Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation

R. F. Snider; R. F. Snider

doi:10.3934/era.2023281

Electronic Research Archive

2023, Volume 31, Issue 9: 5541-5558. doi: 10.3934/era.2023281

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Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation

R. F. Snider ^,

Department of Chemitry, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada

Received: 15 June 2023 Revised: 16 July 2023 Accepted: 02 August 2023 Published: 14 August 2023

The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction).

Keywords:

Citation: R. F. Snider. Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation[J]. Electronic Research Archive, 2023, 31(9): 5541-5558. doi: 10.3934/era.2023281

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Abstract

1. Introduction

The 3D incompressible resistive Hall-Magnetohydrodynamics system (Hall-MHD in short) is the following system of PDEs for $(u, p, B)$ :

$\begin{align} &u_{t} +u\cdot \nabla u - B\cdot \nabla B+\nabla p-\mu\Delta u = 0, \end{align}$

(1a)

$\begin{align} &B_{t} + u\cdot \nabla B -B\cdot \nabla u +{\rm{curl}}\left(\left({\rm{curl}} \;B\right)\times B\right)- \nu\Delta B = 0, \end{align}$

(1b)

$\begin{align} &{\rm{div}}\; u = 0, \quad {\rm{div}}\; B = 0, \end{align}$

(1c)

where $u = (u_{1}, u_{2}, u_{3})$ is the plasma velocity field, $p$ is the pressure, and $B = (B_{1}, B_{2}, B_{3})$ is the magnetic field. $\mu$ and $\nu$ are the viscosity and the resistivity constants, respectively. The Hall-MHD is important in describing many physical phenomena [2,17,19,23,26,27,33]. In particular, the Hall MHD explains magnetic reconnection on the Sun which is very important role in acceleration plasma by converting magnetic energy into bulk kinetic energy.

The Hall-MHD recently has been studied intensively. The Hall-MHD can be derived from either two fluids model or kinetic models in a mathematically rigorous way [1]. Global weak solution, local classical solution, global solution for small data, and decay rates are established in [4,5,6]. There have been many follow-up results of these papers; see [7,8,12,13,14,15,16,18,29,30,31,32,34,35] and references therein.

1.1. The Hall equation with fractional Laplacian

We note that the Hall term ${\rm{curl}}\left(\left({\rm{curl}} \;B\right)\times B\right)$ is dominant in mathematical analysis of (1) and so we only consider the Hall equations ( $(u, p) = 0$ in (1)). Also motivated by [7], we consider the Hall equation with fractional Laplacian:

$\begin{eqnarray} B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\Lambda^{\beta}B = 0, \quad {\rm{div}}\; B = 0, \end{eqnarray}$

(2)

where we take $\nu = 1$ for simplicity. (2) is locally well-posed [7] when $\beta>1$ . But, we do not know whether (2) is locally well-posed when $\beta = 1$ :

$\begin{eqnarray} B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\Lambda B = 0, \quad {\rm{div}}\; B = 0. \end{eqnarray}$

(3)

However, we can show the existence of solutions globally in time if initial data is sufficiently small.

Theorem 1.1. Let $B_{0} \in H^{k}$ with $k>\frac{5}{2}$ and ${\rm{div}}\; B_{0} = 0$ . There exists a constant $\epsilon_{0}>0$ such that if $\|B_{0}\|_{H^{k}} \leq \epsilon_{0}$ , there exists a unique global-in-time solution of (3) satisfying

$\left\|B(t)\right\|^{2}_{H^{k}}+(1-C\epsilon_{0})\int^{t}_{0}\left\|\Lambda^{\frac{1}{2}}B(s)\right\|^{2}_{H^{k}}ds\leq \left\|B_{0}\right\|^{2}_{H^{k}} \quad {{for\; all \; t > 0 .}}$

Moreover, $B$ decays in time

$\begin{eqnarray} \left\|\Lambda^{l}B(t)\right\|_{L^{2}}\leq \frac{C_{0}}{(1+t)^{l}}, \quad 0 < l\leq k, \end{eqnarray}$

(4)

where $C_{0}$ depends on $\|B_{0}\|_{H^{k}}$ which is expressed in (27) explicitly.

Remark 1. The decay rate (4) is consistent with the decay rates of the linear part of (3).

Remark 2. After this work was completed, the referee pointed out that the same result is proved in [37,Theorem 1.1]. Compared to the proof in [37] where they use the Littlewood-Paley decomposition, we use the standard energy energy estimates and classical commutator estimates.

As one of a minimal modification of (3) to show the existence of unique local in time solutions, we now take a logarithmic correction of (3):

$\begin{eqnarray} B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\ln(2+\Lambda)\Lambda B = 0, \end{eqnarray}$

(5)

where the Fourier symbol of $\ln(2+\Lambda)\Lambda$ is $\ln(2+|\xi|)|\xi|$ .

Theorem 1.2. Let $B_{0} \in H^{k}$ with $k>\frac{5}{2}$ and ${\rm{div}}\; B_{0} = 0$ . There exists $T_\ast = T_{\ast}(\|B_{0}\|_{H^{k}})>0$ such that there exists a unique local-in-time solution of (5) satisfying

$\begin{eqnarray} \|B(t)\|_{H^{k}}\leq \ln \left(\frac{1}{e^{-\|B_{0}\|_{H^{k}}}-Ct}\right), \quad 0 < t < T^{\ast} = \frac{\exp\left(-\|B_{0}\|_{H^{k}}\right)}{C}. \end{eqnarray}$

(6)

1.2. 2D models

In this paper, we also deal with 2D models closely related to the $2\frac{1}{2}$ dimensional (3). If we take $B$ of the form

$\begin{eqnarray} B(t, x, y) = \left(-\psi_{y}(t, x, y), \psi_{x}(t, x, y), Z(t, x, y)\right), \end{eqnarray}$

(7)

we can rewrite (3) as

$\begin{align} & \psi_{t}+\Lambda\psi = [\psi, Z], \end{align}$

(8a)

$\begin{align} & Z_{t}+\Lambda Z = [\Delta \psi, \psi], \end{align}$

(8b)

where $[f, g] = \nabla f\cdot \nabla^{\perp}g = f_{x}g_{y}-f_{y}g_{x}$ . (7) is used to show a finite-time collapse to a current sheet [3,20,21,24] and is used in [10] to study regularity of stationary weak solutions.

1.2.1. Case 1

Although (8) is defined in 2D and has nice cancellation properties (18), the local well-posedness seems unreachable. But, suppose that we redistribute the power of the fractional Laplacians in (8) in such a way that (8b) has the full Laplacian and (8a) is inviscid:

$\begin{eqnarray} \psi_{t} = [\psi, Z], \quad Z_{t}-\Delta Z = [\Delta \psi, \psi]. \end{eqnarray}$

(9)

(9) has no direct link to (2), but we may interpret (9) as the $2\frac{1}{2}$ dimensional model of the Hall equations where only $B_{3}$ has the full Laplacian in (2). In this case, we can show that (9) is locally well-posed. Let

$\begin{eqnarray} \mathcal{E}(t) = \left\|\psi(t)\right\|^{2}_{H^{4}}+\left\|Z(t)\right\|^{2}_{H^{3}}, \quad \mathcal{E}_{0} = \left\|\psi_{0}\right\|^{2}_{H^{4}}+\left\|Z_{0}\right\|^{2}_{H^{3}}. \end{eqnarray}$

(10)

Theorem 1.3. There exists $T_{\ast} = T_{\ast}(\mathcal{E}_{0})>0$ such that there exists a unique solution of (9) satisfying

$\mathcal{E}(t)\leq \frac{\mathcal{E}_{01}}{1-Ct\mathcal{E}_{0}} \quad {{for \;all}}\; \ 0 < t\leq T_{\ast} < \frac{1}{C \mathcal{E}_{0}}.$

Moreover, we have the following blow-up criterion:

$\mathcal{E}(t)+ \int^{t}_{0}\left\|\nabla Z(s)\right\|^{2}_{H^{2}}ds < \infty\iff \int^{t}_{0}\left(\left\|\nabla^{2} Z(s)\right\|_{L^{\infty}} +\left\|\nabla^{2}\psi(s)\right\|^{2}_{L^{\infty}} \right)ds < \infty.$

Since there is no dissipative effect in the equation of $\psi$ in (9), we only have the local in tim result in Theorem 1.3. Among the possible conditions for the global existence, we find that adding a damping term to the equation of $\psi$ works. More precisely, we deal with the following

$\begin{eqnarray} \psi_{t}+\psi = [\psi, Z], \quad Z_{t}-\Delta Z = [\Delta \psi, \psi]. \end{eqnarray}$

(11)

In this case, we can show the existence of global in time solutions with small initial data having regularity one higher than the regularity in Theorem 1.3. Moreover, we can find decay rates of $\psi$ by using the structure of equation of $\psi$ which is a damped transport equation, and this is also the reason why the same method cannot be applied to $Z$ . Let

$\begin{split} &\mathcal{F}(t) = \left\|\psi(t)\right\|^{2}_{H^{5}}+\left\|Z(t)\right\|^{2}_{H^{4}}, \quad \mathcal{F}_{0} = \left\|\psi_{0}\right\|^{2}_{H^{5}}+\left\|Z_{0}\right\|^{2}_{H^{4}}, \\ & \mathcal{N}_{1}(t) = \left\|\nabla\psi(t)\right\|^{2}_{H^{4}}+\left\|\nabla Z(t)\right\|^{2}_{H^{4}}. \end{split}$

Theorem 1.4. There exists a constant $\epsilon_{0}>0$ such that if $\mathcal{F}_{0} \leq \epsilon_{0}$ , there exists a unique global-in-time solution of (11) satisfying

$\mathcal{F}(t)+(1-C \epsilon_{0})\int^{t}_{0}\mathcal{N}_{1}(s)ds\leq \mathcal{F}_{0} \quad {{for\; all\; t > 0 .}}$

Moreover, $\psi$ decays exponentially in time

$\left\|\psi(t)\right\|_{L^{2}}\leq \left\|\psi_{0}\right\|_{L^{2}} e^{-t}, \quad \left\|\Lambda^{k}\psi(t)\right\|_{L^{2}}\leq \mathcal{F}^{\frac{k-1}{8}}_{0}\left\|\nabla \psi_{0}\right\|^{\frac{5-k}{4}}_{L^{2}} e^{-\frac{(5-k)(1-C\epsilon_{0})}{4}t}$

with $1\leq k<5.$

1.2.2. Case 2

As another way to redistribute the derivatives in (8), we also deal with

$\begin{eqnarray} \psi_{t}-\Delta \psi = [\psi, Z], \quad Z_{t} = [\Delta \psi, \psi]. \end{eqnarray}$

(12)

Let $\mathcal{E}(t)$ and $\mathcal{E}_{0}$ be defined as before (10).

Theorem 1.5. There exists $T_{\ast}>0$ , which is depending on $\mathcal{E}_{0}$ , such that there exists a unique solution of (12) satisfying

$\mathcal{E}(t)\leq \frac{\mathcal{E}_{0}}{1-Ct\mathcal{E}_{0}} \quad {{for \;all}}\; \ 0 < t\leq T_{\ast} < \frac{1}{C \mathcal{E}_{0}}.$

Moreover, we have the following blow-up criterion

$\mathcal{E}(t)+\int^{t}_{0}\left\|\nabla \psi\right\|^{2}_{H^{4}}ds < \infty \iff \int^{t}_{0}\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}ds.$

We now add a damping term to the equation of $Z$ in (12):

$\begin{eqnarray} \psi_{t}-\Delta \psi = [\psi, Z], \quad Z_{t} + Z = [\Delta \psi, \psi]. \end{eqnarray}$

(13)

In this case, we can use the same regularity used in Theorem 1.5 because the dissipative effect in $\psi$ helps to control $\Delta \psi$ in the equation of $Z$ . Let $\mathcal{N}_{2}(t) = \left\|\nabla\psi(t)\right\|^{2}_{H^{5}}+\left\|Z(t)\right\|^{2}_{H^{3}}$ .

Theorem 1.6. There exists a constant $\epsilon_{0}>0$ such that if $\mathcal{E}_{0} \leq \epsilon_{0}$ , there exists a unique global-in-time solution of (13) satisfying

$\mathcal{E}(t)+(1-C \epsilon_{0})\int^{t}_{0}\mathcal{N}_{2}(s)ds\leq \mathcal{E}_{0} \quad {{for \;all \; t > 0 .}}$

Remark 3. Compared to Theorem 1.3, we only need one term in the blow-up criterion in Theorem 1.5 which is due to the dissipative effect in the equation of $\psi$ . Compared to Theorem 1.4, the proof of Theorem 1.6 is simpler, but we are not able to derive decay rates of $\psi$ and $Z$ .

2. Preliminaries

All constants will be denoted by $C$ and we follow the convention that such constants can vary from expression to expression and even between two occurrences within the same expression. And repeated indices are summed over.

The fractional Laplacian $\Lambda^{\beta} = (\sqrt{-\Delta})^{\beta}$ has the Fourier transform representation

$\widehat{\Lambda^{\beta} f}(\xi) = |\xi|^{\beta}\widehat{f}(\xi).$

For $s>0$ , $H^{s}$ is a energy space equipped with

$\|f\|_{H^{s}} = \|f\|_{L^{2}}+\|f\|_{\dot{H}^{s}}, \quad \|f\|_{\dot{H}^{s}} = \left\|\Lambda^{s}f\right\|_{L^{2}}.$

In the energy spaces, we have the following interpolations: for $s_{0}<s<s_{1}$

$\begin{eqnarray} \left\|f\right\|_{\dot{H}^{s}}\leq \left\|f\right\|^{\theta}_{\dot{H}^{s_{0}}}\left\|f\right\|^{1-\theta}_{\dot{H}^{s_{1}}}, \quad s = \theta s_{0}+(1-\theta)s_{1}. \end{eqnarray}$

(14)

2.1. Inequalities

We begin with two inequalities in 3D:

$\begin{align} & \left\|f\right\|_{L^{\infty}}\leq C\|f\|_{H^{s}}, \quad s > \frac{3}{2}, \end{align}$

(15a)

$\begin{align} & \left\|f\right\|_{L^{p}}\leq C\|f\|_{\dot{H}^{s}}, \quad \frac{1}{p} = \frac{1}{2}-\frac{s}{3}. \end{align}$

(15b)

We also provide the following inequalities in 2D

$\left\|f\right\|_{L^{4}}\leq C\left\|f\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla f\right\|^{\frac{1}{2}}_{L^{2}}, \quad \left\|f\right\|_{L^{\infty}} \leq C\left\|f\right\|^{\frac{1}{2}}_{L^{2}} \left\|\Delta f\right\|^{\frac{1}{2}}_{L^{2}}$

which will be used repeatedly in the proof of Theorem 1.3, Theorem 1.4, Theorem 1.5, and Theorem 1.6. We also recall

$\left\|\nabla^{2}f\right\|_{L^{2}} = \left\|\Delta f\right\|_{L^{2}}$

which holds in any dimension.

We finally provide the Kato-Ponce commutator cstimate [22]

$\begin{eqnarray} \begin{split} \left\|\left[\Lambda^{k}, f\right]g\right\|_{L^{2}}& = \left\|\Lambda^{k}(fg)-f\Lambda^{k}g\right\|_{L^{2}}\\ &\leq C \left\|\nabla f\right\|_{L^{\infty}}\left\|\Lambda^{k-1}g\right\|_{L^{2}}+C \left\|g\right\|_{L^{\infty}}\left\|\Lambda^{k}f\right\|_{L^{2}} \end{split} \end{eqnarray}$

(16)

and the fractional Leibniz rule [11]: for $1\leq p<\infty$ and $p_{i}, q_{i} \ne 1$ ,

$\begin{eqnarray} \begin{split} &\left\|\Lambda^{s}(fg)\right\|_{L^{p}} \le C\left\|\Lambda^{s}f\right\|_{L^{p_{1}}} \|g\|_{L^{q_{1}}} +C\|f\|_{L^{p_{2}}} \left\|\Lambda^{s}g\right\|_{L^{q_{2}}}, \\ & \frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{q_{1}} = \frac{1}{p_{2}}+\frac{1}{q_{2}}. \end{split} \end{eqnarray}$

(17)

2.2. Commutator

We recall the commutator $[f, g] = \nabla f\cdot \nabla^{\perp}g = f_{x}g_{y}-f_{y}g_{x}$ . Then, the commutator has the following properties:

$\begin{align} & \Delta [f, g] = [\Delta f, g] +[f, \Delta g] + 2[f_{x}, g_{x}] +2[f_{y}, g_{y}], \end{align}$

(18a)

$\begin{align} & \int f[f, g] = 0, \end{align}$

(18b)

$\begin{align} & \int f[g, h] = \int g [h, f]. \end{align}$

(18c)

3. Proof of Theorem 1.1 and Theorem 1.2

3.1. Proof of Theorem 1.1

We recall (3):

$\begin{eqnarray} B_{t}+{\rm{curl}} (\left({\rm{curl}} \;B)\times B\right)+\Lambda B = 0. \end{eqnarray}$

(19)

3.1.1. Approximation

We first approximate (19) by putting $\epsilon \Delta B$ to the right-hand side of (19):

$\begin{eqnarray} B_{t}+{\rm{curl}} (\left({\rm{curl}} \;B\right)\times B)+\Lambda B = \epsilon \Delta B. \end{eqnarray}$

(20)

We then mollify (20) as follows

$\begin{eqnarray} \begin{split} &\partial_{t} B^{(\epsilon)}+{\rm{curl}} \left(\mathcal{J}_{\epsilon}\left({\rm{curl}} \mathcal{J}_{\epsilon} B^{(\epsilon)}\right)\times \mathcal{J}_{\epsilon} B^{(\epsilon)}\right)+\Lambda \mathcal{J}^{2}_{\epsilon} B^{(\epsilon)} = \epsilon \mathcal{J}^{2}_{\epsilon}\Delta B^{(\epsilon)}, \\ & B^{(\epsilon)}_{0} = \mathcal{J}_{\epsilon} B_{0}, \end{split} \end{eqnarray}$

(21)

where $\mathcal{J}_{\epsilon}$ is the standard mollifier described in [25,Chapter 3.2]. Then, as proved in [4,Proposition 3.1], there exists a unique global-in-time solution $\left\{B^{(\epsilon)}\right\}$ of (21). Since the bounds in Section 3.1.2 are independent of $\epsilon>0$ , we can pass to the limit in a subsequence and show the existence of smooth solutions globally in time when $B_{0}\in H^{k}$ , $k>\frac{5}{2}$ , is sufficiently small as in [37,Section 3.2].

3.1.2. A priori estimates

We begin with the $L^{2}$ bound:

$\begin{eqnarray} \frac{1}{2}\frac{d}{dt}\left\|B\right\|^{2}_{L^{2}}+\left\|\Lambda^{\frac{1}{2}}B\right\|^{2}_{L^{2}} = 0. \end{eqnarray}$

(22)

We now take $\Lambda^{k}$ to (19) and take the inner product of the resulting equation with $\Lambda^{k}B$ . Then,

$\begin{split} &\frac{1}{2}\frac{d}{dt}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}}+\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}} = -\int\Lambda^{k}{\rm{curl}} (\left({\rm{curl}} \;B)\times B\right) \cdot \Lambda^{k}B\\ & = \int \left(\left[\Lambda^{\frac{1}{2}+k}, B\right]\times {\rm{curl}} \;B\right)\cdot \Lambda^{k-\frac{1}{2}}{\rm{curl}} \;B\leq \left\|\left[\Lambda^{\frac{1}{2}+k}, B\right]\times {\rm{curl}} \;B\right\|_{L^{2}}\left\|\Lambda^{\frac{1}{2}+k}B\right\|_{L^{2}}. \end{split}$

By (16) and (15a) with $k>\frac{5}{2}$ ,

$\begin{eqnarray} \begin{split} \left\|\left[\Lambda^{\frac{1}{2}+k}, B\right]\times {\rm{curl}} \;B\right\|_{L^{2}}&\leq C \left\|\nabla B\right\|_{L^{\infty}}\left\|\Lambda^{k-\frac{1}{2}}{\rm{curl}} \;B\right\|_{L^{2}}\\ &\leq C \left\|B\right\|_{H^{k}}\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}. \end{split} \end{eqnarray}$

(23)

So, we obtain

$\begin{eqnarray} \frac{d}{dt}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}}+\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}\leq C \left\|B\right\|_{H^{k}}\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}. \end{eqnarray}$

(24)

By (22) and (24),

$\frac{d}{dt}\left\|B\right\|^{2}_{H^{k}}+\left\|\Lambda^{\frac{1}{2}}B\right\|^{2}_{H^{k}}\leq C \left\|B\right\|_{H^{k}}\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}.$

If $\left\|B_{0}\right\|_{H^{k}} = \epsilon_{0}$ is sufficiently small, we can derive a uniform bound

$\begin{eqnarray} \left\|B(t)\right\|^{2}_{H^{k}}+(1-C\epsilon_{0})\int^{t}_{0}\left\|\Lambda^{\frac{1}{2}}B(s)\right\|^{2}_{H^{k}}ds\leq \left\|B_{0}\right\|^{2}_{H^{k}} \quad {{\rm{for}} \;{\rm{all}} \;t > 0 .} \end{eqnarray}$

(25)

3.1.3. Uniqueness

Let $B_{1}$ and $B_{2}$ be two solutions of (19). Then, $B = B_{1}-B_{2}$ satisfies

$\begin{eqnarray} B_{t}+\Lambda B+{\rm{curl}} \left(\left({\rm{curl}} \;B_{1}\right)\times B\right) - {\rm{curl}} \left(\left({\rm{curl}} \;B\right)\times B_{2}\right) = 0 \end{eqnarray}$

(26)

with $B_{0} = 0$ . We take the inner product of (26) with $B$ . By (17) with $k>\frac{5}{2}$ ,

$\begin{equation*} \begin{split} \frac{1}{2}\frac{d}{dt}\|B\|^{2}_{L^{2}}+\left\|\Lambda^{\frac{1}{2}} B\right\|^{2}_{L^{2}}& = -\int \left({\rm{curl}}\left(\left({\rm{curl}} \;B_{1}\right)\times B\right)\right) \cdot B\\ & = -\int \Lambda^{\frac{1}{2}}\left(\left(\left({\rm{curl}} \;B_{1}\right)\times B\right)\right) \cdot \Lambda^{-\frac{1}{2}}{\rm{curl}} \;B\\ &\leq C\left\|\nabla B_{1}\right\|_{L^{\infty}}\left\|\Lambda^{\frac{1}{2}} B\right\|^{2}_{L^{2}} +C\left\|\nabla \Lambda^{\frac{1}{2}}B_{1}\right\|_{L^{6}}\|B\|_{L^{3}}\left\|\Lambda^{\frac{1}{2}} B\right\|_{L^{2}}\\ &\leq C\left\|\nabla B_{1}\right\|_{L^{\infty}}\left\|\Lambda^{\frac{1}{2}} B\right\|^{2}_{L^{2}} +C\left\|\Lambda^{\frac{5}{2}}B_{1}\right\|_{L^{2}}\left\|\Lambda^{\frac{1}{2}} B\right\|^{2}_{L^{2}} \\ &\leq C\left\|B_{1}\right\|_{H^{k}} \left\|\Lambda^{\frac{1}{2}} B\right\|^{2}_{L^{2}}, \end{split} \end{equation*}$

where we use (15b) to control $L^{6}$ and $L^{3}$ terms. If $C\epsilon_{0}<1$ , (25) implies $B = 0$ in $L^{2}$ which gives the uniqueness of a solution.

3.1.4. Decay rates

By (14), it is enough to derive the decay rate with $k = l$ to show (4). Since

$\left\|\Lambda^{k}B\right\|^{\frac{2k+1}{k}}_{L^{2}}\leq \left\|B\right\|^{\frac{1}{k}}_{L^{2}}\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}} \leq \left\|B_{0}\right\|^{\frac{1}{k}}_{L^{2}}\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}$

by (14) and (22), we create the following ODE from (24)

$\frac{d}{dt}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}}+\frac{1-C\epsilon_{0}}{\left\|B_{0}\right\|^{\frac{1}{k}}_{L^{2}}}\left\|\Lambda^{k}B\right\|^{\frac{2k+1}{k}}_{L^{2}}\leq 0.$

By solving this ODE, we find the following decay rate

$\begin{eqnarray} \left\|\Lambda^{k}B(t)\right\|_{L^{2}}\leq \frac{\left((2k)^k \left\|B_{0}\right\|_{L^{2}}\left\|\Lambda^{k}B_{0}\right\|_{L^{2}}\right)}{\left(2k \left\|B_{0}\right\|^{\frac{1}{k}}_{L^{2}} +(1-C\epsilon_{0})\left\|\Lambda^{k}B_{0}\right\|^{\frac{1}{k}}_{L^{2}}t\right)^k}. \end{eqnarray}$

(27)

3.2. Proof of Theorem 1.2

We recall (5):

$B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\ln(2+\Lambda)\Lambda B = 0,$

The the uniqueness part of Theorem 1.2 is the same as that of Theorem 1.1 and we only derive a priori bounds. Let

$\left\|\sqrt{\ln(2+\Lambda)}\Lambda^{s}f\right\|^{2}_{L^{2}} = \int \left(\ln(2+|\xi|)\right)|\xi|^{2s}\left|\widehat{f}(\xi)\right|^{2}d\xi.$

We begin with the $L^{2}$ bound:

$\begin{eqnarray} \frac{1}{2}\frac{d}{dt}\left\|B\right\|^{2}_{L^{2}}+\left\|\sqrt{\ln(2+\Lambda)}\Lambda^{\frac{1}{2}}B\right\|^{2}_{L^{2}} = 0. \end{eqnarray}$

(28)

Following the computations in the proof of Theorem 1.1, we also have

$\begin{eqnarray} \frac{d}{dt}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}}+\left\|\sqrt{\ln(2+\Lambda)}\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}\leq C \left\|B\right\|_{H^{k}}\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}. \end{eqnarray}$

(29)

For each $N\in \mathbb{N}$ , we have

$\begin{equation*} \begin{split} &\left\|\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}} = \int_{|\xi|\leq 2^{N}} |\xi|^{2k+1}\left|\widehat{B}(\xi)\right|^{2}d\xi+\int_{|\xi|\ge 2^{N}} |\xi|^{2k+1}\left|\widehat{B}(\xi)\right|^{2}d\xi\\ &\leq 2^{N}\int_{|\xi|\leq 2^{N}} |\xi|^{2k}\left|\widehat{B}(\xi)\right|^{2}d\xi+\frac{1}{\ln(2+2^{N})}\int_{|\xi|\ge 2^{N}} \ln(2+|\xi|)|\xi|^{2k+1}\left|\widehat{B}(\xi)\right|^{2}d\xi\\ &\leq 2^{N}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}} +\frac{1}{\ln(2+2^{N})}\left\|\sqrt{\ln(2+\Lambda)}\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}. \end{split} \end{equation*}$

So, (29) is replaced by

$\begin{split} &\frac{d}{dt}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}}+\left\|\sqrt{\ln(2+\Lambda)}\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}} \\ &\leq C 2^{N}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}}\|B\|_{H^{k}} + \frac{C\|B\|_{H^{k}}}{\ln(2+2^{N})}\left\|\sqrt{\ln(2+\Lambda)}\Lambda^{\frac{1}{2}+k}B\right\|^{2}_{L^{2}}. \end{split}$

We now choose $N>0$ such that

$\frac{1}{2}\ln(2+2^{N}) < C\left\|B\right\|_{H^{k}} < \ln(2+2^{N})$

and so $N \sim \|B\|_{H^{k}}$ . Then, (29) is reduced to

$\begin{eqnarray} \frac{d}{dt}\left\|\Lambda^{k}B\right\|^{2}_{L^{2}} \leq C \exp\left(\|B\|_{H^{k}}\right)\|B\|_{H^{k}} \left\|\Lambda^{k}B\right\|_{L^{2}}. \end{eqnarray}$

(30)

By (28) and (30), we obtain

$\frac{d}{dt}\left\|B\right\|^{2}_{H^{k}}\leq C \exp\left(\|B\|_{H^{k}}\right)\|B\|^{2}_{H^{k}}$

and so we have

$\frac{d}{dt}\left\|B\right\|_{H^{k}}\leq C \exp\left(\|B\|_{H^{k}}\right)\|B\|_{H^{k}} \leq C \exp\left(\|B\|_{H^{k}}\right).$

By solving this ODE, we can derive (6).

4. Proof of Theorem 1.3 and Theorem 1.4

4.1. Proof of Theorem 1.3

We recall (9):

$\begin{align} & \psi_{t} = [\psi, Z], \end{align}$

(31a)

$\begin{align} & Z_{t}-\Delta Z = [\Delta \psi, \psi]. \end{align}$

(31b)

4.1.1. Approximation

We first approximate (31a) by putting $\epsilon \Delta \psi$ to the right-hand side and mollify the resulting equations as (21). Then, we have

$\begin{eqnarray} \begin{split} &\partial_{t}\psi^{(\epsilon)} = \mathcal{J}_{\epsilon}[\mathcal{J}_{\epsilon}\psi^{(\epsilon)}, \mathcal{J}_{\epsilon}Z^{(\epsilon)}]+\epsilon \mathcal{J}^{2}_{\epsilon}\Delta \psi^{(\epsilon)}, \\ & \partial_{t}Z^{(\epsilon)}-\Delta \mathcal{J}^{2}_{\epsilon}Z^{(\epsilon)} = \mathcal{J}_{\epsilon}[\Delta \mathcal{J}_{\epsilon}\psi^{(\epsilon)}, \mathcal{J}_{\epsilon}\psi^{(\epsilon)}] \end{split} \end{eqnarray}$

(32)

with $\psi^{(\epsilon)}_{0} = \mathcal{J}_{\epsilon}\psi_{0}$ and $Z^{(\epsilon)}_{0} = \mathcal{J}_{\epsilon}Z_{0}$ . Since (32) is defined in $\mathbb{R}^{2}$ , the proof of the existence of a unique global-in-time solution of (32) is relatively easier than the one to (21). Moreover, the bounds in Section 4.1.2 are independent of $\epsilon>0$ and so we can pass to the limit in a subsequence and show the existence of smooth solutions locally in time when $\psi_{0}\in H^{4}$ and $Z_{0}\in H^{3}$ .

4.1.2. A priori estimates

We first note that

$\begin{eqnarray} \frac{1}{2}\frac{d}{dt}\left\|\psi\right\|^{2}_{L^{2}} = \int \psi[\psi, Z] = 0. \end{eqnarray}$

(33)

We next multiply (31a) by $-\Delta \psi$ , (31b) by $Z$ , and integrate over $\mathbb{R}^{2}$ . By (18c),

$\begin{eqnarray} \frac{1}{2}\frac{d}{dt} \left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+\left\|Z\right\|^{2}_{L^{2}}\right) + \left\|\nabla Z\right\|^{2}_{L^{2}} = \int \left(-\Delta \psi[\psi, Z]+Z[\Delta \psi, \psi]\right) = 0. \end{eqnarray}$

(34)

We also multiply (31a) by $\Delta^{4} \psi$ , (31b) by $-\Delta^{3} Z$ and integrate over $\mathbb{R}^{2}$ . Then,

$\begin{eqnarray} \begin{split} \frac{1}{2}\frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\Delta^{2} Z\right\|^{2}_{L^{2}} & = \int \Delta^{4} \psi [\psi, Z] -\int\Delta^{3} Z[\Delta \psi, \psi]\\ & = \mathcal{R}. \end{split} \end{eqnarray}$

(35)

We now compute the right-hand side of (35). By (18a), (18b), and (18c),

$\begin{equation} \begin{split} \mathcal{R}& = 2\int\Delta^{2}\psi[\Delta \psi, \Delta Z]+4\int \Delta^{2} \psi[\psi_{x}, \Delta Z_{x}]+4\int \Delta^{2} \psi[\psi_{y}, \Delta Z_{y}]\\ & +4\int \Delta^{2} \psi[\Delta\psi_{x}, Z_{x}]+4\int \Delta^{2} \psi[\Delta \psi_{y}, Z_{y}]+4\int \Delta^{2} \psi[\psi_{xx}, Z_{xx}]\\ &+8\int \Delta^{2} \psi[\psi_{xy}, Z_{xy}]+4\int \Delta^{2} \psi[\psi_{yy}, Z_{yy}]-2\int\Delta^{2}Z[\Delta \psi_{x}, \psi_{x}]\\ &-2\int\Delta^{2}Z[\Delta \psi_{y}, \psi_{y}]. \end{split} \end{equation}$

(36)

So, we find that the number of derivatives acting on $(\psi, \psi, Z)$ are $(4, 4, 2)$ , $(3, 4, 3)$ , and $(4, 2, 4)$ up to multiplicative constants. Hence,

$\begin{equation*} \begin{split} & \frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\Delta^{2} Z\right\|^{2}_{L^{2}}\\ & \leq C\int \left|\nabla^{4}\psi\right| \left|\nabla^{4}\psi\right| \left|\nabla^{2} Z\right|+ C\int \left|\nabla^{3}\psi\right| \left|\nabla^{4}\psi\right| \left|\nabla^{3} Z\right|+ C\int \left|\nabla^{4}\psi\right| \left|\nabla^{2}\psi\right| \left|\nabla^{4} Z\right|\\ & \leq C\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}} \left\|\nabla^{2} Z\right\|_{L^{\infty}}+ C\left\|\nabla^{3}\psi\right\|_{L^{4}} \left\|\Delta^{2}\psi\right\|_{L^{2}} \left\|\nabla^{3} Z\right\|_{L^{4}}\\ &+ C\left\|\Delta^{2}\psi\right\|_{L^{2}} \left\|\nabla^{2}\psi\right\|_{L^{\infty}} \left\|\Delta^{2} Z\right\|_{L^{2}}\\ & \leq C\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}} \left\|\nabla^{2} Z\right\|_{L^{\infty}}+ C\left\|\Delta^{2}\psi\right\|^{\frac{3}{2}}_{L^{2}} \left\|\nabla\Delta\psi\right\|^{\frac{1}{2}}_{L^{2}} \left\|\Delta^{2} Z\right\|_{L^{2}}\\ &+C\left\|\Delta^{2}\psi\right\|_{L^{2}} \left\|\nabla^{2}\psi\right\|_{L^{\infty}} \left\|\Delta^{2} Z\right\|_{L^{2}}\\ &\leq C \mathcal{E}^{2}+\frac{1}{4}\left\|\Delta^{2} Z\right\|^{2}_{L^{2}}+\delta \left\|\nabla^{2} Z\right\|^{2}_{L^{\infty}} \leq C \mathcal{E}^{2}+\frac{1}{2}\left\|\Delta^{2} Z\right\|^{2}_{L^{2}}+\frac{1}{4}\left\|\nabla Z\right\|^{2}_{L^{2}}, \end{split} \end{equation*}$

where we use

$\begin{split} \left\|\nabla^{2} Z\right\|^{2}_{L^{\infty}}&\leq C \left\|\Delta Z\right\|_{L^{2}}\left\|\Delta^{2} Z\right\|_{L^{2}}\leq C \left\|\nabla Z\right\|^{\frac{2}{3}}_{L^{2}}\left\|\Delta^{2} Z\right\|^{\frac{4}{3}}_{L^{2}}\\&\leq C \left\|\nabla Z\right\|^{2}_{L^{2}}+C\left\|\Delta^{2} Z\right\|^{2}_{L^{2}} \end{split}$

with $\delta$ satisfying $4C\delta = 1$ . So, we have

$\begin{eqnarray} \frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\Delta^{2} Z\right\|^{2}_{L^{2}} \leq C \mathcal{E}^{2}+\frac{1}{2}\left\|\nabla Z\right\|^{2}_{L^{2}}. \end{eqnarray}$

(37)

By (33), (34), and (37), we derive $\mathcal{E}' \leq C \mathcal{E}^{2}$ from which we deduce

$\begin{eqnarray} \mathcal{E}(t)\leq \frac{\mathcal{E}_{0}}{1-Ct\mathcal{E}_{0}} \quad {{\rm{for}}\; {\rm{all}}} \;\ 0 < t\leq T_{\ast} < \frac{1}{C \mathcal{E}_{0}}. \end{eqnarray}$

(38)

4.1.3. Uniqueness

Let $(\psi_{1}, Z_{1})$ and $(\psi_{2}, Z_{2})$ be two solutions of (31) and let $\psi = \psi_{1}-\psi_{2}$ and $Z = Z_{1}-Z_{2}$ . Then, $(\psi, Z)$ satisfies the following equations:

$\psi_{t} = [\psi, Z_{1}]+[\psi_{2}, Z], \quad Z_{t}-\Delta Z = [\Delta\psi, \psi_{1}]+[\Delta\psi_{2}, \psi]$

with $\psi(0, x) = Z(0, x) = 0$ . For these equations, we have

$\begin{equation*} \begin{split} &\frac{1}{2}\frac{d}{dt}\left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+ \|Z\|^{2}_{L^{2}}\right)+ \left\|\nabla Z\right\|^{2}_{L^{2}}\\ & = -\int \Delta \psi [\psi, Z_{1}]-\int \Delta \psi [\psi_{2}, Z]+\int Z [\Delta\psi, \psi_{1}]+\int Z[\Delta\psi_{2}, \psi]\\ & = \text{(I)+(II)+(III)+(IV)}. \end{split} \end{equation*}$

The first term is bounded using the definition of $[f, g]$ and ${\rm{div}}\; \nabla^{\perp}Z_{1} = 0$ :

$\text{(I)} = \int \left(\nabla^{\perp}Z_{1}\cdot \nabla \psi\right) \Delta \psi = -\int \left(\nabla^{\perp}\partial_{l}Z_{1}\cdot \nabla \psi\right) \partial_{l} \psi\leq C \left\|\nabla^{2}Z_{1}\right\|_{L^{\infty}}\left\|\nabla \psi\right\|^{2}_{L^{2}}.$

We next bound $\text{(II)+(III)}$ as

$\begin{split} \text{(II)+(III)}& = -\int Z[\Delta \psi, \psi]\leq C \left\|\nabla^{2}\psi\right\|_{L^{\infty}}\left\|\nabla \psi\right\|_{L^{2}}\left\|\nabla Z\right\|_{L^{2}} \\ &\leq C \left(\left\|\nabla^{2}\psi_{1}\right\|^{2}_{L^{\infty}}+\left\|\nabla^{2}\psi_{2}\right\|^{2}_{L^{\infty}}\right)\left\|\nabla \psi\right\|^{2}_{L^{2}}+\frac{1}{4}\left\|\nabla Z\right\|^{2}_{L^{2}}. \end{split}$

The last term is bounded as

$\text{(IV)}\leq C \left\|\nabla^{2}\psi_{2}\right\|_{L^{\infty}}\left\|\nabla \psi\right\|_{L^{2}}\left\|\nabla Z\right\|_{L^{2}} \leq C \left\|\nabla^{2}\psi_{2}\right\|^{2}_{L^{\infty}}\left\|\nabla \psi\right\|^{2}_{L^{2}}+\frac{1}{4}\left\|\nabla Z\right\|^{2}_{L^{2}}.$

So, we have

$\begin{eqnarray} \begin{split} &\frac{d}{dt}\left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+ \|Z\|^{2}_{L^{2}}\right)\\ &\leq C\left( \left\|\nabla^{2}Z_{1}\right\|_{L^{\infty}}+\left\|\nabla^{2}\psi_{1}\right\|^{2}_{L^{\infty}}+\left\|\nabla^{2}\psi_{2}\right\|^{2}_{L^{\infty}}\right)\left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+ \|Z\|^{2}_{L^{2}} \right). \end{split} \end{eqnarray}$

(39)

By (38), $\left\|\nabla^{2}\psi_{1}\right\|^{2}_{L^{\infty}}+\left\|\nabla^{2}\psi_{2}\right\|^{2}_{L^{\infty}}$ is integrable in time. Integrating (34) and (35) in time, we have

$\int^{t}_{0}\left(\left\|\nabla Z(s)\right\|^{2}_{L^{2}}+\left\|\Delta^{2} Z(s)\right\|^{2}_{L^{2}}\right)ds < \infty \quad {{\rm{for}}} \; 0 < t\leq \frac{T_{\ast}}{2}$

which gives the integrability of the first term in the parentheses on the right-hand side of (39). By repeating the same argument one more time, we have the uniqueness up to $T_{\ast}$ .

4.1.4. Blow-up criterion

Let

$\mathcal{B}(s) = \left\|\nabla^{2}Z(s)\right\|_{L^{\infty}} +\left\|\nabla^{2}\psi(s)\right\|^{2}_{L^{\infty}} .$

We first deal with

$\begin{split} &\frac{1}{2}\frac{d}{dt} \left(\left\|\Delta\psi\right\|^{2}_{L^{2}}+\left\|\nabla Z\right\|^{2}_{L^{2}} \right) +\left\| \Delta Z\right\|^{2}_{L^{2}} = \int \Delta^{2} \psi [\psi, Z] -\int\Delta Z[\Delta \psi, \psi]\\ & = 2\int \Delta \psi[\psi_{x}, Z_{x}]+2\int \Delta \psi[\psi_{y}, Z_{y}]\leq C \left\|\nabla^{2}Z\right\|_{L^{\infty}}\left\| \Delta \psi\right\|^{2}_{L^{2}} \end{split}$

and so we have

$\frac{d}{dt} \left(\left\|\Delta\psi\right\|^{2}_{L^{2}}+\left\|\nabla Z\right\|^{2}_{L^{2}} \right) +\left\| \Delta Z\right\|^{2}_{L^{2}} \leq C \left\|\nabla^{2}Z\right\|_{L^{\infty}}\left\| \Delta \psi\right\|^{2}_{L^{2}}.$

This implies

$\begin{eqnarray} \begin{split} &\left\|\Delta \psi(t)\right\|^{2}_{L^{}}+ \left\|\nabla Z(t)\right\|^{2}_{L^{2}}+\int^{t}_{0}\left\| \Delta Z(s)\right\|^{2}_{L^{2}} ds < \infty \\ &\iff \int^{t}_{0}\left\|\nabla^{2}Z(s)\right\|_{L^{\infty}}ds < \infty. \end{split} \end{eqnarray}$

(40)

We also deal with

$\begin{split} &\frac{1}{2}\frac{d}{dt} \left(\left\|\nabla \Delta\psi\right\|^{2}_{L^{2}}+\left\|\Delta Z\right\|^{2}_{L^{2}} \right) +\left\| \nabla \Delta Z\right\|^{2}_{L^{2}} = -\int \Delta^{3} \psi [\psi, Z] +\int\Delta^{2} Z[\Delta \psi, \psi]\\ & = -\int \Delta^{2}\psi [\Delta \psi, Z] -2\int \Delta^{2}\psi \left([\psi_{x}, Z_{x}]+[\psi_{y}, Z_{y}]\right) \\ &- 2\int \Delta\psi \left([\psi_{x}, \Delta Z_{x}]+[\psi_{y}, \Delta Z_{y}]\right) = \text{(I)+(II)+(III)}. \end{split}$

As in Section 4.1.3,

$\begin{eqnarray} \text{(I)} = \int \left(\nabla \nabla^{\perp}Z\cdot \nabla \Delta \psi\right)\cdot \nabla \Delta\psi \leq C\left\|\nabla^{2}Z\right\|_{L^{\infty}} \left\|\nabla\Delta \psi \right\|^{2}_{L^{2}}. \end{eqnarray}$

(41)

We next estimate $\text{(II)}+\text{(III)}$ :

$\begin{equation} \begin{split} &\text{(II)}+\text{(III)} = -4\int \Delta\psi \left([\Delta \psi_{y}, Z_{y}]+[\psi_{y}, \Delta Z_{y}]+[\psi_{xy}, Z_{xy}]+ [\psi_{yy}, Z_{yy}]\right)\\ &\leq C\int \left|\nabla^{2}Z\right| \left|\nabla^{3}\psi \right|^{2}+C\int \left|\nabla^{2}\psi \right| \left|\nabla^{3}\psi \right|\left|\nabla^{3}Z\right|\\ &\leq C\left\|\nabla^{2}Z\right\|_{L^{\infty}} \left\|\nabla\Delta \psi \right\|^{2}_{L^{2}}+C\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}} \left\|\nabla \Delta \psi \right\|^{2}_{L^{2}}+\frac{1}{2}\left\|\nabla \Delta Z \right\|^{2}_{L^{2}}. \end{split} \end{equation}$

(42)

By (41) and (42), we have

$\frac{d}{dt}\left(\left\|\nabla \Delta \psi\right\|^{2}_{L^{2}}+\left\|\Delta Z\right\|^{2}_{L^{2}}\right) +\left\|\nabla \Delta Z\right\|^{2}_{L^{2}}\leq C\left(\left\|\nabla^{2} Z\right\|_{L^{\infty}} +\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}} \right)\left\|\nabla\Delta \psi \right\|^{2}_{L^{2}}$

which implies

$\begin{eqnarray} \left\|\nabla \Delta \psi(t)\right\|^{2}_{L^{2}}+\left\|\Delta Z(t)\right\|^{2}_{L^{2}}+\int^{t}_{0} \left\|\nabla \Delta Z(s)\right\|^{2}_{L^{2}}ds < \infty \iff \int^{t}_{0}\mathcal{B}(s)ds < \infty. \end{eqnarray}$

(43)

We finally deal with

$\frac{1}{2}\frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\Delta^{2}Z\right\|^{2}_{L^{2}} = \int \Delta^{4} \psi [\psi, Z] -\int\Delta^{3} Z[\Delta \psi, \psi] = \mathcal{R}$

with the same $\mathcal{R}$ in (36). So, we have

$\begin{equation*} \begin{split} &\frac{1}{2} \frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\Delta^{2} Z\right\|^{2}_{L^{2}}\\ & \leq C\left\|\nabla^{2} Z\right\|_{L^{\infty}}\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+ C\left\|\nabla^{2}\psi\right\|_{L^{\infty}}\left\|\Delta^{2} Z\right\|_{L^{2}} \left\|\Delta^{2}\psi\right\|_{L^{2}} \\ &+C\left\|\nabla\Delta Z\right\|_{L^{4}}\left\|\nabla\Delta \psi\right\|_{L^{4}}\left\|\Delta^{2}\psi\right\|_{L^{2}}\\ &\leq C\left(\left\|\nabla^{2}Z\right\|_{L^{\infty}} +\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}} +\left\|\nabla \Delta Z\right\|^{\frac{3}{2}}_{L^{2}}\left\|\nabla \Delta \psi\right\|^{\frac{3}{2}}_{L^{2}}\right)\left\|\Delta^{2} \psi \right\|^{2}_{L^{2}}+\frac{1}{2}\left\|\Delta^{2} Z\right\|^{2}_{L^{2}} \end{split} \end{equation*}$

which gives

$\begin{eqnarray} \begin{split} &\frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\Delta^{2} Z\right\|^{2}_{L^{2}}\\ &\leq C\left(\mathcal{B}(s)+\left\|\nabla \Delta Z\right\|^{\frac{3}{2}}_{L^{2}}\left\|\nabla \Delta \psi\right\|^{\frac{3}{2}}_{L^{2}}\right)\left\|\Delta^{2} \psi \right\|^{2}_{L^{2}}. \end{split} \end{eqnarray}$

(44)

By (40) and (43), (44) implies

$\left\|\Delta^{2} \psi(t)\right\|^{2}_{L^{2}}+\left\|\nabla \Delta Z(t)\right\|^{2}_{L^{2}}+\int^{t}_{0} \left\|\Delta ^{2}Z(s)\right\|^{2}_{L^{2}}ds < \infty \iff \int^{t}_{0}\mathcal{B}(s)ds < \infty.$

4.2. Proof of Theorem 1.4

We recall (11):

$\psi_{t}+\psi = [\psi, Z], \quad Z_{t}-\Delta Z = [\Delta \psi, \psi]$

Since the uniqueness is already proved in Section 4.1.3 even without the damping term, we only focus on the a priori bounds and the decay rates.

4.2.1. A priori estimates

We first have

$\begin{equation} \begin{split} &\frac{1}{2}\frac{d}{dt}\left\|\psi\right\|^{2}_{L^{2}}+\left\|\psi\right\|^{2}_{L^{2}} = 0, \\ &\frac{1}{2}\frac{d}{dt} \left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+\left\|Z\right\|^{2}_{L^{2}}\right) +\left\|\nabla \psi\right\|^{2}_{L^{2}}+ \left\|\nabla Z\right\|^{2}_{L^{2}} = 0. \end{split} \end{equation}$

(45)

We now consider the highest order part:

$\begin{split} &\frac{1}{2}\frac{d}{dt} \left(\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\Delta^{2} Z\right\|^{2}_{L^{2}} \right)+\left\|\nabla\Delta^{2}\psi\right\|^{2}_{L^{2}} +\left\|\nabla\Delta^{2} Z\right\|^{2}_{L^{2}} \\ & = -\int \Delta^{5} \psi [\psi, Z] +\int\Delta^{4} Z[\Delta \psi, \psi]. \end{split}$

We compute the right-hand side of this. By (18a), (18b), and (18c),

$\begin{equation} \begin{split} &-\int \Delta^{5} \psi [\psi, Z] +\int\Delta^{4} Z[\Delta \psi, \psi]\\ & = 2\int \Delta^{3}Z\left[\Delta \psi_{x}, \psi_{x}\right]+2\int \Delta^{3}Z\left[\Delta \psi_{y}, \psi_{y}\right]+2\int \Delta^{2}Z\left[\Delta^{2} \psi_{x}, \psi_{x}\right]\\ &+2\int \Delta^{2}Z\left[\Delta^{2} \psi_{y}, \psi_{y}\right] +2\int \Delta Z\left[\Delta^{2} \psi_{x}, \Delta\psi_{x}\right]+2\int \Delta Z\left[\Delta^{2} \psi_{y}, \Delta\psi_{y}\right] \\ &-\int \Delta^{3}\psi[\Delta\psi, \Delta Z]-2\int\Delta^{3}\psi[\Delta \psi_{x}, Z_{x}] -2\int\Delta^{3}\psi[\Delta \psi_{y}, Z_{y}] -2\int\Delta^{3}\psi[ \psi_{x}, \Delta Z_{x}] \\ &-2\int\Delta^{3}\psi[ \psi_{y}, \Delta Z_{y}] -2\int\Delta^{4}\psi[ \psi_{x}, Z_{x}] -2\int\Delta^{4}\psi[ \psi_{y}, Z_{y}] -\int \Delta^{3}\psi[\Delta^{2}\psi, Z]. \end{split} \end{equation}$

(46)

We now count the number of derivatives hitting on $(Z, \psi, \psi)$ using the integration by parts and (18b) and (18c) up to multiplicative constants. Except for the last integral, we have

$\begin{split} &(6, 2, 4) \mapsto (5, 2, 5), \ (5, 3, 4)\, \quad (4, 2, 6) \mapsto (5, 5, 2), \ (4, 3, 5)\\ &(2, 2, 8) \mapsto (3, 2, 7) \mapsto (4, 2, 6), \ (3, 3, 6) \mapsto (5, 5, 2), \ (4, 3, 5)\\ & (2, 4, 6) \mapsto (2, 5, 5), \ (3, 4, 5). \end{split}$

The last integral is

$\int \left(\nabla^{\perp}Z\cdot \nabla \Delta^{2}\psi\right)\Delta^{3}\psi = -\int \left(\nabla^{\perp}\partial_{l}Z\cdot \nabla \Delta^{2}\psi\right)\partial_{l}\Delta^{2}\psi$

and so this gives $(2, 5, 5)$ . So, the combinations of the numbers of derivatives taken on $(Z, \psi, \psi)$ are

$(2, 5, 5), \ (3, 4, 5), \ (4, 3, 5), \ (5, 2, 5), \ (5, 3, 4).$

The first and the fourth cases are bounded by

$\begin{split} &C\left\|\nabla^{2}Z\right\|_{L^{\infty}}\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}}\leq C\left\|\nabla^{2}Z\right\|^{2}_{L^{\infty}}\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}}+\frac{1}{6}\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}}, \\ & C\left\|\nabla^{2}\psi\right\|_{L^{\infty}}\left\|\nabla \Delta^{2}Z\right\|^{2}_{L^{2}}\leq C\left\|\nabla^{2}\psi\right\|_{L^{\infty}}\left\|\nabla \Delta^{2}Z\right\|^{2}_{L^{2}}+\frac{1}{4}\left\|\nabla \Delta^{2}Z\right\|^{2}_{L^{2}}. \end{split}$

The second case is bounded by

$\begin{split} &C\left\|\nabla^{3}Z\right\|_{L^{4}}\left\|\nabla^{4}\psi\right\|_{L^{4}}\left\|\nabla\Delta^{2}\psi\right\|_{L^{2}}\leq C\left\|\Delta Z\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|^{\frac{1}{2}}_{L^{2}} \left\|\Delta^{2}\psi\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla\Delta^{2}\psi\right\|^{\frac{3}{2}}_{L^{2}}\\ & \leq C\left\|\Delta Z\right\|^{2}_{L^{2}}\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|^{2}_{L^{2}} +\frac{1}{6}\left\|\nabla\Delta^{2}\psi\right\|^{2}_{L^{2}}. \end{split}$

The third case is bounded by

$\begin{split} &C\left\|\nabla^{4}Z\right\|_{L^{4}}\left\|\nabla^{3}\psi\right\|_{L^{4}}\left\|\nabla\Delta^{2}\psi\right\|_{L^{2}}\leq C\left\|\Delta^{2}Z\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|^{\frac{1}{2}}_{L^{2}} \left\|\Delta \psi\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla\Delta^{2}\psi\right\|^{\frac{3}{2}}_{L^{2}}\\ & \leq C\left\|\Delta \psi\right\|^{2}_{L^{2}}\left\|\Delta^{2}Z\right\|^{2}_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|^{2}_{L^{2}} +\frac{1}{6}\left\|\nabla\Delta^{2}\psi\right\|^{2}_{L^{2}}. \end{split}$

The last one is bounded by

$\begin{split} &C\left\|\nabla^{3}\psi\right\|_{L^{4}}\left\|\nabla^{4}\psi\right\|_{L^{4}}\left\|\nabla\Delta^{2}Z\right\|_{L^{2}}\leq C\left\|\nabla\Delta \psi\right\|^{\frac{1}{2}}_{L^{2}}\left\|\Delta^{2}\psi\right\|_{L^{2}}\left\|\nabla\Delta^{2}\psi\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|_{L^{2}}\\ & \leq C\left\|\nabla \Delta \psi\right\|_{L^{2}}\left\|\nabla \Delta^{2}\psi\right\|_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|_{L^{2}} \leq C\left\|\nabla \Delta \psi\right\|^{2}_{L^{2}}\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}} +\frac{1}{4}\left\|\nabla\Delta^{2}Z\right\|^{2}_{L^{2}}. \end{split}$

So, we obtain

$\begin{equation} \begin{split} &\frac{d}{dt} \left(\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\Delta^{2} Z\right\|^{2}_{L^{2}} \right)+\left\|\nabla\Delta^{2}\psi\right\|^{2}_{L^{2}} +\left\|\nabla\Delta^{2} Z\right\|^{2}_{L^{2}}\\ & \leq C\left\|\nabla^{2}Z\right\|^{2}_{L^{\infty}}\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}}+ C\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\left\|\nabla \Delta^{2}Z\right\|^{2}_{L^{2}}+C\left\|\nabla \Delta \psi\right\|^{2}_{L^{2}}\left\|\nabla \Delta^{2}\psi\right\|^{2}_{L^{2}}\\ &+C\left\|\Delta Z\right\|^{2}_{L^{2}}\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|^{2}_{L^{2}}+C\left\|\Delta \psi\right\|^{2}_{L^{2}}\left\|\Delta^{2}Z\right\|^{2}_{L^{2}}\left\|\nabla\Delta^{2}Z\right\|^{2}_{L^{2}} \end{split} \end{equation}$

(47)

By (45) and (47),

$\mathcal{F}'(t)+\mathcal{N}_{1}(t)\leq C \left(\mathcal{F}(t)+\mathcal{F}^{2}(t)\right)\mathcal{N}_{1}(t).$

So, if $\mathcal{F}_{0} = \epsilon_{0}$ is sufficiently small, we obtain

$\label{A priori damping dd} \mathcal{F}(t)+(1-C \epsilon_{0})\int^{t}_{0}\mathcal{N}_{1}(s)ds\leq \mathcal{F}_{0} \quad {{\rm{for}} \;{\rm{all}} \;t > 0 .}$

4.2.2. Decay rates

From (45), $\left\|\psi(t)\right\|_{L^{2}}\leq \left\|\psi_{0}\right\|_{L^{2}} e^{-t}$ . Since

$\begin{split} \frac{1}{2}\frac{d}{dt}\left\|\nabla \psi\right\|^{2}_{L^{2}}+\left\|\nabla \psi\right\|^{2}_{L^{2}}& = -\int \Delta \psi[\psi, Z] = \int \left(\nabla^{\perp}Z\cdot \nabla \psi\right)\Delta \psi\\ & = -\int \left(\partial_{l}\nabla^{\perp}Z\cdot \nabla \psi\right)\partial_{l} \psi\\ &\leq \left\|\nabla^{2}Z\right\|_{L^{\infty}}\left\|\nabla \psi\right\|^{2}_{L^{2}} \leq C\epsilon_{0}\left\|\nabla \psi\right\|^{2}_{L^{2}}, \end{split}$

we have

$\left\|\nabla \psi(t)\right\|_{L^{2}}\leq \left\|\nabla \psi_{0}\right\|_{L^{2}} e^{-(1-C\epsilon_{0})t}.$

By using (14), we also obtain

$\left\|\Lambda^{k}\psi(t)\right\|_{L^{2}}\leq \mathcal{F}^{\frac{k-1}{8}}_{0}\left\|\nabla \psi_{0}\right\|^{\frac{5-k}{4}}_{L^{2}} e^{-\frac{(5-k)(1-C\epsilon_{0})}{4}t}, \quad 1\leq k < 5.$

5. Proof of Theorem 1.5 and Theorem 1.6

5.1. Proof of Theorem 1.5

We recall (12):

$\psi_{t}-\Delta \psi = [\psi, Z], \quad Z_{t} = [\Delta \psi, \psi].$

By applying the same approximation and mollification methods in Section 4.1.1, we can show the existence of smooth solutions locally in time when $\psi_{0}\in H^{4}$ and $Z_{0}\in H^{3}$ .

5.1.1. A priori estimates

We first have

$\begin{equation} \begin{split} &\frac{1}{2}\frac{d}{dt}\left\|\psi\right\|^{2}_{L^{2}}+\left\|\nabla \psi\right\|^{2}_{L^{2}} = 0, \\ & \frac{1}{2}\frac{d}{dt} \left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+\left\|Z\right\|^{2}_{L^{2}}\right) + \left\|\Delta \psi\right\|^{2}_{L^{2}} = 0. \end{split} \end{equation}$

(48)

We next deal with

$\frac{1}{2}\frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\nabla \Delta^{2} \psi\right\|^{2}_{L^{2}} = \int \Delta^{4} \psi [\psi, Z] -\int\Delta^{3} Z[\Delta \psi, \psi] = \mathcal{R}$

with the same $\mathcal{R}$ in (36). In this case, we choose the the number of derivatives acting on $(\psi, \psi, Z)$ different from Section 4.1.2, which are given by $(3, 5, 2)$ , $(2, 5, 3)$ , and $(4, 4, 2)$ after several integration by parts. Hence, we have

and so we have the following bound

$\begin{eqnarray} \frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\nabla \Delta^{2} \psi\right\|^{2}_{L^{2}} \leq C \mathcal{E}^{2}. \end{eqnarray}$

(49)

By (48) and (49), we derive $\mathcal{E}' \leq C \mathcal{E}^{2}$ from which we deduce

$\begin{eqnarray} \mathcal{E}(t)\leq \frac{\mathcal{E}_{0}}{1-Ct\mathcal{E}_{0}} \quad {{\rm{for}}\; {\rm{all}}} \;\ 0 < t\leq T_{\ast} < \frac{1}{C \mathcal{E}_{0}}. \end{eqnarray}$

(50)

5.1.2. Uniqueness

Let $(\psi_{1}, Z_{1})$ and $(\psi_{2}, Z_{2})$ be two solutions and let $\psi = \psi_{1}-\psi_{2}$ and $Z = Z_{1}-Z_{2}$ . As in Section 4.1.3

$\begin{equation*} \begin{split} &\frac{1}{2}\frac{d}{dt}\left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+ \|Z\|^{2}_{L^{2}}\right)+ \left\|\Delta \psi\right\|^{2}_{L^{2}}\\ & = -\int \Delta \psi [\psi, Z_{1}]-\int \Delta \psi [\psi_{2}, Z]+\int Z [\Delta\psi, \psi_{1}]+\int Z[\Delta\psi_{2}, \psi]\\ & = \text{(I)+(II)+(III)+(IV)}. \end{split} \end{equation*}$

The first term three terms are bounded as

$\begin{split} \text{(I)}&\leq \left\|\nabla Z_{1}\right\|_{L^{\infty}}\left\|\nabla \psi\right\|_{L^{2}}\left\|\Delta \psi\right\|_{L^{2}} \leq C\left\|\nabla Z_{1}\right\|^{2}_{L^{\infty}}\left\|\nabla \psi\right\|^{2}_{L^{2}}+\frac{1}{3}\left\|\Delta \psi\right\|^{2}_{L^{2}}, \\ \text{(II)+(III)}& = -\int Z[\Delta \psi, \psi]\leq C \left\|\nabla Z\right\|_{L^{\infty}}\left\|\nabla \psi\right\|_{L^{2}}\left\|\Delta \psi\right\|_{L^{2}} \\ &\leq C \left(\left\|\nabla Z_{1}\right\|^{2}_{L^{\infty}}+\left\|\nabla Z_{2}\right\|^{2}_{L^{\infty}}\right)\left\|\nabla \psi\right\|^{2}_{L^{2}}+\frac{1}{3}\left\|\Delta \psi\right\|^{2}_{L^{2}} \end{split}$

The last term is bounded as

$\begin{split} \text{(IV)}&\leq C \left\|\nabla^{3}\psi_{2}\right\|_{L^{4}}\left\|\nabla \psi\right\|_{L^{4}}\left\|Z\right\|_{L^{2}} \leq C \left\|\nabla^{3}\psi_{2}\right\|_{L^{4}}\left\|\nabla \psi\right\|^{\frac{1}{2}}_{L^{2}}\left\|\Delta \psi\right\|^{\frac{1}{2}}_{L^{2}}\left\|Z\right\|_{L^{2}} \\ & \leq C \left\|\nabla^{3}\psi_{2}\right\|^{\frac{4}{3}}_{L^{4}}\left\|\nabla \psi\right\|^{\frac{2}{3}}_{L^{2}}\left\|Z\right\|^{\frac{4}{3}}_{L^{2}} +\frac{1}{3}\left\|\Delta \psi\right\|^{2}_{L^{2}} \leq C \left\|\nabla^{3}\psi_{2}\right\|^{4}_{L^{4}}\left\|\nabla \psi\right\|^{2}_{L^{2}}\\ &+ C \left\|Z\right\|^{2}_{L^{2}} +\frac{1}{3}\left\|\Delta \psi\right\|^{2}_{L^{2}}\\ & \leq C \left\|\nabla\Delta \psi_{2}\right\|^{2}_{L^{2}}\left\|\Delta^{2}\psi_{2}\right\|^{2}_{L^{2}} \left\|\nabla \psi\right\|^{2}_{L^{2}}+ C \left\|Z\right\|^{2}_{L^{2}} +\frac{1}{3}\left\|\Delta \psi\right\|^{2}_{L^{2}}. \end{split}$

So, we have

$\begin{split} &\frac{d}{dt}\left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+ \|Z\|^{2}_{L^{2}}\right)\\&\leq C\left(\left\|\nabla Z_{1}\right\|^{2}_{L^{\infty}}+\left\|\nabla Z_{2}\right\|^{2}_{L^{\infty}} +\left\|\nabla\Delta \psi_{2}\right\|^{2}_{L^{2}}\left\|\Delta^{2}\psi_{2}\right\|^{2}_{L^{2}}\right)\left(\left\|\nabla \psi\right\|^{2}_{L^{2}}+ \|Z\|^{2}_{L^{2}}\right). \end{split}$

By (50), the terms in the parentheses are integrable up to $\frac{T_{\ast}}{2}$ . By repeating the same argument one more time, we have the uniqueness up to $T_{\ast}$ .

5.1.3. Blow-up criterion

To derive the blow-up criterion, we first bound

$\begin{split} &\frac{1}{2}\frac{d}{dt} \left(\left\|\Delta\psi\right\|^{2}_{L^{2}}+\left\|\nabla Z\right\|^{2}_{L^{2}} \right) +\left\| \nabla \Delta \psi\right\|^{2}_{L^{2}} = \int \Delta^{2} \psi [\psi, Z] -\int\Delta Z[\Delta \psi, \psi]\\ & = 2\int \Delta \psi[\psi_{x}, Z_{x}]+2\int \Delta \psi[\psi_{y}, Z_{y}]\leq C \left\|\nabla^{2}\psi\right\|_{L^{\infty}}\left\|\nabla Z\right\|_{L^{2}}\left\|\nabla \Delta \psi\right\|_{L^{2}}\\ &\leq C \left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\left\|\nabla Z\right\|^{2}_{L^{2}}+\frac{1}{2}\left\|\nabla \Delta \psi\right\|^{2}_{L^{2}} \end{split}$

and so we have

$\frac{d}{dt} \left(\left\|\Delta\psi\right\|^{2}_{L^{2}}+\left\|\nabla Z\right\|^{2}_{L^{2}} \right) +\left\| \nabla \Delta \psi\right\|^{2}_{L^{2}} \leq C \left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\left\|\nabla Z\right\|^{2}_{L^{2}}.$

This implies

$\begin{eqnarray} \begin{split} &\left\|\Delta \psi(t)\right\|^{2}_{L^{}}+ \left\|\nabla Z(t)\right\|^{2}_{L^{2}}+\int^{t}_{0}\left\| \nabla \Delta \psi(s)\right\|^{2}_{L^{2}} ds < \infty\\& \iff \int^{t}_{0}\left\|\nabla^{2}\psi(s)\right\|^{2}_{L^{\infty}}ds < \infty \end{split} \end{eqnarray}$

(51)

We also take

$\begin{split} &\frac{1}{2}\frac{d}{dt} \left(\left\|\nabla \Delta\psi\right\|^{2}_{L^{2}}+\left\|\Delta Z\right\|^{2}_{L^{2}} \right) +\left\| \Delta^{2} \psi\right\|^{2}_{L^{2}} = -\int \Delta^{3} \psi [\psi, Z] +\int\Delta^{2} Z[\Delta \psi, \psi]\\ & = -\int \Delta^{2}\psi [\Delta \psi, Z] -2\int \Delta^{2}\psi \left([\psi_{x}, Z_{x}]+[\psi_{y}, Z_{y}]\right) \\&\qquad- 2\int \Delta\psi \left([\psi_{x}, \Delta Z_{x}]+[\psi_{y}, \Delta Z_{y}]\right)\\ & = \text{(I)+(II)+(III)}. \end{split}$

By using the computation in (41),

$\begin{split} \text{(I)}& = \int \left(\nabla \nabla^{\perp}Z\cdot \nabla \Delta \psi\right)\cdot \nabla \Delta\psi \leq C\left\|\nabla^{2}Z\right\|_{L^{2}} \left\|\nabla^{3}\psi \right\|^{2}_{L^{4}} \\ &\leq C\left\|\nabla^{2}Z\right\|^{2}_{L^{2}} \left\|\nabla\Delta \psi \right\|^{2}_{L^{2}}+\frac{1}{6}\left\|\Delta^{2}\psi \right\|^{2}_{L^{2}}. \end{split}$

We next estimate $\text{(II)}+\text{(III)}$ using (42):

$\begin{equation*} \begin{split} \text{(II)}+\text{(III)}&\leq C\int \left|\nabla^{2}Z\right| \left|\nabla^{3}\psi \right|^{2}+C\int \left|\nabla^{2}\psi \right| \left|\nabla^{4}\psi \right|\left|\nabla^{2}Z\right|\\ &\leq C\left\|\Delta Z\right\|^{2}_{L^{2}} \left\|\nabla\Delta \psi \right\|^{2}_{L^{2}}+C\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}} \left\|\Delta Z \right\|^{2}_{L^{2}}+\frac{1}{3}\left\|\Delta^{2}\psi \right\|^{2}_{L^{2}} \end{split} \end{equation*}$

So, we have

$\begin{eqnarray} \begin{split} &\frac{d}{dt}\left(\left\|\nabla \Delta \psi\right\|^{2}_{L^{2}}+\left\|\Delta Z\right\|^{2}_{L^{2}}\right) +\left\|\Delta^{2} \psi\right\|^{2}_{L^{2}}\\&\leq C\left(\left\|\nabla\Delta \psi \right\|^{2}_{L^{2}}+\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\right)\left\|\Delta Z\right\|^{2}_{L^{2}}. \end{split} \end{eqnarray}$

(52)

By (51), (52) implies

$\begin{eqnarray} \begin{split} &\left\|\nabla \Delta \psi(t)\right\|^{2}_{L^{2}}+\left\|\Delta Z(t)\right\|^{2}_{L^{2}}+\int \left\|\Delta^{2} \psi(s)\right\|^{2}_{L^{2}}ds < \infty \\ &\iff \int^{t}_{0}\left\|\nabla^{2}\psi(s)\right\|^{2}_{L^{\infty}}ds < \infty. \end{split} \end{eqnarray}$

(53)

We finally deal with

where we count the number of derivatives acting on $(\psi, \psi, Z)$ in (46) as $(3, 5, 2)$ , $(2, 5, 3)$ , and $(4, 4, 2)$ . Then, we obtain

$\begin{equation*} \begin{split} &\frac{1}{2} \frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\nabla\Delta^{2} \psi\right\|^{2}_{L^{2}}\\ & \leq C\left\|\Delta Z\right\|^{2}_{L^{2}}\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}\\&\qquad+ C\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\left\|\nabla \Delta Z\right\|^{2}_{L^{2}} +C\left\|\Delta Z\right\|^{2}_{L^{4}}\left\|\nabla^{3}\psi\right\|^{2}_{L^{4}}+\frac{1}{2}\left\|\nabla \Delta^{2} \psi\right\|^{2}_{L^{2}}\\ &\leq C\left\|\Delta Z\right\|^{2}_{L^{2}}\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+ C\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\left\|\nabla \Delta Z\right\|^{2}_{L^{2}} +C\left\|\Delta Z\right\|^{2}_{L^{2}}\left\|\nabla\Delta Z\right\|^{2}_{L^{2}}\\ &+C\left\|\nabla \Delta \psi\right\|^{2}_{L^{2}}\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\frac{1}{2}\left\|\nabla \Delta^{2} \psi\right\|^{2}_{L^{2}} \end{split} \end{equation*}$

and so we have

$\begin{equation} \begin{split} &\frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\nabla\Delta^{2} \psi\right\|^{2}_{L^{2}}\\ &\leq C\left(\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}+\left\|\Delta Z\right\|^{2}_{L^{2}}\right)\left\|\nabla \Delta Z\right\|^{2}_{L^{2}} +C\left(\left\|\nabla \Delta \psi\right\|^{2}_{L^{2}}+\left\|\Delta Z\right\|^{2}_{L^{2}}\right)\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}} \end{split} \end{equation}$

(54)

By (51) and (53), (54) implies

$\label{BW 5}\begin{split} &\left\|\Delta^{2} \psi(t)\right\|^{2}_{L^{2}}+\left\|\nabla \Delta Z(t)\right\|^{2}_{L^{2}}+\int^{t}_{0} \left\|\nabla\Delta^{2} \psi(s)\right\|^{2}_{L^{2}}ds < \infty\\& \iff \int^{t}_{0}\left\|\nabla^{2}\psi(s)\right\|^{2}_{L^{\infty}}ds < \infty. \end{split}$

5.2. Proof of Theorem 1.6

We recall (13):

$\psi_{t}-\Delta \psi = [\psi, Z], \quad Z_{t}+Z = [\Delta \psi, \psi].$

Since the uniqueness is already proved in Section 5.1.2 even without the damping term, we only focus on the a priori bounds.

We first have

(55)

We also have

$\begin{split} \frac{1}{2}\frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) &+\left\|\nabla \Delta^{2} \psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \\ & = \int \Delta^{4} \psi [\psi, Z] -\int\Delta^{3} Z[\Delta \psi, \psi] = \mathcal{R} \end{split}$

with the same $\mathcal{R}$ in (36). In this case, we also choose the the number of derivatives acting on $(\psi, \psi, Z)$ as $(3, 5, 2)$ , $(2, 5, 3)$ , and $(4, 4, 2)$ . Hence,

$\begin{equation*} \begin{split} & \frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\nabla\Delta^{2} \psi\right\|^{2}_{L^{2}} +\left\|\nabla\Delta Z\right\|^{2}_{L^{2}}\\ & \leq C \left\|\nabla^{2} Z\right\|_{L^{2}}\left\|\Delta^{2}\psi\right\|^{2}_{L^{4}} + C\left\|\nabla \Delta Z\right\|_{L^{2}} \left\|\nabla^{2}\psi\right\|_{L^{\infty}} \left\|\nabla \Delta^{2} \psi\right\|_{L^{2}}\\ &+ C\left\|\Delta Z\right\|_{L^{4}}\left\|\nabla^{3}\psi\right\|_{L^{4}} \left\|\nabla \Delta^{2}\psi\right\|_{L^{2}} \\ & \leq C \left\|\nabla Z\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla\Delta Z\right\|^{\frac{1}{2}}_{L^{2}} \left\|\nabla \Delta\psi\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla \Delta^{2}\psi\right\|^{\frac{3}{2}}_{L^{2}}+C\left\|\nabla \Delta Z\right\|_{L^{2}} \left\|\nabla^{2}\psi\right\|_{L^{\infty}} \left\|\nabla \Delta^{2} \psi\right\|_{L^{2}}\\ &+ C\left\|\Delta Z\right\|^{\frac{1}{2}}_{L^{2}}\left\|\nabla\Delta Z\right\|^{\frac{1}{2}}_{L^{2}} \left\|\Delta\psi\right\|^{\frac{1}{2}}_{L^{2}} \left\|\nabla \Delta^{2}\psi\right\|^{\frac{3}{2}}_{L^{2}}\\ & \leq C\left(\left\|\nabla Z\right\|^{2}_{L^{2}}\left\|\nabla \Delta\psi\right\|^{2}_{L^{2}}+\left\|\Delta Z\right\|^{2}_{L^{2}}\left\|\Delta\psi\right\|^{2}_{L^{2}}+\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\right)\left\|\nabla\Delta Z\right\|^{2}_{L^{2}}\\&\qquad +\frac{1}{2} \left\|\nabla \Delta^{2} \psi\right\|_{L^{2}}. \end{split} \end{equation*}$

So, we obtain

$\begin{equation} \begin{split} &\frac{d}{dt} \left(\left\|\Delta^{2}\psi\right\|^{2}_{L^{2}}+\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \right) +\left\|\nabla\Delta^{2} \psi\right\|^{2}_{L^{2}} +\left\|\nabla\Delta Z\right\|^{2}_{L^{2}}\\ & \leq C\left(\left\|\nabla Z\right\|^{2}_{L^{2}}\left\|\nabla \Delta\psi\right\|^{2}_{L^{2}}+\left\|\Delta Z\right\|^{2}_{L^{2}}\left\|\Delta\psi\right\|^{2}_{L^{2}}+\left\|\nabla^{2}\psi\right\|^{2}_{L^{\infty}}\right)\left\|\nabla\Delta Z\right\|^{2}_{L^{2}} \end{split} \end{equation}$

(56)

By (55) and (56),

$\mathcal{E}'(t)+\mathcal{N}_{2}(t)\leq C \left(\mathcal{E}(t)+\mathcal{E}^{2}(t)\right)\mathcal{N}_{2}(t).$

So, if $\mathcal{E}_{0} = \epsilon_{0}$ is sufficiently small, we obtain

$\mathcal{E}(t)+(1-C \epsilon_{0})\int^{t}_{0}\mathcal{N}_{2}(s)ds\leq \mathcal{E}_{0} \quad {\rm{for}}\; {\rm{all}}\; t > 0 .$

Acknowledgments

H.B. was supported by NRF-2018R1D1A1B07049015. H. B. acknowledges the Referee for his/her valuable comments that highly improve the manuscript.

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Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation

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Abstract

1. Introduction

1.1. The Hall equation with fractional Laplacian

1.2. 2D models

1.2.1. Case 1

1.2.2. Case 2

2. Preliminaries

2.1. Inequalities

2.2. Commutator

3. Proof of Theorem 1.1 and Theorem 1.2

3.1. Proof of Theorem 1.1

3.1.1. Approximation

3.1.2. A priori estimates

3.1.3. Uniqueness

3.1.4. Decay rates

3.2. Proof of Theorem 1.2

4. Proof of Theorem 1.3 and Theorem 1.4

4.1. Proof of Theorem 1.3

4.1.1. Approximation

4.1.2. A priori estimates

4.1.3. Uniqueness

4.1.4. Blow-up criterion

4.2. Proof of Theorem 1.4

4.2.1. A priori estimates

4.2.2. Decay rates

5. Proof of Theorem 1.5 and Theorem 1.6

5.1. Proof of Theorem 1.5

5.1.1. A priori estimates

5.1.2. Uniqueness

5.1.3. Blow-up criterion

5.2. Proof of Theorem 1.6

Acknowledgments

References

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通讯作者: 陈斌, bchen63@163.com

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