdot (\frac{m+A_{\theta }[u]}{r})f(r)rdr=-\frac{1}{2}\int (m+A_{\theta }[u]){\partial }_r(|f{|}^2)dr=\frac{1}{2}\int {\partial }_r{A}_{\theta }[u]|f{|}^2=-\frac{1}{4}{\int }_0^{\infty }|f{|}^{4}rdr.$$

$$D_{Q}Q=0.$$

$$Q(r)=\sqrt{8}(m+1)\frac{r^m}{1+r^{2m+2}},m\ge 0.$$

$$[\mathcal{C}u](t,r)=\frac{1}{|t|}u(-\frac{1}{t},\frac{r}{|t|})e^{i{r}^2/4t},\forall t\ne 0.$$

$$S(t,r)=\frac{1}{|t|}Q(\frac{r}{|t|})e^{-i\frac{r^2}{4t}},t<0.$$

$$\Vert u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}.$$

$$\Vert u_0{\Vert }_{L^2}^2<8\pi (m+1).$$

$$\Vert Q{\Vert }_{L^2}^2=16\pi (m+1{)}^2{\int }_0^{\infty }\frac{r^{2m+1}}{(1+r^{2m+2}{)}^2}dr=8\pi (m+1){\int }_0^{\infty }\frac{du}{(1+u{)}^2}=8\pi (m+1).$$

$$u(t,\cdot )-Q_{\lambda (t),\gamma (t)}\to z^{\ast },\text{in}{L}^2,\text{as}t\nearrow T.$$

$$\underset{t\in I}{\sup }{\int }_{|x|\ge R}|u(t,x){|}^{2}dx<\eta ,$$

$$u_{\lambda }(t,r)=\frac{1}{\lambda }u(\frac{t}{{\lambda }^2},\frac{x}{\lambda }),\lambda >0,$$

$$u_t+u_{xxx}+{\partial }_x(u^5)=0,$$

$$u(t,\cdot )-Q_{\lambda (t),\gamma (t)}-e^{it{\Delta }^{(-m-2)}}{u}^{\ast }\to 0,\text{in}{L}^2,\text{as}t\to \infty .$$

$$i{\partial }_{t}u+{\partial }_r^{2}u+\frac{1}{r}{\partial }_{r}u-\frac{(m+2{)}^2}{r^2}u=0.$$

$$\Vert u(t){\Vert }_{L^2}=\Vert Q{\Vert }_{L^2}.$$

$$\underset{t\in I}{\sup }{\int }_{|x|\ge R}|u(t,x){|}^{2}dx<\eta ,$$

$${\partial }_t\int Im(\bar{u}\cdot r{\partial }_{r}u)=4E[u].$$

$$\underset{t\in I}{\sup }\int \psi (\frac{r}{R})Im[\bar{u}\cdot r{\partial }_{r}u]{\lesssim }_{E[u],M[u]}R.$$

Full text

1. Introduction

The self-dual Chern–Simons–Schrödinger equation with $m$-equivariance is

[TABLE]

where

[TABLE]

The Chern–Simons–Schrödinger equation was introduced in [JP90a] as a nonrelativistic planar quantum electromagnetic model that exhibits self-duality. It is a gauge covariant nonlinear Schrödinger equation on $\mathbb{R}^{2}$. See also [Dun09], [JP90b], [JP91]. The model $(\ref{1.1})$ is derived after fixing the Coulomb gauge condition and imposing the equivariant symmetry on the scalar field $\phi$:

[TABLE]

See [KKO20], [KK19], and [KK20].

Remark 1**.**

The non-equivariant Chern–Simons–Schrödinger equation will not be discussed here. See [BdBS95], [Huh13], [Lim18], [LST14], and [OP15] for more information.

The solution to $(\ref{1.1})$ has the conserved quantities mass and energy.

[TABLE]

where we denote $\int f(r)=2\pi\int f(r)rdr$. Indeed, $(\ref{1.1})$ is the Hamiltonian PDE for the energy in $(\ref{1.3})$. We also use the inner product

[TABLE]

Integrating by parts,

[TABLE]

Thus,

[TABLE]

where $\nabla$ (acting on a functional) is the Frechet derivative with respect to the inner product $\langle\cdot,\cdot\rangle$. Also,

[TABLE]

The energy functional can be written in the self-dual form

[TABLE]

where $D_{u}$ is the covariant Cauchy–Riemann operator defined by

[TABLE]

Indeed,

[TABLE]

Definition 1** (Bogomol’nyi operator).**

The operator $u\mapsto D_{u}$ is called the Bogomol’nyi operator. Due to $(\ref{1.4.3})$ and the Hamiltonian structure, any static solutions to $(\ref{1.1})$ are given by solutions to the Bogomol’nyi equation

[TABLE]

For $m\geq 0$, there is an explicit $m$-equivariant static solution to the Bogomol’nyi equation, the Jackiw–Pi vortex. This solution is unique up to the symmetries of the equation ([LL20]):

[TABLE]

Equation $(\ref{1.1})$ has the pseudoconformal transform $\mathcal{C}$,

[TABLE]

Since the soliton solution is non-scattering, applying the pseudoconformal transform to $Q$ gives an explicit, finite-time blowup solution,

[TABLE]

It is conjectured that any blowup solution must contain either $(\ref{1.5})$ or $(\ref{1.7})$. Indeed, [LS16] proved global well-posedness and scattering for $(\ref{1.1})$ with initial data with mass below the mass of the ground state,

[TABLE]

Theorem 1**.**

Let $m\in\mathbb{Z}_{+}$. Let $\phi_{0}\in L_{m}^{2}$ with $\|\phi_{0}\|_{L_{m}^{2}}$ and

[TABLE]

Then $(\ref{1.1})$ is globally well-posed in $L_{m}^{2}$ and scatters both forward and backward in time.

Proof.

See Theorem $1.3$ of [LS16]. ∎

Making a $u$-substitution,

[TABLE]

Remark 2**.**

A function $u_{0}\in L_{m}^{2}$ if $u_{0}\in L^{2}$ and $u_{0}$ has the form $(\ref{1.2.1})$. A function $u_{0}\in H_{m}^{1}$ if $u_{0}\in H^{1}$ and has the form $(\ref{1.2.1})$.

More recently, [KKO20] and [KKO22] proved a decomposition for finite time blowup solutions to $(\ref{1.1})$ with finite energy and initial data in a weighted Sobolev space.

Theorem 2**.**

If $m\in\mathbb{Z}_{+}$ and $u$ is a $H_{m}^{1}$-solution to $(\ref{1.1})$ that blows up forward in time at $T<+\infty$, then $u(t)$ admits the decomposition

[TABLE]

Moreover, using the pseudoconformal transformation in $(\ref{1.6})$, it is possible to obtain a similar decomposition for a solution that exists globally forward in time, but fails to scatter forward in time, for initial data that also lies in a weighted $L^{2}$-space.

Proof.

See [KKO22]. ∎

In this paper, we prove a Liouville theorem for solutions to $(\ref{1.1})$ that are global in at least one time direction.

Theorem 3** (Liouville theorem).**

Suppose $u_{0}\in H_{m}^{1}$ is an initial data for $(\ref{1.1})$ that has a solution on the maximal interval of existence $I$. Furthermore, suppose that $I=(-\infty,t_{0})$, where $t_{0}$ could be $+\infty$, or $(t_{0},\infty)$, where $t_{0}$ could be $-\infty$. Also suppose that for any $\eta>0$, there exists $R(\eta)<\infty$ such that

[TABLE]

where $I$ is the interval of existence for a solution to $(\ref{1.1})$. Then $u$ is equal to the soliton solution $(\ref{1.5})$, up to the scaling symmetry,

[TABLE]

and multiplication by $e^{i\gamma}$ for some $\gamma\in\mathbb{R}$.

This result was inspired by the Liouville theorem of [MM00]. There, [MM00] proved that for a solution to the mass-critical generalized Korteweg de-Vries equation,

[TABLE]

with initial data close to the rescaled soliton in $H_{x}^{1}(\mathbb{R})$, and with $H^{1}(\mathbb{R})$ norm uniformly bounded, then the solution to $(\ref{1.13})$ must be the soliton. For the mass-critical generalized KdV equation, it is expected that multi-soliton solutions occur, which necessitates additional constraints on the size of the initial data than we have here.

Unlike the generalized KdV equation, the structure of the self-dual Chern–Simons–Schrödinger equation is defocusing outside of a soliton. For this reason, it is unnecessary to require a uniform bound on $\|u(t)\|_{H^{1}}$ on $I$. Also, since $u_{0}$ need not be close to the soliton, we do not assume an a priori bound on $\|u_{0}\|_{L^{2}}$.

Remark 3**.**

For a solution to $(\ref{1.1})$, $u\in H_{m}^{1,1}$, that exists globally forward in time, then either $u(t)$ scatters forward in time, or $u(t)$ admits the decomposition

[TABLE]

Here, $e^{it\Delta^{(-m-2)}}u^{\ast}$ is the solution to the free, $(-m-2)$–equivariant Schrödinger flow,

[TABLE]

The space $H_{m}^{1,1}$ is the space of $m$-equivariant functions, $(\ref{1.2.1})$, that lie in $H^{1}$ and the weighted $L^{2}$-space, $\||x|u\|_{L^{2}}<\infty$.

If $(\ref{1.13.1})$ could be proved for any $u_{0}\in H_{m}^{1}$, then Theorem 3 would likely follow fairly easily, since $(\ref{1.13.1})$ would at least imply that $u^{\ast}=0$, and thus $\|u\|_{L^{2}}=\|Q\|_{L^{2}}$. This is due to the fact that a scattering solution, or a solution with a scattering piece could not satisfy $(\ref{1.11})$.

The proof of Theorem 3 may be broken down into three steps. First, using a virial identity combined with $(\ref{1.11})$, we prove that any solution to $(\ref{1.1})$ that satisfies $(\ref{1.11})$ must have the mass of the soliton,

[TABLE]

Next, using an argument analogous to the argument in [Mer93], we prove that a solution to $(\ref{1.1})$ satisfying $(\ref{1.11})$ and $(\ref{1.14})$ must be global in both time directions. Indeed, any finite time blowup solution with $\|u_{0}\|_{L^{2}}=\|Q\|_{L^{2}}$ must be a rescaled version of $(\ref{1.7})$, which clearly does not satisfy $(\ref{1.11})$.

Combining $(\ref{1.11})$, $(\ref{1.14})$, $u_{0}\in H^{1}$, and the virial identity shows that $u(t)$ is the soliton.

2. Mass above the soliton

In this section, we prove that if $(\ref{1.11})$, then $u$ has the same mass as the soliton.

Theorem 4**.**

Suppose $u_{0}\in H^{1}$ is an initial data for $(\ref{1.1})$ that has a solution on the maximal interval of existence $I$. Also suppose that for any $\eta>0$, there exists $R(\eta)<\infty$ such that

[TABLE]

where $I$ is the interval of existence for a solution to $(\ref{1.1})$. Then $\|u\|_{L^{2}}=\|Q\|_{L^{2}}$, where $Q$ is the soliton, $(\ref{1.5})$.

Proof.

We prove this using the virial identity

[TABLE]

Lemma 1**.**

For any solution $u$, $0<R<\infty$,

[TABLE]

Here $\psi(r)\in C^{\infty}(\mathbb{R}^{2})$ is a radially symmetric function, $\psi(r)=1$ for $r\leq 1$, $\psi(r)=\frac{3}{2r}$ for $r\geq 2$. Moreover,

[TABLE]

where $\phi(r)$ is a positive, smooth function, $\phi(r)=1$ for $r\leq 1$, $\phi(r)$ supported on $r\leq 2$, and $\phi(r)=\chi(r)^{2}$ for some $\chi\in C_{0}^{\infty}(\mathbb{R}^{2})$.

Proof of Lemma.

Consider two cases separately, when $\|u(t)\|_{H^{1}}$ is uniformly bounded, and the case when $\|u(t)\|_{H^{1}}$ is not uniformly bounded.

Case 1:

[TABLE]

In this case, $I=\mathbb{R}$. Now then,

[TABLE]

Case 2: Since $\|\nabla u(t)\|_{L^{2}}$ is continuous in time, if $\sup_{t\in I}\|\nabla u(t)\|_{L^{2}}=\infty$, then there exists a sequence $t_{n}$ such that

[TABLE]

Set

[TABLE]

Plugging $\lambda(t_{n})$ into $(\ref{1.12})$, let

[TABLE]

By direct computation,

[TABLE]

Now, recall Proposition $4.1$ from [KKO22].

Proposition 1** (Decomposition).**

Let $Z_{1},Z_{2}\in C_{c,m}^{\infty}$ be profiles that satisfy

[TABLE]

Here, $\Lambda$ is the operator $r\partial_{r}+1$. Then for any $M<\infty$, there exists $0<\alpha^{\ast}\ll 1$ such that the following properties hold for all $u\in H_{m}^{1}$ with $\|u\|_{L^{2}}\leq M$ satisfying the small energy condition $\sqrt{E[u]}\leq\alpha^{\ast}\|u\|_{H_{m}^{1}}$.

There exists a unique $(\lambda,\gamma)\in\mathbb{R}_{+}\times\mathbb{R}/2\pi\mathbb{Z}$ such that $\epsilon\in H_{m}^{1}$, defined by the relation

[TABLE]

satisfies the orthogonality conditions,

[TABLE]

and smallness

[TABLE]

Remark 4**.**

The space $\mathcal{H}_{m}^{1}$ is a function space adapted to the linear coercivity of the energy. When $m\geq 1$, as is true in this paper, the spaces $\mathcal{H}_{m}^{1}$ and $H_{m}^{1}$ are equivalent.

Proof.

The proof in [KKO22] relies on the uniqueness of the soliton as a function with zero energy, the nonlinear coercivity of energy in [KKO22], and the implicit function theorem.

Proposition 2** (Nonlinear coercivity of energy).**

For any $M>0$, there exists $\eta>0$ such that the nonlinear coercivity

[TABLE]

holds for any $\epsilon\in H_{m}^{1}$ with $\|\epsilon\|_{L^{2}}\leq M$ satisfying the orthogonality conditions $(\ref{2.12})$ and smallness $\|\epsilon\|_{\dot{\mathcal{H}}_{m}^{1}}\leq\eta$.

Proof of Proposition 2.

We follow the argument in [KKO22]. Observe that, by $(\ref{1.4.4})$,

[TABLE]

Now then, since $D_{Q}Q=0$,

[TABLE]

Here,

[TABLE]

and

[TABLE]

Using the coercivity of $L_{Q}$ proved in [KKO20], [KK19],

Lemma 2** (Coercivity of $L_{Q}$).**

Let $m\geq 0$. Let $\mathcal{Z}_{1}$, $\mathcal{Z}_{2}\in C_{c,m}^{\infty}$ satisfy $(\ref{2.10})$. Then,

[TABLE]

Now then, by Hardy’s inequality,

[TABLE]

Also, by Hardy’s inequality,

[TABLE]

Therefore, we have proved

[TABLE]

Now decompose $\epsilon=\chi_{R}\epsilon+(1-\chi_{R})\epsilon$, where $\chi_{R}(r)=\chi(\frac{r}{R})$ is the function defined in Lemma 1. Then decompose

[TABLE]

By direct computation,

[TABLE]

Next, decompose

[TABLE]

Using the decay of $Q$,

[TABLE]

Therefore,

[TABLE]

Decompose

[TABLE]

By the support of $\chi_{R}$ and $1-\chi_{R}$,

[TABLE]

Using the nonlinear Hardy inequality in [KKO22],

[TABLE]

Therefore, we have proved

[TABLE]

Taking $R\ll\frac{1}{\|\epsilon\|_{\dot{H}_{m}^{1}}}$ and averaging over $\log(\|\epsilon\|_{\dot{\mathcal{H}}_{m}^{1}})$ intervals of the form $\frac{R}{2}\leq r\leq R$, the proof of Proposition 2 is complete. ∎

Now then, suppose there exists a sequence $\|v_{n}\|_{\mathcal{H}_{m}^{1}}$ constant, $E[v_{n}]\rightarrow 0$. By the uniqueness of $Q$ (up to scaling) as a solution to $E[Q]=0$ and the fact that $E[u]\geq 0$, and thus $Q$ is an energy minimizer, $v_{n}$ converges in $\mathcal{H}_{m}^{1}$, and thus, $v_{n}\rightarrow Q$ in $\mathcal{H}_{m}^{1}$. Therefore, $\|v_{n}-Q\|_{\dot{\mathcal{H}}_{m}^{1}}\rightarrow 0$ as $n\rightarrow\infty$.

Next, since

[TABLE]

and

[TABLE]

combining the implicit function theorem with $(\ref{2.10})$, for $E[v_{n}]$ sufficiently small, we can find a unique $\lambda$ and $\gamma$ such that $(\ref{2.12})$ holds.

Then by Proposition 2, $(\ref{2.13})$ holds. ∎

Therefore, for $n(M)$ sufficiently large,

[TABLE]

and we can make the decomposition

[TABLE]

Now, note that $(\ref{1.12})$ implies

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $Q$ is real-valued, $(\ref{2.17})=0$. Next,

[TABLE]

Next,

[TABLE]

Finally, integrating by parts,

[TABLE]

This proves the Lemma. ∎

Now then, compute

[TABLE]

[TABLE]

Integrating by parts,

[TABLE]

Therefore,

[TABLE]

Using $(\ref{1.3})$,

[TABLE]

Next, since $A_{\theta}[u]\lesssim M^{2}$,

[TABLE]

By direct computation,

[TABLE]

so $|A_{\theta}[u]-A_{\theta}[\chi_{R}u]|$ is supported on $r\geq R$. Therefore,

[TABLE]

Finally, for

[TABLE]

consider two cases separately, as in Lemma 1. For $\|u\|_{\dot{H}^{1}}\lesssim 1$, by interpolation,

[TABLE]

where $o_{R}(1)$ is a quantity that approaches [math] as $R\rightarrow\infty$. For the last step, $(\ref{2.1})$ is used. When $\|u\|_{\dot{H}^{1}}\gg 1$, then by Proposition 2,

[TABLE]

with $\lambda(t)\gg 1$. Then by direct computation,

[TABLE]

Also, by Proposition 2 and $(\ref{2.1})$,

[TABLE]

Therefore,

[TABLE]

Therefore,

[TABLE]

Taking $R=T^{1/3}$, there exists a sequence $t_{n}^{\prime}\rightarrow\infty$, $R_{n}\rightarrow\infty$, satisfying

[TABLE]

Therefore,

[TABLE]

and for $n$ sufficiently large,

[TABLE]

Therefore, $\|\epsilon\|_{L^{2}(|x|\leq R_{n}^{\prime})}\rightarrow 0$ for some $R_{n}^{\prime}\rightarrow\infty$, and

[TABLE]

The bounds $(\ref{2.1})$ on the mass implies that there exists $\lambda_{0}>0$ such that $\lambda(t_{n}^{\prime})\geq\lambda_{0}>0$. Therefore,

[TABLE]

as $n\rightarrow\infty$. ∎

Remark 5**.**

Note that the estimate in $(\ref{2.26})$ utilizes that $T\rightarrow\infty$. We cannot use this argument if $u$ blows up in finite time in both directions.

3. Rigidity for finite time blowup solutions at the soliton

Now we duplicate the result of [Mer93] for the Chern–Simons–Schrödinger equation, showing that if $u$ is a blowup solution to $(\ref{1.1})$ with $\|u_{0}\|_{L^{2}}=\|Q\|_{L^{2}}$ and $u_{0}\in H^{1}$, the soluton $u$ must be of the form $(\ref{1.7})$. Such a solution would violate $(\ref{2.1})$ in the scattering time direction.

Theorem 5**.**

If $u$ is a finite time blowup solution with $\|u_{0}\|_{L^{2}}=\|Q\|_{L^{2}}$ and $u_{0}\in H^{1}$, then $u$ is equal to a pseudoconformal transformation of a soliton.

Proof.

By time translation symmetry and the scaling symmetry, suppose that $u$ blows up at time $t=0$, and let $u_{0}$ be the data for $u(t,r)$ at $t=-1$.

Lemma 3**.**

Fix $R>0$ large. Let $\phi\in C_{0}^{\infty}(\mathbb{R}^{2})$ be a smooth cut-off, $\phi(x)=1$ for $|x|\leq 1$, and $\phi(x)=0$ for $|x|>2$. Then,

[TABLE]

Proof.

If $u$ blows up in finite time, $\lim_{t\nearrow 0}\|u(t)\|_{\dot{H}^{1}}=\infty$. Now let

[TABLE]

and let

[TABLE]

Then by $(\ref{2.9})$, $E(v(t))\rightarrow 0$, and $\|v(t)\|_{\dot{H}^{1}}$ and $\|v(t)\|_{L^{2}}$ are uniformly bounded for $-1<t<0$. Therefore, by Proposition 1, for $t$ sufficiently close to [math],

[TABLE]

and furthermore,

[TABLE]

By $(\ref{3.4})$, $(\ref{3.5})$, and $\|v\|_{\dot{H}^{1}}=1$, $\tilde{\lambda}(t)\sim 1$, so

[TABLE]

and since $\|v\|_{L^{2}}=\|Q\|_{L^{2}}$, $(\ref{3.6})$ can be upgraded to convergence in $L^{2}$. Since $\lambda(t)\nearrow\infty$ as $t\nearrow 0$, $(\ref{3.1})$ holds. ∎

Lemma 4**.**

For any $R>0$,

[TABLE]

Proof.

The argument is identical to the argument proving Lemma 1, except that now, insert $\|\epsilon(t)\|_{L^{2}}\rightarrow 0$ into $(\ref{2.17})$–$(\ref{2.19})$, proving $(\ref{3.7})$. ∎

Returning to the proof of Theorem 1, by direct computation,

[TABLE]

Integrating by parts,

[TABLE]

Also, by direct computation and integrating by parts,

[TABLE]

Therefore, by Lemmas 3 and 4, taking $R\nearrow\infty$, for any $-1<t<0$,

[TABLE]

Making a pseudoconformal transformation of the solution, let

[TABLE]

By $(\ref{3.12})$,

[TABLE]

and by $(\ref{3.9})$,

[TABLE]

Therefore, $E[v]=0$, and thus, $v$ is a soliton, so $u$ is a pseudoconformal transformation of a soliton. ∎

4. The Liouville theorem

Now we have proved that the only solution to $(\ref{1.1})$ that satisfies $(\ref{2.1})$ has mass $\|u_{0}\|_{L^{2}}=\|Q\|_{L^{2}}$ and is global in both time directions. Then we complete the proof of the Liouville theorem by showing that $u$ is a soliton.

Theorem 6**.**

The solution satisfying $\|u\|_{L^{2}}=\|Q\|_{L^{2}}$ and $(\ref{1.12})$ is the soliton.

Proof.

We again use the virial identity in $(\ref{2.23})$ and $(\ref{2.24})$, only this time we integrate from $-T$ to $T$. Integrating by parts,

[TABLE]

where again $\phi(r)=\partial_{r}(r\psi(r))$ and $\phi(r)=\chi(r)^{2}$ for some $\chi\in C_{0}^{\infty}$. Now then,

[TABLE]

Next, for

[TABLE]

since $\|u\|_{L^{2}}=\|Q\|_{L^{2}}$, by standard linear algebra,

[TABLE]

Therefore,

[TABLE]

By Hölder’s inequality,

[TABLE]

Now let

[TABLE]

Since $Q(r)\lesssim\frac{1}{r^{4}}$ for $r$ large and $Q(r)\leq 1$ for all $r$, $\tilde{Q}(r)\in L^{2}(\mathbb{R}^{2})$. Moreover,

[TABLE]

Therefore,

[TABLE]

Integrating by parts,

[TABLE]

Therefore,

[TABLE]

and by $(\ref{4.2})$ and $(\ref{4.11})$,

[TABLE]

Also by $(\ref{4.2})$,

[TABLE]

By standard perturbation theory and the fact that the $L_{t,x}^{4}$ norm is invariant under the scaling $(\ref{1.12})$,

[TABLE]

Plugging in $(\ref{4.11})$, since $\lambda(t)\sim\lambda(t_{0})$ for $t\in[t_{0},t_{0}+\frac{1}{\lambda(t_{0})^{2}}]$,

[TABLE]

Making an averaging argument, for a fixed $R_{0}$,

[TABLE]

Therefore, for any $\delta>0$ there exists some $R_{0}\leq R_{\ast}\leq C(\delta)R_{0}$ such that

[TABLE]

Therefore, we have proved

[TABLE]

Taking $R=T^{1/4}$, averaging $(\ref{4.17})$, and plugging in $(\ref{4.11})$, we have proved

[TABLE]

Now for any $j\in\mathbb{Z}$, $j\geq 0$, let

[TABLE]

Then let $T_{j}=t_{j}^{+}-t_{j}^{-}$, $I_{j}=[t_{j}^{-},t_{j}^{+}]$. Then $T_{j}\rightarrow\infty$ as $j\rightarrow\infty$. Then by $(\ref{4.11})$,

[TABLE]

Therefore, by $(\ref{4.17})$,

[TABLE]

Therefore,

[TABLE]

Then, by the dominated convergence theorem, taking $j\rightarrow\infty$,

[TABLE]

which implies $E[u]\equiv 0$, and thus $u$ is a soliton.

∎

5. Acknowledgements

During the writing of this paper, the author was partially supported by NSF grant DMS-2153750.