# Graviton Mass and Entanglement Islands in Low Spacetime Dimensions

Hao Geng  Center for the Fundamental Laws of Nature, Harvard University, 17
Oxford St., Cambridge, MA, 02138, USA.

###### Abstract

It has been conjectured and proven that entanglement island is not consistent
with long-range (massless) gravity in a large class of spacetimes, including
typical asymptotically anti-de Sitter spacetimes, in high spacetime
dimensions. The conjecture and its proof are motivated by the observation that
existing constructions of entanglement islands in high dimensions are all in
gravitational theories where the graviton is massive for which the standard
gravitational Gauss’ law doesn’t apply. In this letter, we show that this
observation persists to lower dimensional cases. We achieve this goal by
providing a unified description of the gravitational Gauss’ law violation in
island models that can work in any dimensions. This unified description
teaches us new lessons on entanglement islands and subregion physics in
quantum gravity. We focus on the case of the (1+1)-dimensional Jackiw-
Teitelboim (JT) gravity for the purpose of demonstration.

††preprint: today

##  I Introduction

The AdS/CFT correspondence [1, 2, 3] states that quantum gravity in a
$(d+1)$-dimensional asymptotically anti-de Sitter space (AdSd+1) duals to a
d-dimensional conformal field theory (CFTd), which is often thought to be
living on the asymptotic boundary of the AdSd+1. This correspondence has
remarkable implications to the black hole information paradox [4, 5] as it
states that the dynamics and states of a black hole in AdSd+1 is fully encoded
in a CFTd which is manifestly unitary. However, to have a proper understanding
of the black hole information paradox, we have to have a detailed study of the
radiation from it and such a study is not easy to be performed in the standard
AdS/CFT. The difficulty is due to several subtleties from the gravitational
nature of the bulk of AdSd+1 [6, 7, 8, 9, 10]. This observation motivates us
to engineer a model in the AdS/CFT correspondence where the radiation from a
black hole can be put in a controlled manner.

Such a model can be constructed by coupling the black hole spacetime to an
external thermal bath. The bath is nongravitational and is described a CFTd+1
(see Fig. 1). It extracts the radiation from the black hole so we can study
the radiation of the black hole by studying the bath. More precisely, we can
dualize the AdSd+1 black hole to a CFTd and think of such a model from the
pure field theory perspective as putting a CFTd on the boundary of a CFTd+1
and this coupling is realized by a double-trace deformation [11, 12]111Also
see [13] for an incomplete early description.

$S_{\text{tot}}=S_{\text{CFT}_{d}}+\int\operatorname{d}\\!^{d}x:\mathcal{O}_{1}(x)\mathcal{O}_{2}(x):+S_{\text{CFT}_{d+1}}\,,$
(1)

where $\mathcal{O}_{1}(x)$ is a single-trace primary operator in CFTd and
$\mathcal{O}_{2}(x)$ is the boundary extrapolation of a single-trace primary
operator of the bath CFTd+1. Recent studies show that a low dimensional
version of this model [14, 15, 16, 17]– the Jackiw-Teitelboim (JT) gravity or
higher dimensional versions with holographic dual– the Karch Randall
braneworld [18, 19, 20, 21, 22, 10] enable us to compute the so-called Page
curve [23] of the black hole radiation with results consistent with unitarity.
In this calculation, there is an emergent new element in quantum gravity–
entanglement island $\mathcal{I}$, which is a gravitational subregion living
in the entanglement wedge of a nongravitational system $\mathcal{R}$
disconnected from it. In the context of an evaporating black hole, the system
$\mathcal{R}$ is the early-time radiation from the black hole and the
subregion $\mathcal{I}$ nonvanishingly overlaps the black hole interior after
the Page time. This suggests that after the Page time the physics in the
interior of the black hole is fully encoded in the early-time radiation from
it.

Nevertheless, it has been known that in such models at higher dimensions
($d\geq 3$) the graviton in the AdSd+1 is massive and this mass is induced
from the bath coupling [24, 25, 26, 11, 12, 27]. This effect has been found to
have profound implications to quantum gravity by noticing that the
gravitational Gauss’ law is modified by the graviton mass [21, 28, 29]. There
exist several ways to understand this mass generating effect from different
descriptions and aspects of the setup. On the one hand, due to the AdS/CFT
correspondence, the graviton mass is dual to the anomalous dimension of the
stress-energy tensor of the CFTd and this anomalous dimension is easy to
understand in the description Eq. (1) as a result of the double-trace
deformation [12, 11]. On the other hand, a direct calculation of the one-loop
graviton propagator [25, 26] shows that the graviton indeed gets a mass as a
result of the bath coupling. Furthermore, the calculation in [25, 26] suggests
that diffeomorphism symmetry is not manifestly broken in the full quantum
gravity theory in AdSd+1 but it is spontaneously broken. As a result, the
graviton mass is generated by a Higgs mechanism. The Higgs mechanism for
graviton mass requires 1) the diffeomorphism symmetry is spontaneously broken,
2) there is a Goldstone vector boson coupled to the graviton in the
Stückelberg form and as a result 3) the graviton obtains a mass by eating this
vector boson. However, instead of showing 1) and 2) with 3) as a result, [25]
showed that there is a composite vector mode induced from the modes of an
AdSd+1 scalar field due to the bath coupling and it propagates in the graviton
self-energy loop generating the graviton mass. The gist of the calculation in
[25, 26] is that the bulk graviton has a propagator at tree-level and the one-
particle irreducible self-energy from the composite vector mode of the bulk
scalar field in the loop will induce a nonzero pole for the graviton
propagator. This makes it difficult to generalize the observation that the
graviton is massive which modifies the gravitational Gauss’ law to lower
dimensional cases ($d\leq 2$) due to the fact that in these cases the bulk
graviton doesn’t have a tree-level propagator. Nevertheless, it is expected
that the modification of the gravitational Gauss’ law in the set up with a
bath coupled should persists in lower dimensional cases. Otherwise there would
be a sharp contradiction between the existence of entanglement island [17] and
the basic requirement in the entanglement wedge reconstruction [29].222See
[30] for string theory evidence that the existence of entanglement island
relies on graviton mass.

In this paper, we focus on the case of (1+1)-dimensional Jackiw-Teitelboim
(JT) gravity studied in [17] where it is shown explicitly that we have
entanglement island if we couple the bulk AdS2 spacetime with a
nongravitational bath. We will show that in this setup even though there is no
kinetic term for the graviton in AdS2 there will be a graviton mass term
induced from the bath coupling which modifies the gravitational Gauss’ law.

$AdS_{d+1}$$Mink_{d+1}$ Figure 1: We couple the gravitational AdSd+1 universe
(the blue shaded region) with a nongravitational bath (the green shaded
region) by gluing them along the asymptotic boundary the AdSd+1 (the red
vertical line). The nongravitational bath is modeled by a (d+1)-dimensional
CFT living on a half Minkowski space which shares the same boundary as the
asymptotic boundary of the AdSd+1. We use the Poincaré coordinates in the
AdSd+1. The coupling is achieved in the dual CFT description by Equ. (1).

##  II Gravitational Gauss’ Law and Entanglement Islands in JT Gravity

In this section, we explain the gravitational Gauss’ law and briefly review
the construction of entanglement island in the context of JT gravity. In the
end, we discuss why they are in tension in a nutshell.

JT gravity [31, 32] is a 1+1-dimensional (2d) dilaton gravity theory whose
bulk action is given by

$S=\frac{1}{16\pi G_{N}}\int
d^{2}x\sqrt{-g}\phi(x)(R[g]+2)+S_{\text{matter}}[\psi;g]\,,$ (2)

where $G_{N}$ is the Newton’s constant, $R[g]$ is the Ricci scalar, $\phi(x)$
is the dilaton field and $S_{\text{matter}}[\psi;g]$ is the action of the
minimally coupled matter fields in the 2d manifold whose metric is
$g_{\mu\nu}(x)$.

The dilaton is a Lagrange multiplier whose path integral fixes the Ricci
scalar to be $R[g]=-2$. Due to the 2d nature of the geometry and
diffeomorphism invariance the metric $g_{\mu\nu}(x)$ is conformally flat and
so we can consider the AdS2 Poincaré patch

$ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}=\frac{1}{x^{2}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{-dt^{2}+dx^{2}}{x^{2}}\,,$
(3)

where the asymptotic boundary is at $x\rightarrow 0$. All other asymptotic AdS
solutions to $R[g]=-2$ are diffeomorphic equivalent to Equ. (3). Hence we can
take the metric to be fixed as Equ. (3) even in quantum theory. In JT gravity
we usually cutoff the geometry at $x=\epsilon$ for $\epsilon\rightarrow 0$ and
a boundary term in the action is important to ensure the well-definiteness of
the action variation (for details see [33]). The variation of Equ. (2) with
respect to the metric gives an equation of motion

$8\pi G_{N}T_{\mu\nu}=-\nabla_{\mu}\nabla_{\nu}\phi+g_{\mu\nu}\nabla^{2}\phi-
g_{\mu\nu}\phi\,,$ (4)

where $T_{\mu\nu}$ is the matter fields’ energy-momentum tensor. Since the
geometry is fixed to be Equ. (3), if we linearize the above equation around a
vacuum solution $\phi^{(0)}$ (i.e. it satisfies Equ. (4) with $T_{\mu\nu}=0$)
we have

$\begin{split}8\pi G\delta
T_{\mu\nu}&=-\nabla_{\mu}\nabla_{\nu}\delta\phi+g_{\mu\nu}\nabla^{2}\delta\phi-
g_{\mu\nu}\delta\phi\,.\end{split}$ (5)

In the case that there is an energy excitation in the bulk we have

$\delta\rho=-\delta T_{0}^{0}=-\frac{1}{8\pi
G_{N}}g^{xx}\nabla_{x}\nabla_{x}\delta\phi+\frac{1}{8\pi G_{N}}\delta\phi\,,$
(6)

which can be written as

$\delta H_{\text{matter}}=\frac{1}{8\pi G_{N}}\int
dx\sqrt{-g}\Bigg{[}-g^{xx}\nabla_{x}\nabla_{x}\delta\phi+\delta\phi\Bigg{]}\,,$
(7)

where $H_{\text{matter}}$ denotes the matter Hamiltonian. Using the metric
Equ. (3), we can see that

$\delta H_{\text{matter}}=-\frac{1}{8\pi G_{N}}\int
dx\partial_{x}\Big{[}\partial_{x}\delta\phi+\frac{\delta\phi}{x}\Big{]}\,,$
(8)

which shows that local energy excitations in the bulk can be detected by a
near boundary measurement of the change in the dilaton field at the same
instant of the excitation. This can be uplifted to a quantum mechanical
statement that the bulk matter field Hamiltonian operator equals to an AdS2
boundary operator [34, 29, 35, 9]. This is the Gauss’s law in JT gravity.

The importance of JT gravity shows up in its recent application to the
calculation of the Page curve for black hole radiation [14, 15]. In this
calculation, the black hole under consideration is a four dimensional near-
extremal charged black hole, for which the s-wave sector of the matter-gravity
coupled theory is described by Equ. (2) [36, 37]. To study the radiation from
such a black hole, a two-dimensional nongravitational bath is glued to the
asymptotic boundary of the reduced geometry Equ. (3). Such a coupling is
achieved by modifying the boundary condition of the AdS2 matter fields
$\psi(x)$ from reflective to transmissive which can be realized by a double-
trace deformation as in Equ. (1) in the dual field theory description. To
simplify the calculation, both the AdS2 matter fields $\psi(x)$ and the
nongravitational bath were modeled by a 2d conformal field theory– the free
massless fermion in [17], for which the formula of the entanglement entropy is
known in a closed form. Now the entropy of the radiation can be studied by
calculating the entanglement entropy of a subregion in the bath (see Fig.2).
The result is that this entanglement entropy obeys the unitary Page curve and
at late times the entanglement wedge [38] of the radiation contains a
disconnected region in the AdS2 bulk which is called an entanglement island.
The holographic interpretation of this result is that the physics in the
island is fully (even nonperturbatively [38]) encoded in the radiation and is
totally independent from its complement $\bar{\mathcal{I}}$ in the bulk AdS2.

From here we can see that there is a direct contradiction between the physics
of the entanglement island and the gravitational Gauss’ law Equ. (8). The
reason is that a near-boundary observer in AdS2 can detect the existence of an
energy excitation inside the island by measuring $\frac{1}{8\pi
G_{N}}\Big{(}\partial_{x}\delta\phi(x)+\frac{\delta\phi(x)}{x}\Big{)}$ at the
same instant of the excitation.333This statement can be uplifted to a quantum
mechanical statement that there exists an AdS2 boundary operator which doesn’t
commute with a local unitary operator in the island region which is made from
the matter fields. This is due to the gravitational Gauss’ law Equ. (8) which
states that the matter Hamiltonian equals to an AdS2 boundary operator. Hence
the physics in the island is not fully independent from its complementary
region in the bulk AdS2.

$\bullet$$\bullet$ Figure 2: The demonstration of the island calculation in
[17]. We draw a Penrose diagram of the setup where the red shaded region is
the bulk AdS2 described by geometry Equ. (3), the blue shaded region is the
bath whose geometry is half of a 2d Minkowski space and the bath is glued to
the bulk along the thick red vertical line ($x=0$). The pink causal diamond in
the bath models the radiation $\mathcal{R}$ and the green causal diamond in
the AdS2 is the island $\mathcal{I}$ of $\mathcal{R}$. The physics in the
island should be independent of that in the orange diamond.

##  III Gravitational Gauss’ Law Violation in the JT Island Model

In this section, we model the bulk matter field by a massive scalar field
$\psi$ with mass square $m^{2}$ for which the transparent boundary condition
with the bath glued on can be explicitly written down as the holographic dual
of the double-trace deformation Equ. (1) [25, 12, 27]. Let’s consider the full
gravitational path integral in the gravitational sector of the system we are
considering

$Z_{\text{Grav}}=\int
D[\phi]D[g_{\mu\nu}]D[\psi]e^{-iS_{\text{JT}}[\phi;g]-iS_{\text{matter}}[\psi;g]}\,,$
(9)

where we can treat the metric fluctuations perturbatively and split it into
two parts– a background part and a perturbative fluctuation

$g_{\mu\nu}(x)=g^{0}_{\mu\nu}(x)+\sqrt{16\pi G_{N}}h_{\mu\nu}(x)\,,$ (10)

where $h_{\mu\nu}(x)$ is symmetric and traceless. In our perturbative
calculation we have to choose a consistent background for the metric and
dilaton $(g^{0}_{\mu\nu}(x),\phi^{0}(x))$. We take the background to satisfy
the vacuum (zero matter source) equation of motion. That is the metric is Equ.
(3) and the dilaton profile $\phi^{(0)}$ satisfies Equ. (4) with
$T_{\mu\nu}=0$. Hence we only have to consider the matter part and its
coupling with $h_{\mu\nu}$ and the dilaton fluctuation $\delta\phi(x)$. Let
focus on the $h_{\mu\nu}$ part for now. To the leading order in $G_{N}$ we
have

$\begin{split}Z_{h}&=\int
D[\psi]D[h_{\mu\nu}]e^{-iS_{\text{matter}}[\psi;g^{0}]-iS_{\text{int}}[\psi,h;g^{0}]}\,.\\\
S_{\text{int}}[\psi,h;g^{0}]&=\sqrt{16\pi G_{N}}\int
d^{2}x\sqrt{-g^{0}(x)}h_{\mu\nu}(x)T^{\mu\nu}_{\text{matter}}(x)\,.\end{split}$
(11)

In this system, the diffeomorphism symmetry

$h_{\mu\nu}(x)\rightarrow
h_{\mu\nu}(x)+\nabla_{\mu}\epsilon_{\nu}(x)+\nabla_{\nu}\epsilon_{\mu}(x)\,,$
(12)

is broken for $\epsilon^{\mu}$ which decays slower than $z^{2}$ as
$z\rightarrow 0$ due to the bath coupling Equ. (1) (see [27] for detailed
explanations).

Nevertheless, it will be useful for us to consider this system as a gauge-
fixed version of a fully diffeomorphism invariant system. This can be realized
by introducing a divergenceless Stückelberg vector field $V^{\mu}(x)$ which
transforms under the diffeomorphism Equ. (12) as

$V^{\mu}(x)\rightarrow V^{\mu}(x)-\epsilon^{\mu}(x)\,,$ (13)

and we define

$\begin{split}h^{\prime}_{\mu\nu}(x)&=h_{\mu\nu}(x)+\nabla_{\mu}V_{\nu}(x)+\nabla_{\nu}V_{\mu}(x)\,,\\\
&\equiv h_{\mu\nu}(x)+\nabla_{(\mu}V_{\nu)}(x).\end{split}$ (14)

We can construct the partition of the fully diffeomorphism invariant system by
integrating in the field $V^{\mu}(x)$

$\begin{split}Z_{\text{full}}&=\int
D[V_{\mu}]D[\psi]D[h_{\mu\nu}]e^{-iS_{\text{matter}}[\psi;g^{0}]-iS_{\text{int}}[\psi,h;g^{0}]}\,\\\
&=\int
D[V_{\mu}]D[\psi]D[h^{\prime}_{\mu\nu}]e^{-iS_{\text{matter}}[\psi;g^{0}]-iS_{\text{int}}[\psi,h^{\prime};g^{0}]}\,\\\
&=\int
D[V_{\mu}]D[\psi]D[h_{\mu\nu}]e^{-iS_{\text{matter}}[\psi;g^{0}]-iS_{\text{int}}[\psi,h^{\prime};g^{0}]}\,,\end{split}$
(15)

which is manifestly invariant under the transform Equ. (12) and Equ. (13) and
fixing the gauge by setting $V^{\mu}(x)=0$ we get back to Equ. (11). 444Here
we note that the diffeomorphism in the system before we introduce the vector
field is also anomalous. The vector field can be used to cancel this anomaly.
We will show later that the diffeomorphism is indeed anomalous and we refer
the readers to [27] to see the details of taming the anomaly using the vector
field.

To understand the physics in the graviton sector, we can integrate out the
matter field $\psi(x)$ in Equ. (15) getting an effective action $S_{eff}[h,V]$
which has to be a function of the diffeomorphism invariant combination Equ.
(14). Therefore, we can focus on the $V^{\mu}(x)$-sector and at the end
replace $\nabla_{\mu}V_{\nu}(x)+\nabla_{\nu}V_{\mu}(x)$ by the symmetric
combination Equ. (14). Since we want to understand the low energy physics, we
only care about the leading order effective action for $V^{\mu}$ in $G_{N}$
expansion which is

$\begin{split}S_{\text{eff}}[V]&=-i\frac{16\pi G_{N}}{2}\int
d^{d+1}x\sqrt{-g_{0}(x)}d^{d+1}y\sqrt{-g_{0}(y)}\times\\\
&\nabla_{(\mu}V_{\nu)}(x)\Pi^{\mu\nu,\rho\sigma}(x,y)\nabla_{(\rho}V_{\sigma)}(y)\,,\end{split}$
(16)

where in our case $d=1$ and we have

$\Pi^{\mu\nu,\rho\sigma}(x,y)=\langle
T\Big{(}T^{\mu\nu}_{\text{matter}}(x)T^{\rho\sigma}_{\text{matter}}(y)\Big{)}\rangle\,.$
(17)

The low energy physics is encoded in the long distance limit that $x$ and $y$
are widely separated. It has been calculated in detail in [25, 26, 12] that in
large distance regime the result is given by

$16\pi
G_{N}\Pi^{\mu\nu,\rho\sigma}(x,y)=2M^{2}\nabla^{\mu}D^{\nu,\sigma}(x,y)\nabla^{\rho}\,,$
(18)

where $D^{\nu,\sigma}(x,y)$ is in the form of the propagator of a
divergenceless vector field satisfying

$(\nabla^{2}-d)D^{\nu,\sigma}(x,y)=-i\frac{g^{\nu\sigma}}{\sqrt{-g}}\delta^{d+1}(x-y)\,,$
(19)

and $M^{2}$ is given in general in [12] as

$M^{2}=-G_{N}\frac{2^{4-d}\pi^{\frac{3-d}{2}}}{(d+2)\Gamma(\frac{d+3}{2})}\frac{a_{\Delta_{1}}a_{\Delta_{2}}\Delta_{1}\Delta_{2}\Gamma[\Delta_{1}]\Gamma[\Delta_{2}]}{\Gamma[\Delta_{1}-\frac{d}{2}]\Gamma[\Delta_{2}-\frac{d}{2}]}\,$
(20)

where $a_{\Delta_{1}}$ and $a_{\Delta_{2}}$ are of opposite signs555They are
both nonzero in bath coupled case and either one of them is zero when there is
no bath [27]. See also [39] for a relevant study from a different perspective.
and we have

$\Delta_{1}=\frac{d}{2}+\sqrt{\frac{d^{2}}{4}+m^{2}}\,,\quad\Delta_{2}=\frac{d}{2}-\sqrt{\frac{d^{2}}{4}+m^{2}}\,.$
(21)

Integrating by part in Equ. (16) we can see that in the low energy regime we
would have

$\begin{split}S_{\text{eff}}[V]&=iM^{2}\int
d^{d+1}x\sqrt{-g_{0}(x)}d^{d+1}y\sqrt{-g_{0}(y)}\times\\\
&V^{\nu}(x)(\nabla^{2}-d)D_{\nu,\sigma}(x,y)\nabla_{\rho}\nabla^{(\rho}V^{\sigma)}(y)\\\
&=-M^{2}\int
d^{d+1}x\sqrt{-g_{0}(x)}V_{\mu}(x)\nabla_{\rho}\nabla^{(\rho}V^{\mu)}(x)\\\
&=\frac{M^{2}}{2}\int
d^{d+1}x\sqrt{-g_{0}(x)}\nabla_{(\mu}V_{\nu)}(x)\nabla^{(\nu}V^{\mu)}(x)\,,\end{split}$
(22)

where we integrated by parts in the first and last steps and we used Equ. (19)
in the second step. Hence the total low energy effective action is

$S_{\text{eff}}[h,V]=\frac{M^{2}}{2}\int
d^{d+1}x\sqrt{-g_{0}(x)}h^{\prime}_{\mu\nu}(x)h^{\prime\mu\nu}(x)\,.$ (23)

If we fix the diffeomorphism gauge Equ. (12) and Equ. (13) by setting
$V^{\mu}=0$ we have

$S_{\text{eff}}[h]=\frac{M^{2}}{2}\int
d^{d+1}x\sqrt{-g(x)}h_{\mu\nu}h^{\mu\nu}\,,$ (24)

which is nothing but a mass term for graviton. Here we notice that this
calculation works in any dimensions and it gives the graviton mass term
without the requirement of having a tree level propagator (i.e. a kinetic
term) of the graviton.

Now we can show that the gravitational Gauss’ law is broken. The reason is
that with the ”graviton mass” term incorporated the full Einstein’s equation
Equ. (4) is modified to

$\small 8\pi
G_{N}T_{\mu\nu}=-\nabla_{\mu}^{(0)}\nabla_{\nu}^{(0)}\phi+g_{\mu\nu}\nabla^{(0)2}\phi-
g_{\mu\nu}^{(0)}\phi-\frac{8\pi G_{N}M^{2}}{2}h_{\mu\nu}\,,$ (25)

where we use $\nabla^{(0)}$ to emphasize that the covariant derivative is
taken in the background geometry Equ. (3). Now as we did in Equ. (5) we can
linearize this equation around the background profile $\phi^{(0)}$ which gives
us

$\small 8\pi G_{N}\delta
T_{\mu\nu}=-\nabla_{\mu}^{(0)}\nabla_{\nu}^{(0)}\delta\phi+g_{\mu\nu}\nabla^{(0)2}\delta\phi-
g_{\mu\nu}^{(0)}\delta\phi-\frac{8\pi G_{N}M^{2}}{2}h_{\mu\nu}\,,$ (26)

from which we get

$\delta H_{\text{matter}}=-\frac{1}{8\pi G_{N}}\int
dx\partial_{x}\Big{[}\partial_{x}\delta\phi+\frac{\delta\phi}{x}\Big{]}+\frac{M^{2}}{2}\int
d^{2}x\sqrt{-g}h^{0}_{0}\,,$ (27)

which is not just a boundary term. As a result, a bulk energy excitation can
be hidden from being detected by the observer near the asymptotic boundary due
to the graviton mass. This is exactly the same result as in the higher
dimensional cases discovered in [29, 40]. This is now consistent with the
existence of entanglement island as in Fig.2.

##  IV Conclusion and Discussions

In this paper, we finished the program initiated in [21] to prove that
gravitational theories in the island models are massive gravity theories by
proving this conjecture in low dimensional cases. We mainly focused on the JT
gravity model where the island model can be analytically solved [17]. To do
this we developed a tool to efficiently extract the graviton mass Equ. (20).
Nevertheless, we emphasize that the tool we developed provides a generic proof
of the conjecture in any dimensions and concretely realizes the fact that the
diffeomorphism symmetry is spontaneously broken by the bath coupling. Our
study can be generalized to the general higher spin gravity for which the bath
breaks the conservation of higher spin currents and we can efficiently show
that the higher spin symmetries are spontaneously broken and the corresponding
higher-spin gauge bosons are massive [41].

Moreover this calculation provides a proof that the AdS bulk diffeomorphism
symmetry is anomalous in the low energy matter sector [27]. This is because
the same calculation could show that

$U^{\nu\rho\sigma}(p)=\int
d^{d+1}x\sqrt{-g}e^{ip_{\alpha}x^{\alpha}}\nabla_{\mu}\langle
T\Big{(}T_{\text{matter}}^{\mu\nu}(x)T_{\text{matter}}^{\rho\sigma}(0)\Big{)}\rangle$
(28)

is nonzero which is a diagnozer of diffeomorphism anomaly [42].666Here we
should notice that we are being schematic as Equ. (28) comes from [42] where
they only consider flat spacetime which has translational symmetry. We should
understand Equ. (28) as merely a statement that the matter stress energy
tensor is not divergenceless inside correlators. Even though the result is
only a contact term, it cannot be removed by any local regularization schemes
which are not sensitive to the long distance effects such as boundary
conditions of the fields. Furthermore, this established a precise realization
of the mechanism proposed in [43] that gauge anomaly in low energy effective
theory of the matter sector signals the spontaneous breaking of the gauge
symmetry. In this mechanism the fact that the gauge boson is massive can be
seen by introducing a Stückelberg field to compensate the anomaly from the
matter sector which is exactly what we did in this paper and in [27]. This
results in a fully covariant anomaly-free description.

Finally, we notice that our study conveys important lessons on entanglement
islands and subregion physics in gravity. It is argued in some papers [44, 45,
46] that the tension between the entanglement island and the long-range nature
of gravity discovered in [21, 28, 29] can be avoided if we consider background
configurations with strong enough features. The essential observations in
these papers are that 1) we can dress local bulk operators to these features
to avoid dressing to the asymptotic boundary or equivalently 2) in these
background configurations the gravitational Gauss’ law would be significantly
modified.777For example, due to the lack of a globally defined time-like
Killing vector. From 2) we can see that these are essentially different ways
to break the long-range nature of gravity. Nevertheless, these ways are highly
non-universal as they would propose that entanglement islands can exist only
in very specific backgrounds and most of the existing models of entanglement
islands don’t satisfy this corollary.888Meanwhile, papers like [44, 45] are in
tension with entanglement wedge reconstruction [38] which states that
operators inside island $\mathcal{I}$ should be nonperturbatively commute with
those from its complement $\bar{\mathcal{I}}$. On the other hand, all existing
models of entanglement island, which is a closed subregion inside a
gravitational universe, are in the context of massive gravity due to the bath
coupling [21, 27]. This observation highly motivates us to believe that there
is a connection between entanglement islands and massive gravity which is
universal. The essential reason is that massive gravity provides a global
feature to dress operators diffeomorphism invariantly and hence entanglement
islands can exist in any backgrounds in this context. This can be seen in the
fully covariant description of massive gravity we outlined in this paper and
[27], where the dynamics of the Stückelberg vector field manifests, by
noticing that the Stückelberg field also transforms under diffeomorphism and
so we can use this field to dress operators diffeomorphism invariantly.999For
example, to the first order in $G_{N}$ expansion which is the order we are
focusing on in this paper, a scalar operator $\hat{\phi}(x)$ can be dressed
diffeomorphism invariantly as $\hat{\Phi}(x)=\hat{\phi}(x+\sqrt{16\pi
G_{N}}V)$. Higher order dressings can be easily done order by order as in [6,
7]. Interestingly, in the gauge-fixed description ($V^{\mu}=0$) this dressing
disappears, i.e. all diffeomorphism invariant operators becomes local, and
meanwhile the gravitational Gauss’ law violation manifests. From here we can
see that information localizes differently in massive gravity theories
compared to the massless gravity theories [40], as in massive gravity
information can be perturbatively hidden in the bulk. This is the essential
reason why the existence of entanglement islands is universally allowed in
massive gravity theories.

## Acknowledgements

We are grateful to Amr Ahmadain, Luis Apolo, Jan de Boer, A. Liam Fitzpatrick,
$\mathring{\text{A}}$smund Folkstad, Eduardo Gonzalo, Brianna Grado-White,
Andreas Karch, Indranil Halder, Ami Katz, Matthew Heydeman, Yangrui Hu, Yikun
Jiang, Juan Maldacena, Carlos Perez-Pardavila, Lisa Randall, Suvrat Raju,
Martin Sasieta, Pushkal Shrivastava, Brian Swingle, Neeraj Tata and Gabriel
Wong for useful discussions at various stages of this project. We thank
Andreas Karch, Carlos Perez-Pardavila, Lisa Randall, Suvrat Raju and Marcos
Riojas for comments on an early version of the draft. The work of HG is
supported by the grant (272268) from the Moore Foundation “Fundamental Physics
from Astronomy and Cosmology” and a grant from Physics Department at Harvard
University.

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# Existence theorem of a weak solution for Navier-Stokes type equations
associated with de Rham complex

Alexander Polkovnikov  Siberian Federal University, Institute of Mathematics
and Computer Science, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia
<EMAIL_ADDRESS>
###### Abstract.

Let $\\{d_{q},\Lambda^{q}\\}$ be de Rham complex on a smooth compact closed
manifold $X$ over $\mathbb{R}^{3}$ with Laplacians $\Delta_{q}$. We consider
operator equations, associated with the parabolic differential operators
$\partial_{t}+\Delta_{2}+N^{2}$ on the second step of complex with nonlinear
bi-differential operator of zero order $N^{2}$. Using by projection on the
next step of complex we show that the equation has unique solution in special
Bochner-Sobolev type functional spaces for some (small enough) time $T^{*}$.

###### Key words and phrases:

elliptic differential complexes, parabolic nonlinear equations, open mapping
theorem

###### 2010 Mathematics Subject Classification:

58J10, 35K45

## Introduction

Consider the de Rham complex on a Riemannian $n$-dimensional smooth compact
closed manifold $X$ with vector bundles $\Lambda^{q}$ of exterior forms of
degree $q$ over $X$,

$0\longrightarrow\varOmega^{0}(X)\xrightarrow{d_{0}}\varOmega^{1}(X)\xrightarrow{d_{1}}\cdots\xrightarrow{d_{n-1}}\varOmega^{n}(X)\longrightarrow
0.$ (0.1)

Here $\varOmega_{q}(X)$ denotes the space of all differential forms of degree
$q$ with smooth coefficients on $X$. In this case the Laplacians
$\Delta_{q}=d_{q}^{*}d_{q}+d_{q-1}d_{q-1}^{*}$, $q=0,1,\dots,n$, of the
complex are second order strongly elliptic differential operators on $X$,
where operator $d_{q}^{*}$ is a formal adjoint to $d_{q}$. As usual, for $q<0$
and $q\geq n$ we assume that $d_{q}=0$.

We want to study the non-linear problems, associated with the complex. With
this purpose, we denote by $M_{i,j}$ two bilinear bi-differential operators of
zero order (see [5] or [25]),

$\begin{array}[]{c}M_{q,1}(\cdot,\cdot):\big{(}\varOmega^{q+1}(X),\
\varOmega^{q}(X)\big{)}\to\varOmega^{q}(X),\\\
M_{q,2}(\cdot,\cdot):\big{(}\varOmega^{q}(X),\
\varOmega^{q}(X)\big{)}\to\varOmega^{q-1}(X).\end{array}$ (0.2)

We set for a differential form $u$ of the degree ${q}$

$N^{q}(u)=:M_{q,1}(d_{q}u,u)+d_{q-1}M_{q,2}(u,u).$ (0.3)

Note, that operator $N^{q}(u)$ is non-linear.

Let now time $T>0$ is finite. Then for any fixed positive number $\mu$ the
operators $\partial_{t}+\mu\Delta_{q}$ are parabolic on the cylinder
$X\times(0,T)$ (see [7]). Consider the following initial problem: given
sufficiently regular differential forms $f$ of the induced bundle
$\Lambda^{q}(t)$ (the variable $t$ enters into this bundle as a parameter) and
$u_{0}$ of the bundle $\Lambda^{q}$, find a differential forms $u$ of the
induced bundle $\Lambda^{q}(t)$ and $p$ of the induced bundle
$\Lambda^{q-1}(t)$ such that

$\begin{cases}\partial_{t}u+\mu\Delta_{q}u+N^{q}(u)+d_{q-1}p=f&\text{in
}X\times(0,T),\\\ d_{q-1}^{*}u=0&\text{in }X\times[0,T],\\\
d_{q-2}^{*}p=0&\text{in }X\times[0,T],\\\ u(x,0)=u_{0}&\text{in
}X,\end{cases}$ (0.4)

For general elliptic complexes this problem was considered in the works [21]
and [29], where the open mapping theorems were proved in the special spaces of
Hölder (see [21]) and Sobolev (see [29]) types. It means that the range of the
non-linear operator ${\mathcal{A}}_{q}$ , related to the problem, is open in
the constructed spaces. However, obtaining an existence theorem for the
solution (even the so-called weak one) and closedness of the range of the
related non-linear operator in such spaces appears to be a more difficult
task.

For example, if we take $q=1$ and a suitable nonlinear term we may treat (0.4)
as the initial problem for the well known Navier-Stokes equations for
incompressible fluid over the manifold $X$ (see, for instance, [16] or [30]).
Note that the equation with respect to $p$ is actually missing in this case,
because $d_{-1}^{*}=0$.

We consider problem (0.4) in the case $n=3$, $q=2$ and a special nonlinearity
$M_{q,1}(d_{q}u,u)=(d_{q}u)u$. It easy to see that in this case we can treat
the de Rham differentials as $d_{2}=\mathrm{div}$, $d_{1}=\mathrm{rot}$,
$d_{2}^{*}=-\nabla$, $d_{1}^{*}=\mathrm{rot}$ and then (0.4) transforms to

$\begin{cases}\partial_{t}u+\mu\Delta_{2}u+N^{2}(u)+\mathrm{rot}\,p=f&\text{in
}X\times(0,T),\\\ \mathrm{rot}\,u=0&\text{in }X\times[0,T],\\\
\mathrm{div}\,p=0&\text{in }X\times[0,T],\\\ u(x,0)=u_{0}&\text{in
}X,\end{cases}$ (0.5)

where

$N^{2}(u)=(\mathrm{div}\,u)u+\mathrm{rot}\,(M_{q,2}(u,u)),$ (0.6)

and Laplacian

$\Delta_{2}u=d_{2}^{*}d_{2}+d_{1}d_{1}^{*}=-\nabla\mathrm{div}\,u+\mathrm{rot}\,\mathrm{rot}\,u=-\Delta
u.$

Here $\Delta u$ is a standard Laplace operator applied componentwise to the
differential form $u$ in the space variable $x$.

Using projection to the next step of complex (0.1), we prove an existence
theorem of weak (distributional) solution in the constructed Bochner-Sobolev
type spaces for some (small enough) time $T^{*}$. Note, that considering
general non-linear perturbations of linear parbolic equations one have to
impose essential restrictions on the non-linear term $N^{2}(u)$ in order to
achieve existence of weak solutions. For example, one of such condition can be
positiveness of non-linear operator $N^{2}(u)$. However, we do not impose such
strong conditions on the non-linear term, but still have an existence of weak
solutions due to special properties of the de Rham complex.

##  1\. Functional spaces

Denote by $L^{p}_{\Lambda^{q}}$, $1\leq p\leq\infty$, space of differential
forms of the degree ${q}$ with coefficients in the Lebesgue space $L^{p}(X)$.
In a similar way we designate the spaces of forms on $X$ whose components are
of Sobolev class or have continuous partial derivatives. We denote it by
$W^{s,p}_{\Lambda^{q}}$ and $C^{s}_{\Lambda^{q}}$ respectively with smoothness
$s$. In particular case, for $p=2$ we designate
$H^{s}_{\Lambda^{q}}:=W^{s,2}_{\Lambda^{q}}$.

For calculations, it is convenient to use the fractional powers of the Laplace
operator. Namely, for differential form $u$ of degree $q$ we denote by

$\nabla^{m}_{q}u:=\begin{cases}\Delta_{q}^{m/2}u,\quad&m\ \text{is even},\\\
(d_{q}\oplus d_{q-1}^{*})\Delta_{q}^{(m-1)/2}u,\quad&m\ \text{is odd}.\\\
\end{cases}$ (1.1)

It easy to see that integration by parts yields

$\sum_{|\alpha|=m}\|\partial^{\alpha}u\|^{2}_{L^{2}_{\Lambda^{q}}}=\|\nabla^{m}_{q}u\|^{2}_{L^{2}_{\Lambda^{q}}}.$

Now, we want to recall the standard Hodge theorem for elliptic complexes. For
this purpose denote by $\mathcal{H}^{q}$ the harmonic space of complex (0.1),
i.e.

$\mathcal{H}^{q}=\left\\{u\in C^{\infty}_{\Lambda^{q}}:d_{q}u=0\mbox{ and
}d_{q-1}^{*}u=0\mbox{ in }X\right\\},$ (1.2)

and by $\Pi^{i}$ the orthogonal projection from $L^{2}_{\Lambda^{q}}$ onto
$\mathcal{H}^{q}$.

######  Theorem 1.1.

Let $0\leq q\leq n$, $s\in\mathbb{Z}_{+}$. Then operator

$\Delta_{q}:H^{s+2}_{\Lambda^{q}}\to H^{s}_{\Lambda^{q}}$ (1.3)

is Fredholm:

(1) the kernel of operator (1.3) equals to the finite-dimensional space
$\mathcal{H}^{q}$;

(2) given $v\in H^{s}_{\Lambda^{q}}$ there is a form $u\in
H^{s+2}_{\Lambda^{q}}$ such that $\Delta_{q}u=v$ if and only if
$(v,h)_{L^{2}_{\Lambda^{q}}}=0$ for all $h\in\mathcal{H}^{q}$;

(3) there exists a pseudo-differential operator $\varphi^{i}$ on $X$ such that
the operator

$\varphi^{q}:H^{s}_{\Lambda^{q}}\to H^{s+2}_{\Lambda^{q}},$ (1.4)

induced by $\varphi^{q}$, is linear bounded and with the identity $I$ we have

$\varphi^{q}\Delta_{q}=I-\Pi^{q}\mbox{ on
}H^{s+2}_{\Lambda^{q}},\quad\Delta_{q}\varphi^{q}=I-\Pi^{q}\mbox{ on
}H^{s}_{\Lambda^{q}}$ (1.5)

###### Proof.

See, for instance, [25, Theorem 2.2.2]. ∎

Denote by $V^{s}_{\Lambda^{q}}:=H^{s}_{\Lambda^{q}}\cap S_{d_{q-1}^{*}}$ the
space of all differential forms $u\in H^{s}_{\Lambda^{q}}$ satisfying
$d_{q-1}^{*}u=0$ in the sense of distributions in $X$. Let now
$L^{2}(I,H^{s}_{\Lambda^{q}})$ be the Bochner space of $L^{2}$-mappings

$u(t):I\to H^{s}_{\Lambda^{q}},$

where $I=[0,T]$, see, for instance, [14]. It is a Banach space with the norm

$\|u\|_{L^{2}(I,H^{s}_{\Lambda^{q}})}^{2}=\int_{0}^{T}\|u\|^{2}_{H^{s}_{\Lambda^{q}}}dt.$

We need to introduce suitable Bocner-Sobolev type spaces, see [24] or [30] for
the de Rham complex and [29] for the general elliptic complexes. Namely, for
$s\in\mathbb{Z}_{+}$ denote by $B^{k,2s,s}_{q,\text{vel}}(X_{T})$ the space of
all differential forms of degree $q$ over $X_{T}:=X\times[0,T]$ with variable
$t\in[0,T]$ as a parameter, such that

$u\in C(I,V_{\Lambda^{q}}^{k+2s})\cap L^{2}(I,V_{\Lambda^{q}}^{k+2s+1})$

and

$\nabla^{m}_{q}\partial^{j}_{t}u\in C(I,V_{\Lambda^{q}}^{k+2s-m-2j})\cap
L^{2}(I,V_{\Lambda^{q}}^{k+2s+1-m-2j})$

for all $m+2j\leq 2s$. It is a Banach space with the norm

$\|u\|^{2}_{B^{k,2s,s}_{q,\text{vel}}}:=\sum_{m+2j\leq 2s\atop 0\leq l\leq
k}\|\nabla^{l}_{q}\nabla^{m}_{q}\partial_{t}^{j}u\|^{2}_{C(I,{L}_{\Lambda^{q}}^{2})}+\|\nabla^{l+1}_{q}\nabla^{m}_{q}\partial_{t}^{j}u\|^{2}_{L^{2}(I,{L}_{\Lambda^{q}}^{2})}.$

Similarly, for $s,k\in{\mathbb{Z}}_{+}$, we define the space
$B^{k,2s,s}_{q,\mathrm{for}}(X_{T})$ to consist of all differential forms

$f\in C(I,{H}_{\Lambda^{q}}^{2s+k})\cap L^{2}(I,{H}_{\Lambda^{q}}^{2s+k+1})$

with the property that

$\nabla^{m}_{q}\partial_{t}^{j}f\in C(I,{H}_{\Lambda^{q}}^{k+2s-m-2j})\cap
L^{2}(I,{H}_{\Lambda^{q}}^{k+2s-m-2j+1})$

for all $m+2j\leq 2s.$ We endow the space $B^{k,2s,s}_{q,\mathrm{for}}(X_{T})$
with the natural norm

$\|f\|^{2}_{B^{k,2s,s}_{q,\mathrm{for}}}:=\sum_{m+2j\leq 2s\atop 0\leq l\leq
k}\|\nabla^{l}_{q}\nabla^{m}_{q}\partial_{t}^{j}f\|^{2}_{C(I,{L}_{\Lambda^{q}}^{2})}+\|\nabla^{l+1}_{q}\nabla^{m}_{q}\partial_{t}^{j}f\|^{2}_{L^{2}(I,{L}_{\Lambda^{q}}^{2})}.$

Lastly, the space for the differential form $p$ we denote by
$B^{k+1,2s,s}_{q-1,\mathrm{pre}}(X_{T})$. This space consists of all forms $p$
from the space $C(I,H_{\Lambda^{q-1}}^{2s+k+1})\cap
L^{2}(I,H_{\Lambda^{q-1}}^{2s+k+2})$ such that $d_{q-1}p\in
B^{k,2s,s}_{q,\mathrm{for}}(X_{T})$, $d_{q-2}^{*}p=0$ and for all
$h\in\mathcal{H}^{q-1}$

$(p,h)_{L^{2}_{\Lambda^{q-1}}}=0.$ (1.6)

It is a Banach space with the norm

$\|p\|_{B^{k+1,2s,s}_{q-1,\mathrm{pre}}}=\|d_{q-1}p\|_{B^{k,2s,s}_{q,\mathrm{for}}}.$

Define now for suitable forms $v$ and $w$ of degree $q$ a bi-differential
operator

$\mathbf{B}_{q}(w,v)=M_{q,1}(d_{q}w,v)+M_{q,1}(d_{q}v,w)+d_{q-1}\big{(}M_{q,2}(w,v)+M_{q,2}(v,w)\big{)},$
(1.7)

with the operators $M_{q,1}$ and $M_{q,2}$ satisfying

$|M_{q,1}(u,v)|\leq c_{q,1}|u|\,|v|,\quad|M_{q,2}(u,v)|\leq
c_{q,2}|u|\,|v|\mbox{ on }X$ (1.8)

with some positive constants $c_{i,j}$. Following theorem allows us to see the
correctness of operators in this spaces.

######  Theorem 1.2.

Suppose that $s\in\mathbb{N}$, $k\in{\mathbb{Z}}_{+}$ and
$2s+k>\frac{n}{2}-1$. Then the mappings

$\begin{array}[]{rrcl}\nabla^{m}_{q}:&B^{k,2(s-1),s-1}_{{q},\mathrm{for}}(X_{T})&\to&B^{k-m,2(s-1),s-1}_{q,\mathrm{for}}(X_{T}),\
m\leq k\\\\[5.69046pt]
\Delta_{q}:&B^{k,2s,s}_{q,\mathrm{vel}}(X_{T})&\to&B^{k,2(s-1),s-1}_{q,\mathrm{for}}(X_{T}),\\\\[5.69046pt]
\partial_{t}:&B^{k,2s,s}_{q,\mathrm{vel}}(X_{T})&\to&B^{k,2(s-1),s-1}_{q,\mathrm{for}}(X_{T}),\\\
\end{array}$

are continuous. Besides, if $w,v\in
B^{k+2,2(s-1),s-1}_{{i},\mathrm{vel}}(X_{T})$ then the mappings

$\begin{array}[]{rrcl}\mathbf{B}_{q}(w,\cdot):&B^{k+2,2(s-1),s-1}_{q,\mathrm{vel}}(X_{T})&\to&B^{k,2(s-1),s-1}_{q,\mathrm{for}}(X_{T}),\\\
\mathbf{B}_{q}(w,\cdot):&B^{k,2s,s}_{q,\mathrm{vel}}(X_{T})&\to&B^{k,2(s-1),s-1}_{q,\mathrm{for}}(X_{T}),\\\
\end{array}$ (1.9)

are continuous, too. In particular, for all $w,v\in
B^{k+2,2(s-1),s-1}_{q,\mathrm{vel}}(X_{T})$ there is positive constant
$c_{s,k}$, independent on $v$ and $w$, such that

$\|\mathbf{B}_{q}(w,v)\|_{B^{k,2(s-1),s-1}_{q,\mathrm{for}}}\leq
c_{s,k}\|w\|_{B^{k+2,2(s-1),s-1}_{q,\mathrm{vel}}}\|v\|_{B^{k+2,2(s-1),s-1}_{q,\mathrm{vel}}}.$
(1.10)

###### Proof.

See, for instance, [30] or [29]. ∎

Let us introduce now the Helmholtz type projection $\mathrm{P}^{q}$ from
$B^{k,2(s-1),s-1}_{q,\mathrm{for}}(X_{T})$ to the kernel of operator
$d_{q}^{*}$.

######  Lemma 1.3.

If $s$, $k\in\mathbb{Z}_{+}$, then for each $q$ the pseudo-differential
operator $\mathrm{P}^{q}=d_{q}^{*}d_{q}\varphi^{q}+\Pi^{q}$ on $X$ induce
continuous map

$\mathrm{P}^{q}:B^{k,2(s-1),s-1}_{q,\mathrm{for}}(X_{T})\to
B^{k,2(s-1),s-1}_{q,\mathrm{vel}}(X_{T}),$ (1.11)

such that

$\mathrm{P}^{q}\circ\mathrm{P}^{q}u=\mathrm{P}^{q}u,\quad(\mathrm{P}^{q}u,v)_{L^{2}_{\Lambda^{q}}(X)}=(u,\mathrm{P}^{q}v)_{L^{2}_{\Lambda^{q}}(X)},\quad(\mathrm{P}^{q}u,(I-\mathrm{P}^{q})u)_{L^{2}_{\Lambda^{q}}(X)}=0$

for all $u,v\in B^{k,2(s-1),s-1}_{q,\mathrm{for}}$.

###### Proof.

See, for instance, [29]. ∎

The following Lemma is just a consequence of Hodge Theorem 1.1.

######  Lemma 1.4.

Let $F\in B^{k,2(s-1),s-1}_{q,\mathrm{for}}(X_{T})$ satisfy
$\mathrm{P}^{q}F=0$ in $X_{T}$. Then there is a unique section $p\in
B^{k+1,2(s-1),s-1}_{q-1,\mathrm{pre}}(X_{T})$ such that (1.6) holds and

$d_{q-1}p=F\mbox{ in }X\times[0,T].$ (1.12)

Now we are ready to go to the main section of this paper.

##  2\. Existence theorem

In order to get existence theorem to the Problem (0.5) we use a projection to
the next step of complex (0.1). Namely, applying an operator
$d_{2}=\mathrm{div}$ to the equation (0.5) we have

$\begin{cases}\partial_{t}\mathrm{div}u-\mu\,\mathrm{div}(\nabla\mathrm{div}\,u)+\mathrm{div}((\mathrm{div}\,u)u)=\mathrm{div}f&\text{in
}X\times(0,T),\\\ \mathrm{div}\,u(x,0)=\mathrm{div}\,u_{0}&\text{in
}X,\end{cases}$ (2.1)

because of $\mathrm{rot}\,u=0$ and $\mathrm{div}\circ\mathrm{rot}\equiv 0$.
Now,

$\mathrm{div}((\mathrm{div}\,u)u)=(\mathrm{div}\,u)^{2}+\Delta u\cdot
u=(\mathrm{div}\,u)^{2}+\nabla\mathrm{div}\,u\cdot u.$

By Theorem 1.1

$u=\varphi^{2}\Delta_{2}u+\Pi^{2}u=\varphi^{2}\nabla\mathrm{div}\,u+\Pi^{2}u.$

Denote

$g=\mathrm{div}\,u,$

then we can rewrite (2.1) by the next way

$\begin{cases}\partial_{t}g-\mu\,\mathrm{div}(\nabla g)+g^{2}+\nabla
g\cdot(\varphi^{2}\nabla g+\Pi^{2}u)=\mathrm{div}f&\text{in }X\times(0,T),\\\
g(x,0)=\mathrm{div}\,u_{0}&\text{in }X.\end{cases}$ (2.2)

######  Theorem 2.1.

Given any pair $(f,u_{0})\in L^{2}(I,(V^{0}_{\Lambda^{2}})^{\prime})\times
V^{1}_{\Lambda^{2}}$. There is time $t_{0}\in(0,T]$ such that for all
$t\in[0,t_{0}]$ there exist a differential form $g\in
C(I,L^{2}_{\Lambda^{3}})\cap L^{2}(I,H_{\Lambda^{3}}^{1})$ with
$\partial_{t}g\in L^{2}(I,(H_{\Lambda^{3}}^{1})^{\prime})$, satisfying

$\left\\{\begin{array}[]{rcl}\displaystyle\frac{d}{dt}(g,v)_{{L}_{{\Lambda^{3}}}^{2}}+\mu(\nabla
g,\nabla v)_{{L}_{{\Lambda^{2}}}^{2}}&=&\langle\mathrm{div}\,f-g^{2}-\nabla
g\cdot(\varphi^{2}\,\nabla g+\Pi^{2}u),v\rangle,\\\
g(\cdot,0)&=&\mathrm{div}\,u_{0}\end{array}\right.$ (2.3)

for all $v\in H_{\Lambda^{3}}^{k}$ with $k\geq 2$.

###### Proof.

Let $\\{u_{m}\\}$ be the sequence of Faedo-Galerkin approximations, namely,

$u_{m}=\sum_{j=1}^{M}c^{(m)}_{j}(t)b_{j}(x),$ (2.4)

then

$g_{m}=\mathrm{div}\,u_{m}=\sum_{j=1}^{M}c^{(m)}_{j}(t)\,\mathrm{div}\,b_{j}(x),$
(2.5)

where the system $\\{b_{j}\\}_{j\in{\mathbb{N}}}$ is a
${L}_{\Lambda^{2}}^{2}(X)$-orthogonal basis in $V^{1}_{\Lambda^{2}}$ and the
functions $u_{m}$ satisfy the following relations

$\frac{d}{dt}(g_{m},\mathrm{div}\,b_{j})_{{L}_{{\Lambda^{3}}}^{2}}+\mu(\nabla
g_{m},\nabla\mathrm{div}\,b_{j})_{{L}_{{\Lambda^{2}}}^{2}}=$ (2.6)
$\langle\mathrm{div}\,f-g_{m}^{2}-\nabla g_{m}\cdot\varphi^{2}\,\nabla
g_{m}-\nabla g_{m}\cdot\Pi^{2}u_{m},\mathrm{div}\,b_{j}\rangle,$
$g_{m}(x,0)=\mathrm{div}\,u_{0,m}(x),$

for all $0\leq j\leq m$ with the initial date $u_{0,m}$ from the linear span
${\mathcal{L}}(\\{b_{j}\\}_{j=1}^{m})$ such that the sequence $\\{u_{0,m}\\}$
converges to $u_{0}$ in $V_{\Lambda^{2}}^{1}$. For instance, as
$\\{u_{0,m}\\}$ we may take the orthogonal projection onto the linear span
${\mathcal{L}}(\\{b_{j}\\}_{j=1}^{m})$.

Multiplying (2.6) by $c_{j}^{(m)}(t)$ and summing by $j$ we have

$(\partial_{t}g_{m},g_{m})_{L_{\Lambda^{3}}^{2}}+\mu(\nabla g_{m},\nabla
g_{m})_{{L}_{{\Lambda^{2}}}^{2}}=\langle\mathrm{div}\,f-g_{m}^{2}-\nabla
g_{m}\cdot\varphi^{2}\,\nabla g_{m}-\nabla
g_{m}\cdot\Pi^{2}u_{m},g_{m}\rangle.$ (2.7)

It follows from Lemma by J.-L. Lions (see, for instance, [27, Ch. III, § 1,
Lemma 1.2]) that

$\frac{d}{dt}\|g_{m}(\cdot,t)\|^{2}_{L_{\Lambda^{3}}^{2}}=2\,\langle\partial_{t}g_{m},g_{m}\rangle.$

Then integrating by $t\in[0,T]$ we see that

$\|g_{m}(\cdot,t)\|^{2}_{{L}_{\Lambda^{3}}^{2}}+2\mu\int_{0}^{t}\|\nabla
g_{m}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt=$ (2.8)
$\|g_{m}(\cdot,0)\|^{2}_{{L}_{\Lambda^{3}}^{2}}+2\int_{0}^{t}\langle\mathrm{div}\,f-g_{m}^{2}-\nabla
g_{m}\cdot\varphi^{2}\,\nabla g_{m}-\nabla g_{m}\cdot\Pi^{2}u_{m},g_{m}\rangle
dt.$

Since $f\in L^{2}(I,L^{2}_{\Lambda^{2}})$ then $\mathrm{div}\,f\in
L^{2}(I,(V^{1}_{\Lambda^{3}})^{\prime})$ and

$2\left|\int_{0}^{t}\langle\mathrm{div}\,f,g_{m}\rangle dt\right|\leq
2\int_{0}^{t}\|\mathrm{div}\,f\|_{(V^{1}_{\Lambda^{3}})^{\prime}}\|g_{m}\|_{V^{1}_{\Lambda^{3}}}dt\leq$
(2.9)
$\frac{4}{\mu}\int_{0}^{t}\|\mathrm{div}\,f\|^{2}_{(V^{1}_{\Lambda^{3}})^{\prime}}dt+\frac{\mu}{4}\int_{0}^{t}\|\nabla
g_{m}\|^{2}_{L^{2}_{\Lambda^{2}}}dt+\frac{\mu}{4}\int_{0}^{t}\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt.$

On the other hand

$2\left|\int_{0}^{t}\langle g_{m}^{2},g_{m}\rangle dt\right|\leq
2\int_{0}^{t}\|g_{m}\|^{3}_{L^{3}_{\Lambda^{3}}}dt.$ (2.10)

Note that in our case $\nabla_{3}=-\nabla$ with $n=3$. Then from the
Gagliardo-Nirenberg inequality (see [20] or [4, Theorem 3.70]) we have

$2\int_{0}^{t}\|g_{m}\|^{3}_{L^{3}_{\Lambda^{3}}}dt\leq$ (2.11)
$c\int_{0}^{t}\left[\Big{(}\|\nabla
g_{m}\|_{L^{2}_{\Lambda^{2}}}+\|g_{m}\|_{L^{2}_{\Lambda^{3}}}\Big{)}^{\frac{1}{2}}\|g_{m}\|^{\frac{1}{2}}_{L^{2}_{\Lambda^{3}}}+\|g_{m}\|_{L^{2}_{\Lambda^{3}}}\right]^{3}dt\leq$
$c_{1}\int_{0}^{t}\left[\|\nabla
g_{m}\|^{\frac{1}{2}}_{L^{2}_{\Lambda^{2}}}\|g_{m}\|^{\frac{1}{2}}_{L^{2}_{\Lambda^{3}}}+\|g_{m}\|_{L^{2}_{\Lambda^{3}}}\right]^{3}dt\leq$
$c_{2}\int_{0}^{t}\left(\|\nabla
g_{m}\|^{\frac{3}{2}}_{L^{2}_{\Lambda^{2}}}\|g_{m}\|^{\frac{3}{2}}_{L^{2}_{\Lambda^{3}}}+\|g_{m}\|^{3}_{L^{2}_{\Lambda^{3}}}\right)dt\leq$
$\frac{\mu}{2}\int_{0}^{t}\|\nabla
g_{m}\|^{2}_{L^{2}_{\Lambda^{2}}}dt+c_{3}\int_{0}^{t}\left(\|g_{m}\|^{3}_{L^{2}_{\Lambda^{3}}}+\|g_{m}\|^{6}_{L^{2}_{\Lambda^{3}}}\right)dt$

with positive constants $c$, $c_{1}$ and $c_{2}$. The last expression is
consequence of standard Young’s inequality. Moreover, there are positive
constants $c$ and $c_{1}$ such that

$\int_{0}^{t}\left(\|g_{m}\|^{3}_{L^{2}_{\Lambda^{3}}}+\|g_{m}\|^{6}_{L^{2}_{\Lambda^{3}}}\right)dt\leq
c\int_{0}^{t}\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}\left(1+\|g_{m}\|_{L^{2}_{\Lambda^{3}}}\right)^{4}dt\leq$
$c_{1}\left(\int_{0}^{t}\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt+\int_{0}^{t}\|g_{m}\|^{6}_{L^{2}_{\Lambda^{3}}}dt\right).$

Then we conclude that

$2\int_{0}^{t}\|g_{m}\|^{3}_{L^{3}_{\Lambda^{3}}}dt\leq\frac{\mu}{2}\int_{0}^{t}\|\nabla
g_{m}\|^{2}_{L^{2}_{\Lambda^{2}}}dt+c\int_{0}^{t}\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt+c\int_{0}^{t}\|g_{m}\|^{6}_{L^{2}_{\Lambda^{3}}}dt$
(2.12)

with some constant $c>0$. Next,

$\int_{0}^{t}\langle\nabla g_{m}\cdot\varphi^{2}\,\nabla g_{m},g_{m}\rangle
dt=\sum_{j=1}^{3}\int_{0}^{t}\int\limits_{X}\partial_{j}g_{m}(\varphi^{2}\partial_{j}g_{m})g_{m}\,dx\,dt=$
$-\sum_{j=1}^{3}\int_{0}^{t}\int\limits_{X}g_{m}(\varphi^{2}\partial_{j}g_{m})\partial_{j}g_{m}\,dx\,dt-\int_{0}^{t}\int\limits_{X}g_{m}^{3}\,dx\,dt,$

because $\varphi^{2}\Delta g_{m}=g_{m}$. It means that

$\int_{0}^{t}\langle\nabla g_{m}\cdot\varphi^{2}\,\nabla g_{m},g_{m}\rangle
dt=-\frac{1}{2}\int_{0}^{t}\int\limits_{X}g_{m}^{3}\,dx\,dt,$

and hence

$2\left|\int_{0}^{t}\langle\nabla g_{m}\cdot\varphi^{2}\,\nabla
g_{m},g_{m}\rangle
dt\right|\leq\int_{0}^{t}\|g_{m}\|^{3}_{L^{3}_{\Lambda^{3}}}dt.$ (2.13)

Finally,

$\int_{0}^{t}\langle\nabla g_{m}\cdot\Pi^{2}u_{m},g_{m}\rangle
dt=\sum_{j=1}^{3}\int_{0}^{t}\int\limits_{X}\partial_{j}g_{m}(\Pi^{2}u_{m}^{j})g_{m}\,dx\,dt=$
$-\sum_{j=1}^{3}\int_{0}^{t}\int\limits_{X}g_{m}(\Pi^{2}u_{m}^{j})\partial_{j}g_{m}\,dx\,dt-\sum_{j=1}^{3}\int_{0}^{t}\int\limits_{X}g^{2}_{m}\partial_{j}(\Pi^{2}u_{m}^{j})\,dx\,dt,$

and then

$\int_{0}^{t}\langle\nabla g_{m}\cdot\Pi^{2}u_{m},g_{m}\rangle dt=0,$ (2.14)

because $\mathrm{div}\,\Pi^{2}u_{m}=0$.

Now, inequalities (2.8) - (2.14) give

$\|g_{m}(\cdot,t)\|^{2}_{L^{2}_{\Lambda^{3}}}+2\mu\int_{0}^{t}\|\nabla
g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt\leq\|g_{m}(\cdot,0)\|^{2}_{L^{2}_{\Lambda^{3}}}+$
(2.15)
$\frac{4}{\mu}\int_{0}^{t}\|\mathrm{div}\,f\|^{2}_{(V^{1}_{\Lambda^{3}})^{\prime}}dt+\mu\int_{0}^{t}\|\nabla
g_{m}\|^{2}_{L^{2}_{\Lambda^{2}}}+\frac{\mu}{4}\int_{0}^{t}\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt+$
$2c\int_{0}^{t}\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt+2c\int_{0}^{t}\|g_{m}\|^{6}_{L^{2}_{\Lambda^{3}}}dt,$

and then

$\|g_{m}(\cdot,t)\|^{2}_{L^{2}_{\Lambda^{3}}}+\mu\int_{0}^{t}\|\nabla
g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt\leq\|g_{m}(\cdot,0)\|^{2}_{{L}^{2}_{\Lambda^{3}}}+$
(2.16)
$\frac{4}{\mu}\|\mathrm{div}\,f\|^{2}_{L^{2}\left(I,{(V^{1}_{\Lambda^{3}})^{\prime}}\right)}+\left(\frac{\mu}{4}+2c\right)\int_{0}^{t}\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}dt+2c\int_{0}^{t}\|g_{m}\|^{6}_{L^{2}_{\Lambda^{3}}}dt.$

It follows from the Gronwall-Perov’s Lemma (see, for instance [18, p. 360])
that there exist a time $t_{0}\in(0,T]$ and a positive constant $C_{t_{0}}$
such that

$\|g_{m}(\cdot,t)\|^{2}_{L^{2}_{\Lambda^{3}}}\leq C_{t_{0}}$ (2.17)

for all $t\in[0,t_{0}]$. Then the sequence $g_{m}$ is bounded in
$L^{\infty}(I_{t_{0}},L^{2}_{\Lambda^{3}})$, where $I_{t_{0}}=[0,t_{0}]$.
Moreover it follows from (2.16) and (2.17) that $\|\nabla
g_{m}(\cdot,t)\|^{2}_{L^{2}(I_{t_{0}},L^{2}_{\Lambda^{3}})}$ is bounded too.
It means that there is a subsequence that converges weakly-$*$ in
$L^{\infty}(I_{t_{0}},L^{2}_{\Lambda^{3}})$ and weakly in
$L^{2}(I_{t_{0}},H^{1}_{\Lambda^{3}})$ to some $g\in
L^{\infty}(I_{t_{0}},L^{2}_{\Lambda^{3}})\cap
L^{2}(I_{t_{0}},H^{1}_{\Lambda^{3}})$. We use the same designation $g_{m}$ for
such a subsequence. Then the standard argument show (see, for instance, [15],
[27] or [13]) that we can to pass to the limit in (2.6) with respect to
$m\to\infty$ and to conclude that the element $g$ satisfies (2.3).

∎

Let us now return to the Problem (0.5). Denoting again $g=\mathrm{div}\,u$ and
multiplying (0.5) scalar in $L^{2}_{\Lambda^{2}}$ by differential form $v\in
V^{k}_{\Lambda^{2}}$ we get

$\left\\{\begin{array}[]{rcl}\displaystyle\frac{d}{dt}(u,v)_{{L}_{{\Lambda^{2}}}^{2}}+\mu(g,\mathrm{div}\,v)_{{L}_{{\Lambda^{3}}}^{2}}&=&\langle
f-gu,v\rangle,\\\ u(x,0)&=&u_{0}.\end{array}\right.$ (2.18)

######  Theorem 2.2.

Let $g\in C(I,L^{2}_{\Lambda^{3}})\cap L^{2}(I,H_{\Lambda^{3}}^{1})$ be the
solution of (2.3) for given any pair $(f,u_{0})\in
L^{2}(I,V^{0}_{\Lambda^{2}})\times V^{1}_{\Lambda^{2}}$. Then there exist a
unique differential form $u\in C(I,V_{\Lambda^{2}}^{1})\cap
L^{2}(I,V_{\Lambda^{2}}^{2})$ satisfying (2.18) for all $v\in
V_{\Lambda^{3}}^{k}$ with $k\geq 2$.

###### Proof.

Indeed, let $\\{u_{m}\\}$ be the sequence of Faedo-Galerkin approximations
(see (2.4)) such that the sequence $\\{g_{m}\\}=\\{\mathrm{div}\,u_{m}\\}$
converges to $g\in C(I,L^{2}_{\Lambda^{3}})\cap L^{2}(I,H_{\Lambda^{3}}^{1})$.
Substituting $u_{m}$ to (2.18) instead of $u$, $v$ and integrating by
$t\in[0,t_{0}]$ we have

$\|u_{m}(\cdot,t)\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}+2\mu\|g_{m}\|^{2}_{L^{2}(I_{t_{0}},L^{2}_{\Lambda^{3}})}=\|u_{m}(x,0)\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}+2\int_{0}^{t_{0}}\langle
f-g_{m}u_{m},u_{m}\rangle dt.$ (2.19)

As usual, we evaluate by Hölder inequality

$2\left|\int_{0}^{t_{0}}\langle f,u_{m}\rangle
dt\right|\leq\int_{0}^{t_{0}}\|u_{m}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt+\int_{0}^{t_{0}}\|f\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt$

and by Gagliardo-Nirenberg inequality

$2\left|\int_{0}^{t_{0}}\langle g_{m}u_{m},u_{m}\rangle
dt\right|\leq\int_{0}^{t_{0}}\|u_{m}\|^{2}_{{L}_{{\Lambda^{2}}}^{4}}\|g_{m}\|_{{L}_{{\Lambda^{3}}}^{2}}dt\leq$
$c\int_{0}^{t_{0}}\|g_{m}\|_{{L}_{{\Lambda^{3}}}^{2}}\left(\|g_{m}\|^{3/2}_{{L}_{{\Lambda^{3}}}^{2}}\|u_{m}\|^{1/2}_{{L}_{{\Lambda^{2}}}^{2}}+\|u_{m}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}\right)dt\leq$
$2\mu\int_{0}^{t_{0}}\|g_{m}\|^{2}_{{L}_{{\Lambda^{3}}}^{2}}dt+c_{1}\int_{0}^{t_{0}}\|g_{m}\|^{4}_{{L}_{{\Lambda^{3}}}^{2}}\|u_{m}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt+c_{2}\int_{0}^{t_{0}}\|g_{m}\|_{{L}_{{\Lambda^{3}}}^{2}}\|u_{m}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt$

with some positive constants $c$, $c_{1}$ and $c_{2}$. Then

$\|u_{m}(\cdot,t)\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}\leq\|u_{0}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}+\int_{0}^{t_{0}}\|f\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt+c\int_{0}^{t_{0}}\|u_{m}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt$
(2.20)

with constant $c>0$, independent on $m$. It follows from Gronwall’s Lemma that

$\|u_{m}(\cdot,t)\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}\leq C,$ (2.21)

where constant $C$ depends on norms
$\|f\|^{2}_{L^{2}(I_{t_{0}},L^{2}_{\Lambda^{2}})}$,
$\|u_{0}\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}$ and
$\|g\|_{C(I_{t_{0}},L^{2}_{\Lambda^{2}})}$, but not on $m$.

It follows that the sequence $u_{m}$ is bounded in
$L^{\infty}(I_{t_{0}},L^{2}_{\Lambda^{3}})$ and there is a subsequence that
converges weakly-$*$ in $L^{\infty}(I_{t_{0}},L^{2}_{\Lambda^{3}})$ to some
$u\in L^{\infty}(I_{t_{0}},L^{2}_{\Lambda^{3}})$. We again use the same
designation $u_{m}$ for such a subsequence. Under hypothesis of this Theorem
the sequence $g_{m}=\mathrm{div}\,u_{m}$ converges to $g\in
C(I,L^{2}_{\Lambda^{3}})\cap L^{2}(I,H_{\Lambda^{3}}^{1})$, then actually
$u\in C(I,V^{1}_{\Lambda^{3}})\cap L^{2}(I,V_{\Lambda^{3}}^{2})$. Passing to
the limit in (2.19) with respect to $m\to\infty$ we conclude that the element
$u$ satisfies (2.18).

Let now $u^{\prime}$ and $u^{\prime\prime}$ are two solutions of (2.18) such
that $\mathrm{div}\,u^{\prime}=\mathrm{div}\,u^{\prime\prime}=g$. Hence
differential form $u=u^{\prime}-u^{\prime\prime}$ satisfies (2.18) with zero
date $(f,u_{0})=(0,0)$. It follows from (2.20) and Gronwall-Perov’s Lemma that
$\|g(\cdot,t)\|_{L^{2}_{\Lambda^{3}}}=0$, then the Problem (2.18) has unique
solution.

Moreover, if $u_{1}$, $u_{2}$ are two solutions of (2.18), corresponding to
the solutions $g_{1}=\mathrm{div}\,u_{1}$ and $g_{2}=\mathrm{div}\,u_{2}$ of
(2.3), then differential form $u=u_{1}-u_{2}$ satisfies

$\left\\{\begin{array}[]{rcl}\displaystyle\frac{d}{dt}(u,v)_{{L}_{{\Lambda^{2}}}^{2}}+\mu(g,\mathrm{div}\,v)_{{L}_{{\Lambda^{3}}}^{2}}&=&\langle-
gu,v\rangle,\\\ u(x,0)&=&0,\end{array}\right.$ (2.22)

where $g=g_{1}-g_{2}$.

$\|u(\cdot,t)\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}+2\mu\|g\|^{2}_{L^{2}(I_{t_{0}},L^{2}_{\Lambda^{3}})}=-2\int_{0}^{t_{0}}\langle
gu,u\rangle dt.$ (2.23)

Applying the Gagliardo-Nirenberg inequality we have

$2\left|\int_{0}^{t_{0}}\langle gu,u\rangle dt\right|\leq
c_{1}\int_{0}^{t_{0}}\|g\|_{L^{2}_{\Lambda^{3}}}\left(\|\nabla
u\|^{3/4}_{{L}_{{\Lambda^{2}}}^{2}}\|u\|^{1/4}_{{L}_{{\Lambda^{2}}}^{2}}+\|u\|_{{L}_{{\Lambda^{2}}}^{2}}\right)^{2}dt\leq$
$c_{2}\int_{0}^{t_{0}}\left(\|g\|^{5/2}_{{L}_{{\Lambda^{2}}}^{2}}\|u\|^{1/2}_{{L}_{{\Lambda^{2}}}^{2}}+\|u\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}\|g\|_{L^{2}_{\Lambda^{3}}}\right)dt\leq$
$2\mu\|g\|^{2}_{L^{2}(I_{t_{0}},L^{2}_{\Lambda^{3}})}+c_{3}\left(\|g\|^{4}_{C(I_{t_{0}},L^{2}_{\Lambda^{3}})}+\|g\|_{C(I_{t_{0}},L^{2}_{\Lambda^{3}})}\right)\int_{0}^{t_{0}}\|u\|_{{L}_{{\Lambda^{2}}}^{2}}dt$

with positive constants $c_{1}$, $c_{2}$ and $c_{3}$. The last inequality,
(2.23) and Gronwall-Perov’s Lemma yields

$\|u(\cdot,t)\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}\leq 0,$

then $u_{1}=u_{2}$ and the Problem (2.18) has unique solution.

∎

######  Corollary 2.3.

Under hypothesis of the Theorem 2.1, let $g\in C(I,L^{2}_{\Lambda^{3}})\cap
L^{2}(I,H_{\Lambda^{3}}^{1})$ is a solution of (2.3) and $u\in
C(I,V_{\Lambda^{2}}^{1})\cap L^{2}(I,V_{\Lambda^{2}}^{2})$ is a corresponding
to $g$ solution of (2.18). If moreover $g\in C(I,H^{1}_{\Lambda^{3}})\cap
L^{2}(I,H_{\Lambda^{3}}^{1})$, then the solution $g$ is unique.

###### Proof.

Indeed, let $g_{1},g_{2}\in C(I,H^{1}_{\Lambda^{3}})\cap
L^{2}(I,H_{\Lambda^{3}}^{1})$ are two solutions of (2.3) with corresponding
forms $u_{1},u_{2}\in C(I,H^{1}_{\Lambda^{3}})\cap
L^{2}(I,H_{\Lambda^{3}}^{1})$ satisfying (2.18). Hence differential form
$g=g_{1}-g_{2}$ satisfies

$\left\\{\begin{array}[]{rcl}\displaystyle\frac{d}{dt}\|g\|^{2}_{{L}_{{\Lambda^{3}}}^{2}}+\mu\|\nabla
g\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}&=&\left\langle-(g_{1}^{2}-g_{2}^{2})-\left(\nabla
g_{1}\cdot(\varphi^{2}\,\nabla g_{1}+\Pi^{2}u_{1})-\right.\right.\\\
&&\left.\left.\nabla g_{2}\cdot(\varphi^{2}\,\nabla
g_{2}+\Pi^{2}u_{2})\right),v\right\rangle,\\\
g(\cdot,0)&=&0.\end{array}\right.$ (2.24)

Integrating by $t\in I_{t_{0}}$ we get

$\|g(\cdot,t\|^{2}_{{L}_{{\Lambda^{3}}}^{2}}+2\mu\int_{0}^{t_{0}}\|\nabla
g\|^{2}_{{L}_{{\Lambda^{2}}}^{2}}dt\leq 2\int_{0}^{t_{0}}\left|\left\langle
g(g_{1}+g_{2})+\right.\right.$ (2.25) $\left.\left.\left(\nabla
g_{1}\cdot(\varphi^{2}\,\nabla g_{1}+\Pi^{2}u_{1})-\nabla
g_{2}\cdot(\varphi^{2}\,\nabla
g_{2}+\Pi^{2}u_{2})\right),v\right\rangle\right|dt,$

We have to estimate right side of (2.25). First, it follows from Gagliardo-
Nirenberg interpolation inequality that

$2\int_{0}^{t_{0}}\left|\left\langle
g(g_{1}+g_{2}),g\right\rangle\right|dt\leq$ (2.26)
$c\|(g_{1}+g_{2})\|_{C(I_{t_{0}},L^{2}_{\Lambda^{3}})}\int_{0}^{t_{0}}\left(\|\nabla
g\|^{3/2}_{L^{2}_{\Lambda^{3}}}\|g\|^{1/2}_{L^{2}_{\Lambda^{3}}}+\|g\|^{2}_{L^{2}_{\Lambda^{3}}}\right)dt\leq$
$\mu\int_{0}^{t_{0}}\|\nabla
g\|^{2}_{L^{2}_{\Lambda^{3}}}dt+c_{1}\int_{0}^{t_{0}}\|g\|^{2}_{L^{2}_{\Lambda^{3}}}dt$

with positive constants $c$, $c_{1}$. Next,

$2\int_{0}^{t_{0}}\left|\left\langle\nabla g_{1}\cdot\varphi^{2}\,\nabla
g_{1}-\nabla g_{2}\cdot\varphi^{2}\,\nabla g_{2}+\nabla
g_{1}\cdot\varphi^{2}\,\nabla g_{2}-\nabla g_{1}\cdot\varphi^{2}\,\nabla
g_{2},g\right\rangle\right|dt\leq$ (2.27)
$2\int_{0}^{t_{0}}\left|\left\langle\nabla g_{1}\cdot\varphi^{2}\,\nabla
g,g\right\rangle\right|dt+2\int_{0}^{t_{0}}\left|\left\langle\nabla
g\cdot\varphi^{2}\,\nabla g_{2},g\right\rangle\right|dt\leq$
$c_{1}\|g_{1}\|_{C(I_{t_{0}},H^{1}_{\Lambda^{3}})}\int_{0}^{t_{0}}\left(\|\nabla
g\|^{3/4}_{L^{2}_{\Lambda^{3}}}\|g\|^{5/4}_{L^{2}_{\Lambda^{3}}}+\|g\|^{2}_{L^{2}_{\Lambda^{3}}}\right)dt+$
$c_{2}\|g_{1}\|_{C(I_{t_{0}},H^{1}_{\Lambda^{3}})}\int_{0}^{t_{0}}\left(\|\nabla
g\|^{7/4}_{L^{2}_{\Lambda^{3}}}\|g\|^{1/4}_{L^{2}_{\Lambda^{3}}}+\|g\|^{2}_{L^{2}_{\Lambda^{3}}}\right)dt\leq$
$\mu\int_{0}^{t_{0}}\|\nabla
g\|^{2}_{L^{2}_{\Lambda^{3}}}dt+c\int_{0}^{t_{0}}\|g\|^{2}_{L^{2}_{\Lambda^{3}}}dt$

with some positive constants $c$, $c_{1}$ and $c_{2}$. Finally,

$2\int_{0}^{t_{0}}\left|\left\langle\nabla g_{1}\cdot\Pi^{2}u_{1}-\nabla
g_{2}\cdot\Pi^{2}u_{2}+\nabla g_{2}\cdot\Pi^{2}u_{1}-\nabla
g_{2}\cdot\Pi^{2}u_{1},g\right\rangle\right|dt\leq$ (2.28)
$2\int_{0}^{t_{0}}\left|\left\langle\nabla
g\cdot\Pi^{2}u_{1},g\right\rangle\right|dt+2\int_{0}^{t_{0}}\left|\left\langle\nabla
g_{2}\cdot\Pi^{2}(u_{1}-u_{2}),g\right\rangle\right|dt$

The Theorem 2.2 implies that $u_{1}=u_{2}$. On the other hand, integrating by
parts we easily see that

$2\int_{0}^{t_{0}}\left|\left\langle\nabla
g\cdot\Pi^{2}u_{1},g\right\rangle\right|dt=0,$

and then (2.28) equals to zero.

Finally, using by (2.25) - (2.28) we get

$\|g(\cdot,t\|^{2}_{{L}_{{\Lambda^{3}}}^{2}}\leq
c\int_{0}^{t_{0}}\|g\|^{2}_{L^{2}_{\Lambda^{3}}}dt$

with some constant $c>0$. Then it follows from Gronwall-Perov’s Lemma that
$\|g(\cdot,t)\|_{L^{2}_{\Lambda^{3}}}=0$, then the Problem (2.3) has unique
solution. ∎

######  Theorem 2.4.

Let $s\in\mathbb{N}$ and $k\in{\mathbb{Z}}_{+}$ with $k>3/2$. Then for all

$(f,u_{0})\in B_{\Lambda^{2},\mathrm{for}}^{k+1,2(s-1),s-1}(X_{T})\times
V_{\Lambda^{2}}^{2s+k+1}$

there exist a time $T_{k}\in(0,T]$ such that the Problem (2.2) has solution

$g\in B_{\Lambda^{3},\mathrm{for}}^{k,2s,s}(X_{T_{k}}).$

Moreover, solution $g$ is unique, if form $u$ in (2.1) satisfied (2.18).

###### Proof.

First of all, denote by

$\Lambda_{r}=\begin{cases}\Lambda^{3},\quad&r\ \text{is even},\\\
\Lambda^{2},\quad&r\ \text{is odd}.\\\ \end{cases}$

As before, let $g_{m}$ be the Faedo-Galerkin approximations, see (2.5). We
start with the following priory estimates.

######  Lemma 2.5.

Under hypothesis of the Theorem 2.4, if $(f,u_{0})\in
B^{k+1,0,0}_{{{2}},\mathrm{for}}(X_{T})\times V^{k+3}_{\Lambda^{2}}$ with some
$k\in{\mathbb{Z}}_{+}$, then there exist a time $T_{k}\in(0,T]$ such that

$\|\nabla^{k^{\prime}}_{2}g_{m}\|^{2}_{C(I_{T_{k}},{L}^{2}_{\Lambda_{k^{\prime}}})}+\mu\,\|\nabla^{k^{\prime}+1}_{2}g_{m}\|^{2}_{L^{2}(I_{T_{k}},{L}^{2}_{\Lambda_{k^{\prime}+1}})}\leq
C_{k^{\prime}}$ (2.29)

for any $0\leq k^{\prime}\leq k+2$, where $I_{T_{k}}=[0,{T_{k}}]$ and the
constants $C_{k^{\prime}}=C_{k^{\prime}}(\mu,f,u_{0})>0$ depending on
$k^{\prime}$, $\mu$ and the norms
$\|f\|_{B^{k+1,0,0}_{2,\mathrm{for}}(X_{T_{k}})}$,
$\|u_{0}\|_{V^{k+3}_{\Lambda^{2}}}$ but not on $m$.

###### Proof.

Indeed, if $k^{\prime}=0$ then (2.29) follows immediately from (2.16) and
Gronwall-Perov’s Lemma. Now, substituting $g_{m}$ and $\nabla_{3}^{2r}g_{m}$
in (2.2) instead of $g$ and $v$ respectively with some $r\in\mathbb{N}$ and
integrating by $t\in[0,T]$ we get

$\|\nabla_{3}^{r}g_{m}(\cdot,t)\|^{2}_{{L}_{\Lambda_{r}}^{2}}+2\mu\int_{0}^{t}\|\nabla_{3}^{r+1}g_{m}\|^{2}_{{L}_{{\Lambda^{3}}}^{2}}dt=$
(2.30)
$\|\nabla_{3}^{r}g_{m}(\cdot,0)\|^{2}_{{L}_{\Lambda_{r}}^{2}}+2\int_{0}^{t}\langle\mathrm{div}\,f-g_{m}^{2}-\nabla
g_{m}\cdot\varphi^{2}\,\nabla g_{m}-\nabla
g_{m}\cdot\Pi^{2}u_{m},\nabla_{3}^{2r}g_{m}\rangle dt.$

We have to estimate the right side of (2.30). First,

$2\left|\int_{0}^{t}\langle\mathrm{div}\,f,\nabla_{3}^{2r}g_{m}\rangle
dt\right|\leq
2\int_{0}^{t}\|\nabla_{3}^{r-1}\mathrm{div}\,f\|_{{L}^{2}_{\Lambda_{r-1}}}\|\nabla_{3}^{r+1}g_{m}\|_{{L}^{2}_{\Lambda_{r+1}}}dt\leq$
(2.31)
$\frac{4}{\mu}\int_{0}^{t}\|\nabla_{3}^{r-1}\mathrm{div}\,f\|^{2}_{{L}^{2}_{\Lambda_{r-1}}}dt+\frac{\mu}{4}\int_{0}^{t}\|\nabla_{3}^{r+1}g_{m}\|^{2}_{{L}^{2}_{\Lambda_{r+1}}}dt.$

Further,

$2\left|\int_{0}^{t}\langle g_{m}^{2},\nabla_{3}^{2r}g_{m}\rangle
dt\right|\leq
2\int_{0}^{t}\|\nabla_{3}^{r-1}(g_{m}^{2})\|_{L^{2}_{\Lambda_{r-1}}}\|\nabla_{3}^{r+1}g_{m}\|_{L^{2}_{\Lambda_{r+1}}}dt.$
(2.32)

Let $r\geq 2$, using by Hölder and Gagliardo-Nirenberg inequalities we get

$\|\nabla_{3}^{r-1}(g_{m}^{2})\|_{L^{2}_{\Lambda_{r-1}}}\leq\sum_{|\alpha|+|\beta|=r-1}c_{\alpha\beta}\|\partial^{\alpha}g_{m}\|_{L^{4}_{\Lambda^{3}}}\|\partial^{\beta}g_{m}\|_{L^{4}_{\Lambda^{3}}}\leq$
(2.33)
$\sum_{|\alpha|+|\beta|=r-1}c_{\alpha\beta}\left(\left(\|\nabla_{3}^{|\alpha|+1}g_{m}\|_{L^{2}_{\Lambda_{|\alpha|+1}}}+\|\nabla_{3}^{|\alpha|}g_{m}\|_{L^{2}_{\Lambda_{|\alpha|}}}\right)^{3/4}\|\nabla_{3}^{|\alpha|}g_{m}\|_{L^{2}_{\Lambda_{|\alpha|}}}^{1/4}+\right.$
$\left.+\|\nabla_{3}^{|\alpha|}g_{m}\|_{L^{2}_{\Lambda_{|\alpha|}}}\right)\left(\left(\|\nabla_{3}^{|\beta|+1}g_{m}\|_{L^{2}_{\Lambda_{|\beta|+1}}}+\|\nabla_{3}^{|\beta|}g_{m}\|_{L^{2}_{\Lambda_{|\beta|}}}\right)^{3/4}\|\nabla_{3}^{|\beta|}g_{m}\|_{L^{2}_{\Lambda_{|\beta|}}}^{1/4}+\right.$
$\left.+\|\nabla_{3}^{|\beta|}g_{m}\|_{L^{2}_{\Lambda_{|\beta|}}}\right)\leq
c\left(\|g_{m}\|^{2}_{H^{r-1}_{\Lambda^{3}}}+\|g_{m}\|^{5/4}_{H^{r-1}_{\Lambda^{3}}}\|\nabla_{3}^{r}g_{m}\|^{3/4}_{L^{2}_{\Lambda_{r}}}\right)$

with some positive constants $c$ and $c_{\alpha\beta}$. For the exception case
$r=1$ the last inequality take the form

$\|\nabla_{3}(g_{m}^{2})\|_{L^{2}_{\Lambda_{r-1}}}\leq
c\left(\|g_{m}\|^{2}_{L^{2}_{\Lambda^{3}}}+\|g_{m}\|^{1/2}_{L^{2}_{\Lambda^{3}}}\|\nabla_{3}g_{m}\|^{3/2}_{L^{2}_{\Lambda^{2}}}\right)$
(2.34)

because of in (2.33) arise a case when $|\alpha|=|\beta|=r-1=0$. It follows
from (2.32), (2.33) and Young’s inequality that

$2\left|\int_{0}^{t}\langle g_{m}^{2},\nabla_{3}^{2r}g_{m}\rangle
dt\right|\leq\frac{\mu}{4}\int_{0}^{t}\|\nabla_{3}^{r+1}g_{m}\|_{L^{2}_{\Lambda_{r+1}}}^{2}dt+$
(2.35)
$c\|g_{m}\|^{4}_{C(I,H^{r-1}_{\Lambda^{3}})}+c\|g_{m}\|^{5/2}_{C(I,H^{r-1}_{\Lambda^{3}})}\int_{0}^{t}\|\nabla_{3}^{r}g_{m}\|^{3/2}_{L^{2}_{\Lambda_{r}}}dt$

for $r\geq 2$ and

$2\left|\int_{0}^{t}\langle g_{m}^{2},\nabla_{3}^{2}g_{m}\rangle
dt\right|\leq\frac{\mu}{4}\int_{0}^{t}\|\nabla_{3}^{2}g_{m}\|_{L^{2}_{\Lambda^{3}}}^{2}dt+$
(2.36)
$c\|g_{m}\|^{4}_{C(I,L^{2}_{\Lambda^{2}})}+c\|g_{m}\|_{C(I,L^{2}_{\Lambda^{2}})}\int_{0}^{t}\|\nabla_{3}g_{m}\|^{3}_{L^{2}_{\Lambda^{2}}}dt$

for $r=1$ with some constant $c>0$.

Next,

$2\left|\int_{0}^{t}\langle\nabla g_{m}\cdot\varphi^{2}\,\nabla
g_{m},\nabla_{3}^{2r}g_{m}\rangle dt\right|\leq$ (2.37)
$2\int_{0}^{t}\|\nabla_{3}^{r-1}\left(\nabla g_{m}\cdot\varphi^{2}\,\nabla
g_{m}\right)\|_{L^{2}_{\Lambda_{r-1}}}\|\nabla_{3}^{r+1}g_{m}\|_{L^{2}_{\Lambda_{r+1}}}dt.$

Analogous by (2.33) we have

$\|\nabla_{3}^{r-1}\left(\nabla g_{m}\cdot\varphi^{2}\,\nabla
g_{m}\right)\|_{L^{2}_{\Lambda_{r-1}}}\leq$ (2.38)
$c\left(\|g_{m}\|_{H^{r-1}_{\Lambda^{3}}}\|\varphi^{2}g_{m}\|_{H^{r+1}_{\Lambda^{3}}}+\|g_{m}\|^{1/4}_{H^{r-1}_{\Lambda^{3}}}\|\varphi^{2}g_{m}\|_{H^{r+1}_{\Lambda^{3}}}\|\nabla_{3}^{r}g_{m}\|^{3/4}_{L^{2}_{\Lambda_{r}}}+\right.$
$\left.\|\nabla_{3}^{r}g_{m}\|^{1/4}_{L^{2}_{\Lambda_{r}}}\|\varphi^{2}g_{m}\|_{H^{r+1}_{\Lambda^{3}}}\|\nabla_{3}^{r+1}g_{m}\|^{3/4}_{L^{2}_{\Lambda_{r}}}\right)$

with $r\in\mathbb{N}$ and some constant $c>0$. Theorem 1.1 imply that
$\|\varphi^{2}g_{m}\|_{H^{r+1}_{\Lambda^{3}}}\leq
c\|g_{m}\|_{H^{r-1}_{\Lambda^{3}}}$ with some positive constant $c$, then

$2\left|\int_{0}^{t}\langle\nabla g_{m}\cdot\varphi^{2}\,\nabla
g_{m},\nabla_{3}^{2r}g_{m}\rangle
dt\right|\leq\frac{\mu}{4}\int_{0}^{t}\|\nabla_{3}^{r+1}g_{m}\|_{L^{2}_{\Lambda_{r+1}}}^{2}dt+$
(2.39)
$c\|g_{m}\|^{4}_{C(I,H^{r-1}_{\Lambda^{3}})}+c\|g_{m}\|^{5/2}_{C(I,H^{r-1}_{\Lambda^{3}})}\int_{0}^{t}\|\nabla_{3}^{r}g_{m}\|^{3/2}_{L^{2}_{\Lambda_{r}}}dt+$
$c\|g_{m}\|^{10}_{C(I,H^{r-1}_{\Lambda^{3}})}\int_{0}^{t}\|\nabla_{3}^{r}g_{m}\|^{2}_{L^{2}_{\Lambda_{r}}}dt$

with $c>0$.

Finally,

$2\left|\int_{0}^{t}\langle\nabla
g_{m}\Pi^{2}u_{m},\nabla_{3}^{2r}g_{m}\rangle dt\right|\leq$ (2.40)
$2\int_{0}^{t}\|\nabla_{3}^{r+1}g_{m}\|_{L^{2}_{\Lambda_{r+1}}}\|\nabla_{3}^{r-1}\left(\nabla
g_{m}\cdot\Pi^{2}u_{m}\right)\|_{L^{2}_{\Lambda_{r-1}}}dt,$

and we have again

$\|\nabla_{3}^{r-1}\left(\nabla
g_{m}\cdot\Pi^{2}u_{m}\right)\|_{L^{2}_{\Lambda_{r-1}}}\leq$ (2.41)
$c\left(\|g_{m}\|_{H^{r-1}_{\Lambda^{3}}}\|\Pi^{2}u_{m}\|_{H^{r}_{\Lambda^{2}}}+\|g_{m}\|^{1/4}_{H^{r-1}_{\Lambda^{3}}}\|\Pi^{2}u_{m}\|_{H^{r}_{\Lambda^{2}}}\|\nabla_{3}^{r}g_{m}\|^{3/4}_{L^{2}_{\Lambda_{r}}}+\right.$
$\left.\|\nabla_{3}^{r}g_{m}\|^{1/4}_{L^{2}_{\Lambda_{r}}}\|\Pi^{2}u_{m}\|_{H^{r}_{\Lambda^{3}}}\|\nabla_{3}^{r+1}g_{m}\|^{3/4}_{L^{2}_{\Lambda_{r}}}\right)$

with positive constant $c$. Operator $\Pi^{2}$ is bounded in
$L^{2}_{\Lambda^{2}}$ by the Hodge Theorem 1.1. On the other hand Theorem 2.2
yields that the sequence $\\{u_{m}\\}$ is bounded in $L^{2}_{\Lambda^{2}}$
(see (2.21)), then $\|\Pi^{2}u_{m}\|_{H^{r}_{\Lambda^{2}}}\leq
c\|g_{m}\|_{H^{r-1}_{\Lambda^{3}}}$ and we get

$2\left|\int_{0}^{t}\langle\nabla
g_{m}\Pi^{2}u_{m},\nabla_{3}^{2r}g_{m}\rangle
dt\right|\leq\frac{\mu}{4}\int_{0}^{t}\|\nabla_{3}^{r+1}g_{m}\|_{L^{2}_{\Lambda_{r+1}}}^{2}dt+$
(2.42)
$c\|g_{m}\|^{4}_{C(I,H^{r-1}_{\Lambda^{3}})}+c\|g_{m}\|^{5/2}_{C(I,H^{r-1}_{\Lambda^{3}})}\int_{0}^{t}\|\nabla_{3}^{r}g_{m}\|^{3/2}_{L^{2}_{\Lambda_{r}}}dt+$
$c\|g_{m}\|^{10}_{C(I,H^{r-1}_{\Lambda^{3}})}\int_{0}^{t}\|\nabla_{3}^{r}g_{m}\|^{2}_{L^{2}_{\Lambda_{r}}}dt$

with $c>0$.

It follows from (2.30) - (2.42) and Gronwall-Perov’s Lemma that if
$(f,u_{0})\in B^{k+1,0,0}_{{{2}},\mathrm{for}}(X_{T})\times
V^{k+3}_{\Lambda^{2}}$ and the norm $\|g_{m}\|_{C(I,H^{r-1}_{\Lambda^{3}})}$
is bounded for some $r\in\mathbb{N}$, $r\leq k+2$, then there exist a time
$t_{r}\in(0,t_{0}]$ and a positive constant $C_{r}$, depending on the norms
$\|f\|_{B^{r+1,0,0}_{2,\mathrm{for}}(X_{T_{k}})}$ and
$\|u_{0}\|_{V^{r+3}_{\Lambda^{2}}}$, such that

$\|\nabla_{3}^{r}g_{m}(\cdot,t)\|^{2}_{{L}_{\Lambda_{r}}^{2}}+\mu\int_{0}^{t_{r}}\|\nabla_{3}^{r+1}g_{m}\|^{2}_{{L}_{{\Lambda^{3}}}^{2}}dt\leq
C_{r}(\mu,f,u_{0}).$ (2.43)

Using by (2.43) consistently for $r=1,\dots,k+2$ we get family of times
$t_{r}$. Denote $T_{k}=\min\limits_{r\leq k+2}t_{r}$, then (2.43) yields that
for any $k\in{\mathbb{Z}}_{+}$ there exist a time $T_{k}$ such that (2.5) is
fulfilled. ∎

Theorem 2.1 imply that there exist a solution $g\in
C(I,L^{2}_{\Lambda^{3}})\cap L^{2}(I,H_{\Lambda^{3}}^{1})$ of (2.3). On the
other hand, it follows from Lemma 2.5 that for each $(f,u_{0})\in
B_{\Lambda^{2},\mathrm{for}}^{k+1,2(s-1),s-1}(X_{T})\times
V_{\Lambda^{2}}^{2s+k+1}$ there exist a time $T_{k}\in(0,T]$ and a subsequence
$\\{g_{m^{\prime}}=\mathrm{div}\,u_{m^{\prime}}\\}$ such that
$\\{g_{m^{\prime}}\\}$ converges weakly in
$L^{2}(I_{T_{k}},{L}^{2}_{\Lambda^{3}})$ and ∗-weakly in
$L^{\infty}(I_{T_{k}},{H}^{k+2}_{\Lambda^{3}})\cap
L^{2}(I,{H}^{k+3}_{\Lambda^{3}})$ to an element $g$, then $g\in
B_{\Lambda^{3},\mathrm{for}}^{k,2s,s}(X_{T_{k}})$. Moreover, the uniqueness of
$g$ immediately follows from Corollary 2.3.

∎

######  Theorem 2.6.

Let $s\in\mathbb{N}$ and $k\in{\mathbb{Z}}_{+}$ with $k\geq 2$. Then for all

$(f,u_{0})\in B_{\Lambda^{2},\mathrm{for}}^{k+1,2(s-1),s-1}(X_{T})\times
V_{\Lambda^{2}}^{2s+k+1}$

there exist a time $T^{*}\in(0,T]$ such that the Problem (0.5) has unique
solution

$(u,p)\in B_{\Lambda^{2},\mathrm{vel}}^{k+1,2s,s}(X_{T_{k}})\times
B_{\Lambda^{2},\mathrm{pre}}^{k+2,2(s-1),s-1}(X_{T_{k}}).$

###### Proof.

Indeed, apply the projection $P^{2}$ (see Lemma 1.3 above) to the equation
(0.5) we have

$\begin{cases}\partial_{t}u+\mu\Delta_{2}u+P^{2}N^{2}(u)=P^{2}f&\text{in
}X\times(0,T),\\\ u(x,0)=u_{0}&\text{in }X,\end{cases}$ (2.44)

then the form $p$ actually has to satisfy the equation

$\mathrm{rot}\,p=(I-P^{2})(f-N^{2}(u))\quad\text{in }X\times(0,T).$ (2.45)

Multiplying (2.44) by $v\in V_{\Lambda^{3}}^{k}$ we get the Problem (2.18),
then the existence and regularity of solution $u$ follows immediately from the
Theorems 2.2 and 2.4. On the other hand, it follows from Lemma 1.4 that there
exist unique differential form $p\in
B_{\Lambda^{2},\mathrm{pre}}^{k+2,2(s-1),s-1}(X_{T_{k}})$, satisfying (2.45).

∎

The work was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics ”BASIS”.

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