# Existence, uniqueness, and long-time behavior of linearized field
dislocation dynamics

Amit Acharya Marshall Slemrod Department of Civil & Environmental Engineering,
and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA
15213, email<EMAIL_ADDRESS>of Mathematics, University of
Wisconsin, Madison, WI 53706, email<EMAIL_ADDRESS>
###### Abstract

This paper examines a system of partial differential equations describing
dislocation dynamics in a crystalline solid. In particular we consider
dynamics linearized about a state of zero stress and use linear semigroup
theory to establish existence, uniqueness, and time asymptotic behavior of the
linear system.

Keywords: dislocations, small deformations, linear contraction semi-groups.

AMS subject classifications: Primary: 35Q74, 37L05, 74E15: Secondary: 74H20,
74H25, 74H40, 74B20

This paper is dedicated to our friend Costas Dafermos on the occasion of his
80th birthday.

##  1 Introduction

Dislocations are topological defects in elasticity and their dynamics and
interaction are of significant scientific and technological interest. They
were introduced in the theory of elasticity by Volterra in 1907 and continue
to be vigorously studied to this day, with a complete dynamical theory within
continuum mechanics of unrestricted material and geometric nonlinearity taking
shape in recent years (see, e.g., [3, 4, 11, 2, 1, 1] with detailed
bibliographic threads to earlier works in the overall subject). Here, we give
a mathematical analysis of the governing nonlinear pde system of dislocation
mechanics, linearized about a state of vanishing stress.

This paper has two additional sections after this Introduction. Section 2
presents the ‘small deformation’ model of nonlinear dislocation mechanics, see
e.g. [3]. We then linearize this system about a state of vanishing stress to
obtain a linear system of partial differential equations which describe the
evolution of a class of deformations of an elastic solid. In Section 3 we use
the Lumer-Phillips Theorem [10] to establish existence and uniqueness of
solutions to our linear evolutionary system. Specifically, the Lumer-Phillips
Theorem yields existence of a $C^{0}$ semigroup of contractions on a Hilbert
space $H$. Furthermore, this evolution is dissipative and allows us to
determine the time-asymptotic behavior via an application of a theorem of S.
Foguel [8]. Here, time asymptotic behavior is only given with respect to weak
convergence in the Hilbert space $H$ as lack of a compact resolvent for the
infinitesimal generator of our $C^{0}$ semigroup appears to preclude
application of a theorem of Dafermos and Slemrod [7] for strong convergence.
Finally, two examples are presented which show that the limit dynamics
predicted by the linearized dissipative theory allows for both a non-trivial
static solution as well as an oscillating motion.

##  2 Linearized dislocation mechanics

We consider the following ‘small deformation’ model of nonlinear dislocation
mechanics:

$\displaystyle\partial_{j}v_{i}-\partial_{t}u_{ij}-J_{ij}$ $\displaystyle=0$
(1) $\displaystyle-\rho\,\partial_{t}v_{i}+\partial_{j}T_{ij}$
$\displaystyle=0$ $\displaystyle\mathbb{C}_{ijkl}u_{kl}$
$\displaystyle=T_{ij}$ $\displaystyle e_{jrs}\alpha_{ir}V_{s}$
$\displaystyle=J_{ij}$ $\displaystyle e_{rjk}\,\partial_{j}u_{ik}$
$\displaystyle=\alpha_{ir}$ $\displaystyle e_{smn}T_{pm}\alpha_{pn}$
$\displaystyle=V_{s}$

where $v$ is the material particle velocity, $u=\varepsilon+\omega$ is the
elastic distortion, $\varepsilon_{ij}=\frac{1}{2}(u_{ij}+u_{ji})$ is the
elastic strain, $\omega_{ij}=\frac{1}{2}(u_{ij}-u_{ji})$ is the elastic
rotation, $J$ is the dislocation flux or plastic strain rate, $\rho$ is the
mass density, $T$ is the stress, $\mathbb{C}$ is the tensor of elastic moduli
with major and minor symmetries, $\alpha$ is the dislocation density tensor,
and $V$ is the dislocation velocity vector.

For any $\alpha$ field that is the curl of a skew symmetric tensor field, and
with $0$-traction boundary conditions (i.e., $T_{ij}n_{j}=0$, where $n_{j}$ is
the outward unit normal field on the body), the unique solution (in a simply
connected body) for stress
$T_{ij}=\mathbb{C}_{ijkl}u_{kl}=\mathbb{C}_{ijkl}\varepsilon_{kl}$
corresponding to the system

$\displaystyle\partial_{j}(\mathbb{C}_{ijkl}u_{kl})$ $\displaystyle=0$
$\displaystyle e_{rjk}\partial_{j}u_{ik}$ $\displaystyle=\alpha_{ir}$

is $T=0$ with $\varepsilon=0$. Thus, for the entire class of such dislocation
density distributions, $J=0,V=0$, and the class
$\left\\{(v_{i},u_{ij})|v_{i}=0,u_{ij}=\frac{1}{2}(u_{ij}-u_{ij}),\,\mbox{i.e.,
skew}\right\\}$ constitutes steady state solutions of (1). Motivated by the
above, we linearize (1) about any state $(v_{i},u_{ij})$ with
$(v_{i}=0,\varepsilon_{ij}=0)$, with the field $\alpha$ defined from (1)5,
regardless of the topology of the body. Such 0-stress steady state solutions
represent (within the confines of the ‘small deformation’ theory (1) being
considered) non-trivial elastic distortion and dislocation density fields
(e.g. arbitrarily fine distributions of dislocation walls) - unlike in
classical linear elasticity.

Henceforth, we will assume $T=0,J=0$, and $V=0$.

We denote the dependent fields of the linearized system with overhead dots
‘$\dot{(\ )}$’ to obtain the system

$\displaystyle\partial_{j}\dot{v}_{i}-\partial_{j}\dot{u}_{ij}-\dot{J}_{ij}$
$\displaystyle=0$ (2)
$\displaystyle-\rho\partial_{t}\dot{v}_{i}+\partial_{j}(\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl})$
$\displaystyle=0$ $\displaystyle\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}$
$\displaystyle=\dot{T}_{ij}$ $\displaystyle e_{jrs}\alpha_{ir}\dot{V}_{s}$
$\displaystyle=\dot{J}_{ij}$ $\displaystyle e_{rjk}\partial_{j}\dot{u}_{ik}$
$\displaystyle=\dot{\alpha}_{ir}$ $\displaystyle
e_{smn}\dot{T}_{pm}\alpha_{pn}$ $\displaystyle=\dot{V}_{s}.$

We then have the following energy equality for the linearized system:

$\displaystyle\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}\left(\partial_{j}\dot{v}_{i}-\partial_{j}\dot{u}_{ij}-\dot{J}_{ij}\right)=0,$
(3)
$\displaystyle-\rho\dot{v}_{i}\partial_{t}\dot{v}_{i}+\dot{v}_{i}\partial_{j}(\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl})=0$
$\displaystyle\Longrightarrow\partial_{j}\left(\dot{v}_{i}\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}\right)-\partial_{t}\left(\frac{1}{2}\rho\dot{v}_{i}\dot{v}_{i}+\frac{1}{2}\mathbb{C}_{ijkl}\dot{\varepsilon}_{ij}\dot{\varepsilon}_{kl}\right)=\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}\dot{J}_{ij}=\dot{T}_{ij}\dot{J}_{ij}.$

From (2)4,6 we have

$\displaystyle\dot{V}_{s}$ $\displaystyle=e_{sjr}\dot{T}_{ij}\alpha_{ir}$ (4)
$\displaystyle\dot{T}_{ij}\dot{J}_{ij}$
$\displaystyle=\dot{T}_{ij}e_{sjr}\alpha_{ir}\dot{V}_{s}=\dot{V}_{s}\dot{V}_{s}\geq
0,$

which yields energy dissipation and will be crucial to our analysis.
Introducing the notation

$\displaystyle\dot{J}_{ij}$
$\displaystyle=e_{sjr}\alpha_{ir}e_{smn}\dot{T}_{pm}\alpha_{pn}=(e_{sjr}\alpha_{ir}e_{smn}\alpha_{pn})\mathbb{C}_{pmkl}\dot{u}_{kl}=\mathbb{B}_{ijkl}\dot{u}_{kl}=\mathbb{B}_{ijkl}\dot{\varepsilon}_{kl}$
(5)

(2) becomes

$\displaystyle\partial_{t}\dot{u}_{ij}-\partial_{j}\dot{v}_{i}+\mathbb{B}_{ijkl}\dot{u}_{kl}$
$\displaystyle=0$ (6)
$\displaystyle\rho\partial_{t}\dot{v}_{i}-\partial_{j}(\mathbb{C}_{ijkl}\dot{u}_{kl})$
$\displaystyle=0,$

with ‘nonnegative dissipation’ (4) in force.

##  3 Analysis of linearized dislocation mechanics

For the analysis of system (2) it is useful to decompose it into symmetric and
skew parts as follows

$\displaystyle\partial_{t}\dot{\varepsilon}_{ij}$
$\displaystyle=\frac{1}{2}(\partial_{j}\dot{v}_{i}+\partial_{i}\dot{v}_{j})-\dot{J}_{(ij)}$
(7) $\displaystyle\rho\partial_{t}\dot{v}_{i}$
$\displaystyle=\partial_{j}\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}$
$\displaystyle\partial_{t}\dot{\omega}_{ij}$
$\displaystyle=\frac{1}{2}(\partial_{j}\dot{v}_{i}-\partial_{i}\dot{v}_{j})-\dot{J}_{[ij]}.$

Here, and in the following, we use the index notation
$A_{\ldots(ij)\ldots}=\frac{1}{2}(A_{\ldots ij\dots}+A_{\ldots ji\ldots})$ and
$A_{\ldots[ij]\dots}=\frac{1}{2}(A_{\ldots ij\ldots}-A_{\ldots ji\ldots})$.

We note from (5) that $\mathbb{B}_{ijkl}=\mathbb{B}_{ij(kl)}$, a property
inherited from the minor symmetries of the elastic moduli $\mathbb{C}$, and

$\dot{J}_{ij}=\mathbb{B}_{ij(kl)}\dot{\varepsilon}_{kl}.$

###  3.1 Existence and Uniqueness

Let $\Omega$ be an open, bounded, connected set in $\mathbb{R}^{3}$. By
$\partial\Omega$ we denote the boundary of $\Omega$ which is $C^{1}$-smooth
and by $n_{j},j=1,2,3$ the unit normal on $\partial\Omega$ pointing towards
the exterior. We denote by $\partial\Omega_{1},\partial\Omega_{2}$ fixed
subsets of $\partial\Omega$, and
$\partial\Omega_{1}^{c}:=\partial\Omega-\overline{\partial\Omega_{1}}$,
$\partial\Omega_{2}^{c}:=\partial\Omega-\overline{\partial\Omega_{2}}$.

Introduce the Sobolev spaces

$\displaystyle L^{2}(\Omega)$
$\displaystyle=\left\\{v:\Omega\to\mathbb{R}^{3}\,|\,\int_{\Omega}dx\,v_{i}v_{i}<\infty\right\\}$
$\displaystyle H^{0}(\Omega)$
$\displaystyle=\left\\{\varepsilon:\Omega\to\mathbb{R}^{3\times
3}_{sym}\,|\,\int_{\Omega}dx\,\varepsilon_{ij}\varepsilon_{ij}<\infty\right\\}\
\mathbb{R}^{3\times 3}_{sym}\ \mbox{is the set of $3\times 3$ symmetric
matrices}$ $\displaystyle H^{1}(\Omega)$
$\displaystyle=\left\\{v:\Omega\to\mathbb{R}^{3}\,|\,\partial_{j}v_{i}\,\mbox{in
the sense of
distributions},\,\int_{\Omega}dx\,\partial_{j}v_{i}\partial_{j}v_{i}+v_{i}v_{i}<\infty\right\\},$

and we insist on

$\displaystyle\mathbb{C}_{ijkl}$
$\displaystyle=\mathbb{C}_{jikl}=\mathbb{C}_{ijlk}=\mathbb{C}_{klij}$
$\displaystyle\mathbb{C}_{ijkl}\varepsilon_{ij}\varepsilon_{kl}$
$\displaystyle\geq
a_{0}\,\varepsilon_{ij}\varepsilon_{kl},\forall\varepsilon,a_{0}>0\ \mbox{ a
constant}.$

Let

$U=(\dot{\varepsilon}_{ij},\dot{v}_{i})$

so that (2) is written as

$\displaystyle\partial_{t}U$ $\displaystyle=A\,U$ (8) $\displaystyle A\,U$
$\displaystyle=\begin{bmatrix}\partial_{(j}\dot{v}_{i)}-\mathbb{B}_{(ij)(kl)}\dot{\varepsilon}_{kl}\\\
\\\ \frac{1}{\rho}\partial_{j}(\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl})\\\
\end{bmatrix}.$

We wish to establish that $A$ is the infinitesimal generator of a $C^{0}$
semi-group of contractions $S(t)$ on the Hilbert space

$H=H^{0}(\Omega)\times L^{2}(\Omega)$

with inner product $\langle\ ,\ \rangle_{H}$ defined as

$\displaystyle\langle(\varepsilon_{ij},v_{i}),(\bar{\varepsilon}_{ij},\bar{v}_{i})\rangle_{H}$
$\displaystyle=\langle\mathbb{C}_{ijkl}\varepsilon_{kl},\bar{\varepsilon}_{ij}\rangle_{H^{0}(\Omega)}+\langle\rho
v_{i},\bar{v}_{i}\rangle_{L^{2}(\Omega)}$ (9) $\displaystyle D(A)$
$\displaystyle=\\{(\varepsilon_{ij},v_{i})\in H|\,AU\in
H,v_{i}(x)=0,\mbox{a.e.}\,x\in\partial\Omega_{1},\mathbb{C}_{ijkl}\varepsilon_{kl}n_{j}=0\
a.e.~{}x\in\partial\Omega_{2}\\}$
$\displaystyle=\left\\{(\varepsilon_{ij},v_{i})\in
H|\,v_{i}\in\widehat{H}^{1}(\Omega),\partial_{j}(\mathbb{C}_{ijkl}\varepsilon_{kl})\in
L^{2}(\Omega),\mathbb{C}_{ijkl}\varepsilon_{kl}n_{j}=0\
a.e.~{}x\in\partial\Omega_{2}\right\\}$ $\displaystyle\widehat{H}^{1}(\Omega)$
$\displaystyle:=\left\\{v_{i}\in H^{1}(\Omega)|v_{i}=0\
a.e~{}x\in\partial\Omega_{1}\right\\}.$

We note that $\overline{D(A)}=H$. Furthermore, we have $\langle
U,AU\rangle_{H}=\langle AU,U\rangle_{H}\leq 0$ for all $U\in D(A)$, i.e., $A$
is _dissipative_. To see this, we compute

$\displaystyle\langle U,AU\rangle_{H}$
$\displaystyle=\int_{\Omega}dx\,\mathbb{C}_{klij}\dot{\varepsilon}_{ij}(\partial_{(l}\dot{v}_{k)}-\mathbb{B}_{(kl)(pm)}\dot{\varepsilon}_{pm})+\dot{v}_{i}\partial_{j}(\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl})$
$\displaystyle=\int_{\partial\Omega}da\,\dot{v}_{i}\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}n_{j}-\int_{\Omega}dx\,\mathbb{C}_{klij}\dot{\varepsilon}_{ij}\mathbb{B}_{(kl)(pm)}\dot{\varepsilon}_{pm}$
$\displaystyle=-\int_{\Omega}dx\,\dot{T}_{kl}\dot{J}_{kl}=-\int_{\Omega}dx\,\dot{V}_{s}\dot{V}_{s}\leq
0.$

By the Lumer-Phillips Theorem [10, p. 250] $A$ will then be the infinitesimal
generator of a $C^{0}$ semigroup of contractions on $H$ if the range condition

$\mbox{range}(\lambda I-A)=H,$

for some $\lambda\in\mathbb{R}$, i.e., we need solvability of the system

$\displaystyle\lambda\dot{\varepsilon}_{ij}-\partial_{(j}\dot{v}_{i)}+\mathbb{B}_{(ij)(kl)}\dot{\varepsilon}_{kl}$
$\displaystyle=f_{ij}$ (10)
$\displaystyle\lambda\rho\dot{v}_{i}-\partial_{j}(\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl})$
$\displaystyle=g_{i},$

for given $(f,g)\in H$.

We ensure this by writing for $\lambda$ sufficiently large

$\dot{\varepsilon}_{ij}=\mathbb{G}_{ijkl}(\partial_{l}\dot{v}_{k}+f_{kl})$

where $\mathbb{G}_{ijkl}$ is the inverse of
$(\lambda\delta_{ik}\delta_{jl}+\mathbb{B}_{(ij)(kl)})$, or more succintly,
with $\mathbb{D}_{ijkl}:=\mathbb{B}_{(ij)(kl)}$,
$\mathbb{G}=\mathcal{R}(-\lambda,\mathbb{D})$ where
$\mathcal{R}(\lambda,\mathbb{D})$ denotes the resolvent of $\mathbb{D}$.

The usual expansion of the resolvent (see, e.g., [10, p. 211]) gives

$\lambda\mathcal{R}(-\lambda,\mathbb{D})=\mathbb{I}+\mathbb{D}\lambda^{-1}\sum_{j=0}^{\infty}(-\lambda)^{-j}\mathbb{D}^{j}.$

Hence for $r=\frac{\|\mathbb{D}\|}{\lambda}<1$, where $\|\mathbb{D}\|$ denotes
the operator norm of $\mathbb{B}$, the series converges and we define

$\displaystyle\mathbb{K}$
$\displaystyle=\mathbb{D}\lambda^{-1}\sum_{j=0}^{\infty}(-\lambda)^{-j}\mathbb{D}^{j}$
(11) $\displaystyle\|\mathbb{K}\|$ $\displaystyle=r(1-r)$

to get

$\lambda\mathbb{G}=\mathbb{I}+\mathbb{K}\Longrightarrow\mathbb{G}=\lambda^{-1}\mathbb{I}+\lambda^{-1}\mathbb{K},$

where $\|\lambda^{-1}\mathbb{K}\|\leq\mbox{const}\,\lambda^{-2}$. Hence
$\mathbb{G}_{ijkl}=\lambda^{-1}\delta_{ik}\delta_{jl}+\lambda^{-1}\mathbb{K}_{ijkl}$,
and

$\dot{\varepsilon}_{ij}=\lambda^{-1}(\partial_{(j}\dot{v}_{i)}+f_{ij})+\lambda^{-1}\mathbb{K}_{ijkl}(\partial_{(l}\dot{v}_{k)}+f_{kl})$
(12)

Now substitute (12) into (10) to obtain

$\displaystyle\lambda^{2}v_{i}-\partial_{j}(\mathbb{C}_{ijkl}(\partial_{l}\dot{v}_{k}+f_{kl}))-\partial_{j}(\mathbb{C}_{ijkl}\mathbb{K}_{klmn}(\partial_{n}\dot{v}_{m}+f_{mn}))=\lambda
g_{i}$ (13)
$\displaystyle\Longrightarrow\lambda^{2}v_{i}-\partial_{j}(\mathbb{C}_{ijkl}\partial_{l}\dot{v}_{k})-\partial_{j}(\mathbb{C}_{ijkl}\mathbb{K}_{klmn}\partial_{n}\dot{v}_{m})=\partial_{j}(\mathbb{C}_{ijkl}f_{kl})+\partial_{j}(\mathbb{C}_{ijkl}\mathbb{K}_{klmn}f_{mn})+\lambda
g_{i}$

Write (13) in weak form

$\displaystyle\int_{\Omega}dx\,\lambda^{2}\overline{v}_{i}\dot{v}_{i}+\partial_{j}\overline{v}_{i}\mathbb{C}_{ijkl}\partial_{l}\dot{v}_{k}$
$\displaystyle+\partial_{j}\overline{v}_{i}(\mathbb{C}_{ijkl}\mathbb{K}_{klmn})\partial_{n}\dot{v}_{m}$
(14)
$\displaystyle=\int_{\Omega}dx\,-\partial_{j}\overline{v}_{i}\mathbb{C}_{ijkl}f_{kl}-\partial_{j}\overline{v}_{i}(\mathbb{C}_{ijkl}\mathbb{K}_{klmn})f_{mn}+\lambda\overline{v}_{i}g_{i}$
$\displaystyle\qquad\forall\ \overline{v}_{i}\in\widehat{H}^{1}(\Omega).$

The left hand side of (14) defines a bilinear form
$\mathcal{B}(v,\overline{v})$ which satisfies

  1. 1.

$\mathcal{B}(v,\overline{v})\leq\mbox{const}\|v\|_{\widehat{H}^{1}(\Omega)}\|\overline{v}\|_{\widehat{H}^{1}(\Omega)}$
(boundedness)

  2. 2.

$\mathcal{B}(v,v)\geq\mbox{const}\|v\|^{2}_{\widehat{H}^{1}(\Omega)}$ for
$\lambda$ sufficiently large by (11) (coercivity).

Here we have assumed that $\mathbb{C}_{ijkl},\alpha_{ij}$ are continuous on
$\Omega$ and used Korn’s inequality (see, e.g, Ciarlet [5]).

As the right hand side of (14) defines a bounded linear functional on
$\widehat{H}^{1}(\Omega)$, the Lax-Milgram Theorem [10, p. 92] yields a unique
solution $\dot{v}_{i}\in\widehat{H}^{1}(\Omega)$ to the weak form (14).
Substituting this $\dot{v}_{i}$ into (12) we have defined
$\dot{\varepsilon}_{ij}$ where the pair

$(\dot{\varepsilon}_{ij},\dot{v}_{i})\in
H^{0}(\Omega)\times\widehat{H}^{1}(\Omega)$

solves (10). From (10) we see that
$\partial_{j}(\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl})\in L^{2}(\Omega)$.
Furthermore, retracing our steps from (14) back to (10) tells us that

$\int_{\partial\Omega_{2}}\overline{v}_{i}\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}\,n_{j}\,da=0$

and hence $(\dot{\varepsilon}_{ij},\dot{v}_{i})\in D(A)$. Thus, the range
condition of the Lumer-Phillips theorem is satisfied and we we have

######  Theorem 3.1.

$A$ is the infinitesimal generator of a $C^{0}$ semi-group of contractions
$S(t)$ on $H$and for $U_{0}\in H$, $S(t)U_{0}$ provides the unique weak
solution to (7)1,2.

######  Corollary 3.1.1.

$(S(t)U_{0},\dot{\omega}_{ij})$ yields the unique weak solution of (7), where
$\dot{\omega}_{ij}$ is defined via the definite integral in $t$ of the right
hand side of (7)3 with initial condition $\dot{\omega}_{ij}(0)\in
H^{0}(\Omega)$ at $t=0$.

###### Proof.

Substitute $S(t)U_{0}=(\dot{\omega}_{ij}(t),\dot{v}_{i}(t))$ into the well-
defined right hand side of (7)3 yielding a continuous in time right hand side.
∎

###  3.2 Time-asymptotic behavior

Since our system (7)1,2 is linear, we are able to exploit the special
properties of linear contraction semigroup on Hilbert space. As it is not
obvious that $(\lambda I-A)^{-1}:H\to H$ is compact, we cannot immediately
apply the results of Dafermos and Slemrod [7] on strong decay in $H$. Instead,
we follow an argument of O’Brien [9] which in turn was based on the following
theorem of S. Foguel [8]:

######  Theorem 3.2.

For a $C^{0}$ semigroup of contractions $S(t)$ on a Hilbert space $H$ define
the isometric subspace $H_{u}$ of $H$ as

$H_{u}=\left\\{U_{1}\in H\ |\
\|S(t)U_{1}\|_{H}=\|U_{1}\|_{H}=\left\|S^{*}(t)U_{1}\right\|_{H},t\geq
0\right\\}.$

Then $H_{u}$ is a closed invariant subspace and $S(t)$ forms a $C^{0}$ semi-
group of unitary operators on $H_{u}$, and for $W_{0}$ orthogonal to $H_{u}$

$S(t)W_{0}=S^{*}(t)W_{0}\rightharpoonup 0$ (15)

as $t\to\infty$, where $\rightharpoonup$ denotes weak convergence in $H$.

We apply Foguel’s theorem as follows:

Decompose

$\displaystyle A=A_{1}+A_{2}$
$\displaystyle(A_{1}U)_{ij}=\begin{bmatrix}\partial_{(j}\dot{v}_{i)}\\\ \\\
\frac{1}{\rho}\partial_{j}(\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl})\end{bmatrix}\qquad(A_{2}U)_{ij}=\begin{bmatrix}-\mathbb{B}_{(ij)(kl)}\dot{\varepsilon}_{kl}\\\
\\\ 0\end{bmatrix}$

with $D(A_{1})=D(A)$. Recall from the dissipation inequality (4) and (9) that

$\langle U,AU\rangle=\langle
U,A_{2}U\rangle=-\|\dot{V}\|^{2}_{L^{2}(\Omega)}.$ (16)

Now decompose the initial data $U_{0}\in H$ as $U_{0}=U_{1}+W_{0}$ with
$U_{1}\in H_{u}$ and $W_{0}\perp H_{u}$. By Foguel’s theorem

$\left(S(t)U_{0}-S(t)U_{1}\right)\rightharpoonup 0\qquad\mbox{as}\
t\to\infty.$ (17)

Hence time-asymptotic behavior of $S(t)U_{0}$ is determined by time-asymptotic
behavior of $S(t)U_{1}$. Furthermore, since $H_{u}$ is invariant under $S(t)$
and $D(A)$ is dense in $H$, then $H_{u}\cap D(A)$ is dense in $H_{u}$. Take
$U_{1}\in H_{u}\cap D(A)$ so that $S(t)U_{1}\in D(A)$. We know from Foguel’s
theorem that

$\|S(t)U_{1}\|^{2}_{H}=\|U_{1}\|^{2}_{H}\qquad\Longrightarrow\qquad\frac{d}{dt}\|S(t)U_{1}\|^{2}_{H}=0.$
(18)

On the other hand, the dissipation inequality (16) gives us

$\frac{d}{dt}\|S(t)U_{1}\|^{2}_{H}=\langle
S(t)U_{1},A_{2}S(t)U_{1}\rangle_{H}=-\|\dot{V}\|^{2}_{L^{2}(\Omega)},$ (19)

where

$\dot{V}_{s}=e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\dot{\varepsilon}_{kl}=:\mathsf{D}_{skl}\dot{\varepsilon}_{kl}=:(\mathsf{D}U)_{s},$
(20)

i.e.,

$\frac{d}{dt}\|S(t)U_{1}\|^{2}_{H}=-\|\mathsf{D}S(t)U_{1}\|^{2}_{L^{2}(\Omega)}\qquad\forall\
t\in\mathbb{R}^{+}.$ (21)

Comparison of (18) and (21) gives

$\mathsf{D}S(t)U_{1}=0\qquad\forall\ t\geq 0.$ (22)

But if $\mathsf{D}S(t)U_{1}=0$ for $t\geq 0$ then $A_{2}S(t)U_{1}=0$ for
$t\geq 0$ and our system reduces to

$\frac{d}{dt}U=A_{1}U,$

$U=S(t)U_{1}$. But $A_{1}$ generates the unitary semigroup $S_{1}(t)$, i.e.,
$S_{1}(t)U_{1}$ is the unique weak solution to the system of linear elasticity
where $\dot{v}_{i}$ now plays the role of the displacement. Hence

$S(t)U_{1}=S_{1}(t)U_{1},\qquad t\geq 0,$ (23)

and from (22)

$DS_{1}(t)U_{1}=0,\qquad t\geq 0.$ (24)

By density of $H_{u}\cap D(A)$ in $H_{u}$ (23) and (24) hold for all $U_{1}\in
H_{u}$. Hence we have proven

######  Theorem 3.3.

$(S(t)U_{0}-S_{1}(t)U_{1})\rightharpoonup 0$ as $t\to\infty$ when
$\mathsf{D}S_{1}(t)U_{1}=0$ for $t\geq 0$, i.e. weak solutions of our system
(7)1,2 weakly approach weak solutions of the equations of linear elasticity
which are constrained by
$e_{sjr}\alpha_{ir}\dot{T}_{ij}=e_{sjr}\alpha_{ir}\mathbb{C}_{ijkl}\dot{\varepsilon}_{kl}=0$.
The limit system is given by

$\displaystyle\partial_{t}\overline{\varepsilon}_{ij}$
$\displaystyle=\frac{1}{2}(\partial_{j}\overline{v}_{i}+\partial_{i}\overline{v}_{j})$
(25) $\displaystyle\rho\,\partial_{t}\overline{v}_{i}$
$\displaystyle=\partial_{j}(\mathbb{C}_{ijkl}\overline{\varepsilon}_{kl})$
$\displaystyle 0$
$\displaystyle=e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\overline{\varepsilon}_{kl}\
\mbox{in}\ \Omega,$ $\displaystyle\mbox{and}\quad\overline{v}_{i}$
$\displaystyle=0\quad\mbox{on}\ \partial\Omega_{1}$
$\displaystyle\mathbb{C}_{ijkl}\overline{\varepsilon}_{kl}n_{j}$
$\displaystyle=0\quad\mbox{on}\ \partial\Omega_{2}.$

######  Corollary 3.3.1.

$\partial_{t}\dot{\omega}_{ij}(t)-\frac{1}{2}(\partial_{j}\overline{v}_{i}-\partial_{i}\overline{v}_{j})(t)\to
0$ as $t\to\infty$ in the sense of distributions, where
$\overline{\epsilon}_{ij},\overline{v}_{i}$ satisfies (25).

###### Proof.

We know from Theorem 3.3 that

$(\dot{\varepsilon}(t),\dot{v}(t))-(\overline{\varepsilon}(t),\overline{v}(t))\rightharpoonup
0$

in $H$ as $t\to\infty$, where $(\overline{\varepsilon},\overline{v})$ satisfy
(25). Hence for all $w\in C^{\infty}_{0}(\Omega)$ we have

$\displaystyle\langle\partial_{[j}\dot{v}_{i]}(t),w\rangle_{L^{2}(\Omega)}-\langle\partial_{[j}\overline{v}_{i]}(t),w\rangle_{L^{2}(\Omega)}$
$\displaystyle\to 0\qquad\mbox{as}\ t\to\infty$
$\displaystyle\langle\dot{J}_{[ij]}(t),w\rangle_{L^{2}(\Omega)}$
$\displaystyle\to 0\qquad\mbox{as}\ t\to\infty.$

That is

$\displaystyle\partial_{[j}\dot{v}_{i]}(t)-\partial_{[j}\overline{v}_{i]}(t)$
$\displaystyle\to 0$ $\displaystyle\dot{J}_{[ij]}(t)$ $\displaystyle\to
0\qquad\mbox{as}\ t\to\infty$

in the sense of distributions.

By Corollary 3.1.1, (7)3 is satisfied in the sense of distributions for
$\dot{\omega}(0)\in H^{0}(\Omega)$. Hence for (7)3 we have

$\partial_{t}\dot{\omega}_{ij}(t)-\frac{1}{2}\partial_{[j}\overline{v}_{i]}(t)\to
0\qquad\mbox{as}\ t\to\infty$

in the sense of distributions, where
$(\overline{\varepsilon}_{ij},\overline{v}_{i})$ satisfy (25). ∎

###  3.3 Limit system analysis

We now provide an analysis of the limit system (25) following the argument of
Dafermos [6].

First assume that the data for the limit system is in
$D(A^{2}_{1})=\\{(\overline{\varepsilon}_{ij},\overline{v}_{i})\in
H|A_{1}(\overline{\varepsilon}_{ij},\overline{v}_{i})\in H\\}$ so that we may
differentiate w.r.t $t$ to obtain
$\rho\partial_{tt}\overline{v}_{i}=\partial_{j}(\mathbb{C}_{ijkl}\partial_{t}\overline{\varepsilon}_{kl})$,
i.e.,

$\rho\partial_{tt}\overline{v}_{i}=\partial_{j}(\mathbb{C}_{ijkl}\partial_{l}\overline{v}_{k})\qquad\mbox{in}\
\Omega,t\in\mathbb{R}^{+},$ (26)

with $\overline{v}_{i}=0$ on $\partial\Omega_{1}$,
$(\mathbb{C}_{ijkl}\partial_{l}\overline{v}_{k})n_{j}=0$ on
$\partial\Omega_{2}$ (the restriction on the data is only for temporary
convenience since our results for the general data in $H$ follow by density of
$D(A^{2}_{1})$ in $H$).

Express solutions of (26) as

$\overline{v}_{i}=\sum_{p}c^{(p)}e^{i\lambda^{(p)}t}\phi^{(p)}_{i},\qquad
c^{(p)}\ \mbox{constants}.$ (27)

Then, (26) implies

$\sum_{p}-\rho{\lambda^{(p)}}^{2}c^{(p)}e^{i\lambda^{(p)}t}\phi^{(p)}_{i}=\sum_{p}c^{(p)}e^{i\lambda^{(p)}t}\partial_{j}\left(\mathbb{C}_{ijkl}\partial_{l}\phi_{k}^{(p)}\right)$

with $\phi^{(p)}_{l}=0$ on $\partial\Omega_{1}$,
$(\mathbb{C}_{ijkl}\partial_{l}\phi_{k})n_{j}=0$ on $\partial\Omega_{2}$.

Hence, for eigenfunctions $\phi^{(p)}_{i}$ satisfying

$\partial_{j}\left(\mathbb{C}_{ijkl}\partial_{l}\phi^{(p)}_{k}\right)=-\rho{\lambda^{(p)}}^{2}\phi_{i}^{(p)}\
\mbox{in}\ \Omega\ \mbox{with}\ \phi^{(p)}_{i}=0\ \mbox{on}\
\partial\Omega_{1},\ (\mathbb{C}_{ijkl}\partial_{l}\phi_{k})n_{j}=0\
\mbox{on}\ \partial\Omega_{2},$ (28)

(27) yields a solution of (25)1,2.

The eigenfunctions exist and are orthogonal in $H^{0}(\Omega)$. Furthermore,
we may assume they are normalized so that
$\lVert\phi\rVert_{H^{0}(\Omega)}=1$. Thus, $c^{(p)}$ may be determined by
initial data $\overline{v}_{i}(0)=\sum_{p}c^{(p)}\phi^{(p)}_{i}$, i.e.
$c^{(p)}=\left(\overline{v}(0),\phi^{(p)}\right)_{H^{0}(\Omega)}$.

Note that since (25)3 holds for all $t\in\mathbb{R}$, we may differentiate
w.r.t $t$ to obtain
$e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\partial_{t}\overline{\varepsilon}_{kl}=0$
for all $t\in\mathbb{R}$, and hence by (25)

$e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\partial_{l}\overline{v}_{k}=0.$ (29)

Substitute (27) into (29) to obtain

$e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\sum_{q}c^{(q)}e^{i\lambda^{(q)}t}\partial_{l}\phi^{(q)}_{k}=0\
\mbox{in}\ \Omega\ \mbox{for all}\ t\in\mathbb{R}^{+}.$ (30)

We assume there are no repeated eigenvalues for our domain $\Omega$. Then the
lhs of (30) defines an almost periodic function and we must have

$e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\,c^{(q)}\,\partial_{l}\phi^{(q)}_{k}=0\qquad\mbox{no
sum on}\ q.$ (31)

Thus there are two cases to consider:

Case 1:

$e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\,\partial_{l}\phi^{(q)}_{k}=0\qquad\mbox{for
some}\ q.$

In this case $c_{q}\neq 0$ for this choice of $q$ and the limit solution (25)
may contain non-trivial oscillations.

Case 2:

$e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\,\partial_{l}\phi^{(q)}_{k}\neq
0\qquad\mbox{for all}\ q.$

In this case $c_{q}=0$ for all $q$ and we have

$\overline{v_{i}}=0\ \mbox{in}\ \Omega\ \mbox{for all}\ t$

and hence by (25)

$\partial_{t}\overline{\varepsilon}_{ij}=0\ \mbox{in}\ \Omega\ \mbox{for all}\
t$

i.e. $\overline{\varepsilon}_{ij}$ is a time-independent equilibrium solution
of (25) with

$\displaystyle\partial_{j}(\mathbb{C}_{ijkl}\overline{\varepsilon}_{kl})$
$\displaystyle=0$ (32) $\displaystyle
e_{smn}\alpha_{pn}\mathbb{C}_{pmkl}\overline{\varepsilon}_{kl}$
$\displaystyle=0$

in $\Omega$. We provide two explicit examples corresponding to Cases 1 and 2.

Examples

Recall that we need the base $\alpha_{ij}$ state to be the curl of a skew-
symmetric tensor field to have $T_{ij}=0$, i.e. a stress-free base state.
Thus, $\alpha_{ij}$ must be of the form

$\alpha_{ij}=e_{jmk}\partial_{m}(e_{ikr}\psi_{r})=-(\partial_{r}\psi_{r})\delta_{ij}+\partial_{i}\psi_{j}.$

We choose the vector field $\psi$ of the form

$\psi_{k}(x_{1},x_{2},x_{3})=A(x_{1},x_{2},x_{3})\delta_{k1}$

which yields

$\alpha_{pn}=-(\partial_{1}A)\delta_{pn}+\partial_{p}A\delta_{n1}.$

This distribution corresponds to a ‘crossed-grid’ of screw dislocations
parallel to the coordinate axes superposed on an edge dislocation distribution
with line direction along $x_{1}$ and Burgers vector in the $x_{2}-x_{3}$
plane.

Furthermore, choose the ansatz of uniaxial stress fields

$\overline{T}_{pm}=\sigma(x_{1},t)\delta_{p1}\delta_{m1}$ (33)

to note that the r.h.s. of (25)3 becomes

$e_{smn}\left((\partial_{1}A)\delta_{pn}+(\partial_{p}A)\delta_{n1}\right)\sigma\delta_{p1}\delta_{m1}=0.$

Thus, by our chosen ansatz for $\alpha_{ij}$ and the (time-dependent, for now)
limit stress field $\overline{T}_{ij}$, (25)3 is satisfied.

Physically, the screw dislocation distributions along $x_{2}$ and $x_{3}$
directions (i.e., $\alpha_{22},\alpha_{23}$) do not have a Burgers vector
favorably aligned to the uniaxial stress field along the $x_{1}$ direction to
produce a non-trivial Peach-Koehler driving force. The edge and screw
distributions with line direction in the $x_{1}$ direction do not lie parallel
to a plane on which the uniaxial stress field ansatz
$\sigma\delta_{p1}\delta_{m1}=\overline{T}_{pm}$ can produce a traction, and
hence these see no driving force for motions as well (the edge distributions
with Burgers vector in the $x_{2}-x_{3}$ plane share both of these ‘null’
driving force attributes).

####  3.3.1 Static solution of limit system

Consider an isotropic elastic material with
$\mathbb{C}_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$,
where $\lambda,\mu$ are the Lamé parameters. Consider a time-independent,
uniaxial stress field of the form (33) with $\sigma(x_{1})$. Now define
$\overline{\varepsilon}_{ij}$ through

$\overline{\varepsilon}_{ij}(x_{1},x_{2},x_{3},t):=\frac{1}{2\mu}\left(\sigma(x_{1})\delta_{i1}\delta_{j1}-\frac{\lambda}{3\lambda+2\mu}\sigma(x_{1})\delta_{ij}\right)$

(which can also be expressed as
$\mathbb{S}_{ijkl}\left(\sigma(x_{1})\delta_{k1}\delta_{l1}\right)$, where
$\mathbb{S}_{ijkl}=\frac{1}{4\mu}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})-\frac{\lambda}{2\mu(3\lambda+2\mu)}\delta_{ij}\delta_{kl}$
is the elastic compliance tensor, the inverse of the elastic stiffness,
$\mathbb{C}_{ijkl}$, on the space of symmetric second-order tensors). Hence,
$\mathbb{C}_{ijkl}\,\overline{\varepsilon}_{kl}(x_{1},x_{2},x_{3},t)=\sigma(x_{1})\delta_{i1}\delta_{j1}$
in this case. Making the choice $\sigma(x_{1})=\sigma_{0}$, where
$\sigma_{0}\in\mathbb{R}$ is an arbitrary constant, and setting

$\displaystyle\overline{\varepsilon}_{ij}$
$\displaystyle=\mathbb{S}_{ijkl}\sigma_{0}\delta_{k1}\delta_{l1}$
$\displaystyle\overline{v}_{i}$ $\displaystyle=0,$

we have a non-trivial static solution to the limit system (25) with
$\partial\Omega_{2}=\phi$.

####  3.3.2 Oscillating solution of the limit system

We consider a time-dependent, uniaxial stress field ansatz corresponding to
isotropic linear elasticity with shear modulus $\mu$ and the other Lamé
parameter set to $0$ in terms of the function $U(x_{1},t)$ given by

$\overline{T}_{ij}(x_{1},t)=\mu(\partial_{j}U(x_{1},t)\delta_{i1}+\partial_{i}U(x_{1},t)\delta_{j1}).$

Comparison with (33) yields the definition

$\sigma(x_{1},t):=2\mu\,\partial_{1}U(x_{1},t).$

We consider a cylinder with a uniform rectangular cross-section normal to the
$x_{1}$ direction as the body $\Omega$, with free (i.e.traction-free) bounding
surfaces perpendicular to $x_{2}$ and $x_{3}$. The lateral surfaces of the
cylinder correspond to $\partial\Omega_{2}$, and $\partial\Omega_{1}$
comprises the boundaries of the cylinder perpendicular to its axis, $x_{1}$.
Let the $x_{1}$-coordinate of the planar surfaces comprising
$\partial\Omega_{1}$ be $x_{l}$ and $x_{r}$. We now set

$\displaystyle\overline{v}_{i}(x_{1},x_{2},x_{3},t)$
$\displaystyle=\partial_{t}U(x_{1},t)\delta_{1i}$ (34)
$\displaystyle\overline{\varepsilon}_{ij}(x_{1},x_{2},x_{3},t)$
$\displaystyle=\,\partial_{1}U(x_{1},t)\delta_{1i}\delta_{1j},$

and require that

$\displaystyle\rho\,\partial_{t}\partial_{t}U$
$\displaystyle=2\mu\,\partial_{1}\partial_{1}U\quad\mbox{in}\ \Omega$
$\displaystyle U(x_{1},t)$
$\displaystyle=0\Rightarrow\partial_{t}U(x_{1},t)=0\quad\mbox{for}\
x_{1}=x_{l},x_{r}.$

Non-trivial solutions to this wave equation exist and define oscillating
solutions to the limit system (25) through the definitions (34) (with
$\mathbb{C}_{ijkl}\varepsilon_{kl}=\overline{T}_{ij}(x_{1},t)$).

In summary, our limit system analysis yields the following consequence of
Theorem 3.3:

######  Corollary 3.3.2.

Assume (28) has no repeated eigenvalues. For initial data $U_{0}\in H$ the
limit system solution $S(t)U_{1}$ must lie in either Case 1: non-trivial
oscillations or Case 2: a non-trivial solution $\overline{\varepsilon}$ to
(32) representing a static solution.

## Acknowledgment

The work of AA was supported by the grant NSF OIA-DMR #2021019.

## References

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  * ZAWB [15] Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak.  A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations.  Journal of the Mechanics and Physics of Solids, 84:145–195, 2015\.


#  blackThe first ground-based detection of the 752 GHz water line in local
ultra-luminous infrared galaxies using APEX-SEPIA

Daysi Quinatoa,1 Chentao Yang,2 Edo Ibar,1 Elizabeth Humphreys,3,4 E-mail:
<EMAIL_ADDRESS>Susanne Aalto,2 Loreto Barcos-Muñoz,5,6 Eduardo
González-Alfonso,7 Violette Impellizzeri,8 Yara Jaffé,9,1 Lijie Liu,10,11
Sergio Martín, 3,4 Axel Weiss,12 Zhi-Yu Zhang13,14
1Instituto de Física y Astronomía, Universidad de Valparaíso, Avda. Gran
Bretaña 1111, Valparaíso, Chile
2Department of Earth and Space Sciences, Chalmers University of Technology,
Onsala Observatory, 439 94 Onsala, Sweden
3Joint ALMA Observatory, Alonso de Córdova 3107 Vitacura, Santiago 763-0355,
Chile
4European Southern Observatory, Alonso de Córdova, 3107, Vitacura, Santiago
763-0355, Chile
5National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA,
22903, USA
6Department of Astronomy, University of Virginia, 530 McCormick Road,
Charlottesville, VA, 22903, USA
7Universidad de Alcalá, Departamento de Física y Matemáticas, Campus
Universitario, 28871 Alcalá de Henares, Madrid, Spain
8Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, The
Netherlands
9Departamento de Física, Universidad Técnica Federico Santa María Casilla
110-V, Valparaíso, Chile
10Cosmic Dawn Center (DAWN), Denmark
11DTU-Space, Elektrovej, Building 328 , 2800, Kgs. Lyngby, Denmark
12Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn,
Germany
13 School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
People’s Republic of China
14 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210023, People’s Republic of China

(Accepted 2023 November 03. Received 2023 October 30; in original form 2023
August 23)

###### Abstract

We report the first ground-based detection of the water line p-H2O
($2_{11}\text{--}2_{02}$) at 752.033 GHz in three $z<0.08$ ultra-luminous
infrared galaxies (ULIRGs): IRAS 06035-7102, IRAS 17207-0014 and IRAS
09022-3615. Using the Atacama Pathfinder EXperiment (APEX), with its Swedish-
ESO PI Instrument for APEX (SEPIA) band-9 receiver, we detect this H2O line
with overall signal-to-noise ratios of 8–10 in all three galaxies. Notably,
this is the first detection of this line in IRAS 06035-7102. Our new APEX-
measured fluxes, between 145 to 705 Jy km s-1, are compared with previous
values taken from Herschel SPIRE FTS. We highlight the great capabilities of
APEX for resolving the H2O line profiles blackwith high spectral resolutions
while also improving by a factor of two the significance of the detection
within moderate integration times. While exploring the correlation between the
p-H2O(211–202) and the total infrared luminosity, our galaxies are found to
follow the trend at the bright end of the local ULIRG’s distribution. The
p-H2O(211–202) line spectra are compared to the mid-$J$ CO black and HCN
spectra, and dust continuum previously observed with ALMA. In the complex
interacting system IRAS 09022-3615, blackthe profile of the water emission
line is offset in velocity with respect to the ALMA CO($J=4-3$) emission. For
IRAS 17207-0014 and IRAS 06035-7102, the profiles between the water line and
the CO lines are spectroscopically aligned. This pilot study demonstrates the
feasibility of directly conducting ground-based high-frequency observations of
this key water line, opening the possibility of detailed follow-up campaigns
to tackle its nature.

###### keywords:

ISM: molecules – galaxies: ISM – infrared: galaxies

††pubyear: 2023††pagerange: blackThe first ground-based detection of the 752
GHz water line in local ultra-luminous infrared galaxies using APEX-SEPIA–6

##  1 Introduction

Ultra-Luminous InfraRed Galaxies (ULIRGs; with total infrared luminosity
$L_{\mathrm{IR}}$ between 1012 $L_{\odot}$ and 1013 $L_{\odot}$) are part of
the most extreme galaxy populations in the nearby Universe (Sanders et al.,
2003). They are characterized to be rich in dust and molecular gas (e.g.
Solomon et al., 1997; Genzel et al., 1998; Greve et al., 2005) and black to
have higher gas fractions compared to normal star-forming galaxies (Gao &
Solomon, 2004). ULIRGs are mostly late-stage mergers (e.g. Sanders & Mirabel,
1996) understood as a transitional phase in the evolution of galaxies, passing
through a brief starburst phase triggered by a major interaction that cause
the in-falling of the interstellar medium (ISM) to the innermost regions,
likely triggering nuclear starburst and active galactic nuclei (AGN) but with
complex evolution links between the two (e.g. Farrah et al., 2001; Lonsdale et
al., 2006). After the molecular gas is consumed, they will probably end up as
passive galaxies hosting supermassive black holes in their centres. Therefore,
ULIRGs offer unique insights into our understanding of galaxy evolution. To
describe their nature, it is imperative to understand why they have such
intense, compact star formation and what powers the nuclear star formation
activity. The majority of the radiation produced in their nuclei, whether by
an AGN or by star formation, is absorbed by dust and re-emitted in the far-
infrared (far-IR). Because of the intense obscuration, identifying the
predominant power source of local ULIRGs is challenging. Using emission lines
from blackmolecules such as OH and H2O, whose excitation depend on the far-IR
continuum photons, could be a potential diagnostic to distinguish AGN from
starburst activity (González-Alfonso et al., 2010; van der Werf et al., 2011;
Pensabene et al., 2022; Decarli et al., 2023).

H2O is the third most abundant molecule in the ISM, either in the gas phase in
warm regions or in the solid phase on dust mantles (e.g. van Dishoeck et al.,
2013), and serves as one of the most important coolants of the cold molecular
gas (Neufeld et al., 1995). black In particular, the thermal para-H2O(211–202)
line is emitted at 752.033 GHz rest-frame and has an upper energy level of
$E_{\mathrm{up}}/k_{\mathrm{B}}=136.9$ K. In Galactic star forming regions,
this line is predominantly originated by shocks that trace high density and
temperature gas (e.g. Mottram et al., 2014; van Dishoeck et al., 2021). On the
other hand, surveys of submillimeter (submm) H2O lines with Herschel in local
galaxies revealed that this water line is one of the brightest submm H2O lines
(Yang et al., 2013; Lu et al., 2017). The modelling of water in galaxies,
assuming different phases of the ISM, with different dust temperatures, shows
that this line comes predominantly from the warm phase with
$T_{\mathrm{dust}}\sim 45\text{--}75$ K (González-Alfonso et al., 2014). The
p-H2O ($2_{11}\text{--}2_{02}$) line can be excited via collision in warm
dense conditions with a column density of
$N_{\mathrm{H_{2}O}}\sim(0.5-2)\times 10^{17}$ cm-2, sharing a common spatial
distribution with those regions traced by mid-$J$ CO lines in star-forming
galaxies (Liu et al., 2017; González-Alfonso et al., 2014). Additionally, H2O
lines can be excited radiatively by far-IR photons. This so-called pumping
mechanism of p-H2O ($2_{11}\text{--}2_{02}$) is induced by the 101$\mu$m
continuum, which excites the H2O molecules from the base energy level 111 to
220. This level 220 then cascades down to 211 and later to 202 producing line
emission at 1229, 752, and 988 GHz respectively. Furthermore, the combination
of far-IR 75 $\mu$m absorption ortho-H2O ($3_{21}\text{--}2_{12}$) and the
$3_{21}\text{--}3_{12}$ 1163 GHz emission enhance the radiative excitation for
the submm lines with $E_{\mathrm{up}}/k_{\mathrm{B}}<300$ K, including the 752
GHz one (González-Alfonso et al., 2022). The p-H2O ($2_{11}\text{--}2_{02}$)
far-IR pumping makes H2O lines a prominent tracer of the conditions of the
far-IR field, especially in highly obscured regions of galaxies (e.g.
González-Alfonso et al., 2014). Previous studies have shown that the p-H2O
($2_{11}\text{--}2_{02}$) is well correlated with blackthe total infrared
luminosity in both local and high-redshift galaxies (Omont et al., 2013; Yang
et al., 2013, 2016; Jarugula et al., 2019; Berta et al., 2023). The nearly
linear correlation blackbetween $L_{\mathrm{H{{}_{2}}O}}$ and
$L_{\mathrm{IR}}$ lines suggest that water can be excited to high energy
levels by far infrared pumping and provides another approach for tracing the
far infrared field in star-forming galaxies. The correlation can be a natural
consequence of far-IR pumping H2O excitation (González-Alfonso et al., 2014).

In dusty star-forming galaxies, H2O lines trace both, the properties of the
molecular gas and the dust content, offering a powerful tool to study their
physical nature. This becomes especially important in ULIRGs as they present
ideal conditions for water emission, intense far-IR radiation fields and warm
dense gas content. In fact, the H2O lines can offer unique insights into the
innermost dust-obscured regions with very high opacity, providing essential
information on the ISM physical conditions of the most extreme ULIRGs (e.g.
Falstad et al., 2017; Liu et al., 2017; Yang et al., 2020; González-Alfonso et
al., 2021), where often the p-H2O ($2_{11}\text{--}2_{02}$) line becomes
optically thick (e.g. González-Alfonso et al., 2014).

Although thermal H2O lines provide powerful line diagnostics, their detection
from ground-based observatories is impeded by the presence of water vapour in
the Earth’s atmosphere, which significantly reduces atmospheric transmission.
From space, the H2O lines have been studied in large samples of far-IR bright
galaxies with limited spectral resolutions with Herschel Space Observatory
SPIRE-FTS (e.g. Yang et al., 2013; Pearson et al., 2016), despite a
blackhandful of sources studied with high-spectral resolution with Herschel
HIFI (Liu et al., 2017). blackIt is known that the p-H2O
($2_{11}\text{--}2_{02}$) line is ubiquitous in the spectra of molecular
clouds in the Milky Way and local galaxies, but it has also been found to be a
prominent emission line amongst the water lines detected in high-redshift
infrared bright galaxies, both unlensed (e.g. Gullberg et al., 2016; Riechers
et al., 2017; Casey et al., 2019; Lehnert et al., 2020; Li et al., 2020;
Stanley et al., 2021; Pensabene et al., 2021) and gravitationally amplified
(e.g. Omont et al., 2011; Lis et al., 2011; Bradford et al., 2011; Combes et
al., 2012; Omont et al., 2013; Riechers et al., 2013; Bothwell et al., 2013;
Yang et al., 2016; Jarugula et al., 2019, 2021; Apostolovski et al., 2019;
Yang et al., 2019a, b, 2020; Berta et al., 2021). The similarity found in
profiles between water and mid-$J$ to high-$J$ level CO transitions strongly
suggests that water emission originates from regions of active star formation
(e.g. see Figure 13 from Yang et al. 2017). Exploring the properties of water
lines then becomes a key aspect for characterizing the cosmic star formation
history of the Universe.

It is important to highlight that most of the detected submm water lines from
local galaxies have been made by Herschel (see Weiß et al. 2010; Yang et al.
2013; Liu et al. 2017), nevertheless as this is black no longer available, and
until the next generation of far-infrared space telescopes, exploiting ground-
based submm facilities are the only possibility to explore high-frequencies to
measure water lines with high spectral resolution and, in the case of ALMA,
unprecedented angular resolution.

In this work, we present high-spectral resolution observations of the p-H2O
($2_{11}\text{--}2_{02}$) black in three low-redshift ULIRGs (IRAS 06035-7102,
IRAS 17208-0014 and IRAS 09022-3615) using the single-dish APEX telescope with
the SEPIA receiver in Band-9 (Baryshev et al., 2015; Belitsky et al., 2018).
Throughout this work, we assumed a $\Lambda$CDM cosmology with $\Omega_{\rm
M}=0.3$, $\Omega_{\Lambda}=0.7$ and $H_{0}=75\,{\rm km\,s^{-1}\,Mpc^{-1}}$.

##  2 Data

###  2.1 blackSource selection

blackThe three sources were selected from a sample of ULIRGs observed in water
lines with Herschel and presented in Yang et al. (2013). From this sample, we
selected the sources below declination $+25^{\circ}$ and, in order to ensure a
good atmospheric transmission, we excluded sources whose observed water line
p-H2O ($2_{11}\text{--}2_{02}$) (752.033 GHz rest-frame) frequencies are
higher than 722 GHz. Eight ULIRGs fulfil the selection criteria. Based on
integration time estimates, considering expected line fluxes and atmospheric
transmission, as a pilot project, we chose to target three sources, IRAS
17207-0014, IRAS 06035-7102, and IRAS 09022-361. These three sources maximised
the detection of the water line within a short integration time.

###  2.2 Observations

We observed IRAS 06035-7102 ($z=0.07946$), IRAS 17208-0014 ($z=0.04281$), and
IRAS 09022-3615 ($z=0.05964$) with APEX SEPIA-660 Band 9 receiver (Project:
E-0103.B-0471A-2018, PI: C. Yang) operating at the observing frequency range
of $\sim$ 690–725 GHz. The APEX beam size at this frequency is
$9\aas@@fstack{\prime\prime}5$. Observations were carried out between May
11-12, June 03, 22-24, and August 21-22, 27-29 in 2019. The ON-OFF
observations were performed in the wobbler-switching symmetric mode with an
amplitude of 40 arcsecs and a wobbling rate of 1.5 Hz. The Doppler correction
to account for the motion of the Earth was applied during the observations.
R-Dor and L2-Pup were used for pointing and focus calibration. The focus was
checked at the beginning of each observing session, and pointing was checked
every approximately 1.5 hours. Spectral setup covers 8 GHz per sideband and a
separation of 8 GHz between the upper- and lower-side bands. The observation
was centred on the redshifted line frequency of interest, and we requested a
line peak-to-rms ratio of S/N$>$5 in velocity channels of $\sim$50 km s-1. The
precipitable water vapour (PWV) during the observations varied between 0.4 and
0.9 mm, corresponding to typical atmospheric transmission of 60% and 20%,
respectively. The total integration times ranged between 4 and 15 hrs per
target (Table 1).

###  2.3 Data reduction

The data were reduced using the Grenoble Image and Line Data Analysis Software
(GILDAS)111https://www.iram.fr/IRAMFR/GILDAS/ \- Continuum and Line Analysis
Single-dish Software (CLASS) package. For each target, we collected all the
scans for different observing dates. For each observing date, we checked all
the individual scans and discarded those affected by anomalous noise levels or
weather conditions as noted by the log-file of the observations. For each
scan, we masked a velocity window from $-$500 to $+$500 km s-1 centred at the
expected line observing frequency to fit a second-order polynomial
subtraction, which accounts better for the baseline shapes, to remove the
baseline out of the spectra. Then we combined all of the individual spectra,
using noise-based weighing and aligning them at the frequency of interest. We
have also explored the baseline subtraction using first-order polynomials and
did not find any major differences compared with the second-order polynomial
fits.

Figure 1: Redshifted para-H2O(211–202) at 752.033 GHz emission line for the
three ULIRGs presented in this work. The spectra have been centred according
to the redshift in Table 1. blackThe histograms show the binned APEX spectra,
while the red lines correspond to Gaussian fits. Overplotted are the
unapodized Herschel SPIRE-FTS spectra in blue. Error bars show a comparison
between uncertainties within a bandwidth of 40 GHz, in both spectra after
considering a convolution of the APEX spectra with the SPIRE-FTS instrumental
line spread function. The upper panel shows the atmospheric transmission for
SEPIA 660 at PWV of 0.5 mm at an elevation of 55∘.

To display the spectra, we use a spectral resolution of 50 km s-1 channel
width (see Figure 1). Considering the antenna temperature $T^{*}_{\mathrm{A}}$
corrected for the atmospheric attenuation, the forward efficiency and signal
band gain, we derive flux densities using a conversion factor of 75 $\pm$ 6 Jy
K-1 measured during the observations period with SEPIA-660222https://www.apex-
telescope.org/telescope/efficiency/. The measured noise root mean squared
(RMS), for all three sources, ranges between 77 and 340 mJy beam-1. The final
spectra were exported to Python for the remaining analysis. IRAS 17208-0014
became the brightest source, so less integration time was needed to reach the
requested signal-to-noise ratios (see Table 1).

###  2.4 SED photometry

Previous analyses indicate that the p-H2O ($2_{11}\text{--}2_{02}$) is likely
to be dominated by far-IR pumping excitation by the photons at 101 $\mu$m
(e.g. González-Alfonso et al., 2022). If this is the case for our sources, we
would expect a tight correlation between H2O emission and far-IR dust
continuum. black In order to further explore this, we re-analyzed the Herschel
Photodetector Array Camera and Spectrometer (PACS) 100 $\mu$m emission of IRAS
17208-0014 (Obslist:1342241375, 1342241376) and IRAS 09022-3615
(Obslist:1342233593, 1342233594). These were retrieved from the Herschel
Public Archive (program OT1, PI: D. Sanders). Each galaxy was observed in
cross-scanning mode at 20 arcsec s-1 and unimap projected maps were
considered. Since PACS simultaneously observes at 100 $\mu$m and at 160
$\mu$m, we performed aperture photometry at both wavelengths following the
pointSourceAperturePhotometry script using the Herschel Interactive Processing
Environment (HIPE) version 15.0.1 software (Ott, 2010). We measured the flux
at different aperture radii black (up to $\sim$ 4 $\times$ full width at half
maximum, FWHM) and checked the aperture at which the encircled energy fraction
stabilizes. This profile is compared to the point spread function profile
finding that both behave similarly, concluding that the two far-IR sources are
point-like at the Herschel PACS beam (FWHM $\sim$
6$\aas@@fstack{\prime\prime}$7 $\times$ 6$\aas@@fstack{\prime\prime}$9 for 100
$\mu$m and $\sim$ 10$\aas@@fstack{\prime\prime}$6 $\times$
12$\aas@@fstack{\prime\prime}$1 for 160 $\mu$m). The photometric error is
estimated using six equal apertures placed in the background around the
source. The results for this aperture photometry are in agreement with those
presented by Chu et al. (2017).

Using photometric points from the Wide-field Infrared Survey Explorer (WISE)
22 $\mu$m, the Infrared Astronomical Satellite (IRAS) 60 $\mu$m, PACS 70
$\mu$m and Herschel SPIRE 250, 350, and 500 $\mu$m, for all three sources we
construct the Spectral Energy Distribution (SED) to estimate their total
infrared luminosities $L_{\mathrm{IR}}$ (see Table 3 for the details).

###  2.5 Ancillary submm data

Public ALMA observations of CO($J$ = 4–3) and 630 $\mu$m dust continuum were
retrieved for IRAS 09022-3615 and IRAS 17208-0014 (project ID: 2018.1.00994.S,
PI: T. Michiyama). For IRAS 06035-7102, we retrieved data from HCN($J$ = 2–1)
(project ID:2017.1.00022.S, PI: M. Imanishi) and from CO($J$ = 3–2) and 850
$\mu$m dust continuum (project ID:2018.1.00503.S, PI: A. Gowardhan). All
observations were reduced using the standard pipeline calibration scripts
provided by ALMA with the Common Astronomy Software Applications (CASA; CASA
Team et al., 2022) for the respective cycles. After obtaining the calibrated
measurement sets, the imaging process was done with the task TCLEAN. We
created an initial datacube where we identified the line-free channels. For
the continuum subtraction, we used the task UVCONTSUB. We ran the cleaning
process down to 1 $\sigma$ using interactive masks located at the source
positions. The spectral resolution of the cubes is set to 50 km s-1, using a
natural weighting and a primary-beam corrected map. To better compare the
p-H2O ($2_{11}\text{--}2_{02}$) line to the ALMA CO($J=4\text{--}3$),
CO($J=3\text{--}2$), and HCN($J=2\text{--}1$) observations, the ALMA data
cubes were degraded in spatial resolution down to the same APEX beam using the
task IMSMOOTH. The final ALMA spectra used for comparison are extracted at the
convolved central brightest pixel in Jy beam-1 units (Figure 1).

Figure 2: Left: Hubble Space Telescope R-band images of the three ULIRGs
presented in this work. We use a logarithmic stretch function to enhance the
faint emission. The beams of Herschel SPIRE (solid circle) and APEX (dashed
line) are overlaid. blackCenter: The contours represent the 20$\%$, 40$\%$,
60$\%$, 90 $\%$ of the maximum peak intensity at 100 $\mu$m from Herschel PACS
(cyan) and CO($J$ = 3-2) (magenta), CO($J$ = 4-3) (magenta) or HCN($J$ = 2-1)
(blue) from ALMA. These maps are at their native resolution and their beams
are on the lower side of the left hand side figure. Right: The observed far-IR
SED for each galaxy, including photometry from WISE, IRAS, and Herschel (red
points). The fitted SED is shown as a solid black line, while the blue shadow
shows $1\sigma$ uncertainties.

##  3 Results and Discussion

Table 1: Observed H2O ($2_{11}\text{--}2_{02}$) line parameters

N | Source | RA (J2000) | Dec (J2000) | z | $\nu_{\mathrm{obs}}$ | $D_{\mathrm{L}}$ | FWHM${}_{\mathrm{H_{2}O}}$ |  $S_{\mathrm{H_{2}O}}\,\Delta$v | $L_{\mathrm{H_{2}O}}$ | $t_{\mathrm{int}}$
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | [GHz] | [Mpc] | [km s-1] | [Jy km s-1] | [$\textup{L}_{\sun}/10^{6}$] | [h]
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
1 | IRAS 06035-7102 | 06:02:54.01 | -71:03:10.2 | 0.07946 | 696.7 | 325.0 | 414.88$\pm$54.01 | 145.73$\pm$16.32 | 10.33$\pm$1.16 | 12.3
2 | IRAS 17208-0014 | 17:23:21.96 | -00:17:00.9 | 0.04281 | 721.2 | 175.68 | 515.90$\pm$68.61 | 704.77$\pm$83.72 | 15.64$\pm$1.86 | 4.2
3 | IRAS 09022-3615 | 09:04:12.71 | -36:27:01.0 | 0.05964 | 709.7 | 238.84 | 414.81$\pm$49.89 | 199.08$\pm$20.84 | 7.90$\pm$0.83 | 14.3

  * •

Note: (1) Source name. (2) Right ascension. (3) Declination. (4) redshift from
the NASA/IPAC Extragalactic Database (NED). (5) Central observed frequency.
(6) Luminosity distance in Mpc from Yang et al. (2013). (7) The H2O
($2_{11}\text{--}2_{02}$) FWHM. (8) The velocity-integrated flux density. (9)
Line luminosity. (10) Total APEX integration time (including overheads).

###  3.1 Water line measurements

Table 2: A comparison of the velocity integrated p-H2O
($2_{11}\text{--}2_{02}$) flux densities S${}_{\mathrm{H_{2}O}}\Delta$v
between our results and previous ones with Herschel SPIRE-FTS taken from the
literature.

Source | APEX | HSO-SPIRE | HSO-SPIRE
---|---|---|---
| [Jy km s-1] | [Jy km s-1] | [Jy km s-1]
|  | Yang et al. (2013) | Pearson et al. (2016)
1 | 145.73 $\pm$ 16.32 | $<411^{*}$ |
2 | $704.77\pm 83.72$ | $697.98\pm 88.70$ | $654.23\pm 57.40^{**}$
3 | $199.08\pm 20.84$ | $184.65\pm 52.32$ | $165.95\pm 40.68^{**}$

Note: ∗ The value is a 3-$\sigma$ upper limit, although the original work
reports a $<$ 3 $\sigma$ tentative detection of $408.63\pm 136.98$ Jy km s-1
with low significance. ∗∗ The conversion factor from W m-2 to Jy km s-1 is
assumed $Q=3\times 10^{22}/\nu_{\mathrm{rest}}$[GHz].

Using APEX SEPIA Band-9 we detected the thermal transition of the water
molecule p-H2O ($2_{11}\text{--}2_{02}$) in all three ULIRGs at $>8\sigma$
significance levels. For IRAS 06035-7102, this is the first clear detection of
p-H2O ($2_{11}\text{--}2_{02}$), superseding the tentative $\sim 3\sigma$
detection reported by Yang et al. (2013). The spectra of p-H2O
($2_{11}\text{--}2_{02}$) observed with APEX toward IRAS 06035-7102, IRAS
17208-0014, and IRAS 09022-3615 are shown in Figure 1, black in comparison
with the low resolution Herschel SPIRE-FTS spectra (the full spectra can be
seen in Figure 5). black We note that the zero velocity is defined by the
redshift of the source, as indicated in Table 1. We have identified a velocity
shift of 100 km/s in the case of IRAS 09022-3615, which will be discussed in
detail in Section 3.4. We confirm the line detection at a higher significance
(at least a factor of two) and higher spectral resolution. The spectra show
that p-H2O ($2_{11}\text{--}2_{02}$) is detected with a peak flux density of
$0.45\pm 0.09$ Jy beam-1 in IRAS 09022-3615, $1.28\pm 0.34$ Jy beam-1 in IRAS
17208-0014 and $0.33\pm 0.08$ Jy beam-1 in IRAS 06035-7102. The velocity-
integrated flux densities and a comparison with previous values taken from the
literature are presented in Table 2. The values agree within the
uncertainties, with the line fluxes and tentative limits measured with the
Herschel SPIRE-FTS by Yang et al. (2013) and Pearson et al. (2016).

The instrumental line spread function of SPIRE-FTS is a function that
redistributes the line flux along the frequency axis (Hopwood et al., 2015),
which correlates the channels across the line profiles. On the other hand, as
SEPIA is a heterodyne instrument, it does not have this line spread function
effect, offering independent channels and demonstrating its power for
resolving the line profiles. Thus APEX not only provides an accurate
measurement of the line flux but also provides a direct way to recover the
true line profiles, revealing further information about the kinematics of the
gas.

The high spectral resolution of APEX allows us to resolve the p-H2O
($2_{11}\text{--}2_{02}$) lines blackand determine the full width at half
maximum (FWHM) in all three galaxies, which are 415, 516 and 415 km s-1 for
IRAS 06035-7102, IRAS 17208-0014 and IRAS 09022-3615, respectively (Table 1).
Within the $9\aas@@fstack{\prime\prime}5$ APEX beam and a spectral resolution
of 50 km s-1, the line profiles are consistent with a single Gaussian,
facilitating clean measurements of their velocity-integrated flux densities.

To compare the spectral noise between the APEX and the previous Herschel
SPIRE-FTS measurements, we convolved the APEX spectra with the line spread
function profile of SPIRE-FTS and then binned these spectra to match the
spectral SPIRE-FTS resolution. The error bars (measured within a 40 GHz
bandwidth for both) obtained for the convolved/binned APEX and SPIRE-FTS are
shown in Figure 1, demonstrating the higher significance reached by APEX to
characterise the p-H2O ($2_{11}\text{--}2_{02}$) line emission.

To compute the H2O luminosities, we follow the equation in Solomon & Vanden
Bout (2005):

$L_{\mathrm{H_{2}O}}=1.04\times 10^{-3}S_{\mathrm{H_{2}O}}\Delta
v\nu_{\mathrm{rest}}(1+z)^{-1}D_{\mathrm{L}}^{2}[L_{\sun}],$ (1)

where $S_{\mathrm{H_{2}O}}\Delta v$ is the velocity integrated flux density in
units of Jy km s-1, the rest-frame frequency of the line $\nu_{\mathrm{rest}}$
is related to the observed frequency $\nu_{\mathrm{obs}}$ as
$\nu_{\mathrm{rest}}=\nu_{\mathrm{obs}}(1+z)$ is in GHz, $D_{\rm L}$ is the
luminosity distance in Mpc and $z$ is the redshift. Derived luminosities are
presented in Table 1.

In Figure 2, we present a comparison between the APEX
($9\aas@@fstack{\prime\prime}5$) and Herschel SPIRE-FTS (35″) beam sizes. As
both spectra show similar intensities, we suggest that most of the water
emission comes from a compact region smaller than the APEX beam. To look at
the optical nature of the sources, we present images from the Hubble Space
Telescope (HST) with the Advanced Camera for Surveys (ACS), the Wield Field
Camera (WFC) and the Wide Field Planetary Camera (WFPC2) from programs 6346
(WFPC2, PI: K. Borne) and 10592 (ACS, PI: A. Evans, see Kim et al. 2013) in
the $I$ filter (F814W, $\lambda=8333$Å). The $I$-band images of our galaxies
have a large field of view ($202\arcsec\times 202^{\prime\prime}$) and capture
the detailed structure of the galaxies and the full extent of each ULIRG
interaction. We can see that IRAS 06035-7102 is an extended interacting double
system, separated by a distance of $\sim$ 10′′ (in WFC2 image; Arribas et al.
2008). IRAS 17208-0014 is an advanced merger with two tidal tails toward the
southeast and northwest. IRAS 09022-3615 is a late-type merger presenting a
tidal tail. While in IRAS 06035-7102 and IRAS 17208-0014, the ALMA mid-$J$ CO
and HCN(2–1) maps show that the molecular gas is concentrated towards the
nuclear region where the far-IR emission peaks, the CO(3–2) emission in IRAS
09022-3615 locates offset from the peak of the HST image and the far-IR dust
emission, indicating complex ISM structure and kinematics. Our APEX beams are
well aligned with the bulk of the compact molecular gas and dust emissions,
which should trace the total H2O emission from our targets.

Figure 3: Correlation between $L_{\rm H_{2}O}$ and $L_{\mathrm{IR}}$ in local
(grey) and high redshift (green and blue) galaxies. The linear correlation by
Yang et al. (2013) is shown as a blue dashed line at
$L_{\mathrm{H_{2}O}}/$$L_{\mathrm{IR}}$ $=$ 5.53 $\times$ 10-6. black Shown in
red are the derived values for the three galaxies studied in this paper. The
local galaxy sample and the high-$z$ ULIRGs are taken from Yang et al. (2013,
2016); Berta et al. (2023)

###  3.2 blackThe IR luminosities

To estimate the total infrared luminosity ($L_{\mathrm{IR}}$ ), we model the
SED of galaxies between 8 and 1000 $\mu$m. We use the Monte Carlo InfraRed SED
(MCIRSED; Drew & Casey 2022), which fits the data with a function consisting
of a mid-infrared (mid-IR) power law and a far-IR modified blackbody. The
model has the form

$S(\lambda)=\begin{cases}N_{\mathrm{pl}}\lambda^{\alpha}&\text{:
}\frac{\partial\log S}{\partial\log\lambda}>\alpha\\\
\frac{N_{\mathrm{bb}}\left(1-e^{-(\lambda_{0}/\lambda)^{\beta}}\right)\lambda^{-3}}{e^{hc/\lambda
kT}-1}&\text{: }\frac{\partial\log
S}{\partial\log\lambda}\leq\alpha,\end{cases}$ (2)

where $\lambda$ is the rest-frame wavelength, $\alpha$ is the mid-IR power-law
slope, $N_{\mathrm{pl}}$ and $N_{\mathrm{bb}}$ are normalization constants,
$\lambda_{0}$ is the wavelength where the dust opacity equals unity, $\beta$
is the dust emissivity index, $T$ is the luminosity-weighted characteristic
dust temperature, $h$, $k$ and $c$ are Planck, Boltzmann and the speed of
light constants. The two functions connect at the wavelength where the slope
of the modified blackbody is equal to the slope of the power law function. In
our model, $\alpha$ characterises the hot dust emission and is constrained to
be between 0 <$\alpha$< 6, while on the other hand, $\beta$ controls the slope
in the Rayleigh-Jeans regime and it is constrained to be between 0.5 and 5.0.
In Figure 2, we show the SED fitting and the estimated fitted values. The
value of $\alpha$ for the three galaxies is steep ($>4.0$), consistent with a
lack of significant hot dust emission coming from a buried powerful AGN. The
value of the emissivity index, less than 2.5, is in agreement with most values
in local star-forming galaxies. As expected, the infrared luminosities are
higher than $10^{12}$ $L_{\odot}$, consistent with being ULIRGs.

###  3.3 The $L_{\rm IR}-L_{\rm H_{2}O}$ correlation

In Figure 3, we present the correlation between $L_{\mathrm{H_{2}O}}$ and
$L_{\mathrm{IR}}$ as found in local and high redshift galaxies. Our sample is
highlighted in red and is part of the brightest low redshift ULIRGs. Previous
studies have shown that the $L_{\mathrm{IR}}$ -$L_{\mathrm{H_{2}O}}$
correlation is slightly superlinear (e.g. Omont et al., 2013; Yang et al.,
2013; Jarugula et al., 2019) compared to other water lines with the exception
of H2O(211–202), which is also excited by 101 $\mu$m photons. This correlation
opens the possibility of using water as a tracer of the far-IR emission and
suggests that the IR pumping mechanism is the main responsible for the
excitation of water molecules producing the p-H2O ($2_{11}\text{--}2_{02}$)
line at 752GHz. Nevertheless, studies of local galaxies show that the low-J
($J<3$) H2O lines blackcan also be both collisionally and far-IR excited (Liu
et al., 2017). One simple approach to hint light on the nature of the exciting
mechanism of the p-H2O ($2_{11}\text{--}2_{02}$) is to compare the spatial
distribution of the water emission to that of the dust continuum near
$101\,\mu$m, compared to other molecular gas tracers, which are well known to
be collisionally excited, such as CO($J$ = 3–2), CO($J$ = 4–3) or HCN($J$ =
2–1). However, given that our APEX observations are only black consistent with
single pointings, we here compare the line profiles as an indication of a
common nature, assuming that kinematic structures are associated with similar
spatial distributions.

Figure 4:  Comparison between the CO(3–2), CO(4–3) line and the p-H2O
($2_{11}\text{--}2_{02}$) for the ULIRGs in our sample. Left: Overplotted APEX
p-H2O ($2_{11}\text{--}2_{02}$) and ALMA CO(3–2), CO(4–3) spectra (smoothed to
the resolution of the APEX beam before extraction). Both spectra are
arbitrarily normalized to the peak of a simple Gaussian fit (red for p-H2O
($2_{11}\text{--}2_{02}$) and blue for CO(4–3)) in order to compare the
profiles. In the lower panel, we show the residuals between the profiles and
the Gaussian fits. Right: Velocity field (moment 1) of CO($J$ = 3–2), CO($J$ =
4–3) (colorbar) compared to the PACS 100 $\mu$m dust emission(orange contours)
and ALMA intensity CO($J$ = 4–3) (dashed magenta) contours, 630 $\mu$m (purple
contours) and 850 $\mu$m (purple contours). blackThe synthesised beams of ALMA
CO emission are shown at the bottom left of the right hand side column.

###  3.4 Spatial and spectral comparison between H2O, CO, HCN and dust

In order to get insights blackon the origin of the excitation mechanisms of
the p-H2O ($2_{11}\text{--}2_{02}$) emission, in Figure 2, we present the APEX
blackpointing overplotted on the ALMA CO($J$ = 3–2), CO($J$ = 4–3) and HCN($J$
= 2–1) integrated line contours. blackBecause 101$\mu$m photons are
predominantly exciting the H2O 220 level that relates to the p-H2O
($2_{11}\text{--}2_{02}$) line (e.g. Liu et al., 2017; González-Alfonso et
al., 2014), we explore the spatial distribution of the 100 $\mu$m dust
continuum using the PACS-100$\mu$m images in relation to that of the H2O line.
We note that the PACS 100 $\mu$m emission in IRAS 17208-0014 and IRAS
09022-3615 is point-like (at $9\aas@@fstack{\prime\prime}5$ fwhm $\sim$ 14
kpc); hence most of the far-IR photons come from the innermost central regions
of these ULIRGs. The interferometric ALMA observations toward IRAS 06035-7102
show the HCN($J$ = 2–1) emission has a deconvolved major axis FWHM of $0.71\pm
0.13$ kpc associated with a central region, while the CO($J$ = 3–2) is
spatially aligned with HCN($J$ = 2–1) presenting a deconvolved major axis FWHM
size of $3.1\pm 1.0$ kpc. In IRAS 17208-0014, the APEX pointing is centred on
the CO($J$ = 4–3) emission, and blackthe PACS-100 $\mu$m emission is centred
on the same region. However, in IRAS 09022-3615, we see a significant offset
between the peak position of the PACS-100 $\mu$m photometry and the CO($J$ =
4–3) blackemission (Figures 2, 4 $\&$ 6), while at the same time, the APEX
pointing is slightly offset with respect to their peaks, located in the
redshifted part of the CO emission.

The left panel of Figure 4 shows a normalised spectroscopic comparison between
the p-H2O ($2_{11}\text{--}2_{02}$), CO($J$ = 3–2) and CO($J$ = 4–3) at 50 km
s-1 channel width. The velocities from ALMA and APEX spectra are under the
same frame LSRK (Doppler correction was already applied during the
observations). In IRAS 06035-7102, blackboth CO($J$ = 3–2) and the p-H2O
($2_{11}\text{--}2_{02}$) emission are spectroscopically aligned. The same is
seen for IRAS 17208-0014 when comparing CO($J$ = 4–3) and p-H2O
($2_{11}\text{--}2_{02}$). Nevertheless, in IRAS 09022-3615, we find that the
water emission is redshifted compared to the gas traced by CO($J$ = 4–3). This
is best illustrated by the Gaussian fits of the p-H2O
($2_{11}\text{--}2_{02}$) (red) and CO($J$ = 4–3) (blue), where we find that
H2O has a peak emission at about $+100$ km s-1 with respect to the systemic
velocity defined by the redshift of the sources (see Table 2). This is evident
from the residual spectrum, where we can see that there is an excess of water
emission in the red part of the spectra at a similar velocity while having a
negative peak at the blue part.

The right panel of Figure 4 shows for IRAS 06035-7102 the intensity and moment
1 map of the CO($J$ = 3–2) together with the contours of the intensity of dust
emission at 850 $\mu$m. We show that the APEX pointing is aligned with the 850
$\mu$m ALMA dust continuum and with the CO($J$ = 3–2), suggesting that all the
different signals come from the same compact region. For IRAS 17208-0014 and
IRAS 09022-3615, we present the intensity and moment 1 map of the CO($J$ =
4–3) with respect to the contours of the intensity of dust emission of PACS
100 $\mu$m and ALMA 630 $\mu$m. In IRAS 17208-0014, the CO($J$ = 4–3), the 100
$\mu$m and 630 $\mu$m dust continuum blackare well aligned each other. This
suggests that the different tracers are revealing the same dense gas from the
very central part of the galaxy. In IRAS 09022-3615, both CO($J$ = 4–3) and
the 630 $\mu$m dust continuum are spatially aligned, nevertheless the PACS-100
$\mu$m emission is spatially offset by $\sim 2\aas@@fstack{\prime\prime}7$
(2.9 kpc). black If such a spatial offset is real, we might be finding an
explanation for the $+100$ km s-1 offset between the p-H2O
($2_{11}\text{--}2_{02}$) and CO(4–3) lines, where possibly the water line is
excited by infrared pumping, via the $101$ $\mu$m pumping path, at a region
different than where the CO line is produced, as the PACS-100 $\mu$m emission
is peaking at the redshifted part of the CO emission. In this case, the offset
between $100$ $\mu$m (PACS) and 630 $\mu$m (ALMA) might be due to
temperature/optical-depth variation across the galaxy. After exploring
photometric data taken with the Spitzer Space Observatory at 8$\mu$m and
24$\mu$m, we confirm a good alignment between ALMA 630 $\mu$m and Spitzer
data. blackConsidering that PACS photometry lies between these two
wavelengths, this suggests instead that the observed offset might be
attributed to issues with the astrometric calibration of the Herschel-PACS
observation. The absolute pointing Herschel error in scan maps can range from
$0\aas@@fstack{\prime\prime}9$ to $2\aas@@fstack{\prime\prime}36$ (see Table 5
of Sánchez-Portal et al. 2014), an uncertainty which is not consistent but
varies at each specific observing epoch subject to unique conditions. Our
observed spatial offset is approximately the upper limit of the astrometric
error. If the offset is indeed caused by the astrometric error, nevertheless,
the origin of the spectroscopic offset remains unclear. However, in the
literature, high spectral resolution observations have indicated the presence
of variations in the line profiles of CO($J$ = 3–2) and p-H2O
($2_{11}\text{--}2_{02}$) in Mrk 232 and NGC 6240 (as depicted in Figure 2
from Liu et al. 2017), nevertheless these differences have not been explored
in detail. It is worth noting that IRAS 09022-3615 has a complex morphology,
as shown in Figure 6, where the peak of MIPS 24 $\mu$m and ALMA 630 $\mu$m is
not aligned with the R-band HST image.

Due to the limited spatial resolution of APEX, to fully understand the origin
and powering source of the submm H2O emission, higher angular resolution
observations, such as ALMA Band-9 observations, are needed. These resolved
observations need to be integrated into detailed photo-dissociation and
radiative transfer modelling to tackle the nature of the p-H2O
($2_{11}\text{--}2_{02}$) emission and provide insights on the usage of this
line to describe the warm ISM in galaxies.

black

##  4 Conclusion

We present the first ground-based detection blackof the p-H2O
($2_{11}\text{--}2_{02}$) emission line at 752.033 GHz in local ULIRGs using
APEX SEPIA Band 9 towards IRAS 06035-7102 ($z$ = 0.07946), IRAS 17208-0014
($z$ = 0.04281) and IRAS 09022-3615 ($z$ = 0.05964). We demonstrate that
despite the low transmission of the atmosphere at these high frequencies,
blackobserving this thermal water line in local galaxies can be done using
ground-based facilities operating at high frequencies, reaching high signal-
to-noise in a reasonable integration time ($\sim$ 5-15 hours per source) black
with high-spectral resolutions. For all three sources, the final APEX spectra
are of high quality, improving the previous measurements done by Herschel
SPIRE-FTS both in signal-to-noise ratios and spectral resolution, resulting in
the first clear detection of this water line in IRAS 06035-7102.

We have spectrally resolved the p-H2O ($2_{11}\text{--}2_{02}$) line and
derived their velocity-integrated flux densities and intrinsic luminosities.
We find that the H2O emission follows the $L_{\mathrm{H{{}_{2}}O}}$ \-
$L_{\mathrm{IR}}$ correlation. We compared the p-H2O ($2_{11}\text{--}2_{02}$)
line emission to the dust continuum at different wavelengths 100 $\mu$m, 630
$\mu$m, 850 $\mu$m and the molecular gas traced by CO($J$ = 3–2), CO($J$ =
4–3) and HCN($J$ = 2–1). blackIn IRAS 06035-7102, the water emission is co-
spatial with the dust emission at 850 $\mu$m and spectrally aligned with the
gas traced by CO($J$ = 3–2). In IRAS 17208-0014, a similar behaviour is seen,
where the continuum dust emission at 100 $\mu$m and 630 $\mu$m are spatially
aligned, while in frequency space the gas traced by CO($J$ = 4–3) is
spectrally aligned with the p-H2O ($2_{11}\text{--}2_{02}$) line. black In the
case of IRAS 09022-3615, the p-H2O ($2_{11}\text{--}2_{02}$) water emission
line is redshifted by 100 km s-1 as compared with the CO($J$ = 4–3) emission
line. The origin of this offset is still inconclusive, and higher angular
resolution observations are needed to understand the reasons behind this
shift.

Our pilot APEX survey opens a new window blackfor studying submm H2O emission
lines in local ULIRGs using ground-based facilities, offering better
sensitivity and spectral resolution than the past space-based facilities, such
as Herschel. Future higher spatial resolution observations, using ALMA Band-9,
together with detailed radiative transfer modelling, can help us explore the
distribution of the molecular gas and black dust continuum in local ULIRGs to
disentangle the origin of the exciting mechanism of the p-H2O
($2_{11}\text{--}2_{02}$) line by reaching down to the scales of giant
molecular clouds.

## Acknowledgements

blackWe are thankful to the referee, Pierre Cox, for his valuable suggestions,
which have significantly enhanced the quality of this paper. DQ acknowledges
support from the National Agency for Research and Development
(ANID)/Scholarship Program/Doctorado Nacional/2021-21212222. C.Y. and S.A.
acknowledge support from ERC Advanced Grant 789410. E.I. acknowledges funding
by ANID FONDECYT Regular 1221846. blackEG-A thanks the Spanish MICINN for
support under projects PID2019-105552RB-C41 and PID2022-137779OB-C41. Y.J.
acknowledges financial support from ANID BASAL project No. FB210003. This
publication is based on data acquired with the Atacama Pathfinder Experiment
(APEX), project ID 103.B-0471. APEX is a collaboration between the Max-Planck-
Institut fur Radioastronomie, the European Southern Observatory, and the
Onsala Space Observatory. This publication makes use of the following ALMA
data: ADS/JAO.ALMA#2017.1.00022.S, ADS/JAO.ALMA#2018.1.00994.S ALMA is a
partnership of ESO (representing its member states), NSF (USA) and NINS
(Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. PACS has been developed by
a consortium of institutes led by MPE (Germany) and including UVIE (Austria);
KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-
IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been
supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium),
CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain).
SPIRE has been developed by a consortium of institutes led by Cardiff
University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA,
LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory
(Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK);
and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been
supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES,
CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and
NASA (USA). This work is based [in part] on observations made with the Spitzer
Space Telescope, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology under a contract with NASA.

## Data Availability

The data underlying this article are available in the ESO Science Archive at
https://archive.eso.org/scienceportal/home, and can be accessed with the
Project ID: 103.B-0471

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##  Appendix A Photometry of the galaxies

The following Table 3 presents the photometric fluxes derived from the PACS
analyses at 100 $\mu$m and 160 $\mu$m (see Section 2.4). Data is presented
together with fluxes taken from the literature.

Table 3: Multi-wavelength infrared photometry for the three galaxies presented
in Table 1. Flux densities are shown in Jy units.

N | WISE | IRAS | PACS | IRAS | PACS | PACS | SPIRE | SPIRE | SPIRE
---|---|---|---|---|---|---|---|---|---
| 22$\mu$m | 60$\mu$m | 70$\mu$m | 100$\mu$m | 100$\mu$m | 160$\mu$m | 250$\mu$m | 350$\mu$m | 500$\mu$m
| [Jy] | [Jy] | [Jy] | [Jy] | [Jy] | [Jy] | [Jy] | [Jy] | [Jy]
1 | 0.214 $\pm$ 0.001 | 5.13 $\pm$ 0.15 |  | 5.65 $\pm$ 0.28 |  |  | 1.23 $\pm$ 0.02 | 0.397 $\pm$ 0.001 | 0.130 $\pm$ 0.008
2 | 0.548 $\pm$ 0.003 | 31.14 $\pm$ 1.87 | 38.09 $\pm$ 1.74 | 34.90 $\pm$ 2.09 | 38.58 $\pm$ 0.02 | 23.31 $\pm$ 0.10 | 7.92 $\pm$ 0.04 | 2.95 $\pm$ 0.01 | 0.954 $\pm$ 0.009
3 | 0.430 $\pm$ 0.002 | 11.47 $\pm$ 0.57 | 12.81 $\pm$ 0.58 |  | 13.25 $\pm$ 0.02 | 8.09 $\pm$ 0.08 | 2.449 $\pm$ 0.007 | 0.823 $\pm$ 0.004 | 0.252 $\pm$ 0.005

Note: WISE emission at 22 $\mu$m from AllWISE Source Catalog (Cutri et al.,
2021). IRAS 60 and 100 $\mu$m measurements again retrieved from NASA/IPAC
Extragalactic Database(NED). PACS emission at 70 $\mu$m from Chu et al.
(2017). SPIRE emission at 250 $\mu$m, 350 $\mu$m and 500 $\mu$m from Clements
et al. (2018)

##  Appendix B Herschel SPIRE-FTS

The following Figure 5 presents the full spectra obtained by Herschel SPIRE-
FTS for IRAS6035-7102, IRAS17208-0014 and IRAS09022-3615

Figure 5: Herschel SPIRE-FTS spectra for the three sources. Black segments
indicate the bandwidth where we measured the RMS. Red squares show the spectra
taken to compare with our observations. The dashed magenta line is p-H2O
($2_{11}\text{--}2_{02}$). In IRAS 6035-7102, the water line was not detected
with Herschel SPIRE-FTS

##  Appendix C Dust continuum

The following Figure 6 compares different dust continuum observations towards
IRAS 09022-3615. A comparison between the emission PACS 100 $\mu$m, Spitzer-
MIPS 24 $\mu$m and ALMA 630 $\mu$m are shown. black Considering the alignment
between the Spitzer-24 $\mu$m and ALMA-630 $\mu$m photometry, and that PACS
photometry lies between these two wavelengths, the figure suggests the
$2\aas@@fstack{\prime\prime}9$ PACS offset (see Figure 4) might be related to
a calibration issue with the Herschel astrometry.

Figure 6: Different dust continuum emissions in IRAS 09022-3615 shown on top
of a Hubble Space Telescope R-band image. The contours represent the 20$\%$,
40$\%$, 60$\%$, 90 $\%$ of the maximum peak intensity at 100 $\mu$m from PACS
(orange) and 630 $\mu$m from ALMA (blue) and Spitzer-MIPS 24$\mu$m (cyan).