Since $E$ is bounded from below and decreasing along weak solutions, we know
that the limit

$\displaystyle E_{\infty}:=\lim_{t\rightarrow\infty}E(u^{L}(t),v^{L}(t))$

exists. Hence, recalling the definition of $\omega^{L}(u_{0},v_{0})$, we infer
that

$\displaystyle E(u_{*},v_{*})=E_{\infty}\quad\text{for
all}\;(u_{*},v_{*})\in\omega^{L}(u_{0},v_{0}).$ (4.24)

Since $E$ is nonincreasing along weak solutions, it holds that
$E(u^{L}(t),v^{L}(t))\geq E_{\infty}$ for all $t\geq 0$.

If there exists a time $t_{*}\geq 0$ such that
$E(u^{L}(t_{*}),v^{L}(t_{*}))=E_{\infty}$, then the existence of a unique
limit $(u_{\infty},v_{\infty})\in\mathcal{N}_{m}$ is a direct consequence of
Lemma 4.6. In particular, we have

$\displaystyle\big{(}u_{0},v_{0}\big{)}=\big{(}u^{L}(t),v^{L}(t)\big{)}=\big{(}u^{L}(t_{*}),v^{L}(t_{*})\big{)}=\big{(}u_{\infty},v_{\infty}\big{)}$

for all $t\geq 0$.

In the following, we will thus assume that $E(u^{L}(t),v^{L}(t))>E_{\infty}$
for all $t\geq 0$. From Lemma 4.12 we infer that for every critical point
$(u_{*},v_{*})\in{\mathcal{W}_{\beta,0}^{2}}$ of $E$ there exist constants
$\gamma(u_{*},v_{*})\in\left(0,\frac{1}{2}\right]$ and $C(u_{*},v_{*})$,
$\sigma(u_{*},v_{*})>0$ such that the the Łojasiewicz–Simon inequality (4.22)
holds for all

$\displaystyle(u,v)\in
B_{\sigma(u_{*},v_{*})}(u_{*},v_{*})=\Big{\\{}(u,v)\in{\mathcal{W}_{\beta,0}^{2}}\,:\,\|(u,v)-(u_{*},v_{*})\|_{{\mathcal{W}_{\beta,0}^{2}}}<\sigma(u_{*},v_{*})\Big{\\}}.$

Obviously, the set $\omega^{L}(u_{0},v_{0})$ is covered by the union of all
open balls $B_{\sigma(u_{*},v_{*})}(u_{*},v_{*})$ with
$(u_{*},v_{*})\in\omega^{L}(u_{0},v_{0})$. Since $\omega^{L}(u_{0},v_{0})$ is
compact in ${\mathcal{W}_{\beta,0}^{2}}$, we can select finitely many points
$(u_{n},v_{n})\in\omega^{L}(u_{0},v_{0})$, $n=1,...,N$ such that

$\displaystyle\omega^{L}(u_{0},v_{0})\subset\left(\,\bigcup_{n=1}^{N}B_{\sigma(u_{n},v_{n})}(u_{n},v_{n})\right)=:\mathcal{U}.$

Recalling (4.24) and arguing as in [13, Theorem 2.3], we can now find
constants $\gamma\in(0,\frac{1}{2}]$ and $C>0$ depending only on $L$, $\beta$
and $\Omega$ and $(u_{0},v_{0})$ but not on $n$ or $N$ such that for all
$(u,v)\in\cal U$,

$\displaystyle\left|E(u,v)-E_{\infty}\right|^{1-\gamma}\leq
C\|E^{\prime}(u,v)\|_{({\mathcal{W}_{\beta,0}^{1}})^{\prime}}\;.$ (4.25)

We point out that $\gamma<\frac{1}{2}$ can be assumed without loss of
generality. Since $\mathcal{U}$ is an open subset of
${\mathcal{W}_{\beta,0}^{2}}$, and
$\omega^{L}(u_{0},v_{0})\subset\mathcal{U}$, there exists a time $t_{0}\geq 1$
such that $\big{(}u^{L}(t),v^{L}(t)\big{)}\subset\mathcal{U}$ for all $t\geq
t_{0}$. In particular, this means that $\big{(}u^{L}(t),v^{L}(t)\big{)}$
satisfies the inequality (4.25) for all $t\geq t_{0}$.

Let now $s\geq t_{0}$ be arbitrary, and without loss of generality we assume
that $\mu^{L}$ and $\theta^{L}$ can be evaluated at the time $s$ with
$\big{(}\mu^{L}(s),\theta^{L}(s)\big{)}\in\mathcal{H}^{1}$. Now, applying
integration by parts on (4.8), we obtain

$\displaystyle\left\langle
E^{\prime}\big{(}u^{L}(s),v^{L}(s)\big{)},(\zeta,\xi)\right\rangle_{{\mathcal{W}_{\beta,0}^{1}}}=\int_{\Omega}\mu^{L}(s)\,\zeta\,\mathrm{d}x+\int_{\Gamma}\theta^{L}(s)\,\xi\,\mathrm{d}\Gamma$
(4.26)

for all $(\zeta,\xi)\in{\mathcal{W}_{\beta,0}^{1}}$. We now define

$\displaystyle
c:=\frac{\beta\int_{\Omega}\mu^{L}(s)\,\mathrm{d}x+\int_{\Gamma}\theta^{L}(s)\,\mathrm{d}\Gamma}{\beta^{2}\left|\Omega\right|+\left|\Gamma\right|},\quad\bar{\mu}:=\mu^{L}(s)-\beta
c,\quad\bar{\theta}:=\theta^{L}(s)-c,$

meaning that $(\bar{\mu}(s),\bar{\theta}(s))\in{\mathcal{W}_{\beta,0}^{1}}$.
For any $(\zeta,\xi)\in{\mathcal{W}_{\beta,0}^{1}}$, we thus obtain

$\displaystyle\int_{\Omega}\mu^{L}(s)\,\zeta\,\mathrm{d}x+\int_{\Gamma}\theta^{L}(s)\,\xi\,\mathrm{d}\Gamma=\int_{\Omega}\bar{\mu}\zeta\,\mathrm{d}x+\int_{\Gamma}\bar{\theta}\xi\,\mathrm{d}\Gamma.$
(4.27)

Applying the bulk-surface Poincaré type inequality presented in [43, Lem. 7.1]
(with $K:=L$ and $\alpha:=\beta$), we conclude that

$\displaystyle\big{\|}E^{\prime}\big{(}u^{L}(s),v^{L}(s)\big{)}\big{\|}_{({\mathcal{W}_{\beta,0}^{1}})^{\prime}}\leq\|(\bar{\mu},\bar{\theta})\|_{\mathcal{L}^{2}}\leq
c_{P}\|(\bar{\mu},\bar{\theta})\|_{L,\beta}=c_{P}\big{\|}\big{(}\mu^{L}(s),\theta^{L}(s)\big{)}\big{\|}_{L,\beta}\;,$
(4.28)

where the constant $c_{P}>0$ depends only on $L$, $\beta$ and $\Omega$.

Recalling that $(u,v)\in C([1,\infty);\mathcal{H}^{2})$ and interpreting
$(u^{L},v^{L},\mu^{L},\theta^{L})$ as a weak solution starting at time $s$
(instead of zero), the energy inequality (3.6) yields that (4.17) holds for
all $t\in\mathbb{R}_{0}^{+}$ with $t\geq s$. As the right-hand side of this
inequality does not depend on $t$, we may pass to the limit $t\to\infty$ on
the left-hand side, which gives

$\displaystyle\dfrac{1}{2}\int_{s}^{\infty}\big{\|}\big{(}\mu^{L}(\tau),\theta^{L}(\tau)\big{)}\big{\|}_{L,\beta}^{2}\,\mathrm{d}\tau\leq
E(u^{L}(s),v^{L}(s))-E_{\infty}.$ (4.29)

Hence, combining (4.25), (4.28) and (4.29), we infer that

$\displaystyle\int_{s}^{\infty}\big{\|}\big{(}\mu^{L}(\tau),\theta^{L}(\tau)\big{)}\big{\|}_{L,\beta}^{2}\,\mathrm{d}\tau\leq
C\big{\|}\big{(}\mu^{L}(s),\theta^{L}(s)\big{)}\big{\|}_{L,\beta}^{1/(1-\gamma)}$

for some constant $C\geq 0$ depending only on $L$, $\beta$ and $\Omega$, and
almost all $s\geq t_{0}$. We recall that $\gamma<\frac{1}{2}$ was assumed
without loss of generality. As the expression on the left hand side can also
be bounded by a constant via the energy inequality (3.6), we invoke [25, Lem.
7.1] to deduce that

$\displaystyle\nabla\mu^{L}\in L^{1}(t_{0},\infty;L^{2}(\Omega)),\text{ \
}\nabla\theta^{L}\in L^{1}(t_{0},\infty;L^{2}(\Gamma)),\text{ \
}\beta\theta^{L}-\mu^{L}\in L^{1}(t_{0},\infty;L^{2}(\Gamma)).$

(Alternatively, one could also argue as in [13, Proof of Thm. 2.3] to obtain
the same result.) Recalling the weak formulations (3.4a) and (3.4b), we
further obtain $(\partial_{t}u^{L},\partial_{t}v^{L})\in
L^{1}\big{(}t_{0},\infty;\mathcal{H}_{\beta}^{-1}\big{)}$. Consequently, for
any $t_{0}<t_{1}<t_{2}$, we have

$\displaystyle\big{\|}\big{(}u^{L}(t_{2}),v^{L}(t_{2})\big{)}-\big{(}u^{L}(t_{1}),v^{L}(t_{1})\big{)}\big{\|}_{L,\beta,*}\leq\int_{t_{1}}^{t_{2}}\big{\|}\big{(}\partial_{t}u^{L}(\tau),\partial_{t}v^{L}(\tau)\big{)}\big{\|}_{L,\beta,*}\,\mathrm{d}\tau\;,$

and the right-hand side converges to zero as $t_{2}>t_{1}\to\infty$. Using the
interpolation inequalities from Lemma 2.4 and Corollary 2.5 along with the
smoothing property (3.7), we conclude that

$\displaystyle\big{\|}\big{(}u^{L}(t_{2}),v^{L}(t_{2})\big{)}-\big{(}u^{L}(t_{1}),v^{L}(t_{1})\big{)}\big{\|}_{\mathcal{H}^{2}}$
$\displaystyle\quad\leq
C\left(\frac{t+1}{t}\right)^{\frac{3}{8}}\big{\|}\big{(}u^{L}(t_{2}),v^{L}(t_{2})\big{)}-\big{(}u^{L}(t_{1}),v^{L}(t_{1})\big{)}\big{\|}_{L,\beta,*}^{\frac{1}{4}}$
$\displaystyle\quad\longrightarrow 0,\quad\text{as $t_{2}>t_{1}\to\infty$}.$

Invoking the Cauchy criterion, this implies the existence of a unique limit
$(u_{\infty},v_{\infty})\in{\mathcal{W}_{\beta,0}^{2}}$ such that the
convergence assertion (4.23) is satisfied. In particular, it holds that
$(u_{\infty},v_{\infty})\in\omega^{L}(u_{0},v_{0})\subset\mathcal{N}_{0}$
which completes the proof. ∎

We will now see that solutions starting in $\mathcal{A}^{L}_{m}$ converge to a
stationary point also for $t\to-\infty$.

######  Theorem 4.14 (Convergence to a single stationary point as
$t\to-\infty$).

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A4}$) hold. Then, for
any $(u_{0},v_{0})\in\mathcal{A}_{m}^{L}$, the corresponding state
$(u^{L}(t),v^{L}(t))$ exists for all $t\in\mathbb{R}$, and there exists a
unique stationary point $(u_{-\infty},v_{-\infty})\in\mathcal{N}_{m}$ such
that

$\displaystyle\lim_{t\to-\infty}\|\big{(}u^{L}(t),v^{L}(t)\big{)}-(u_{-\infty},v_{-\infty})\|_{\mathcal{H}^{2}}=0.$
(4.30)

In particular, this means that
$\alpha^{L}(u_{0},v_{0})=\\{(u_{-\infty},v_{-\infty})\\}\subset\mathcal{N}_{m}$.

  * Proof.

As in the proof of Theorem 4.13, it suffices to address the case $m=0$. Let
$(u_{0},v_{0})\in\mathcal{A}_{0}^{L}$ be arbitrary, and let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding weak solution of
the system (3.1).

We recall that the state $\big{(}u^{L}(t),v^{L}(t)\big{)}$ associated with
$(u_{0},v_{0})$ exists for all times $t\in\mathbb{R}$ and lies in $A^{L}_{0}$.
According to Definition 4.7 and Theorem 4.9, the global attractor
$\mathcal{A}^{L}_{0}$ is a compact subset of ${\mathcal{W}_{\beta,0}^{2}}$.
Hence, since $E$ is continuous, we know that
$E(\mathcal{A}^{L}_{0})\subset\mathbb{R}$ is compact. As $E$ is nonincreasing
along weak solutions, we infer that the limit

$\displaystyle E_{-\infty}:=\lim_{t\rightarrow-\infty}E(u^{L}(t),v^{L}(t))$

exists. Recalling the definition of $\alpha^{L}(u_{0},v_{0})$, we conclude
that

$\displaystyle E(u_{*},v_{*})=E_{-\infty}\quad\text{for all
$(u_{*},v_{*})\in\alpha^{L}(u_{0},v_{0})$}.$

Based on this, the claim of Theorem 4.14 can be established by proceeding
analogously to the proof of Theorem 4.13. ∎

As a consequence, we obtain the following result which provides further
information of the geometric structure of the global attractor.

######  Corollary 4.15.

Under the assumptions ($\mathrm{A1}$)-($\mathrm{A4}$), the global attractor
${\cal{A}}_{m}^{L}$ of the dynamical system
$({\mathcal{W}_{\beta,m}^{1}},S_{m}^{L}(t))$ can be defined as the union of
the unstable manifolds of all stationary points, i.e.,

$\displaystyle{\cal{A}}_{m}^{L}=\underset{(u_{*},v_{*})\in\mathcal{N}_{m}}{\bigcup}\mathcal{M}^{L}_{\text{u}}\big{(}\\{(u_{*},v_{*})\\}\big{)}.$
(4.31)

The assertion follows directly from Theorem 4.14 and Definition 4.8.

######  Remark 4.16.

If the set of stationary points were finite, Corollary 4.15 could also be
established without the Łojasiewicz–Simon inequality. However, as the set of
stationary points is usually infinite, we would merely get
${\cal{A}}_{m}^{L}=\mathcal{M}^{L}_{\text{u}}(\mathcal{N}_{m})$ (see (4.19))
instead of (4.31) if we disregard the Łojasiewicz–Simon inequality.

###  4.2 Long-time dynamics of the GMS model

Based on a result from [53] on closed semigroups, the existence of a global
attractor in the “finite-energy phase space” has already been established for
the GMS model ($L=0$) in [36, Cor. 3.11].

However, it is also possible to obtain the existence of a global attractor for
the dynamical system $\big{(}{\mathcal{W}_{\beta,m}^{1}},S_{m}^{0}(t)\big{)}$
by using the same arguments as in Section 4.1. It is worth mentioning that the
GMS model has the same set of stationary points $\mathcal{N}_{m}$ as the KLLM
model. Hence, we already know from Subsection 4.1.1 that the stationary point
set of the GMS model is nonempty and bounded in ${\mathcal{W}_{\beta,m}^{1}}$.
The smoothing property (3.66) implies the asymptotic compactness of the
dynamical system $\big{(}{\mathcal{W}_{\beta,m}^{1}},S_{m}^{0}(t)\big{)}$,
meaning that an analogue of Lemma 4.5 can be established. Furthermore, it can
be shown that the energy functional $E$ is a strict Lyapunov function for the
dynamical system $\big{(}{\mathcal{W}_{\beta,m}^{1}},S_{m}^{0}(t)\big{)}$ and
consequently, the existence of a unique global attractor $\mathcal{A}_{m}^{0}$
can be proved analogously as in Subsection 4.1.2.

It was further established in [36, Thm. 3.22] that the $\omega$-limits set
consists of one single stationary point. We point out that Theorem 4.14 could
be adapted to the case $L=0$ and thus, Corollary 4.15 remains true for the GMS
model.

##  5 Stability of the global attractor for the GMS model

We intend to study the stability of the global attractor $\mathcal{A}^{0}_{m}$
of the GMS model ($L=0$) with respect to perturbations $\mathcal{A}^{L}_{m}$
with small $L>0$. At least formally, this means we have to investigate the
asymptotic limit $L\to 0$ of the family $\\{\mathcal{A}^{L}_{m}\\}_{L>0}$ of
global attractors for the KLLM model. To provide a rigorous notion of such a
limit, we recall the following definition which is included in [15, Thm.
7.2.8].

######  Definition 5.1 (Upper semicontinuity of global attractors).

Let $X$ be a Banach space, and let $\Lambda$ be a metric space. Suppose that
for any $\lambda\in\Lambda$, $(X,\\{S^{\lambda}(t)\\}_{t\geq 0})$ is a
dynamical system possessing a global attractor $\mathcal{A}^{\lambda}\subset
X$. Then, the family $\\{\mathcal{A}^{\lambda}\\}_{\lambda\geq 0}$ is called
_upper semicontinuous at the point $\lambda_{*}\in\Lambda$_ if

$\displaystyle\lim_{\lambda\rightarrow\lambda_{*}}\mathrm{dist}_{X}\big{(}\mathcal{A}^{\lambda},\mathcal{A}^{\lambda_{*}}\big{)}=0.$

To prove that the family $\\{\mathcal{A}_{m}^{L}\\}_{L\geq 0}$ is upper
semicontinuous at $L=0$, we need to know how a weak solution
$(u^{L},v^{L},\mu^{L},\theta^{L})$ to the KLLM model behaves in the asymptotic
limit $L\to 0$.

######  Lemma 5.2 (The asymptotic limit $L\to 0$).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold, let $L>0$ and
$m\in\mathbb{R}$ be arbitrary, and let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum. Let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding unique weak
solution to the KLLM model, and let $(u^{0},v^{0},\mu^{0},\theta^{0})$ denote
the corresponding weak solution of the GMS model. Then, for any $T_{*}>0$, it
holds that

$\displaystyle(u^{L},v^{L})\to(u^{0},v^{0})\quad\text{strongly in
$C([0,T_{*}];\mathcal{L}^{2})$ as $L\to 0$}.$ (5.1)

Moreover, there exist constants $A,B>0$ depending (monotonically increasingly)
on $\|(u_{0},v_{0})\|_{\mathcal{H}^{1}}$ but not on $L$ such that for all
$L>0$,

$\displaystyle\big{\|}(u^{L},v^{L})-(u^{0},v^{0})\big{\|}_{C([1,T_{*}];\mathcal{L}^{2})}\leq
Ae^{BT_{*}}\,L^{\frac{1}{4}}.$ (5.2)

  * Proof.

The convergence (5.1) has already been established in [42, Thm. 4.1].
Moreover, is was also shown that there exists a positive constant $c(T_{*})$
depending (monotonically increasingly) on
$\|(u_{0},v_{0})\|_{\mathcal{H}^{1}}$ and $T_{*}$ but not on $L$ such that for
all $L>0$,

$\displaystyle\big{\|}(u^{L},v^{L})-(u^{0},v^{0})\big{\|}_{C([1,T_{*}];\mathcal{L}^{2})}\leq
c(T_{*})\,L^{\frac{1}{4}}.$

Studying the proof of [42, Thm. 4.1] carefully, we find that the constant
$c(T_{*})$ depends (at most) exponentially on $T_{*}$. This results from the
application of Gronwall’s lemma. We can thus find constants $A,B>0$
independent of $L$ such that $c(T_{*})=Ae^{BT_{*}}$. Thus, the proof is
complete. ∎

By means of the smoothing properties (3.7) and (3.66), we can further
generalize this convergence result to higher order norms.

######  Corollary 5.3 (Convergence rates for the limit $L\to 0$).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold, let $L>0$ and
$m\in\mathbb{R}$ be arbitrary, and let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum. Let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding unique weak
solution to the KLLM model, and let $(u^{0},v^{0},\mu^{0},\theta^{0})$ denote
the corresponding weak solution of the GMS model. Then, for any $T_{*}>1$,

$\displaystyle(u^{L},v^{L})\to(u^{0},v^{0})\quad\text{strongly in
$C([1,T_{*}];\mathcal{H}^{2})$ as $L\to 0$}.$ (5.3)

Moreover, for any $T_{*}>1$, there exist constants
$C_{1}(T_{*}),C_{2}(T_{*})>0$ depending (monotonically increasingly) on
$\|(u_{0},v_{0})\|_{\mathcal{H}^{1}}$ but not on $L$ such that

$\displaystyle\big{\|}(u^{L},v^{L})-(u^{0},v^{0})\big{\|}_{C([1,T_{*}];\mathcal{H}^{1})}\leq
C_{1}(T_{*})\,L^{\frac{1}{6}},$ (5.4)
$\displaystyle\big{\|}(u^{L},v^{L})-(u^{0},v^{0})\big{\|}_{C([1,T_{*}];\mathcal{H}^{2})}\leq
C_{2}(T_{*})\,L^{\frac{1}{12}}.$ (5.5)

  * Proof.

Let $T_{*}>1$ be arbitrary. We first recall that
$(u^{L},v^{L}),(u^{0},v^{0})\in C([1,T_{*}];\mathcal{H}^{2})$ according to
Lemma 3.4 and Lemma 3.9. Using the interpolation inequality (2.20) from Lemma
2.4, and the smoothing properties (3.7) and (3.66), we deduce that

$\displaystyle\underset{t\in[1,T_{*}]}{\sup}\;\|(u^{L},v^{L})(t)-(u^{0},v^{0})(t)\|_{\mathcal{H}^{1}}$
$\displaystyle\quad\leq
C\underset{t\in[1,T_{*}]}{\sup}\;\|(u^{L},v^{L})(t)-(u^{0},v^{0})(t)\|_{\mathcal{L}^{2}}^{\frac{2}{3}}\left[(C_{*}+C_{0})(1+T_{*})^{\frac{1}{2}}\right]^{\frac{1}{3}}.$

Invoking the convergence rate (5.2) we directly conclude (5.4) from the above
estimate. The estimate (5.5) can be established similarly. In particular, this
proves (5.3) and thus, the proof is complete. ∎

We now intend to establish the following stability result which is the main
result of this section. Here, the term “stability” is to be understood as
semicontinuity of the family of perturbed global attractors.

######  Theorem 5.4 (Stability of the global attractor $\mathcal{A}^{0}_{m}$).

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A4}$) hold and let
$m\in\mathbb{R}$ be arbitrary. Then, the family of global attractors
$\\{\mathcal{A}_{m}^{L}\\}_{L\geq 0}$ is upper semicontinuous at $L=0$ in the
sense of Definition 5.1.

To prove this theorem, we will exploit the following abstract result which can
be found in [15, Thm. 7.2.8].

######  Proposition 5.5 (A criterion for upper semicontinuity of global
attractors).

Let $X$ be a Banach space, and let $\Lambda$ be a metric space. Suppose that
for any $\lambda\in\Lambda$, $(X,\\{S^{\lambda}(t)\\}_{t\geq 0})$ is a
dynamical system possessing a global attractor $\mathcal{A}^{\lambda}\subset
X$. We further assume that the following conditions hold:

  1. $\mathrm{(i)}$

There exists a compact set $K\subset X$ such that
$\mathcal{A}^{\lambda}\subset K$ for all $\lambda\geq 0$.

  2. $\mathrm{(ii)}$

If $(x_{k})_{k\in\mathbb{N}}\subset X$ and $\lambda_{k}\in\Lambda$ are
sequences satisfying

     * •

$x_{k}\in\mathcal{A}^{\lambda_{k}}$ for all $k\in\mathbb{N}$,

     * •

$x_{k}\to x_{*}$ as $k\to\infty$,

     * •

$\lambda_{k}\to\lambda_{*}$ as $k\to\infty$,

then there exists $t_{*}>0$ such that $S^{\lambda_{k}}(t)x_{k}\rightarrow
S^{\lambda_{*}}(t)x_{*}$ in $X$ for all $t>t_{*}$.

Then the family $\\{\mathcal{A}^{\lambda}\\}_{\lambda\geq 0}$ is upper
semicontinuous at the point $\lambda_{*}$.

  * Proof of Theorem 5.4.

In order to apply Proposition 5.5, we need to verify the conditions (i) and
(ii) imposed therein.

Step 1: To verify Condition (i), we show that there exists a compact set
$\mathcal{K}_{m}\subset{\mathcal{W}_{\beta,m}^{1}}$ independent of $L$ such
that $\mathcal{A}^{L}_{m}\subset\mathcal{K}_{m}$ for all $L\geq 0$.

For the KLLM model ($L>0$), Lemma 4.11 implies that for every
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$, the set
$\omega^{L}(u_{0},v_{0})$ is included in the set of stationary points. As
discussed Subsection 4.2, the same holds true for the GMS model ($L=0$). We
further recall that the set $\mathcal{N}_{m}$ is a bounded subset of
${\mathcal{W}_{\beta,m}^{2}}$ which does not depend on the parameter $L$.
Hence, we can choose an open, bounded set $\mathcal{U}\subset\mathcal{H}^{2}$
such that for all $L\geq 0$,

$\displaystyle\omega^{L}(u_{0},v_{0})\subseteq\mathcal{N}_{m}\subset\mathcal{U}$

for all $(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$. Consequently, for every
$L\geq 0$ and every initial datum
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$, there exists a time
$t^{L}_{(u_{0},v_{0})}\geq 1$ such that

$\displaystyle
S^{L}_{m}\left(t\right)(u_{0},v_{0})\in\mathcal{U}\quad\text{for all $t\geq
t^{L}_{(u_{0},v_{0})}$.}$ (5.6)

Now, we define the set

$\displaystyle\mathcal{K}_{m}:=\overline{\underset{L\in[0,\infty)}{\bigcup}\;\underset{t\geq
1}{\bigcup}\;S^{L}_{m}\left(t\right)\mathcal{U}}^{\mathcal{H}^{1}}.$ (5.7)

Since $\mathcal{U}$ is bounded in ${\mathcal{W}_{\beta,m}^{2}}$ uniformly in
$L\geq 0$, the smoothing property (3.7) of the KLLM model implies that
$S^{L}_{m}\left(t\right)\mathcal{U}$ is bounded in $\mathcal{H}^{3}$ uniformly
in $L>0$ and $t\geq 1$. Moreover, using the smoothing property (3.66) of the
GMS model, we conclude that $S^{\infty}_{m}\left(t\right)\mathcal{U}$ is
bounded in $\mathcal{H}^{3}$ uniformly in $t\geq 1$. This entails that
$\mathcal{K}_{m}\subset{\mathcal{W}_{\beta,m}^{1}}$ is bounded in
$\mathcal{H}^{3}$, closed in $\mathcal{H}^{1}$, and thus compact in
${\mathcal{W}_{\beta,m}^{1}}$. We further infer from (5.6) that
$\mathcal{K}_{m}$ is an _absorbing set_ for every semigroup $S^{L}_{m}(t)$
with $L\geq 0$ (i.e., for every $L\geq 0$ and every bounded set
$B\subset{\mathcal{W}_{\beta,m}^{1}}$, there exists a time $t_{B}^{L}\geq 1$
such that $S^{L}_{m}(t)B\subset\mathcal{K}_{m}$ for all $t\geq t_{B}^{L}$).
Recall that for all $L\geq 0$, the global attractor $A_{m}^{L}$ is a bounded
subset of ${\mathcal{W}_{\beta,m}^{1}}$. We thus conclude that there exists a
time $t^{L}\geq 1$ such that

$\displaystyle\mathcal{A}_{m}^{L}=S_{m}^{L}(t^{L})\mathcal{A}_{m}^{L}\subset\mathcal{K}_{m}$

due to the invariance property of the global attractor. This directly proves
that $\mathcal{A}^{L}_{m}\subset\mathcal{K}_{m}$ for all $L\geq 0$.

Step 2: To verify Condition (ii) of Proposition 5.5, let
$\\{(u_{k},v_{k})\\}_{k\in\mathbb{N}}\subset{\mathcal{W}_{\beta,m}^{1}}$ be
any sequence with $(u_{k},v_{k})\in\mathcal{A}^{L_{k}}_{m}$ and
$(u_{k},v_{k})\to(u_{*},v_{*})$ in $\mathcal{H}^{1}$ as $k\to\infty$, and let
$\\{L_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{R}_{0}^{+}$ be any sequence with
$L_{k}\to 0$ as $k\to\infty$.

Let now $t\geq t_{*}:=1$ and $\varepsilon>0$ be arbitrary. Using the
continuous dependence estimate from Lemma 3.9, we deduce that

$\displaystyle\big{\|}S^{L_{k}}_{m}(t)(u_{k},v_{k})-S^{0}_{m}(t)(u_{*},v_{*})\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\leq\big{\|}S^{L_{k}}_{m}(t)(u_{k},v_{k})-S^{0}_{m}(t)(u_{k},v_{k})\big{\|}_{\mathcal{H}^{1}}+\big{\|}S^{0}_{m}(t)(u_{k},v_{k})-S^{0}_{m}(t)(u_{*},v_{*})\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\leq\underset{(u,v)\in\mathcal{K}_{m}}{\sup}\;\big{\|}S^{L_{k}}_{m}(\cdot)(u,v)-S^{0}_{m}(\cdot)(u,v)\big{\|}_{C([0,t];\mathcal{H}^{1})}+\Lambda_{1}^{0}(t)\big{\|}(u_{k},v_{k})-(u_{*},v_{*})\big{\|}_{L,\beta,*}.$
(5.8)

Since $L_{k}\to 0$ as $k\to\infty$, and since $\mathcal{K}_{m}$ is bounded,
Corollary 5.3 implies the existence of a number $K_{1}\in\mathbb{N}$ such that
for all $k\geq K_{1}$, the first summand in (5.8) is smaller than
$\varepsilon/2$. Furthermore, the convergence $(u_{k},v_{k})\to(u_{*},v_{*})$
in $\mathcal{H}^{1}$ directly implies that

$\displaystyle\big{\|}(u_{k},v_{k})-(u_{*},v_{*})\big{\|}_{L,\beta,*}\to
0\quad\text{as $k\to\infty$}.$

Hence, there exists a number $K_{2}\in\mathbb{N}$ such that for all $k\geq
K_{2}$, the second summand in (5.8) is smaller than $\varepsilon/2$. In
summary, we get

$\displaystyle\big{\|}S^{L_{k}}_{m}(t)(u_{k},v_{k})-S^{0}_{m}(t)(u_{*},v_{*})\big{\|}_{\mathcal{H}^{1}}<\varepsilon\quad\text{for
all $k\geq K:=\max\\{K_{1},K_{2}\\}$,}$

and since $\varepsilon>0$ was arbitrary, this verifies Condition (ii) of
Proposition 5.5.

Eventually, Proposition 5.5 can be applied on the family
$\\{\mathcal{A}^{L}_{m}\\}_{L\geq 0}$ which proves the assertion of Theorem
5.4. ∎

######  Remark 5.6.

We point out that Proposition 5.5 cannot be used to prove the stability of the
global attractor associated with $L=\infty$. This because the semigroups
associated with $L<\infty$ and the semigroup associated with $L=\infty$ would
have to be defined on the same domain $X$. However, due to the different mass
conservation law of the LW model, its solution operator
$S^{0}_{(m_{1},m_{2})}$ would have to be defined on the linear subspace

$\displaystyle{\mathcal{V}}_{(m_{1},m_{2})}=\big{\\{}(u,v)\in{\mathcal{V}}^{1}\;|\;\langle
u\rangle_{\Omega}=m_{1}\;\text{and}\;\langle
v\rangle_{\Gamma}=m_{2}\big{\\}}.$

This entails that the only reasonable choice for $X$ in Proposition 5.5 would
be $X:={\mathcal{W}_{\beta,m}^{1}}\cap{\mathcal{V}}_{(m_{1},m_{2})}$. Thus,
$X$ is nonempty only if $m=\beta m_{1}+m_{2}$. However, even in this case, we
cannot ensure that $S^{L}_{m}X\subseteq X$ for any $L<\infty$, which means
that $S^{L}_{m}$ does not define a semigroup on the space $X$. Consequently,
Proposition 5.5 is not applicable and thus, the results of Section 5 cannot be
transferred to the scenario $L\to\infty$.

In particular, for the aforementioned reasons, we cannot find a compact set
$\mathcal{K}_{m}$ which acts as an absorbing set the semigroup corresponding
to $L=\infty$ but also for the semigroups associated with $L<\infty$. (Note
that if $\mathcal{K}_{m}$ were a compact absorbing set, it also would have to
absorb itself.) As such a set $\mathcal{K}_{m}$ will play a crucial role in
the subsequent section, it is also not possible to adapt the results of
Section 6 to the situation $L\to\infty$.

##  6 Existence of a robust family of exponential attractors

In this section, for every $L\in[0,1]$, we intend to construct an exponential
attractor for the dynamical system
$(\mathcal{W}^{1}_{\beta,m},\\{S^{L}_{m}\left(t\right)\\}_{t\geq 0})$. We
further show that the exponential attractor associated with $L=0$ is robust
against perturbations $L>0$ in some certain sense.

We first recall the definition of exponential attractors (see, e.g., [21, Def.
4.1] in the discrete case and [15, Def. 7.4.4] in the continuous case).

######  Definition 6.1 (Exponential attractors).

Let $M$ be a metric space, let $I=[0,\infty)$ or $I=\mathbb{N}_{0}$, and let
$\\{S(i)\\}_{i\geq 0}$ be a semigroup on $M$. Then, a compact set
${\mathfrak{M}}\subset M$ is called an exponential attractor for the dynamical
system $\big{(}M,\\{S(i)\\}_{i\in I}\big{)}$ if the following properties hold:

  1. $\mathrm{(i)}$

The set ${\mathfrak{M}}$ is compact in $M$ and has finite fractal dimension,
i.e.,

$\dim_{\mathrm{frac},M}(\mathfrak{M})<\infty.$

  2. $\mathrm{(ii)}$

The set ${\mathfrak{M}}$ is forward invariant under the semigroup
$\\{S(i)\\}_{i\geq 0}$, i.e.,

$S(i){{\mathfrak{M}}}\subset{{\mathfrak{M}}}\quad\text{for all $i\in I$.}$

  3. $\mathrm{(iii)}$

The set ${\mathfrak{M}}$ is an exponentially attracting set for the dynamical
system $\big{(}M,\\{S(i)\\}_{i\in I}\big{)}$, i.e., for every bounded subset
$B\subset M$, there exist constants $C,a>0$ such that

$\displaystyle\mathrm{dist}_{M}\left(S(i)B,{{\mathfrak{M}}}\right)\leq
Ce^{-ai}\quad\text{for all $i\in I$.}$

In the proof of Theorem 5.4, we obtained the existence of a compact absorbing
set $\mathcal{K}_{m}$ (that does not depend on $L$) such that for all $L\geq
0$ and every bounded set $B\subset{\mathcal{W}_{\beta,m}^{1}}$ there exists a
time $t_{0}^{L}(B)\geq 1$ such that

$\displaystyle S_{m}^{L}(t)B\subset\mathcal{K}_{m}\quad\text{for all $t\geq
t_{0}^{L}(B)$.}$

As for any $t\geq 0$ the state $S_{m}^{L}(t)$ depends continuously on the
parameter $L$, we infer that also the time $t_{0}^{L}(B)$ depends continuously
on $L$. We thus obtain

$\displaystyle\bigcup_{L\in[0,1]}S_{m}^{L}(t)B\subset\mathcal{K}_{m}\quad\text{for
all $t\geq t_{0}(B):=\underset{L\in[0,1]}{\max}\,t_{0}^{L}(B)$.}$

It is worth mentioning that the set $\mathcal{K}_{m}$ might not be forward
invariant under the semigroup $\\{S_{m}^{L}(t)\\}_{t\geq 0}$ for any $L\geq
0$. However, as $\mathcal{K}_{m}$ is a bounded absorbing set, it also absorbs
itself. Thus, defining $T_{0}:=t_{0}(\mathcal{K}_{m})\geq 1$, we conclude that

$\displaystyle\bigcup_{L\in[0,1]}S_{m}^{L}(t)\mathcal{K}_{m}\subset\mathcal{K}_{m}\quad\text{for
all $t\geq T_{0}$.}$ (6.1)

The next step is to construct a family
$\\{\mathfrak{M}^{L}_{D}\\}_{L\in[0,1]}$ of exponential attractors for the
discrete dynamical systems

$\displaystyle\left(\mathcal{K}_{m},\\{D_{m}^{L}(n)\\}_{n\in\mathbb{N}_{0}}\right),\quad\text{where}\quad
D^{L}_{m}(n):=S^{L}_{m}(nT_{0})\quad\text{for all $n\in\mathbb{N}_{0}$},$
(6.2)

with $L\in[0,1]$. We further show that the exponential attractor associated
with $L=0$ is robust against perturbations of the parameter $L$. Eventually,
we will extend these results to the continuous dynamical systems
$\big{(}\mathcal{K}_{m},S_{m}^{L}(t)\big{)}$ with $L\in[0,1]$.

It obviously holds that for all $(u,v)\in\mathcal{K}_{m}$,

$\displaystyle D^{L}_{m}(n)(u,v)=\big{[}\underbrace{D^{L}_{m}(1)\circ...\circ
D^{L}_{m}(1)}_{\text{$n$
times}}\big{]}(u,v)=:\big{[}D^{L}_{m}(1)\big{]}^{n}(u,v).$ (6.3)

Therefore, the operator $D^{L}_{m}(1)=S^{L}_{m}(T_{0})$ will play a crucial
role in the analysis.

###  6.1 Robust exponential attractors for discrete dynamical systems

To prove the existence of a robust family of exponential attractors for the
discrete dynamical system
$(\mathcal{K}_{m},\\{D_{m}^{L}(n)\\}_{n\in\mathbb{N}_{0}})$ we first present a
simple generalization of the abstract result [21, Thm. 4.4.], which can be
established by a rescaling argument.

######  Lemma 6.2 (Robust exponential attractors for general discrete
dynamical systems).

Suppose that $X$ and $Y$ are Banach spaces such that $Y$ is compactly embedded
in $X$, and let $M$ be a bounded subset of $X$. Let $L_{*}>0$ be arbitrary. We
further assume that there exists a family
$\\{\mathcal{F}^{L}\\}_{L\in[0,L_{*}]}$ of operators $\mathcal{F}^{L}:M\to
M\cap Y$ which satisfies the following assumptions.

  1. $\mathrm{(i)}$

There exist a constant $\Lambda\geq 0$ such that for all $x_{1},x_{2}\in M$
and all $L\in[0,L_{*}]$,

$\displaystyle\big{\|}\mathcal{F}^{L}(x_{1})-\mathcal{F}^{L}(x_{2})\big{\|}_{Y}\leq\Lambda\|x_{1}-x_{2}\|_{X}\;.$
(6.4)

  2. $\mathrm{(ii)}$

There exists constants $\Theta>0$ and $\zeta\in(0,1]$ such that for all $x\in
M$, $L\in[0,L_{*}]$ and $n\in\mathbb{N}_{0}$,

$\displaystyle\big{\|}\big{[}\mathcal{F}^{L}\big{]}^{n}(x)-\big{[}\mathcal{F}^{0}\big{]}^{n}(x)\big{\|}_{X}\leq\Theta^{n}\,L^{\zeta}\;,$
(6.5)

where $\big{[}\mathcal{F}^{L}\big{]}^{n}$ denotes the $n$-fold composition
$\mathcal{F}^{L}\circ...\circ\mathcal{F}^{L}$.

Then, for every $L\in[0,L_{*}]$, there exists an exponential attractor
$\mathcal{M}^{L}\subset M$ for the discrete semigroup
$(M,\\{\mathcal{D}^{L}(n)\\}_{n\in\mathbb{N}_{0}})$ with
$\mathcal{D}^{L}(n):=[\mathcal{F}^{L}]^{n}$, $n\in\mathbb{N}_{0}$ in the sense
of Definition 6.1.

Moreover, these exponential attractors can be chosen in such a way that the
following properties hold:

  1. $\mathrm{(a)}$

There exists constants $C_{1}>0$ and $0<\gamma<1$ such that for all $L\geq 0$,

$\displaystyle\mathrm{dist}_{\mathrm{sym},X}(\mathcal{M}^{L},\mathcal{M}^{0})\leq
C_{1}L^{\zeta\gamma}.$ (6.6)

  2. $\mathrm{(b)}$

The fractal dimension of $\mathcal{M}^{L}$ is uniformly bounded, i.e., there
exists a constant $C_{2}>0$ such that for all $L\in[0,L_{0}]$:

$\displaystyle\dim_{\mathrm{frac},X}(\mathcal{M}^{L})\leq C_{2}.$ (6.7)

  3. $\mathrm{(c)}$

There exist constants $C_{3},A>0$ such that for all $L\in[0,L_{*}]$ and all
$n\in\mathbb{N}_{0}$,

$\displaystyle\mathrm{dist}_{X}\big{(}\mathcal{D}^{L}(n)M\,,\,\mathcal{M}^{L}\big{)}\leq
C_{3}e^{-An},$ (6.8)

meaning that the rate of convergence to these attractors is uniform in $L$.

  * Proof.

We first assume that (ii) holds with $\zeta=1$. In this case, the assertions
of Lemma 6.2 have already been established in [21, Thm. 4.4.]. We point out
that the assertion (c) is not explicitly stated in [21, Thm. 4.4.] but follows
directly from its proof, in which the authors showed that

$\displaystyle\mathrm{dist}_{X}\big{(}\mathcal{D}^{L}(n)M\,,\,\mathcal{M}^{L}\big{)}=\mathrm{dist}_{X}\big{(}[\mathcal{F}^{L}]^{n}M\,,\,\mathcal{M}^{L}\big{)}\leq
2^{1-n}R\quad\text{for all $n\in\mathbb{N}_{0}$.}$

Here, $R>0$ denotes the radius of a fixed ball $B_{R}(x_{0})$ in $X$ with some
center $x_{0}\in M$ such that $M\subset B_{R}(x_{0})$. Hence, choosing
$C_{3}=2R$ and $A=\ln(2)$, we obtain (c).

Let us now assume that (ii) holds with $\zeta\in(0,1)$. In this case, the
assertions can be established by slightly modifying the proof of [21, Thm.
4.4.]. Alternatively, the change of variables $L\mapsto K:=L^{\zeta}$ could be
used to transform back to the already known situation $\zeta=1$. ∎

We now want to apply this abstract result on the discrete dynamical system
$(\mathcal{K}_{m},\\{D^{L}_{m}(n)\\}_{n\in\mathbb{N}_{0}})$. To this end, we
first need a compact Lipschitz estimate that is uniform in $L\in[0,1]$.

######  Lemma 6.3 (Uniform compact Lipschitz estimate).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold, and let $m\in\mathbb{R}$
and $L\in[0,1]$ be arbitrary. Moreover, for any $i\in\\{1,2\\}$, let
$(u_{0,i},v_{0,i})\in{\mathcal{W}_{\beta,m}^{1}}$ be an arbitrary initial
datum, and let $(u_{i}^{L},v_{i}^{L},\mu_{i}^{L},\theta_{i}^{L})$ denote the
corresponding weak solution of the system (3.1).

Then there exists a positive, non-decreasing function $\Lambda\in
C(\mathbb{R}_{0}^{+})$ depending only on $\Omega$, $\beta$, $F$ and $G$, such
that for all $L\in[0,1]$,

$\displaystyle\big{\|}\big{(}u^{L}_{2},v_{2}^{L}\big{)}(t)-\big{(}u^{L}_{1},v_{1}^{L}\big{)}(t)\big{\|}_{\mathcal{H}^{1}}\leq\Lambda(t)\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{(\mathcal{H}^{1})^{\prime}}\,,\quad\text{for
all $t\geq 1.$}$

  * Proof.

Recall that the functions $\Lambda_{1}^{*}$ in Proposition 3.4 and
$\Lambda_{1}^{0}$ in Proposition 3.9 are independent of $L$. Along with Lemma
2.2, we obtain

$\displaystyle\big{\|}\big{(}u^{L}_{2},v_{2}^{L}\big{)}(t)-\big{(}u^{L}_{1},v_{1}^{L}\big{)}(t)\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\leq\max\\{\Lambda^{*}_{1}(t),\Lambda^{0}_{1}(t)\\}\,\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{L,\beta,*}$
$\displaystyle\leq
C\max\\{\Lambda^{*}_{1}(t),\Lambda^{0}_{1}(t)\\}\,\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{(\mathcal{H}^{1})^{\prime}},$

for all $L\in[0,1]$ and all $t\geq 1$. This directly proves the claim. ∎

Lemma 6.2 and Lemma 6.3 can now be used to establish the following result.

######  Corollary 6.4 (Robust exponential attractors for
$(\mathcal{K}_{m},\\{D^{L}_{m}(n)\\}_{n\in\mathbb{N}_{0}})$).

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A3}$) are satisfied and
let $m\in\mathbb{R}$ be arbitrary.

Then, for every $L\in[0,1]$, there exists an exponential attractor
${\mathfrak{M}}^{L}_{D}\subset\mathcal{K}_{m}$ for the dynamical system
$(\mathcal{K}_{m},D_{m}^{L}(n))_{n\in\mathbb{N}_{0}}$ in the sense of
Definition 6.1, where $M=\mathcal{K}_{m}$ is to be understood as a metric
subspace of $X=(\mathcal{H}^{1})^{\prime}$.

Moreover, these exponential attractors can be chosen in such a way that the
following properties hold:

  1. $\mathrm{(a)}$

The fractal dimension of $\mathfrak{M}^{L}_{D}$ is uniformly bounded, i.e.,
there exists a constant $c_{1}\geq 0$ such that for all $L\in[0,1]$:

$\displaystyle\dim_{\mathrm{frac},{(\mathcal{H}^{1})^{\prime}}}(\mathfrak{M}^{L}_{D})\leq
c_{1}.$ (6.9)

  2. $\mathrm{(b)}$

There exist constants $c_{2}>0$ and $0<\gamma<1$ such that for all
$L\in[0,1]$,

$\displaystyle\mathrm{dist}_{\mathrm{sym},{(\mathcal{H}^{1})^{\prime}}}(\mathfrak{M}^{L}_{D},\mathfrak{M}^{0}_{D})\leq
c_{2}L^{\gamma/4}.$ (6.10)

  3. $\mathrm{(c)}$

There exist constants $c_{3},a>0$ such that for all $L\in[0,1]$ and all
$n\in\mathbb{N}_{0}$,

$\displaystyle\mathrm{dist}_{{(\mathcal{H}^{1})^{\prime}}}\big{(}D^{L}(n)\mathcal{K}_{m}\,,\,\mathfrak{M}^{L}_{D}\big{)}\leq
c_{3}e^{-an},$ (6.11)

meaning that the rate of convergence to these attractors is uniform in $L$.

  * Proof.

To prove the assertion we intend to apply Lemma 6.2 with
$Y={\mathcal{W}_{\beta,m}^{1}}$, $X=(\mathcal{H}^{1})^{\prime}$,
$M=\mathcal{K}_{m}$, $\mathcal{F}^{L}=D^{L}_{m}(1)$ and
$\mathcal{D}^{L}=D^{L}_{m}$. This means that only the conditions (i) and (ii)
of Lemma 6.2 need to be verified.

Condition (i): We first recall that $D^{L}_{m}(1)=S^{L}_{m}(T_{0})$ with
$T_{0}\geq 1$ as constructed in (6.1). Hence, Lemma 6.3 implies that

$\displaystyle\big{\|}D^{L}_{m}(1)(u_{2},v_{2})-D^{L}_{m}(1)(u_{1},v_{1})\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\leq\Lambda(T_{0})\big{\|}(u_{2},v_{2})-(u_{1},v_{1})\big{\|}_{(\mathcal{H}^{1})^{\prime}}$

for all $L\in[0,1]$. This verifies condition (i) of Lemma 6.2.

Condition (ii): Furthermore, using the continuous embedding
$\mathcal{L}^{2}\hookrightarrow(\mathcal{H}^{1})^{\prime}$ and applying Lemma
5.2 with $T_{*}=nT_{0}$, we infer that for all $L\in[0,1]$ and
$(u,v)\in\mathcal{K}_{m}$,

$\displaystyle\big{\|}D^{L}_{m}(n)(u,v)-D^{0}_{m}(n)(u,v)\big{\|}_{(\mathcal{H}^{1})^{\prime}}\leq
C\big{\|}D^{L}_{m}(n)(u,v)-D^{0}_{m}(n)(u,v)\big{\|}_{\mathcal{L}^{2}}$
$\displaystyle\quad\leq
Ce^{CnT_{0}}\,L^{\frac{1}{4}}\leq\big{[}(C+1)e^{CT_{0}}\big{]}^{n}\,L^{\frac{1}{4}}.$

Here, $C$ is a positive constant independent of $L$. Hence, condition (ii) of
Lemma 6.2 with $\zeta=1/4$ and $\Theta=(C+1)e^{CT_{0}}$ is fulfilled.

This allows us to apply Lemma 6.2 as described above which directly proves the
assertions and thus, the proof is complete. ∎

###  6.2 Robust exponential attractors for the continuous dynamical system

We are now ready to present the main result of this section, which is the
existence of a robust family of exponential attractors for the continuous
dynamical system $(\mathcal{K}_{m},\\{S^{L}_{m}(t)\\}_{t\geq 0})$.

######  Theorem 6.5 (Robust exponential attractors for
$(\mathcal{K}_{m},\\{S^{L}_{m}(t)\\}_{t\geq 0})$).

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A3}$) are satisfied,
and let $m\in\mathbb{R}$ be arbitrary.

Then, for every $L\in[0,1]$, there exists an exponential attractor
${\mathfrak{M}}^{L}\subset\mathcal{K}_{m}$ for the dynamical system
$(\mathcal{K}_{m},\\{S^{L}_{m}(t)\\}_{t\geq 0})$ in the sense of Definition
6.1, where $M=\mathcal{K}_{m}$ is to be understood as a metric subspace of
$\mathcal{H}^{1}$.

Moreover, these exponential attractors can be chosen in such a way that the
following properties hold:

  1. $\mathrm{(a)}$

The fractal dimension of $\mathfrak{M}^{L}$ is uniformly bounded, i.e., there
exists a constant $C_{1}\geq 0$ such that for all $L\in[0,1]$:

$\displaystyle\dim_{\mathrm{frac},\mathcal{H}^{1}}(\mathfrak{M}^{L})\leq
C_{1}.$ (6.12)

  2. $\mathrm{(b)}$

There exist constants $C_{2}>0$ and $0<\gamma<1$ such that for all
$L\in[0,1]$,

$\displaystyle\mathrm{dist}_{\mathrm{sym},\mathcal{H}^{1}}(\mathfrak{M}^{L},\mathfrak{M}^{0})\leq
C_{2}\big{(}L^{1/6}+L^{\gamma/4}).$ (6.13)

  3. $\mathrm{(c)}$

There exists constants $C_{3},\alpha>0$ such that for all $L\in[0,1]$ and all
$t\geq 0$,

$\displaystyle\mathrm{dist}_{\mathcal{H}^{1}}\big{(}S^{L}_{m}(t)\mathcal{K}_{m},\mathfrak{M}^{L}\big{)}\leq
C_{3}\,e^{-\alpha t},$ (6.14)

meaning that the rate of convergence to these attractors is uniform in $L$.

  * Proof.

To establish the assertion, we proceed similarly as in the proof of [21, Thm.
5.1] (which in turn is based on the proof of the general result [20, Thm.
3.1]). Let $T_{0}\geq 1$ denote the number introduced in (6.1). Since
$\mathcal{K}_{m}$ is a compact subset of $\mathcal{H}^{1}$, we can find a
radius $K>0$ (independent of $L$) such that $\|(u,v)\|_{\mathcal{H}^{1}}\leq
K$ for all $(u,v)\in\mathcal{K}_{m}$. Let now $C$ denote a generic positive
constant, depending only on $T_{0}$ and $K$ that may change its value from
line to line. For any $L\in[0,1]$, we set

$\displaystyle\mathfrak{M}^{L}:=\bigcup_{t\in[T_{0},2T_{0}]}S_{m}^{L}(t)\mathfrak{M}^{L}_{D}.$
(6.15)

We now intend to show that $\\{\mathfrak{M}^{L}\\}_{L\in[0,1]}$ is a family of
exponential attractors with respect to the $\mathcal{H}^{1}$-metric which
satisfies the assertions (a), (b) and (c).

Invoking the compact Lipschitz estimate from Lemma 6.3, we infer that
$\mathfrak{M}^{L}$ is compact in $\mathcal{H}^{1}$ for every $L\in[0,1]$. We
next show that the fractal dimension of $\mathfrak{M}^{L}$ is bounded
uniformly in $L$. To this end, we consider the mapping

$\displaystyle\mathcal{F}^{L}:[T_{0},2T_{0}]\times\mathcal{K}_{m}\to\mathcal{K}_{m},\quad(t,u,v)\mapsto
S^{L}(t)(u,v).$

It obviously holds that
$\mathfrak{M}^{L}=\mathcal{F}^{L}([T_{0},2T_{0}]\times\mathfrak{M}^{L}_{D})$.
Hence, invoking Proposition 3.4, Proposition 3.5, Proposition 3.9, Proposition
3.10 and Lemma 6.3, we infer that there exists some constant $h\geq 0$
depending only on $T_{0}$ and $K$, such that

$\displaystyle\big{\|}\mathcal{F}^{L}(t_{1},u_{1},v_{1})-\mathcal{F}^{L}(t_{2},u_{2},v_{2})\big{\|}_{\mathcal{H}^{1}}=\big{\|}S_{m}^{L}(t_{1})(u_{1},v_{1})-S_{m}^{L}(t_{2})(u_{2},v_{2})\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\quad\leq
C\big{(}\big{\|}(u_{1},v_{1})-(u_{1},v_{2})\big{\|}_{\mathcal{H}^{1}}+\left|t_{1}-t_{2}\right|^{\frac{3}{16}}\big{)}\leq
h\big{\|}(t_{1},u_{1},v_{1})-(t_{2},u_{1},v_{2})\big{\|}_{\mathbb{R}\times\mathcal{H}^{1}}^{\frac{3}{16}}$

for all $t_{1},t_{2}\in[T_{0},2T_{0}]$ and all
$(u_{1},v_{1}),(u_{2},v_{2})\in\mathcal{K}_{m}$, where $\mathbb{R}$ is endowed
with the Euclidean norm. This means that $\mathcal{F}^{L}$ is Hölder
continuous with exponent $3/16$ and Hölder constant $h$.

Note that $hr^{3/16}\leq r^{1/8}$ if $r>0$ is sufficiently small. Hence,
recalling the definition of the fractal dimension (see (2.3)), we obtain

$\displaystyle\dim_{\mathrm{frac},\mathcal{H}^{1}}\big{(}\mathfrak{M}^{L}\big{)}=\dim_{\mathrm{frac},\mathcal{H}^{1}}\big{(}\mathcal{F}^{L}([T_{0},2T_{0}]\times\mathfrak{M}^{L}_{D})\big{)}$
$\displaystyle\quad=\underset{r\to
0}{\lim\sup}\;\frac{\mathcal{N}_{hr^{3/16}}\big{(}\mathcal{F}^{L}([T_{0},2T_{0}]\times\mathfrak{M}^{L}_{D})\,;\,\mathcal{H}^{1}\big{)}}{-\ln\big{(}h\,r^{3/16}\big{)}}$
$\displaystyle\quad\leq\underset{r\to
0}{\lim\sup}\;\frac{\mathcal{N}_{r}\big{(}[T_{0},2T_{0}]\times\mathfrak{M}^{L}_{D}\,;\,\mathbb{R}\times\mathcal{H}^{1}\big{)}}{-\ln\big{(}r^{1/8}\big{)}}$
$\displaystyle\quad=8\dim_{\mathrm{frac},\mathbb{R}\times\mathcal{H}^{1}}\big{(}[T_{0},2T_{0}]\times\mathfrak{M}^{L}_{D}\big{)}$
$\displaystyle\quad\leq
8\big{(}\dim_{\mathrm{frac},\mathcal{H}^{1}}\big{(}\mathfrak{M}^{L}_{D}\big{)}+1\big{)}\leq
8\big{(}c_{1}+1\big{)},$

where $c_{1}>0$ is the constant from Corollary 6.4(a). This verifies (a).

Proceeding as in the proof of [20, Thm. 3.1], it is straightforward to check
that the set $\mathfrak{M}^{L}$ is forward invariant, i.e.,
$S^{L}(t)\mathfrak{M}^{L}\subset\mathfrak{M}^{L}$ for all $t\geq 0$.

Let now $L\in(0,1]$, $t>0$ and $(u,v)\in\mathcal{K}_{m}$ be arbitrary. Without
loss of generality, we assume that $t\geq T_{0}$. Then there exist
$s\in[T_{0},2T_{0})$ and $n\in\mathbb{N}_{0}$ such that $t=nT_{0}+s$. Due to
Corollary 6.4, and since $\mathfrak{M}^{L}_{D}$ is compact in
$(\mathcal{H}^{1})^{\prime}$, we can further find
$(\bar{u},\bar{v})\in\mathfrak{M}^{L}_{D}$ such that

$\displaystyle\big{\|}D^{L}(n)(u,v)-(\bar{u},\bar{v})\big{\|}_{(\mathcal{H}^{1})^{\prime}}\leq
c_{3}e^{-an}.$

Then, recalling $1\leq T_{0}\leq s<2T_{0}$ and $n=(t-s)/T_{0}$, we can use the
compact Lipschitz estimate from Lemma 6.3 to conclude that

$\displaystyle\big{\|}S^{L}_{t}(u,v)-S^{L}_{s}(\bar{u},\bar{v})\big{\|}_{\mathcal{H}^{1}}=\big{\|}S^{L}_{s}D^{L}(n)(u,v)-S^{L}_{s}(\bar{u},\bar{v})\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\quad\leq
C\big{\|}D^{L}(n)(u,v)-(\bar{u},\bar{v})\big{\|}_{(\mathcal{H}^{1})^{\prime}}\leq
Ce^{-an}$ $\displaystyle\quad=Ce^{-at/T_{0}}e^{as/T_{0}}\leq Ce^{-\alpha t}$

with $\alpha:=a/T_{0}$. Since $S_{s}^{L}(\bar{u},\bar{v})\in\mathfrak{M}^{L}$,
this implies

$\displaystyle\mathrm{dist}_{X}\big{(}S^{L}_{m}(t)\mathcal{K}_{m},\mathfrak{M}^{L}\big{)}\leq
Ce^{-\alpha t}$

which proves (c). In particular, we have thus shown that for all $L\in[0,1]$,
$\mathfrak{M}^{L}$ is indeed an exponential attractor for the dynamical system
$(\mathcal{K}_{m},S_{m}^{L}(t))$ with respect to the $\mathcal{H}^{1}$-metric.

It remains to prove (b). Therefore, let $L\in(0,1]$ and
$(u_{*},v_{*})\in\mathfrak{M}^{L}$ be arbitrary. Hence, there exist
$t\in[T_{0},2T_{0}]$ and $(u_{0},v_{0})\in\mathfrak{M}^{L}_{D}$ such that
$(u_{*},v_{*})=S^{L}(t)(u_{0},v_{0})$. According to Corollary 6.4(b), there
exists $(\bar{u}_{0},\bar{v}_{0})\in\mathfrak{M}^{0}_{D}$ such that

$\displaystyle\big{\|}(u_{0},v_{0})-(\bar{u}_{0},\bar{v}_{0})\big{\|}_{(\mathcal{H}^{1})^{\prime}}\leq
c_{2}L^{\gamma/4}.$

We now set
$(\bar{u}_{*},\bar{v}_{*}):=S^{0}(t)(\bar{u}_{0},\bar{v}_{0})\in\mathfrak{M}^{0}$.
Hence, using the Hölder estimate from Proposition 3.9, the convergence
estimate from Corollary 5.3, and Lemma 2.2, we conclude that

$\displaystyle\big{\|}(u_{*},v_{*})-(\bar{u}_{*},\bar{v}_{*})\big{\|}_{\mathcal{H}^{1}}=\big{\|}S^{L}(t)(u_{0},v_{0})-S^{0}(t)(\bar{u}_{0},\bar{v}_{0})\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\quad\leq\big{\|}S^{L}(t)(u_{0},v_{0})-S^{0}(t)(u_{0},v_{0})\big{\|}_{\mathcal{H}^{1}}+\big{\|}S^{0}(t)(u_{0},v_{0})-S^{0}(t)(\bar{u}_{0},\bar{v}_{0})\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\quad\leq\big{\|}S^{L}(t)(u_{0},v_{0})-S^{0}(t)(u_{0},v_{0})\big{\|}_{\mathcal{H}^{1}}+C\big{\|}(u_{0},v_{0})-(\bar{u}_{0},\bar{v}_{0})\big{\|}_{(\mathcal{H}^{1})^{\prime}}$
$\displaystyle\quad\leq CL^{1/6}+CL^{\gamma/4}.$

This directly implies

$\displaystyle\mathrm{dist}_{\mathcal{H}^{1}}\big{(}\mathfrak{M}^{L},\mathfrak{M}^{0}\big{)}\leq
C\big{(}L^{1/6}+L^{\gamma/4}\big{)}.$

Proceeding analogously, we derive the same estimate for
$\mathrm{dist}_{\mathcal{H}^{1}}(\mathfrak{M}^{0},\mathfrak{M}^{L})$.
Combining both estimates, we obtain (b). Thus, the proof is complete. ∎

## Acknowledgement

Sema Yayla was supported by the Scientific and Technological Research Council
of Turkey (TUBITAK). Harald Garcke and Patrik Knopf were partially supported
by the RTG 2339 “Interfaces, Complex Structures, and Singular Limits” of the
German Science Foundation (DFG). The support is gratefully acknowledged.

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#  Thermodynamics of the 3-dimensional Einstein-Maxwell system

Shoichiro Miyashita

Department of Physics, National Dong Hwa University, Hualien, Taiwan, R.O.C.

e-mail address : s-miyashita”at”gms.ndhu.edu.tw

( )

###### Abstract

Recently, I studied the thermodynamical properties of the Einstein-Maxwell
system with a box boundary in 4-dimensions [1]. In this paper, I investigate
those in 3-dimensions. Similar to the 4-dimensional case, the system is
thermodynamically well-behaved when $\Lambda<0$ (due to the contribution of
the “bag of gold” saddles). However, when $\Lambda=0$, a crucial difference to
the 4-dimensional case appears, i.e. the 3-dimensional system turns out to be
thermodynamically unstable, while the 4-dimensional one is thermodynamically
stable.

##  1 Introduction

Recently, I studied the thermodynamical properties of the Einstein-Maxwell
system in 4-dimensions with box boundary condition [1]. There I found that
there exists a new type of saddle point geometries, which I called “bag of
gold (BG)”, and that they contribute to the partition function. As a result,
the peculiar behavior of the free energy, observed in [2], was resolved and it
is shown that the system has a well-defined thermodynamic description for
$\Lambda\leq 0$. During the investigation, I thought that generalizations to
other dimensions might not change the properties qualitatively. Therefore, I
concentrated only on the 4-dimensional case. However, after finishing the
work, I noticed the paper [3] which discusses the thermodynamical properties
of the Einstein-Maxwell system in 3-dimensions with box boundary condition.
Although they did not consider the contribution of BG saddles, their results
indicate that the properties are qualitatively different from those of 4-(or
higher) dimensions. Therefore, I investigate how their results are modified by
including the BG saddle contributions and how they differ from the case of
4-dimensions.

The organization of this paper is as follows: In Section 2, I study the
thermodynamical properties of the Einstein-Maxwell system with box boundary
condition in 3-dimensions when $\Lambda<0$. In subsection 2.1, I review the
analysis of Huang and Tao [3], in which they investigated the detailed
properties of the empty saddles and the black hole (BH) saddles of the system.
Although the analysis on these saddles was correct, their two additional
assumptions lead to the incorrect results on the phase structure of the
system. In subsection 2.2, I point out that the Bertotti-Robinson (BR) saddles
and the BG saddles were missing in their analysis and investigate their
properties. I then show the complete phase structure of the system. In Section
3, I point out a qualitative difference on the behavior of phase structure
between the 3-dimensional and the 4-dimensional system when $\Lambda
r_{b}^{2}$ is close to zero. In Section 4, I investigate the thermodynamical
properties of the Einstein-Maxwell system with box boundary condition in
3-dimensions when $\Lambda=0$. Throughout this paper, I concentrate only on
the grand canonical ensembles.

##  2 $\Lambda<0$ case

In this section, I study the thermodynamical properties of the 3-dimensional
Einstein-Maxwell system with box boundary and $\Lambda<0$. In 3-dimensions, it
is often said that there is no BH when $\Lambda\geq 0$, and only when
$\Lambda<0$, BTZ BHs can exist. Therefore it can be a good starting point for
studying thermodynamics in 3-dimensions. And indeed, in [3], they considered
this case and investigated thermodynamic properties. In subsection 2.1, the
basic setups and the properties of empty saddles and BH saddles are reviewed,
which are systematically studied in [3]. For empty saddles and BH saddles,
their results are correct. But for other states, they made wrong statements.
As a result, some part of their results on the phase diagram are incorrect. In
subsection 2.2, I point out that BR saddles and BG saddles were missing in
their investigation and examine their properties. Finally, I show the complete
phase diagram.

###  2.1 empty saddles and BH saddles

The Euclidean action for grand canonical ensembles is

$\displaystyle I^{E}[\boldsymbol{g},\boldsymbol{A}]=\frac{-1}{16\pi
G}\int_{\mathcal{M}}d^{3}x\sqrt{g}(\mathcal{R}-2\Lambda)+\frac{1}{16\pi}\int_{\mathcal{M}}d^{3}x\sqrt{g}F_{\mu\nu}F^{\mu\nu}\hskip
71.13188pt$ $\displaystyle+\frac{-1}{8\pi
G}\int_{\partial\mathcal{M}}d^{2}y\sqrt{\gamma}\Theta+\frac{1}{8\pi
G}\int_{\partial\mathcal{M}}d^{2}y\sqrt{\gamma}\sqrt{|\Lambda|}$ (2.1)

I focus only on the metric of the form

$\displaystyle\boldsymbol{g}(r)=f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\phi^{2}$
(2.2)

and assume that they are the dominant contributions to the Euclidean path
integral representation of the grand canonical partition function
$Z(\beta,\mu,r_{b})$ of gravity. Here, $r$ is the areal radius and the
boundary at $r=r_{b}$. Within this class of metrics, there are at least two
types of solutions. One is the empty saddle;

$\displaystyle{\rm Empty~{}saddle:}~{}~{}f(r)=1-\Lambda r^{2}\hskip
105.2751pt$ (2.3) $\displaystyle A_{t}(r)=-i\tilde{\mu}\hskip 119.50148pt$
(2.4)

The other is the black hole (BH) saddle. 111 Although in the standard
parametrization of solutions I should say that there is another type of
saddles  $\displaystyle f(r)=-\Lambda r^{2}$ $\displaystyle
A_{t}(r)=-i\tilde{\mu}$ which corresponds to the Poincare patch of AdS, we can
obtain this from $r_{H}\to 0$ limit of BH saddles in the parametrization in
the grand canonical ensembles. As I will explain shortly, the parameter
$\tilde{Q}$ in (2.5) and (2.6) can be written as
$\displaystyle\tilde{Q}=\frac{\mu}{2}\sqrt{\frac{-\Lambda(r_{b}^{2}-r_{H}^{2})}{\left(2G\mu^{2}+\log\frac{r_{b}}{r_{H}}\right)\log\frac{r_{b}}{r_{H}}}}$
With this expression, we can show that  $\displaystyle({\rm
the~{}last~{}term~{}in~{}(\ref{BHf})})\to 0$ $\displaystyle
A_{t}\to-i\sqrt{-\Lambda}r_{b}\mu=-i\tilde{\mu}$ In addition, unlike BH
saddles, this saddle can have any temperature. However, the free energy of
this state is always larger than that of the empty saddle (i.e.
$F_{Poincare}=0,~{}F_{empty}=-\frac{\sqrt{1-\Lambda
r_{b}^{2}}}{4G}+\frac{\sqrt{|\Lambda|}}{4G}r_{b}<0$.) So I will ignore this
saddle in this paper.

$\displaystyle{\rm BH~{}saddle:}~{}~{}f(r)=\Lambda r_{H}^{2}-\Lambda
r^{2}-8G\tilde{Q}^{2}\log\frac{r}{r_{H}}$ (2.5) $\displaystyle
A_{t}(r)=-2i\tilde{Q}\log\frac{r}{r_{H}}\hskip 59.75095pt$ (2.6)

where $\tilde{\mu},\tilde{Q}$ and $r_{H}\in(0,r_{b})$ are parameters that
depend on $\beta,\mu,r_{b}$ through the boundary condition of the path
integral for the grand canonical partition function $Z(\beta,\mu,r_{b})$;

$\displaystyle{\rm\underline{boundary~{}condition}}\hskip 318.67078pt$
$\displaystyle{\rm
Empty~{}saddle:}~{}~{}\frac{\tilde{\mu}}{\sqrt{f(r_{b})}}=\mu\hskip
216.2411pt$ (2.7) $\displaystyle{\rm
BH~{}saddle:}~{}~{}\frac{4\pi}{f^{\prime}(r_{H})}\sqrt{f(r_{b})}=\beta,~{}~{}\frac{2\tilde{Q}}{\sqrt{f(r_{b})}}\log\frac{r_{b}}{r_{H}}=\mu\hskip
81.09052pt$ (2.8)

In the following, the function $f$ will be used only for the BH (and BG)
saddles. Substituting these field configurations to the action (2.1), we get

$\displaystyle{\rm\underline{free~{}energy}}\hskip 361.3499pt$
$\displaystyle{\rm Empty~{}saddle:}~{}~{}F=-\frac{\sqrt{1-\Lambda
r_{b}^{2}}}{4G}+\frac{\sqrt{|\Lambda|}}{4G}r_{b}\hskip 187.78836pt$ (2.9)
$\displaystyle{\rm
BH~{}saddle:}~{}~{}F=-\frac{\sqrt{f(r_{b})}}{4G}+\frac{\sqrt{|\Lambda|}}{4G}r_{b}+\frac{\Lambda
r_{H}^{2}}{4G\sqrt{f(r_{b})}}+\frac{\tilde{Q}^{2}}{\sqrt{f(r_{b})}}-\frac{2\tilde{Q}^{2}}{\sqrt{f(r_{b})}}\log\frac{r_{b}}{r_{H}}$
(2.10)

Again, I want to emphasize that the parameters $r_{H}$ and $\tilde{Q}$ are
functions of $\beta,\mu,r_{b}$. For the empty saddle, since it has no $T$
dependence and $\mu$ dependence, the thermodynamical quantities are given by

$\displaystyle{\rm\underline{thermodynamical~{}quantity}}\hskip 284.52756pt$
$\displaystyle{\rm Empty~{}saddle:}~{}~{}E=-\frac{\sqrt{1-\Lambda
r_{b}^{2}}}{4G}+\frac{\sqrt{|\Lambda|}}{4G}r_{b}\hskip 176.407pt$ (2.11)
$\displaystyle S=0\hskip 278.83708pt$ (2.12) $\displaystyle Q=0\hskip
278.83708pt$ (2.13)

For the BH saddle, we can obtain them by using the relations
$\displaystyle\left(\frac{\partial
F}{\partial\beta}\right)_{\mu,r_{b}}=-E,\displaystyle\left(\frac{\partial
F}{\partial\mu}\right)_{\beta,r_{b}}=-Q$ and $S=\beta(E-\mu Q-F)$.
Alternatively, since we also have finer-grained information about the saddle
point geometries, we can obtain them by

$\displaystyle E=\int d\phi\sqrt{\sigma}u_{i}\tau^{ij}\xi_{j}$ (2.14)
$\displaystyle Q=\int d\phi\sqrt{\sigma}j^{i}u_{i}$ (2.15)

where $\tau_{ij}$ and $j^{i}$ are the boundary currents of the metric and the
gauge field, defined by
$\displaystyle\tau_{ij}\equiv\frac{2}{\sqrt{\gamma}}\frac{\delta
I^{E}}{\delta\gamma^{ij}},~{}~{}j^{i}\equiv\frac{i}{\sqrt{\gamma}}\frac{\delta
I^{E}}{\delta A_{i}}$. (The former current is known as the Brown-York tensor
[4].) $\xi^{i}$ is the normalized Killing vector of the $U(1)$ isometry on the
boundary and $u^{i}$ is the normal vector of the integration surface. Using
these fine-grained definition and the relation $S=\beta(E-\mu Q-F)$, we get

$\displaystyle{\rm\underline{thermodynamical~{}quantity}}\hskip 284.52756pt$
$\displaystyle{\rm
BH~{}saddle:}~{}~{}E=-\frac{\sqrt{f(r_{b})}}{4G}+\frac{\sqrt{|\Lambda|}}{4G}r_{b}\hskip
184.9429pt$ (2.16) $\displaystyle S=\frac{\pi r_{H}}{2G}\hskip 258.92024pt$
(2.17) $\displaystyle Q=\tilde{Q}\hskip 271.72389pt$ (2.18)

Although they did not insist on this point in [3], they showed that there
exists a maximum temperature for BHs that can be reached by the limit of the
BH horizon approaching the boundary, and that there are no BHs above this
temperature. Since they did not show the analytical expression of the
temperature, I will derive it here. From the boundary conditions, the
temperature can be rewritten as a function of $\mu,r_{H}$, and its square is

$\displaystyle
T(r_{H})^{2}=\frac{-\Lambda\left\\{-G\mu^{2}(r_{b}^{2}-r_{H}^{2})+\left(2G\mu^{2}+\log\frac{r_{b}}{r_{H}}\right)r_{H}^{2}\log\frac{r_{b}}{r_{H}}\right\\}^{2}}{4\pi^{2}r_{H}^{2}(r_{b}^{2}-r_{H}^{2})\left(\log\frac{r_{b}}{r_{H}}\right)^{3}\left(2G\mu^{2}+\log\frac{r_{b}}{r_{H}}\right)}$
(2.19)

By defining $p$ and $q$ with

$\displaystyle
p(r_{H})=-\Lambda\left\\{-G\mu^{2}(r_{b}^{2}-r_{H}^{2})+\left(2G\mu^{2}+\log\frac{r_{b}}{r_{H}}\right)r_{H}^{2}\log\frac{r_{b}}{r_{H}}\right\\}^{2}$
(2.20) $\displaystyle
q(r_{H})=4\pi^{2}r_{H}^{2}(r_{b}^{2}-r_{H}^{2})\left(\log\frac{r_{b}}{r_{H}}\right)^{3}\left(2G\mu^{2}+\log\frac{r_{b}}{r_{H}}\right)\hskip
45.52458pt$ (2.21)

we can show that

$\displaystyle\lim_{r_{H}\to
r_{b}}\frac{p^{\prime\prime\prime\prime}(r_{H})}{q^{\prime\prime\prime\prime}(r_{H})}=\frac{-\Lambda(1-2G\mu^{2})^{2}}{16\pi^{2}G\mu^{2}}$
(2.22)

Therefore, for fixed $\mu$, $r_{H}\to r_{b}$ limit of temperature is

$\displaystyle
T(r_{b})=\frac{\sqrt{-\Lambda}}{4\pi}\frac{1-2G\mu^{2}}{\sqrt{G}\mu}$ (2.23)

The existence of a maximum temperature of charged BHs in the Einstein-Maxwell
system in a box was pointed out in [2, 1] for 4-dimensions, and in [5] for
higher dimensions. Similarly, we can derive the maximum charge of BH saddles;

$\displaystyle Q(r_{b})=\frac{r_{b}}{2}\sqrt{\frac{-\Lambda}{G}}$ (2.24)

Note that $T(r_{b})$ does not depend on $r_{b}$. On the other hand, $Q(r_{b})$
does not depend on $\mu$.

In addition, they have made two claims in [3] that I would like to claim to be
false in the next subsection.

Wrong statements in [3]

  * •

When taking the $r_{H}\to r_{b}$ limit of the BH saddles, it corresponds to
the “M state”. Its entropy is $S=\frac{\pi r_{b}}{2G}$, its charge is
$Q=\frac{r_{b}}{2}\sqrt{\frac{-\Lambda}{G}}$, and its energy is
$E=\frac{\sqrt{-\Lambda}}{4G}r_{b}$.
And its temperature $T$ and its chemical potential $\mu$ can be arbitrary
values.

  * •

There is a critical temperature, beyond which there are no BH saddles. Above
this temperature,  “M states” give the dominant contribution.

According to their paper, the free energy of the “M state” is

$\displaystyle F_{Mstate}(T,\mu)=\frac{\sqrt{-\Lambda}}{4G}r_{b}-T\frac{\pi
r_{b}}{2G}-\mu\frac{r_{b}}{2}\sqrt{\frac{-\Lambda}{G}}$ (2.25)

They expressed this state as a state of “the black hole merging with the
boundary”. An example of the behavior of the free energies of the empty, BH,
and M states, and the phase diagram are shown in Fig. 1. In the next
subsection, I will show that the horizon does not merge with the boundary in
the $r_{H}\to r_{b}$ limit and can even be larger than the boundary.

Figure 1: Incorrect $F-T$ diagram and phase diagram. (For the empty and BH
phases, they are correct.) In [3], they claimed that there are two phase
transitions; one is the Hawking-Page transition between the empty and BH
phases, the other is a second order phase transition between the BH phase and
the M state. The transition point is the same as the termination point of the
BH branch and it is given by (2.23) and marked with the red dot in the left
figure. (2.23) also represents the boundary between the BH phase and the M
states in the right figure. The boundary between the empty phase and the M
state is given by the relation $T=\frac{\sqrt{1-\Lambda r_{b}^{2}}}{2\pi
r_{b}}-\frac{\mu}{\pi}\sqrt{-\Lambda G}$.

###  2.2 BR saddles, BG saddles, and thermodynamics

Instead of their two claims, I would like to claim that the following two are
correct;

My claim

  * •

When taking the $r_{H}\to r_{b}$ limit of the BH saddles, it corresponds to
the Bertotti-Robinson(BR) saddles. Its entropy is $S=\frac{\pi r_{b}}{2G}$,
its charge is $Q=\frac{r_{b}}{2}\sqrt{\frac{-\Lambda}{G}}$, and its energy is
$E=\frac{\sqrt{-\Lambda}}{4G}r_{b}$.
And its temperature $T$ and its chemical potential $\mu$ are NOT arbitrary
values.

  * •

There is a critical temperature, beyond which there are no BH saddles. Above
this temperature,  “bag of gold(BG)” saddles give the dominant contribution.

Firstly, let’s show the limit $r_{H}\to r_{b}$ does not corresponds to the
situation where “the black hole merges with the cavity” but to the Bertotti-
Robinson (BR) geometry. Define a small parameter $\varepsilon$ by
$r_{H}=r_{b}-\varepsilon$. The first and second derivatives of $f(r)$ at
$r=r_{H}=r_{b}-\varepsilon$ is

$\displaystyle
f^{\prime}(r_{H})=\frac{-\Lambda}{G\mu^{2}}(1-2G\mu^{2})\varepsilon+O(\varepsilon^{2})$
(2.26) $\displaystyle f^{\prime\prime}(r_{H})=-4\Lambda+O(\varepsilon)\hskip
65.44142pt$ (2.27)

By defining a new coordinate
$r=r_{H}+\varepsilon\tilde{r},t=\frac{\tilde{t}}{\varepsilon}$, the metric
becomes

$\displaystyle\boldsymbol{g}=\left[\frac{-\Lambda}{G\mu^{2}}(1-2G\mu^{2})\tilde{r}-2\Lambda\tilde{r}^{2}+O(\varepsilon)\right]d\tilde{t}^{2}+\frac{1}{\left[\frac{-\Lambda}{G\mu^{2}}(1-2G\mu^{2})\tilde{r}-2\Lambda\tilde{r}^{2}+O(\varepsilon)\right]}d\tilde{r}^{2}+(r_{b}+O(\varepsilon))^{2}d\phi^{2}$
(2.28)

Then, $\varepsilon\to 0$ limit (i.e. near horizon and near extremal limit)
leads to

$\displaystyle\boldsymbol{g}=\left[\frac{-\Lambda}{G\mu^{2}}(1-2G\mu^{2})\tilde{r}-2\Lambda\tilde{r}^{2}\right]d\tilde{t}^{2}+\frac{1}{\left[\frac{-\Lambda}{G\mu^{2}}(1-2G\mu^{2})\tilde{r}-2\Lambda\tilde{r}^{2}\right]}d\tilde{r}^{2}+r_{b}^{2}d\phi^{2}$
(2.29) $\displaystyle\tilde{r}\in[0,1]$ (2.30)

which is the Euclidean version of the 3-dimensional Bertotti-Robinson (BR)
geometry. The gauge field can be obtained in a similar way, i.e,

$\displaystyle\boldsymbol{A}=A_{t}(r)dt\hskip 71.13188pt$
$\displaystyle=\left[\sqrt{\frac{-\Lambda}{G}}\varepsilon\tilde{r}+O(\varepsilon^{2})\right]\frac{d\tilde{t}}{\varepsilon}$
$\displaystyle\to-i\sqrt{\frac{-\Lambda}{G}}\tilde{r}d\tilde{t}\hskip
45.52458pt$ (2.31)

As with the standard argument for BHs, the absence of a conical singularity at
$\tilde{r}=0$ leads to the unique temperature;

$\displaystyle
T_{BR}=\frac{\sqrt{-\Lambda}}{4\pi}\frac{1-2G\mu^{2}}{\sqrt{G}\mu}$ (2.32)

which of course is the same as (2.23). We can also check that the parameter
$\mu$ is actually the chemical potential of this saddle
$\displaystyle\frac{A_{\tilde{t}}(\tilde{r}=1)}{\sqrt{g_{\tilde{t}\tilde{t}}}}=\mu$.
Free energy and other thermodynamical quantities are given by

$\displaystyle{\rm\underline{free~{}energy}}\hskip 361.3499pt$
$\displaystyle{\rm
BR~{}saddle:}~{}~{}F=\frac{\sqrt{-\Lambda}}{4G}r_{b}-\frac{r_{b}}{8G\mu}\sqrt{\frac{-\Lambda}{G}}-\frac{r_{b}\mu}{4}\sqrt{\frac{-\Lambda}{G}}\hskip
142.26378pt$ (2.33)
$\displaystyle{\rm\underline{thermodynamical~{}quantity}}\hskip 284.52756pt$
$\displaystyle{\rm BR~{}saddle:}~{}~{}E=\frac{\sqrt{-\Lambda}}{4G}r_{b}\hskip
267.45572pt$ (2.34) $\displaystyle S=\frac{\pi r_{b}}{2G}\hskip 285.95027pt$
(2.35) $\displaystyle Q=\frac{r_{b}}{2}\sqrt{\frac{-\Lambda}{G}}\hskip
263.18799pt$ (2.36)

Figure 2: Spatial section of the BH saddle, BR saddle, and BG saddle. They are
characterized by the radius of the horizon (or bolt); whether it is smaller or
larger than the radius of the boundary.

So far, we have seen that the $r_{H}\to r_{b}$ limit of BH saddles is not a
singular geometry of zero size but a BR saddle, whose circumference of the
transverse circle remains constant in the radial direction. Since the limit is
not singular, one might wonder whether a further deformation of saddles (i.e.
the extension of the parameter range of $r_{H}$ beyond $r_{b}$ of BH saddles)
is possible. In fact, it is possible. [1] (Fig. 2)

For the metric and gauge field (2.5), (2.6), we still have regular geometries
when $r_{H}>r_{b}$. These are “bag of gold(BG)” saddles. The role of these
saddles in pure gravity is discussed in [6, 7, 8] and their importance for the
4-dimensional Einstein-Maxwell system was recently shown in [1]. To
distinguish it from the BH, let’s use $r_{G}$ for the horizon radius of BG
saddles. The expression of boundary conditions, thermodynamical quantities,
and free energy are all the same except for the following two points [1]; (i)
Since $r$ is increasing toward the bolt, $f^{\prime}(r_{G})$ is negative. So
there is a minus sign in the relation between $\beta$ and $f^{\prime}(r_{G})$
(2.37). (ii) Since the $r$ component of the normal vector of the boundary is
negative for the same reason as before, $Q$ is not $\tilde{Q}$, but
$Q=-\tilde{Q}$ (2.41). Summarize,

Figure 3: Behavior of the free energies. The terminal point of the BH branch
corresponds to the BR saddle. Above the terminal temperature, there are BG
phases which are thermodynamically stable. For low $\mu$, there is a Hawking-
Page phase transition between BH/BG phases as shown in the left figure. For
high $\mu$, the Hawking-Page phase transition does not occur and the system is
in the BH/BG phase for all $T$ as in the right figure. The left figure is the
correct version of Fig. 1.

$\displaystyle{\rm\underline{boundary~{}condition}}\hskip 318.67078pt$
$\displaystyle{\rm
BG~{}saddle:}~{}~{}\frac{-4\pi}{f^{\prime}(r_{G})}\sqrt{f(r_{b})}=\beta,~{}~{}\frac{2\tilde{Q}}{\sqrt{f(r_{b})}}\log\frac{r_{b}}{r_{G}}=\mu\hskip
83.93553pt$ (2.37) $\displaystyle{\rm\underline{free~{}energy}}\hskip
361.3499pt$ $\displaystyle{\rm
BG~{}saddle:}~{}~{}F=\frac{\sqrt{f(r_{b})}}{4G}+\frac{\sqrt{|\Lambda|}}{4G}r_{b}-\frac{\Lambda
r_{G}^{2}}{4G\sqrt{f(r_{b})}}-\frac{\tilde{Q}^{2}}{\sqrt{f(r_{b})}}+\frac{2\tilde{Q}^{2}}{\sqrt{f(r_{b})}}\log\frac{r_{b}}{r_{G}}\hskip
8.5359pt$ (2.38)
$\displaystyle{\rm\underline{thermodynamical~{}quantity}}\hskip 284.52756pt$
$\displaystyle{\rm
BG~{}saddle:}~{}~{}E=\frac{\sqrt{f(r_{b})}}{4G}+\frac{\sqrt{|\Lambda|}}{4G}r_{b}\hskip
204.85974pt$ (2.39) $\displaystyle S=\frac{\pi r_{G}}{2G}\hskip 270.30118pt$
(2.40) $\displaystyle Q=-\tilde{Q}\hskip 274.56891pt$ (2.41)

The parameter range of $r_{G}$ is
$r_{G}\in\left[r_{b},r_{b}e^{2G\mu^{2}}\right)$ and the $r_{G}\to
r_{b}e^{2G\mu^{2}}$ limit corresponds to $T\to\infty$. Therefore, the BG
branch is connected to the BH branch and is always present and is dominant
above the BR temperature. The examples the free energy behavior are shown in
Fig. 3. The behavior of the phase diagram depends on the value $-\Lambda
r_{b}^{2}$ and are classified into three cases; the case of $0<-\Lambda
r_{b}^{2}<1$, $-\Lambda r_{b}^{2}=-1$, and $1<-\Lambda r_{b}^{2}$. They are
shown in Fig. 4.

Figure 4: Examples of phase diagrams. The behavior of phase diagram depends on
the value $-\Lambda r_{b}^{2}$ are classified into three cases; the case of
$0<-\Lambda r_{b}^{2}<1$ (Left), $-\Lambda r_{b}^{2}=-1$ (Middle), and
$1<-\Lambda r_{b}^{2}$ (Right). The left figure is the correct version of Fig.
1.

##  3 Difference between 3-dim. and 4-dim. ($\Lambda<0$)

There are many qualitative and quantitative differences between the Einstein-
Maxwell system in 3-dimensions and that in 4-dimensions. 222 For example,  •
There is no upper bound on $\mu$ for the BH and BG saddles in 3-dim., but
$\sqrt{G}\mu=1$ is the upper bound in 4-dim. • In both cases, there exists the
maximum horizon radius for fixed $\mu$. $r_{G,max}=r_{b}e^{2G\mu^{2}}$ for
3-dim. and $r_{G,max}=r_{b}\frac{1}{1-G\mu^{2}}$ for 4-dim. • In 4-dim., there
can exist another small BH phase in the low temperature region, separated from
the main BH phase, when $-\Lambda r_{b}^{2}$ is in a certain range. Compare
Fig 4 and Fig. 11 in [1]. • When $\mu=0$, the transition temperature is always
given by $T=\frac{1}{2\pi r_{b}}$ in 3-dim. and does not depend on $\Lambda$,
as shown in [3]. In 4-dim. it also depends on $\Lambda$. What I want to
emphasize here the most is the behavior of the BH phase (and the BG phase) in
the phase diagram versus the value of $\Lambda$. In 4-dimensions, when we take
$\Lambda\to 0$ limit or $\Lambda=0$, the phase diagram is still similar to the
left figure in Fig. 4, i.e. there is a BH phase. However, in 3-dimensions, if
we decrease the value of $-\Lambda r_{b}^{2}$, the BH phase shrinks. This can
be seen from the fact that the boundary curve between the BH phase and the BG
phase is given by (2.32) and the curve becomes close to the $T$ axis. This
indicates that the BH phase disappear when $\Lambda=0$. The disappearance of
the BH phase is not surprising, since, under some suitable conditions,
including the dominant energy condition, it is proved that there are no BHs in
3-dimensions when $\Lambda=0$. [9, 10] Even for the system with a box
boundary, this statement still be true. What may be surprising is the
existence of the BG phase even for the $\Lambda\to 0$ limit. It seems to imply
that even for $\Lambda=0$, the BG phase exists, possibly as a
thermodynamically stable state. However, I will show in the next section that
BG saddles do exist, but they cause thermodynamic instability.

##  4 $\Lambda=0$ case

In this case, there exist two types of saddles, the one is the empty saddle,

$\displaystyle\boldsymbol{g}(r)=dt^{2}+dr^{2}+r^{2}d\phi^{2}$ (4.1)

And the other saddle can be obtained by setting $\Lambda=0$ in (2.5), (and
replacing the parameter $r_{H}$ with $r_{G}$,)

$\displaystyle f(r)=-8GQ^{2}\log\frac{r}{r_{G}}$ (4.2) $\displaystyle
A_{t}(r)=2iQ\log\frac{r}{r_{G}}$ (4.3)

Since for the coordinate range $r\in[r_{b},r_{G})$, $f(r)>0$ and $f(r_{G})=0$,
this is the BG saddle, which is not singular. And here I also set
$\tilde{Q}=-Q$ since these are BGs. The boundary conditions are similar to
those in subsection 2.2. From the boundary conditions, we get the following
expression;

$\displaystyle\frac{1}{2G}\log\frac{r_{G}}{r_{b}}=\mu^{2}$ (4.4)
$\displaystyle T=\frac{Q}{2\pi r_{b}}\frac{1}{\mu e^{2G\mu^{2}}}$ (4.5)

At this point, we know that, for fixed $\mu$, the entropy does not change no
matter how we increase or decrease the temperature. This implies that the heat
capacity is zero. Therefore, if these saddles give dominant contribution, it
leads to thermodynamical instability of the system. 333 Precisely, it is
neither stable nor unstable, it is marginal. However, if it is not stable, the
system may not reach thermal equilibrium. It is in this sense that I refer to
this situation as unstable.  Explicitly, free energy and thermodynamical
quantities are given by

$\displaystyle{\rm\underline{free~{}energy}}\hskip 361.3499pt$
$\displaystyle{\rm empty~{}saddle:}~{}~{}F=\frac{-1}{4G}\hskip 254.65207pt$
(4.6) $\displaystyle{\rm
BG~{}saddle:}~{}~{}F=-\frac{Q^{2}}{\sqrt{f(r_{b})}}-\frac{2Q^{2}}{\sqrt{f(r_{b})}}\log\frac{r_{b}}{r_{G}}\hskip
142.26378pt$ $\displaystyle=-\frac{\pi
r_{b}T}{2G}\left(1+4G\mu^{2}\right)e^{2G\mu^{2}}\hskip 153.6447pt$ (4.7)
$\displaystyle{\rm\underline{thermodynamical~{}quantity}}\hskip 284.52756pt$
$\displaystyle{\rm empty~{}saddle:}~{}~{}E=\frac{-1}{4G}\hskip 254.65207pt$
(4.8) $\displaystyle S=0\hskip 266.03343pt$ (4.9) $\displaystyle Q=0\hskip
266.03343pt$ (4.10) $\displaystyle{\rm BG~{}saddle:}~{}~{}E=0\hskip
267.45572pt$ (4.11) $\displaystyle S=\frac{\pi r_{G}}{2G}=\frac{\pi
r_{b}}{2G}e^{2G\mu^{2}}\hskip 193.47882pt$ (4.12) $\displaystyle Q=2\pi
r_{b}T\mu e^{2G\mu^{2}}\hskip 211.97293pt$ (4.13)

Since, in this case, the BG free energy (4.7) is a linear function of $T$, the
“transition temperature” can be easily found as

$\displaystyle T_{tr}=\frac{1}{2\pi
r_{b}\left(1+4G\mu^{2}\right)e^{2G\mu^{2}}}$ (4.14)

The “phase diagram” is shown in Fig. 5.

Figure 5: “Phase diagram” of the case where $\Lambda=0$. Since the BG “phase”
is not thermodynamically stable, neither is the system itself.

Finally, let’s discuss what happens when we turn off $\mu$. From (4.4), we see
that $\mu\to 0$ limit corresponds to $r_{G}\to r_{b}$ limit. Again, this limit
does not lead to the merging of horizon and boundary but leads to a regular
geometry, which is the direct product of the 2-dimensional Rindler space and
$S^{1}$. By defining a small parameter $\varepsilon$ by
$\varepsilon=G\mu^{2}$, the first derivative of $f$ at $r=r_{G}$ is

$\displaystyle
f^{\prime}(r_{G})=-\frac{8GQ^{2}}{r_{G}}=-32\pi^{2}r_{b}T^{2}\varepsilon+O(\varepsilon^{2})$
(4.15)

Since
$r_{G}=r_{b}e^{2\varepsilon}=r_{b}+2r_{b}\varepsilon+O(\varepsilon^{2})$,
let’s define new coordinates by
$(r_{G}-r)=2r_{b}\varepsilon\tilde{r},~{}t=\frac{\tilde{t}}{2r_{b}\varepsilon}$.
Then, the ranges of these new coordinates will be
$\tilde{r}\in[0,1],~{}\tilde{t}\in\left[0,\frac{1}{4\pi T^{2}}\right]$ in the
$\varepsilon\to 0$ limit. Then, in the $\varepsilon\to 0$ limit, the metric
and gauge field become 444 An alternative way is to take a double scaling
limit of the BR geometry (2.29):  $\displaystyle 4\pi
T=\sqrt{\frac{-\Lambda}{G\mu^{2}}}{\rm:fixed,}~{}~{}~{}~{}\Lambda\to
0,~{}G\mu^{2}\to 0$

$\displaystyle\boldsymbol{g}=\left[64\pi
r_{b}^{2}T^{2}\varepsilon^{2}\tilde{r}+O(\varepsilon^{3})\right]\frac{d\tilde{t}^{2}}{4r_{b}^{2}\varepsilon^{2}}+\frac{4r_{b}^{2}\varepsilon^{2}}{\left[64\pi
r_{b}^{2}T^{2}\varepsilon^{2}\tilde{r}+O(\varepsilon^{3})\right]}d\tilde{r}^{2}+(r_{b}+O(\varepsilon))^{2}d\phi^{2}$
$\displaystyle\to
16\pi^{2}T^{2}\tilde{r}d\tilde{t}^{2}+\frac{1}{16\pi^{2}T^{2}\tilde{r}}d\tilde{r}^{2}+r_{b}^{2}d\phi^{2}\hskip
184.9429pt$ (4.16) $\displaystyle\boldsymbol{A}\to 0\hskip 345.7008pt$ (4.17)

If we further define the coordinates $\rho$ and $\tau$ by
$\tilde{r}=4\pi^{2}T^{2}\rho^{2},~{}\tau=8\pi^{2}T^{2}\tilde{t}$, the metric
takes the standard Rindler form

$\displaystyle\boldsymbol{g}=\rho^{2}d\tau^{2}+d\rho^{2}+r_{b}^{2}d\phi^{2}$
(4.18) $\displaystyle\tau\in[0,2\pi],~{}\rho\in[0,2\pi T]$ (4.19)

The free energy of the $Rindler\times S^{1}$ saddles is given by simply
setting $\mu=0$ in the free energy of the BG saddles (4.7), and we can easily
check that it will be dominant above the temperature $T=\frac{1}{2\pi r_{b}}$.
(Fig. 5)

At first glance, the phase structures of the Einstein-Maxwell system in
3-dimensions and 4-dimensions are similar when $-\Lambda r_{b}^{2}$ is
sufficiently large. There are three phases, the empty phase, the BH phase, and
the BG phase (Fig. 4). However, when it is close to zero, differences appear;
For 4-dimensions, the BH phase exists when $\Lambda\to 0$ or $\Lambda=0$, and
the system is thermodynamically stable. For 3-dimensions, however, it shrinks
in the $\Lambda\to 0$ limit and ceases to exist at $\Lambda=0$. So in
3-dimensions, when a negative $\Lambda$ is switched off, the boxed systems no
longer have the saddles with the smaller horizon (i.e. BH saddles), as
previously proved [9, 10], but still have the ones with the same size or
larger horizon (i.e. BR saddles and BG saddles). But these saddles make the
system thermodynamically unstable. 555 When $\Lambda=0$ and $\mu>0$, the BG
saddle is not completely unstable, but it is a kind of marginal point, in the
sense that its heat capacity is not negative, but zero. It would be
interesting to study the existence of hairy extensions of BG saddles and their
thermodynamical stability when $\Lambda=0$.  When we also switch off $\mu$,
the system does not allow the saddle with the larger horizon (BG saddle), but
still allows the ones with the same size horizon (BR saddle, or $Rindler\times
S^{1}$ saddle), which again makes the system thermodynamically unstable. 666
For $\Lambda>0$ case, there are the empty phase and the BG phase. The form of
the metric and gauge field are the same as in the $\Lambda<0$ case. The
coordinate range of $r$ can be classified by the value of $\mu$;
$r\in[r_{b},\infty)$ for $2G\mu^{2}\leq 1$, and
$r\in\left[r_{b}e^{2G\mu^{2}},\infty\right)$ for $1<2G\mu^{2}$. But in both
cases, the BG saddles are thermodynamically unstable and so is the system.
This situation is the same as in the 4-dimensional case [1].

## Acknowledgement

This work is supported in part by the National Science and Technology Council
(No. 111-2112-M-259-016-MY3).

##  Appendix A Off-shell geometries for M states

This appendix is purely an appendix and has no direct relation to the result
in this paper. In [3], they claimed that the $r_{H}\to r_{b}$ limit of the BH
leads to the merging of the horizon and the boundary. Even for such non-
geometrical states, they claimed that there are thermodynamical meaning and
assigned the free energy of the form

$\displaystyle F_{Mstate}(T,\mu)=\frac{\sqrt{-\Lambda}}{4G}r_{b}-T\frac{\pi
r_{b}}{2G}-\mu\frac{r_{b}}{2}\sqrt{\frac{-\Lambda}{G}}$ (A.1)

But in subsection 2.2, I showed that the limit corresponds to the BR geometry,
which is a regular geometry and it does not have arbitrary temperature and
chemical potential, but they have a relationship (2.32). Although the
geometric picture of the M states was wrong, the free energy of the M states
at the BR point was correct.

$\displaystyle F_{Mstate}(T_{BR}(\mu),\mu)=F_{BR}(\mu)$ (A.2)

In other words, at this BR point, the M state has a geometric picture, which
is the on-shell BR geometry. So one might expect that other M states deformed
from the M state at the BR point might have a geometric picture. Here, I show
that BR geometries with a conical singularity

$\displaystyle\boldsymbol{g}=\left[\frac{-\Lambda}{G\mu^{2}}(1-2G\mu^{2})\tilde{r}-2\Lambda\tilde{r}^{2}\right]d\tilde{t}+\frac{1}{\left[\frac{-\Lambda}{G\mu^{2}}(1-2G\mu^{2})\tilde{r}-2\Lambda\tilde{r}^{2}\right]}d\tilde{r}^{2}+r_{b}^{2}d\phi^{2}$
(A.3)
$\displaystyle\boldsymbol{A}=-i\sqrt{\frac{-\Lambda}{G}}\tilde{r}d\tilde{t}\hskip
277.41437pt$ (A.4)
$\displaystyle\tilde{r}\in[0,1],~{}~{}~{}~{}~{}~{}~{}\tilde{t}\in\left[0,\beta\mu\sqrt{\frac{G}{-\Lambda}}\right]\hskip
128.0374pt$

can be a family of off-shell geometries whose free energy is given by (A.1).
777 Of course, there are infinitely many families of off-shell geometries
satisfying (A.1). This is just a simple example, and probably the simplest.
The parameters of this geometry are $r_{b},\mu$, and $\beta$, where $\beta$ is
the circumference of the Euclidean time circle at the boundary. When
$\displaystyle\beta\neq\beta_{BR}=\frac{4\pi}{\sqrt{-\Lambda}}\frac{\sqrt{G}\mu}{1-2G\mu^{2}}$,
this geometry has a conical singularity at $\tilde{r}=0$, thus it is off-
shell. The action functional for geometries with a conical singularity is
given by [11]

$\displaystyle I^{E}[\boldsymbol{g},\boldsymbol{A}]=\frac{-1}{16\pi
G}\int_{\mathcal{M}\setminus\Sigma}d^{3}x\sqrt{g}(\mathcal{R}-2\Lambda)+\frac{1}{16\pi}\int_{\mathcal{M}\setminus\Sigma}d^{3}x\sqrt{g}F_{\mu\nu}F^{\mu\nu}\hskip
113.81102pt$ $\displaystyle+\frac{-1}{8\pi
G}\int_{\partial\mathcal{M}}d^{2}y\sqrt{\gamma}\Theta+\frac{1}{8\pi
G}\int_{\partial\mathcal{M}}d^{2}y\sqrt{\gamma}\sqrt{|\Lambda|}+\frac{-1}{8\pi
G}\int_{\Sigma}d\phi\sqrt{\sigma}(2\pi-\theta(\phi))$ (A.5)

The last term is for conical singularities and $\Sigma$ represents the
codimension-two surface of a conical singularity and $2\pi-\theta(\phi)$
represents the (position dependent) conical deficit on the surface. The bulk
and boundary terms are the same as for a regular BR, i.e. they are given by
$\beta$ times $F_{BR}$ (2.33). The additional contribution comes from the last
term. Substituting (A.3) for it gives

$\displaystyle\frac{-1}{8\pi
G}\int_{\Sigma}d\phi\sqrt{\sigma}(2\pi-\theta(\phi))=\frac{-r_{b}}{4G}\left(2\pi+\frac{\Lambda}{2G\mu^{2}}(1-2G\mu^{2})\beta\mu\sqrt{\frac{G}{-\Lambda}}\right)$
$\displaystyle=-\frac{\pi
r_{b}}{2G}+\frac{r_{b}}{8G\mu}\sqrt{\frac{-\Lambda}{G}}\beta-\frac{r_{b}}{4}\sqrt{\frac{-\Lambda}{G}}\beta\mu\hskip
22.76228pt$ (A.6)

Therefore, the “off-shell” free energy of these singular BR geometries is
given by

$\displaystyle F_{singularBR}=F_{BR}+T\times(\ref{FS})\hskip 248.96161pt$
$\displaystyle=\frac{\sqrt{-\Lambda}}{4G}r_{b}-\frac{r_{b}}{8G\mu}\sqrt{\frac{-\Lambda}{G}}-\frac{r_{b}\mu}{4}\sqrt{\frac{-\Lambda}{G}}-\frac{\pi
r_{b}}{2G}T+\frac{r_{b}}{8G\mu}\sqrt{\frac{-\Lambda}{G}}-\frac{r_{b}}{4}\sqrt{\frac{-\Lambda}{G}}\mu$
$\displaystyle=\frac{\sqrt{-\Lambda}}{4G}r_{b}-\frac{\pi
r_{b}}{2G}T-\frac{r_{b}}{2}\sqrt{\frac{-\Lambda}{G}}\mu\hskip 192.05609pt$
(A.7)

which is equivalent to the free energy of the M state (A.1).

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