On the local and global existence of the Hall equations with fractional Laplacian and related equations

1.   Introduction

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

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:

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 $\beta = 1$:

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

Moreover, $B$ decays in time

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):

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

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

we can rewrite (3) as

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:

(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

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

Moreover, we have the following blow-up criterion:

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

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

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

Moreover, $\psi$ decays exponentially in time

with $1\leq k<5.$

1.2.2.   Case 2

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

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

Moreover, we have the following blow-up criterion

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

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

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

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

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

2.1.   Inequalities

We begin with two inequalities in 3D:

We also provide the following inequalities in 2D

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

which holds in any dimension.

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

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

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:

3.   Proof of Theorem 1.1 and Theorem 1.2

3.1.   Proof of Theorem 1.1

We recall (3):

3.1.1.   Approximation

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

We then mollify (20) as follows

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:

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

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

So, we obtain

By (22) and (24),

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

3.1.3.   Uniqueness

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

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

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

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

By solving this ODE, we find the following decay rate

3.2.   Proof of Theorem 1.2

We recall (5):

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

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

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

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

So, (29) is replaced by

We now choose $N>0$ such that

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

By (28) and (30), we obtain

and so we have

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):

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

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

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

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

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

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,

where we use

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

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

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:

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

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

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

The last term is bounded as

So, we have

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

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

We first deal with

and so we have

This implies

We also deal with

As in Section 4.1.3,

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

By (41) and (42), we have

which implies

We finally deal with

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

which gives

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

4.2.   Proof of Theorem 1.4

We recall (11):

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

We now consider the highest order part:

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

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

The last integral is

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

The first and the fourth cases are bounded by

The second case is bounded by

The third case is bounded by

The last one is bounded by

So, we obtain

By (45) and (47),

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

4.2.2.   Decay rates

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

we have

By using (14), we also obtain

5.   Proof of Theorem 1.5 and Theorem 1.6

5.1.   Proof of Theorem 1.5

We recall (12):

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

We next deal with

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

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

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

The first term three terms are bounded as

The last term is bounded as

So, we have

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

and so we have

This implies

We also take

By using the computation in (41),

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

So, we have

By (51), (52) implies

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

and so we have

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

5.2.   Proof of Theorem 1.6

We recall (13):

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

We also have

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,

So, we obtain

By (55) and (56),

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

Acknowledgments

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