Corresponding author<EMAIL_ADDRESS>aff1]Moscow State University,
Moscow, 119991, Russian Federation.

# On nonstrictly hyperbolic systems and models of natural sciences reducible
to them

Olga Rozanova  [

###### Abstract

We show that many important models of natural science in their mathematical
formulation can be reduced to non-strictly hyperbolic systems of the same
kind. This allows the same methods to be applied to them, so that some
important results concerning a particular model can be obtained as corollaries
of general theorems. However, in each case, the models have their own
characteristics. The article contains an overview of both methods that are
potentially applicable to such cases, as well as known results obtained for a
particular model. In addition, we introduce new models and obtain results for
them.

##  1 INTRODUCTION

This paper is devoted to a special case of inhomogeneous non-strictly
hyperbolic systems and related parabolic systems. We first present general
methods that can be applied to these systems. We start with a theorem on the
criterion for the existence of a globally smooth solution of the Cauchy
problem, then we discuss the existence of a simple and traveling waves, then
we describe the stochastic regularization method, which allows one to obtain
formulas for representing continuous solutions, and finally, we define a
generalized solution for non-strictly hyperbolic systems of a given type,
which are written in a non-divergence form.

Further, we give an example of a non-strictly hyperbolic system of two
equations, which corresponds to an oscillatory process. We show that the
following models can be reduced to this system and its parabolic counterpart:

  * •

model of a cold plasma;

  * •

model of a stratified fluid in the gravity field;

  * •

model of heat conduction in Reyleigh-Bénard convection;

  * •

model of oscillatory flow of blood in arteries;

  * •

Euler-Poisson equations arising in astrophysics and physics of semi-
conductors.

Let us denote ${\bf V}=(V_{1},V_{2},\dots,V_{n})$, $V_{i}=V_{i}(t,x)$, $t\geq
0$, $x\in\mathbb{R}$, $t\geq 0$, and consider the system

$\displaystyle{\partial{\bf V}\over\partial t}+A({\bf
V})\displaystyle{\partial{\bf V}\over\partial x}=Q{\bf V},$ (1)

where $A({\bf V})=V_{1}\mathbb{E}$, $\mathbb{E}$ is the $n\times n$ unit
matrix and $Q={Q_{ij}}$ is a $n\times n$ constant matrix. The matrix $A$ has
the eigenvalues $\lambda_{i}({\bf V})=V_{1}$, the respective eigenvectors
${\bf v}_{i}=(0,...,\underbrace{1}_{i-th\,place},...,0)$, $i=1,\dots,n,$ form
the basis of the ${\mathbb{R}}^{n}$ space.

The initial conditions are

${\bf V}(0,x)={\bf V}_{0}(x)\in C^{1}(\mathbb{R}).$ (2)

According to the standard classification (1) is non-strictly hyperbolic (see
[8]) and has a local in time solution as smooth as initial data.

We consider also a parabolic counterpart of (1)

$\displaystyle{\partial{\bf V}\over\partial t}+V_{1}\displaystyle{\partial{\bf
V}\over\partial x}=Q{\bf V}+B\displaystyle{\partial^{2}{\bf V}\over\partial
x^{2}},$ (3)

where $B={b_{ij}}$ is a $n\times n$ constant matrix.

For some choice of $B$ system (3) can be considered as a regularization of
(1), which solution allows shocks waves, therefore there arise a problem of
justification of convergence of the viscosity solution to a shocks. However,
we do not touch here this complicated question. In our applications the
components of $B$ are fixed constants.

##  2 1\. CRITERION OF A SINGULARITY FORMATION

The main advantage of (1) is a possibility to study its solution along
characteristics $\dot{x}(t)=V_{1}$, i.e. to reduce the dynamics to a system of
ODEs. Thus, the components of solution obey linear system

$\displaystyle{d{\bf V}\over dt}=Q{\bf V},\quad\displaystyle{dx\over
dt}=V_{1},$ (4)

subject to initial data

${\bf V}(0)={\bf V}_{0}(x_{0}),\quad x(0)=x_{0},\quad x_{0}\in\mathbb{R}.$ (5)

If we differentiate (1) with respect to $x$ and obtain along characteristics
$\displaystyle{dx\over dt}=V_{1}$, $x(0)=x_{0}$, a quadratically nonlinear
system for ${\bf v}=(v_{1},...,v_{n})$, $v_{i}=(V_{i})_{x}$, $i=1,...,n$:

$\displaystyle{d{\bf v}\over dt}=-v_{1}{\bf v}+Q{\bf v}.$ (6)

This system can be linearized using Radon’s lemma [18], [14].

######  Theorem 1 (The Radon lemma)

A matrix Riccati equation

$\dot{W}=M_{21}(t)+M_{22}(t)W-WM_{11}(t)-WM_{12}(t)W,$ (7)

($W=W(t)$ is a matrix $(n\times m)$, $M_{21}$ is a matrix $(n\times m)$,
$M_{22}$ is a matrix $(m\times m)$, $M_{11}$ is a matrix $(n\times n)$,
$M_{12}$ is a matrix $(m\times n)$) is equivalent to the homogeneous linear
matrix equation

$\dot{Y}=M(t)Y,\quad M=\left(\begin{array}[]{cc}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{array}\right),$ (8)

($Y=Y(t)$ is a matrix $(n\times(n+m))$, $M$ is a matrix $((n+m)\times(n+m))$ )
in the following sense.

Let on some interval ${\mathcal{J}}\in\mathbb{R}$ the matrix-function
$\,Y(t)=\left(\begin{array}[]{c}\mathfrak{Q}(t)\\\
\mathfrak{P}(t)\end{array}\right)$ ($\mathfrak{Q}$ is a matrix $(n\times n)$,
$\mathfrak{P}$ is a matrix $(n\times m)$)  be a solution of (8) with the
initial data

$\displaystyle Y(0)=\left(\begin{array}[]{c}I\\\ W_{0}\end{array}\right)$

($I$ is the identity matrix $(n\times n)$, $W_{0}$ is a constant matrix
$(n\times m)$)  and $\det\mathfrak{Q}\neq 0$ on ${\mathcal{J}}$. Then
$W(t)=\mathfrak{P}(t)\mathfrak{Q}^{-1}(t)$ is the solution of (7) with
$W(0)=W_{0}$ on ${\mathcal{J}}$.

System (6) can be written as (7) with

$\displaystyle W={\bf v},\quad M_{11}=(0),\quad M_{12}=(1,0,...,0),\quad
M_{21}=(0,0,...,0)^{T},\quad M_{22}=Q.$

Thus, we obtain the linear Cauchy problem

$\left(\begin{array}[]{c}\dot{q}\\\ \dot{u}_{1}\\\ \dots\\\
\dot{u}_{n}\end{array}\right)=\left(\begin{array}[]{ccccc}0&1&0&\ldots&0\\\
0&Q_{11}&Q_{12}&\ldots&Q_{1n}\\\ \vdots&\vdots&\ddots&\vdots\\\
0&Q_{n1}&Q_{n2}&\ldots&Q_{nn}\end{array}\right)\,\left(\begin{array}[]{c}q\\\
u_{1}\\\ \dots\\\ u_{n}\end{array}\right),\quad\left(\begin{array}[]{c}q(0)\\\
u_{1}(0)\\\ \dots\\\ u_{n}(0)\end{array}\right)=\left(\begin{array}[]{c}1\\\
v_{1}(0)\\\ \dots\\\ v_{n}(0)\end{array}\right),$ (9)

where $v_{i}(0)=(V_{0i})_{x}(x_{0})$, $x_{0}\in\mathbb{R}$, $i=1,...,n$.

Let us take into account that the singularity formation for the hyperbolic
systems means a finite time blow up either solution itself or its first
derivatives, then we can conclude that the solution itself which is described
by linear system (4) with data (5), do not blows up. Further, it follows from
Radon’s theorem that $v_{i}(t)=\frac{u_{i}(t)}{q(t)}$. Therefore the
derivatives blow up at the moment, when $q(t)$, a part of solution to (9),
vanishes.

We summarize this result as follows.

###### Theorem 2

  1. 1.

The behavior of the solution of the Cauchy problem (1), (2) along the
characteristics can be completely described by linear dynamics with the help
of the systems (4) and (9).

  2. 2.

If for every point $x_{0}\in\mathbb{R}$ the scalar function $q(t)$, the
component of solution to (9), does not vanish for all $t>0$, then the solution
to (1), (2) keeps classical smoothness for all $t>0$.

  3. 3.

Otherwise, if there exists a point $x_{0}\in\mathbb{R}$ such that the solution
to (9) is such that $q(t_{*})=0$ for a finite $t_{*}>0$, then the solution to
(1), (2) blows up is a finite time. The exact time $T_{*}$ of the blow up is

$\displaystyle
T_{*}=\min\limits_{x_{0}\in\mathbb{R}}\left(t_{*}>0\,\Big{|}\,q(t_{*})=0\right).$

###### Remark 1

Although $q(t)$ can always be found explicitly, it is a quasi-polynomial, the
solution to the transcendental equation $q(t)=0$ for $n>2$ can usually be
found only numerically.

##  3 2\. SIMPLE WAVES

###### Definition 1

A solution of (1) or (3) is called a simple wave if there exists a functional
dependence $V_{i}=V_{i}(U(t,x))$, $i=2,...,n$, for some sufficiently smooth
function $U(t,x)$.

For example, we can consider a simple wave of the form $V_{i}=V_{i}(V_{1})$,
$i=2,...,n$. A particular important class of simple waves is the class of
traveling waves, when $U(t,x)=\xi=x-wt$, $w=\rm const$.

If we substitute $V_{i}=V_{i}(V_{1})$ to (1), we get

$\begin{array}[]{l}\displaystyle{dV_{i}\over
dV_{1}}=\frac{Q_{i1}V_{1}+\sum\limits_{j=2}^{n}Q_{ij}V_{j}}{Q_{11}V_{1}+\sum\limits_{j=2}^{n}Q_{1j}V_{j}},\quad
i=2,...,n.\end{array}$ (10)

Traveling waves $V_{i}=V_{i}(\xi)$ for (1) satisfy the system

$\displaystyle\begin{array}[]{l}\displaystyle{dV_{i}\over
d\xi}=\frac{\sum\limits_{j=1}^{n}Q_{ij}V_{j}}{V_{1}-w},\quad
i=1,2,...,n.\end{array}$

We can see that for $n=2$ the relation between $V_{1}$ and $V_{2}$ can be
found explicitly in elementary functions, since the system reduced to one
fractional linear equation. For some matrices $Q$ the number of equations in
(10) reduces, it signifies that the first integrals exists and
$V_{i}=V_{i}(V_{1},..,V_{j}),$ $1<j<n$, $i=j+1,...,n$. If $j=n-1$, then there
is a full set of first integrals.

###### Remark 2

The examples show that properties of simple waves can drastically differ from
the properties of solutions of general form.

The construction of simple waves for a parabolic system (3) is more
complicated, since it reduces to higher order ODE systems, usually non-
integrable. However, as we will show by examples, there are happy exceptions.

###  3.1 3\. STOCHASTIC REGULARIZATION

The method of stochastic regularization was applied first to the Hopf
equation, including multidimensional case, in [1], and then it turned out that
the scope of its application is sufficiently wider. Namely, it was used for
the regularization of the solution of the scalar a scalar conservation laws
[2], for a system in Riemann invariants [19], for two-dimensional transport
equations on a rotating plane [20]. The main idea is to consider a stochastic
perturbation of intensity $\sigma>0$ along characteristics and then introduce
a joint density of distribution of the particle position and ”velocity” (the
solution of initial equation not looking at its physical sense), and consider
new system of PDEs describing averaged density and averaged ”velocity”. This
system is integro-differential and its solution tends as $\sigma\to 0$ to the
solution of the initial equation provided it is continuous, and to the
solution in the sense of ”free particles”, otherwise. Thus, the method of
stochastic regularization is very convenient instrument to construct
rarefaction waves, but the limit shock wave is different from the standard
entropy shock.

For system (1), the stochastic regularization method can be applied, and its
justification does not differ from the case of the Hopf equation [1], although
it is associated with technical difficulties.

As a first stem we consider a stochastic differential system which describes a
stochastic perturbation of the trajectory $x(t)$, such that
$\dot{x}(t)=V_{1}(t,x(t))$, $x(0)=x_{0}$:

$\displaystyle dX(t)$ $\displaystyle=$
$\displaystyle\mathfrak{V}_{1}(t)dt+\sigma dW,$ $\displaystyle
d\mathfrak{V}(t)$ $\displaystyle=$ $\displaystyle Q\mathfrak{V}(t),$

where $W$ is a standard Wiener process, $\sigma>0$, $\mathfrak{V}(t)={\bf
V}(t,X(t))$, $\mathfrak{V}_{1}(t)=V_{1}(t,X(t))$. Here $X(t)$ and
$\mathfrak{V}=(\mathfrak{V}_{1},...,\mathfrak{V}_{n})$ are random processes
starting from the initial point $x_{0},{\bf V}_{0}(x_{0})$.

Let $P(t,x,v_{1},..,v_{n})$ be the joint probability density function for the
processes $(X,\mathfrak{V}(t))$ satisfying the Cauchy problem for the Fokker-
Planck equation,

$\displaystyle\frac{\partial P}{\partial t}\,+\,v_{1}\frac{\partial
P}{\partial x}\,$ $\displaystyle+$
$\displaystyle\,\sum\limits_{i=1}^{n}\,\frac{\partial}{\partial
v_{i}}\,\left(\sum\limits_{j=1}^{n}Q_{ij}v_{j}\,P\right)\,+\frac{1}{2}\,\sigma^{2}\,\frac{\partial^{2}P}{\partial
x^{2}}=0,$ $\displaystyle P_{0}(t,x,v_{1},\dots,v_{n})$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\delta(v_{i}-V_{i}(0,x))\,f_{0}(x),\quad
f_{0}\in L_{1}(\mathbb{R}).$

We introduce the following functions

$\displaystyle\rho(t,x)$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}P(t,x,v_{1},\dots,v_{n})\,dv_{1}\dots
dv_{n},$ $\displaystyle\hat{\bf V}(t,x)$ $\displaystyle=$
$\displaystyle\frac{1}{\rho(t,x)}\int\limits_{\mathbb{R}}vP(t,x,v_{1},\dots,v_{n})\,dv_{1}\dots
dv_{n},\quad v=(v_{1},...,v_{n}).$

After computations we obtain the system

$\displaystyle\displaystyle{\partial\rho\over\partial t}$ $\displaystyle+$
$\displaystyle\displaystyle{\partial\rho\hat{V}_{1}\over\partial
x}=\frac{\sigma^{2}}{2}\displaystyle{\partial^{2}\rho\over\partial x^{2}},$
$\displaystyle\displaystyle{\partial\rho\hat{\bf V}\over\partial t}$
$\displaystyle+$ $\displaystyle\displaystyle{\partial\rho\hat{V}_{1}\hat{\bf
V}\over\partial x}=Q\rho\hat{\bf
V}+\frac{\sigma^{2}}{2}\displaystyle{\partial^{2}\rho\hat{\bf V}\over\partial
x^{2}}-\int\limits_{\mathbb{R}}\,(v-\hat{\bf V})(v_{1}-\hat{V}_{1})P_{x}\,dv.$

Similarly to [1], one can prove that $\hat{\bf V}\to{\bf V},$ $\sigma\to 0,$
$t>0,$ $x\in\mathbb{R},$ for a continuous ${\bf V}$.

##  4 4\. GENERALIZED SOLUTION TO THE CAUCHY PROBLEM

If we still want to consider the solution to the Cauchy problem (1), (2) after
the moment of blowup, we encounter sufficient difficulties. The problem is
that system (1) is not written in a divergence form, so we cannot transfer
derivatives to test functions, as is always done when determining the
generalized solution of conservation laws. We use a trick, hinted at in the
previous section, where the density function naturally arose. We will have to
supplement system (1) with a new equation, after which we can already write it
in a conservative form. The solution of (1) turns out to be part of the
solution of the new augmented system. Note that the new system is no longer
hyperbolic, which leads to the appearance of a delta function in the velocity
component, just as it was for the system of gas dynamics ”without pressure”
[24].

As usual, we start with smooth solutions and complete the system with the
continuity equation:

$\displaystyle\displaystyle{\partial\rho\over\partial t}$ $\displaystyle+$
$\displaystyle\displaystyle{\partial\rho V_{1}\over\partial x}=0,$
$\displaystyle\displaystyle{\partial\rho{\bf V}\over\partial t}$
$\displaystyle+$ $\displaystyle\displaystyle{\partial\rho V_{1}{\bf
V}\over\partial x}=\rho Q{\bf V}.$

This system has a divergent form, and we can define the usual generalized
solution in the sense of an integral identity.

In fact, we modify to our case the definition of strong singular solution
according to V.M. Shelkovich [24]. Let $\Gamma$ be a $C^{1}$ be a smooth curve
given as $S(t,x)=0$, $S_{x}\neq 0$, dividing the $(t,x)$ – plane into two
part, such that $\mathbb{R}^{2}=\Omega_{-}\bigcup\Gamma\bigcup\Omega_{+}$.
Further, let $\delta(\Gamma)$ be the $\delta$ \- function, concentrated on
$\Gamma$, $e(t,x)\Big{|}_{\Gamma}$ be its amplitude,
$v_{\delta}(t,x)=-{S_{t}}/{S_{x}}$ be its velocity,
$\mathbb{I}=v_{\delta}(1,...,1)^{T}$, $\frac{\delta\phi(t,x)}{\delta
t}\Big{|}_{\Gamma}=\left(\displaystyle{\partial\phi\over\partial
t}+v_{\delta}\displaystyle{\partial\phi\over\partial
t}\right)\Big{|}_{S(t,x)=0}$, [17].

###### Definition 2

We call a vector-function ${\bf V}\in C^{1}(\Omega_{-})\cap
C(\bar{\Omega}_{-})\cap C^{1}(\Omega_{+})\cap C(\bar{\Omega}_{+})$ the
generalized solution of system (1), if for every test function
$\phi(t,x)\in\mathcal{D}(\mathbb{R}^{2})$ and every couple of functions
$(\hat{\rho}(t,x),e(t,x))$, $\hat{\rho}\in C^{1}(\Omega_{-})\cap
C(\bar{\Omega}_{-})\cap C^{1}(\Omega_{+})\cap C(\bar{\Omega}_{+})$, $e(t,x)\in
C(\mathbb{R}^{2})$, such that

$\displaystyle\begin{array}[]{l}P(t)=\int\limits_{\mathbb{R}}\hat{\rho}(t,x)\,dx+e(t,x)\Big{|}_{\Gamma}=\rm
const,\end{array}$

the identity

$\displaystyle\begin{array}[]{l}\int\limits_{\mathbb{R}^{2}}\hat{\rho}\left[{\bf
V}(\phi_{t}+V_{1}\phi_{x})-Q{\bf
V}\phi\right]\,dx\,dt+\int\limits_{\Gamma}e(t,x)\mathbb{I}\frac{\delta\phi}{\delta
t}\frac{dl}{\sqrt{1+v^{2}_{\delta}}}=0\end{array}$

holds.

As always, when defining a generalized solution, the question of its
uniqueness arises. Traditionally, to single out a unique generalized solution,
they use the geometric entropy condition

$\displaystyle\min\\{{V_{1}}_{-},{V_{1}}_{+}\\}\leq\dot{\Phi}(t)\leq\max\\{{V_{1}}_{-},{V_{1}}_{+}\\},$

where $x=\Phi(t)$ is a result of resolving the inexplicit relation $S(t,x)=0$,
${V_{1}}_{\pm}$ is the limit values of $V_{1}(t,x)$ as $x\to\Phi(t)\pm 0.$ It
implies that characteristics from both sides of the shock come to the shock.

Solving the Riemann problem for a specific system (1) is a very difficult
problem even for the case $n=2$. We will not touch it here and only note that
non-uniqueness can arise here not only for the shock wave, but also for the
rarefaction wave. This phenomenon is connected precisely with the non-strict
hyperbolicity.

##  5 5\. MAIN EXAMPLE

Now we restrict ourself by a particular case of system (1), and are going to
show that many important models of physics can be reduced to it or its
parabolic counterparts. In this example $n=2$, ${\bf V}=(V,U)$,
$Q=\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right)$, such that the
system takes the form

$\displaystyle\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}=-U,\quad\displaystyle{\partial U\over\partial
t}+V\displaystyle{\partial U\over\partial x}=V.$ (14)

Along characteristics given by equation $\dot{x}(t)=V(t,x(t)$ we have

$\displaystyle\dot{V}=-U,\qquad\dot{U}=V,$

therefore

$\displaystyle\displaystyle{d(V^{2}+U^{2})\over dt}=\rm const.$

Thus, the solution is bounded provided it is smooth and

$\displaystyle\max\limits_{t>0,\,x\in\mathbb{R}}{\sqrt{V^{2}+U^{2}}}=\max\limits_{x\in\mathbb{R}}{\sqrt{V_{0}^{2}+U_{0}^{2}}}.$

###  5.1 5.1. Singularities formation

The dynamics along characteristics of derivatives $(V_{x},U_{x})=(v,u)$ is as
follows:

$\dot{v}=-u-v^{2},\qquad\dot{u}=v(1-u).$ (15)

On the phase plane $u,v$ we have one equilibrium point $(0,0)$, a center.
System (15) has the first integral

$\displaystyle v^{2}+2u-1=C(u-1)^{2},$

corresponding to a second-order curve. The type of this curve depends on the
sign of $D=v^{2}+2e-1.$ If $D<0$, then the phase curve is an ellipse, the
derivatives for $t>0$, the period is $2\pi$. Otherwise, the phase curve is a
parabola ($D=0$) or a hyperbola ($D>0$), the derivatives become infinite in a
finite time.

This results can be summarised as a theorem [21].

###### Theorem 3

For the existence and uniqueness of continuously differentiable $2\pi-$
periodic in time solution $(V,U)$ of

$\displaystyle\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}$ $\displaystyle=$
$\displaystyle-U,\quad\displaystyle{\partial U\over\partial
t}+V\displaystyle{\partial U\over\partial x}=V,$ $\displaystyle(V,U)|_{t=0}$
$\displaystyle=$ $\displaystyle(V_{0},U_{0})\in C^{1}({\mathbb{R}})$

it is necessary and sufficient that inequality

$\displaystyle(V^{\prime}_{0}(x))^{2}+2U^{\prime}_{0}(x)-1<0$

holds at each point $x\in\mathbb{R}$.

If there exists at least one point $x_{0}$ for which the opposite inequality
holds, then the derivatives of the solution become infinite in a finite time.

###### Remark 3

In this simple case we obtained the criterion of singularity formation
independently, but it also follows from Theorem 2 as a corollary.

###  5.2 5.2. Traveling waves for $B\neq 0$

To find traveling waves we have to substitute ${\bf V}=\mathbb{V}(\xi)$,
$\xi=x-wt$, to (3).

The resulting ODE system is

$\displaystyle(-w{\mathbb{E}}+A({\mathbb{V}})){\mathbb{V}}^{\prime}=Q{\mathbb{V}}+B{\mathbb{V}}^{\prime\prime}$

or

$\displaystyle(-w+{\mathbb{V}}_{1}){\mathbb{V}}^{\prime}=Q{\mathbb{V}}+B{\mathbb{V}}^{\prime\prime}.$

It our particular 2D case ${\mathbb{V}}=({\mathcal{V},\mathcal{U}})$,
therefore

$\displaystyle(-w+{\mathcal{V}}){\mathcal{V}}^{\prime}$ $\displaystyle=$
$\displaystyle-{\mathcal{U}}+b_{11}{\mathcal{V}}^{\prime\prime}+b_{12}{\mathcal{U}}^{\prime\prime},$
$\displaystyle(-w+{\mathcal{V}}){\mathcal{U}}^{\prime}$ $\displaystyle=$
$\displaystyle{\mathcal{V}}+b_{21}{\mathcal{V}}^{\prime\prime}+b_{22}{\mathcal{U}}^{\prime\prime}.$

As we noticed in Sec.2, for $B=0$ the system can always be integrated, the
integral found in [21]. For $B\neq 0$ it is not integrable in the general
case.

##  6 6\. A “COLD” (ELECTRON) PLASMA

The equations of electron plasma consist of hydrodynamic equations together
with Maxwell’s equations [15], [3]:

$\displaystyle\begin{array}[]{l}\displaystyle{\partial n\over\partial
t}+\mbox{div}\,(n{\bf V})=0\,,\quad\displaystyle{\partial{\bf P}\over\partial
t}+\left({\bf V}\cdot\nabla\right){\bf p}=e\,\left({\bf
E}+\displaystyle{1\over c}\left[{\bf V}\times{\bf B}\right]\right)-q{\bf
P}+\nu\Delta{\bf P},\vspace{0.5em}\\\ \gamma=\sqrt{1+\displaystyle{|{\bf
P}|^{2}\over m^{2}c^{2}}}\,,\quad{\bf V}=\displaystyle{{\bf P}\over
m\gamma}\,,\vspace{0.5em}\\\ \displaystyle{1\over c}\frac{\partial{\bf
E}}{\partial t}=-\displaystyle{4\pi\over c}en{\bf V}+{\mbox{curl}\,}\,{\bf
B}\,,\quad\displaystyle{1\over c}\frac{\partial{\bf B}}{\partial
t}=-{\mbox{curl}\,}\,{\bf E}\,,\quad\mbox{div}\,{\bf B}=0\,,\end{array}$

where $e,m$ is the charge and mass of the electron ( $e<0$), $c$ is the speed
of light, $n,{\bf P},{\bf V}$ are density, momentum and speed of electrons,
$\gamma$ is the Lorentz factor, ${\bf E},{\bf B}$ are vectors of electric and
magnetic fields. It this model it is assumed that ions are immobile, $q>0$ is
a coefficient of intensity of electron-ion interactions.

In dimensionless quantities under the following assumptions:

  * •

$\gamma=1$, ${\bf P}={\bf V}$ (non-relativistic case);

  * •

${\bf B}=0$ (electrostaticity);

  * •

$\nu=0$ (non-viscous case);

  * •

$q=0$ (no interaction with ions);

  * •

${\bf V}=(V(t,x),0,0)$, ${\bf E}=(E(t,x),0,0)$;

we obtain the following system [7]

$\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}=-E,\qquad\displaystyle{\partial E\over\partial
t}+V\displaystyle{\partial E\over\partial x}=V,$ (17)

where in the notation of Sec.5 $(V,U)=(V,E)$. It coincides with (14).

Thus, as a corollary of Theorem 2 we obtain a criterion for the singularity
formation. The singularity means the “gradient catastrophe” for $V$ and $E$
and the delta-singularity for the density.

If we do not neglect viscosity, we cannot apply the previous methods of study
of singularity formation. Moreover, we can expect that the presence of viscous
term would eliminate the singularity at all. It is a complicated issue, but,
as follows from our consideration below, this hypothesis does not seem to be
correct. Indeed, the first step for the justification of the viscosity method
for the shock solution of the Riemann problem is the construction of a
traveling wave $(V,E)=({\mathcal{V}}(\xi),{\mathcal{E}}(\xi))$, $\xi=x-wt$,
coinciding the left and right states, such that
${\mathcal{V}}^{\prime}(\pm\infty)={\mathcal{E}}^{\prime}(\pm\infty)=0$.
However, the traveling wave does not exist on the whole axis
$\xi\in\mathbb{R}$, and this is the main obstacle. Moreover, the presence of
viscosity does not simplify the structure of the traveling wave.

Thus, system (8) for the traveling wave takes the form

$\displaystyle(-w+{\mathcal{V}})\,{\mathcal{V}}^{\prime}=-{\mathcal{E}}+\nu{\mathcal{V}}^{\prime\prime},\qquad(-w+{\mathcal{V}})\,{\mathcal{E}}^{\prime}={\mathcal{V}},$

which can be rewritten as one equation

$\displaystyle\nu\,{\mathcal{V}}^{\prime\prime\prime}-({\mathcal{V}}-w)\,{\mathcal{V}}^{\prime\prime}-({\mathcal{V}}^{\prime})^{2}-\frac{{\mathcal{V}}}{{\mathcal{V}}-w}=0.$
(18)

Its solution cannot be found explicitly, however we can study small
perturbations of the zero steady state. To this aim we assume
${\mathcal{V}}=\varepsilon{v}+o(\varepsilon)$, $\varepsilon<<1$, which implies
the linear equation

$\displaystyle\nu w\,v^{\prime\prime\prime}+w^{2}\,v^{\prime\prime}+v=0.$

Roots of the characteristic equation are the following:

  * •

$\nu=0$: $\lambda_{1,2}=\pm\,\i\,w$, the traveling wave is periodic with
respect to $\xi$;

  * •

$\nu>0$: $\lambda_{1,2}=\alpha\pm\,\i\,\beta$, $\lambda_{3}\in\mathbb{R}$, the
traveling wave is not periodic.

Now we present the picture, obtained numerically, which show a drastic
difference between viscous and non-viscous traveling waves for solutions of
(18). The computations are made by the Runge-Kutta method in the MAPLE packet
for $w=2$, ${\mathcal{V}}(0)=1$, ${\mathcal{V}}^{\prime}(0)=0$.

Figure 1: Periodic traveling wave for $\nu=0$ (left). Traveling wave for
$\nu=0.2$, the solution exists on a bounded interval (right).

##  7 7\. MODEL OF PERTURBATION OF THE BASIC STATE OF REST FOR HEAT CONDUCTION
IN REYLEIGH-BÉNARD CONVECTION AND FOR STRATIFIED FLUID IN THE GRAVITY FIELD

1\. The model of perturbation of the basic state of rest with heat conduction
in Reyleigh-Bénard convection was introduced by P. Drazin, 2002 [12].

In dimensionless variables it has the form

$\displaystyle\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}=-\Theta+\nu\displaystyle{\partial^{2}V\over\partial
x^{2}},\qquad\displaystyle{\partial\Theta\over\partial
t}+V\displaystyle{\partial\Theta\over\partial
x}=V+\kappa\displaystyle{\partial^{2}\Theta\over\partial x^{2}},$ (19)

where $x$ is the vertical coordinate, $V$ and $\Theta$ are perturbations of
the vertical component of velocity and temperature, $\nu$ and $\kappa$ are
coefficients of kinematic viscosity and of the heat diffusion.

2\. The model of a stratified fluid near a state of rest in a gravitational
field was considered by V.G. Baidulov in 2014 [5].

It has the form

$\displaystyle\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}=-gS+\bar{\nu}\displaystyle{\partial^{2}V\over\partial
x^{2}},\qquad\displaystyle{\partial S\over\partial t}+V\displaystyle{\partial
S\over\partial
x}=\frac{1}{\Lambda}V+\bar{\kappa}\displaystyle{\partial^{2}S\over\partial
x^{2}},$

where $x$ is the vertical coordinate, $V$ and $S$ are perturbations of the
vertical component of velocity and salinity, $g$ is the acceleration of
gravity, $\Lambda$ is the stratification scale ($N=\sqrt{g/\Lambda}$ is the
buoyancy frequency), $\nu$ and $\kappa$ are coefficients of kinematic
viscosity and of the diffusion of salt.

In dimensionless variables the system takes the form

$\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}=-S+\nu\displaystyle{\partial^{2}V\over\partial
x^{2}},\qquad\displaystyle{\partial S\over\partial t}+V\displaystyle{\partial
S\over\partial x}=V+\kappa\displaystyle{\partial^{2}S\over\partial x^{2}},$
(20)

where $\nu={\bar{\nu}N}/{\bar{V}^{2}}$ and
$\kappa={\bar{\kappa}N}/{\bar{V}^{2}}$ are analogous to the Reynolds number
and Péclet number.

We see that systems (19) and (20) coincide up to notation. Further,
$(V,U)=(V,\Theta)$ for the first case and $(V,U)=(V,S)$ for the second case.

Thus, in the non-dissipative limit $\nu=\kappa=0$ both systems coincide with
(14), and therefore we get a criterion for the singularity formation as a
corollary of Theorem 3.

Let us study traveling waves in the dissipative case, for example, for the 1D
stratified fluid. The system for traveling waves is

$\displaystyle(-w+{\mathcal{V}})\,{\mathcal{V}}^{\prime}=-{\mathcal{S}}+\nu{\mathcal{V}}^{\prime\prime},\qquad(-w+{\mathcal{V}})\,{\mathcal{S}}^{\prime}={\mathcal{V}}+\kappa{\mathcal{S}}^{\prime\prime},$

it can be written as one nonlinear ODE of the 4th order for $\mathcal{S}$, it
is non-integrable.

Small perturbations of the zero steady state under the assumption
${\mathcal{S}}=\varepsilon{s}+o(\varepsilon),\quad\varepsilon<<1,$ satisfy the
following linear equation

$\displaystyle\nu\kappa\,s^{\prime\prime\prime\prime}+(\nu+\kappa)w\,s^{\prime\prime\prime}+w^{2}\,s^{\prime\prime}+s=0.$

Roots of the characteristic equation are the following:

  * •

for $\nu=\kappa=0$: $\lambda_{1,2}=\pm\,\i\,w$, the traveling wave is periodic
with respect to $\xi$;

  * •

for $\nu>0$, $\kappa>0$: $\lambda_{1,2}=\alpha\pm\,\i\,\beta$,

$\lambda_{3,4}\in\mathbb{R}$, the same sign, for large $w$,

or $\lambda_{3,4}=\gamma\pm\,\i\,\delta$, for small $w$,

the traveling wave is not periodic.

  * •

for $\nu>0$, $\kappa=0$ or $\nu=0$, $\kappa>0$:
$\lambda_{1,2}=\alpha\pm\,\i\,\beta$, $\lambda_{3}\in\mathbb{R}$, the
traveling wave is not periodic (this situation was considered in Sec.6).

The numerical computations show that the traveling waves for the nonlinear
equation exist only on a bounded interval, the picture is very similar to
Fig.1, right.

##  8 8\. QUASI-ONE-DIMENSIONAL MODELS OF BLOOD FLOW IN VESSELS

A commonly used model of quasi-one-dimensional blood flow in vessels was
introduced in [6], see also [25], [4]. In fact it is a gas-dynamics-like
system of conservation laws

$\displaystyle\displaystyle{\partial S\over\partial t}+\displaystyle{\partial
SV\over\partial x}=0,$ $\displaystyle\displaystyle{\partial SV\over\partial
t}+\displaystyle{\partial SV^{2}\over\partial
x}+\frac{S}{\rho}\displaystyle{\partial P\over\partial x}=F,$ $\displaystyle
P(S)=P_{0}+D(S-S_{0}),$

where $S(t,x)$ is the cross-sectional area of the vessel, $V(t,x)$ is the
velocity of blood flow, $P(t,x)$ is the pressure, $F(t,x,S,V)$ is some
external force, $\rho=\rm const$ is the density of blood, constants $P_{0}$,
$S_{0}$ are the average pressure and average cross-sectional area, $D$ is the
rigidity of the wall.

The main problem of this model is that for imitation of the blood pulsation it
is necessary to add a periodical with respect to $t$ exterior force $F$. In
other words, oscillations are imposed from outside, and are not determined by
the properties of the system itself. This situation can be corrected by
modeling the response of the vessel wall to the passage of a fluid flow. It
turns out that spontaneous pulsations can be maintained if the walls of the
vessel contract according to a certain law [16].

Thus, we assume that $F=SE,$ where
$E=-\int\limits_{-\infty}^{x}\,(S-S_{0})\,dx<\infty$. The sense of this
assumption is the following: the acceleration of velocity at a given point is
proportional to the cross-sectional area of the vessel at that point and to
the total volume of fluid that can flow through that point.

Then

$\displaystyle\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}=-E+\mu\displaystyle{\partial^{2}E\over\partial
x^{2}},\qquad\displaystyle{\partial E\over\partial t}+V\displaystyle{\partial
E\over\partial x}=S_{0}V,$

where $\mu=\frac{D}{\rho},$ $P=P_{0}-DE_{x}.$

If we set $S_{0}=1$, then in the limit $D\to 0$ (a small rigidity) we get
system (14), where $(V,U)=(V,E)$, therefore as a corollary of Theorem 3 we
obtain a criterion for the singularity formation in a smooth solution.

However, our aim is to obtain conditions for a normal cardiac activity, in
other words, conditions for existence of traveling waves. For
$(V,E)=({\mathcal{V}}(\xi),{\mathcal{E}}(\xi)),$ $\xi=x-wt,$ we get

$\displaystyle(-w+{\mathcal{V}})\,{\mathcal{V}}^{\prime}=-{\mathcal{E}}+\mu{\mathcal{E}}^{\prime\prime},\qquad(-w+{\mathcal{V}})\,{\mathcal{E}}^{\prime}=S_{0}{\mathcal{V}}.$

It can be transformed to

$\displaystyle{\mathcal{E}}^{\prime}=-\frac{S_{0}\mathcal{V}}{{\mathcal{V}}-w},\qquad{\mathcal{V}}^{\prime}=\frac{{\mathcal{E}}({\mathcal{V}}-w)^{2}}{({\mathcal{V}}-w)^{3}+\mu
wS_{0}}.$ (21)

System (21) on the phase plane $({\mathcal{E}},{\mathcal{V}})$ has the only
equilibrium ${\mathcal{E}}={\mathcal{V}}=0$. If $w^{2}>\mu S_{0}$ it is a
center, what means that periodic small perturbations exist. If $w^{2}<\mu
S_{0}$ it is a saddle, what imply nonexistence of a periodic solution.

In contrast with the cases considered in Secs.6 and 7, system (21) has the
first integral

$\displaystyle{\mathcal{E}}^{2}+S_{0}{\mathcal{V}}^{2}-{w\mu
S_{0}^{2}}\,\frac{2{\mathcal{V}}-w}{({\mathcal{V}}-w)^{2}}=\rm const.$

Fig.2 represents phase curves on the plane $({\mathcal{E}},{\mathcal{V}})$
depending on the proximity of the initial deviation to the origin.

Figure 2: Periodic traveling wave for $w^{2}>\mu S_{0}$ ($\mu=S_{0}=1$,
$w=2$). Red: ${\mathcal{E}}(0)=0,{\mathcal{V}}(0)=-1$, blue:
${\mathcal{E}}(0)=0,{\mathcal{V}}(0)=0.5$. Periodic traveling wave exists for
${\mathcal{E}}(0)=0,0<{\mathcal{V}}(0)<w-(\mu S_{0}w)^{1/3}$, $w>0$.

If a traveling wave exists, then we can close the vessel to obtain a simplest
model of vicious circle of blood circulation. Let us compute a diameter $L$ of
the closed vessel.

We denote

$\displaystyle\Psi({\mathcal{E}},{\mathcal{V}})={\mathcal{E}}^{2}+S_{0}{\mathcal{V}}^{2}-{w\mu
S_{0}^{2}}\,\frac{2{\mathcal{V}}-w}{({\mathcal{V}}-w)^{2}}.$

Assume that the initial data $({\mathcal{E}}(0),{\mathcal{V}}(0))$ are such
that the motion is periodic. Let $V_{-}<0$ and $V_{+}>0$ be the closest to
zero roots of

$\displaystyle S_{0}{\mathcal{V}}^{2}-{w\mu
S_{0}^{2}}\,\frac{2{\mathcal{V}}-w}{({\mathcal{V}}-w)^{2}}=\Psi({\mathcal{E}}(0),{\mathcal{V}}(0)).$

Then the period of motion along the phase curve is

$\displaystyle L$ $\displaystyle=$ $\displaystyle
2\int\limits_{V_{-}}^{V_{+}}\,\frac{({\mathcal{V}}-w)^{2}}{({\mathcal{V}}-w)^{3}+\mu
wS_{0}}\,{\mathcal{E}}({\mathcal{V}})\,d{\mathcal{V}},$
$\displaystyle{\mathcal{E}}({\mathcal{V}})$ $\displaystyle=$
$\displaystyle\sqrt{\Psi({\mathcal{E}}(0),{\mathcal{V}}(0))-S_{0}{\mathcal{V}}^{2}+{w\mu
S_{0}^{2}}\,\frac{2{\mathcal{V}}-w}{({\mathcal{V}}-w)^{2}}}.$

In addition, the opposite result can be obtained: for each closed vessel of
perimeter $L$, there exists a constant $w$, corresponding to the velocity of
the traveling wave inside this vessel.

##  9 9\. PRESSURELESS EULER-POISSON EQUATIONS

The pressureless Euler-Poisson system is one of the most important models
describing compressible media in the presence of a potential force, provided
that the Laplacian of the potential depends on the density[13]. It has the
form

$\displaystyle\displaystyle{\partial n\over\partial t}+\mbox{div}\,(n{\bf
V})=0,\qquad\displaystyle{\partial{\bf V}\over\partial t}+\left({\bf
V}\cdot\nabla\right){\bf V}=\,k\,\nabla\Phi-q{\bf V}+\nu\Delta{\bf
V},\qquad\Delta\Phi=n-n_{0},$ (22)

where $n$ is the density, $\Phi$ is the force potential, the vector ${\bf V}$
is the velocity, $n_{0}={\rm const}\geq 0$ is the density background, $q={\rm
const}\geq 0$ is the friction coefficient, $k=\rm const$, the sign of which
corresponds to the type of force. Namely, if $k>0$, the force is repulsive (it
arises in models of plasma and semi-conductors), if $k<0$, the force is
attractive (it arises in astrophysics). If $k=0$, system (22) decomposes and
its part that relates to density and velocity corresponds to the pressureless
gas dynamics.

System (22) can be written in other terms. Namely, let us introduce ${\bf E}$
as $n=n_{0}-\mbox{div}\,{\bf E},$ and remove $n$.

The resulting system is

$\displaystyle\displaystyle{\partial{\bf V}\over\partial t}+\left({\bf
V}\cdot\nabla\right){\bf V}=\,-k{\bf E}-q{\bf V}+\Delta{\bf
V},\qquad\frac{\partial{\bf E}}{\partial t}+{\bf V}\mbox{div}\,{\bf E}={\bf
V}.$

In 1D case it takes the form

$\displaystyle\displaystyle{\partial V\over\partial t}+V\displaystyle{\partial
V\over\partial x}=-kE-qV+\nu\displaystyle{\partial^{2}V\over\partial
x^{2}},\qquad\displaystyle{\partial E\over\partial t}+V\displaystyle{\partial
E\over\partial x}=n_{0}V.$

Let us dwell on the non-dissipative case $\nu=0$. Then

  * •

for $k>0$, $n_{0}>0$, $q=0$ we get the equations of a cold plasma
($k=n_{0}=1$), just considered in Sec.6;

  * •

for $k>0$, $n_{0}>0$, $q>0$ we get the equations of a frictional cold plasma
[23];

  * •

for $k<0$, $n_{0}=0$,

for $k>0$, $n_{0}>0$,

for $k>0$, $n_{0}=0$,

we also obtain non-strictly hyperbolic systems of equations, but their
behavior differs from one of cold plasma equations see [13].

In all these cases Theorem 3 can be applied for obtaining the criterion of the
singularity formation for solutions to the Cauchy problem, the generalized
solution can be defined after the moment of singularities formation and simple
and traveling waves can be constructed.

##  10 10\. DAVIDSON’S MODEL

The influence of the magnetic field to the cold (collisional) plasma in the
simplest case can be described by the model, introduced by R. C. Davidson and
P. P. Schram in [9] and became popular after the appearance of the book [10]
in 1972. It belongs to the type (1) and consists of three equations:

$\displaystyle\displaystyle{\partial V_{1}\over\partial
t}+V_{1}\displaystyle{\partial V_{1}\over\partial x}$ $\displaystyle=$
$\displaystyle-qV_{1}-E_{1}-B_{0}\,V_{2},$
$\displaystyle\displaystyle{\partial V_{2}\over\partial
t}+V_{1}\displaystyle{\partial V_{2}\over\partial x}$ $\displaystyle=$
$\displaystyle-qV_{2}+B_{0}\,V_{1},$ $\displaystyle\displaystyle{\partial
E_{1}\over\partial t}+V_{1}\displaystyle{\partial E_{1}\over\partial x}$
$\displaystyle=$ $\displaystyle V_{1}.$

Here ${\bf V}=(V_{1}(t,x),V_{2}(t,x),0),\quad{\bf E}(t,x)=(E_{1}(t,x),0,0),$
where $(x,y,z)$ are the Cartesian coordinate, ${\rm rot}\,{\bf E}=0$ (the
electric field is irrotational), the magnetic field does not depend on time
and space, directed along $z$, ${\bf B}(x,t)=(0,0,B_{0})$, $B_{0}\equiv{\rm
const}$. Davidson’s model as a non-strictly hyperbolic system with
$Q=\left(\begin{array}[]{ccc}-q&-B_{0}&-1\\\ B_{0}&-q&0\\\
1&0&0\end{array}\right).$

For non-collisional case $q=0$ the following results were obtained in [22]:

  * •

A criterion for a singularity formation:

###### Theorem 4

For the existence of a $C^{1}$ – smooth $\frac{2\pi}{\sqrt{1+B_{0}^{2}}}$ \-
periodic solution $(V_{1},V_{2},E)$ it should be

$\displaystyle\left((V_{1}^{0})^{\prime}\right)^{2}+2\,(E^{0})^{\prime}+2B_{0}\,(V_{2}^{0})^{\prime}-B_{0}^{2}-1<0.$

at any point $x\in\mathbb{R}$.

  * •

The increasing of $|B_{0}|$ basically leads to an expansion of the class of
initial data providing the global smoothness.

  * •

The traveling waves exist (the explicit formula).

For collisional case $q>0$ the following results were obtained in [11]:

  * •

A criterion for maintaining the global smoothness of the solution in terms of
the initial data and the coefficients $B_{0}$ and $q$.

  * •

The initial data are divided into two classes: one leads to stabilization to
the equilibrium, and the other leads to the destruction of the solution in a
finite time.

  * •

For small $|B_{0}|$ an increase in the intensity factor first leads to a
change in the oscillatory behavior of the solution to monotonic damping, which
is then again replaced by oscillatory damping.

  * •

At large values of $|B_{0}|$, the solution is characterized by oscillatory
damping regardless of the value of the intensity factor $q$.

  * •

Both the presence of an external magnetic field of strength $B_{0}$ and a
sufficiently large collisional factor $q$ help to suppress the formation of a
finite-dimensional singularity.

##  11 DISCUSSION

This paper is a survey of methods that can be applied to nonstrictly
hyperbolic systems of a particular type. Initially, such systems arose in the
study of equations simulating cold plasma, but later it turned out that there
are many other models of this type. We tried to mention all that we know
today. For some of these models, in particular those describing cold plasmas,
the behavior of smooth solutions (or solutions up to the moment of loss of
smoothness) for the inviscid case has been studied quite well; we provide
references to the corresponding works. However, the study of generalized
solutions even for the simplest models is a completely open question. The same
applies to the study of the method of vanishing viscosity, and in general the
influence of the viscosity matrix on the behavior of the solution. Therefore,
we can consider that the paper contains a program of further research, some
parts of which may turn out to be very difficult.

##  12 ACKNOWLEDGMENTS

Supported by the Moscow Center for Fundamental and Applied Mathematics under
the agreement N 075-15-2019-1621.

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