# Well-posedness for a class of pseudo-differential hyperbolic equations on
the torus

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS>,  Julio Delgado  Julio Delgado: Departmento de
Matematicas Universidad del Valle Cali-Colombia E-mail address
<EMAIL_ADDRESS>and  Michael Ruzhansky  Michael
Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University, Belgium and School of Mathematical Sciences Queen Mary
University of London United Kingdom E-mail address<EMAIL_ADDRESS>
###### Abstract.

In this paper we establish the well-posedness of the Cauchy problem for a
class of pseudo-differential hyperbolic equations on the torus. The class
considered here includes a space-like fractional order Laplacians. By applying
the toroidal pseudo-differential calculus we establish regularity estimates,
existence and uniqueness in the scale of the standard Sobolev spaces on the
torus.

###### Key words and phrases:

Fractional hyperbolic equations, periodic pseudo-differential operators,
energy estimates, microlocal analysis.

###### 2020 Mathematics Subject Classification:

Primary 35S10, 35S05; Secondary 35R11, 35A01.

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N:Analysis and
Partial Differential Equations and by the Methusalem programme of the Ghent
University Special Research Fund(BOF)(Grant number 01M01021). D. Cardona is
also supported by the Research Foundation-Flanders (FWO) under the
postdoctoral grant No 1204824N. J. Delgado is also supported by Vice. Inv.
Universidad del Valle Grant CI 71329, MathAmSud and Minciencias-Colombia under
the project MATHAMSUD21-MATH-03. Michael Ruzhansky is also supported by EPSRC
grants EP/R003025/2 and EP/V005529/1.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
    2. 1.2 Periodic vs Euclidean models
    3. 1.3 State-of-the-art
    4. 1.4 Main result and organisation of the paper
  2. 2 Pseudo-differential calculus on the torus
    1. 2.1 Scalar-valued classes on the torus
    2. 2.2 Matrix-valued classes on the torus
  3. 3 Pseudo-differential hyperbolic equations on the torus

##  1\. Introduction

###  1.1. Outline

In this work we analyse the well-posedness for hyperbolic periodic problems
associated to positive elliptic pseudo-differential operators on the torus
$\mathbb{T}^{n}\cong[0,1)^{n}$. Although the analysis carried out here can be
extended to general compact Lie groups, here we concentrate our attention to
the commutative case, namely, the case of the torus, due to the simplified
formulation of the pseudo-differential calculus in this context, established
in terms of the periodic Fourier analysis as developed in [26, 27]. The case
of general compact Lie groups will be considered in a subsequent work with the
periodic case as a fundamental model.

To precise the notation, let $P(x,D)$ be an elliptic positive pseudo-
differential operator of order $\nu>0$ on the torus. In terms of the periodic
pseudo-differential calculus developed in [26, 27], to the operator $P(x,D)$
one can associate a symbol $p:=p(x,\xi)$ globally defined on the phase space
$\mathbb{T}^{n}\times\mathbb{Z}^{n},$ allowing the integral representation

$\displaystyle
P(x,D)f(x)=\sum_{\xi\in\mathbb{Z}^{n}}\int\limits_{\mathbb{T}^{n}}e^{2\pi
i(x-y)\cdot\xi}p(x,\xi)f(y)dy,\,f\in C^{\infty}(\mathbb{T}^{n}).$ (1.1)

We have used the notation $x\cdot\xi=x_{1}\xi_{1}+\cdots+x_{n}\xi_{n}$ for the
standard inner product. In this setting the ellipticity condition means that
the symbol satisfies the estimate

$C_{1}\lvert\xi\rvert^{\nu}\leq\lvert p(x,\xi)\rvert\leq
C_{2}\lvert\xi\rvert^{\nu},\,\xi\in\mathbb{Z}^{n},\xi\neq
0,\,x\in\mathbb{T}^{n},$ (1.2)

for some positive constants $C_{1},C_{2}>0.$ Then, with the notation above,
for a given time $T>0,$ our main goal is to establish the well-posedness for
the following Cauchy problem :

$\left\\{\begin{array}[]{rl}{\displaystyle\frac{\partial^{2}v}{\partial
t^{2}}}=&-P(x,D)v+\omega,\,\,\mbox{ (in the sense of
}\mathcal{D}^{\prime}(]0,T[\times\mathbb{T}^{n}))\\\ {\displaystyle
v}(0)=&f,\\\ {\displaystyle\frac{\partial v}{\partial t}}(0)=&g,\\\
\end{array}\right.$ (1.3)

where $f\in H^{s}(\mathbb{T}^{n}),\,g\in H^{s-\frac{\nu}{2}}(\mathbb{T}^{n}),$
and $\omega\in L^{2}([0,T],H^{s-\frac{\nu}{2}}(\mathbb{T}^{n}))$, for some
$s\in\mathbb{R}$. We clarify the contributions of this note in Theorem 1.1
below, but first, we discuss the main differences between the Euclidean
diffusion operators and the ones in the periodic setting. For instance, a
first look at the models of the form (1.3) could give the impression that
their analysis can be carried out by the standard periodisation techniques,
however, this is not the case as we will explain in the simplest case of the
fractional Laplacian on the torus. Then, the situation changes dramatically if
one considers non-local perturbations of this operator even of lower order as
well as perturbations with variable coefficients. For the main aspects about
the periodisation of Euclidean models we refer the reader to [26, 27].

###  1.2. Periodic vs Euclidean models

Our setting includes the case of the fractional Laplacian
$P(x,D)=(-\Delta)^{\frac{\nu}{2}}.$ Let us observe that the toroidal
quantization formula in (1.1) and the pseudo-differential calculus developed
in [26, 27] will allow us to consider the diffusion operator
$(-\Delta)^{\frac{\nu}{2}}$ as a pseudo-differential operator for any $\nu>0.$
Indeed, the functional calculus for the Laplacian on the torus implies the
integral formula

$\displaystyle(-\Delta)^{\frac{\nu}{2}}f(x)=\sum_{\xi\in\mathbb{Z}^{n}}\int_{\mathbb{T}^{n}}e^{2\pi
i(x-y)\cdot\xi}(2\pi)^{\nu}\lvert\xi\rvert^{\nu}f(y)dy,\,f\in
C^{\infty}(\mathbb{T}^{n}).$ (1.4)

From this formula one can deduce the non-locality of the operator if e.g.
$\nu\notin 2\mathbb{N}_{0}.$ This “global pseudo-differential representation”
is not available for the fractional Laplacian on ${\mathbb{R}}^{n}$, where it
is only a pseudo-diferential operator when $\nu$ is an even integer, due to
the lack of smoothness of the symbol at the origin.

One reason to emphasize about the role of the fractional Laplacian on the
torus as a crucial model in the analysis of the hyperbolic problems as in
(1.3) came from the following fact. One may think that the analysis of non-
local operators like the power $(-\Delta)^{\frac{\nu}{2}}$ can be obtained by
periodisation techniques of the results for the fractional Laplacian on
${\mathbb{R}}^{n}$. However, the analysis of periodic models changes
dramatically due to the non-locality of the operators. Another obstruction
came with the structure of distributions on the torus. Indeed, observe that
the class of smooth functions, as well as the class of $L^{p}$-functions on
$\mathbb{T}^{n}$, cannot be identified with the Schwartz class or with the
$L^{p}$-spaces on ${\mathbb{R}}^{n},$ respectively, even with assumptions of
extensions from the torus to the real line under the periodicity condition.

On the other hand, a more distinctive fact arises with the behavior of the
singularities of conormal distributions. Indeed, the kernel of the fractional
Laplacian on the Euclidean space is not an integrable function. Then, as it
was pointed out in [24], its “periodisation” in principle has just a formal
meaning. It is not clear that the fractional Laplacian acting on periodic
functions coincides with the fractional Laplacian on the torus as defined as a
Fourier multiplier of Fourier series. We refer the reader to Roncal and Stinga
[24] for the analysis of the fractional Laplacian on the torus and for its
properties in comparison with the ones of the fractional Laplacian on the
Euclidean space.

###  1.3. State-of-the-art

The research in evolution equations governed by the fractional Laplacian and
other elliptic pseudo-differential operators on different structures has been
intensive in the last decades. However the literature on the fractional
hyperbolic case is less known compared with the ones involving models of
fractional diffusion. Fractional hyperbolic equations have been treated for
instance in [12] within the framework of Weyl-Hörmander calculus. We refer the
reader to Remark 1.2 for a discussion about hyperbolic problems for pseudo-
differential terms in the Euclidean space as well as in the periodic setting.
For the analysis of boundary value problems for the Euclidean fractional
Laplacian we refer to Grubb [17].

On the other hand, the wide variety of applications of the fractional
diffusion is spread throughout fluid mechanics [9], [22], mathematical finance
[10], fractional dynamics [18], [21], strange kinetics and anomalous
transport, see [19],[11],[3] and the references therein. Also, drift-diffusion
equations with fractional diffusion have intensely attracted the interest
during the last 12 years starting with the works of Caffarelli and Vasseur in
[3] and their subsequent developments. In a particular but crucial setting,
the fractional Laplacian, which is the model of the operators considered in
this work, is an interesting object in its own, and on the torus it has been
studied in [24]. Several aspects of the harmonic analysis of the fractional
Laplacian on lattices also have been investigated in [7] and in [8]. Recent
works on the fractional Laplacian and its different generalizations can be
found in [16], [4], [2], [17], [14], [28] and the references therein. An
accessible presentation of the fractional Sobolev spaces and the fractional
Laplacian can be found in [29].

###  1.4. Main result and organisation of the paper

We will apply the point of view of the global quantisation for pseudo-
differential operators on the torus as developed in [26, 27]. In order to
present our main result we will employ the following additional notation.

  * •

In the periodic hyperbolic model (1.3), the symbol $p:=p(x,\xi)$ of the
operator will be considered in the Hörmander classes
$S^{\nu}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$ see Section 2 for
the definition of these classes.

  * •

The class of pseudo-differential operators
$\Psi^{\nu}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ denotes the
family of pseudo-differential operators with symbols in the toroidal class
$S^{\nu}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$ We refer to this
class as the toroidal/periodic $(\rho,\delta)$-Hörmander class of order $\nu.$

  * •

We will denote by $H^{s}(\mathbb{T}^{n})$ the standard Sobolev space of order
$s\in\mathbb{R},$ on the torus $\mathbb{T}^{n},$ see Remark 2.14 for details.

Our main theorem can be stated in the following way.

######  Theorem 1.1.

Let $\nu>0$, a given time $T>0$, and let $0\leq\delta<\rho\leq 1$. Let
$s\in\mathbb{R}$, $f_{0}\in H^{s}(\mathbb{T}^{n}),\,f_{1}\in
H^{s-\frac{\nu}{2}}(\mathbb{T}^{n})$ and let $\omega\in
L^{2}([0,T],H^{s-\frac{\nu}{2}}(\mathbb{T}^{n}))$. Let
$P(x,D)\in\Psi^{\nu}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ be a
positive elliptic pseudo-differential operator in the
$(\rho,\delta)$-Hörmander class of order $\nu$.

Then, there exists a unique solution $u\in C([0,T],H^{s}(\mathbb{T}^{n}))$ of
the Cauchy problem

$\left\\{\begin{array}[]{rl}{\displaystyle\frac{\partial^{2}u}{\partial
t^{2}}}\,\,\,\,=&-P(x,D)u+w,\,\,\mbox{ (in the sense of
}\mathcal{D}^{\prime}(]0,T[\times\mathbb{T}^{n}))\\\ {\displaystyle
u}(0)=&f_{0},\\\ {\displaystyle\frac{\partial u}{\partial
t}}(0)=&f_{1},\end{array}\right.$ (1.5)

satisfying the following energy estimate

$\|u(t)\|_{H^{s}(\mathbb{T}^{n})}^{2}\leq
Ce^{Ct}\left(\|f_{0}\|_{H^{s}(\mathbb{T}^{n})}^{2}+\|f_{1}\|_{H^{s-\frac{\nu}{2}}(\mathbb{T}^{n})}^{2}+\int\limits_{0}^{t}\|w(\tau)\|^{2}_{H^{s-\frac{\nu}{2}}(\mathbb{T}^{n})}d\tau\right).$
(1.6)

Consequently, if $w\in C^{\infty}([0,T]\times\mathbb{T}^{n})$ and every
$f_{i}\in C^{\infty}(\mathbb{T}^{n})$ is smooth, then the solution $u$ belongs
to the class $C^{\infty}([0,T]\times\mathbb{T}^{n})$.

######  Remark 1.2.

Fractional hyperbolic equations have been treated for instance in [12], where
in particular a version of Theorem 1.1 has been proved within the framework of
the Weyl-Hörmander calculus. In particular, such a result provides the well-
posedness for hyperbolic equations in the Euclidean Hörmander classes.
However, in view of the discussion above, our main Theorem 1.1 cannot be
recovered from its Euclidean analogue just by periodisation techniques, see
Subsection 1.2 for details.

######  Remark 1.3.

In order to prove the energy estimate in Theorem 1.1, the hyperbolic problem
is simplified to a vector-valued diffusion model. In this context, we will
employ a matrix-valued version of the toroidal calculus developed in [26, 27],
see Subsection 2.2 for details. The energy estimate obtained from Theorem 1.1
applied to the fractional Laplacian on the torus can be found in Corollary
3.7. In this case we use the property
$(-\Delta)^{\frac{\nu}{2}}\in\Psi^{\nu}_{1,0}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$
However, other examples in the $(\rho,\delta)$-setting appear e.g. if
$\delta=0$ with oscillating multipliers

$P(x,D)=(-\Delta)^{\frac{\nu}{2}}e^{i(-\Delta)^{\frac{1-\rho}{2}}}\in\Psi^{\nu}_{\rho,0}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$
(1.7)

with the oscillating parameter $\rho$ satisfying the inequality $0<\rho\leq
1.$

######  Remark 1.4.

We also observe that when localising the Hörmander classes from the Euclidean
space to the torus one can construct the localised classes
$\Psi^{\nu}_{\rho,\delta,loc}(\mathbb{T}^{n})$ but due to the structure of the
asymptotic expansions on these classes, the conditions $0\leq\delta<\rho\leq
1,$ and $\rho\geq 1-\delta$ arise. In particular these two restrictions imply
the inequality $\rho>1/2.$ Our Theorem 1.1 operates below the range $\rho\leq
1/2,$ where the construction under local coordinate systems is not available.
For instance, our approach allow the analysis of the operators in (1.7) when
$\rho\leq 1/2.$

This work is organised as follows:

  * •

in Section 2 we give a brief review of basic preliminaries on periodic pseudo-
differential operators.

  * •

In Section 3 we develop our analysis for the proof of our main result. We
first obtain energy estimates for first-order systems, see Theorem 3.5. Then,
this analysis is used in the proof of Theorem 1.1.

##  2\. Pseudo-differential operators on the torus

In this section we review some basic elements of the pseudo-differential
calculus on the torus. For a comprehensive study on this theory we refer to
[26, 27] and to [26] for the quantisation on the torus.

###  2.1. Scalar-valued classes on the torus

The use of global representations by Fourier series instead of local
representations in coordinate charts as it is more customary when working on
manifolds has some advantages. In particular, it will lead us to improve some
results regarding the range of the exponent in the fractional Laplacian on the
torus.

For our further analysis let us fix the required notation. Let
$\mathbb{T}^{n}={\mathbb{R}}^{n}/\mathbb{Z}^{n}$ be the $n$-dimensional torus.
As it is well known, the Schwartz space $\mathcal{S}({\mathbb{R}}^{n})$ is
convenient to define the Euclidean Fourier transform. In the periodic setting,
its counterpart is the corresponding Schwartz space
$\mathcal{S}(\mathbb{Z}^{n})$ with respect to the dual variable. We introduce
it as follows.

######  Definition 2.1.

Let $\mathcal{S}(\mathbb{Z}^{n})$ denote the space of rapidly decaying
functions $\varphi:\mathbb{Z}^{n}\rightarrow\mathbb{C}$. That is,
$\varphi\in\mathcal{S}(\mathbb{Z}^{n})$ if for any $M>0$ there exists a
constant $C_{\varphi,\,M}$ such that

$\forall\xi\in\mathbb{Z}^{n},\,\,\lvert\varphi(\xi)\rvert\leq
C_{\varphi,\,M}\langle\xi\rangle^{-M}$

for all $\xi\in\mathbb{Z}^{n}$. The topology on $\mathcal{S}(\mathbb{Z}^{n})$
is given by the seminorms $p_{k}$, where $k\in\mathbb{N}_{0}$ and
$p_{k}(\varphi):=\sup\limits_{\xi\in\mathbb{Z}^{n}}\langle\xi\rangle^{k}\lvert\varphi(\xi)\rvert.$

We can now define the toroidal or periodic Fourier transform.

######  Definition 2.2.

We denote by $\mathcal{F}_{\mathbb{T}^{n}}$ the toroidal Fourier transform
$\mathcal{F}_{\mathbb{T}^{n}}:C^{\infty}(\mathbb{T}^{n})\rightarrow\mathcal{S}(\mathbb{Z}^{n}),\,\,\phi\mapsto\widehat{\phi}\,\,$
by

$(\mathcal{F}_{\mathbb{T}^{n}}\varphi)(\xi)=\int\limits_{\mathbb{T}^{n}}e^{-i2\pi
x\cdot\xi}\varphi(x)dx,\;\,\xi\in\mathbb{Z}^{n}.$

######  Remark 2.3.

Note that the notion of Fourier transform is related to the spectral
decomposition of $L^{2}(\mathbb{T}^{n})$ with respect to the collection
$\\{e^{2\pi ix\cdot k}\\}_{k\in\mathbb{Z}^{n}},$ which is an orthonormal basis
of $L^{2}(\mathbb{T}^{n})$. Also, we sometimes write $\widehat{\phi}(\xi)$
instead of $(\mathcal{F}_{\mathbb{T}^{n}}\varphi)(\xi)$. The toroidal Fourier
transform is a bijection and its inverse
$\mathcal{F}_{\mathbb{T}^{n}}^{-1}:\mathcal{S}(\mathbb{Z}^{n})\rightarrow
C^{\infty}(\mathbb{T}^{n})$ is given by

$\forall
x\in\mathbb{T}^{n},\,\,\phi(x)=\sum\limits_{\xi\in\mathbb{Z}^{n}}e^{i2\pi
x\cdot\xi}\widehat{\phi}(\xi).$

Thus, for every $h\in\mathcal{S}(\mathbb{Z}^{n})$ we have

$\forall
x\in\mathbb{T}^{n},\,\,(\mathcal{F}_{\mathbb{T}^{n}}^{-1}h)(x)=\sum\limits_{\xi\in\mathbb{Z}^{n}}e^{i2\pi
x\cdot\xi}h(\xi).$

A fundamental notion for introducing the Hörmander classes on the torus is the
one of difference operators. It will allow us to “differentiate” symbols with
respect to the discrete variable $\xi\in\mathbb{Z}.$ We record this notion as
follows.

######  Definition 2.4.

(Finite differences $\Delta_{\xi}^{\alpha}$). Let
$\sigma:\mathbb{Z}^{n}\rightarrow\mathbb{C}$ and $1\leq i,j\leq n.$ Let
$\delta_{j}\in\mathbb{N}_{0}^{n}$ be defined by

$(\delta_{j})_{i}:=\left\\{\begin{array}[]{rl}1,&\mbox{ if }i=j\\\ 0,&\mbox{
if }i\neq j.\end{array}\right.$ (2.1)

We define the forward partial difference operator $\Delta_{\xi_{j}}$ by

$\forall\xi\in\mathbb{Z}^{n},\,\,\Delta_{\xi_{j}}\sigma(\xi)=\sigma(\xi+\delta_{j})-\sigma(\xi),$

and for $\alpha\in\mathbb{N}_{0}^{n}$ define

$\Delta_{\xi}^{\alpha}:=\Delta_{\xi}^{\alpha_{1}}\cdots\Delta_{\xi}^{\alpha_{n}}.$

######  Proposition 2.5.

(Formulae for $\Delta_{\xi}^{\alpha}$) Let
$\phi:\mathbb{Z}^{n}\rightarrow\mathbb{C}$. We have

$\forall\xi\in\mathbb{Z}^{n},\,\forall\alpha\in\mathbb{N}_{0}^{n},\,\,\Delta_{\xi}^{\alpha}\phi(\xi)\,=\,\sum\limits_{\beta\leq\alpha}(-1)^{\lvert\alpha-\beta\rvert}{\alpha\choose\beta}\phi(\xi+\beta).$
(2.2)

We now recall the definition of the toroidal symbol classes. We define the
discrete Japanese bracket
$\langle\xi\rangle:=(1+\lvert\xi\rvert^{2})^{\frac{1}{2}}$ for
$\xi\in\mathbb{Z}^{n}$, with
$\lvert\xi\rvert^{2}=\xi_{1}^{2}+\cdots+\xi_{n}^{2},$ denoting the Euclidean
norm.

######  Definition 2.6.

Let $m\in\mathbb{R}$, $0\leq\delta,\rho\leq 1.$ Then the toroidal symbol class
$S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ consists of those
functions $a:=a(x,\xi)$ which are smooth in $x\in\mathbb{T}^{n},$ for all
$\xi\in\mathbb{Z}^{n}$, and which satisfy the toroidal symbol inequalities

$\lvert\Delta_{\xi}^{\alpha}\partial_{x}^{\beta}a(x,\xi)\rvert\leq
C_{\alpha\beta}\langle\xi\rangle^{m-\rho\lvert\alpha\rvert+\delta\lvert\beta\rvert}$
(2.3)

for every $x\in\mathbb{T}^{n},\xi\in\mathbb{Z}^{n}$ and for all
$\alpha,\beta\in\mathbb{N}_{0}^{n}$.

######  Remark 2.7.

The family of seminorms

$p_{\alpha,\beta,\rho,\delta,m}(a):=\sup_{(x,\xi)\in\mathbb{T}^{n}\times\mathbb{Z}^{n}}\langle\xi\rangle^{\rho\lvert\alpha\rvert-\delta\lvert\beta\rvert-m}\lvert\partial_{x}^{\beta}\Delta_{\xi}^{\alpha}a(x,\xi)\rvert<\infty,$
(2.4)

defines a Fréchet structure on every toroidal class
$S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$

######  Definition 2.8.

Let $m\in\mathbb{R}$ and let $0\leq\delta,\rho\leq 1.$ The toroidal
quantisation associated to a symbol $a\in
S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ is the densely
defined operator

$A=a(x,D):=\textnormal{Op}(a):C^{\infty}(\mathbb{T}^{n})\rightarrow
C^{\infty}(\mathbb{T}^{n}),$ (2.5)

defined by

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Af(x)=\sum\limits_{\xi\in\mathbb{Z}^{n}}e^{2\pi
ix\cdot\xi}a(x,\xi)\widehat{f}(\xi),\,\,\,f\in
C^{\infty}(\mathbb{T}^{n}),\,x\in\mathbb{T}^{n}.$

######  Remark 2.9.

Note that if one has a continuous linear operator
$A:C^{\infty}(\mathbb{T}^{n})\rightarrow C^{\infty}(\mathbb{T}^{n})$, its
symbol $a:=a(x,\xi)$ can be recovered from the following formula

$a(x,\xi)=e^{-i2\pi x\cdot\xi}Ae_{\xi}(x),$ (2.6)

where for every $\xi\in\mathbb{Z}^{n}$, $e_{\xi}(x)=e^{i2\pi x\cdot\xi},$
$x\in\mathbb{T}^{n},$ is the canonical trigonometric polynomial.

######  Definition 2.10.

The family of pseudo-differential operators defined by the toroidal
quantisation formula in (2.5) corresponding to the class of symbols $a\in
S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ will be denoted by
$\text{\rm Op}S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$.

The toroidal quantisation has been extensively analysed in [26, 27] for the
general case of $\mathbb{T}^{n}$ and on compact Lie groups. For the toroidal
Hörmander class of order $m\in\mathbb{R},$ one has
$\Psi^{m}(\mathbb{T}^{n},\textnormal{loc})=\\{\sigma(x,D):\sigma\in
S^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})\\}$ (cf. [26, 27]) for the class of
symbols $\Psi^{m}(\mathbb{T}^{n},\textnormal{loc})$ defined by local
coordinate systems.

######  Remark 2.11.

The family of seminorms

$p_{\alpha,\beta,\rho,\delta,m}(A):=\sup_{(x,\xi)\in\mathbb{T}^{n}\times\mathbb{Z}^{n}}\langle\xi\rangle^{\rho\lvert\alpha\rvert-\delta\lvert\beta\rvert-m}\lvert\partial_{x}^{\beta}\Delta_{\xi}^{\alpha}a(x,\xi)\rvert<\infty,\,A=\textnormal{Op}(a),$
(2.7)

defines a Fréchet structure on every class
$\textnormal{Op}(S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})).$

The following proposition gives the equivalence between Euclidean Hörmander’s
symbols and toroidal symbols. It corresponds to Corollary 4.6.13 in [26, 27].

######  Proposition 2.12.

For $m\in\mathbb{R}$, $0\leq\delta\leq 1$ and $0<\rho\leq 1$ we have

$\Psi^{m}_{\rho,\delta}:=\text{\rm
Op}S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{R}^{n})=\text{\rm
Op}S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$

i.e., classes of $1$-periodic pseudo-differential operators with Euclidean
(Hörmander’s) symbols in $\text{\rm
Op}S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{R}^{n})$ and toroidal
symbols in $\text{\rm
Op}S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ coincide.

In the next theorem we describe some fundamental properties of the calculus of
pseudo-differential operators on the torus.

######  Theorem 2.13.

Let $0\leqslant\delta<\rho\leqslant 1,$ and let $m\in\mathbb{R}.$ Then:

  * -

the mapping $A\mapsto
A^{*}:\Psi^{m}_{\rho,\delta}\rightarrow\Psi^{m}_{\rho,\delta},$ that assigns
to each operator $A$ its formal adjoint $A^{*},$ is a continuous linear
mapping between Fréchet spaces and the symbol $\sigma_{A^{*}}(x,\xi)$ of
$A^{*},$ satisfies the asymptotic expansion,

$\sigma_{A^{*}}(x,\xi)\sim\sum_{\alpha}\Delta_{\xi}^{\alpha}\partial_{x}^{\alpha}(\overline{\sigma_{A}(x,\xi)}),\,(x,\xi)\in\mathbb{T}^{n}\times\mathbb{Z}^{n}.$

This means that, for every $N\in\mathbb{N},$ and for all $\ell\in\mathbb{N},$

$\displaystyle\Delta_{\xi}^{\alpha_{\ell}}\partial_{x}^{\beta}\left(\sigma_{A^{*}}(x,\xi)-\sum_{\lvert\alpha\rvert\leqslant
N}\Delta_{\xi}^{\alpha}\partial_{x}^{\alpha}(\overline{\sigma_{A}(x,\xi)})\right)$
$\displaystyle\hskip
56.9055pt\in{S}^{m-(\rho-\delta)(N+1)-\rho\ell+\delta\lvert\beta\rvert}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$

where $\alpha_{\ell}\in\mathbb{N}_{0}^{n}$ is such that
$\lvert\alpha_{\ell}\rvert=\ell.$

  * -

The mapping $(A_{1},A_{2})\mapsto A_{1}\circ
A_{2}:\Psi^{m_{1}}_{\rho,\delta}\times\Psi^{m_{2}}_{\rho,\delta}\rightarrow\Psi^{m_{1}+m_{2}}_{\rho,\delta}$
is a continuous bilinear mapping between Fréchet spaces, and the symbol of
$A=A_{1}\circ A_{2}$ satisfies the asymptotic formula

$\sigma_{A}(x,\xi)\sim\sum_{\alpha}(\Delta_{\xi}^{\alpha}\sigma_{A_{1}}(x,\xi))(\partial_{x}^{\alpha}\sigma_{A_{2}}(x,\xi)),$

which, in particular, means that, for every $N\in\mathbb{N},$ and for all
$\ell\in\mathbb{N},$

$\displaystyle\Delta_{\xi}^{\alpha_{\ell}}\partial_{x}^{\beta}\left(\sigma_{A}(x,\xi)-\sum_{\lvert\alpha\rvert\leqslant
N}(\Delta_{\xi}^{\alpha}\sigma_{{A_{1}}}(x,\xi))(\partial_{x}^{\alpha}\sigma{A_{2}}(x,\xi))\right)$
$\displaystyle\hskip
56.9055pt\in{S}^{m_{1}+m_{2}-(\rho-\delta)(N+1)-\rho\ell+\delta\lvert\beta\rvert}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$

for all $\alpha_{\ell}\in\mathbb{N}_{0}^{n}$ with
$\lvert\alpha_{\ell}\rvert=\ell.$

  * -

For $0\leqslant\delta\leqslant\rho\leqslant 1,$ $\delta\neq 1,$ let us
consider a continuous linear operator
$A:C^{\infty}(\mathbb{T}^{n})\rightarrow\mathscr{D}^{\prime}(\mathbb{T}^{n})$
with symbol
$\sigma\in{S}^{0}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. Then $A$
extends to a bounded operator from $L^{2}(\mathbb{T}^{n})$ to
$L^{2}(\mathbb{T}^{n}).$

######  Remark 2.14.

We now recall the definition of the Sobolev space of order $s\in\mathbb{R}$ on
the torus. For $u\in\mathcal{D}^{\prime}(\mathbb{T}^{n})$ we define the norm
$\|\cdot\|_{H^{s}(\mathbb{T}^{n})}$ by

$\|u\|_{H^{s}(\mathbb{T}^{n})}:=\left(\sum\limits_{\xi\in\mathbb{Z}^{n}}(1+\lvert\xi\rvert^{2})^{s}\lvert\hat{u}(\xi)\rvert^{2}\right)^{1/2}.$

The Sobolev space $H^{s}(\mathbb{T}^{n})$, is the space of the $1$-periodic
distributions $u$ such that $\|u\|_{H^{s}(\mathbb{T}^{n})}<\infty.$

###  2.2. Matrix-valued classes on the torus

In order to study our problem (1.3), we will reduce it to a first-order system
with respect to the time variable. For this reduced system, we will require
the notion of a matrix-valued symbolic calculus on the torus which we
introduce it as follows:

######  Definition 2.15.

Let $\ell\in\mathbb{N}.$ An $\ell\times\ell$-matrix symbol

$\mathfrak{a}:=[a_{ij}]_{1\leq
i,j\leq\ell},\,\,a_{ij}:\mathbb{T}^{n}\times\mathbb{Z}^{n}\rightarrow\mathbb{C},$
(2.8)

belongs to the matrix-valued class
$S^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ if
each one of its entries $a_{ij}$ is a symbol in
$S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. The
$\ell\times\ell$-matrix-valued pseudo-differential operator
$\mathfrak{A}:=\textnormal{Op}(\mathfrak{a})$ associated to the symbol
$\mathfrak{a},$ is defined via

$\textnormal{Op}(\mathfrak{a}):=[\textnormal{Op}(a_{ij})]_{1\leq
i,j\leq\ell},\,\,\textnormal{Op}(a_{ij})\in\Psi^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$
(2.9)

######  Remark 2.16.

We will use the notation

$\Psi^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}):=\textnormal{Op}(S^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})),$

for the class of operators $\textnormal{Op}(\mathfrak{a})$ with a symbol
$\mathfrak{a}$ belonging to the matrix-valued class
$S^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$

######  Remark 2.17.

The family of seminorms

$p_{\alpha,\beta,\rho,\delta,m}(\mathfrak{a}):=\sup_{(x,\xi)\in\mathbb{T}^{n}\times\mathbb{Z}^{n}}\sup_{1\leq
i,j\leq\ell}\langle\xi\rangle^{\rho\lvert\alpha\rvert-\delta\lvert\beta\rvert-m}\lvert\partial_{x}^{\beta}\Delta_{\xi}^{\alpha}a_{ij}(x,\xi)\rvert<\infty,$
(2.10)

defines a Fréchet structure on every class
$S_{\ell\times\ell;\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$
Moreover, the family of seminorms

$p_{\alpha,\beta,\rho,\delta,m}(\textnormal{Op}(\mathfrak{a})):=p_{\alpha,\beta,\rho,\delta,m}(\mathfrak{a}),\,A=\textnormal{Op}(a),$
(2.11)

defines a Fréchet structure on every class
$\textnormal{Op}(S_{\ell\times\ell;\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})).$

######  Remark 2.18.

The basic properties of the calculus for
$S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ also hold for the
matrix-valued classes
$S^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$.
Indeed, the classes of symbols
$S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$ $m\in\mathbb{R},$
are closed under the addition

$\mathfrak{a}_{1}+\mathfrak{a}_{2}=[a_{1,ij}+a_{2,ij}]_{i,j=1}^{\ell},\,\mathfrak{a}_{k}=[a_{k,ij}]_{i,j=1}^{\ell},\,k=1,2,$

of matrix-valued symbols. Indeed, note that

$S^{m_{1}}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})\times
S^{m_{2}}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})\ni(\mathfrak{a}_{1},\mathfrak{a}_{2})\mapsto\mathfrak{a}_{1}+\mathfrak{a}_{2}\in
S^{m_{1}+m_{2}}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).\,$
(2.12)

Is clear that, in this case,

$\textnormal{Op}(\mathfrak{a}_{1}+\mathfrak{a}_{2})=\textnormal{Op}(\mathfrak{a}_{1})+\textnormal{Op}(\mathfrak{a}_{2})\in\Psi^{m_{1}+m_{2}}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$

Also, observe that if we define on $L^{2}(\mathbb{T}^{n};\mathbb{C}^{\ell})$
the inner product

$(u,v)_{L^{2}(\mathbb{T}^{n};\mathbb{C}^{\ell})}=\sum_{i=1}^{\ell}(u_{i},v_{i})_{L^{2}(\mathbb{T}^{n})},\,u:=(u_{1},\cdots,u_{\ell}),\,v:=(v_{1},\cdots,v_{\ell})\in
L^{2}(\mathbb{T}^{n};\mathbb{C}^{\ell}),$

we have that

$\displaystyle(\textnormal{Op}(\mathfrak{a})u,v)_{L^{2}(\mathbb{T}^{n};\mathbb{C}^{\ell})}$
$\displaystyle=\sum_{i=1}^{\ell}((\textnormal{Op}(\mathfrak{a})u)_{i},v_{i})_{L^{2}(\mathbb{T}^{n})}=\sum_{i=1}^{\ell}\sum_{k=1}^{\ell}(\textnormal{Op}(\mathfrak{a})_{ik}u_{k},v_{i})_{L^{2}(\mathbb{T}^{n})}$
$\displaystyle=\sum_{k=1}^{\ell}\sum_{i=1}^{\ell}(u_{k},\textnormal{Op}(\mathfrak{a})_{ik}^{*}v_{i})_{L^{2}(\mathbb{T}^{n})}.$

So, defining

$\textnormal{Op}(\mathfrak{a}^{*})_{ki}:=\textnormal{Op}(\mathfrak{a})_{ik}^{*}$
(2.13)

the formal adjoint $\textnormal{Op}(\mathfrak{a})^{*}$ of
$\textnormal{Op}(\mathfrak{a})$ is determined by the entries

$\textnormal{Op}(\mathfrak{a})^{*}=[\textnormal{Op}(\mathfrak{a}^{*})_{ki}]_{k,i=1}^{\ell}.$
(2.14)

Indeed, observe that

$\displaystyle(\textnormal{Op}(\mathfrak{a})u,v)_{L^{2}(\mathbb{T}^{n};\mathbb{C}^{\ell})}$
$\displaystyle=\sum_{k=1}^{\ell}\sum_{i=1}^{\ell}(u_{k},\textnormal{Op}(\mathfrak{a})_{ik}^{*}v_{i})_{L^{2}(\mathbb{T}^{n})}=\sum_{k=1}^{\ell}\sum_{i=1}^{\ell}(u_{k},\textnormal{Op}(\mathfrak{a}^{*})_{ki}v_{i})_{L^{2}(\mathbb{T}^{n})}$
$\displaystyle=\sum_{k=1}^{\ell}(u_{k},(\textnormal{Op}(\mathfrak{a}^{*})v)_{k})_{L^{2}(\mathbb{T}^{n};\mathbb{C}^{\ell})}$
$\displaystyle=(u,\textnormal{Op}(\mathfrak{a}^{*})v)_{L^{2}(\mathbb{T}^{n};\mathbb{C}^{\ell})}.$

The previous analysis also proves that if
$\textnormal{Op}(\mathfrak{a})\in\Psi^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$
then
$\textnormal{Op}(\mathfrak{a})^{*}\in\Psi^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$

In view of the previous remark and as a direct consequence of Theorem 2.13 the
calculus for matrix-valued pseudo-differential operators can be summarised in
the following result.

######  Theorem 2.19.

Let $0\leqslant\delta<\rho\leqslant 1,$ and for every $m\in\mathbb{R}.$ Then:

  * -

the mapping $A\mapsto
A^{*}:\Psi^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})\rightarrow\Psi^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$
that assigns to each operator $A$ its formal adjoint $A^{*},$ is a continuous
linear mapping between Fréchet spaces.

  * -

The mapping $(A_{1},A_{2})\mapsto A_{1}\circ
A_{2}:\Psi^{m_{1}}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})\times\Psi^{m_{2}}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})\rightarrow\Psi^{m_{1}+m_{2}}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$
is a continuous bilinear mapping between Fréchet spaces.

######  Remark 2.20.

Consider the positive Laplacian
$\mathcal{L}_{\mathbb{T}^{n}}=-\sum_{i=1}^{n}\partial_{x_{i}}^{2}$ on the
torus $\mathbb{T}^{n}.$ In view of the functional calculus for elliptic
operators one has that

$\Lambda^{s}:=(1+\mathcal{L}_{\mathbb{T}^{n}})^{\frac{s}{2}}\in\Psi^{s}_{1,0}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),\,s\in\mathbb{R}.$
(2.15)

Note that on $C^{\infty}(\mathbb{T}^{n},\mathbb{C}^{\ell})$ the operator

$\Lambda^{s}_{\ell\times\ell}:=[\Lambda^{s}\delta_{ij}]_{i,j=1}^{\ell},$
(2.16)

where $\delta_{ij}$ is the Delta-Kronecker, defines an operator in the class
$\Psi^{s}_{\ell\times\ell;1,0}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$ Since,
for any $u=(u_{1},\cdots,u_{\ell})\in
C^{\infty}(\mathbb{T}^{n},\mathbb{C}^{\ell})$ one has that

$\displaystyle\Lambda^{s}_{\ell\times\ell}u=(\Lambda^{s}u_{1},\cdots,\Lambda^{s}u_{\ell}),$

in order to simplify the notation we will always write
$\Lambda^{s}:=\Lambda^{s}_{\ell\times\ell}$ omitting the sub-index
$\ell\times\ell.$

Now, the simplified notation above allows us to introduce the vector-valued
Sobolev space $H^{s}(\mathbb{T}^{n},\mathbb{C}^{\ell})$, defined by the
$1$-periodic distributions $u$ such that

$\|u\|_{H^{s}(\mathbb{T}^{n},\mathbb{C}^{\ell})}=\|\Lambda^{s}u\|_{L^{2}(\mathbb{T}^{n},\mathbb{C}^{\ell})}=\left(\sum_{i=1}^{\ell}\|\Lambda^{s}u_{i}\|_{L^{2}(\mathbb{T}^{n})}^{2}\right)^{\frac{1}{2}}<\infty.$

##  3\. Pseudo-differential hyperbolic equations on the torus

In this section we establish the well-posedness for the problem (1.3). We
first recall the notions of ellipticity and strong ellipticity in the setting
of the $S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$-classes. We
then obtain energy estimates and the corresponding consequences on the well-
posedness for first order systems and its consequence on Fractional hyperbolic
equations. Some aspects on the the fractional Laplacian are briefly reviewed.
For a comprehensive study on this theory we refer to [26, 27] and for a more
detailed discussion on the fractional Laplacian on the torus we refer to [24].

The classical notion of ellipticity for pseudo-differential operators on
${\mathbb{R}}^{n}$ extends into the toroidal setting in an analogous way. We
point out that some further restrictions on $\rho,\delta$ have to be imposed
in order to obtain an useful definition.

######  Definition 3.1.

Let $m\in\mathbb{R}$, $0\leq\delta<\rho\leq 1$ and $\sigma$ be a symbol in
$\sigma\in S_{\rho,\delta}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$.

  * •

We say that the corresponding pseudo-differential operator $\sigma(x,D)$ is
elliptic of order $m$, if $\sigma$ satisfies

$\exists
n_{0}\in\mathbb{N},\,\forall(x,\xi)\in\mathbb{T}^{n}\times\mathbb{Z}^{n}:\,\,\lvert\xi\rvert\geq
n_{0}\implies\lvert\sigma(x,\xi)\rvert\geq C_{0}\langle\xi\rangle^{m},$ (3.1)

for some constants $n_{0},C_{0}>0$.

  * •

We will also say that $\sigma(x,D)$ is a strongly elliptic operator if
$\sigma$ satisfies:

$\exists
n_{0}\in\mathbb{N},\,\forall(x,\xi)\in\mathbb{T}^{n}\times\mathbb{Z}^{n}:\,\,\lvert\xi\rvert\geq
n_{0}\implies{\mbox{R}e}\,\sigma(x,\xi)\geq C_{0}\langle\xi\rangle^{m},$ (3.2)

for some constants $n_{0},C_{0}>0$.

######  Remark 3.2.

We now specialize into the case of the fractional Laplacian. We first state a
basic property of this interesting operator on the torus that distinguish it
from the case of ${\mathbb{R}}^{n}$. As a consequence of this fact and the
above theorem we obtain some corollaries for the fractional diffusion.

Let $\nu$ be a strictly positive real number, the fractional Laplacian
$(-\Delta)^{\frac{\nu}{2}}$ on the torus $\mathbb{T}^{n}$ is defined as the
Fourier multiplier corresponding to $\lvert\xi|^{\nu}$, that is

$\widehat{(-\Delta)^{\frac{\nu}{2}}u}(\xi)=(2\pi)^{\nu}\lvert\xi|^{\nu}\widehat{u}(\xi),$
(3.3)

for every $\xi\in\mathbb{Z}^{n}$ and $u\in C^{\infty}(\mathbb{T}^{n})$. The
Fourier inversion formula on the torus also allows the integral representation
in (1.4) for $(-\Delta)^{\frac{\nu}{2}}$. Other alternative definitions are
possible as in the case of ${\mathbb{R}}^{n}$. We refer the reader to [20] for
a recent review on the most common ones and the equivalence between them on
${\mathbb{R}}^{n}$. We point out that on ${\mathbb{R}}^{n}$, the fractional
Laplacian $(-\Delta)^{\frac{\nu}{2}}$ has not a symbol in a class of pseudo-
differential operators unless $\frac{\nu}{2}$ be an integer, however in our
toroidal setting the fractional Laplacian will be a pseudo-differential
operator for every $\nu>0.$ We first state a mild lemma on the torus
clarifying such property.

######  Lemma 3.3.

Let $\nu>0.$ Then $(-\Delta)^{\frac{\nu}{2}}\in\text{\rm
Op}S_{1,0}^{\nu}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$.

###### Proof.

We should show that the function $\sigma:\mathbb{Z}^{n}\rightarrow\mathbb{R}$
defined by $\sigma(\xi)=\lvert\xi|^{\nu}$ belongs to
$S_{1,0}^{\nu}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. We choose a function
$\chi\in C^{\infty}({\mathbb{R}}^{n})\,$ such that

$\chi(\xi):=\left\\{\begin{array}[]{rl}1,&\mbox{ if }\lvert\xi\rvert\geq 1,\\\
0,&\mbox{ if }\lvert\xi\rvert\leq\frac{1}{2}.\end{array}\right.$ (3.4)

We observe that $\widetilde{\sigma}(\xi):=\chi(\xi)\lvert\xi\rvert^{\nu}$ is
smooth on ${\mathbb{R}}^{n}$ and $\widetilde{\sigma}\in
S_{1,0}^{\nu}(\mathbb{T}^{n}\times{\mathbb{R}}^{n})$. Hence
$\widetilde{\sigma}(x,D)\in\text{\rm
Op}S_{1,0}^{\nu}(\mathbb{T}^{n}\times{\mathbb{R}}^{n})$ and by Proposition
2.12 we obtain that $\widetilde{\sigma}(x,D)\in\text{\rm
Op}S_{1,0}^{\nu}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. On the other hand we
have that $\chi(\xi)\lvert\xi\rvert^{\nu}=\lvert\xi\rvert^{\nu}=\sigma(\xi)$
for all $\xi\in\mathbb{Z}^{n}$, and therefore $\sigma\in
S_{1,0}^{\nu}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. ∎

######  Remark 3.4.

The fractional Laplacian $(-\Delta)^{\frac{\nu}{2}}$ is an element of the
class $\text{\rm Op}S_{1,0}^{\nu}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ for any
$\nu>0$. Then it is clear that it is also a strongly elliptic operator of
order $\nu$. This property is an advantage over the classical case of the
fractional Laplacian on ${\mathbb{R}}^{n}$ where one can only refer to it as a
pseudo-differential operator when $\nu$ is an even integer.

In the following theorems we will write $K(t,x,\xi)\in
S^{m}_{\ell\times\ell}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$, which should be
understood in the sense that for each $t\in\mathbb{R}$ fixed,
$K(t,\cdot,\cdot)\in
S^{m}_{\ell\times\ell;1,0}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. We will also
require of the spaces

$H^{+\infty}:=C^{\infty}(\mathbb{T}^{n})=\bigcap\limits_{s\in\mathbb{R}}H^{s}\text{
and
}H^{-\infty}:=\mathscr{D}^{\prime}(\mathbb{T}^{n})=\bigcup\limits_{s\in\mathbb{R}}H^{s}.$

The following theorem establishes a fundamental energy estimate is the
toroidal version within the setting of the pseudo-differential calculus as
introduced in [26, 27].

######  Theorem 3.5.

Let $m>0$ and $K(t,x,\xi)\in
S^{m}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$
depending smoothly on $t$. Assume that $K^{*}(t,x,D_{x})+K(t,x,D_{x})\in
OpS^{0}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$,
where for $t$ fixed, $K(t,x,D_{x})=K(t)$ denotes the pseudo-differential
operator corresponding to $K(t,x,\xi)$. Let $s\in\mathbb{R}$, $T>0$. Let

$v\in C([0,T],H^{s+1}(\mathbb{T}^{n};\mathbb{C}^{\ell}))\bigcap
C^{1}([0,T],H^{s}(\mathbb{T}^{n};\mathbb{C}^{\ell}))$

and $Q:=\partial_{t}-K(t).$ Then $v\in
C^{\infty}(\mathbb{T}^{n},\mathbb{C}^{\ell})$ satisfies

$\|v(t)\|_{H^{s}(\mathbb{T}^{n},\mathbb{C}^{\ell})}^{2}\leq
e^{Ct}\left(\|v(0)\|_{H^{s}(\mathbb{T}^{n},\mathbb{C}^{\ell})}^{2}+\int\limits_{0}^{t}\|\omega(\tau)\|^{2}_{H^{s}(\mathbb{T}^{n},\mathbb{C}^{\ell})}d\tau\right)$
(3.5)

for all $t\in[0,T].$ Moreover, we can replace $v(0)$ by $v(T)$ on the right-
hand side of (3.5). The same conclusion holds for the operator $Q^{*}$.

###### Proof.

For any $s\in\mathbb{R},$ let us write
$H^{s}:=H^{s}(\mathbb{T}^{n},\mathbb{C}^{\ell}).$ We set $\Lambda^{s}$ be the
Fourier multiplier given by
$\Lambda(\xi)=(1+\lvert\xi\rvert^{2})^{\frac{s}{2}}$. Let us follow the
notation in Remark 2.20. It is clear that $u$ belongs to
$H^{s}(\mathbb{T}^{n},\mathbb{C}^{\ell})$ if and only if $\Lambda^{s}u\in
L^{2}(\mathbb{T}^{n},\mathbb{C}^{\ell})$. We assume that $v\in
C([0,T],H^{s+1})\bigcap C^{1}([0,T],H^{s})$ and write $\omega=Qv$. Since
$\partial_{t}v=(\partial_{t}-K(t))v+K(t)v=Qv+K(t)v=\omega+K(t)v$, we observe
that

$\displaystyle\frac{d}{dt}\|v(t)\|_{H^{s}}^{2}$
$\displaystyle=\frac{d}{dt}\langle\Lambda^{s}v,\Lambda^{s}v\rangle$
$\displaystyle=2Re\langle\Lambda^{s}v_{t},\Lambda^{s}v\rangle$
$\displaystyle=2Re\langle\Lambda^{s}(K(t)v+\omega),\Lambda^{s}v\rangle$
$\displaystyle=2Re\langle\Lambda^{s}K(t)v,\Lambda^{s}v\rangle$
$\displaystyle-2Re\langle K(t)\Lambda^{s}v,\Lambda^{s}v\rangle$
$\displaystyle+2Re\langle K(t)\Lambda^{s}v,\Lambda^{s}v\rangle$
$\displaystyle+2Re\langle\Lambda^{s}\omega,\Lambda^{s}v\rangle$
$\displaystyle=2Re\langle[\Lambda^{s},K(t)]v,\Lambda^{s}v\rangle$
$\displaystyle+2Re\langle K(t)\Lambda^{s}v,\Lambda^{s}v\rangle$ (3.6)
$\displaystyle+2Re\langle\Lambda^{s}\omega,\Lambda^{s}v\rangle.$

We note that the term (3.6) can be written in the following way

$\displaystyle 2Re\langle K(t)\Lambda^{s}v,\Lambda^{s}v\rangle$
$\displaystyle=\langle K(t)\Lambda^{s}v,\Lambda^{s}v\rangle+\overline{\langle
K(t)\Lambda^{s}v,\Lambda^{s}v\rangle}$ $\displaystyle=\langle
K(t)\Lambda^{s}v,\Lambda^{s}v\rangle+\langle\Lambda^{s}v,K(t)\Lambda^{s}v\rangle$
$\displaystyle=\langle K(t)\Lambda^{s}v,\Lambda^{s}v\rangle+\langle
K(t)^{*}\Lambda^{s}v,\Lambda^{s}v\rangle$
$\displaystyle=\langle(K(t)+K(t)^{*})\Lambda^{s}v,\Lambda^{s}v\rangle.$

Now, we get $A(t)=[\Lambda^{s},K(t)]\in
OpS^{s}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$.
Indeed, observe that

$A(t)=[\Lambda^{s},K(t)]=[A(t)_{ij}]_{i,j=1}^{\ell}$

where

$\displaystyle A(t)_{ij}$ $\displaystyle=([\Lambda^{s},K(t)])_{ij}$
$\displaystyle=[\Lambda^{s}K(t)]_{ij}-[K(t)\Lambda^{s}]_{ij}$
$\displaystyle=\sum_{k=1}^{\ell}\Lambda^{s}\delta_{ik}K(t)_{kj}-\sum_{k=1}^{\ell}K(t)_{ik}\Lambda^{s}\delta_{kj}$
$\displaystyle=\Lambda^{s}K(t)_{ij}-K(t)_{ij}\Lambda^{s}$
$\displaystyle=[\Lambda^{s},K(t)_{ij}].$

The commutator properties of the scalar-valued calculus on the torus implies
that any entry $[\Lambda^{s},K(t)_{ij}]$ belong to the class
$OpS^{s+1-1}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})=OpS^{s}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$
Consequently, we have that $A(t)=[\Lambda^{s},K(t)]\in
OpS^{s}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ as
claimed. Since we also have $K(t)+K(t)^{*}\in
OpS^{0}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$, it
follows that

$\frac{d}{dt}\|v(t)\|_{H^{s}}^{2}\leq$
$\displaystyle\leq\|A(t)v\|_{L^{2}}\|v\|_{H^{s}}+C_{1}\|v\|^{2}_{H^{s}}+C_{2}\|\omega\|_{H^{s}}\|v\|_{H^{s}}$
$\displaystyle\leq
C\|v\|_{H^{s}}\|v\|_{H^{s}}+C_{1}\|v\|^{2}_{H^{s}}+C_{2}\|\omega\|_{H^{s}}\|v\|_{H^{s}}$
$\displaystyle\leq C\|v\|^{2}_{H^{s}}+C\|\omega\|^{2}_{H^{s}}.$

An application of the Gronwall inequality gives us the energy inequality

$\|v(t)\|_{H^{s}}^{2}\leq
e^{Ct}\left(\|v(0)\|_{H^{s}}^{2}+\int\limits_{0}^{t}\|\omega(\tau)\|^{2}_{H^{s}}d\tau\right).$
(3.7)

We can also prove an analogous estimate with $v(T)$ instead of $v(0)$ on the
right-hand side of the inequality (3.7). The conclusion for $Q^{*}$ follows
analogously. ∎

We now obtain a consequence regarding the existence, uniqueness and regularity
as an application of the above estimates.

######  Theorem 3.6.

Let $m>0$ and $K(t,x,\xi)\in
S^{m}_{\ell\times\ell}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ depending smoothly
on $t$. Assume that $K^{*}(t,x,D_{x})+K(t,x,D_{x})\in
OpS^{0}_{\ell\times\ell;\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. Let
$s\in\mathbb{R}$, $T>0$, $f\in H^{s},\,\omega\in L^{2}([0,T],H^{s})$. Then,
there exists a unique $v\in C([0,T],H^{s})$ such that

$\left\\{\begin{array}[]{rl}{\displaystyle\frac{\partial v}{\partial
t}}=&K(t)v+\omega,\,\,\mbox{ (in the sense of
}\mathcal{D}^{\prime}(]0,T[\times\mathbb{T}^{n})\\\ {\displaystyle
v}(0)=&f.\end{array}\right.$ (3.8)

Moreover, the solution $v$ satisfies the energy estimate (3.5). If $\omega\in
C^{\infty}([0,T],H^{+\infty})$ and $f\in H^{+\infty}$ then $v\in
C^{\infty}([0,T],H^{+\infty})$.

###### Proof.

We will now prove the existence of a solution $v$ of (3.8) in
$C([0,T],H^{s})$. The proof is an adaptation of the corresponding part in the
proof of Theorem 4.5 in [6]. We write $Q=\frac{\partial}{\partial t}-K$ and we
introduce the space $E=\\{\varphi\in
C^{\infty}([0,T],H^{-\infty}):\varphi(T)=0\\}$. We will see that we can define
a linear form $\beta$ on $Q^{*}E$ by

$Q^{*}\varphi\rightarrow\beta(Q^{*}\varphi)=\int\limits_{0}^{T}(\omega(t,\cdot),\varphi(t,\cdot))dt+\frac{1}{i}(f,\varphi(0,\cdot)).$

We note that the energy estimate (3.5) holds for $-s$ for the Cauchy problem
(3.8) corresponding to the operator $Q^{*}$ with $\|v(T,\cdot)\|_{H^{s}}$ on
the right hand side of (3.5). Thus, for $\varphi\in E$ we have

$\|\varphi(t,\cdot)\|_{H^{-s}}^{2}\leq
C\int\limits_{0}^{T}\|Q^{*}\varphi(t^{\prime},\cdot)\|_{H^{-s}}^{2}dt^{\prime},\,\,\,t\in[0,T]\,\,,$

so that

$\lvert\beta(Q^{*}\varphi)\rvert^{2}\leq
C^{\prime}\int\limits_{0}^{T}\|Q^{*}\varphi(t^{\prime},\cdot)\|_{H^{-s}}^{2}dt^{\prime}.$

We deduce that $\beta$ is well defined and continuous with respect to the
topology induced on $Q^{*}E$ by $L^{2}([0,T,],H^{-s})$. An application of the
Hahn-Banach theorem implies the existence of an element
$v\in(L^{2}([0,T,],H^{-s})^{\prime}=L^{2}([0,T,],H^{s})$ such that

$(v,Q^{*}\varphi)=\int\limits_{0}^{T}(\omega(t,\cdot),\varphi(t,\cdot))dt+\frac{1}{i}(f,\varphi(0,\cdot))$
(3.9)

for all $\varphi\in E$. In particular, if $\varphi\in
C^{\infty}(]0,T[\times\mathbb{T}^{n})$, (3.9) implies that $Qv=\omega$ in
$\mathcal{D}^{\prime}([0,T]\times\mathbb{T}^{n}))$. Thus
$\frac{\partial}{\partial t}v=Kv+\omega\in L^{2}([0,T,],H^{s-1})$. An
integration by parts with respect to $t$ in (3.9) implies that
$(v(0,\cdot),\varphi(0,\cdot))=(f,\varphi(0,\cdot))$ for all $\varphi\in E$
and consequently $v(0)=v(0,\cdot)=f$.

Now, if $\omega\in C^{\infty}([0,T],H^{+\infty})$ and $f\in H^{+\infty}$, the
above argument shows that $v\in C([0,T],H^{+\infty})$. Moreover, since
$\frac{\partial}{\partial t}v=Kv+\omega$, one can deduce step by step that
$v\in C^{k}([0,T],H^{+\infty})$ for all $k\geq 0$. Consequently $v\in
C^{\infty}([0,T],H^{+\infty})$.

We will now prove that $v\in C([0,T],H^{s})$ and that it satisfies the energy
estimate (3.5). Suppose we have sequences $(\omega_{j})$ in
$C^{\infty}([0,T]\times\mathbb{T}^{n})$ and $(f_{j})$ in
$C_{0}^{\infty}(\mathbb{T}^{n})$ such that $\omega_{j}\rightarrow\omega$ in
$L^{2}([0,T,],H^{s})$ and $f_{j}\rightarrow f$ in $H^{s}$. Let $v_{j}\in
C^{\infty}([0,T],H^{+\infty})$ be the solution of
$Qv_{j}=\omega_{j},v_{j}(0,\cdot)=f_{j}$. The inequality (3.5) applied to the
$v_{j}-v_{k}$ shows that $v_{j}$ is a Cauchy sequence in $C([0,T],H^{s})$ so
that $v_{j}\rightarrow\widetilde{v}$ in $C([0,T],H^{s})$. In the limit, we
have $Q\widetilde{v}=\omega,\,\widetilde{v}(0,\cdot)=f$; consequently, the
uniqueness shows that $\widetilde{v}=v$.

The corresponding inequality (3.5) for $v$ is obtained passing to the limit in
this inequality applied to $v_{j}$. In this way we conclude the proof of the
Theorem. The uniqueness of the solution $v$ follows from the energy inequality
(3.5). ∎

We can now establish the well-posedness for the Cauchy problem (1.3) as a
consequence of Theorem 3.6. We are ready to prove our main theorem.

###### Proof of Theorem 1.1.

In order to apply Theorem 3.6, we define first

$A=(I+P)^{\frac{1}{2}}.$

Observe that

$\frac{\partial}{\partial t}\underbrace{\frac{\partial u}{\partial t}}_{\rm
v_{2}}=-P(x,D)A^{-1}\underbrace{Au}_{\rm v_{1}},$ (3.10)

with $v_{1}=Au,\,v_{2}=\frac{\partial u}{\partial t}$. In view of the
functional calculus for elliptic operators we have that $A\in
OpS^{\frac{\nu}{2}}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$ Hence, we have the
matrix identity

$\frac{\partial}{\partial t}\left[{\begin{array}[]{c}v_{1}\\\ v_{2}\\\
\end{array}}\right]=\left[{\begin{array}[]{cc}0&A\\\ -P(x,D)A^{-1}&0\\\
\end{array}}\right]\left[{\begin{array}[]{c}v_{1}\\\ v_{2}\\\
\end{array}}\right]+\left[{\begin{array}[]{c}0\\\ w\\\ \end{array}}\right].$

Indeed, the identity in the first component is reduced to the fact that

$\displaystyle\frac{\partial}{\partial t}v_{1}=\frac{\partial}{\partial
t}Au=A\frac{\partial}{\partial t}u=av_{2},$

where we have used that $A$ is independent of $t,$ and then the operators $A$
and $\frac{\partial}{\partial t}$ commute. On the other hand, the identity in
the second component is consequence of the fact that $\frac{\partial}{\partial
t}v_{2}=u_{tt},$ and the decomposition in (3.10).

It is clear that $K(t)\in OpS^{\frac{\nu}{2}}_{2\times
2,\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$, where $K(t)$ is the
constant in $t,$ $2\times 2$ matrix-valued operator:

$K(t)=\left[{\begin{array}[]{cc}0&A\\\ -P(x,D)A^{-1}&0\\\
\end{array}}\right]\,.$

On the other hand, observe that $K(t)+K(t)^{*}\in OpS_{2\times
2}^{0}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. To prove this, let us compute
$K(t)^{*}$ as follows

$\displaystyle K(t)^{*}=\left[{\begin{array}[]{cc}0&(-P(x,D)A^{-1})^{*}\\\
A&0\\\ \end{array}}\right]\,.$

Since the operator $A^{-1}$ commutes with $P$, we have that

$(-P(x,D)A^{-1})^{*}=-P(x,D)A^{-1},$

in view of the positivity of $P(x,D).$ Note also that

$K(t)+K(t)^{*}=\left[{\begin{array}[]{cc}0&A-P(x,D)A^{-1}\\\
A-P(x,D)A^{-1}&0\\\ \end{array}}\right].\,$

Indeed

$A^{*}A=I+P(x,D),$

and

$A^{*}=A^{-1}+P(x,D)A^{-1}=P(x,D)A^{-1}+R,$

with

$R=A^{-1}\in
OpS^{-\frac{\nu}{2}}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})\subset
OpS^{0}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}).$

Since $A$ is positive, we have that

$A-P(x,D)A^{-1}=A^{*}-P(x,D)A^{-1}\in
OpS^{0}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$

and so, we have proved that $K(t)+K(t)^{*}\in OpS_{2\times
2;\rho,\delta}^{0}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$. Now, since $Af_{0}\in
H^{s-\frac{\nu}{2}}$ and applying Theorem 3.6 to

$v(0)=f=\left[{\begin{array}[]{c}Af_{0}\\\ f_{1}\\\ \end{array}}\right]\in
H^{s-\frac{\nu}{2}},\,\,\omega=\left[{\begin{array}[]{c}0\\\ w\\\
\end{array}}\right]\in L^{2}([0,T],H^{s-\frac{\nu}{2}}),$ (3.11)

we obtain that $v\in C([0,T],H^{s-\frac{\nu}{2}})$. Since $u=A^{-1}v_{1}\in
H^{s}$ we deduce that $u\in C([0,T],H^{s})$. The uniqueness of $u$ follows
from the uniqueness of $v$ and the invertibility of $A$ since $u=A^{-1}v_{1}$.
The inequality (3.13) is an immediate consequence of the inequality (3.5)
applied to the data (3.11). The last conclusion of the theorem also follows
from the analogous part of Theorem 3.6. ∎

Observe that in view of Theorem 1.1 applied to the fractional Laplacian
$P(x,D)=(-\Delta)^{\frac{\nu}{2}}$ we have the following energy estimate.

######  Corollary 3.7.

Let $\nu>0$, $T>0$. If $s\in\mathbb{R}$, $f_{0}\in H^{s},\,f_{1}\in
H^{s-\frac{\nu}{2}},\,\omega\in L^{2}([0,T],H^{s-\frac{\nu}{2}})$. Then, there
exists a unique solution $u\in C([0,T],H^{s})$ of the Cauchy problem

$\left\\{\begin{array}[]{rl}{\displaystyle\frac{\partial^{2}u}{\partial
t^{2}}}\,\,\,\,=&-(-\Delta)^{\frac{\nu}{2}}u+w,\,\,\mbox{ (in the sense of
}\mathcal{D}^{\prime}(]0,T[\times\mathbb{T}^{n}))\\\ {\displaystyle
u}(0)=&f_{0},\\\ {\displaystyle\frac{\partial u}{\partial
t}}(0)=&f_{1}.\end{array}\right.$ (3.12)

Moreover, the solution $u$ satisfies the following energy estimate

$\|u(t)\|_{H^{s}}^{2}\leq
Ce^{Ct}\left(\|f_{0}\|_{H^{s}}^{2}+\|f_{1}\|_{H^{s-\frac{\nu}{2}}}^{2}+\int\limits_{0}^{t}\|w(\tau)\|^{2}_{H^{s-\frac{\nu}{2}}}d\tau\right).$
(3.13)

Moreover, if $w\in C^{\infty}([0,T],H^{+\infty})$ and $f\in H^{+\infty}$ then
$u\in C^{\infty}([0,T],H^{+\infty})$.

## References

  * [1] F. Nazarov A. Kiselev and A. Volberg.  Global well-posedness for the critical 2D dissipative quasi-geostrophic equation.  Invent. Math., (167):445––453, 2007.
  * [2] S. Bhattacharyya, T. Ghosh, and G. Uhlmann.  Inverse problems for the fractional-Laplacian with lower order non-local perturbations.  Trans. Amer. Math. Soc., 374(5):3053––3075, 2021.
  * [3] L. A. Caffarelli and A. Vasseur.  Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation.  Ann. of Math., 171(3):1903––1930, 2010.
  * [4] D. Cao, W. Dai, and Q. Guolin.  Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians.  Trans. Amer. Math. Soc., 374(7):4781––4813, 2021.
  * [5] D. Cardona, J. Delgado, and M. Ruzhansky.  Analytic functional calculus and Gårding inequality on graded Lie groups with applications to diffusion equations. arxiv:2111.07469.  2021\.
  * [6] J. Chazarain and A. Piriou.  Introduction to the Theory of Linear Partial Differential Equations.  North-Holland, 1982.
  * [7] O. Ciaurri, T. Alastair, L. Roncal, J. Torrea, and J. Varona.  Harmonic analysis associated with a discrete Laplacian.  J. Anal. Math., 132:109–131, 2017.
  * [8] O. Ciaurri, L. Roncal, P. Stinga, J. Torrea, and J. Varona.  Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications.  Adv. Math., 330:688––738, 2018.
  * [9] P. Constantin.  Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, 1.  43 Lecture Notes in Mathematics 1871, Springer. Springer, Berlin, 2006\.
  * [10] R. Cont and P. Tankov.  Financial Modelling with Jump Processes.  Financial Mathematics Series. Chapmann and Hall/CRC, Boca Raton, FL, 2004\.
  * [11] M. Dabkowski.  Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equations.  Geom. Funct. Anal, 21:1–13, 2011.
  * [12] J. Delgado.  On a class of hyperbolic equations in Weyl-Hörmander calculus.  J. Math. Anal. Appl., 436(1):339–354, 2016.
  * [13] J. Delgado.  On on the well-posedness of the Cauchy problem for a class of degenerate parabolic equations.  Int. Equ. Oper. Theor., 90(1):19pp, 2018.
  * [14] E. Di Nezza, G. Palatucci, E. Valdinoci.  Hitchhiker’s guide to the fractional Sobolev spaces.  Bull. Sci. Math., 136(5):521––573, 2012.
  * [15] Ch. Epstein and C. Pop.  Regularity for the supercritical fractional Laplacian with drift.  J. Geom. Anal., 26(2):1231––1268, 2016.
  * [16] R. Frank and P. Ivanisvili.  Hypercontractivity of the semigroup of the fractional Laplacian on the n-sphere.  J. Funct. Anal., 281(8):109––145, 2021.
  * [17] G. Grubb.  Exact Green’s formula for the fractional Laplacian and perturbations.  Math. Scand., 126(3):568–592, 2020.
  * [18] J. Klafter I. M. Sokolov and A. Blumen.  Fractional kinetics.  Physics Today, November., pages 48–54, 2002.
  * [19] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasigeostrophic equation, Invent. math. 167 (2007), 445–453.
  * [20] M. Kwasnicki.  Ten equivalent definitions of the fractional Laplace operator.  Fract. Calc. Appl. Anal., 20(1):7–51, 2017.
  * [21] N. Laskin.  Principles of fractional quantum mechanics.  Fractional dynamics, World Sci. Publ., pages 393––427, 2012.
  * [22] A. J. Majda and A. L. Bertozzi.  Vorticity and Incompressible Flow.  Cambridge Texts in Applied Mathematics. Cambridge Univ. Press, Cambridge, 2002.
  * [23] A. Petrosyan and C.A. Pop.  Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift.  J. Funct. Anal., 268(2):417––472, 2015.
  * [24] L. Roncal and P. Stinga.  Fractional laplacian on the torus.  Commun. Contemp. Math., 18(3), 2016.
  * [25] M. Ruzhansky, V. Turunen, and J. Wirth.  Hörmander class of pseudo-differential operators on compact lie groups and global hypoellipticity.  J. Fourier Anal. Appl., 20(3):476–499, 2014.
  * [26] M. Ruzhansky, V. Turunen.  Quantization of pseudo-differential operators on the torus, J. Fourier Anal. Appl., 16, 943–982, 2010.
  * [27] M. Ruzhansky, V. Turunen.  Pseudo-differential operators and symmetries. Background analysis and advanced topics, volume 2 of Pseudo-Differential Operators. Theory and Applications.  Birkhäuser Verlag, Basel, 2010.
  * [28] R. Servadei, E. Valdinoci.  Fractional Laplacian equations with critical Sobolev exponent.  Rev. Mat. Complut., 28(3):655–676, 2015.
  * [29] R. Servadei and E. Valdinoci.  The Brezis-Nirenberg result for the fractional Laplacian.  Trans. Amer. Math. Soc.., 367(1):67––102, 2015.
  * [30] M. E. Taylor.  Pseudodifferential operators, volume 34 of Princeton Mathematical Series.  Princeton University Press, Princeton, N.J., 1981.


# Study of Silicon Photomultiplier Performance at Different Temperatures

N. Anfimov<EMAIL_ADDRESS>D. Fedoseev  A. Rybnikov  A. Selyunin
<EMAIL_ADDRESS>S. Sokolov  A. Sotnikov  Joint Institute for Nuclear
Research, Joliot-Curie 6, Dubna, Russia, 141980

###### Abstract

Decreasing the operation temperature of a Silicon Photo-Multiplier (SiPM)
leads to a drop in its dark noise. Some experiments consider cold temperatures
as an option for low noise applications of SiPM. One of those is the TAO
detector, which requires operation at $T\approx-50~{}^{\circ}$C. A significant
dependence of the Photon Detection Efficiency (PDE) of a SiPM on different
temperatures was reported with a drastic drop around this temperature. In this
paper, we present studies of performance for two samples of SiPMs from
Hamamatsu and AdvanSID (FBK) companies in a broad temperature range. No
significant difference for the PDE was observed.

###### keywords:

Silicon Photomultiplier - SiPM, Photon Detection Efficiency - PDE, Dark Count
Rate - DCR, PDE temperature dependence.

††journal: Nuclear Instruments and Methods in Physics Research A

##  1 Introduction

A Silicon photomultiplier (SiPM) is an array of small avalanche photodiodes,
called pixels, that operate above the breakdown voltage (overvoltage (OV)) in
a Geiger mode. The breakdown voltage is the minimum reverse bias voltage that
leads to self-sustaining avalanche multiplication in every SiPM’s pixel. Such
a structure allows for detecting and working with very low-intensity light in
a proportional mode with a range limited by the total number of pixels. SiPMs
are being introduced to many different applications. Nowadays they are widely
used in experimental physics and medical diagnostic techniques. The main
advantages with respect to conventional photomultipliers are higher detection
efficiency, less sensitivity to magnetic fields, compact dimensions, and gain
of up to 106. On the other hand, there are drawbacks of SiPMs: high dark rate,
optical crosstalk and temperature dependence of some parameters.

Currently, a satellite experiment for the JUNO neutrino experiment called TAO
is under consideration[1]. It is proposed that TAO will precisely measure the
primary neutrino spectrum from the nuclear reactors. The TAO detector is a
spherical vessel filled with a liquid scintillator covered by photosensors. To
achieve the required energy resolution of less than $2~{}\%$ at 1 MeV for the
TAO experiment one needs to maximize the light detection efficiency and
minimize the noise level. SiPM is a good candidate as a photosensor for TAO
since it has a higher photon detection efficiency(PDE) compared to
conventional PMTs. To reduce the high noise level of the SiPM it is proposed
to operate the TAO detector at low temperature $T\approx-50~{}^{\circ}$C [2].
But there is a relevant study [3] where a significant drop of the PDE for
1$\times$1 mm2 FBK-IRST SiPM type T6-V1-PD at
$T\approx-50~{}^{\circ}$C$~{}(\approx 220~{}$K) was observed (see fig. 1).
Such a drop may totally wash out the advantage of SiPM usage. Assuming the
effect might exist for other SiPM models we have studied two SiPM samples:
Hamamatsu MPPC-S13360-6025CS [4] (single element), AdvanSID ASD-NUV4S-P-4x4TD
[5] (SiPM array).

The paper is organized as follows. In section 2 we describe experimental setup
and methods that we used in our study. Next section 3 presents results for
both SiPM samples at different temperatures. And finally, in section 4 we draw
our conclusions.

Figure 1: Normalized photon detection efficiency as a function of temperature
for fixed overvoltage of 2V and three wavelengths (300 nm, 500 nm, and 800 nm)
for 1$\times$1 mm2 FBK-IRST SiPM type T6-V1-PD [3].

##  2 Experimental setup

In our measurements, a Dewar’s vessel is used as a cryostat to keep Liquid
Nitrogen (LN). Nitrogen vapor produces a gradient of temperature at different
heights inside the Dewar. By moving the SiPM one can provide different
temperatures in a broad range from room conditions down to LN environment. To
prevent any light leakage into the Dewar’s vessel it is placed inside a dark
room that is completely insulated from exterior light.

###  2.1 Incident light stability

To provide a valid study one needs to guarantee that light intensity is stable
in time and with environmental changes. In our measurements, we use a
stabilized LED light source with $\lambda=$ 425 nm wavelength produced by the
HVSys company [6]. The light from the LED is transported to the SiPM by means
of a plastic optical fiber. The light traveling through the optical fiber may
vary at level of about 10 % with temperature changes. It might be driven by
changing the optical properties between the fiber’s core and the cladding. To
reduce such a behavior, temperature stabilization is implemented along the
fiber’s length.

(a) (b)

Figure 2: Schematic view of the experimental setup: a - Dipped probe into the
Dewar’s vessel, b - Probe schematic with SiPM’s bed and light guides are
mounted in a copper tube.

###  2.2 Setup description

The cryostat vessel of $\sim$30 liters is filled to about 1/3 with Liquid
Nitrogen (see fig. 2a). An assembly of a light delivery system (see fig. 2b)
with the SiPM is moving along the cryostat depth. The light delivery system is
an optical fiber bundle of 6 fibers which is placed inside a copper pipe that
screens it from temperature changes. The light is sent to the SiPM or
reflective plate (calibration) through the central fiber. Then the light
reflects back from the SiPM and travels to the other fibers which are used to
monitor light stability with temperature changes. The pipe is wound by a
heating cable with feedback provided by a thermal sensor that is placed inside
the pipe. The heating cable keeps the fiber bundle at a stable temperature.
The assembly is additionally thermally insulated on the outside.

A reflective plate that replaces the silicon dice (SiPM) for the light
stability calibration is placed in front of the fiber bundle in order to
reflect the light back through the other 5 fibers. In this measurement, we use
quite high light intensity which is enough to monitor the reflected light by
means of a PIN-photodiode ThorLabs FDS100-CAL [7] and a Keithley 6487
Picoammeter/Voltage source. The measurement gives an estimation of the light
stability for different environmental temperatures. Overall light stability in
our system is reached at the level of 1%.

The reflection plate is then replaced with a single SiPM which is stuck to the
aluminum substrate (bed) with a thermal grease with an embedded thermosensor
on its backside. For the AdvanSID SiPM array we use a copper tub. The SiPM and
thermosensor are placed inside the tub which is wound with heating cable(see
fig. 3). In both cases, the temperature stability is guaranteed with precision
better than $1^{\circ}$C.

Figure 3: Copper tub is wound with heating cable and PCB with ASD-
NUV4S-P-4x4TD

###  2.3 Readout chain and data acquisition

A readout chain in our measurements consists of a SiPM power supply source, an
amplifier and a high-speed digitizer(see fig. 4). Apart from the readout
chain, two power supply sources are utilized to supply the heating cables. All
of this equipment is located outside the darkroom, except the warm amplifier
that sits close to the Dewar.

To provide a bias voltage to the SiPM, the Keithley 6487 is used. The source
automatically measures the current-voltage (I-V) curves of the SiPM that are
used for a breakdown voltage determination.

A signal from the SiPM is digitized by means of DRS4 [8] evaluation board that
can sample a signal within a range of (0.7-5) GS/s with about 11.5 effective
bits precision. It operates in external-trigger mode and is synchronized with
the LED trigger signal. The saved data are analyzed by software using the ROOT
package [9]. We put a high-gain amplifier into the readout chain to increase
the sensitivity of the ADC by 225 times in order to distinguish single
photoelectrons.

Figure 4: Readout chain circuit.

###  2.4 Data analysis

By integrating waveforms within a fixed window (gate) we can extract
charge111A SiPM’s charge spectrum usually provides better single electron
resolution compared to its amplitude spectrum (see fig. 5b). Usage of the
charge spectra allows better extraction of pedestal events. spectrum of a
SiPM. We acquire $N$-events for signal - (LED is ON), see fig. 5a, and
$D$-events for dark (LED is OFF) spectra. Using the number of events in the
pedestal for signal $N_{0}$ (blue area under the pedestal peak) and for dark
$D_{0}$ spectra (gray area given in the same range) the estimation of the
average number of photoelectrons $\hat{\mu}$ can be found [10]

$\hat{\mu}=-\ln{\left(\frac{N_{0}}{N}\cdot\frac{D}{D_{0}}\right)}=-\ln{\left(\frac{\hat{p}_{\xi_{0}}}{\hat{p}_{\lambda_{0}}}\right)}=\hat{\xi}-\hat{\lambda},$
(1)

where $\hat{p}_{\xi_{0}}$, $\hat{p}_{\lambda_{0}}$ \- estimators of the
pedestal probability in signal and dark spectra, and
$\hat{\xi}=-\ln(\hat{p}_{\xi_{0}})$,
$\hat{\lambda}=-\ln(\hat{p}_{\lambda_{0}})$ \- estimators of the average
number of photoelectrons in signal and dark spectra, respectively. Statistical
error in this case could be evaluated [10]

$\frac{\hat{\sigma}_{\hat{\mu}}}{\hat{\mu}}\approx\frac{1}{\sqrt{N}}\sqrt{\frac{e^{\hat{\xi}}+e^{\hat{\lambda}}-2}{(\hat{\xi}-\hat{\lambda})^{2}}}.$
(2)

(a) (b)

Figure 5: Typical charge and amplitude spectra of SiPMs: a - charge spectrum
illustrate the pedestal method for evaluation of average number of
photoelectrons. Blue area and $N_{0}$ \- pedestal events on signal spectrum,
gray area and $D_{0}$ \- pedestal events on dark spectrum (incorporated
picture); b - amplitude spectrum of SiPM and an illustration of the threshold
extraction method.

To compare parameters at different temperatures one should operate SiPM with
similar overvoltage. A way to find the breakdown voltage is to analyze the IV-
curve of a SiPM. There are several approaches of breakdown voltage extraction.
In our study, we use the Inverse Logarithmic Derivative (ILD) test described
in [11]. The ILD for IV-curve reads

$f(V)=\left[\frac{1}{I}\cdot\frac{dI}{dV}\right]^{-1}$ (3)

and the minimum of the $f(V)$ yields the inflection point which defines the
breakdown voltage. We also test the quadratic fit for AndvanSid SiPM which was
used in the paper [3]. Both approaches demonstrate good consistency of the
breakdown voltage extraction (see table 1). Measured voltages for both SiPMs
using the ILD test are shown in fig. 6.

Figure 6: Breakdown voltages for Hamamatsu and AdvanSid SiPMs vs temperature
using ILD test (see eq. 3). Table 1: Breakdown voltages for the AdvanSid SiPM
at different temperature points extracted by ILD test and quadratic fit

SiPM type | Breakdown voltage $V_{bd}$ at different temperatures
---|---
ASD-NUV4S-P-4x4TD (ILD test) | 26.57 V[ $23~{}^{o}$C] | 26.02 V[ $2~{}^{o}$C] | 25.39 V[$-22~{}^{o}$C]
24.58 V[$-52~{}^{o}$C] | 23.83 V[$-84~{}^{o}$C] |
ASD-NUV4S-P-4x4TD (quadratic fit) | 26.55 V[ $23~{}^{o}$C] | 26.00 V[ $2~{}^{o}$C] | 25.39 V[$-22~{}^{o}$C]
24.57 V[$-52~{}^{o}$C] | 23.84 V[$-84~{}^{o}$C] |

We evaluated the Dark Count Rate (DCR) of the SiPM using the following
procedure. By processing raw waveforms we count a number of dark pulses (LED
is OFF) that exceed some fixed threshold. We processed $w=10000$ waveforms
within $W=1.024~{}\mu$s duration window222In our case, SiPM pulses $\sim 100$
ns (FWHM) which can introduce a small fraction of pulses miscounting. We took
it into account shrinking the window by the pulse width. (1024 samples at 1
GS/s sampling rate). The overall counting window is $w\times W$. The DCR rate
$R$ could be found from the number of dark pulses $N$ that have been counted
in the overall window as

$R=\frac{N}{w\times W}$ (4)

The number of the random pulses $N$ follows Poisson distribution333Standard
deviation $\sigma_{N}=\sqrt{N}$ and a standard deviation of the DCR using eq.
4 is

$\sigma_{R}=\frac{\sqrt{N}}{w\times W}.$ (5)

The threshold is obtained as an empirical compromise between efficiency and
noise pick up. By measuring SiPM’s amplitude spectrum (see. fig. 5b) we
defined the optimal threshold444We assume signal and noise pulses are of
similar amplitude. $T$ as

$\text{T}=A_{1}-2\sigma_{A_{1}},$ (6)

where $A_{1}$ and $\sigma_{A_{1}}$ \- the amplitude and its standard deviation
of the single photoelectron signal, respectively. Decreasing the threshold
leads to significant noise pick up at a small gain (overvoltage). In our case
(see eq. 6) only $\sim$2,5 % of SiPM’s signals are miscounted in the dark rate
evaluation procedure.555Assuming single pixel’s amplitudes follow Normal
distribution

(a) (b)

Figure 7: PDE as a function of wavelength: a – for Hamamatsu SiPM at
overvoltage OV = 5 V[4], b – for AdvanSid SiPM at OV = 6 V[5]

Measurements were performed for both SiPM samples at different temperatures
and in the range of operating overvoltages. To present the PDE more
illustratively we normalized measured relative PDE ($\hat{\mu}$) to 25 % (see
fig. 7) for Hamamatsu SiPM ($V_{OV}=5.0~{}V$ at $T=23.0~{}^{\circ}$C) and to
42 % 666On fig. 7b the PDE is a bit higher than 40 % at $V_{OV}$=6 V. For our
sample, we were not able to reach $V_{OV}$ above 5.68 V because of high noise
level. And we assumed the PDE=42 % at this point. for ASD-NUV4S-P-4x4TD
($V_{OV}=5.68~{}V$ at $T=23.0~{}^{\circ}$C).

##  3 Results

The PDE and the DCR dependencies on overvoltage for two SiPMs were studied at
the wide range of temperatures from $T=23.0~{}^{\circ}$C (room temperature)
and even below the $T=-52.0~{}^{\circ}$C which is close to the TAO operational
temperature.

To compare the results of the PDE behavior for all temperatures we plot the
PDE as a function of overvoltage (see. fig. 8). One can see that there is no
significant temperature dependence of the PDE for both samples.

The DCR for SiPMs(see. fig. 9) decreases rapidly and reaches a drop by a
factor of several orders of magnitude for low temperatures compared to room
conditions. This fact is one of the major considerations for the SiPM usage at
$T=-50~{}^{\circ}$C for the TAO detector. As can be seen from fig. 9 for low
overvoltages at negative temperatures we cannot obtain DCR values777Small
number of dark pulses comparing to a noise pick up at a very low threshold.

(a) (b)

Figure 8: PDE as a function of overvoltage: a – for Hamamatsu SiPM, b – for
AdvanSid SiPM

(a) (b)

Figure 9: Evaluated DCR as a function of overvoltage: a – for Hamamatsu SiPM,
b – for AdvanSid SiPM

##  4 Summary

In general, the setup and technique demonstrated in this paper allows for
measurements with different types of SiPMs in a wide range of temperatures
from Liquid Nitrogen (77 K) up to the Room temperature. We studied the Photon
Detection Efficiency of SiPMs in a broad temperature range including
$T=-52.0~{}^{\circ}$C ($\sim 220$ K) which is in the considered region of the
TAO detector. As a result, two types of SiPMs - Hamamatsu S13360-6025CS and
AdvanSID ASD-NUV4S-P-4x4TD have no significant difference in the PDE. These
measurements do not exclude the behavior of the PDE in a broader temperature
range and demonstrate an absence of difference for particular temperature
points.888And incident light wavelength $\lambda=$425 nm.

The result presented in the study [3] that demonstrates the PDE drop in the
temperature region around $T\sim$ 200 K to 220 K have not been found for our
SiPM samples. Therefore, the SiPM usage in the TAO experiment is relevant and
might have advantages comparing to the conventional PMTs. The choice of the
proper type of SiPM for TAO requires a complex study of the SiPM’s parameters
at the TAO operational temperature.

##  5 Acknowledgments

The authors are indebted to Dr. Dmitriy Naumov, prof. Alexander Olshevskiy and
Christopher Kullenberg from the Joint Institute for Nuclear Research for their
great help in the preparation of this paper.

## References

  * Perrot [2018] F. Perrot, Status and physics potential of the JUNO experiment, Bari, 2018. Symposium DISCRETE18.
  * Cao [2019] G. Cao, JUNO-TAO Experiment with Large Area High Performance SiPMs, Vienna, 2019. SiPM Workshop.
  * Collazuol et al. [2011] G. Collazuol, M. Bisogni, S. Marcatili, C. Piemonte, A. D. Guerra,  Studies of silicon photomultipliers at cryogenic temperatures,  Nucl. Instrum. Meth. A628 (2011) 389 – 392. VCI 2010.
  * Hamamatsu [2019] Hamamatsu, MPPC S13360-6025CS, 2019\. https://www.hamamatsu.com/eu/en/product/type/S13360-6025CS/index.html.
  * AdvanSiD [2019] AdvanSiD, ASD-NUV4S-P-4x4TD, 2019\. http://advansid.com/products/product-detail/asd-sipm4s-p-4x4td.
  * HVSys [2019] HVSys, Calibrated LED sources of light flashes, 2019. http://hvsys.ru/images/data/news/5_small_1368802948.pdf.
  * ThorLabs [2019] ThorLabs, FDS100-CAL - Calibrated Si Photodiode, 2019. https://www.thorlabs.com/thorproduct.cfm?partnumber=FDS100-CAL.
  * Institute [2019] P. S. Institute, DRS4 evaluation board, 2019\. https://www.psi.ch/en/drs/evaluation-board.
  * Brun and Rademakers [1997] R. Brun, F. Rademakers,  ROOT - an object oriented data analysis framework,  Nucl. Instr. and Meth. A 389 (1997) 81–86.
  * Anfimov et al. [2019] N. Anfimov, A. Rybnikov, A. Sotnikov,  Optimization of the light intensity for photodetector calibration,  Nucl. Instrum. Meth. A939 (2019) 61 – 65.
  * Chmill et al. [2017] V. Chmill, E. Garutti, R. Klanner, M. Nitschke, J. Schwandt,  Study of the breakdown voltage of SiPMs,  Nucl. Instrum. Meth. A845 (2017) 56–59.