# Electroweak Baryogenesis From Dark CP Violation

Marcela Carena  Theoretical Physics Department, Fermilab, P.O. Box 500,
Batavia, IL 60510, USA  Enrico Fermi Institute, University of Chicago,
Chicago, Illinois, 60637, USA  Kavli Institute for Cosmological Physics,
University of Chicago, Chicago, Illinois, 60637, USA  Mariano Quirós
Department of Physics, University of Notre Dame, 225 Nieuwland Hall, Notre
Dame, IN 46556, USA  Institut de Física d’Altes Energies (IFAE), The Barcelona
Institute of Science and Technology (BIST), Campus UAB, 08193 Bellaterra
(Barcelona) Spain  Yue Zhang  Theoretical Physics Department, Fermilab, P.O.
Box 500, Batavia, IL 60510, USA  Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60208, USA

###### Abstract

We present a novel mechanism of electroweak baryogenesis where CP violation
occurs in a dark sector, comprised of standard model gauge singlets, thereby
evading the strong electric dipole moment constraints. In this framework, the
background of time-like component of a new gauge boson $Z^{\prime}_{\mu}$,
generated at electroweak temperatures, drives the electroweak sphaleron
processes to create the required baryon asymmetry. We first discuss the
crucial ingredients for this mechanism to work, and then show that all of them
can be elegantly embedded in ultraviolet completions with spontaneously broken
gauged lepton number. The models under consideration have a rich phenomenology
and can be experimentally probed in leptophilic $Z^{\prime}$ searches, dark
matter searches, heavy Majorana neutrino searches, as well as through hunting
for new Higgs portal scalars in multi-lepton channels at colliders.

††preprint: FERMILAB-PUB-18-636-T

The observed matter-antimatter asymmetry in the universe is widely believed to
yield strong evidence for new phenomena beyond the standard model (SM) of
particle physics. Electroweak baryogenesis (EWBG) is an elegant mechanism
Kuzmin  _et al._ (1985); Cohen  _et al._ (1990, 1991); Farrar and Shaposhnikov
(1993); Quiros (1994); Huet and Nelson (1995, 1996); Riotto (1996); Carena
_et al._ (1997); Quiros (1999); Carena  _et al._ (2001); Cline  _et al._
(2000); Carena  _et al._ (2003); Lee  _et al._ (2005); Cline (2006) that
generates the observed baryon asymmetry at the electroweak phase transition
(EWPT). This demands new physics close to the electroweak scale, to account
for CP violating effects larger than those present in the SM. Moreover, the
requirement of a sufficiently strong EWPT, along with the precision
measurements of the Higgs boson properties, demands an extended scalar sector
affecting the out-of-equilibrium processes.

The impressive recent progress in electric dipole moment (EDM) experiments
Andreev  _et al._ (2018); Baron  _et al._ (2014); Griffith  _et al._ (2009);
Baker  _et al._ (2006) imposes strong constraints on the required new sources
of CP violation in SM extensions, such as two-Higgs-doublet models and
supersymmetry Shu and Zhang (2013); Ipek (2014); Jung and Pich (2014); Abe
_et al._ (2014); Inoue  _et al._ (2014); Cheung  _et al._ (2014); Bian  _et
al._ (2015); Chen  _et al._ (2015); Fuyuto  _et al._ (2016); Jiang  _et al._
(2016); Blinov  _et al._ (2015); Balazs  _et al._ (2017); Bian and Chen
(2017); Chen  _et al._ (2018); Cesarotti  _et al._ (2018); Egana-Ugrinovic and
Thomas (2018). This provides a strong motivation to consider CP violation
triggered in dark sectors through SM gauge singlets Cline  _et al._ (2017),
which may naturally suppress contributions to EDMs. Such an enticing idea,
however, leaves the challenging task of finding the suitable mechanism to
transfer the CP violation from the dark sector to the visible sector, to
successfully create the baryon asymmetry at electroweak temperatures.

In this Letter, we propose a new mechanism of EWBG where the transfer of CP
violation to the visible sector is achieved by means of a vector boson
$Z^{\prime}_{\mu}$ which couples to the SM leptons, and to dark fermions with
CP violating Yukawa interactions involving additional SM singlet scalars. Such
scalars may provide, through the Higgs portal, a sufficiently strong first-
order EWPT. The time-like component of the new gauge boson, $Z^{\prime}_{0}$,
is CP odd and can transfer CP violation to the visible sector. During EWPT,
the CP violating source yields a nonzero background $\langle
Z^{\prime}_{0}\rangle$, which acts as a chemical potential for the SM leptons,
providing a thermal equilibrium lepton number asymmetry. In the absence of any
primordial asymmetries, such source term, through the electroweak sphaleron
processes, generates an equal amount of baryon and lepton number asymmetries,
which freeze in after the EWPT is completed and the sphalerons become
inactive. We point out that the current which couples to $Z^{\prime}_{\mu}$
must be anomalous with respect to the SM weak interaction during the EWBG
epoch.

In the following, we will present the main ingredients of the proposed EWBG
mechanism, and briefly explore its phenomenological consequences. We refer to
the companion paper Carena  _et al._ for a more detailed, quantitative
presentation.

The Need of An Anomalous Current Coupled to $Z^{\prime}$. The effective
Lagrangian relevant to our discussion contains the couplings of a vector boson
$Z^{\prime}_{\mu}$ to a current involving SM leptons and quarks, $J^{\mu}$,

$\displaystyle\mathcal{L}_{eff}=\mathcal{L}_{\rm
SM}-\frac{1}{4}Z_{\mu\nu}^{\prime}Z^{\prime\mu\nu}+\frac{1}{2}M_{Z^{\prime}}^{2}Z_{\mu}^{\prime}Z^{\prime\mu}+g^{\prime}Z^{\prime}_{\mu}J^{\mu}\
,$ (1) $\displaystyle
J^{\mu}=\sum_{i=1}^{3}\left[q_{L_{L_{i}}}\bar{L}_{L_{i}}\gamma^{\mu}L_{L_{i}}+q_{e_{R_{i}}}\bar{e}_{R_{i}}\gamma^{\mu}e_{R_{i}}+q_{\nu_{R_{i}}}\bar{\nu}_{R_{i}}\gamma^{\mu}\nu_{R_{i}}\right.$
$\displaystyle\ \ \
+\left.q_{Q_{L_{i}}}\bar{Q}_{L_{i}}\gamma^{\mu}Q_{L_{i}}+q_{u_{R_{i}}}\bar{u}_{R_{i}}\gamma^{\mu}u_{R_{i}}+q_{d_{R_{i}}}\bar{d}_{R_{i}}\gamma^{\mu}d_{R_{i}}\right],$
(2)

where $L_{L_{i}},e_{R_{i}}$ and $\nu_{R_{i}}$ are the SM leptons and right-
handed neutrinos, $Q_{L_{i}},u_{R_{i}}$ and $d_{R_{i}}$ are SM quarks, and
$q_{F}$ is the charge of the corresponding fermion $F$.

We assume that the above effective Lagrangian describes a period of the early
universe when EWBG occurs. Moreover, we further assume that the vector field
$Z^{\prime}_{\mu}$ develops a time-like background, $\langle
Z^{\prime}_{0}\rangle\neq 0$, sourced by a $U(1)$ charge density, whose origin
will be addressed later on. Through Eq. (1) the $Z^{\prime}_{0}$ background
acts as a chemical potential for the fermions. Of particular interest to us
are the SM quark and lepton doublets. In the presence of a chemical potential,
if the lepton or quark number were allowed to change independently, the
particle-antiparticle number density asymmetry, defined as $\Delta n_{F}\equiv
n_{F}-n_{\bar{F}}$ with $F=L_{L_{i}},Q_{L_{i}}$, will be generated and evolve
toward its thermal equilibrium value

$\Delta n_{F}^{\rm EQ}=\frac{2N_{c}}{3}T_{c}^{2}g^{\prime}q_{F}\left\langle
Z^{\prime}_{0}\right\rangle\ ,$ (3)

where $N_{c}=3\ (1)$ for quarks (leptons) is the color factor, whereas $T_{c}$
is the EWPT critical temperature. This expression is exact in the electroweak
symmetric phase where all SM fermions are massless.

Within the context of EWBG, there is only one process where the lepton ($L$)
and baryon ($B$) numbers are simultaneously violated – the electroweak
sphalerons. Each sphaleron process violates $B+L$ but conserves $B-L$ among
the left-handed $SU(2)_{L}$ doublets, where the baryon and lepton asymmetries
are defined as

$\Delta n_{B_{L}}\equiv\frac{1}{3}\sum_{i=1}^{3}\Delta n_{Q_{L_{i}}},\ \ \
\Delta n_{L_{L}}\equiv\sum_{i=1}^{3}\Delta n_{L_{L_{i}}}\ .$ (4)

The $(B\pm L)_{L}$ asymmetries satisfy the Boltzmann equations

$\displaystyle\begin{split}&\frac{\partial\Delta n_{(B+L)_{L}}}{\partial
t}=\Gamma_{\rm sph}\left(\mathcal{S}-\Delta
n_{(B+L)_{L}}\rule{0.0pt}{11.38109pt}\right)\ ,\\\ &\frac{\partial\Delta
n_{(B-L)_{L}}}{\partial t}=0,\ \ \ \ \ \mathcal{S}=\sum_{i=1}^{3}\left(\Delta
n_{L_{L_{i}}}^{\rm EQ}+\Delta n_{Q_{L_{i}}}^{\rm EQ}\right)\ ,\end{split}$ (5)

where $\Gamma_{\rm sph}\simeq 120\alpha_{w}^{5}T_{c}$ is the sphaleron rate in
the electroweak unbroken phase Bodeker  _et al._ (2000) and it is
exponentially suppressed in the broken phase. $\mathcal{S}$ serves as the
source for creating a net $B+L$ asymmetry, with $\Delta n_{L_{L_{i}}}^{\rm
EQ}$ and $\Delta n_{Q_{L_{i}}}^{\rm EQ}$ contributing to it democratically,
the same way as sphalerons act on every $SU(2)_{L}$ doublet.

Starting from a primordially symmetric universe implies, $\Delta
n_{B_{L}}=\Delta n_{L_{L}}=(1/2)\Delta n_{(B+L)_{L}}$. With this, the first
equation in (5) simplifies to

$\displaystyle\frac{\partial\Delta n_{B_{L}}}{\partial t}=\Gamma_{\rm
sph}\left(\frac{1}{2}\mathcal{S}-\Delta
n_{B_{L}}\rule{0.0pt}{11.38109pt}\right)\ ,$ (6)

and from Eq. (3) it is straightforward to derive

$\displaystyle\mathcal{S}=\frac{2}{3}T^{2}g^{\prime}\,\sum_{i=1}^{3}\left(q_{L_{L_{i}}}+3q_{Q_{L_{i}}}\rule{0.0pt}{11.38109pt}\right)\left\langle
Z^{\prime}_{0}\right\rangle\,.$ (7)

Remarkably Eq. (7) is proportional to the non-conservation of the current
$J^{\mu}$, i.e., the coefficient appearing its chiral anomaly with respect to
$SU(2)_{L}^{2}$,

$\displaystyle\partial_{\mu}J^{\mu}\propto\sum_{i=1}^{3}\left(q_{L_{L_{i}}}+3q_{Q_{L_{i}}}\right){\rm
tr}(W\widetilde{W})\ ,$ (8)

where $W$ ($\widetilde{W}$) is the $SU(2)_{L}$ field (dual) strength. Hence we
have found a necessary condition for the proposed EWBG mechanism to work,
namely, the current to which the $Z^{\prime}_{\mu}$ couples must be anomalous
with respect to $SU(2)_{L}^{2}$. Had the charges $q_{F}$ in Eq. (2) been
arranged such that the current $J^{\mu}$ were conserved, the source term in
Eq. (6) would have vanished and, in turn, no net baryon asymmetry would have
been created.

The effective Lagrangian of Eq. (1) can be obtained from a UV complete $U(1)$
gauge theory whose gauge boson is $Z^{\prime}_{\mu}$ and $q_{F}$ are the
corresponding SM fermion $U(1)$ charges. In the case of a non-conserved
$J^{\mu}$, additional fermions (anomalons) are required to render the $U(1)$
anomaly free. The total current of the $U(1)$ gauge symmetry is the sum of
$J^{\mu}$ in Eq. (2) and that of the anomalons, $J^{\mu}_{a}$, such that the
anomaly cancellation condition imposes,
$\partial_{\mu}(J^{\mu}+J^{\mu}_{a})=0$. The anomalon fields, once introduced,
will also contribute to the source term $\mathcal{S}$, Eq. (7). Here, however,
we assume that the $U(1)$ gauge symmetry is spontaneously broken above the
electroweak scale (e.g., at TeV scales), and that the anomalons get symmetry
breaking masses. If the anomalons have masses much larger than the EWPT
temperature, their population, as well as their impact on the electroweak
sphalerons, will become Boltzmann suppressed. In such case, the anomalons,
although canceling the gauge anomalies, have negligible contribution to
$\mathcal{S}$.

Good candidates for such a $U(1)$ symmetry include gauged lepton number,
baryon number, or any flavor dependent combination of the two that remains
anomalous, within the SM, with respect to $SU(2)_{L}^{2}$. In contrast, the
proposed EWBG mechanism cannot work if the $U(1)$ is already anomaly free
given the SM fermion content (plus right-handed neutrinos), for example,
$B-L$, $L_{\mu}-L_{\tau}$, etc. In the following we discuss the realization of
our EWBG mechanism in a UV complete model with gauged lepton number,
$U(1)_{\ell}$. As we shall see, a dark matter candidate also naturally emerges
in the theory.

As another useful remark, interestly the above discussion remains valid even
if the $\langle Z^{\prime}_{0}\rangle$ background is space-time inhomogeneous.
Indeed, we will consider a first order EWPT which temporarily creates $\langle
Z^{\prime}_{0}\rangle$ in front of the expanding bubble walls.

CP Violation and the Electroweak Phase Transition. We will now address the
origin of the $Z^{\prime}_{0}$ field background, as well as the dynamics of
the EWPT. In analogy to a static electric potential, the $\langle
Z^{\prime}_{0}\rangle$ background is C, CP and CPT odd, and can be generated
by a net $U(1)_{\ell}$ charge distribution near the bubble wall. To this end
we introduce a fermionic particle $\chi$ with CP violating microscopic
interactions with the bubble wall. Since $\chi$ is a SM gauge singlet that
cannot couple to the Higgs field through renormalizable interactions, we will
introduce a SM scalar singlet $S$ to interact with it,

$\displaystyle\bar{\chi}_{L}(m_{0}+\lambda
e^{i\theta_{\lambda}}S)\chi_{R}+{\rm h.c.}$ (9)

Within the bubble wall, the Higgs VEV turns on, while the $S$ VEV
simultaneously turns off. Such a transition to the electroweak broken phase
has been studied and involves a two-step process from the original vacuum with
$\langle S\rangle=\langle H\rangle=0$ Espinosa  _et al._ (2012); Patel  _et
al._ (2013); Cheung and Zhang (2013); Curtin  _et al._ (2014); Cline  _et al._
(2017). The necessary ingredient to allow for a strongly first order EWPT is a
sizable scalar quartic term, $|S|^{2}|H|^{2}$, by which the Higgs becomes a
portal to the dark sector.

We will consider the following ansatz for the $S$ profile across the bubble
wall,
$|S(z)|=s_{0}\left[1+\tanh(z/L_{\omega})\rule{0.0pt}{8.53581pt}\right]/2$. The
coordinate $z$ is defined in the rest frame of the bubble wall which is
located at $z=0$, whereas $L_{\omega}$ is the wall width. Observe that to
accommodate a physical CP violating effect through Eq. (9), we need a scalar
potential that fixes the phase of $S$. During the EWPT, the VEV of $S$
contributes to the $\chi$ mass through Eq. (9), whereas the bare mass term
$m_{0}$ has its origin in the spontaneous breaking of $U(1)_{\ell}$. If the
two mass terms carry different, space-time dependent phases, the dispersion
relations of $\chi_{L},\chi_{R}$ and their antiparticles will be modified in a
CP violating way. This affects the phase space distributions of such particles
and yields a non-trivial solution to the corresponding diffusion equations,
leading to net number density asymmetries in $\chi_{L},\chi_{R}$,

$\displaystyle\Delta n_{\chi}(z)\equiv
n_{\chi_{L}}-n_{\chi^{c}_{L}}=-(n_{\chi_{R}}-n_{\chi^{c}_{R}})\neq 0\ .$ (10)

The spatial distribution of $\Delta n_{\chi}(z)$ will peak around the bubble
wall. For details on solving the diffusion equations and the numerical
computation of $\Delta n_{\chi}(z)$, we refer the reader to the companion
paper Carena  _et al._ . If $\chi_{L}$ and $\chi_{R}$ carry different
$U(1)_{\ell}$ quantum numbers, the above chiral asymmetries will give a net
$U(1)_{\ell}$ charge density distribution around the bubble wall,

$\rho_{\ell}(z)=(q_{\chi_{L}}-q_{\chi_{R}})\Delta n_{\chi}(z)\ .$ (11)

Neglecting the curvature of the bubble wall, the $\langle
Z^{\prime}_{0}\rangle$ background sourced by $\rho_{\ell}$ can be calculated
in cylindrical coordinates to be

$\langle
Z^{\prime}_{0}(z)\rangle=\frac{g^{\prime}}{2M_{Z^{\prime}}}\int_{-\infty}^{\infty}dy\
\rho_{\ell}(y)\ e^{-M_{Z^{\prime}}|z-y|}\ .$ (12)

Given this $\langle Z^{\prime}_{0}\rangle$ background, the final baryon
asymmetry generated can be obtained by solving Eq. (6),

$\Delta n_{B}=\frac{\Gamma_{\rm sph}}{v_{\omega}}\int^{\infty}_{0}dz\
\mathcal{S}(z)\ e^{-\Gamma_{\rm sph}z/v_{\omega}}\ ,$ (13)

where $v_{\omega}$ is the bubble wall expansion velocity. The parametric
dependence in today’s baryon to entropy ratio is, $\eta_{B}={\Delta
n_{B}}/{s}\sim g^{\prime
2}N_{g}^{2}T_{c}^{3}L_{\omega}\alpha_{W}^{5}/(M_{Z^{\prime}}^{2}v_{\omega})$.

UV Complete Models. Next, we discuss unifying all the above ingredients for
EWBG into a UV complete framework with gauged (anomaly free) lepton number
symmetry, $U(1)_{\ell}$. There are several choices to define the lepton number
$\ell$. The most obvious one is to gauge all the three SM families universally
by taking $\ell=L_{e}+L_{\mu}+L_{\tau}$. Alternatively, one could also gauge
only two lepton flavors such as $\ell=L_{\mu}+L_{\tau}$. We will consider
these two cases as benchmark models. The minimal set of new fermion content is
given in Table 1 Fileviez Perez and Wise (2010); Schwaller  _et al._ (2013),
where $\mathtt{q}$ is an arbitrary real number. The index $i$ runs through
$e,\mu,\tau$ ($\mu,\tau$) in the first (second) model, and $N_{g}=3\ (2)$
defines the number of families charged under the $U(1)_{\ell}$,
correspondingly.

Particle | $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{\ell}$
---|---|---|---|---
$L_{L_{i}}$ | 1 | 2 | -1/2 | 1
$e_{R_{i}}$ | 1 | 1 | -1 | 1
$\nu_{R_{i}}$ | 1 | 1 | 0 | 1
$L^{\prime}=(\nu_{L}^{\prime},e_{L}^{\prime})^{T}$ | 1 | 2 | -1/2 | $\mathtt{q}$
$e^{\prime}_{R}$ | 1 | 1 | -1 | $\mathtt{q}$
$\chi_{R}$ | 1 | 1 | 0 | $\mathtt{q}$
$R^{\prime}=(\nu_{R}^{\prime\prime},e_{R}^{\prime\prime})^{T}$ | 1 | 2 | -1/2 | $\mathtt{q}+N_{g}$
$e^{\prime\prime}_{L}$ | 1 | 1 | -1 | $\mathtt{q}+N_{g}$
$\chi_{L}$ | 1 | 1 | 0 | $\mathtt{q}+N_{g}$
$\Phi$, $S$ | 1 | 1 | 0 | $N_{g}$
Table 1: UV completion of the effective theory. Figure 1: We scan broadly over
the model parameters to find points that allow for successful EWBG as proposed
in this work (blue points). We show various experimental constraints (LEP,
BaBar, beam dump, electron and muon $g-2$, CCFR) by the correspondingly
labeled shaded regions, in the gauged $L_{e}+L_{\mu}+L_{\tau}$ (left) and
$L_{\mu}+L_{\tau}$ (right) models.

To spontaneously break the $U(1)_{\ell}$, and at the same time give masses to
the new fermions, we introduce the complex scalar $\Phi$ carrying
$U(1)_{\ell}$ number $N_{g}$. We assume that $\Phi$ picks up a VEV,
$v_{\Phi}/\sqrt{2}$, above the electroweak scale. This VEV gives mass to the
gauge boson $Z^{\prime}$, $M_{Z^{\prime}}=N_{g}g^{\prime}v_{\Phi}/2$, which
can still be light if the gauge coupling $g^{\prime}$ is sufficiently small.
We can also write down Yukawa couplings of the form,

$\left(c_{L}\bar{R}^{\prime}L^{\prime}+c_{e}\bar{e}^{\prime}_{L}e^{\prime\prime}_{R}+c_{\chi}\bar{\chi}_{L}\chi_{R}\rule{0.0pt}{9.95845pt}\right)\Phi+{\rm
h.c.}\ ,$ (14)

which will give vectorlike masses (with respect to the SM) to the new
fermions. We assume $c_{L}\sim c_{e}$ to be large enough so that
$L^{\prime},R^{\prime},e^{\prime\prime}_{L},e^{\prime}_{R}$ are sufficiently
heavy in comparison with the critical temperature of the EWPT. As noted
earlier, the fermions $L^{\prime}$ and $R^{\prime}$ are needed to cancel the
$U(1)_{\ell}\otimes SU(2)_{L}^{2}$ gauge anomaly, whereas decoupling them from
the thermal bath provides the necessary condition for our EWBG mechanism to
work. On the other hand, we assume the parameter $c_{\chi}$ to be sufficiently
small so that $\chi$ is light and remains populated in the thermal bath during
the EWPT. Their $U(1)_{\ell}$ charges are fixed in the UV theory:
$q_{\chi_{L}}=\mathtt{q}+N_{g}$, $q_{\chi_{R}}=\mathtt{q}$, and their
difference does not depend on $\mathtt{q}$. This helps to eliminate a free
parameter from Eq. (11), so that $\rho_{\ell}(z)=N_{g}\Delta n_{\chi}$.

The fermion $\chi$ will source CP violation when it interacts with the
expanding bubble wall. The $\chi$-$\Phi$ interaction in Eq. (14) is
responsible for generating the $m_{0}=c_{\chi}v_{\Phi}/\sqrt{2}$ mass term in
Eq. (9). In addition, as discussed before, another complex scalar $S$ with the
same quantum numbers as $\Phi$ and a scalar potential that fixes its phase is
required to yield a physical CP phase, barring the redefinitions of fermion
fields. Moreover, such a complex scalar will also be responsible for the
strong EWPT through a two-step phase transition.

To summarize, we have argued that all the key ingredients for our EWBG
mechanism can be elegantly accommodated in a UV theory of gauged lepton
number. The UV completion is useful in shedding light into the parameters of
the low energy effective theory.

Neutrino Cosmology. It is worth commenting on the neutrino sector and
implications of cosmological measurements on additional neutrino degrees of
freedom, $\Delta N_{eff}$ Barger  _et al._ (2003); Ghosh  _et al._ (2011);
Barger  _et al._ (2011); Hamann  _et al._ (2011), for the two benchmark
models. The $U(1)_{\ell}$ gauge interaction could thermalize, in the very
early universe, all the new fermions charged under it, and in particular the
right-handed neutrinos $\nu_{R_{i}}$. To avoid an excessive contribution to
$\Delta N_{eff}$, one option is to make the $U(1)_{\ell}$ interaction decouple
early enough, preferably above the QCD phase transition temperature, $T_{\rm
QCD}\sim 100\,$MeV. This implies $v_{\Phi}\gtrsim 10\,$TeV if
$M_{Z^{\prime}}\gg T_{\rm QCD}$, or $g^{\prime}\lesssim 10^{-5}$ if
$M_{Z^{\prime}}\ll T_{\rm QCD}$. The other option is to implement the seesaw
mechanism by giving Majorana masses to $\nu_{R_{i}}$. If all the $\nu_{R_{i}}$
are heavier than $\sim 500\,$MeV, they will decay before the big-bang
nucleosynthesis and have no effect in $\Delta N_{eff}$ Berryman  _et al._
(2017). Interestingly, this option could easily be achieved in the gauged
$L_{\mu}+L_{\tau}$ model where both $\Phi$ and $S$ have charge $N_{g}=2$. The
experimental search for heavy Majorana neutrinos is of great phenomenological
interests Atre  _et al._ (2009), especially as the new $U(1)_{\ell}$ gauge
interaction allows them to be more copiously produced. We will investigate
this exciting opportunity in a future work.

Experimental Probes. Here we summarize the phenomenological predictions unique
to our EWBG mechanism and discuss the present experimental bounds in the two
benchmark models presented above.

– The main motivation for this work is to provide the necessary amount of CP
violation for baryogenesis, without being in tension with EDM measurements. In
the gauged $U(1)_{\ell}$ UV complete models presented here, the fermion field
$\chi$ responsible for CP violation is a SM singlet, thereby eliminating its
Barr-Zee type Barr and Zee (1990) contribution to EDMs. In Ref. Carena  _et
al._ , we will show that the leading contribution arises at the four-loop
level.

– In our EWBG mechanism, the $Z^{\prime}_{\mu}$ gauge boson transmits the dark
CP violation to the SM sector, and thereby the generation of the observed
baryon asymmetry, $\eta_{B}\simeq 0.9\times 10^{-10}$, restricts the values of
$M_{Z^{\prime}}$ as a function of its gauge coupling $g^{\prime}$. In Fig. 1,
the blue points are obtained from a scan over the parameter space that can
account for the correct $\eta_{B}$. The allowed $Z^{\prime}$ masses are in the
MeV to TeV range, with decreasing $g^{\prime}$ values for lighter
$Z^{\prime}$, in agreement with the parametric dependence estimated below Eq.
(13). In the left (right) panel of Fig. 1, we show the gauged
$L_{e}+L_{\mu}+L_{\tau}$ ($L_{\mu}+L_{\tau}$) benchmark models.

– The search for $Z^{\prime}$ provides a handle on these EWBG scenarios. In
the gauged $L_{e}+L_{\mu}+L_{\tau}$ model, the $Z^{\prime}$ has a coupling to
the electron, which is subject to constraints from electron $g-2$,
$e^{+}e^{-}$ colliders (LEP, BaBar) and electron beam dump experiments
Alexander  _et al._ (2016), as shown by the correspondingly labeled shaded
regions in the left panel of Fig. 1. On the other hand, the gauged
$L_{\mu}+L_{\tau}$ model is free from the above constraints. There is,
however, a relevant constraint from neutrino trident production Altmannshofer
_et al._ (2014) which excludes the magenta region (labeled by CCFR) in the
right panel of Fig. 1. Interestingly, Fig. 1 shows that there is a region of
parameter space that can explain the muon $g-2$ anomaly (yellow band), and in
the case of the $L_{\mu}+L_{\tau}$ model, such a region is allowed and favored
by EWBG, with $M_{Z^{\prime}}\lesssim 200\,$MeV.

– The models considered here provide a dark matter candidate, $\chi$. The
$U(1)_{\ell}$ gauge invariance implies that the SM singlet fermion $\chi$ can
only interact with the SM particles via the $Z^{\prime}$ exchange, making
$\chi$ a leptophilic dark matter candidate Fox  _et al._ (2011). Its thermal
relic density and detection prospects will be discussed in detail in Ref.
Carena  _et al._ .

– It is possible to search for the dark scalar $S$ at high energy colliders,
where it can be pair produced through the Higgs portal interaction. If $S$ is
lighter than twice the $\chi$ mass, it must decay via a $\chi$ loop into a
pair of $Z^{\prime}$ bosons, yielding four leptons in the final state. The
$S$-$\chi$ interaction is inherently CP violating and such decay can provide a
test of dark CP violation via interference effects in the golden
$4\ell$-channel.

Summary. We have proposed a novel mechanism for EWBG in which the CP violation
occurs in a dark sector and is transmitted to the observable sector via the
time-like background of a $Z^{\prime}_{\mu}$ vector boson during a strong
first-order EWPT. The $Z^{\prime}_{\mu}$ is the gauge boson of a $U(1)_{\ell}$
gauge symmetry, and couples to an anomalous SM lepton number current. After
the spontaneous $U(1)_{\ell}$ symmetry breaking, the new $SU(2)_{L}$ doublet
fermions required to render the theory anomaly free become massive and
decouple from the thermal bath before the EWPT. Because the CP violating
interactions are active in the dark sector, its effects on EDMs are highly
suppressed and evade present bounds. We show two benchmark scenarios with
gauged $U(1)_{L_{e}+L_{\mu}+L_{\tau}}$ and $U(1)_{L_{\mu}+L_{\tau}}$
symmetries which provide concrete examples of UV completions. The models under
consideration provide a rich phenomenology that can be probed in searches for
leptophilic $Z^{\prime}$, dark matter, heavy Majorana neutrinos, and new
scalars in multi-lepton channels at the LHC or prospective high energy
colliders.

###### Acknowledgements.

We thank Zackaria Chacko, James Cline, Bogdan Dobrescu, Paddy Fox, Ian Low,
David Morrissey, Tim Tait and Carlos Wagner for useful discussions and
comments. This manuscript has been authored by Fermi Research Alliance, LLC
under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics. Work at University of
Chicago is supported in part by U.S. Department of Energy grant number DE-
FG02-13ER41958. The work of M.Q. is partly supported by Spanish MINEICO under
Grant CICYT-FEDER-FPA2014-55613-P and FPA2017-88915-P, and by the Severo Ochoa
Excellence Program of MINEICO under Grant SEV-2016-0588. The work of Y.Z. is
also supported by the DoE under contract number DE-SC0007859. Y.Z. would like
to thank the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1607611, where part of this work was performed, and
Colegio De Fisica Fundamental E Interdisciplinaria De Las Americas (COFI) for
a travel support during the completion of this work.

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# Stability estimate for a partial data inverse problem for the convection-
diffusion equation

Soumen Senapati† and Manmohan Vashisth∗ † TIFR Centre for Applicable
Mathematics, Bangalore 560065, India.
E-mail<EMAIL_ADDRESS>∗ Department of Mathematics, Indian Institute of
Technology, Jammu 181221, India.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article, we study the stability in the inverse problem of determining
the time-dependent convection term and density coefficient appearing in the
convection-diffusion equation, from partial boundary measurements. For
dimension $n\geq 2$, we show the convection term (modulo the gauge term)
admits log-log stability, whereas log-log-log stability estimate is obtained
for the density coefficient.

Keywords: Inverse problems, partial Dirichlet to Neumann map, parabolic
equation, Carleman estimates, stability estimate.

Mathematics subject classification 2010: 35R30, 35K20.

##  1\. Introduction

For $n\geq 2$, let $\Omega\subset\mathbb{R}^{n}$ be a bounded simply connected
domain having $C^{2}$ smooth boundary $\Gamma=\partial\Omega$. For $T>0$, let
us introduce the parabolic operator $\mathcal{L}_{A,q}$ in the cylinder
$Q:=(0,T)\times\Omega$ by

$\displaystyle\mathcal{L}_{A,q}:=\partial_{t}-\sum_{j=1}^{n}\left(\partial_{j}+{A_{j}(t,x)}\right)^{2}+q(t,x),$
(1.1)

where
$A(t,x):=\left(A_{1}(t,x),A_{2}(t,x),...,A_{n}(t,x)\right)\in(W^{1,\infty}(Q))^{n}$
and $q\in L^{\infty}(Q)$. We consider an initial boundary value problem (IBVP)
known for the convection-diffusion equation which models physical processes
like mass or heat transfer within a body and, also appears in probabilistic
study of diffusion process (like, the Fokker-Planck and Kolmogorov equations),
finance (like, the Black–Scholes or the Ornstein-Uhlenbeck processes) and
chemical engineering (for describing the movement of macro-particles)

$\displaystyle\begin{aligned} \begin{cases}\mathcal{L}_{A,q}u(t,x)=0,\ \
(t,x)\in Q,\\\ u(0,x)=0,\ \ x\in\Omega,\\\ u(t,x)=f(t,x),\ \
(t,x)\in\Sigma:=(0,T)\times\Gamma.\end{cases}\end{aligned}$ (1.2)

We first briefly discuss some well-posedness results regarding the above IBVP.
Following [16], one can consider suitable spaces for the forward problem
(1.2). However, in the context of our article, we assume more regularity on
the coefficients in the operator (1.1). Therefore, we define the admissible
set of coefficients for a given $m>0$ by

$\displaystyle\mathcal{M}(m)=\big{\\{}(A,q)\in(W^{2,\infty}(Q))^{n}\times
W^{1,\infty}(Q);\|A\|_{W^{2,\infty}(Q)}+\|q\|_{W^{1,\infty}(Q)}\leq
m\big{\\}}.$

Now, we introduce some time-dependent Sobolev spaces for $p,q$ being non-
negative real numbers and $M=\Omega$ or, $\Gamma$

$\displaystyle H^{p,q}\left((0,T)\times
M\right):=L^{2}\left(0,T;H^{p}(M)\right)\cap H^{q}\left(0,T;L^{2}(M)\right),$

equipped with the norm

$\displaystyle\|u\|_{H^{p,q}\left((0,T)\times
M\right)}=\|u\|_{L^{2}\left(0,T;H^{p}(M)\right)}+\|u\|_{H^{q}\left(0,T;L^{2}(M)\right)}.$

Further, we denote $H^{p,q}_{0}(\Sigma):=\\{f\in H^{p,q}(\Sigma);f(0,x)=0,\
x\in\Omega\\}$. The existence of unique solution $u\in H^{2,1}(Q)$ to the IBVP
(1.2) for a given Dirichlet data $f\in
H_{0}^{\frac{3}{2},\frac{3}{4}}(\Sigma)$ follows from [46]. Also then, we have
some $C>0$ depending only on $m$ and $Q$ such that

$\displaystyle\|u\|_{H^{2,1}(Q)}\leq
C\|f\|_{H_{0}^{\frac{3}{2},\frac{3}{4}}(\Sigma)}.$

We can define the Dirichlet to Neumann (DN) map
$\Lambda^{*}_{A,q}:H_{0}^{\frac{3}{2},\frac{3}{4}}(\Sigma)\to
H^{\frac{1}{2},\frac{1}{4}}(\Sigma)$ as

$\displaystyle\Lambda^{*}_{A,q}(f)=\left(\partial_{\nu}u+2(\nu\cdot
A)u\right)|_{\Sigma},$

where $u$ solves (1.2) and $\nu(x)$ denotes the unit outward normal at
$x\in\Gamma$.

The inverse problem under consideration here is the stable recovery of time-
evolving properties of a homogeneous medium such as $A$ and $q$, by applying
heat source on $\Sigma$ and measuring the heat flux on a part of $\Sigma$. To
be precise, we study the stability aspects for unique recovery of $(A,q)$ from
a partial DN map which measures the Neumann outputs on a part of $\Sigma$
related to a small open neighborhood of the $\omega_{0}$-illuminated face
which is defined in (2.1). It is known that the convection term can be
recovered only under the divergence free condition (with respect to space
variables) because of the non-uniqueness associated to the gauge transform
(see [49, 53]). We derive a stability estimate for the divergence free
convection term $A$. In doing so, Vessella’s conditional stability result [55]
will be used crucially. Also we borrow an important construction for the
principal term in the geometric optics solutions from [42] which was
originally used in the framework of dynamical Schrödinger equation. The decay
in the remainder terms of the geometric optics solutions follows from a
Carleman estimate. For stability of the density coefficient, we again use
Vessella’s result [55] in combination with the stability estimate of $A$.

The issues regarding unique and stable determination of coefficients appearing
in parabolic PDEs from boundary measurements have attracted much attention
during last several decades. Motivated by the seminal work [54] by Sylvester
and Uhlmann, Isakov in [31] uniquely determined the time-dependent coefficient
when $A=0$, by using an argument based on completeness of the product of
solutions. The stability issues of the same problem has been resolved by
Choulli in [22]. In [2], Avdonin and Seidman used boundary control (BC) method
pioneered by Belishev which is further developed by Katchalov, Kurylev and
Lassas (see [4, 34] and references therein), to establish uniqueness result
for time-independent $q$. In the absence of any zeroth order term, Cheng and
Yamamoto proved in [17, 18, 19] uniqueness of convection term which belongs to
some Lebesgue spaces from single measurement when $n=2$. Gaitan and Kian in
[29] obtained stable determination result for time-dependent $q$ in a bounded
cylindrical domain when $A=0$ which was further generalized in the article
[44] by Kian and Yamamoto proving analogous results in time-fractional
diffusion equation settings. In [24], Choulli and Kian derived logarithmic
stability estimates for time-dependent term $q$ working only with partial DN
map, in the absence of first order coefficients. In [11], Bellassoued and
Rassas stably determined the convection term $A$ and density coefficient $q$
both of which are time-independent. Vashisth and Sahoo in [53] obtained unique
determination result for time-dependent convection term (modulo gauge
equivalence) and density coefficient from full Dirichlet and partial Neumann
data. In this work, we have proved the stability estimate for determining the
time-dependent convection term and the density coefficients from the knowledge
of full Dirichlet data and the Neumann data measured on a portion which is
slightly bigger than half of the lateral boundary. We refer to [9, 20, 21, 23,
25, 22, 26, 29, 32, 33, 48] for more works in inverse problems related to
parabolic PDEs. Also, there have been a considerable amount of work done in
the context of hyperbolic and dynamical Schrödinger equations (see [1, 3, 6,
10, 13, 14, 28, 40, 41, 42, 43, 12, 50, 51, 7, 8, 30, 38, 36, 37, 39, 45, 52,
47, 35] and references therein).

The article is organized as follows. In §2, the main result of the article is
stated. Then boundary and interior Carleman estimates for the operator
$\mathcal{L}_{A,q}$ blue are derived in §3, followed by the construction of
geometric optics solutions in §4. Finally, we discuss the stable determination
results for the convection term $A$ and density coefficient $q$ in §5.

##  2\. Statement of the main result

We begin this section by introducing some notations. Following [15], fix an
$\omega_{0}\in\mathbb{S}^{n-1}$ and define the $\omega_{0}$-shadowed and
$\omega_{0}$-illuminated faces by

$\displaystyle\partial\Omega_{+}(\omega_{0}):=\left\\{x\in\partial\Omega:\
\nu(x)\cdot\omega_{0}\geq 0\right\\},\ \
\partial\Omega_{-}(\omega_{0}):=\left\\{x\in\partial\Omega:\
\nu(x)\cdot\omega_{0}\leq 0\right\\}$

of $\partial\Omega$ where $\nu(x)$ is the outward unit normal to
$\partial\Omega$ at $x\in\partial\Omega$. For a given small $\epsilon>0$, we
define the small open neighborhoods of $\partial\Omega_{+}(\omega_{0})$ and
$\partial\Omega_{-}(\omega_{0})$ by

$\displaystyle\partial\Omega_{+,{\epsilon}/{2}}(\omega_{0}):=\left\\{x\in\partial\Omega;\
\nu(x)\cdot\omega_{0}>\frac{\epsilon}{2}\right\\},\quad\mbox{and}\quad\partial\Omega_{-,{\epsilon}/{2}}(\omega_{0}):=\left\\{x\in\partial\Omega;\
\nu(x)\cdot\omega_{0}<\frac{\epsilon}{2}\right\\}.$ (2.1)

respectively. Corresponding to $\partial\Omega_{\pm}(\omega_{0})$ and
$\partial\Omega_{\pm,{\epsilon}/{2}}(\omega_{0})$, we denote the lateral
boundary parts by
$\Sigma_{\pm}(\omega_{0}):=(0,T)\times\partial\Omega_{\pm}(\omega_{0})$ and
$\Sigma_{\pm,{\epsilon}/{2}}(\omega_{0}):=(0,T)\times\partial\Omega_{\pm,{\epsilon}/{2}}(\omega_{0})$
respectively. Let us define the partial Dirichlet to Neumann map denoted by
$\Lambda_{A,q}:H_{0}^{\frac{3}{2},\frac{3}{4}}(\Sigma)\to
H^{\frac{1}{2},\frac{1}{4}}(\Sigma_{-,{\epsilon}/{2}}(\omega_{0}))$ as

$\displaystyle\Lambda_{A,q}(f)=\left(\partial_{\nu}u+2(A\cdot\nu)u\right)\big{|}_{\Sigma_{-,{\epsilon}/{2}}(\omega_{0})}$
(2.2)

We now state the main result of this article.

######  Theorem 2.1.

Let $(A_{i},q_{i})\in\mathcal{M}(m),\ i=1,2$ and $T>\textnormal{diam}\
\Omega$, where $\Omega\subset\mathbb{R}^{n}$ is a bounded $C^{2}$ smooth
domain for $n\geq 2$. We denote by $\Lambda_{i}$ the partial DN map
corresponding to $\mathcal{L}_{A_{i},q_{i}}$ as defined in (2.2). Under the
assumption $A_{1}|_{\Sigma}=A_{2}|_{\Sigma}$ and $\nabla_{x}\cdot
A_{1}=\nabla_{x}\cdot A_{2}$ in $Q$, we have the following estimates for some
positive constants $C,\alpha_{1},\alpha_{2},\beta_{1}$ and $\beta_{2}$
depending on $m$ and $Q$

$\displaystyle\|A_{1}-A_{2}\|_{L^{2}(Q)}\leq
C\left(\|\Lambda_{1}-\Lambda_{2}\|^{\alpha_{1}}+\left|\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right|^{\alpha_{2}}\right),$
$\displaystyle\|q_{1}-q_{2}\|_{L^{2}(Q)}\leq
C\left(\|\Lambda_{1}-\Lambda_{2}\|^{\beta_{1}}+\left|\log\left|\log|\log\|\Lambda_{1}-\Lambda_{2}\|\right||\right|^{\beta_{2}}\right).$

We remark here that, in the recent work [5], Bellassoued and Ben Fraj
discussed the stability aspects of determining the time-dependent coefficients
appearing in the convection-diffusion equation and proved logarithmic and
double logarithmic stability results for the convection term and density
coefficient respectively. Moreover the Neumann measurements there are taken on
any arbitrary part of the lateral boundary $\Sigma$; but the coefficients in
[5] are assumed to be known in an open set containing $\Sigma$ which is
essential to apply a local unique continuation result near the boundary. In
contrast to [5], we consider coefficients which agree only on the lateral
boundary. Although we work with the Neumann data measured on a particular
subset of $\Sigma$, which is slightly more than half of the boundary and
obtain double and triple logarithmic stability estimates for $A$ and $q$
respectively.

##  3\. Boundary and interior Carleman estimates

We prove here a boundary Carleman estimate involving for the operator
$\mathcal{L}_{A,q}$ which will be used to control the boundary terms appearing
in the integral identity given by (5.6) where no information is given. Now, we
choose $x_{0}\in\mathbb{R}^{n}$ such that
$\inf\limits_{x\in\overline{\Omega}}\left(x+x_{0}\right)\cdot\omega>0$. For
this choice of $\omega$ and $x_{0}$, the derivation of Carleman estimate goes
as follows.

######  Theorem 3.1.

Let $\phi(t,x)=\lambda^{2}t+\lambda x\cdot\omega$ and $u\in
C^{2}(\overline{Q})$ with $u(0,\cdot)=0$ and $u|_{\Sigma}=0$. For
$(A,q)\in\mathcal{M}(m)$ there exist $\lambda_{1},C>0,$ depending only on $m$
and $Q$ such that

$\displaystyle\begin{aligned} &\int_{Q}e^{-2\phi(t,x)}\left(\lambda^{2}\lvert
u(t,x)\rvert^{2}+\lvert\nabla_{x}u(t,x)\rvert^{2}\right)\
\mathrm{d}x\mathrm{d}t+\int_{\Omega}e^{-2\phi(T,x)}\left(\lambda\lvert
u(T,x)\rvert^{2}+\lvert\nabla_{x}u(T,x)\rvert^{2}\right)\ \mathrm{d}x\\\
&\quad+\lambda\int_{\Sigma_{+,\omega}}e^{-2\phi(t,x)}\omega\cdot\nu(x)|\partial_{\nu}u(t,x)|^{2}\
\mathrm{d}S_{x}\mathrm{d}t\leq
C\int_{Q}e^{-2\phi(t,x)}\lvert\mathcal{L}_{A,q}u(t,x)\rvert^{2}\
\mathrm{d}x\mathrm{d}t\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+C\lambda\int_{\Sigma_{-,\omega}}e^{-2\phi(t,x)}\lvert\omega\cdot\nu(x)\rvert\lvert\partial_{\nu}u(t,x)\rvert^{2}\
\mathrm{d}S_{x}\mathrm{d}t\end{aligned}$ (3.1)

holds for all $\lambda\geq\lambda_{1}$.

###### Proof.

We have to convexify the Carleman weight $\phi$ appropriately due to the
presence of first order derivatives in $\mathcal{L}_{A,q}$. For a proof of the
boundary Carleman estimate, we refer to [16] where the following convexified
weight has been considered for $s>0$

$\displaystyle\phi_{s}(t,x):=\lambda^{2}t+\lambda
x\cdot\omega-\frac{s\left((x+x_{0})\cdot\omega\right)^{2}}{2}$

where $x_{0}\in\mathbb{R}^{n}$ is as mentioned in the line just before the
statement of Theorm 3.1. ∎

We write down the interior Carleman estimate which easily follows from Theorem
3.1 and will be used to construct the geometric optics solutions.

######  Corollary 3.2.

(Interior Carleman estimate). For $(A,q)\in\mathcal{M}(m)$ there exist
$\lambda_{1},C>0$ depending only on $m$ and $Q$ such that the following
estimate holds for $u\in C_{c}^{\infty}(Q)$ and $\lambda\geq\lambda_{1}$

$\displaystyle\begin{aligned} \int_{Q}e^{-2\phi(t,x)}\left(\lambda^{2}\lvert
u(t,x)\rvert^{2}+\lvert\nabla_{x}u(t,x)\rvert^{2}\right)\
\mathrm{d}x\mathrm{d}t\leq
C\int_{Q}e^{-2\phi(t,x)}\lvert\mathcal{L}_{A,q}u(t,x)\rvert^{2}\
\mathrm{d}x\mathrm{d}t.\end{aligned}$

##  4\. Construction of geometric optics solutions

In this section, we construct the geometric optics solutions for the parabolic
operator $\mathcal{L}_{A,q}$ and its formal $L^{2}$ adjoint
$\mathcal{L}^{*}_{A,q}=\mathcal{L}_{-A,\overline{q}}$. For $\lambda>0$, let
$\phi(t,x):=\lambda^{2}t+\lambda x\cdot\omega$ be the weight function. Then we
construct the geometric optics solutions $u$ and $v$ for the operators
$\mathcal{L}_{A,q}$ and $\mathcal{L}^{*}_{A,q}$ respectively which have the
following forms

$\displaystyle\begin{aligned}
&u(t,x)=e^{\phi(t,x)}\left(B_{g}+R_{g}\right)(t,x),\ \ \ \\\ \textnormal{and}\
\ &v(t,x)=e^{-\phi(t,x)}\left(B_{d}+R_{d}\right)(t,x).\end{aligned}$ (4.1)

Next we show that for $\lambda$ large enough, the remainder terms $R_{g}$ and
$R_{d}$ can be estimated in terms of their principal terms $B_{g}$ and $B_{d}$
respectively. The decay of $R_{d}$ and $R_{g}$ in $\lambda$ will be crucial to
derive stability results from an integral identity obtained by using the
solution to an adjoint problem and the given data.

We start with some definition and notations. For $m\in\mathbb{R}$, we define
$L^{2}(0,T;H^{m}_{\lambda}(\mathbb{R}^{n}))$ by

$L^{2}(0,T;H^{m}_{\lambda}(\mathbb{R}^{n})):=\\{u(t,\cdot)\in\mathcal{S}^{\prime}(\mathbb{R}^{n});\left(\lambda^{2}+|\xi|^{2}\right)^{m/2}\mathcal{F}_{x}u(t,\xi)\in
L^{2}((0,T)\times\mathbb{R}^{n})\\},$

equipped with the norm

$\|u\|^{2}_{L^{2}(0,T;H^{m}_{\lambda}(\mathbb{R}^{n}))}=\int_{0}^{T}\int_{\mathbb{R}^{n}}\left(\lambda^{2}+|\xi|^{2}\right)^{m}|\mathcal{F}_{x}u(t,\xi)|^{2}\
\mathrm{d}\xi dt,$

where $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ denote the space of tempered
distributions on $\mathbb{R}^{n}$ and $\mathcal{F}_{x}$ is the Fourier
transform with respect to the space variables. For $0<\delta<<1$, we consider
a sequence $\eta_{\delta}\in C_{c}^{\infty}(\delta,T-\delta)$ such that

$\mbox{$\eta_{\delta}\equiv 1$ on $[2\delta,T-2\delta]$ and, }\
\|\eta_{\delta}\|_{W^{k,\infty}(\mathbb{R})}\leq
C\delta^{-k},\quad\textnormal{for }k\in\mathbb{N}.$

######  Theorem 4.1.

Let $\delta\in(0,T/4)$, $\mathcal{L}_{A,q}$ be as in (1.1) and for
$\omega\in\mathbb{S}^{n-1}$, let $\phi(t,x)=\lambda^{2}t+\lambda
x\cdot\omega$.

  1. (1)

(Exponentially growing solutions) For $(\tau,\xi)\in\mathbb{R}^{1+n}$ such
that $\omega\cdot\xi=0$, $(A,q)\in\mathcal{M}(m)$ and $D\in
W^{2,\infty}(Q)^{n}$ with $\|D\|_{W^{2,\infty}(Q)}\leq C_{0}$, there exists
$\lambda_{0}>0$ depending on $m,C_{0}$ and $Q$ such that for
$\lambda\geq\lambda_{0}$, we can find $v_{g}\in
H^{2,1}\left((0,T)\times\Omega\right)$ solution to

$\displaystyle\begin{aligned}
\begin{cases}\mathcal{L}_{A,q}v(t,x)=0,\quad(t,x)\in Q,\\\ v(0,x)=0,\quad
x\in\Omega\end{cases}\end{aligned}$

taking the form

$\displaystyle v_{g}(t,x)=e^{\phi(t,x)}\left(B_{g}(t,x)+R_{g}(t,x)\right)$
(4.2)

where $B_{g}(t,x)$ is given by

$\displaystyle
B_{g}(t,x)=\eta_{\delta}(t)\frac{\xi}{|\xi|}\cdot\nabla_{x}\left(e^{-i(\tau,\xi)\cdot(t,x)}e^{\left(\int_{\mathbb{R}}\omega\cdot
D(t,x+s\omega)\
\mathrm{d}s\right)}\right)e^{\left(\int_{0}^{\infty}\omega\cdot
A(t,x+s\omega)\ \mathrm{d}s\right)}$ (4.3)

and $R_{g}$ satisfies the following estimate

$\displaystyle\|R_{g}\|_{L^{2}(0,T;H^{k}(\Omega))}\leq
C\lambda^{-1+k}\delta^{-3}\langle\tau,\xi\rangle^{3},\ \ \mbox{for
$k\in\\{0,1,2\\}$}.$ (4.4)

  2. (2)

(Exponentially decaying solutions) For $(\mathcal{A},q)\in\mathcal{M}(m)$,
there exists $\lambda_{0}>0$ depending on $m$ and $Q$ such that for
$\lambda\geq\lambda_{0}$, we can find $v_{d}\in
H^{2,1}\left((0,T)\times\Omega\right)$ solution to

$\displaystyle\begin{aligned}
\begin{cases}\mathcal{L}^{*}_{A,q}v(t,x)=0,\quad(t,x)\in Q,\\\ v(T,x)=0,\quad
x\in\Omega.\end{cases}\end{aligned}$

taking the form

$\displaystyle v_{d}(t,x)=e^{-\phi(t,x)}\left(B_{d}(t,x)+R_{d}(t,x)\right)$
(4.5)

where $B_{d}(t,x)$ is given by

$\displaystyle
B_{d}(t,x)=\eta_{\delta}(t)e^{\left(\int_{0}^{\infty}\omega\cdot
A(t,x+s\omega)\ \mathrm{d}s\right)}$ (4.6)

and $R_{d}$ satisfies the following estimates

$\displaystyle\|R_{d}\|_{L^{2}(0,T;H^{k}(\Omega))}\leq
C\lambda^{-1+k}\delta^{-3},\ \ \mbox{for $k\in\\{0,1,2\\}$}.$

The proof of the above Theorem relies mainly on some arguments from functional
analysis as we need to consider appropriate functional which would be extended
and identified by Hahn-Banach and Riesz representation theorems. But
continuity of such functional would be possible once we obtain suitable
negative order Carleman estimates. Thus we state the following Carleman
estimate in a negative order Sobolev space and proof of this estimate follows
from very standard arguments (see [5, 7, 8, 16, 27] and references therein).
To state the Carleman estimate, we define the conjugated operator
$\mathcal{P}_{\lambda}$ by

$\mathcal{P}_{\lambda}:=e^{-\phi}\mathcal{L}_{A,q}e^{\phi}.$

######  Proposition 4.2.

(Shifting the index). Let $A,q,\phi$ and $\mathcal{L}_{A,q}$ be as in Theorem
4.1.

  1. (1)

Let $\mathcal{P}_{\lambda}:=e^{-\phi}\mathcal{L}_{A,q}e^{\phi},$ then there
exists $\lambda_{0}$ and $C>0$ such that for $u\in
C^{1}([0,T];C_{c}^{\infty}(\Omega))$ with $u(T,\cdot)=0$ and
$\lambda\geq\lambda_{0}$, we have

$\displaystyle\|u\|_{L^{2}(0,T;H^{-1}_{\lambda}(\mathbb{R}^{n}))}\leq
C\|\mathcal{P}_{\lambda}u\|_{L^{2}(0,T;H^{-2}_{\lambda}(\mathbb{R}^{n}))}.$
(4.7)

  2. (2)

Let $\mathcal{P}^{*}_{\lambda}:=e^{\phi}\mathcal{L}^{*}_{A,q}e^{-\phi},$ then
there exists $\lambda_{0}$ and $C>0$ such that for $u\in
C^{1}([0,T];C_{c}^{\infty}(\Omega))$ with $u(0,\cdot)=0$ and
$\lambda\geq\lambda_{0}$, we have

$\displaystyle\|u\|_{L^{2}(0,T;H^{-1}_{\lambda}(\mathbb{R}^{n}))}\leq
C\|\mathcal{P}^{*}_{\lambda}u\|_{L^{2}(0,T;H^{-2}_{\lambda}(\mathbb{R}^{n}))}.$
(4.8)

For the sake of completeness, we prove the following standard proposition
which will lead to the proof of Theorem 4.1.

######  Proposition 4.3.

For $f\in L^{2}\left(0,T;H^{1}(\mathbb{R}^{n})\right)$ there exists $w\in
H^{2,1}\left((0,T)\times\mathbb{R}^{n}\right)$ solving the IVP

$\displaystyle\begin{aligned}
\begin{cases}\mathcal{P}_{\lambda}w=f,\quad\textnormal{in }Q,\\\ w(0,x)=0,\
x\in\Omega\end{cases}\end{aligned}$

and it satisfies, $\|w\|_{L^{2}(0,T;H^{2}_{\lambda}(\Omega))}\leq
C\|f\|_{L^{2}(0,T;H^{1}_{\lambda}(\Omega))}$ for some $C>0$ depending only on
$m$ and $Q$.

###### Proof.

Consider the space $\mathcal{W}:=\\{\mathcal{P}_{\lambda}^{*}u:u\in
C^{1}\left([0,T];C^{\infty}_{c}(\Omega)\right)\mbox{and}\ u(T,\cdot)=0\\}$
equipped with the norm
$\|\cdot\|_{L^{2}(0,T;H^{-2}_{\lambda}(\mathbb{R}^{n}))}$. Now for $f\in
L^{2}\left(0,T;H^{1}(\mathbb{R}^{n})\right)$, define a functional $T_{f}$ on
$\mathcal{W}$ by

$\displaystyle
T_{f}(\mathcal{P}_{\lambda}^{*}u):=\int_{\mathbb{R}^{1+n}}f(t,x)u(t,x)\
\mathrm{d}x\mathrm{d}t.$ (4.9)

Now using (4.7), we have that $T_{f}$ is a continuous linear functional on
$\mathcal{W}$ with

$\displaystyle\|T_{f}\|_{\mathcal{W}}\leq
C\|f\|_{L^{2}(0,T;H^{1}_{\lambda}(\mathbb{R}^{n}))}$ (4.10)

and hence by the Hahn-Banach Theorem, $T_{f}$ can be extended to
${L^{2}(0,T;H^{-2}_{\lambda}(\mathbb{R}^{n}))}$ which will be still denoted by
$T_{f}$ and satisfy (4.10). Finally using the Riesz representation theorem,
there exists $w\in{L^{2}(0,T;H^{2}_{\lambda}(\mathbb{R}^{n}))}$ such that for
$v\in{L^{2}(0,T;H^{-2}_{\lambda}(\mathbb{R}^{n}))}$ we have

$\displaystyle T_{f}(v)=\int_{\mathbb{R}^{1+n}}v(t,x)w(t,x)\
\mathrm{d}x\mathrm{d}t.$ (4.11)

Now combining (4.9) and (4.11) we obtain

$\displaystyle\int_{\mathbb{R}^{1+n}}\mathcal{P}_{\lambda}^{*}v(t,x)w(t,x)\
\mathrm{d}x\mathrm{d}t=\int_{\mathbb{R}^{1+n}}v(t,x)f(t,x)\
\mathrm{d}x\mathrm{d}t,$ (4.12)

for $v\in C^{1}\left([0,T];C^{\infty}_{c}(\Omega)\right)$ such that
$v(T,\cdot)=0$ in $\Omega$. This gives us $\mathcal{P}_{\lambda}w=f$ in $Q$.
Since $f\in L^{2}\left(0,T;H^{1}(\mathbb{R}^{n})\right)$ we have $w\in
H^{1}\left(0,T;L^{2}(\mathbb{R}^{n})\right)$. Now for $v\in
C^{1}\left([0,T];C^{\infty}_{c}(\Omega)\right)$ satisfying $v(T,\cdot)=0$, we
use the integration by parts to (4.12) to obtain

$\int_{\Omega}w(0,x)v(0,x)\ \mathrm{d}x=0.$

Hence $w(0,\cdot)=0$ in $\Omega$. Finally we use (4.10) and (4.11) to get

$\|T_{f}\|_{L^{2}(0,T;H^{-2}_{\lambda}(\Omega))}=\|w\|_{L^{2}(0,T;H^{2}_{\lambda}(\Omega))}\leq
C\|f\|_{L^{2}(0,T;H^{1}_{\lambda}(\Omega))}.$

∎

###  4.1. Proof of Theorem 4.1

First observe that

$\displaystyle\mathcal{L}_{A,q}\left(e^{\phi}v\right)=e^{\phi}\left(\mathcal{L}_{A,q}v-2\lambda\omega\cdot(\nabla_{x}+A)v\right).$

Now using the expressions for $v_{g}$, $B_{g}$ and $\mathcal{L}_{A,q}v_{g}=0$,
we have that the remainder terms $R_{g}$ solves

$\displaystyle\mathcal{P}_{\lambda}R_{g}=-\mathcal{L}_{A,q}B_{g}$ (4.13)

where $\mathcal{P}_{\lambda}:=e^{-\phi}\mathcal{L}_{A,q}e^{\phi}$ is the
conjugated operator as defined earlier and $B_{g}$ solves the following
transport equation

$\omega\cdot\left(\nabla_{x}+A\right)B_{g}=0.$

Hence from (4.13) and Proposition 4.3, it is clear that the remainder term
$R_{g}$ satisfies

$\displaystyle\|R_{g}\|_{L^{2}(0,T;H^{k}(\Omega))}\leq
C\lambda^{-1+k}\|B_{g}\|_{H^{3}(Q)},\ \textnormal{for }k\in\\{0,1,2\\}.$

This completes the proof for the construction of exponentially growing
solutions to $\mathcal{L}_{A,q}v=0$. One can carry out exactly same set of
arguments to prove the existence of exponentially decaying solutions having
the form given by equation (4.5) and solution to $\mathcal{L}_{A,q}^{*}v=0$.
This complete the proof of Theorem 4.1. Hence from (4.13) and Proposition 4.3,
it is clear that the remainder term $R_{g}$ satisfies

$\displaystyle\|R_{g}\|_{L^{2}(0,T;H^{k}(\Omega))}\leq
C\lambda^{-1+k}\|B_{g}\|_{H^{3}(Q)},\ \textnormal{for }k\in\\{0,1,2\\}.$

This completes the proof for the construction of exponentially growing
solutions to $\mathcal{L}_{A,q}v=0$. One can carry out exactly same set of
arguments to prove the existence of exponentially decaying solutions having
the form given by equation (4.5) and solution to $\mathcal{L}_{A,q}^{*}v=0$.
This complete the proof of Theorem 4.1.

##  5\. Proof of theorem 2.1

In this section, we prove the main result on stability for the first and
zeroth order coefficients. But first we derive an integral identity using
Green’s formula where we will plug in the geometric optics solutions
constructed before. We simultaneously consider exponentially growing and
decaying solutions to avoid any exponential term or boundary terms at initial
or final time. To be precise, we construct $u_{2}$ and $v$ as the
exponentially growing and decaying solutions for the operators
$\mathcal{L}_{A_{2},q_{2}}$ and $\mathcal{L}_{-A_{1},\overline{q}_{1}}$
respectively by using Theorem 4.1. Taking $0<\delta<<1$,
$(\tau,\xi)\in\mathbb{R}^{1+n}$ with $\xi\cdot\omega=0$ and
$D(t,x)=A(t,x):=\left(A_{1}-A_{2}\right)(t,x)$ we have

$\displaystyle u_{2}(t,x)=e^{\phi(t,x)}\left(B_{2}+R_{2}\right)(t,x),$ (5.1)
$\displaystyle\textnormal{and}\ \ v(t,x)=e^{-\phi(t,x)}\left(B+R\right)(t,x)$
(5.2)

where

$\displaystyle
B_{2}(t,x)=\eta_{\delta}(t)\frac{\xi}{|\xi|}\cdot\nabla_{x}\left(e^{-i\left(t\tau+x\cdot\xi\right)}e^{\left(\int_{\mathbb{R}}\omega\cdot
A(t,x+s\omega)\
\mathrm{d}s\right)}\right)e^{\left(\int_{0}^{\infty}\omega\cdot
A_{2}(t,x+s\omega)\ \mathrm{d}s\right)},$ $\displaystyle
B(t,x)=\eta_{\delta}(t)e^{\left(-\int_{0}^{\infty}\omega\cdot
A_{1}(t,x+s\omega)\ \mathrm{d}s\right)}$

and $R_{2},R\in L^{2}(0,T;H^{2}_{\lambda}(\Omega))$ satisfy

$\displaystyle\|R_{2}\|_{L^{2}(0,T;H^{k}_{\lambda}(\Omega))}\leq
C\lambda^{-1+k}\delta^{-3}\langle\tau,\xi\rangle^{3}\textnormal{ and}\
\|R\|_{L^{2}(0,T;H^{k}_{\lambda}(\Omega))}\leq C\lambda^{-1+k}\delta^{-3},\
\textnormal{for }k\in\\{0,1,2\\}.$ (5.3)

Hence, we have for some $\beta>0$

$\displaystyle\|u_{2}\|_{L^{2}(\Sigma)}\leq
e^{\beta\lambda}\delta^{-3}\langle\tau,\xi\rangle^{3}.$ (5.4)

Also, observe that $u_{2}(0,x)=v(T,x)=0$ for $x\in\Omega$. Now, consider
$u_{1}$ to be the solution of the IBVP

$\displaystyle\begin{cases}\mathcal{L}_{A_{1},q_{1}}w(t,x)=0,\quad(t,x)\in
Q,\\\ w(0,x)=0,\ \ \ x\in\Omega\\\ \ w(t,x)=u_{2}(t,x),\ \ \
(t,x)\in\Sigma\end{cases}$

and define $u:=u_{1}-u_{2}$ in $Q$. Then we get

$\displaystyle\begin{aligned}
\begin{cases}\mathcal{L}_{A_{1},q_{1}}u(t,x)=2A(t,x)\cdot\nabla_{x}u_{2}(t,x)+\widetilde{q}(t,x)u_{2}(t,x),\
\ \ (t,x)\in Q,\\\ u(0,x)=0,\ \ \ x\in\Omega\\\ u(t,x)=0,\ \ \
(t,x)\in\Sigma\end{cases}\end{aligned}$ (5.5)

where

$\displaystyle\begin{aligned} &A(t,x)\equiv\\{A^{j}(t,x)\\}_{1\leq j\leq
n}:=A_{1}(t,x)-A_{2}(t,x),\ \ \
\widetilde{q}(t,x):=\widetilde{q}_{1}(t,x)-\widetilde{q}_{2}(t,x)\ \
\mbox{and}\\\ &q(t,x):=q_{1}(t,x)-q_{2}(t,x).\end{aligned}$

Using Green’s formula, we have

$\displaystyle\begin{aligned}
&\int_{Q}(\mathcal{L}_{A_{1},q_{1}}u)(t,x)\overline{v}(t,x)\
\mathrm{d}x\mathrm{d}t-\int_{Q}u(t,x){\mathcal{L}_{-A_{1},\overline{q}_{1}}\overline{v}(t,x)}\
\mathrm{d}x\mathrm{d}t=\int_{\Omega}u(T,x)\overline{v}(T,x)\ \mathrm{d}x\\\ &\
\ \ \ \ \ \ -\int_{\Omega}u(0,x)\overline{v}(0,x)\
\mathrm{d}x-\int_{\Sigma}\overline{v}(t,x)\partial_{\nu}u(t,x)\
\mathrm{d}S_{x}\mathrm{d}t+\int_{\Sigma}u(t,x)\overline{\partial_{\nu}v(t,x)}\
\mathrm{d}S_{x}\mathrm{d}t,\end{aligned}$

which after using (5.5) becomes

$\displaystyle 2\int_{Q}(A(t,x)\cdot\nabla_{x}u_{2}(t,x))\overline{v}(t,x)\
\mathrm{d}x\mathrm{d}t+\int_{Q}\widetilde{q}(t,x)u_{2}(t,x)\overline{v}(t,x)\
\mathrm{d}x\mathrm{d}t=-\int_{\Sigma}\overline{v}(t,x)\partial_{\nu}u(t,x)\
\mathrm{d}S_{x}\mathrm{d}t.$ (5.6)

We observe from (5.1)

$\displaystyle
A(t,x)\cdot\nabla_{x}u_{2}(t,x)=e^{\phi}\left(\lambda\omega\cdot
A(t,x)B_{2}(t,x)+m(t,x)\right),$ (5.7)

for some $m\in L^{2}(0,T;H^{1}(\Omega))$ satisfying the following estimate

$\displaystyle\|m\|_{L^{2}(0,T;H^{k}(\Omega))}\leq
C\lambda^{k}\delta^{-3}\langle\tau,\xi\rangle^{3},\textnormal{ for
}k\in\\{0,1\\}\ \mbox{and}\ \lambda\geq\lambda_{0}.$ (5.8)

An application of the Cauchy-Schwartz inequality together with
(5.2),(5.3),(5.7) and (5.8) give us

$\displaystyle\left(A\cdot\nabla_{x}u_{2}\right)(t,x)\overline{v}(t,x)=\left(\lambda(\omega\cdot
A)B_{2}B+n\right)(t,x)$ (5.9)

for some $n\in L^{1}(Q)$ satisfying

$\displaystyle\|n\|_{L^{1}(Q)}\leq C\delta^{-6}\langle\tau,\xi\rangle^{3}.$
(5.10)

Also from (5.1) and (5.2), it is clear that for $\lambda\geq\lambda_{0}$

$\displaystyle\left|\int_{Q}\widetilde{q}(t,x)u_{2}(t,x)\overline{v}(t,x)\
\mathrm{d}x\mathrm{d}t\right|\leq C\delta^{-6}\langle\tau,\xi\rangle^{3}.$
(5.11)

Now, we find an upper bound for the right hand side of (5.6) using the
boundary Carleman estimate (3.1). We observe

$\displaystyle\int_{\Sigma}\overline{v}(t,x)\partial_{\nu}u(t,x)\
\mathrm{d}S_{x}\mathrm{d}t=\int_{\Sigma_{-,{\epsilon}/{2}}(\omega_{0})}\overline{v}(t,x)\partial_{\nu}u(t,x)\
\mathrm{d}S_{x}\mathrm{d}t+\int_{\Sigma_{+,{\epsilon}/{2}}(\omega_{0})}\overline{v}(t,x)\partial_{\nu}u(t,x)\
\mathrm{d}S_{x}\mathrm{d}t$

where $\Sigma_{+,{\epsilon}/{2}}(\omega_{0})$ is the part of lateral boundary
where we do not have any knowledge of Neumann measurements. Although the
contribution from that part can be estimated by using the boundary Carleman
estimate. Meanwhile for $\omega\in\mathbb{S}^{n-1}$ satisfying
$|\omega-\omega_{0}|\leq\frac{\epsilon}{2}$, we have
$\Sigma_{+,{\epsilon}/{2}}(\omega_{0})\subseteq\Sigma_{+}(\omega)$ and
$\Sigma_{-}(\omega)\subseteq\Sigma_{-,{\epsilon}/{2}}(\omega_{0})$. Thus, we
get from (5.4)

$\displaystyle\|\overline{v}\partial_{\nu}u\|_{L^{1}\left(\Sigma_{-,{\epsilon}/{2}}(\omega_{0})\right)}\leq
Ce^{\beta\lambda}\delta^{-3}\left\|\partial_{\nu}u\right\|_{{L^{2}}\left(\Sigma_{-,{\epsilon}/{2}}(\omega_{0})\right)}\leq
Ce^{\beta\lambda}\delta^{-6}\langle\tau,\xi\rangle^{3}\|\Lambda_{1}-\Lambda_{2}\|$
(5.12)

and

$\displaystyle\begin{aligned}
\|\overline{v}\partial_{\nu}u\|_{L^{1}\left(\Sigma_{+,{\epsilon}/{2}}(\omega_{0})\right)}&\leq
C\delta^{-3}\|e^{-\phi}\partial_{\nu}u\|_{L^{2}\left(\Sigma_{+,{\epsilon}/{2}}(\omega_{0})\right)}\leq\frac{2C\delta^{-3}}{\sqrt{\epsilon}}\sqrt{\int_{\Sigma_{+,{\epsilon}/{2}}(\omega_{0})}e^{-2\phi}|\omega\cdot\nu(x)||\partial_{\nu}u|^{2}\
\mathrm{d}S_{x}}\\\
&\leq\frac{2C\delta^{-3}}{\sqrt{\epsilon}}\sqrt{\int_{\Sigma_{+}(\omega)}e^{-2\phi}|\omega\cdot\nu(x)||\partial_{\nu}u|^{2}\
\mathrm{d}S_{x}}\\\
&\leq\frac{2C\delta^{-3}}{\sqrt{\epsilon\lambda}}\left(\|e^{-\phi}\mathcal{L}_{A_{1},q_{1}}(u)\|_{L^{2}(Q)}+\sqrt{\lambda}\|e^{-\phi}\partial_{\nu}u\|_{L^{2}\left(\Sigma_{-}(\omega)\right)}\right).\end{aligned}$

Using (3.1), we have

$\displaystyle\|\overline{v}\partial_{\nu}u\|_{L^{1}\left(\Sigma_{+,{\epsilon}/{2}}(\omega_{0})\right)}\leq\frac{2C\delta^{-3}}{\sqrt{\epsilon\lambda}}\left(\|e^{-\phi}\left(2A\cdot\nabla_{x}u_{2}+\widetilde{q}u_{2}\right)\|_{L^{2}(Q)}+\sqrt{\lambda}\|e^{-\phi}\partial_{\nu}u\|_{L^{2}\left(\Sigma_{-,{\epsilon}/{2}}(\omega_{0})\right)}\right).$

Finally using (5.3),(5.4),(5.7) and (5.8), we get

$\displaystyle\|\overline{v}\partial_{\nu}u\|_{L^{1}\left(\Sigma_{+,{\epsilon}/{2}}(\omega_{0})\right)}\leq
C\delta^{-6}\langle\tau,\xi\rangle^{3}\left(\sqrt{\lambda}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right).$
(5.13)

After dividing the integral identity (5.6) by large $\lambda$ and using
Equations (5.7) to (5.13), we obtain

$\displaystyle\left|\int_{Q}(\omega\cdot A)(t,x)B_{2}(t,x)B(t,x)\
\mathrm{d}x\mathrm{d}t\right|\leq
C\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)\delta^{-6}\langle\tau,\xi\rangle^{3}.$
(5.14)

Next, we relate the integral in the left hand side of (5.14) with the Fourier-
transform of $A$ as is done in [42].

$\displaystyle\int_{Q}(\omega\cdot A)(t,x)B_{2}(t,x)B(t,x)\
\mathrm{d}x\mathrm{d}t$ $\displaystyle\ \ \
=\int_{\mathbb{R}^{1+n}}(\omega\cdot
A)(t,x)\eta^{2}_{\delta}(t)e^{\left(-\int_{0}^{\infty}\omega\cdot
A(t,x+s\omega)\
\mathrm{d}s\right)}\frac{\xi}{|\xi|}\cdot\nabla_{x}\left(e^{\int_{\mathbb{R}}\omega\cdot
A(t,x+s\omega)\ \mathrm{d}s}e^{-i\left(t\tau+x\cdot\xi\right)}\right)\
\mathrm{d}x\mathrm{d}t.$

Now, using the decomposition
$\mathbb{R}^{n}=\mathbb{R}\omega\oplus\omega^{\perp}$, we write
$x:=x_{\perp}+s\omega$ such that $x_{\perp}\in\omega^{\perp}$ and denote

$f(s,t,x_{\perp}):=\int_{s}^{\infty}\omega\cdot A(t,x_{\perp}+\mu\omega)\
\mathrm{d}\mu.$

Using these in the previous equation, we have

$\displaystyle\begin{aligned} &\int_{Q}(\omega\cdot A)(t,x)B_{2}(t,x)B(t,x)\
\mathrm{d}x\mathrm{d}t\\\ &\ \ \
=-\int_{\mathbb{R}}\eta^{2}_{\delta}(t)\int_{\omega^{\perp}}\frac{\xi}{|\xi|}\cdot\nabla_{x}\left(e^{\int_{\mathbb{R}}\omega\cdot
A(t,x_{\perp}+s\omega)\
\mathrm{d}s}e^{-i\left(t\tau+x_{\perp}\cdot\xi\right)}\right)\left(\int_{\mathbb{R}}f^{\prime}(s,t,x_{\perp})e^{-f(s,t,x_{\perp})}\
\mathrm{d}s\right)\mathrm{d}x_{\perp}\mathrm{d}t\\\ \end{aligned}$

where $\mathrm{d}x_{\perp}$ stands for the surface measure on
$\omega^{\perp}$. Now using the Fundamental theorem of calculus and
integration by parts, we get

$\displaystyle\int_{Q}(\omega\cdot A)(t,x)B_{2}(t,x)B(t,x)\
\mathrm{d}x\mathrm{d}t$
$\displaystyle=\int_{\mathbb{R}}\eta^{2}_{\delta}(t)\int_{\omega^{\perp}}\frac{\xi}{|\xi|}\cdot\nabla_{x}\left(e^{\int_{\mathbb{R}}\omega\cdot
A(t,x_{\perp}+s\omega)\
ds}e^{-i\left(t\tau+x_{\perp}\cdot\xi\right)}\right)\left(e^{\left(-\int_{\mathbb{R}}\omega\cdot
A(t,x_{\perp}+\mu\omega)\
d\mu\right)}-1\right)\mathrm{d}x_{\perp}\mathrm{d}t,$
$\displaystyle=\int_{\mathbb{R}}\eta^{2}_{\delta}(t)\int_{\omega^{\perp}}e^{-i\left(t\tau+x_{\perp}\cdot\xi\right)}\frac{\xi}{|\xi|}\cdot\nabla_{x}\left(\int_{\mathbb{R}}\omega\cdot
A(t,x_{\perp}+s\omega)\ \mathrm{d}s\right)\mathrm{d}x_{\perp}\mathrm{d}t$
$\displaystyle=\int_{\mathbb{R}}\eta^{2}_{\delta}(t)\int_{\mathbb{R}^{n}}e^{-i\left(t\tau+x\cdot\xi\right)}\frac{\xi}{|\xi|}\cdot\nabla_{x}(\omega\cdot
A)(t,x)\
\mathrm{d}x\mathrm{d}t=i|\xi|\int_{\mathbb{R}}\eta^{2}_{\delta}(t)\int_{\mathbb{R}^{n}}e^{-i\left(t\tau+x\cdot\xi\right)}\omega\cdot
A(t,x)\ \mathrm{d}x\mathrm{d}t.$

Finally, we obtain

$\displaystyle\int_{Q}(\omega\cdot A)(t,x)B_{2}(t,x)B(t,x)\
\mathrm{d}x\mathrm{d}t=i|\xi|\ \widehat{\eta^{2}_{\delta}\omega\cdot
A}(\tau,\xi),$ (5.15)

where $\omega\cdot\xi=0$.

Let us consider the spherical cap
$\mathcal{C}_{\omega_{0}}:=\\{\omega\in\mathbb{S}^{n-1};|\omega-\omega_{0}|<\frac{\epsilon}{2}\\}$
and the set
$\mathcal{H}=\cup_{\omega\in\mathcal{C}_{\omega_{0}}}\mathcal{H}_{\omega}$
where $\mathcal{H}_{\omega}$ is the plane passing through origin and
perpendicular to $\omega$. Now for $(\tau,\xi)\in\mathbb{R}\times\mathcal{H}$,
$\lambda\geq\lambda_{0}$ and choosing $\omega(\xi)\in\mathcal{C}_{\omega_{0}}$
such that $\omega(\xi)\cdot\xi=0$ in (5.14) and (5.15), we get

$\displaystyle\left|\left(\eta_{\delta}^{2}\partial_{k}\omega(\xi)\cdot
A\right)\vspace*{-.5mm}\widehat{\ }\ (\tau,\xi)\right|\leq
C\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)\delta^{-6}\langle\tau,\xi\rangle^{3}$
(5.16)

where $\partial_{k}$ denote the partial derivative with respect to the space
variable $x_{k}$ for $k\in\\{1,2,\cdots,n\\}$. With the help of (5.16), we aim
to establish the desired stability estimate via the Fourier inversion. Because
of the Vessella’s conditional stability result [55], it is enough to derive a
uniform estimate for the Fourier transform of $\eta_{\delta}^{2}A$ over an
open cone only. This we do in the following lemma. We have crucially used the
divergence free assumption on $A$ to prove the following lemma.

######  Lemma 5.1.

If $div_{x}(A)=0$ then for $(\tau,\xi)\in\mathbb{R}\times\mathcal{C}$ and
$k\in\\{1,2,...,n\\}$ we have

$\displaystyle|\widehat{\eta_{\delta}^{2}\partial_{k}A}(\tau,\xi)|\leq
C\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)\delta^{-6}\langle\tau,\xi\rangle^{3}$
(5.17)

where, $\mathcal{C}\subseteq\mathcal{H}$ is an open cone in $\mathbb{R}^{n}$.

###### Proof.

For a fixed nonzero $\xi\in\mathcal{H}$ and $k\in\\{1,2,...,n\\}$, we choose a
set of $(n-1)$ linearly independent vectors from $\mathcal{C}_{\omega_{0}}$
which are perpendicular to $\xi$ and denoted by $\\{\omega_{i}(\xi)\\}_{1\leq
i\leq n-1}$. Then, consider the following set of $n$ linear equations

$\displaystyle\ \ \
\sum_{j=1}^{n}{\omega}_{i}^{j}(\xi)\widehat{\eta_{\delta}^{2}\partial_{k}A^{j}}(\tau,\xi)=G_{i}(\tau,\xi),\quad
i\in\\{1,2,...,n-1\\},$ (5.18) and $\displaystyle\ \ \
\sum_{j=1}^{n}\xi_{j}\widehat{\eta_{\delta}^{2}\partial_{k}A^{j}}(\tau,\xi)=0,\
\left(\textnormal{since}\ \nabla_{x}\cdot A=0\ \textnormal{gives}\
\nabla_{x}\cdot(\eta_{\delta}^{2}\ \partial_{k}A)=0\right).$ (5.19)

Here, $\\{G_{i}(\tau,\xi)\\}_{1\leq i\leq n-1}$ are real numbers with a upper
bound given by (5.17). Now, consider the matrix $M_{\xi}$ related to the
system of equations (5.18) and (5.19) as follow

$\displaystyle
M_{\xi}=\begin{pmatrix}\omega_{1}^{1}(\xi)&\omega_{1}^{2}(\xi)&\cdots&\omega_{1}^{n}(\xi)\\\
\omega_{2}^{1}(\xi)&\omega_{2}^{2}(\xi)&\cdots&\omega_{2}^{n}(\xi)\\\
\vdots&\vdots&\ddots&\vdots\\\
\omega_{n-1}^{1}(\xi)&\omega_{n-1}^{2}(\xi)&\cdots&\omega_{n-1}^{n}(\xi)\\\
\frac{\xi_{1}}{|\xi|}&\frac{\xi_{2}}{|\xi|}&\cdots&\frac{\xi_{n}}{|\xi|}\end{pmatrix}.$

From our assumption of $\\{\omega_{i}(\xi)\\}_{1\leq i\leq n-1}$, it is clear
that $M_{\xi}$ is a non-singular matrix which is homogeneous of order zero in
$\xi$. Hence, we take some $\xi_{0}\in\mathbb{S}^{n-1}\cap\mathcal{H}$ then,
$0<|\det M_{\xi_{0}}|$. Now as we vary $\xi$ in a neighborhood of $\xi_{0}$ in
$\mathbb{S}^{n-1}$ say $\widetilde{\mathcal{C}}$ we see the plane
$\mathcal{H}_{\xi}$ changes continuously giving a linearly independent set of
vectors $\\{\omega_{i}(\xi)\\}_{1\leq k\leq n-1}$ which depend continuously on
$\xi$. Then we have $c>0$ which is independent of
$\xi\in\widetilde{\mathcal{C}}$ such that

$\displaystyle 0<c\leq|\det M_{\xi}|.$ (5.20)

For any $r>0$ and $(\tau,\xi)\in\mathbb{R}\times\widetilde{\mathcal{C}}$,
using the uniform positive lower bound for the matrix $M_{\xi}$ in the system
of linear equations (5.18) and (5.19), we get

$\displaystyle\lvert\widehat{\eta_{\delta}^{2}\partial_{k}A^{j}}(r\tau,r\xi)\rvert\leq
C\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)\delta^{-6}\langle
r\tau,r\xi\rangle^{3}$ (5.21)

where $k,j\in\\{1,2,\cdots,n\\}$. Define
$\mathcal{C}\equiv\cup_{r>0}r\widetilde{\mathcal{C}},$ which is an open cone
in $\mathbb{R}^{n}$. Thus, for $(\tau,\xi)\in\mathbb{R}\times\mathcal{C}$, we
get (5.17). ∎

######  Lemma 5.2.

For $R\geq 1$ and $\delta\in(0,T/4)$, there exist $C>0$ and $\theta\in(0,1)$
such that the following estimate holds

$\displaystyle\left\||\xi|\widehat{\eta_{\delta}^{2}A}\right\|_{L^{\infty}(B(0,R))}\leq
Ce^{R(1-\theta)}\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)^{\theta}\delta^{-6\theta}R^{3\theta}.$
(5.22)

###### Proof.

Fix $k\in\\{1,2,\cdots,n\\}$ and consider the analytic function $f_{R,k}$
given by

$f_{R,k}(t,x)=\widehat{\eta_{\delta}^{2}\partial_{k}A}(Rt,Rx),\quad\mbox{ for
}R>0\mbox{ and }(t,x)\in\mathbb{R}^{n+1}.$

For any multi-index $\gamma$ in $\mathbb{R}^{1+n}$ we observe

$\displaystyle\quad\left|\partial_{(t,x)}^{\gamma}f_{R,k}(t,x)\right|$
$\displaystyle=\
\left|\partial_{(t,x)}^{\gamma}\widehat{\eta_{\delta}^{2}\partial_{k}A}(Rt,Rx)\right|=\left|\int_{\mathbb{R}^{1+n}}e^{-iR(s,y)\cdot(t,x)}(-i)^{|\gamma|}R^{|\gamma|}(s,y)^{\gamma}(\eta_{\delta}^{2}\partial_{k}A)(s,y)\
\mathrm{d}s\mathrm{d}y\right|$
$\displaystyle\leq\int_{\mathbb{R}^{1+n}}R^{|\gamma|}\left(s^{2}+|y|^{2}\right)^{\frac{|\gamma|}{2}}(\eta_{\delta}^{2}\partial_{k}A)(s,y)\
\mathrm{d}s\mathrm{d}y,$
$\displaystyle\leq(2T^{2})^{\frac{|\gamma|}{2}}R^{|\gamma|}\int_{\mathbb{R}^{1+n}}|(\eta_{\delta}^{2}\partial_{k}A)(s,y)|\
\mathrm{d}s\mathrm{d}y.$

Now using diam$(\Omega)<T$ and apriori bound of $A$, we have

$\displaystyle\quad\left|\partial_{(t,x)}^{\gamma}f_{R,k}(t,x)\right|\leq
C_{*}\ (2T^{2})^{\frac{|\gamma|}{2}}\
R^{|\gamma|}=C_{*}(\sqrt{2}T)^{|\gamma|}|\gamma|!\
\frac{R^{|\gamma|}}{|\gamma|!},$

which immediately gives

$\displaystyle\left|\partial_{(t,x)}^{\gamma}f_{R,k}(t,x)\right|\leq
C_{*}e^{R}\frac{|\gamma|!}{(T^{-1})^{|\gamma|}},\ \mbox{ for
}(t,x)\in\mathbb{R}^{n+1}\mbox{ and multi-index }\gamma.$ (5.23)

Since $f_{R,k}$ satisfies (5.23), therefore using the Vessella’s conditional
stability result [55] to $f_{R,k}$, we obtain

$\displaystyle\|f_{R,k}\|_{L^{\infty}(B(0,1))}\leq Ce^{R(1-\theta)}\
\|f_{R,k}\|_{L^{\infty}((\mathbb{R}\times\mathcal{C})\cap
B(0,1))}^{\theta},\quad\mbox{for some}\ \theta\in(0,1).$ (5.24)

Since
$\|f_{R,k}\|_{L^{\infty}(B(0,1))}=\left\|\widehat{\eta_{\delta}^{2}\partial_{k}A}\right\|_{L^{\infty}(B(0,R))}$,
therefore using lemma (5.1) and equation (5.24), we get

$\displaystyle\left\|\widehat{\eta_{\delta}^{2}\partial_{k}A}\right\|_{L^{\infty}(B(0,R))}\leq
Ce^{R(1-\theta)}\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)^{\theta}\delta^{-6\theta}(1+R^{2})^{\frac{3\theta}{2}}.$

which can be expressed for $R\geq 1$ in the following form

$\displaystyle\left\||\xi|\widehat{\eta_{\delta}^{2}A}\right\|_{L^{\infty}(B(0,R))}\leq
Ce^{R(1-\theta)}\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)^{\theta}\delta^{-6\theta}R^{3\theta}.$

∎

Now combining (5.22) and the apriori assumption on the potentials, we
establish a Sobolev bound of $A$ in terms of the partial DN map. The main
argument here is to set a comparison between the large parameters $\lambda$
and $R$. Also, we have to choose the small parameter $\delta$ accordingly. But
first we observe the following estimate

$\displaystyle\left\|\eta_{\delta}^{2}A\right\|_{L^{2}(Q)}^{\frac{2}{\theta}}$
$\displaystyle=\left(\int_{\mathbb{R}^{1+n}}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y\right)^{\frac{1}{\theta}}=\left(\int_{B(0,R)}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y+\int_{B(0,R)^{c}}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y\right)^{\frac{1}{\theta}}$ $\displaystyle\ \ \ \ \ \ \
\leq\left(\underset{T_{1}}{\underbrace{\int_{B(0,R)}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y}}\right)^{\frac{1}{\theta}}+\left(\underset{T_{2}}{\underbrace{\int_{B(0,R)^{c}}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y}}\right)^{\frac{1}{\theta}}.$ (5.25)

$T_{2}$ can be easily estimated after using the apriori assumptions for the
potentials. We see

$\displaystyle\begin{aligned}
T_{2}&=\int_{B(0,R)^{c}}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y\\\ &\ \ \ \ \
\leq\frac{1}{R^{2}}\int_{\mathbb{R}^{1+n}}\langle\tau,\xi\rangle^{2}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y\leq\
\frac{1}{R^{2}}\|\eta_{\delta}^{2}A\|_{H^{1}(Q)}^{2}\leq\
\frac{C}{\delta^{2}R^{2}}.\end{aligned}$ (5.26)

To estimate $T_{1}$, we use lemma 5.2 . We break $T_{1}$ into two parts. We
consider $T_{1}=T_{11}+T_{12}$, where

$\displaystyle\begin{aligned} T_{11}&:=\int_{B(0,R)\cap\\{(s,y);\ |y|\leq
R^{-\frac{3}{n}}\\}}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y\\\ &\ \ \ \ \ \ \ \ \leq\
\|\widehat{\eta_{\delta}^{2}A}\|_{L^{\infty}(\mathbb{R}^{n+1})}\int_{-R}^{R}\int_{|y|\leq
R^{-\frac{3}{n}}}\ \mathrm{d}s\mathrm{d}y\leq\ CR^{-2}\end{aligned}$ (5.27)

and

$\displaystyle\begin{aligned} T_{12}&:=\int_{B(0,R)\cap\left\\{(s,y);\
|y|>R^{-\frac{3}{n}}\right\\}}|\widehat{\eta_{\delta}^{2}A}|^{2}(s,y)\
\mathrm{d}s\mathrm{d}y\\\ &\ \ \ \ \ \ \ \ \leq
e^{2R(1-\theta)}\left(\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)^{2\theta}\delta^{-12\theta}R^{6\theta+n+1+\frac{6}{n}}.\end{aligned}$
(5.28)

Because of the support condition of $\\{\eta_{\delta}\\}_{\delta>0}$, we have

$\displaystyle\|A\|_{L^{2}(Q)}^{2}\leq\|\eta_{\delta}^{2}A\|_{L^{2}(Q)}^{2}+\
C\delta.$ (5.29)

Taking $\alpha=6+\frac{n^{2}+n+6}{n\theta}$ and combining (5.25)-(5.28), we
obtain

$\displaystyle\|A\|_{L^{2}(Q)}^{\frac{2}{\theta}}\leq$ $\displaystyle
C\left(\frac{R^{\alpha}}{\delta^{12}}e^{\frac{2R(1-\theta)}{\theta}}\left(\frac{1}{\lambda}+e^{2\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|^{2}\right)+\frac{1}{\delta^{\frac{2}{\theta}}R^{\frac{2}{\theta}}}+\delta^{\frac{1}{\theta}}\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\underset{I}{\underbrace{\frac{R^{\alpha}e^{\frac{2R(1-\theta)}{\theta}}}{\lambda\delta^{12}}}}+\underset{II}{\underbrace{\frac{R^{\alpha\theta}e^{\frac{2R(1-\theta)}{\theta}+2\beta\lambda}}{\delta^{12}}\|\Lambda_{1}-\Lambda_{2}\|^{2}}}+\underset{III}{\underbrace{\frac{1}{\delta^{\frac{2}{\theta}}R^{\frac{2}{\theta}}}}}+\underset{IV}{\underbrace{\delta^{\frac{1}{\theta}}}}\right).$
(5.30)

We now choose $\lambda,\delta$ and $R$ in a way so that the terms (I), (III)
and (IV) in (5.30) are comparable. That is when

$\displaystyle\delta=\frac{1}{R^{\frac{2}{3}}}\ \ \textnormal{and}\ \
\lambda=R^{\alpha+8+\frac{2}{3\theta}}e^{\frac{2R(1-\theta)}{\theta}}$ (5.31)

and hence there exists $\kappa>0$ (independent of $R$) such that II of (5.30)
can be bounded by

$\displaystyle e^{e^{\kappa R}}\|\Lambda_{1}-\Lambda_{2}\|^{2}.$ (5.32)

Combining (5.30)-(5.32), it is clear that

$\displaystyle\|A\|_{L^{2}(Q)}^{\frac{2}{\theta}}\leq
C\left(\frac{1}{R^{\frac{2}{3\theta}}}+e^{e^{\kappa
R}}\|\Lambda_{1}-\Lambda_{2}\|^{2}\right).$ (5.33)

We now choose $R>0$ large enough (which in turn depends on the smallness of
partial DN map) which is
$R=\frac{1}{\kappa}\log\big{|}\log\|\Lambda_{1}-\Lambda_{2}\|\big{|}$, so that
(5.33) becomes

$\displaystyle\|A\|_{L^{2}(Q)}^{\frac{2}{\theta}}\leq
C\left(\|\Lambda_{1}-\Lambda_{2}\|+\left(\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right)^{-\frac{2}{3\theta}}\right).$
(5.34)

We note that, (5.34) can be easily derived for the case when
$\|\Lambda_{1}-\Lambda_{2}\|$ is not so small. This concludes the proof for
stability of first order coefficients from the partial DN map.

Now we establish the stability result for the zeroth order term. There will be
no zeroth order term left in (5.6) once we divide it by large $\lambda$. So we
have to make necessary changes for deriving Fourier estimates. We explicitly
use here the stability result for the first order terms (5.34). We consider a
different exponentially growing solutions for $\mathcal{L}_{A_{2},q_{2}}$,
whereas the geometric optics for $\mathcal{L}_{-A_{1},\overline{q}_{1}}$ is
same as before which is (5.2). We have

$\displaystyle u_{2}(t,x)=e^{\phi(t,x)}\left(B_{2}+R_{2}\right)(t,x)$

where

$\displaystyle
B_{2}(t,x)=e^{-i\left(t\tau+x\cdot\xi\right)}\eta_{\delta}(t)e^{\left(\int_{0}^{\infty}\omega\cdot
A_{2}(t,x+s\omega)\ \mathrm{d}s\right)},$

and $R_{2}\in L^{2}\left(0,T;H^{2}(\Omega)\right)$ satisfying for
$k\in\\{0,1,2\\}$

$\displaystyle\|R_{2}\|_{L^{2}(0,T;H^{k}_{\lambda}(\Omega))}\leq
C\lambda^{-1+k}\delta^{-3}\langle\tau,\xi\rangle^{3}.$ (5.35)

For convenience, we rewrite the integral inequality

$\displaystyle 2\int_{Q}(A\cdot\nabla_{x}u_{2})(t,x)\overline{v}(t,x)\
\mathrm{d}x\mathrm{d}t+\int_{Q}\widetilde{q}(t,x)u_{2}(t,x)\overline{v}(t,x)\
\mathrm{d}x\mathrm{d}t=\int_{\Sigma}\overline{v}(t,x)\partial_{\nu}u(t,x)\
\mathrm{d}S_{x}\mathrm{d}t.\quad$ (5.36)

First, we simplify all the terms present in left hand side of (5.36). We
observe

$\displaystyle\begin{aligned}
\widetilde{q}(t,x)u_{2}(t,x)\overline{v}(t,x)=\widetilde{q}e^{-it\tau-
ix\cdot\xi}\eta_{\delta}^{2}(t)e^{-\int_{0}^{\infty}\omega\cdot
A(t,x+s\omega)\mathrm{d}s}+B_{2}(t,x)R(t,x)+B(t,x)R_{2}(t,x).\end{aligned}$
(5.37)

We use Cauchy-Schwarz inequality alongwith (5.3) and (5.35) to obtain

$\displaystyle\|B_{2}R\|_{L^{1}(Q)}+\|BR_{2}\|_{L^{1}(Q)}\leq\frac{C}{\lambda}\delta^{-3}\langle\tau,\xi\rangle^{3}$
(5.38)

The other term present in the L.H.S of (5.36) is

$\displaystyle
2(A\cdot\nabla_{x}u_{2})(t,x)\overline{v}(t,x)=2\left(\lambda\omega\cdot
AB_{2}+\lambda\omega\cdot
AR_{2}+A\cdot\nabla_{x}B_{2}+A\cdot\nabla_{x}R_{2}\right)(t,x)\left(\overline{B}+\overline{R}\right)(t,x)$
(5.39)

which is estimated by using the remainder term estimates given in (5.3) and
(5.35)

$\displaystyle\left|\int_{Q}(A\cdot\nabla_{x}u_{2})(t,x)\overline{v}(t,x)\
\mathrm{d}x\mathrm{d}t\right|\leq
C\lambda\|A\|_{L^{2}(Q)}\delta^{-3}\langle\tau,\xi\rangle^{3}.$ (5.40)

To estimate the boundary term in the R.H.S of (5.36), we proceed as before and
obtain

$\displaystyle\left|\int_{\Sigma}\overline{v}(t,x)\partial_{\nu}u(t,x)\
\mathrm{d}S_{x}\mathrm{d}t\right|\leq
C\left(\sqrt{\frac{\mathcal{K}}{\lambda}}+\delta^{-6}\langle\tau,\xi\rangle^{3}e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)$
(5.41)

Here $\mathcal{K}$ is the R.H.S of the boundary Carleman estimate (3.1)
applied to $\mathcal{L}_{A_{1},q_{1}}$ for $u$,

$\displaystyle\mathcal{K}=\int_{Q}e^{-2\phi}|\mathcal{L}_{A_{1},q_{1}}u|^{2}\
\mathrm{d}x\mathrm{d}t+\lambda\int_{\Sigma_{-}(\omega)}e^{-2\phi}|\omega\cdot\nu(x)||\partial_{\nu}u|^{2}\
\mathrm{d}S_{x}\mathrm{d}t.$ (5.42)

From (5.5) we have,
$\mathcal{L}_{A_{1},q_{1}}u=2A\cdot\nabla_{x}u_{2}+\tilde{q}u_{2}$. Hence we
see

$\displaystyle
e^{-\phi}\mathcal{L}_{A_{1},q_{1}}u=2\left(A\cdot\nabla_{x}B_{2}+\lambda\omega\cdot
AR_{2}+A\cdot\nabla_{x}B_{2}+A\cdot\nabla_{x}R_{2}\right)+\tilde{q}(B_{2}+R_{2}).$
(5.43)

Thus, we use (5.35) to obtain

$\displaystyle\mathcal{K}\leq
C\left(\lambda^{2}\|A\|_{L^{2}(Q)}^{2}+1+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|^{2}\right)\delta^{-12}\langle\tau,\xi\rangle^{6}.$
(5.44)

Combining (5.36)-(5.44), we conclude for
$(\tau,\xi)\in\mathbb{R}\times\mathcal{H}$

$\displaystyle\begin{aligned}
\left|\widehat{\eta_{\delta}^{2}\widetilde{q}}(\tau,\xi)\right|&=\left|\int_{Q}e^{-i\left(t\tau+x\cdot\xi\right)}\eta_{\delta}^{2}(t)\widetilde{q}(t,x)\
\mathrm{d}x\mathrm{d}t\right|\\\ &\ \ \ \ \ \leq
C\left(\lambda\|A\|_{L^{2}(Q)}+\frac{1}{\sqrt{\lambda}}+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|\right)\delta^{-6}\langle\tau,\xi\rangle^{3}.\end{aligned}$
(5.45)

Basically (5.45) gives estimate for the Fourier transform of
$\eta_{\delta}^{2}\widetilde{q}$ over the cone $\mathbb{R}\times\mathcal{H}$
in $\mathbb{R}^{1+n}$. So we apply Vessella’s conditional stability result
[55] as done before to obtain this estimate over arbitrary large balls.
Mimicing the arguments presented before, we get the following estimate similar
to (5.30)

$\displaystyle\|\widetilde{q}\|_{L^{2}(Q)}^{\frac{2}{\theta}}\leq
C\left(\frac{R^{\alpha^{\prime}}}{\delta^{12}}e^{\frac{2R(1-\theta)}{\theta}}\left(\lambda^{2}\|A\|_{L^{2}(Q)}^{2}+1+e^{\beta\lambda}\|\Lambda_{1}-\Lambda_{2}\|^{2}\right)+\frac{1}{\delta^{\frac{2}{\theta}}R^{\frac{2}{\theta}}}+\delta^{\frac{1}{\theta}}\right)$
$\displaystyle\ \leq
C\left(\underset{I}{\underbrace{\frac{R^{\alpha^{\prime}}e^{\frac{2R(1-\theta)}{\theta}}}{\delta^{12}}\lambda^{2}\|A\|_{L^{2}(Q)}^{2}}}+\underset{II}{\underbrace{\frac{R^{\alpha^{\prime}}e^{\frac{2R(1-\theta)}{\theta}}}{\delta^{12}}}}+\underset{III}{\underbrace{\frac{R^{\alpha\theta}e^{\frac{2R(1-\theta)}{\theta}+\beta\lambda}}{\delta^{12}}\|\Lambda_{1}-\Lambda_{2}\|^{2}}}+\underset{IV}{\underbrace{\frac{1}{\delta^{\frac{2}{\theta}}R^{\frac{2}{\theta}}}}}+\underset{V}{\underbrace{\delta^{\frac{1}{\theta}}}}\right)$
(5.46)

We choose $\delta$ and $R$ such that (II),(IV) and (V) of (5.46) are
comparable. That is when

$\displaystyle\delta=\frac{1}{R^{\frac{2}{3}}}\ \textnormal{and}\
\lambda=R^{\alpha^{\prime}+8+\frac{2}{3\theta}}e^{\frac{2R(1-\theta)}{\theta}}$

here $\alpha^{\prime}=6+\frac{n+1}{\theta}$. Now using the stability result
(5.34), we obtain

$\displaystyle\|\widetilde{q}\|_{L^{2}(Q)}^{\frac{2}{\theta}}\leq
C\left({e^{\kappa R}}\|\Lambda_{1}-\Lambda_{2}\|^{2\mu_{1}}+{e^{\kappa
R}}\left|\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right|^{-2\mu_{2}}+e^{e^{\kappa
R}}\|\Lambda_{1}-\Lambda_{2}\|^{2}+\frac{1}{R^{\frac{2}{3\theta}}}\right)$
(5.47)

for some constants $\kappa>0$ and $\mu_{1},\mu_{2}>0$. Taking
$R=\frac{\mu_{1}}{\kappa}\log\log|\log\|\Lambda_{1}-\lambda_{2}\||$, we get
from (5.47) that $\|\widetilde{q}\|_{L^{2}(Q)}^{\frac{2}{\theta}}$ has the
upper bound

$\displaystyle\begin{aligned}
&\|\Lambda_{1}-\Lambda_{2}\|^{2\mu_{1}}\left(\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right)^{-2\mu_{2}}+\left(\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right)^{-\mu_{2}}\\\
&\ \ \ \ \ \ \ \ \ \ \
+\|\Lambda_{1}-\Lambda_{2}\|^{2}|\log\|\Lambda_{1}-\Lambda_{2}\||+\left(\log\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right)^{-\frac{2}{3\theta}}.\end{aligned}$
(5.48)

We note that our choice of $R$ related to the smallness of $\delta$ and
largeness of $\lambda$. Hence, the estimate (5.48) is valid only when
$\|\Lambda_{1}-\Lambda_{2}\|$ is small enough. The other case follows easily.
Also, we need smallness of $\|\Lambda_{1}-\Lambda_{2}\|$ such that the
following hold

$\displaystyle\|\Lambda_{1}-\Lambda_{2}\|^{\mu_{1}}\left(\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right)^{-2\mu_{2}}+\|\Lambda_{1}-\Lambda_{2}\||\log\|\Lambda_{1}-\Lambda_{2}\||\leq
C.$

Thus for both the cases, we arrive at the following estimate where
$C,\alpha_{1}$ and $\alpha_{2}>0$

$\displaystyle\|\widetilde{q}\|_{L^{2}(Q)}\leq
C\left(\|\Lambda_{1}-\Lambda_{2}\|_{L^{2}(Q)}^{\alpha_{1}}+\left|\log\left|\log|\log\|\Lambda_{1}-\lambda_{2}\|\right|\right|^{-\alpha_{2}}\right).$
(5.49)

Now we want to prove the stability result for $q:=q_{1}-q_{2}$. We recall

$\displaystyle q(t,x)=\widetilde{q}(t,x)+\nabla_{x}\cdot
A(t,x)+\left(|A_{1}|^{2}-|A_{2}|^{2}\right)(t,x).$

Hence, we obtain the following

$\displaystyle\|q\|_{L^{2}(Q)}\leq\|\widetilde{q}\|_{L^{2}(Q)}+(2m+1)\|A\|_{L^{2}(0,T;H^{1}(\Omega))}.$
(5.50)

Since our assumptions on the first order perturbations are more than $H^{1}$,
we can translate the $L^{2}$ norm estimates to that of $H^{1}$ using
logarithmic convexity for Sobolev norms. Thus there exist $C>0$ and
$\theta\in(0,1)$ depending only on $m$ and $Q$ so that we have

$\displaystyle\|A\|_{H^{1}(Q)}\leq C\|A\|_{L^{2}(Q)}^{\theta}$ (5.51)

Using (5.51), the $L^{2}$ stability results in (5.34) and (5.49) in (5.50), we
obtain

$\displaystyle\|q\|_{L^{2}(Q)}\leq
C\left(\|\Lambda_{1}-\Lambda_{2}\|^{\beta_{1}}+\left|\log|\log|\log\|\Lambda_{1}-\Lambda_{2}\||\right|^{-\beta_{2}}\right).$

for some $C,\beta_{1}$ and $\beta_{2}>0$. This completes the proof of Theorem
2.1.

## Acknowledgments

The authors would like to express their sincere gratitude and thanks to Venky
Krishnan for many enlightening discussions and useful comments on this
project.

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