# Hyperbolic systems with non-diagonalisable principal part and variable
multiplicities, III: singular coefficients

Claudia Garetto  Claudia Garetto: School of Mathematical Sciences Queen Mary
University of London Mile End Road, London, E1 4NS United Kingdom E-mail
address<EMAIL_ADDRESS>and  Bolys Sabitbek  Bolys Sabitbek: School of
Mathematical Sciences Queen Mary University of London Mile End Road, London,
E1 4NS United Kingdom E-mail address<EMAIL_ADDRESS>
(Date: May 2022)

###### Abstract.

In this paper we continue the analysis of non-diagonalisable hyperbolic
systems initiated in [25, 26]. Here we assume that the system has
discontinuous coefficients or more in general distributional coefficients.
Well-posedness is proven in the very weak sense for systems with singularities
with respect to the space variable or the time variable. Consistency with the
classical theory is proven in the case of smooth coefficients.

###### Key words and phrases:

Hyperbolic systems, very weak solutions, regularisation

###### 2010 Mathematics Subject Classification:

Primary 35L40: 35L81; Secondary 35D99;

The authors are supported by the EPSRC grant EP/V005529/2

On behalf of all authors, the corresponding author states that there is no
conflict of interest. This manuscript has no associated data.

##  1\. Introduction

This paper continues the analysis of hyperbolic systems with non-
diagonalisable principal part and variable multiplicities started by Gramchev
and Ruzhansky in [27] for $2\times 2$ hyperbolic systems and continued by the
first author, Jäh and Ruzhansky in [25, 26] for systems of arbitrary size.
Here we assume that the system coefficients are singular, i.e., distributional
in $x$ but still continuous with respect to $t$ or when smoothness is assumed
in the space variable we allow singularities in time. Let us briefly summarise
what is known about hyperbolic systems with non-diagonalisable principal part.
Let

$\begin{cases}D_{t}u=A(t,x,D_{x})u+L(t,x,D_{x})u+f(t,x),\qquad(t,x)\in[0,T]\times\mathbb{R}^{n},\\\
u(0,x)=g(x),\end{cases}$

where $A$ is an upper triangular $m\times m$ matrix with real eigenvalues and
$L$ is the matrix of the lower order terms. Note that the eigenvalues might
coincide and therefore we are dealing here in general with a hyperbolic system
with multiplicities and non-diagonalisable principal part. It is also not
restrictive to assume that the matrix $A$ is upper-triangular since conditions
are given in [25] which allow the reduction into upper-triangular form of the
system above. For a non exhausting overview on hyperbolic problems with
multiplicities we refer the reader to [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 16, 17, 18, 19, 21, 22, 24, 32, 33, 36, 37, 39].

Combining the results proven in [25, 26, 27] we have that the Cauchy problem
above is well-posed in anisotropic Sobolev spaces of any order provided that
the entries of $A=(a_{ij})$ are symbols in $C([0,T],S^{1}(\mathbb{R}^{2n}))$
and the entries of $L=(\ell_{ij})$ are symbols in
$C([0,T],S^{0}(\mathbb{R}^{2n})$ fulfilling the following Levi type condition:

$\ell_{ij}\in C([0,T],S^{j-i}(\mathbb{R}^{2n}))$

for $i>j$. In addition, when $A$ depends only on $x$ and assuming intersection
of finite order at points of multiplicity one can obtain a representation
formula for the solution which leads to results of propagation of
singularities. See [26] for more details. The results above heavily rely on
the fact that the system coefficients are smooth in space and at least
continuous on time. This is due to the fact that the system is solved making
use of symbolic calculus and Fourier integral operators methods [30].

When the system coefficients are less than continuous in time or less than
smooth in space it might be difficult even to define a notion of solution.
Indeed, as a consequence of Schwartz’s impossibility result such system might
fail to have a distributional solution [31, 34, 35]. For this reason, we want
to prove here well-posedness in a _very weak sense_. Note that the desire to
drop the regularity assumption of the system coefficients has physical
motivation, since this kind of systems naturally arise in wave propagation
into a multi-layered medium, like the Subsoil, or in conical refraction in
crystals, see [19] and references therein.

The notion of a _very weak solution_ has been introduced by the first author
and Ruzhansky in [19]. It is based on the idea of replacing the original
problem with a regularised one, where the coefficients and initial data are
regularised via convolution with suitable mollifiers. This generates a net of
hyperbolic problems with smooth coefficients depending on the regularising
parameter $\varepsilon\in(0,1)$. The corresponding net of solutions
$(u_{\varepsilon})_{\varepsilon}$ is said to be a very weak solution if it
satisfies some specific behaviour with respect to the parameter, that we will
refer to as _moderate_. Note that the notion of very weak solution is inspired
by the theory of algebras of generalised functions [14, 15, 23, 28, 35]
however, here we work with nets of functions rather than equivalence classes
and we have full flexibility of regularising coefficients and initial data
with different mollifiers and scales, basing our choice on the intrinsic
properties of the problem we are studying. Moderateness will also be measured
according to different norms or seminorms which are determined by the function
space where we aim to prove well-posedness: $C^{\infty}$, Sobolev spaces,
Gevrey classes, etc.

In the sequel we list the main results obtained in this paper.

We begin by focusing on the Cauchy problem

$D_{t}u=A(t,x,D_{x})u+L(t,x,D_{x})u+f(t,x),\qquad(t,x)\in[0,T]\times\mathbb{R}^{n},$

with initial data $u(0,x)=g(x)$, where,

$A(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j}(t,x)D_{x_{j}}&a_{12}(t,x,D_{x})\\\
0&\sum_{j=1}^{n}\lambda_{2j}(t,x)D_{x_{j}}\end{bmatrix},$

and

$L(t,x,D_{x})=\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
\ell_{21}(t,x)\langle D_{x}\rangle^{-1}&\ell_{22}(t,x)\end{bmatrix}.$

We assume that the matrices $A$ and $L$ are defined by pseudo-differential
operators of order $1$ and $0$ respectively, with symbols which are not smooth
in $x$. Namely, we assume that the system coefficients are continuous in $t$
but distributional with compact support in $x$. As a toy model one could think
about a discontinuous function in $x$ or a Dirac distribution in $x$. After
regularisation, i.e. convolution with a suitable mollifier

$\psi_{\varepsilon}(x)=\omega(\varepsilon)^{-n}\psi(\omega(\varepsilon)^{-1}x),$

where $\omega(\varepsilon)\to 0$ as $\varepsilon\to 0$, we work on

$\begin{cases}D_{t}u_{\varepsilon}=A_{\varepsilon}(t,x,D_{x})u_{\varepsilon}+L_{\varepsilon}(t,x,D_{x})u_{\varepsilon}+f_{\varepsilon}(t,x),&\quad(t,x)\in[0,T]\times\mathbb{R}^{n},\\\
u_{\varepsilon}(0,x)=g_{\varepsilon}(x),&\quad x\in\mathbb{R}^{n},\end{cases}$

where $u_{\varepsilon}(0,x)=[g_{1,\varepsilon}(x),g_{2,\varepsilon}(x)]^{T}$
and
$f_{\varepsilon}(t,x)=[f_{1,\varepsilon}(t,x),f_{2,\varepsilon}(t,x)]^{T}$,
and the system matrices are

$A_{\varepsilon}(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j,\varepsilon}(t,x)D_{x_{j}}&a_{12,\varepsilon}(t,x,D_{x})\\\
0&\sum_{j=1}^{n}\lambda_{2j,\varepsilon}(t,x)D_{x_{j}}\end{bmatrix},$

and

$L_{\varepsilon}(t,x,D_{x})=\begin{bmatrix}\ell_{11,\varepsilon}(t,x)&\ell_{12,\varepsilon}(t,x)\\\
\ell_{21,\varepsilon}(t,x)\langle
D_{x}\rangle^{-1}&\ell_{22,\varepsilon}(t,x)\end{bmatrix}.$

By the theory of Fourier integral operators [25, 26] we know that the solution
$u_{\varepsilon}=[u_{1,\varepsilon},u_{2,\varepsilon}]^{T}$ can be written
explicitly via action of FIOs or integrated FIOs on the initial data, that we
will denote here with $G^{0}_{i,\varepsilon}$ and $G_{i,\varepsilon}$,
respectively, with $i=1,2$. The corresponding phase functions,
$\phi_{i,\varepsilon}$ can be also explicitly calculated by solving the
Eikonal equation determined by the system eigenvalues. It turns out that to
guarantee Sobolev mapping properties of the operators involved in our proof
(see [38]) we need nets of symbols of order $0$ as well as some bound on the
phase function, namely,

$\displaystyle|\partial_{\xi}\phi_{i,\varepsilon}(t,x,\xi)-\partial_{\xi}\phi_{i,\varepsilon}(t,y,\xi)|$
$\displaystyle\geq C(\varepsilon)|x-y|\,\,\,\,\text{ for
}\,\,x,y\in\mathbb{R}^{n},\,\,\xi\in\mathbb{R}^{n},$
$\displaystyle|\partial_{y}\phi_{i,\varepsilon}(t,y,\xi)-\partial_{y}\phi_{i,\varepsilon}(t,y,\eta)|$
$\displaystyle\geq C(\varepsilon)|\xi-\eta|\,\,\,\,\text{ for
}\,\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

for $i=1,2$. In particular, it is crucial to study the $\varepsilon$-behaviour
of the composition operator

$\mathcal{G}^{0}_{1,\varepsilon}:=G_{1,\varepsilon}\circ(a_{12,\varepsilon}+\ell_{12,\varepsilon})\circ
G_{2,\varepsilon}\circ\ell_{21,\varepsilon}\langle D_{x}\rangle^{-1}.$

Summarising, we identify the following set of hypotheses which guarantees the
hyperbolicity of the system as well as its solvability in the very weak sense:

  * (H1)

the coefficients $(\lambda_{1j,\varepsilon})_{\varepsilon}$ and
$(\lambda_{2j,\varepsilon})_{\varepsilon}$ are real valued and
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $j=1,\ldots,n$ with compact
support in $x$;

  * (H2)

$(\partial_{k}\lambda_{1j,\varepsilon})_{\varepsilon}$,
$(\partial_{k}\lambda_{2j,\varepsilon})_{\varepsilon}$ and
$(\ell_{ii,\varepsilon})_{\varepsilon}$ are logarithmic moderate for
$k,j=1,\ldots,n$ and $i=1,2$;

  * (H3)

the nets of constants $(C(\varepsilon))_{\varepsilon}$ and
$(C^{-1}(\varepsilon))_{\varepsilon}$ are moderate and

$||\mathcal{G}^{0}_{1,\varepsilon}||_{H^{s}\rightarrow H^{s}}=O(1).$

In detail, we prove that

  * •

if (H1), (H2) and (H3) hold then the Cauchy problem above has a very weak
solution (a solution $(u_{\varepsilon})_{\varepsilon}$ which is
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate) and it is well-posed in the
very weak sense (Theorem 3.2, Corollary 3.3);

  * •

if $A$ is diagonal and $L$ upper-triangular then (H1) and (H2) are sufficient
to get very weak well-posedness (Corollary 3.4);

  * •

if $A$ and $L$ are both upper-triangular then the Cauchy problem is well-posed
in the very weak sense provided that (H1), (H2) hold and the nets of constants
$(C(\varepsilon))_{\varepsilon}$ and $(C^{-1}(\varepsilon))_{\varepsilon}$ are
moderate (Corollary 3.5);

  * •

the last two results are easily extendable to matrices of size $m\times m$.

This paper is organised as follows: in Section 2 we collect some preliminaries
on regularisation and very weak solutions for hyperbolic equations. The main
results are proven in Section 3. In Section 4 we prove that very weak
solutions recover the classical solutions in the limit as $\varepsilon\to 0$
in the case of smooth coefficients. We also discuss some examples and we show
how to adapt our methods to coefficients which are singular in time rather
than in space. The paper ends with an Appendix devoted to the
$L^{2}$-boundedness of Fourier integral operators [38]. In particular, we
formulate precise bounds for FIOs depending on $\varepsilon$ of the type
encountered in the paper.

##  2\. Preliminaries: regularisation and very weak solutions

In this section we recall the notion of very weak solution as introduced in
[19]. The main idea, which goes back to algebras of generalised functions,
consists in regularising the equation coefficients via a Friedrichs type
mollifier $\psi$ ($\psi\in C^{\infty}_{c}(\mathbb{R}^{n})$, $\psi\geq 0$ with
$\int\psi=1)$. In other words, given a compactly supported distribution $u$ on
$\mathbb{R}^{n}$, we generate a net of smooth functions
$(u_{\varepsilon})_{\varepsilon}$, where $\varepsilon\in(0,1]$, by writing

$u_{\varepsilon}=u*\psi_{\omega(\varepsilon)},$

where

$\psi_{\omega(\varepsilon)}(x)=\omega(\varepsilon)^{-n}\psi(x/\omega(\varepsilon)),$

and $\omega(\varepsilon)$ is a positive function converging to $0$ as
$\varepsilon\rightarrow 0$ with $\omega(\varepsilon)\geq c\varepsilon^{a}$ for
some $c,a>0$, uniformly with respect to $\varepsilon$. It is clear that
$u_{\varepsilon}$ converges to $u$ as $\varepsilon\to 0$. In addition, the net
$(u_{\varepsilon})_{\varepsilon}$ fulfills some extra properties with respect
to the parameter $\varepsilon$ that are illustrated in the proposition below.
For a detailed proof of we refer the reader to [19, 28, 35] and reference
therein.

######  Proposition 2.1.

  * (i)

If $u\in\mathcal{E^{\prime}}(\mathbb{R}^{n})$ then there exists
$N\in\mathbb{N}_{0}$ and for all $\alpha\in\mathbb{N}_{0}^{n}$ there exists
$c>0$ such that

$|\partial^{\alpha}(u*\psi_{\omega(\varepsilon)})(x)|\leq
c\omega(\varepsilon)^{-N-|\alpha|},$

for all $x\in\mathbb{R}^{n}$ and $\varepsilon\in(0,1]$.

  * (ii)

If $f\in C_{c}^{\infty}(\mathbb{R}^{n})$ then for all
$\alpha\in\mathbb{N}_{0}^{n}$ there exists $c>0$ such that

$|\partial^{\alpha}(f*\psi_{\omega(\varepsilon)})(x)|\leq c,$

for all $x\in\mathbb{R}^{n}$ and $\varepsilon\in(0,1]$.

  * (iii)

If $f\in C_{c}^{\infty}(\mathbb{R}^{n})$ and mollifier $\psi$ has all the
moments vanishing, i.e., $\int_{\mathbb{R}^{n}}\psi(x)dx=1$ and
$\int_{\mathbb{R}^{n}}x^{\alpha}\psi(x)dx=0$ for all multi-index $\alpha$ with
$|\alpha|>1$, then for all $\alpha\in\mathbb{N}_{0}^{\alpha}$ and for all
$q\in\mathbb{N}_{0}$ there exists $c>0$ such that

$|\partial^{\alpha}(f*\psi_{\omega(\varepsilon)}(x)-f(x))|\leq
c\omega(\varepsilon)^{q},$

for all $x\in\mathbb{R}^{n}$ and $\varepsilon\in(0,1]$.

It follows that the nets obtained via convolution with a mollifier are
_moderate_ or _negligible_ with respect to the seminorms of
$C^{\infty}(\mathbb{R}^{n})$ (see Definition 2.1). Analogously one can replace
$C^{\infty}(\mathbb{R}^{n})$ with $C^{\infty}([0,T]\times\mathbb{R}^{n})$. In
the sequel, $K\Subset\mathbb{R}^{n}$ stands for $K$ is a compact set in
$\mathbb{R}^{n}$.

######  Definition 2.1.

  * (i)

A net $(v_{\varepsilon})_{\varepsilon}\in C^{\infty}(\mathbb{R}^{n})^{(0,1]}$
is $C^{\infty}$-moderate if for all $K\Subset\mathbb{R}^{n}$ and for all
$\alpha\in\mathbb{N}_{0}^{n}$ there exists $N\in\mathbb{N}_{0}$ and $c>0$ such
that

$|\partial^{\alpha}v_{\varepsilon}(x)|\leq c\varepsilon^{-N},$

uniformly in $x\in K$ and $\varepsilon\in(0,1]$.

  * (ii)

A net $(v_{\varepsilon})_{\varepsilon}\in C^{\infty}(\mathbb{R}^{n})^{(0,1]}$
is $C^{\infty}$-negligible if for all $K\Subset\mathbb{R}^{n}$,
$\alpha\in\mathbb{N}_{0}^{n}$ and $q\in\mathbb{N}_{0}$ there exists $c>0$ such
that

$|\partial^{\alpha}v_{\varepsilon}(x)|\leq c\varepsilon^{q},$

uniformly in $x\in K$ and $\varepsilon\in(0,1]$.

  * (iii)

A net $(v_{\varepsilon})_{\varepsilon}\in
C^{\infty}([0,T]\times\mathbb{R}^{n})^{(0,1]}$ is $C^{\infty}$-moderate if for
all $K\Subset\mathbb{R}^{n}$ and for all $l\in\mathbb{N}_{0}$ and
$\alpha\in\mathbb{N}_{0}^{n}$ there exists $N\in\mathbb{N}_{0}$ and $c>0$ such
that

$|\partial^{k}_{t}\partial^{\alpha}v_{\varepsilon}(t,x)|\leq
c\varepsilon^{-N},$

uniformly in $t\in[0,T]$, $x\in K$ and $\varepsilon\in(0,1]$.

  * (iv)

A net $(v_{\varepsilon})_{\varepsilon}\in
C^{\infty}([0,T]\times\mathbb{R}^{n})^{(0,1]}$ is $C^{\infty}$-negligible if
for all $K\Subset\mathbb{R}^{n}$, $k\in\mathbb{N}_{0}$,
$\alpha\in\mathbb{N}_{0}^{n}$ and $q\in\mathbb{N}_{0}$ there exists $c>0$ such
that

$|\partial^{k}_{t}\partial^{\alpha}v_{\varepsilon}(t,x)|\leq
c\varepsilon^{q},$

uniformly in $t\in[0,T]$, $x\in K$ and $\varepsilon\in(0,1]$.

  * (v)

A net $(v_{\varepsilon})_{\varepsilon}\in
C([0,T],C^{\infty}(\mathbb{R}^{n})^{(0,1]}$ is
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate if for all
$K\Subset\mathbb{R}^{n}$ and for all $\alpha\in\mathbb{N}_{0}^{n}$ there
exists $N\in\mathbb{N}_{0}$ and $c>0$ such that

$|\partial^{\alpha}_{x}v_{\varepsilon}(t,x)|\leq c\varepsilon^{-N},$

uniformly in $t\in[0,T]$, $x\in K$ and $\varepsilon\in(0,1]$.

  * (vi)

A net $(v_{\varepsilon})_{\varepsilon}\in
C^{\infty}([0,T]\times\mathbb{R}^{n})^{(0,1]}$ is
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-negligible if for all
$K\Subset\mathbb{R}^{n}$, $\alpha\in\mathbb{N}_{0}^{n}$ and
$q\in\mathbb{N}_{0}$ there exists $c>0$ such that

$|\partial^{\alpha}_{x}v_{\varepsilon}(t,x)|\leq c\varepsilon^{q},$

uniformly in $t\in[0,T]$, $x\in K$ and $\varepsilon\in(0,1]$.

######  Remark.

If we are dealing with nets of numbers then we will measure moderateness and
negligibility in terms of the module in $\mathbb{C}$ or $\mathbb{R}$ and we
will simply talk of moderate and negligible nets. Analogously, we can state
the definitions above for vectors by replacing $C^{\infty}(\mathbb{R}^{n})$
with $C^{\infty}(\mathbb{R}^{n})^{m}$. We will still talk of
$C^{\infty}$-moderatenss and $C^{\infty}$-negligibility.

###  2.1. State of the art on hyperbolic equations with low regular
coefficients

In this subsection we give a brief state of the art on hyperbolic equations
with low regular coefficients, with particular focus on the very weak well-
posedness of the corresponding Cauchy problem.

We begin by recalling that the Cauchy problem for ordinary differential
equations has been investigated already in [28] in the context of Colombeau
algebras of generalised functions. The results obtained there (Theorem 1.5.2
[28]) can be rewritten at the net level as follows, for the Cauchy problem

$\begin{split}\dot{x_{\varepsilon}}(t)&=F_{\varepsilon}(t,x_{\varepsilon}(t)),\\\
x_{\varepsilon}(t_{0})&=x_{0,\varepsilon},\end{split}$ (1)

where $t\in\mathbb{R}$, $(F_{\varepsilon})_{\varepsilon}$ is a net in
$(C^{\infty}(\mathbb{R}^{1+n}))^{n}$ and $(x_{\varepsilon})_{\varepsilon}$ is
a net of initial conditions. In the sequel we are making use of the standard
norm in $\mathbb{R}^{1+n}$ that for the sake of simplicity we will continue to
denote with $|\cdot|$.

######  Proposition 2.2.

Let $(F_{\varepsilon})_{\varepsilon}$ be a net in
$(C^{\infty}(\mathbb{R}^{1+n}))^{n}$ such that

  * (i)

for all $k\in\mathbb{N}_{0}$ and $\alpha\in\mathbb{N}_{0}^{n}$ there exists
$N\in\mathbb{N}_{0}$ and $c>0$ such that

$|\partial^{k}_{t}\partial^{\alpha}_{x}F_{\varepsilon}(t,x)|\leq
c\varepsilon^{-N}(1+|(t,x)|)^{N},$

for all $(t,x)\in\mathbb{R}\times\mathbb{R}^{n}$ and $\varepsilon\in(0,1]$.

  * (ii)

there exists $c>0$ such that

$\|\nabla_{x}F_{\varepsilon}(t,x)\|_{L^{\infty}(\mathbb{R}^{1+n})}\leq
c|\log\varepsilon|,$

for all $\varepsilon\in(0,1]$.

For all $\varepsilon\in(0,1]$ the Cauchy problem (1) has a unique solution
$x_{\varepsilon}\in C^{\infty}(\mathbb{R}^{1+n})^{n}$ such that if
$(x_{0,\varepsilon})_{\varepsilon}$ is moderate then
$(x_{\varepsilon})_{\varepsilon}$ is $C^{\infty}(\mathbb{R})$-moderate as
well. The solution $(x_{\varepsilon})_{\varepsilon}$ is unique modulo,
negligible perturbation, i.e., if we perturb the initial data with a
negligible net (i.e., $x_{0,\varepsilon}+n_{\varepsilon}$, where
$(n_{\varepsilon})_{\varepsilon}$ is negligible) then the corresponding
solution will differ from $(x_{\varepsilon})_{\varepsilon}$ by a
$C^{\infty}$-negligible net.

For the sake of simplicity we say that
$(\nabla_{x}F_{\varepsilon})_{\varepsilon}$ is logarithmic moderate. Note that
one can reformulate the proposition above on a finite time interval $[0,T]$
and replace the polynomial bound in (i) with an assumption of compact support
with respect to the $x$-variable.

Proposition 2.2 can also be stated for nets that are only continuous with
respect to $t$. This means to set $k=0$ and replace $C^{\infty}$ with
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$. Continuity with respect to $t\in[0,T]$
and smoothness with respect to $x$ will be the standard assumptions throughout
this paper for our nets of functions.

Let us now consider the scalar case, i.e., the Cauchy problem for first order
hyperbolic equations. In detail, let

$\displaystyle D_{t}w_{\varepsilon}$
$\displaystyle=\sum_{j=1}^{n}a_{j,\varepsilon}(t,x)D_{j}w_{\varepsilon}+a_{0,\varepsilon}(t,x)w_{\varepsilon},$
(2) $\displaystyle w_{\varepsilon}(0,x)$ $\displaystyle=w_{0,\varepsilon},$

where $D_{j}=D_{x_{j}}$, the coefficients $a_{j,\varepsilon}$ are real-valued
for all $j=1,\ldots,n$. The equation coefficients are compactly supported
distributions with respect to $x$ regularised by convolution with a mollifier
$\psi$ that we assume non-negative and with integral $1$. In detail,

  * -

$a_{j}\in C([0,T]\times\mathcal{E^{\prime}}(\mathbb{R}^{n}))$ with $a_{j}\geq
0$ for all $j=1,\ldots,n$,

  * -

$a_{0}\in C([0,T]\times\mathcal{E^{\prime}}(\mathbb{R}^{n}))$

  * -

$w_{0}\in\mathcal{E^{\prime}}(\mathbb{R}^{n})$.

and

  * -

$a_{j,\varepsilon}(t,x)=(a_{j}(t,\cdot)*\psi_{\omega(\varepsilon)})(x)$ for
$j=1,\ldots,n$,

  * -

$a_{0,\varepsilon}(t,x)=(a_{0}(t,\cdot)*\psi_{\omega(\varepsilon)})(x)$,

  * -

$w_{0},\varepsilon(x)=w_{0}*\psi_{\varepsilon}(x)$.

######  Definition 2.2.

We say that the Cauchy problem

$\begin{split}D_{t}w&=\sum_{j=1}^{n}a_{j}(t,x)D_{j}w+a_{0}(t,x)w,\\\
w(0,x)&=w_{0}\end{split}$

admits a _very weak solution_ $(w_{\varepsilon})_{\varepsilon}$ if the net
$(w_{\varepsilon})_{\varepsilon}$ solves the regularised problem (2) and it is
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate.

The following result, is the net-version of the well-posedness results
obtained in [14, 29, 34]. It shows that a transport equation with singular
coefficients admits a very weak solution under suitable choice of the
regularising scale. In the sequel we say that a net of function is logarithmic
moderate if its $L^{\infty}([0,T]\times\mathbb{R}^{n})$-norm is estimated by a
logarithmic scale.

######  Proposition 2.3.

Let the coefficients $(a_{j,\varepsilon})_{\varepsilon}$ be
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $j=1,\ldots,n$ with compact
support in $x$. Suppose that $(a_{j,\varepsilon})_{\varepsilon}$ are real
valued and $(\partial_{k}a_{j,\varepsilon})_{\varepsilon}$ as well as
$(a_{0,\varepsilon})_{\varepsilon}$ are logarithmic moderate
$(k,j=1,\ldots,n)$. Then,

  * (i)

the Cauchy problem above admits a very weak solution
$(w_{\varepsilon})_{\varepsilon}$ which is compactly supported with respect to
$x$.

  * (ii)

If $a_{0,\varepsilon}$ as well as $w_{0,\varepsilon}$ are real-valued then the
solution $(w_{\varepsilon})_{\varepsilon}$ is real valued as well.

  * (iii)

Negligible perturbations of the coefficients and initial data lead to a
negligible perturbation in the very weak solution.

In [14] the authors investigated further the properties of the very weak
solution $(w_{\varepsilon})_{\varepsilon}$ and proved that it can be written
as the action of a Fourier integral operator (FIO) on the initial data
$(w_{0,\varepsilon})_{\varepsilon}$. We summarise here the main steps of their
argument, mainly to solve the Eikonal equation to determine the phase function
of the FIO and to solve a certain transport equation to determine its symbol.
As expected, we will work with nets of FIO’s, phase functions and symbols
depending on the regularising parameter $\varepsilon\in(0,1]$.

###  2.2. Nets of phase functions

We begin by constructing the phase function of the FIO making use of the
characteristic curves associated to the principal part of the equation. This
means to solve the linear Cauchy problem

$\begin{split}\partial_{t}\phi_{\varepsilon}(t,x,\xi)&=\sum_{j=1}^{n}a_{j,\varepsilon}(t,x)\partial_{j}\phi_{\varepsilon}(t,x,\xi)\\\
\phi_{\varepsilon}(0,x,\xi)&=x\cdot\xi.\end{split}$ (3)

Under the assumptions on coefficients $(a_{j,\varepsilon})_{\varepsilon}$ in
Proposition 2.3, the Cauchy problem above is solved by a
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate net
$(\phi_{\varepsilon})_{\varepsilon}$ which can be written as

$\phi_{\varepsilon}(t,x,\xi)=\sum_{l=1}^{n}\omega_{l,\varepsilon}(t,x)\xi_{l},$

where $\omega_{l,\varepsilon}$ with $l=1,\ldots,n$ are solutions of the Cauchy
problem

$\displaystyle\partial_{t}\omega_{l,\varepsilon}(t,x)$
$\displaystyle=\sum_{j=1}^{n}a_{j,\varepsilon}(t,x)\partial_{j}\omega_{l,\varepsilon}(t,x),$
(4) $\displaystyle\omega_{l,\varepsilon}(0,x)$ $\displaystyle=x_{l}.$

We recall that the characteristic equations associated with the Cauchy problem
(4) are given by

$\displaystyle\frac{d}{ds}\gamma_{l,\varepsilon}(x,t;s)$
$\displaystyle=-a_{l,\varepsilon}(s,\gamma_{1,\varepsilon}(x,t;s),\ldots,\gamma_{n,\varepsilon}(x,t;s)),$
(5) $\displaystyle\gamma_{l,\varepsilon}(x,t;t)$
$\displaystyle=x_{l}\quad\text{ for }\quad l=1,\ldots,n,$

The solutions $\gamma_{1,\varepsilon},\ldots,\gamma_{n,\varepsilon}$ are the
components of the characteristic curve

$\gamma_{\varepsilon}=(\gamma_{1,\varepsilon},\ldots,\gamma_{n,\varepsilon})$

associated the differential operator
$\sum_{j=1}^{n}a_{j,\varepsilon}(t,x)D_{j}$. The following proposition has
been proven in [14] in the context of generalised functions of Colombeau type.
For the advantage of the reader we present here a detailed proof at the net
level.

######  Proposition 2.4.

Let the coefficients $(a_{j,\varepsilon})_{\varepsilon}$ be real valued
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate, compactly supported in $x$ and
let $(\partial_{k}a_{j,\varepsilon})_{\varepsilon}$ be logarithmic moderate
$(k,j=1,\ldots,n)$. Then, the solution $\omega_{l,\varepsilon}(t,x)$ of the
Cauchy problem (4) can be written as

$\omega_{l,\varepsilon}(x,t)=\gamma_{l,\varepsilon}(x,t;0)=x_{l}+\int_{0}^{t}a_{l,\varepsilon}(\tau,\gamma_{1,\varepsilon}(x,t;\tau),\ldots,\gamma_{n,\varepsilon}(x,t;\tau))d\tau,$

for $l=1,\ldots,n$.

###### Proof.

From Proposition 2.3 it is easy to show that there exists a real valued
solution $\omega_{l,\varepsilon}$. It remains to prove that
$\omega_{l,\varepsilon}(t,x)=\gamma_{l,\varepsilon}(x,t;0)$. This comes from
the fact that $\omega_{l,\varepsilon}$ is constant along the characteristic
curves $\gamma_{\varepsilon}(x,t;s)$

$\displaystyle\frac{d}{ds}\omega_{l,\varepsilon}(s,\gamma_{1,\varepsilon}(x,t;s),\ldots,\gamma_{n,\varepsilon}(x,t;s))=0.$

Hence,

$\displaystyle\omega_{l,\varepsilon}(t,\gamma_{1,\varepsilon}(x,t;t),\ldots,\gamma_{n,\varepsilon}(x,t;t))=\omega_{l,\varepsilon}(0,\gamma_{1,\varepsilon}(x,t;0),\ldots,\gamma_{n,\varepsilon}(x,t;0))$

for each $t\in\mathbb{R}$. This implies
$\omega_{l,\varepsilon}(t,x)=\gamma_{l,\varepsilon}(x,t;0)$. By solving
directly the characteristic equations and making use of Proposition 2.2 we
know that $\gamma_{l,\varepsilon}(x,t;0)$ is $C^{\infty}$-moderate and it has
the following form

$\gamma_{l,\varepsilon}(x,t;s)=x_{l}-\int_{t}^{s}a_{l,\varepsilon}(\tau,\gamma_{1,\varepsilon}(x,t;\tau),\ldots,\gamma_{n,\varepsilon}(x,t;\tau))d\tau,$

then

$\omega_{l,\varepsilon}(x,t)=\gamma_{l,\varepsilon}(x,t;0)=x_{l}+\int_{0}^{t}a_{l,\varepsilon}(\tau,\gamma_{1,\varepsilon}(x,t;\tau),\ldots,\gamma_{n,\varepsilon}(x,t;\tau))d\tau.$

This completes the proof. ∎

As a straightforward consequence we obtain the following result for the
Eikonal Cauchy problem (3)

######  Corollary 2.5.

If the coefficients $(a_{j,\varepsilon})_{\varepsilon}$ are real valued
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate with compact support in $x$
with $(\partial_{k}a_{j,\varepsilon})_{\varepsilon}$ is logarithmic moderate
$(k,j=1,\ldots,n)$ then the phase function

$\displaystyle\phi_{\varepsilon}(t,x,\xi)$
$\displaystyle=\sum_{l=1}^{n}\omega_{l,\varepsilon}(t,x)\xi_{l}=\sum_{l=1}^{n}\gamma_{l,\varepsilon}(x,t;0)\xi_{l},$
$\displaystyle=\sum_{l=1}^{n}x_{l}\xi_{l}+\xi_{l}\int_{0}^{t}a_{l,\varepsilon}(\tau,\gamma_{1,\varepsilon}(x,t;\tau),\ldots,\gamma_{n,\varepsilon}(x,t;\tau))d\tau,$

solves the Eikonal Cauchy problem (3).

###  2.3. The transport equation in this context

We now pass to solve the transport equation, in other words to solve the
Cauchy problem (2) with initial condition set to $1$. i.e.,

$\displaystyle\begin{cases}D_{t}b_{\varepsilon}(t,x)=\sum_{j=1}^{n}a_{j,\varepsilon}(t,x)D_{j}b_{\varepsilon}(t,x)+a_{0,\varepsilon}(t,x)b_{\varepsilon}(t,x),\\\
b_{\varepsilon}(0,\cdot)=1.\end{cases}$ (6)

The characteristic equations of the Cauchy problem (6) have the form

$\displaystyle\frac{d}{ds}\gamma_{l,\varepsilon}(x,t;s)$
$\displaystyle=-a_{l,\varepsilon}(s,\gamma_{1,\varepsilon}(x,t;s),\ldots,\gamma_{n,\varepsilon}(x,t;s)),$
$\displaystyle\gamma_{l,\varepsilon}(x,t;t)$ $\displaystyle=1.$

and therefore

$\gamma_{l,\varepsilon}(x,t;s)=1-\int_{t}^{s}a_{l,\varepsilon}(\tau,\gamma_{1,\varepsilon}(x,t;\tau),\ldots,\gamma_{n,\varepsilon}(x,t;\tau))d\tau$

for $l=1,\ldots,n$. By solving

$\displaystyle\frac{d}{ds}z(s)$
$\displaystyle=a_{0,\varepsilon}(s,\gamma_{1,\varepsilon}(x,t;s),\ldots,\gamma_{n,\varepsilon}(x,t;s))z(s),$
$\displaystyle z(0)$ $\displaystyle=1,$

where

$\displaystyle
z(s)=b_{\varepsilon}(s,\gamma_{1,\varepsilon}(x,t;s),\ldots,\gamma_{n,\varepsilon}(x,t;s)),$

we get

$b_{\varepsilon}(t,x)=e^{i\int_{0}^{t}a_{0,\varepsilon}(\tau,\gamma_{1,\varepsilon}(x,t;\tau),\ldots,\gamma_{n,\varepsilon}(x,t;\tau))d\tau}.$
(7)

Using Proposition 2.3 and integrating along the characteristics we have the
following result.

######  Proposition 2.6.

Let the coefficients $(a_{j,\varepsilon})_{\varepsilon}$ be
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $j=1,\ldots,n$ with compact
support in $x$. Suppose that $(a_{j,\varepsilon})_{\varepsilon}$ are real
valued and $(\partial_{k}a_{j,\varepsilon})_{\varepsilon}$ as well as
$(a_{0,\varepsilon})_{\varepsilon}$ are logarithmic moderate
$(k,j=1,\ldots,n)$, then the solution $(b_{\varepsilon})_{\varepsilon}$ of the
Cauchy problem (6) is expressed by

$b_{\varepsilon}(t,x)=e^{i\int_{0}^{t}a_{0,\varepsilon}(s,\gamma_{1,\varepsilon}(t,x;s),\ldots,\gamma_{n,\varepsilon}(t,x;s))ds}$
(8)

and is $C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate.

We are now ready to write the solution of the Cauchy problem for first order
hyperbolic equations in terms of a FIO formula. In other words we prove that
the Cauchy problem in Definition 2.2 admits a very weak solution which is the
action of a family of parameterised FIO’s on the initial data. The results of
the following subsection are extracted from [14] where were originally stated
in the context of generalised functions of Colombeau type.

###  2.4. Fourier Integral Operator formula

######  Proposition 2.7.

Let the coefficients $(a_{j,\varepsilon})_{\varepsilon}$ be
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $j=1,\ldots,n$ with compact
support in $x$. Suppose that $(a_{j,\varepsilon})_{\varepsilon}$ are real
valued and $(\partial_{k}a_{j,\varepsilon})_{\varepsilon}$ as well as
$(a_{0,\varepsilon})_{\varepsilon}$ are logarithmic moderate
$(k,j=1,\ldots,n)$. Then the solution $(w_{\varepsilon})_{\varepsilon}$ of the
Cauchy problem (2) is $C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate and can
be written as

$w_{\varepsilon}(t,x)=F_{\phi_{\varepsilon}}(b_{\varepsilon})(w_{0,\varepsilon})(t,x):=\int_{\mathbb{R}^{n}}e^{i\phi_{\varepsilon}(t,x,\xi)}b_{\varepsilon}(t,x)\hat{w}_{0,\varepsilon}(\xi)d\xi,$
(9)

where the phase function $\phi_{\varepsilon}$ is defined by Corollary 2.5 and
symbol $b_{\varepsilon}$ by Proposition 2.6.

The solution of the inhomogeneous Cauchy problem

$\displaystyle D_{t}w_{\varepsilon}(t,x)$
$\displaystyle=\sum_{j=1}^{n}a_{j,\varepsilon}(t,x)D_{j}w_{\varepsilon}(t,x)+a_{0,\varepsilon}(t,x)w_{\varepsilon}(t,x)+f_{\varepsilon}(t,x),$
(10) $\displaystyle w_{\varepsilon}(0,\cdot)$
$\displaystyle=w_{0,\varepsilon}(x),$

where the net $(f_{\varepsilon})_{\varepsilon}$ is
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate with compact support in $x$,
can also be expressed into FIO form. We begin by noting that Fourier integral
operator $F_{\phi_{\varepsilon}}(b_{\varepsilon})$ of Proposition 2.7 is given
by

$\displaystyle
F_{\phi_{\varepsilon}}(b_{\varepsilon})(w_{0,\varepsilon})(t,x)$
$\displaystyle=\int_{\mathbb{R}^{n}}e^{i\phi_{\varepsilon}(t,x,\xi)}b_{\varepsilon}(t,x)\hat{w}_{0,\varepsilon}(\xi)d\xi$
$\displaystyle=b_{\varepsilon}(t,x)\int_{\mathbb{R}^{n}}e^{i\gamma_{\varepsilon}(x,t;0)\cdot\xi}\hat{w}_{0,\varepsilon}(\xi)d\xi$
$\displaystyle=b_{\varepsilon}(t,x)w_{0,\varepsilon}(\gamma_{\varepsilon}(x,t;0)).$

It defines, for each $t\in\mathbb{R}$, a map

$U_{\varepsilon}(t)=F_{\phi_{\varepsilon}}(b_{\varepsilon})(t):w_{0,\varepsilon}\rightarrow
F_{\phi_{\varepsilon}}(b_{\varepsilon})(w_{0,\varepsilon})(t,\cdot),$

such that $U_{\varepsilon}(0)=I$ and

$U_{\varepsilon}(t)^{-1}:v\rightarrow\frac{1}{b_{\varepsilon}(t,\gamma_{\varepsilon}(x,t;0))}v(\gamma_{\varepsilon}(x,t;0)).$

Hence, we obtain the following statement.

######  Theorem 2.8.

Let the coefficients $(a_{j,\varepsilon})_{\varepsilon}$ be
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $j=1,\ldots,n$ with compact
support in $x$. Suppose that $(a_{j,\varepsilon})_{\varepsilon}$ are real
valued and $(\partial_{k}a_{j,\varepsilon})_{\varepsilon}$ as well as
$(a_{0,\varepsilon})_{\varepsilon}$ are logarithmic moderate
$(k,j=1,\ldots,n)$. Then the solution $(w_{\varepsilon})_{\varepsilon}$ of the
Cauchy problem (10) is $C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate and can
be written as

$w_{\varepsilon}(t,x)=F_{\phi}(b_{\varepsilon})(t)\left(w_{0,\varepsilon}+i\int_{0}^{t}\frac{1}{b_{\varepsilon}(\tau,\gamma_{\varepsilon}(\cdot,\tau;0))}f_{\varepsilon}(\tau,\gamma_{\varepsilon}(\cdot,\tau;0))d\tau\right)(x),$
(11)

where the phase function $\phi_{\varepsilon}$ is defined Corollary 2.5 and
symbol $b_{\varepsilon}$ by Proposition 2.6.

In the rest of the paper we will focus on hyperbolic systems with irregular
coefficients. More precisely, we will formulate conditions on the principal
part and the lower order terms matrix (Levi conditions) which will guarantee
the existence of a very weak solution. Our analysis will also investigate
different levels of singularity of the system coefficients and how they relate
to the corresponding very weak solution.

##  3\. Hyperbolic systems with irregular coefficients

We want to study the Cauchy problem for $2\times 2$ systems of the type

$D_{t}u=A(t,x,D_{x})u+L(t,x,D_{x})u+f(t,x),\qquad(t,x)\in[0,T]\times\mathbb{R}^{n},$

with initial data $u(0,x)=g(x)$, where,

$A(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j}(t,x)D_{x_{j}}&a_{12}(t,x,D_{x})\\\
0&\sum_{j=1}^{n}\lambda_{2j}(t,x)D_{x_{j}}\end{bmatrix},$

and

$L(t,x,D_{x})=\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
\ell_{21}(t,x)\langle D_{x}\rangle^{-1}&\ell_{22}(t,x)\end{bmatrix},$

are matrices of pseudo-differential operators of order 1 and 0, respectively,
and $\lambda_{ij}\in\mathbb{R}$ for $i=1,2$ and $j=1,\ldots,n$. Differently
from [25] we drop here the assumption of smoothness with respect to $x$, so
the entries of the matrices above are assumed to be low regular in $x$, namely
discontinuous or in general distributional. Passing to regularisation via
convolution with a mollifier we are lead to study the regularised Cauchy
problem

$\begin{cases}D_{t}u_{\varepsilon}=A_{\varepsilon}(t,x,D_{x})u_{\varepsilon}+L_{\varepsilon}(t,x,D_{x})u_{\varepsilon}+f_{\varepsilon}(t,x),&\quad(t,x)\in[0,T]\times\mathbb{R}^{n},\\\
u_{\varepsilon}(0,x)=g_{\varepsilon}(x),&\quad x\in\mathbb{R}^{n},\end{cases}$
(12)

where $u_{\varepsilon}(0,x)=[g_{1,\varepsilon}(x),g_{2,\varepsilon}(x)]^{T}$
and
$f_{\varepsilon}(t,x)=[f_{1,\varepsilon}(t,x),f_{2,\varepsilon}(t,x)]^{T}$,
with

$A_{\varepsilon}(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j,\varepsilon}(t,x)D_{x_{j}}&a_{12,\varepsilon}(t,x,D_{x})\\\
0&\sum_{j=1}^{n}\lambda_{2j,\varepsilon}(t,x)D_{x_{j}}\end{bmatrix},$

and

$L_{\varepsilon}(t,x,D_{x})=\begin{bmatrix}\ell_{11,\varepsilon}(t,x)&\ell_{12,\varepsilon}(t,x)\\\
\ell_{21,\varepsilon}(t,x)\langle
D_{x}\rangle^{-1}&\ell_{22,\varepsilon}(t,x)\end{bmatrix},$

where
$(\lambda_{1j,\varepsilon})_{\varepsilon},(\lambda_{2j,\varepsilon})_{\varepsilon}\in
C([0,T],C^{\infty}(\mathbb{R}^{n}))^{(0,1]}$ for $j=1,\ldots,n$, are real-
valued and $(a_{12,\varepsilon})_{\varepsilon}$ and
$(\ell_{ik,\varepsilon})_{\varepsilon}$ are nets of symbols of order $1$ and
$0$ respectively, i.e., $(a_{12,\varepsilon})_{\varepsilon}\in
C([0,T],S^{1}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n}))^{(0,1]}$ and
$(\ell_{ik,\varepsilon})_{\varepsilon}\in
C([0,T],S^{0}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n}))^{(0,1]}$ for
$i,k=1,2$.

We will prove that the original Cauchy problem admits a very weak solution by
carefully analysing the Sobolev well-posedness of the regularised Cauchy
problem (12). This will require some preliminary work inspired by [25] and the
notions of Sobolev moderateness and negligibility.

###  3.1. Preliminaries: Fourier integral operators and Sobolev norms

For each $\lambda_{ij,\varepsilon}$, $i=1,2$, $j=1,\ldots,n$ we will be
denoting by $G_{i,\varepsilon}^{0}w_{0,\varepsilon}$ the solution to

$\displaystyle\begin{cases}D_{t}w_{\varepsilon}=\sum_{j=1}^{n}\lambda_{ij,\varepsilon}(t,x)D_{x_{j}}w_{\varepsilon}+\ell_{ii,\varepsilon}(t,x,D_{x})w_{\varepsilon},\\\
w_{\varepsilon}(0,x)=w_{0,\varepsilon}(x),\end{cases}$ (13)

and by $G_{i,\varepsilon}f_{\varepsilon}$ the solution to

$\displaystyle\begin{cases}D_{t}w_{\varepsilon}=\sum_{j=1}^{n}\lambda_{ij,\varepsilon}(t,x)D_{x_{j}}w_{\varepsilon}+\ell_{ii,\varepsilon}(t,x,D_{x})w_{\varepsilon}+f_{\varepsilon}(t,x),\\\
w_{\varepsilon}(0,x)=0.\end{cases}$

These are Fourier integral operators where the phase function
$\phi_{i,\varepsilon}$ is defined by Corollary 2.5 and symbol

$b_{i,\varepsilon}(t,x)=e^{i\int_{0}^{t}\ell_{ii,\varepsilon}(s,\gamma_{1,\varepsilon}(t,x;s),\ldots,\gamma_{n,\varepsilon}(t,x;s))ds},$

by Proposition 2.6 for $i=1,2$. Namely, for

$\phi_{i,\varepsilon}(t,x,\xi)=x\xi+\sum_{j=1}^{n}\xi_{j}\int_{0}^{t}\lambda_{ij,\varepsilon}(\tau,\gamma_{\varepsilon}(x,t;\tau))d\tau=\gamma_{i,\varepsilon}(x,t;0)\xi,$

the operators $G^{0}_{i,\varepsilon}$ and $G_{i,\varepsilon}$, $i=1,2$, are
given by

$\begin{split}G^{0}_{i,\varepsilon}w_{0,\varepsilon}(t,x)=F_{\phi_{i,\varepsilon}}(b_{i,\varepsilon})(w_{0,\varepsilon})(t,x)&:=\int_{\mathbb{R}^{n}}e^{{\rm
i}\phi_{i,\varepsilon}(t,x,\xi)}b_{i,\varepsilon}(t,x)\hat{w}_{0,\varepsilon}(\xi)d\xi,\\\
&=b_{i,\varepsilon}(t,x)w_{0,\varepsilon}(\gamma_{i,\varepsilon}(x,t;0))\end{split}$

and

$\displaystyle
G_{i,\varepsilon}f_{\varepsilon}(t,x)=F_{\phi_{i,\varepsilon}}(b_{i,\varepsilon})(t)\left(i\int_{0}^{t}\frac{1}{b_{i,\varepsilon}(s,\gamma_{i,\varepsilon}(\cdot,s;0))}f_{\varepsilon}(s,\gamma_{i,\varepsilon}(\cdot,s;0))ds\right)(x)$
$\displaystyle=b_{i,\varepsilon}(t,x)\left(i\int_{0}^{t}\frac{1}{b_{i,\varepsilon}(s,\gamma_{i,\varepsilon}(\gamma_{i,\varepsilon}(x,t;0),s;0)}f_{\varepsilon}(s,\gamma_{i,\varepsilon}(\gamma_{i,\varepsilon}(x,t;0),s;0)))ds\right)$
$\displaystyle=i\int_{0}^{t}\int_{\mathbb{R}^{2n}}{\rm
e}^{i\gamma_{i,\varepsilon}(\gamma_{i,\varepsilon}(x,t;0),s;0)\xi-
iy\xi}\frac{1}{b_{i,\varepsilon}(s,\gamma_{i,\varepsilon}(\gamma_{i,\varepsilon}(x,t;0),s;0))}b_{i,\varepsilon}(t,x)f_{\varepsilon}(s,y)dyd{}^{-}\xi
ds$ $\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{2n}}{\rm
e}^{i\gamma_{i,\varepsilon}(\gamma_{i,\varepsilon}(x,t;0),s;0)\xi-
iy\xi}A_{i,\varepsilon}(s,t,x)f_{\varepsilon}(s,y)dyd{}^{-}\xi ds.$

Combining Proposition 2.7 with Theorem 2.8 we easily obtain the following
proposition which describes the mapping properties of the operators
$G^{0}_{i,\varepsilon}$ and $G_{i,\varepsilon}$.

######  Proposition 3.1.

Assume that

  * (H1)

The coefficients $(\lambda_{ij,\varepsilon})_{\varepsilon}$ are
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $i=1,2$ and $j=1,\ldots,n$
with compact support in $x$.

  * (H2)

The coefficients $(\lambda_{ij,\varepsilon})_{\varepsilon}$ are real valued
and $(\partial_{k}\lambda_{ij,\varepsilon})_{\varepsilon}$ as well as
$(\ell_{ii,\varepsilon})_{\varepsilon}$ are logarithmic moderate for
$k,j=1,\ldots,n$ and $i=1,2$.

The operators $G^{0}_{i,\varepsilon}$ and $G_{i,\varepsilon}$ map
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate nets with compact support in
$x$ into nets of the same type. The same holds with moderate replaced by
negligible.

We recall that pseudo-differential operators are a special kind of Fourier
integral operators so in the sequel we will also deal with nets of pseudo-
differential operators which will therefore have the same mapping properties
of $G^{0}_{i,\varepsilon}$. More precisely, we will work with the following
nets of symbols.

######  Definition 3.1.

  * (i)

A net $(a_{\varepsilon})_{\varepsilon}\in
C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})^{(0,1]}$ is
$S^{m}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n})$-moderate if for all
$K\Subset\mathbb{R}^{n}$ and for all $\alpha,\beta\in\mathbb{N}_{0}^{n}$ and
there exists $N\in\mathbb{N}_{0}$ and $c>0$ such that

$\langle\xi\rangle^{|\beta|-m}|\partial^{\alpha}_{x}\partial^{\beta}_{\xi}a_{\varepsilon}(x,\xi)|\leq
c\varepsilon^{-N},$

uniformly in $\xi\in\mathbb{R}^{n}$, $x\in K$ and $\varepsilon\in(0,1]$.

  * (ii)

A net $(a_{\varepsilon})_{\varepsilon}\in
C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})^{(0,1]}$ is
$S^{m}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n})$-negligible if for all
$K\Subset\mathbb{R}^{n}$ and for all $\alpha,\beta\in\mathbb{N}_{0}^{n}$ and
there exists $q\in\mathbb{N}_{0}$ and $c>0$ such that

$\langle\xi\rangle^{|\beta|-m}|\partial^{\alpha}_{x}\partial^{\beta}_{\xi}a_{\varepsilon}(x,\xi)|\leq
c\varepsilon^{q},$

uniformly in $\xi\in\mathbb{R}^{n}$, $x\in K$ and $\varepsilon\in(0,1]$.

Finally, we denote by
$C([0,T],S^{m}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n}))^{(0,1]}$ the space
of net of all symbols $(a_{\varepsilon}(t,x,\xi))_{\varepsilon}\in
S^{m}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n})^{(0,1]}$ which are continuous
with respect to $t$ and in analogy with the definition above we can define
moderate and negligible nets in
$C([0,T],S^{m}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n}))^{(0,1]}$ by
employing uniform estimates with respect to $t\in[0,T]$.

We can now go back to our regularised Cauchy problem. As explained in detail
in [25] the solution of the Cauchy problem (12) can be written as

$\displaystyle u_{1,\varepsilon}$ $\displaystyle=$ $\displaystyle
U^{0}_{1,\varepsilon}+G_{1,\varepsilon}((a_{12,\varepsilon}+\ell_{12,\varepsilon})u_{2,\varepsilon}),$
(14) $\displaystyle u_{2,\varepsilon}$ $\displaystyle=$ $\displaystyle
U^{0}_{2,\varepsilon}+G_{2,\varepsilon}(\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}u_{1,\varepsilon}),$ (15)

where

$U_{i,\varepsilon}^{0}=G_{i,\varepsilon}^{0}g_{i,\varepsilon}+G_{i,\varepsilon}(f_{i,\varepsilon}),\quad
i=1,2.$ (16)

We want to prove that a smooth solution exists by proving that it belongs to
every Sobolev space $H^{s}$. This means to make use of the Sobolev mapping
properties of all the operators involved above.

Plugging (15) in (14), we obtain

$u_{1,\varepsilon}=\tilde{U}^{0}_{1,\varepsilon}+G_{1,\varepsilon}(a_{12,\varepsilon}G_{2,\varepsilon}(\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}u_{1,\varepsilon}))+G_{1,\varepsilon}(\ell_{12,\varepsilon}G_{2,\varepsilon}(\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}u_{1,\varepsilon})),$ (17)

where

$\tilde{U}^{0}_{1,\varepsilon}=G_{1,\varepsilon}^{0}g_{1,\varepsilon}+G_{1,\varepsilon}(f_{1,\varepsilon})+G_{1,\varepsilon}((a_{12,\varepsilon}+\ell_{12,\varepsilon})U^{0}_{2,\varepsilon}).$
(18)

By employing Proposition 5.2 in the Appendix we have that

$\begin{split}\|G_{i,\varepsilon}v\|_{H^{s}}&\leq
TC^{\prime}(\varepsilon)\sup_{t,s\in[0,T]}\sup_{|\beta|\leq
2n+1}||\partial_{x}^{\beta}A_{i,\varepsilon}(s,t,x)||_{L^{\infty}(\mathbb{R}^{n}_{x})}\|v\|_{H^{s}},\\\
\|G^{0}_{i,\varepsilon}v\|_{H^{s}}&\leq
C^{\prime}(\varepsilon)\sup_{t\in[0,T]}\sup_{|\beta|\leq
2n+1}||\partial_{x}^{\beta}b_{i,\varepsilon}(t,x)||_{L^{\infty}(\mathbb{R}^{n}_{x})}\|v\|_{H^{s}},\end{split}$
(19)

where $C^{\prime}(\varepsilon)$ depends on the positive nets $C(\varepsilon)$,
$C_{\alpha,i}(\varepsilon)$ and $C_{\beta,i}(\varepsilon)$ appearing below:

$\displaystyle|\partial_{\xi}\phi_{i,\varepsilon}(t,x,\xi)-\partial_{\xi}\phi_{i,\varepsilon}(t,y,\xi)|$
$\displaystyle\geq C(\varepsilon)|x-y|\,\,\,\,\text{ for
}\,\,x,y\in\mathbb{R}^{n},\,\,\xi\in\mathbb{R}^{n},$
$\displaystyle|\partial_{y}\phi_{i,\varepsilon}(t,y,\xi)-\partial_{y}\phi_{i,\varepsilon}(t,y,\eta)|$
$\displaystyle\geq C(\varepsilon)|\xi-\eta|\,\,\,\,\text{ for
}\,\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

and

$\begin{split}|\partial_{y}^{\alpha}\partial_{\xi}\phi_{i,\varepsilon}(t,y,\xi)|&\leq
C_{\alpha,i}(\varepsilon),\\\
|\partial_{y}\partial_{\xi}^{\beta}\phi_{i,\varepsilon}(t,y,\xi)|&\leq
C_{\beta,i}(\varepsilon),\end{split}$

for all $t\in[0,T]$, for $1\leq|\alpha|,|\beta|\leq 2n+2$ and $i=1,2$.

In the sequel, we introduce the concept of $H^{s}$-moderate,
$H^{s}$-negligible nets, and a very weak solution of Sobolev order $s$.

######  Definition 3.2.

  * (i)

A net $(v_{\varepsilon})_{\varepsilon}\in H^{s}(\mathbb{R}^{n})^{(0,1]}$ is
$H^{s}$-moderate if there exists $N\in\mathbb{N}_{0}$ and $c>0$ such that

$||v_{\varepsilon}(x)||_{H^{s}(\mathbb{R}^{n})}\leq c\varepsilon^{-N},$

uniformly in $\varepsilon\in(0,1]$.

  * (ii)

A net $(v_{\varepsilon})_{\varepsilon}\in H^{s}(\mathbb{R}^{n})^{(0,1]}$ is
$H^{s}$-negligible if for all $q\in\mathbb{N}_{0}$ there exists $c>0$ such
that

$||v_{\varepsilon}(x)||_{H^{s}(\mathbb{R}^{n})}\leq c\varepsilon^{q},$

uniformly in $\varepsilon\in(0,1]$.

Analogously, one can replace $H^{s}(\mathbb{R}^{n})$ with
$C([0,T],H^{s}(\mathbb{R}^{n}))$ and state the corresponding notions of
$C([0,T],H^{s}(\mathbb{R}^{n}))$-moderate and
$C([0,T],H^{s}(\mathbb{R}^{n}))$-negligible net. This will appear in the
following notion of _very weak solution of Sobolev order s_.

######  Definition 3.3.

We say that the Cauchy problem

$\begin{cases}D_{t}u=A(t,x,D_{x})u+L(t,x,D_{x})u+f(t,x),\\\
u(0,x)=g_{0}(x)\end{cases}$

admits a _very weak solution_ $(u_{\varepsilon})_{\varepsilon}$ of _Sobolev
order s_ if the net $(u_{\varepsilon})_{\varepsilon}$ solves the regularised
problem (12) and it is $C([0,T],H^{s}(\mathbb{R}^{n}))$-moderate.

We are now ready to investigate the Cauchy problem (12) more in detail.

###  3.2. Hyperbolic systems with non-diagonalisable principal part

Consider the regularised hyperbolic system of type

$\begin{cases}D_{t}u_{\varepsilon}=A_{\varepsilon}(t,x,D_{x})u_{\varepsilon}+L_{\varepsilon}(t,x,D_{x})u_{\varepsilon}+f_{\varepsilon}(t,x),&\quad(t,x)\in[0,T]\times\mathbb{R}^{n},\\\
u_{\varepsilon}(0,x)=g_{\varepsilon}(x),&\quad x\in\mathbb{R}^{n},\end{cases}$

where $u_{\varepsilon}(0,x)=[g_{1,\varepsilon}(x),g_{2,\varepsilon}(x)]^{T}$
and
$f_{\varepsilon}(t,x)=[f_{1,\varepsilon}(t,x),f_{2,\varepsilon}(t,x)]^{T}$,
with the $A_{\varepsilon}(t,x,D_{x})$ and $L_{\varepsilon}(t,x,D_{x})$ given
by

$A_{\varepsilon}(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j,\varepsilon}(t,x)D_{x_{j}}&a_{12,\varepsilon}(t,x,D_{x})\\\
0&\sum_{j=1}^{n}\lambda_{2j,\varepsilon}(t,x)D_{x_{j}}\end{bmatrix},$

and

$L_{\varepsilon}(t,x,D_{x})=\begin{bmatrix}\ell_{11,\varepsilon}(t,x)&\ell_{12,\varepsilon}(t,x)\\\
\ell_{21,\varepsilon}(t,x)\langle
D_{x}\rangle^{-1}&\ell_{22,\varepsilon}(t,x)\end{bmatrix},$

where
$(\lambda_{1j,\varepsilon})_{\varepsilon},(\lambda_{2j,\varepsilon})_{\varepsilon}\in
C([0,T],C^{\infty}(\mathbb{R}^{n}))^{(0,1]}$ for $j=1,\ldots,n$ and
$(a_{12,\varepsilon})_{\varepsilon}$ and
$(\ell_{ik,\varepsilon})_{\varepsilon}$ are nets of symbols of order $1$ and
$0$ respectively, i.e., $(a_{12,\varepsilon})_{\varepsilon}\in
C([0,T],S^{1}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n}))^{(0,1]}$ and
$(\ell_{ik,\varepsilon})_{\varepsilon}\in
C([0,T],S^{0}_{0,1}(\mathbb{R}^{n}\times\mathbb{R}^{n}))^{(0,1]}$ for
$i,k=1,2$.

The net of solutions $(u_{\varepsilon})_{\varepsilon}$ of this hyperbolic
system can be formulated in terms of operators $G_{i,\varepsilon}^{0}$ and
$G_{i,\varepsilon}$ introduced in Subsection 3.1 if the following hypotheses
hold:

  * (H1)

The coefficients $(\lambda_{1j,\varepsilon})_{\varepsilon}$ and
$(\lambda_{2j,\varepsilon})_{\varepsilon}$ are real valued and
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $j=1,\ldots,n$ with compact
support in $x$.

  * (H2)

Also, $(\partial_{k}\lambda_{1j,\varepsilon})_{\varepsilon}$,
$(\partial_{k}\lambda_{2j,\varepsilon})_{\varepsilon}$ and
$(\ell_{ii,\varepsilon})_{\varepsilon}$ are logarithmic moderate for
$k,j=1,\ldots,n$ and $i=1,2$.

Note that these assumptions allow to write $(u_{1,\varepsilon})_{\varepsilon}$
and $(u_{2,\varepsilon})_{\varepsilon}$ as in (14) and (15). More details can
be found in Theorem 2.8.

In order to prove the existence of a very weak solution we start from the
component $u_{1,\varepsilon}$ and the equation

$u_{1,\varepsilon}=\widetilde{U}^{0}_{1,\varepsilon}+G_{1,\varepsilon}(a_{12,\varepsilon}G_{2,\varepsilon}(\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}u_{1,\varepsilon}))+G_{1,\varepsilon}(\ell_{12,\varepsilon}G_{2,\varepsilon}(\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}u_{1,\varepsilon})),$

where

$\widetilde{U}^{0}_{1,\varepsilon}=G_{1,\varepsilon}^{0}g_{1,\varepsilon}+G_{1,\varepsilon}(f_{1,\varepsilon})+G_{1,\varepsilon}((a_{12,\varepsilon}+\ell_{12,\varepsilon})U^{0}_{2,\varepsilon}).$

We make use of the Banach fixed point theorem assuming the additional
hypothesis:

  * (H3)

The operators $G^{0}_{i,\varepsilon}$ and $G_{i,\varepsilon}$ map
$H^{s}$-moderate and $C([0,T],H^{s}(\mathbb{R}^{n}))$-moderate nets into
themselves, respectively and

$\mathcal{G}^{0}_{1,\varepsilon}:=G_{1,\varepsilon}\circ(a_{12,\varepsilon}+\ell_{12,\varepsilon})\circ
G_{2,\varepsilon}\circ\ell_{21,\varepsilon}\langle D_{x}\rangle^{-1}$

has the operator norm in $H^{s}$ strictly less than 1

$||\mathcal{G}^{0}_{1,\varepsilon}||_{H^{s}\rightarrow H^{s}}=O(1).$

Note that $\mathcal{G}^{0}_{1,\varepsilon}$ acts continuously on $H^{s}$ since
it is of order $0$.

Let us now work on better understanding the hypothesis (H3). We have that (H3)
holds if

$\displaystyle||G_{1,\varepsilon}\circ a_{12,\varepsilon}\circ
G_{2,\varepsilon}\circ\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}||_{H^{s}\rightarrow H^{s}}\leq
TC_{1}(\varepsilon)A_{1}(\varepsilon)a_{12}(\varepsilon)C_{2}(\varepsilon)A_{2}(\varepsilon)L_{21}(\varepsilon)=O(1),$

and

$\displaystyle||G_{1,\varepsilon}\circ\ell_{12,\varepsilon}\circ
G_{2,\varepsilon}\circ\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}||_{H^{s}\rightarrow H^{s}}\leq
TC_{1}(\varepsilon)A_{1}(\varepsilon)L_{12}(\varepsilon)C_{2}(\varepsilon)A_{2}(\varepsilon)L_{21}(\varepsilon)=O(1),$

where

$\displaystyle A_{i}(\varepsilon)$
$\displaystyle:=\sup_{t,s\in[0,T]}\sup_{|\beta|\leq
2n+1}||\partial_{x}^{\beta}A_{i,\varepsilon}(s,t,x)||_{L^{\infty}(\mathbb{R}^{n}_{x})}\,\,\text{
for }\,\,i=1,2,$ $\displaystyle a_{12}(\varepsilon)$
$\displaystyle:=\sup_{t\in[0,T]}\sup_{|\alpha|,|\beta|\leq
2n+1}||\partial_{\xi}^{\alpha}\partial_{x}^{\beta}a_{12,\varepsilon}(t,x,\xi)||_{L^{\infty}(\mathbb{R}^{n}_{x}\times\mathbb{R}^{n}_{\xi})},$
$\displaystyle L_{12}(\varepsilon)$
$\displaystyle:=\sup_{t\in[0,T]}\sup_{|\beta|\leq
2n+1}||\partial_{x}^{\beta}\ell_{12,\varepsilon}(t,x)||_{L^{\infty}(\mathbb{R}^{n}_{x})},$
$\displaystyle L_{21}(\varepsilon)$
$\displaystyle:=\sup_{t\in[0,T]}\sup_{|\beta|\leq
2n+1}||\partial_{x}^{\beta}\ell_{21,\varepsilon}(t,x)||_{L^{\infty}(\mathbb{R}^{n}_{x})},$

and $C_{1}(\varepsilon)$ and $C_{2}(\varepsilon)$ include the combination of
constants $C(\varepsilon)$, $C_{\alpha,i}(\varepsilon)$ and
$C_{\beta,i}(\varepsilon)$ from

$\displaystyle|\partial_{\xi}\phi_{i,\varepsilon}(t,x,\xi)-\partial_{\xi}\phi_{i,\varepsilon}(t,y,\xi)|$
$\displaystyle\geq C(\varepsilon)|x-y|\,\,\,\,\text{ for
}\,\,x,y\in\mathbb{R}^{n},\,\,\xi\in\mathbb{R}^{n},$
$\displaystyle|\partial_{y}\phi_{i,\varepsilon}(t,y,\xi)-\partial_{y}\phi_{i,\varepsilon}(t,y,\eta)|$
$\displaystyle\geq C(\varepsilon)|\xi-\eta|\,\,\,\,\text{ for
}\,\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

and that

$|\partial_{y}^{\alpha}\partial_{\xi}\phi_{i,\varepsilon}(t,y,\xi)|\leq
C_{\alpha,i}(\varepsilon),\,\,|\partial_{y}\partial_{\xi}^{\beta}\phi_{i,\varepsilon}(t,y,\xi)|\leq
C_{\beta,i}(\varepsilon),$

for $1\leq|\alpha|,|\beta|\leq 2n+2$ and $i=1,2$.

We refer the reader to the appendix at the end of the paper for the Sobolev
mapping properties of Fourier integral operators that we have employed above.
Note that the nets of constants $C_{\alpha,i}(\varepsilon)$,
$C_{\beta,i}(\varepsilon)$ are automatically moderate while $C(\varepsilon)$
and $C^{-1}(\varepsilon)$ are assumed to be moderate to make sure that our
operators have the right Sobolev mapping properties. The stronger hypothesis
(H3) is required to allow a fixed point argument independent of the parameter
$\varepsilon$. We therefore conclude that (H3) can be written as

  * (H3)

the nets of constants $(C(\varepsilon))_{\varepsilon}$ and
$(C^{-1}(\varepsilon))_{\varepsilon}$ are moderate and

$||\mathcal{G}^{0}_{1,\varepsilon}||_{H^{s}\rightarrow H^{s}}=O(1).$

If (H3) holds, then we will apply Banach’s fixed point theorem in the space
$X(t):=C([0,t],H^{s}(\mathbb{R}^{n}))$ for $t\in[0,T]$ with the norm

$||u_{1,\varepsilon}||_{X(t)}=\sup_{0\leq\tau\leq
t}||u_{1,\varepsilon}(\tau,\cdot)||_{H^{s}}.$

Note that we can rewrite the equation for $u_{1,\varepsilon}$ as

$u_{1,\varepsilon}={\tilde{U}}_{1,\varepsilon}^{0}+\mathcal{G}^{0}_{1,\varepsilon}u_{1,\varepsilon}.$

Using composition of Fourier integral operators and hypothesis (H3) we have
that the 0-order Fourier integral operator $\mathcal{G}^{0}_{1,\varepsilon}$
maps $X(t)$ continuously into itself and for small time interval it is a
contraction, in the sense that there exists $T^{*}\in[0,T]$ such that

$||\mathcal{G}^{0}_{1,\varepsilon}(u-v)||_{X(T^{*})}\leq
CT^{*}||u-v||_{X(T^{*})},$ (20)

with $CT^{*}<1$. The existence of a unique fixed point $u_{1,\varepsilon}$ for
the map $\mathcal{G}^{0}_{1,\varepsilon}$ is provided by Banach’s fixed point
theorem and is equivalent to say that the operator
$I-\mathcal{G}^{0}_{1,\varepsilon}$ is invertible on $[0,T^{*}]$ for all
values of $\varepsilon$. It therefore follows that
$(u_{1,\varepsilon})_{\varepsilon}$ inherits the moderateness properties of
$({\tilde{U}}_{1,\varepsilon}^{0})_{\varepsilon}$. Note that as already
observed in [25] the constant $C$ does not depend on the initial data so the
argument can be iterated to cover the full interval $[0,T]$.

We have therefore proven the following general result.

######  Theorem 3.2.

Let us consider the Cauchy problem

$\begin{cases}D_{t}u=A(t,x,D_{x})u+L(t,x,D_{x})u+f(t,x),&\quad(t,x)\in[0,T]\times\mathbb{R}^{n},\\\
u(0,x)=g(x),&\quad x\in\mathbb{R}^{n},\end{cases}$ (21)

where

$A(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j}(t,x)D_{x_{j}}&a_{12}(t,x,D_{x})\\\
0&\sum_{j=1}^{n}\lambda_{2j}(t,x)D_{x_{j}}\end{bmatrix}$

is an upper triangular matrix of first order differential operator and

$L(t,x,D_{x})=\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
\ell_{21}(t,x)\langle D_{x}\rangle^{-1}&\ell_{22}(t,x)\end{bmatrix},$

is a matrix of pseudo-differential operator of order 0 with $\ell_{12}$ of
order $-1$, continuous with respect to $t$ and such that

  * (i)

all coefficients of $A(t,x,D_{x})$ and $L(t,x,D_{x})$ are in
$C([0,T],\mathcal{E}^{\prime}(\mathbb{R}^{n}))$ with compact support in $x$,

  * (ii)

the initial data $g_{i}(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ and the
source term $f_{i}(t,x)\in C([0,T],\mathcal{E}^{\prime}(\mathbb{R}^{n}))$ with
compact support in $x$ for $i=1,2$.

Assume that (H1), (H2), (H3) are satisfied. Let the nets of regularised
initial data $(g_{i,\varepsilon})_{\varepsilon}$ and the right-hand side
$(f_{i,\varepsilon})_{\varepsilon}$ be $H^{s+i-1}$-moderate and
$C([0,T],H^{s+i-1})$-moderate for $i=1,2$, respectively. Then the net of
solutions $(u_{\varepsilon})_{\varepsilon}$ is a very weak solution of
anisotropic Sobolev type, i.e. $u_{i,\varepsilon}$ is
$C([0,T],H^{s+i-1})$-moderate for $i=1,2$.

######  Remark.

Note that when regularising a distribution with compact support with a
mollifier which is also compactly supported we automatically get a moderate
net of smooth functions with compact support and therefore a moderate net of
any Sobolev order.

The existence of a very weak solution in the sense of Definition 2.2 follows
from the following corollary.

######  Corollary 3.3.

If (H1) and (H2), (H3) hold then the Cauchy problem (21) has a very weak
solution $(u_{\varepsilon})_{\varepsilon}$, i.e., a net
$(u_{\varepsilon})_{\varepsilon}$ which solves the regularised Cauchy problem
and it is $C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate.

######  Remark.

Negligible perturbations of the system coefficients and the initial data leads
to a negligible perturbation in the solution. This follows from the fact that
our nets of operators maps negligible nets into negligible nets. For this
reason we say that our Cauchy problem is well-posed in a very weak sense.

###  3.3. Special cases of the hyperbolic system in (21)

In this subsection, we analyse some special cases of Theorem 3.2. First, we
assume that the matrix $L(t,x,D_{x})$ of the lower order terms is upper
triangular and that $A(t,x,D_{x})$ is either diagonal or in upper triangular
form. In both these cases the hypotheses (H1) and (H2) are sufficient to prove
Theorem 3.2.

######  Corollary 3.4.

Let us consider the Cauchy problem (21) with a diagonal matrix of first order
differential operators $A(t,x,D_{x})$ and an upper triangular matrix of
0-order pseudo-differential operator $L(t,x,D_{x})$ as

$A(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j}(t,x)D_{x_{j}}&0\\\
0&\sum_{j=1}^{n}\lambda_{2j}(t,x)D_{x_{j}}\end{bmatrix},$

and

$L(t,x,D_{x})=\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
0&\ell_{22}(t,x)\end{bmatrix},$

where

  * (i)

all coefficients of $A(t,x,D_{x})$ and $L(t,x,D_{x})$ are in
$C([0,T],\mathcal{E}^{\prime}(\mathbb{R}^{n}))$ with compact support in $x$,

  * (ii)

the initial data $g_{i}(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ and the
source term $f_{i}(t,x)\in C([0,T],\mathcal{E}^{\prime}(\mathbb{R}^{n}))$ with
compact support in $x$ for $i=1,2$.

Assume that (H1) and (H2) are satisfied. Then the net of solutions
$(u_{\varepsilon})_{\varepsilon}$ of the regularised problem is a very weak
solution, i.e. $(u_{i,\varepsilon})_{\varepsilon}$ is
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $i=1,2$ and can be written
as

$\displaystyle u_{1,\varepsilon}$
$\displaystyle=G_{1,\varepsilon}^{0}g_{1,\varepsilon}+G_{1,\varepsilon}f_{1,\varepsilon},$
$\displaystyle u_{2,\varepsilon}$
$\displaystyle=G_{2,\varepsilon}^{0}g_{2,\varepsilon}+G_{2,\varepsilon}f_{2,\varepsilon}.$

######  Corollary 3.5.

Let the matrices $A(t,x,D_{x})$ and $L(t,x,D_{x})$ in the Cauchy problem (21)
be both upper triangular. Let

  * (i)

all coefficients of $A(t,x,D_{x})$ and $L(t,x,D_{x})$ are in
$C([0,T],\mathcal{E}^{\prime}(\mathbb{R}^{n}))$ with compact support in $x$,

  * (ii)

the initial data $g_{i}(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ and the
source term $f_{i}(t,x)\in C([0,T],\mathcal{E}^{\prime}(\mathbb{R}^{n}))$ with
compact support in $x$ for $i=1,2$.

Assume that (H1) and (H2) are satisfied. Assume that there exist moderate nets
of constants $(C(\varepsilon))_{\varepsilon}$ and
$(C^{-1}(\varepsilon))_{\varepsilon}$ such that

$\displaystyle|\partial_{\xi}\phi_{i,\varepsilon}(t,x,\xi)-\partial_{\xi}\phi_{i,\varepsilon}(t,y,\xi)|$
$\displaystyle\geq C(\varepsilon)|x-y|\,\,\,\,\text{ for
}\,\,x,y\in\mathbb{R}^{n},\,\,\xi\in\mathbb{R}^{n},$ (22)
$\displaystyle|\partial_{y}\phi_{i,\varepsilon}(t,y,\xi)-\partial_{y}\phi_{i,\varepsilon}(t,y,\eta)|$
$\displaystyle\geq C(\varepsilon)|\xi-\eta|\,\,\,\,\text{ for
}\,\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

for all $i=1,2$. Then the net of solutions $(u_{\varepsilon})_{\varepsilon}$
of the regularised problem is a very weak solution of anisotropic Sobolev type
for all $s\in\mathbb{R}$, i.e. $(u_{i,\varepsilon})_{\varepsilon}$ is
$C([0,T],H^{s+i-1})$-moderate for $i=1,2$ and can be written as

$\displaystyle u_{1,\varepsilon}$
$\displaystyle=G_{1,\varepsilon}^{0}g_{1,\varepsilon}+G_{1,\varepsilon}f_{1,\varepsilon}+G_{1,\varepsilon}(a_{12,\varepsilon}U^{0}_{2,\varepsilon}),$
$\displaystyle u_{2,\varepsilon}$
$\displaystyle=U^{0}_{2,\varepsilon}=G_{2,\varepsilon}^{0}g_{2,\varepsilon}+G_{2,\varepsilon}f_{2,\varepsilon},$

where $U^{0}_{2,\varepsilon}$ is defined in (16).

######  Remark.

It is important to note that Corollary 3.4 is independent of the hypothesis
(H3) and that Corollary 3.5 only requires the hypothesis on the phase function
which guarantees the right Sobolev mapping properties for our Fourier integral
operators. This is due to the fact that the Banach fixed point theorem is not
required for the proof of Corollaries 3.4 and 3.5. It is also clear that
Corollary 3.4 and 3.5 hold for $m\times m$-systems as well. It is indeed
sufficient to formulate the hypotheses (H1) and (H2) with $i=1,\ldots,m$ and
assume that (22) holds for $i=1,\ldots,m$.

We now focus on $2\times 2$-systems where the hypothesis (H3) is needed to
prove the existence of a very weak solution. For the sake of the reader we
reformulate Proposition 5.2 for the operators involved in (H3). This will
apply to a scenario where the matrices $A(t,x,D_{x})$ and $L(t,x,D_{x})$ are
not as in the previous two corollaries. Let
$\phi_{i,\varepsilon}(t,x,\xi)=\gamma_{i,\varepsilon}(x,t;0)\xi$. We have that
if there exists constants $C,C_{\alpha},C_{\beta}$ independent of
$\varepsilon\in(0,1]$ such that

$\displaystyle|\gamma_{i,\varepsilon}(t,x;0)-\gamma_{i,\varepsilon}(t,y;0)|$
$\displaystyle\geq C|x-y|\,\,\,\,\text{ for
}\,\,t\in[0,T],\,x,y\in\mathbb{R}^{n},$
$\displaystyle|\partial_{y}\gamma_{i,\varepsilon}(t,y;0)\cdot(\xi-\eta)|$
$\displaystyle\geq C|\xi-\eta|\,\,\,\,\text{ for
}\,\,t\in[0,T],\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

and

$|\partial_{y}^{\alpha}\gamma_{i,\varepsilon}(t,y;0)|\leq
C_{\alpha},\,\,|\partial_{y}\partial_{\xi}^{\beta}(\gamma_{i,\varepsilon}(t,y;0)\cdot\xi)|\leq
C_{\beta},\,\,\,\text{for}\,t\in[0,T],\,y\in\mathbb{R}^{n},$

for $1\leq|\alpha|,|\beta|\leq 2n+2$ and $i=1,2$, and the lower order terms
$\ell_{12,\varepsilon}$ and $\ell_{21,\varepsilon}$ have derivatives up to
order $2n+1$ which are bounded with respect to the parameter $\varepsilon$ as
well, then the hypothesis (H3) is automatically fulfilled. This corresponds to
a case of higher regularity of the coefficients which is treated in the next
corollary.

######  Corollary 3.6.

Let us consider the Cauchy problem (21) with given a diagonal matrix of first
order differential operators $A(t,x,D_{x})$ and a matrix of 0-order pseudo-
differential operator $L(t,x,D_{x})$ as

$A(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j}(t,x)D_{x_{j}}&0\\\
0&\sum_{j=1}^{n}\lambda_{2j}(t,x)D_{x_{j}}\end{bmatrix},$

and

$L(t,x,D_{x})=\begin{bmatrix}0&\ell_{12}(t,x)\\\
\ell_{21}(t,x)&0\end{bmatrix},$

where

  * (i)

the diagonal elements $\lambda_{1j}$, $\lambda_{2j}\in
C([0,T],C^{k+1}(\mathbb{R}^{n}))$ for $j=1,\ldots,n$ and $k\geq 2n+1$ with
compact support in $x$;

  * (ii)

the initial data $g(x)\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ and the source
term $f_{i}(t,x)\in C([0,T],\mathcal{E}^{\prime}(\mathbb{R}^{n}))$ with
compact support in $x$, for $i=1,2$;

  * (iii)

the non-diagonal elements $\ell_{12}(t,x),\ell_{21}(t,x)\in
C([0,T],C^{k}(\mathbb{R}^{n}))$ with compact support with respect to $x$.

Then the net of solutions $(u_{\varepsilon})_{\varepsilon}$ of the regularised
problem is a very weak solution of Sobolev type for all $s\in\mathbb{R}$, i.e.
$(u_{i,\varepsilon})_{\varepsilon}$ is $C([0,T],H^{s})$-moderate for $i=1,2$.

###### Proof of Corollary 3.6.

The enhanced regularity of the coefficients allows (H1), (H2), and (H3) to be
satisfied automatically with constants that do not depend on $\varepsilon$.
Under assumptions (H1) and (H2), we can express the
$(u_{\varepsilon})_{\varepsilon}$ components as:

$\displaystyle u_{1,\varepsilon}$
$\displaystyle=G^{0}_{1,\varepsilon}g_{1,\varepsilon}+G_{1,\varepsilon}f_{1,\varepsilon}+G_{1,\varepsilon}(\ell_{12,\varepsilon}u_{2,\varepsilon}),$
$\displaystyle u_{2,\varepsilon}$
$\displaystyle=G^{0}_{2,\varepsilon}g_{2,\varepsilon}+G_{2,\varepsilon}f_{2,\varepsilon}+G_{2,\varepsilon}(\ell_{21,\varepsilon}u_{1,\varepsilon}).$

By substituting $u_{2,\varepsilon}$ into the equation of $u_{1,\varepsilon}$,
we get

$u_{1,\varepsilon}=\widetilde{U}_{1,\varepsilon}^{0}+\mathcal{G}^{0}_{1,\varepsilon}u_{1,\varepsilon},$

where

$\displaystyle\widetilde{U}_{1,\varepsilon}^{0}$
$\displaystyle:=G^{0}_{1,\varepsilon}g_{1,\varepsilon}+G_{1,\varepsilon}f_{1,\varepsilon}+G_{1,\varepsilon}(\ell_{12}\circ
G^{0}_{2,\varepsilon}g_{2,\varepsilon})+G_{1,\varepsilon}(\ell_{12}\circ
G_{2,\varepsilon}f_{2,\varepsilon}),$
$\displaystyle\mathcal{G}^{0}_{1,\varepsilon}$
$\displaystyle:=G_{1,\varepsilon}\circ\ell_{12,\varepsilon}\circ
G_{2,\varepsilon}\circ\ell_{21,\varepsilon}.$

Note that $\mathcal{G}^{0}_{1,\varepsilon}$ is a zero-order operator. In other
words it is not necessary in this specific case to assume that the entry below
the diagonal in the matrix $L(t,x,D_{x})$ is of order $-1$. Thus, one can
prove that $\mathcal{G}^{0}_{1,\varepsilon}$ has the operator norm in $H^{s}$
strictly less than 1 due to (H3). Moreover,
$\widetilde{U}^{0}_{1,\varepsilon}$ is $H^{s}$-moderate. By applying the
Banach fixed-point theorem, we prove the existence of very weak solution
$(u_{1,\varepsilon})_{\varepsilon}$, i.e., $(u_{1,\varepsilon})_{\varepsilon}$
is $C([0,T],H^{s})$-moderate. Finally, we obtain $u_{2,\varepsilon}$ by
substituting $u_{1,\varepsilon}$ into the equation for $u_{2,\varepsilon}$. ∎

##  4\. Consistency and Applications

In this section, we prove that our result is consistent with the classical
well-posedness result obtained in [25] when the system is regular enough. In
detail, we show that every very weak solution converges to the classical
solution when it exists, namely when the system has smooth coefficients with
respect to $x$. In addition, we discuss some examples of physical relevance.

######  Theorem 4.1.

Consider the Cauchy problem (21), where $A(t,x,D_{x})$ is an upper triangular
matrix of first-order pseudo-differential operators and $L(t,x,D_{x})$ is a
matrix of zero-order pseudo-differential operators, continuous with respect to
$t$, of the form

$A(t,x,D_{x})=\begin{bmatrix}\sum_{j=1}^{n}\lambda_{1j}(t,x)D_{x_{j}}&a_{12}(t,x,D_{x})\\\
0&\sum_{j=1}^{n}\lambda_{2j}(t,x)D_{x_{j}}\end{bmatrix}$

and

$L(t,x,D_{x})=\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
\ell_{21}(t,x)\langle D_{x}\rangle^{-1}&\ell_{22}(t,x)\end{bmatrix}.$

Assume that all coefficients of $A(t,x,D_{x})$ and $L(t,x,D_{x})$ are
continuous in $t$, smooth and compactly supported with respect to $x$, the
initial data $g_{i}(x)\in C^{\infty}_{c}(\mathbb{R}^{n})$ and $f_{i}(t,x)\in
C([0,T],C^{\infty}_{c}(\mathbb{R}^{n}))$ for $i=1,2$ and $s\in\mathbb{R}$.
Hence,

  * (i)

the Cauchy problem (21) has a unique solution $u$ with components $u_{i}\in
C([0,T],C^{\infty}_{c}(\mathbb{R}^{n}))$ for $i=1,2$;

  * (ii)

the Cauchy problem (21) has a very weak solution
$(u_{\varepsilon})_{\varepsilon}$, i.e., the components
$(u_{i,\varepsilon})_{\varepsilon}$ are
$C([0,T],C^{\infty}(\mathbb{R}^{n}))$-moderate for $i=1,2$.

  * (iii)

the net $u_{\varepsilon}(t,\cdot)$ converges to $u(t,\cdot)$ as
$\varepsilon\to 0$ in $L^{2}(\mathbb{R}^{n}_{x})$ uniformly with respect to
$t\in[0,T]$.

###### Proof of Theorem 4.1.

(i) The well-posedness of the Cauchy problem can be derived from the results
presented by the first author, Jäh, and Ruzhansky in [25].

(ii) The existence of a very weak solution of anisotropic Sobolev type is
obtained from Theorem 3.2. Note that since the coefficients are smooth the
hypotheses (H1)-(H3) are fulfilled with constants independent of
$\varepsilon$.

(iii) Let $(u_{\varepsilon})_{\varepsilon}$ and $u$ be a very weak solution
and the classic solution of our Cauchy problem, respectively. Our argument is
independent of the choice of the mollifier and the scale we are using in our
regularisation, so we namely prove that every very weak solution produced in
this way is convergent to the unique classical solution.

We now compare the classical Cauchy problem with solution $u$, with the
regularised one with solution $(u_{\varepsilon})_{\varepsilon}$. It follows
that

$\begin{cases}D_{t}(u_{\varepsilon}-u)=A_{\varepsilon}(u_{\varepsilon}-u)+L_{\varepsilon}(u_{\varepsilon}-u)+(f_{\varepsilon}-f)+R_{\varepsilon},&\quad(t,x)\in[0,T]\times\mathbb{R}^{n},\\\
(u_{\varepsilon}-u)(0,x)=(g_{\varepsilon}-g)(x),&\quad
x\in\mathbb{R}^{n},\end{cases}$

where

$R_{\varepsilon}(t,x)=[(A_{\varepsilon}-A)(t,x,D_{x})+(L_{\varepsilon}-L)(t,x,D_{x})]u(t,x).$

Repeating the argument in Theorem 3.2, we arrive at

$\displaystyle
V_{1,\varepsilon}=U^{0}_{1,\varepsilon}+G_{1,\varepsilon}\circ(a_{12,\varepsilon}+\ell_{12,\varepsilon})\circ
G_{2,\varepsilon}\circ\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}V_{1,\varepsilon},$

where

$\displaystyle V_{1,\varepsilon}$ $\displaystyle=u_{1,\varepsilon}-u_{1},$
$\displaystyle U^{0}_{1,\varepsilon}$
$\displaystyle=G_{1,\varepsilon}^{0}(g_{1,\varepsilon}-g_{1})+G_{1,\varepsilon}([f_{1,\varepsilon}-f_{1}]+R_{1,\varepsilon})$
$\displaystyle+G_{1,\varepsilon}\circ(a_{12,\varepsilon}+\ell_{12,\varepsilon})\circ[G_{2,\varepsilon}^{0}(g_{2,\varepsilon}-g_{2})+G_{2,\varepsilon}(f_{2,\varepsilon}-f_{2})+G_{2,\varepsilon}R_{2,\varepsilon}],$

and

$\displaystyle R_{1,\varepsilon}$
$\displaystyle:=[(\lambda_{1,\varepsilon}-\lambda_{1})(t,x,D_{x})u_{1}+(a_{12,\varepsilon}-a_{12})u_{2}+(\ell_{11,\varepsilon}-\ell_{11})u_{1}+(\ell_{12,\varepsilon}-\ell_{12})u_{2}],$
$\displaystyle R_{2,\varepsilon}$
$\displaystyle:=[(\lambda_{2,\varepsilon}-\lambda_{2})(t,x,D_{x})u_{2}+(\ell_{21,\varepsilon}-\ell_{21})\langle
D_{x}\rangle^{-1}u_{1}+(\ell_{22,\varepsilon}-\ell_{22})u_{2}].$

By using the hypothesis (H3) and the Banach fixed-point theorem with $CT\leq
1$, we get

$\displaystyle||V_{1,\varepsilon}||_{L^{2}(\mathbb{R}^{n}_{x})}$
$\displaystyle=||U^{0}_{1,\varepsilon}||_{L^{2}(\mathbb{R}^{n}_{x})}+||G_{1,\varepsilon}\circ(a_{12,\varepsilon}+\ell_{12,\varepsilon})\circ
G_{2,\varepsilon}\circ\ell_{21,\varepsilon}\langle
D_{x}\rangle^{-1}V_{1,\varepsilon}||_{L^{2}(\mathbb{R}^{n}_{x})}$
$\displaystyle\leq||U^{0}_{1,\varepsilon}||_{L^{2}(\mathbb{R}^{n}_{x})}+CT||V_{1,\varepsilon}||_{L^{2}(\mathbb{R}^{n}_{x})}$
$\displaystyle\leq||U^{0}_{1,\varepsilon}||_{L^{2}(\mathbb{R}^{n}_{x})}.$

Note that because of the regularity of the coefficients the operators involved
above fulfill estimates which are independent of the parameter
$\varepsilon\in(0,1]$. We therefore have that
$||U^{0}_{1,\varepsilon}||_{L^{2}(\mathbb{R}^{n}_{x})}\rightarrow 0$ as
$\varepsilon\rightarrow 0$ since
$||g_{1,\varepsilon}-g_{1}||_{L^{2}(\mathbb{R}^{n}_{x})}\rightarrow 0$,
$||f_{1,\varepsilon}-f_{1}||_{L^{2}(\mathbb{R}^{n}_{x})}\rightarrow 0$,
$||\lambda_{i,\varepsilon}-\lambda_{i}||_{L^{\infty}(\mathbb{R}^{n}_{x})}\rightarrow
0$,
$\langle{x}\rangle^{-1}||a_{12,\varepsilon}-a_{12}||_{L^{\infty}(\mathbb{R}^{n}_{x})}\rightarrow
0$, and
$||\ell_{ij,\varepsilon}-\ell_{ij}||_{L^{\infty}(\mathbb{R}^{n}_{x})}\rightarrow
0$ for $i,j=1,2$ as $\varepsilon\rightarrow 0$. This gives that
$||u_{1,\varepsilon}-u_{1}||_{L^{2}(\mathbb{R}^{n}_{x})}\rightarrow 0$. From
the continuity assumption we have that all the limits above hold uniformly
with respect to $t\in[0,T]$. This yields to
$\|u_{\varepsilon}-u\|_{L^{2}(\mathbb{R}^{n}_{x})}\to 0$ as $\varepsilon\to 0$
uniformly with respect to $t\in[0,T]$. ∎

We conclude this section with the analysis of few examples where the system
coefficients are less than continuous or in general distributions with compact
support.

###  4.1. Examples

For the sake of simplicity we work in space dimension 1. Let us consider the
Cauchy problem

$\begin{cases}D_{t}u=\begin{bmatrix}H(x)D_{x}&a_{12}(t,x,D_{x})\\\
0&H(x)D_{x}\end{bmatrix}u+\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
\ell_{21}(t,x)\langle D_{x}\rangle^{-1}&\ell_{22}(t,x)\end{bmatrix}u,\\\
u(0,x)=g(x),\end{cases}$

where $(t,x)\in[0,T]\times\mathbb{R}$, $u(0,x)=[g_{1}(x),g_{2}(x)]^{T}$ with
$g_{1},g_{2}\in\mathcal{E}^{\prime}(\mathbb{R})$ and the operators $a_{12}$
and $\ell_{ij}$ are pseudo-differential operators of order $1$ and $0$
respectively continuous with respect to $t$. For the moment we assume that
only the coefficients on the diagonal of the matrix $A$ are non-regular, i.e.,
$H$ is the Heaviside function ($H(x)=1$ for $x\geq 0$ and $H(x)=0$ for $x<0$).
One could in principle replace $H$ with any discontinuous but bounded
function. Note that already in this situation the system is not treatable
within the classical theory of hyperbolic systems with multiplicities that as
in [25, 26] requires smoothness with respect to the variable $x$. It is
therefore meaningful to look at the regularised problem and at the net
$(u_{\varepsilon})_{\varepsilon}$ of its solutions. In detail,

$\begin{cases}D_{t}u_{\varepsilon}=\begin{bmatrix}H_{\varepsilon}(x)D_{x}&a_{12}(t,x,D_{x})\\\
0&H_{\varepsilon}(x)D_{x}\end{bmatrix}u_{\varepsilon}+\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
\ell_{21}(t,x)\langle
D_{x}\rangle^{-1}&\ell_{22}(t,x)\end{bmatrix}u_{\varepsilon},\\\
u_{\varepsilon}(0,x)=g_{\varepsilon}(x),\end{cases}$

where

$\displaystyle H_{\varepsilon}(x)$
$\displaystyle=(H*\psi_{\omega(\varepsilon)})(x),$ $\displaystyle
g_{1,\varepsilon}(x)$ $\displaystyle=(g_{1}*\psi_{\varepsilon})(x),$
$\displaystyle g_{2,\varepsilon}(x)$
$\displaystyle=(g_{2}*\psi_{\varepsilon})(x).$

We will choose the scale $\omega(\varepsilon)$ later in order to make sure
that the hypothesis (H2) holds.

We can reformulate the regularised problem in terms of Fourier integral
operators follows

$\displaystyle u_{1,\varepsilon}$
$\displaystyle=G_{1,\varepsilon}^{0}g_{1,\varepsilon}+G_{1,\varepsilon}((a_{12}+\ell_{12})u_{2,\varepsilon}),$
$\displaystyle u_{2,\varepsilon}$
$\displaystyle=G_{2,\varepsilon}^{0}g_{2,\varepsilon}+G_{2,\varepsilon}(\ell_{21}\langle
D_{x}\rangle^{-1}u_{1,\varepsilon}).$

where

$\displaystyle G_{j,\varepsilon}^{0}g_{j,\varepsilon}(t,x)$
$\displaystyle=\int_{\mathbb{R}}{\rm e}^{i(x+H_{\varepsilon}(x)t)\xi}{\rm
e}^{\int_{0}^{t}\ell_{jj}(\tau,x+H_{\varepsilon}(x)\tau)d\tau}\hat{g}_{j,\varepsilon}(\xi)d\xi$
$\displaystyle={\rm
e}^{\int_{0}^{t}\ell_{jj}(\tau,x+H_{\varepsilon}(x)\tau)d\tau}g_{j,\varepsilon}(x+H_{\varepsilon}(x)t),$
$\displaystyle G_{j,\varepsilon}f_{\varepsilon}(t,x)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}{\rm
e}^{i(x+H_{\varepsilon}(x)t)\xi}{\rm
e}^{\int_{s}^{t}\ell_{jj}(\tau,x+H_{\varepsilon}(x)\tau)d\tau}\hat{f}_{\varepsilon}(s,\xi)d\xi
ds$ $\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{2}}{\rm
e}^{i(x+H_{\varepsilon}(x)t)\xi-
iy\xi}A_{j,\varepsilon}(s,t,x)f_{\varepsilon}(s,y)\,dyd{}^{-}\xi ds$
$\displaystyle=\int_{0}^{t}{\rm
e}^{\int_{s}^{t}\ell_{jj}(\tau,x+H_{\varepsilon}(x)\tau)d\tau}f_{\varepsilon}(s,x+H_{\varepsilon}(x)(t-s))ds,$

for $j=1,2$.

Let us now set $\omega^{-1}(\varepsilon)=\log(\varepsilon^{-1})$. Note that in
this case the assumptions (H1) and (H2) are immediately fulfilled. Indeed,
$(H_{\varepsilon})_{\varepsilon}$ is real-valued and moderate and
$H^{\prime}_{\varepsilon}=\psi_{\omega(\varepsilon)}$ is logarithmic moderate.
In addition $\|H_{\varepsilon}\|_{L^{\infty}}\leq 1$.

In the special case when both the matrices of the principal part and the lower
order terms are diagonal the assumption (H3) is not needed. In detail, if
$a_{12}=0$ and $\ell_{12}=\ell_{21}=0$, then

$\displaystyle u_{1,\varepsilon}$
$\displaystyle=e^{\int_{0}^{t}\ell_{11}(\tau,x+H_{\varepsilon}(x)\tau,D_{x})d\tau}g_{1,\varepsilon}(x+H_{\varepsilon}(x)t),$
$\displaystyle u_{2,\varepsilon}$
$\displaystyle=e^{\int_{0}^{t}\ell_{22}(\tau,x+H_{\varepsilon}(x)\tau,D_{x})d\tau}g_{2,\varepsilon}(x+H_{\varepsilon}(x)t).$

In general for a full matrix $L$ one would need the assumption (H3) to be
fulfilled as well. This is investigated in the following example where we also
handle singularities in time.

Let

$\begin{cases}D_{t}u=\begin{bmatrix}\lambda_{1}(t)D_{x}&a_{12}(t,x,D_{x})\\\
0&\lambda_{2}(t)D_{x}\end{bmatrix}u+\begin{bmatrix}\ell_{11}(t,x)&\ell_{12}(t,x)\\\
\ell_{21}(t,x)\langle D_{x}\rangle^{-1}&\ell_{22}(t,x)\end{bmatrix}u,\\\
u(0,x)=g(x),\end{cases}$

where $(t,x)\in[0,T]\times\mathbb{R}$, $\lambda_{i}$, $i=1,2$, are
distributions with compact support in $[0,T]$, and the coefficients of the
matrix of the lower order terms are distributions with compact support in $t$
and smooth in $x$. For the sake of simplicity we assume that $a_{12}\in
C([0,T],S^{1}(\mathbb{R}^{2n}))$. As usual we have that
$u(0,x)=[g_{1}(x),g_{2}(x)]^{T}$ with
$g_{1},g_{2}\in\mathcal{E}^{\prime}(\mathbb{R})$. Instead of regularising in
$x$ this time we regularise in $t$ and restrict the domain of our nets to the
interval $[0,T]$ after regularisation. We obtain

$\begin{cases}D_{t}u_{\varepsilon}=\begin{bmatrix}\lambda_{1,\varepsilon}(t)D_{x}&a_{12}(t,x,D_{x})\\\
0&\lambda_{2,\varepsilon}(t)D_{x}\end{bmatrix}u_{\varepsilon}+\begin{bmatrix}\ell_{11,\varepsilon}(t,x)&\ell_{12,\varepsilon}(t,x)\\\
\ell_{21,\varepsilon}(t,x)\langle
D_{x}\rangle^{-1}&\ell_{22,\varepsilon}(t,x)\end{bmatrix}u_{\varepsilon},\\\
u_{\varepsilon}(0,x)=g_{\varepsilon}(x),\end{cases}$

where

$\lambda_{i,\varepsilon}(t)=\lambda_{i}\ast\psi_{\varepsilon},$

for $t\in[0,T]$. In order to guarantee that (H1) and (H2) are fulfilled we
assume that the coefficients $\lambda_{i}$ are real valued for $i=1,2$ and we
regularise the coefficients $\ell_{ii}$, $i=1,2$ via a logarithmic scale,
i.e., $\omega^{-1}(\varepsilon)=\ln(\varepsilon^{-1})$. It is easy to solve
the corresponding Eikonal equation and compute the phase functions
$\phi_{i,\varepsilon}$. Indeed,

$\phi_{i,\varepsilon}(t,x,\xi)=x\xi+\xi\int_{0}^{t}\lambda_{i,\varepsilon}(s)\,ds.$

It follows that

$\displaystyle|\partial_{\xi}\phi_{i,\varepsilon}(t,x,\xi)-\partial_{\xi}\phi_{i,\varepsilon}(t,y,\xi)|$
$\displaystyle=|x-y|\,\,\,\,\text{ for
}\,\,x,y\in\mathbb{R}^{n},\,\,\xi\in\mathbb{R}^{n},$
$\displaystyle|\partial_{y}\phi_{i,\varepsilon}(t,y,\xi)-\partial_{y}\phi_{i,\varepsilon}(t,y,\eta)|$
$\displaystyle=|\xi-\eta|\,\,\,\,\text{ for
}\,\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

and therefore the net $C(\varepsilon)$ required in (H3) is identically equal
to $1$. In addition, since the regularisation is not effecting the variable
$x$ all the constants involved in (H3) are independent of the parameter
$\varepsilon$ are therefore (H3) is trivially satisfied. It follows that our
Cauchy problem is well-posed in the very weak sense.

##  5\. Appendix: $L^{2}$-boundedness of Fourier integral operators

In this appendix we state an important theorem by Ruzhansky-Sugimoto [38] on
the global $L^{2}$ boundedness of FIO for amplitudes which are independent of
the variable $x$. This theorem will be later applied to integrated FIOs.

######  Theorem 5.1 (Ruzhansky-Sugimoto [38]).

Let operator $T$ be defined by

$Tu(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i(x\cdot\xi+\phi(y,\xi))}A(y,\xi)u(y)dyd\xi,$
(23)

where $A(y,\xi)\in C^{\infty}(\mathbb{R}^{n}_{y}\times\mathbb{R}^{n}_{\xi})$,
and $\phi(y,\xi)\in C^{\infty}(\mathbb{R}^{n}_{y}\times\mathbb{R}^{n}_{\xi})$
is a real-valued function. Assume that
$|\partial_{y}^{\alpha}\partial_{\xi}^{\beta}A(y,\xi)|$ is uniformly bounded
for $|\alpha|,|\beta|\leq 2n+1$. Also assume that

$\displaystyle|\partial_{\xi}\phi(x,\xi)-\partial_{\xi}\phi(y,\xi)|$
$\displaystyle\geq C|x-y|\,\,\,\,\text{ for
}\,\,x,y\in\mathbb{R}^{n},\,\,\xi\in\mathbb{R}^{n},$ (24)
$\displaystyle|\partial_{y}\phi(y,\xi)-\partial_{y}\phi(y,\eta)|$
$\displaystyle\geq C|\xi-\eta|\,\,\,\,\text{ for
}\,\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

and that

$|\partial_{y}^{\alpha}\partial_{\xi}\phi(y,\xi)|\leq
C_{\alpha},\,\,|\partial_{y}\partial_{\xi}^{\beta}\phi(y,\xi)|\leq C_{\beta},$
(25)

for $1\leq|\alpha|,|\beta|\leq 2n+2$. Then the operator $T$ is
$L^{2}(\mathbb{R}^{n})$-bounded, and satisfies

$||T||_{L^{2}\rightarrow L^{2}}\leq C^{\prime}\sup_{|\alpha|,|\beta|\leq
2n+1}||\partial_{y}^{\alpha}\partial_{\xi}^{\beta}A(y,\xi)||_{L^{\infty}(\mathbb{R}^{n}_{y}\times\mathbb{R}^{n}_{\xi})},$

where $C^{\prime}$ depends on the constants in (24) and (25).

Note that since $\Gamma_{x}=\Gamma_{y}=\mathbb{R}^{n}$ in Theorem 5.1, we may
write the following assumption

$|\det\partial_{x}\partial_{y}\phi(x,y)|\geq C>0,\,\,\,\text{ for all
}x,y\in\mathbb{R}^{n},$

instead of (24).

Let us now consider operators of the type

$\displaystyle Gu(t,x)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{n}}e^{i\phi(t,x,\xi)}A(t,s,x)\hat{u}(s,\xi)d\xi
ds$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\phi(t,x,\xi)-iy\cdot\xi}A(t,s,x)u(s,y)dyd\xi
ds$ $\displaystyle=\int_{0}^{t}Ku(t,s,x)ds,$

where

$\displaystyle
Ku(s,x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\phi(t,x,\xi)-iy\cdot\xi}A(t,s,x)u(s,y)dyd\xi.$

It follows that

$\displaystyle||Gu(t)||_{L^{2}(\mathbb{R}^{n}_{x})}=\int_{0}^{t}||Kg(s)||_{L^{2}(\mathbb{R}^{n}_{x})}ds$
$\displaystyle\leq t||K||_{L^{2}\rightarrow
L^{2}}||u||_{L^{\infty}(\mathbb{R}_{s})L^{2}(\mathbb{R}^{n}_{x})}$
$\displaystyle=t||K^{*}||_{L^{2}\rightarrow
L^{2}}||u||_{L^{\infty}(\mathbb{R}_{s})L^{2}(\mathbb{R}^{n}_{x})}.$

In order to apply Theorem 5.1, we compute the adjoint of operator $K$

$\displaystyle(Ku,v)$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\phi(t,x,\xi)-iy\cdot\xi}A(t,s,x)u(s,y)dyd\xi\overline{v}(s,x)dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\overline{\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\phi(t,x,\xi)-iy\cdot\xi}A(t,s,x)\overline{v}(s,x)dxd\xi}u(s,y)dy$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{-i\phi(t,x,\xi)+iy\cdot\xi}A(t,s,x)v(s,x)dxd\xi
u(s,y)dy$ $\displaystyle=(u,K^{*}v),$

where

$\displaystyle
K^{*}v(t,s,x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi-i\phi(t,y,\xi)}A(t,s,y)v(s,y)dyd\xi.$

Now the adjoint operator $K^{*}$ has the same form as the operator $T$ in
Theorem 5.1.

######  Proposition 5.2.

Let operator $K^{*}$ be defined by

$K^{*}u(t,s,x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi-i\phi(t,y,\xi)}A(t,s,y)u(s,y)dyd\xi,$
(26)

where $A(t,s,y)\in C^{\infty}(\mathbb{R}^{n}_{y})$ is continuous with respect
to $t,s\in[0,T]$ and $\phi(t,y,\xi)\in
C^{\infty}(\mathbb{R}^{n}_{y}\times\mathbb{R}^{n}_{\xi})$ is a real-valued
function continuous with respect to $t\in[0,T]$. Assume that
$|\partial_{y}^{\gamma}A(s,t,y)|$ is uniformly bounded for $|\gamma|\leq
2n+1$. Also assume that

$\displaystyle|\partial_{\xi}\phi(t,x,\xi)-\partial_{\xi}\phi(t,y,\xi)|$
$\displaystyle\geq C|x-y|\,\,\,\,\text{ for
}\,\,t\in[0,T],\,x,y\in\mathbb{R}^{n},\,\,\xi\in\mathbb{R}^{n},$ (27)
$\displaystyle|\partial_{y}\phi(t,y,\xi)-\partial_{y}\phi(t,y,\eta)|$
$\displaystyle\geq C|\xi-\eta|\,\,\,\,\text{ for
}\,\,t\in[0,T],\,y\in\mathbb{R}^{n},\,\,\xi,\eta\in\mathbb{R}^{n},$

and that

$|\partial_{y}^{\alpha}\partial_{\xi}\phi(t,y,\xi)|\leq
C_{\alpha},\,\,|\partial_{y}\partial_{\xi}^{\beta}\phi(t,y,\xi)|\leq
C_{\beta},$ (28)

for $1\leq|\alpha|,|\beta|\leq 2n+2$, uniformly with respect to all the
variables. Then the operator $K^{*}(t,s,\cdot)$ is
$L^{2}(\mathbb{R}^{n})$-bounded, and satisfies

$\sup_{t,s\in[0,T]}||K^{*}(t,s,\cdot)||_{L^{2}\rightarrow L^{2}}\leq
C^{\prime}\sup_{t,s\in[0,T]}\sup_{|\gamma|\leq
2n+1}||\partial_{y}^{\gamma}A(t,s,y)||_{L^{\infty}(\mathbb{R}^{n}_{y})},$

where $C^{\prime}$ depends on the constants in (27) and (28).

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