#  On the properties of multidimensional electrostatic oscillations of an
electron plasma

Olga S. Rozanova  Mathematics and Mechanics Department, Lomonosov Moscow State
University, Leninskie Gory, Moscow, 119991, Russian Federation,
<EMAIL_ADDRESS>
###### Abstract.

We consider the classical Cauchy problem for a system of equations describing
3D arbitrary electrostatic oscillations of the cold plasma and introduce an
iteration procedure that allows estimating the blow-up time from below. This
procedure is constructive provided one succeed to obtain a two-sided estimate
of an additional quantity depending on the solution. We show that this is
possible in the case of one and two dimensions, as well as for solutions with
zero vorticity. For the particular case of two-dimensional initial data with
the radial symmetry, refined sufficient conditions for destruction and
preservation of smoothness in the first period of oscillations are obtained.
Moreover, we give the example of estimating the blow-up time for the data for
which results of numerics exist and discuss a roughness of our estimate.

###### Key words and phrases:

Quasilinear hyperbolic system, cold plasma, electrostatic oscillations, blow
up

###### 1991 Mathematics Subject Classification:

Primary 35Q60; Secondary 35L60, 35L67, 34M10

##  1\. Introduction

The equations of hydrodynamics of ”cold” or electron plasma in the non-
relativistic approximation in dimensionless quantities take the form(see,
e.g., [1], [8])

(1) $\displaystyle\displaystyle{\partial n\over\partial t}+\mbox{div}\,(n{\bf
V})=0,\quad\displaystyle{\partial{\bf V}\over\partial t}+\left({\bf
V}\cdot\nabla\right){\bf V}=\,-{\bf E}-\left[{\bf V}\times{\bf B}\right],$ (2)
$\displaystyle\frac{\partial{\bf E}}{\partial t}=n{\bf V}+{\rm rot}\,{\bf
B},\qquad\frac{\partial{\bf B}}{\partial t}=-{\rm rot}\,{\bf
E},\qquad\mbox{div}\,{\bf B}=0,$

$n$ and ${\bf V}=(V_{1},V_{2},V_{3})$ are the density and velocity of
electrons, ${\bf E}=(E_{1},E_{2},E_{3})$ and ${\bf B}=(B_{1},B_{2},B_{3})$ are
vectors of electric and magnetic fields. All components of solution depends on
$t\in{\mathbb{R}}_{+}$ and $x\in{\mathbb{R}}^{3}$.

At present, much attention is paid to the study of cold plasma in connection
with the possibility of accelerating electrons in the wake wave of a powerful
laser pulse [7]; nevertheless, there are very few theoretical results in this
area.

It is commonly known that the plasma oscillations described by (1), (2), tend
to break. Mathematically, the breaking process means a blow-up of the
solution, and the appearance of a delta-shape singularity of the electron
density [6]. Among main interests is a study of possibility of existence of a
smooth solution as long as possible.

Let us assume that the oscillations are electrostatic, i.e. ${\rm rot}\,{\bf
E}$ and the magnetic field ${\bf B}$ does not change with time. For
simplicity, we set ${\bf B}=0$.

From the first equations of (1) and (2) under the assumption that the solution
is sufficiently smooth and that the steady-state density is equal to 1, it
follows

(3) $\displaystyle n=1-\mbox{div}\,{\bf E},$

thus, $n$ can be removed from the system. Thus, we get

(4) $\displaystyle\displaystyle{\partial{\bf V}\over\partial t}+\left({\bf
V}\cdot\nabla\right){\bf V}=\,-{\bf E},\quad\frac{\partial{\bf E}}{\partial
t}+{\bf V}\mbox{div}\,{\bf E}={\bf V},$

together with

(5) $\mbox{curl}\,{\bf E}=0,\quad\mbox{curl}\,((1-\mbox{div}\,{\bf E}){\bf
V})=0.$

Consider the initial data

(6) $({\bf V},{\bf E})|_{t=0}=({\bf V}_{0},{\bf E}_{0})(x)\in
C^{2}({\mathbb{R}}^{3}),\quad x\in{\mathbb{R}}^{3}.$

with properties (5).

######  Definition 1.

We will say that the solution to the problem (4), (6) does not blow up (or the
oscillations do not break) on $[0,T),\,T>0,$ if the density $n$, found as (3)
remains bounded for all $t\in T.$

The main problem is that the hyperbolic system (4), (6) has locally in time a
smooth solution [5], however, it not necessary satisfies (5). In the general
case system (4), (5) is overdetermined.

Nevertheless, as it will be shown below, there exist important classes of
solutions such that (5) automatically hold. It is affine solutions and
radially symmetric solutions (see Sec.3.2). Therefore below we assume  that we
deal only with solutions with property (5).

We denote ${\mathcal{D}}=\mbox{div}\,{\bf V}$,
${\bf\Xi}=(\xi_{1},\xi_{2},\xi_{3})=\mbox{curl}\,{\bf V}$,
$\lambda=\mbox{div}\,{\bf E}$, $J_{1}=\det(\|\partial_{x_{i}}{V_{j}}\|),$
$i,j=1,2$, $J_{2}=\det(\|\partial_{x_{i}}{V_{j}}\|),$ $i,j=2,3$,
$J_{3}=\det(\|\partial_{x_{i}}{V_{j}}\|),$ $i,j=1,3$, $J=J_{1}+J_{2}+J_{3}.$

System (4), (5) implies

$\displaystyle\displaystyle{\partial{\mathcal{D}}\over\partial t}$
$\displaystyle+$ $\displaystyle({\bf
V}\cdot\nabla{\mathcal{D}})=-{\mathcal{D}}^{2}+2J-\lambda,\qquad\displaystyle{\partial\lambda\over\partial
t}+({\bf V}\cdot\nabla\lambda)={\mathcal{D}}(1-\lambda),$

Thus, along the characteristics $\displaystyle{dx_{i}\over dt}=V_{i},\,i=1,2,$
starting from the point $x_{0}$ we obtain the Cauchy problem for the nonlinear
system of three ODEs (non-closed due to $J$):

(7)
$\displaystyle\dot{\mathcal{D}}=-{\mathcal{D}}^{2}+2J-\lambda,\quad\dot{\lambda}={\mathcal{D}}(1-\lambda),$
$\displaystyle({\mathcal{D}},\lambda(t,x_{0})|_{t=0}=({\mathcal{D}}_{0},\lambda_{0}(x_{0}).$

######  Definition 2.

We will say that the breaking of oscillations does not occur in the first
period for the initial data (6), if the projection of each phase trajectory
starting at $(\lambda_{0},{\mathcal{D}}_{0})=$ $({\rm div}{\bf
E}_{0}(x_{0}),{\rm div}{\bf V}_{0}(x_{0}))$, $x_{0}\in{\mathbb{R}}^{3}$ on the
plane $(\lambda,\mathcal{D})$, intersects the semi-axis ${\mathcal{D}}=0$,
$\lambda<0$ at least at one point (the starting point of the trajectory does
not count).

In other words, all trajectories make at least one revolution around the
coordinate origin on the plane $(\lambda,\mathcal{D})$.

In Sec.2 we show that if a two-sided estimate

$J_{-}(\lambda,\mathcal{D}^{2})\leq J\leq J_{+}(\lambda,\mathcal{D}^{2})$

is known, it is possible to construct an iterative procedure that allows one
to find the guaranteed number of oscillations before the blow-up (Sec.2.3) and
to estimate from below the lifetime of a solution with bounded density
(Sec.2.4). In Sec.3 we consider particular cases of the problem (4), (6) and
show that in some cases the solution remains smooth for all $t>0$. In Sec.4 is
dedicated to the cases, where the estimate of $J$ can be obtained. For the
case of plain oscillations (Sec.4.1) and the case of irrotational oscillations
in any dimensions (Sec.4.2), we prove a sufficient condition for the
boundedness of the density in the first period of oscillations. In Sec.5 we
construct two-sided estimates of the projection of the integral trajectory to
the plane $(\lambda,{\mathcal{D}})$ for the case of radially symmetric
oscillations in any dimensions. In Sec.6 we consider the initial data in the
form of a standard laser pulse corresponding to the 2D (plain) axisymmetric
solution and prove refined estimates allowing us to obtain, in particular,
sufficient conditions for the preservation of smoothness by the solution to
the Cauchy problem on several periods of oscillations and compare it with the
results of direct numerics. In Sec.7, we summarize the results obtained and
discuss further hypotheses and open problems.

##  2\. Method for obtaining a sufficient conditions for boundedness of
density

We denote for convenience $s=\lambda-1\leq 0$ (the inequality follows from
(3), since $n\geq 0$). Thus, we get

$\displaystyle\frac{d\mathcal{D}}{ds}=\frac{{\mathcal{D}}^{2}+2s+2J+1}{{\mathcal{D}}s}.$

Further, replacing $Z={\mathcal{D}}^{2}\geq 0$, we obtain

(8) $\displaystyle\frac{dZ}{ds}=\frac{2(Z+s+1)}{s}+\frac{4J}{s}\equiv
Q(s,Z,J).$

Let us assume that the following inequality is known:

(9) $\displaystyle 2J_{-}(Z,s)\leq 2J\leq 2J_{+}(Z,s).$

Then, taking into account the sign of $s$, we obtain the estimate:

(10) $\displaystyle Q_{1}(s,Z)\leq Q(s,Z,J)\leq Q_{2}(s,Z),$

where

(11) $\displaystyle Q_{1}(s,Z)=\frac{2(Z+s+2J_{+}(s,Z)+1)}{s},$ (12)
$\displaystyle Q_{2}(s,Z)=\frac{2(Z+s+2J_{-}(s,Z)+1)}{s}.$

Now we can apply Chaplygin’s theorem on differential inequalities, according
to which the solution $Z(s)$ of the Cauchy problem for (8) with initial
conditions $Z(s_{0})=Z_{0}$ for $s>s_{0}$ satisfies the inequality

(13) $\displaystyle Z_{1}(s)\leq Z(s,J)\leq Z_{2}(s),$

and for $s<s_{0}$ the inverse inequality

(14) $\displaystyle Z_{2}(s)\leq Z(s,J)\leq Z_{1}(s),$

where $Z_{k}(s)$ are the solutions to problems $\frac{dZ}{ds}=Q_{k}(s,Z)$,
$Z(s_{0})=Z_{0}$, $k=1,2.$

Since $Z\geq 0,$ then the phase trajectory outgoing from the point
$(s_{0},Z_{0})$, is bounded, if the solution $Z_{1}(s)\to-\infty$ as
$s\to-\infty$. If $Z_{2}(s)\to+\infty$ as $s\to-\infty$, without becoming
zero, then the phase trajectory outgoing from this point is unbounded.

Thus, to obtain sufficient conditions for the global in time boundedness of
the density and sufficient blow-up conditions, it is enough for us to solve
the equations $\frac{dZ}{ds}=Q_{1}(s,Z)$ and $\frac{dZ}{ds}=Q_{2}(s,Z)$.

First of all, note that on the plane $(s,{\mathcal{D}})$ the motion is
clockwise (for positive $\mathcal{D}$, the value of $s$ increases). Assume
that the equation $Z_{2}(s)=0$ has two roots on the semiaxis $s<0$, we denote
them $S_{-}$ and $S_{+}$, $S_{-}\leq S_{+}$, or one root $S_{+}$.

In the latter case $Z_{1}(s)\to+\infty$ for $s\to-\infty$.

If the equation $Z_{2}(s)=0$ has two roots on the semiaxis $s<0$, we denote
them $s_{-}$ and $s_{+}$, $s_{-}\leq s_{+}$.

The trajectory ${\mathcal{D}}(s,t)$ lies between the curves $\sqrt{Z_{2}(s)}$
and $\sqrt{Z_{1}(s)}$. Denote by $s^{k}_{*}$ the point at which
${\mathcal{D}}(s,t)$ intersects the axis ${\mathcal{D}}=0$ for the $k$-th
time. If ${\mathcal{D}}>0$, then
$\sqrt{Z_{1}(s)}\leq{\mathcal{D}}(s,t)\leq\sqrt{Z_{2}(s)}$, $s$ increases. If
${\mathcal{D}}<0$, then
$-\sqrt{Z_{1}(s)}\leq{\mathcal{D}}(s,t)\leq-\sqrt{Z_{2}(s)}$, $s$ decreases.

Let us compose the curve $L$ as follows: it consists of the part of trajectory
$\sqrt{Z_{2}(s)}$ if ${\mathcal{D}}>0$ and switches to the part of trajectory
$-\sqrt{Z_{1}(s)}$ if ${\mathcal{D}}<0$ at the axis ${\mathcal{D}}=0$. If
$-\sqrt{Z_{1}}(s)=0$ has two roots at $s<0$, then $L$ switches once again the
the axis ${\mathcal{D}}=0$ to the trajectory $\sqrt{Z_{2}(s)}$, and so on.
Denote by $L^{k}$ the point at which $L$ intersects the axis ${\mathcal{D}}=0$
for the $k$-th time for ${\mathcal{D}}_{0}<0$ and in the $k-1$-th time for
${\mathcal{D}}_{0}\geq 0$. Thus, the trajectory ${\mathcal{D}}(s,t)$ makes at
least one revolution if the number of points $L^{k}$ is more then 2
($L^{2},L^{3},\dots$) for ${\mathcal{D}}_{0}\geq 0$ and more then 1
($L^{1}\dots$) for ${\mathcal{D}}_{0}<0$.

In its turn, we compose the curve $l$ as follows: it consists of the part of
trajectory $\sqrt{Z_{1}(s)}$ if ${\mathcal{D}}>0$, switches to the part of
trajectory $-\sqrt{Z_{2}(s)}$ if ${\mathcal{D}}<0$ at the axis
${\mathcal{D}}=0$ and switches once again the the axis ${\mathcal{D}}=0$ to
the trajectory $\sqrt{Z_{1}(s)}$, and so on. Denote by $l^{k}$ the point at
which $l$ intersects the axis ${\mathcal{D}}=0$ for the $k$-th time for
${\mathcal{D}}_{0}<0$ and in the $k-1$-th time for ${\mathcal{D}}_{0}\geq 0$.

We note that $s_{*}^{k}$ for all $k\in\mathbb{N}$ lies between $l^{k}$ and
$L^{k}$.

###  2.1. General estimates of $J$

######  Lemma 2.1.

(15) $-|\tilde{\bf\Xi}|^{2}-\tilde{\mathcal{D}}^{2}\leq
2J\leq{\mathcal{D}}^{2}+|{\bf\Xi}|^{2},$

where $\tilde{\bf\Xi}$ and $\tilde{\mathcal{D}}$ are defined as ${\bf\Xi}$ and
${\mathcal{D}}$ with the change of all derivatives $\partial_{i}V_{j}$ to
$|\partial_{i}V_{j}|$.

###### Proof.

The estimate (15) is a corollary of the elementary inequality $-(a-b)^{2}\leq
2ab\leq(a+b)^{2},$ for every $a,b\in\mathbb{R}$. $\Box$

Estimate (15) does not yet provide an opportunity to find explicit bounds, but
we will use it for particular cases.

###  2.2. Behavior in the upper hyperplane ${\mathcal{D}}>0$

Now we are going to prove that if ${\mathcal{D}}(t)$ becomes unbounded, it
happens when ${\mathcal{D}}<0.$

######  Lemma 2.2.

The projection of the trajectory ${\mathcal{D}}={\mathcal{D}}(s,t)$ to the
plane $(s,{\mathcal{D}})$ is bounded in the upper hyperplane
${\mathcal{D}}>0$.

###### Proof.

As follows from the second equation of (7), the motion along the projection
${\mathcal{D}}={\mathcal{D}}(s,t)$ onto the plane $(s,{\mathcal{D}})$ is
clockwise for $s<0$. Therefore, in the upper hyperplane ${\mathcal{D}}>0$
there are two possibilities for a trajectory starting from any point
$(s_{0},{\mathcal{D}}_{0}>0)$:

1\. to cross the axis ${\mathcal{D}}=0$ and fall into the lower hyperplane,
which means that the projection of the phase trajectory ${\mathcal{D}}(s,t)$
was bounded for ${\mathcal{D}}>0$,

or

2\. to tend to $+\infty$ as $s\to-0$ and $t\to t_{*}\leq\infty$. (This would
mean that a vacuum point is formed along the characteristic).

Assume that the latter possibility realizes, i.e. ${\mathcal{D}}(t)\to+\infty$
and $s(t)\to-0$ as $t\to t_{*}\leq\infty$. We see from (7) that
$\dot{\mathcal{D}}=-{\mathcal{D}}^{2}+2J-s-1$, therefore $J\to+\infty$, if
$t\to t_{*}$ and $s(t)\to-0$. Therefore $J>K>0$ for sufficiently small $s<0$.
Then (12) implies

$\displaystyle\frac{dZ}{ds}\leq Q_{2}(s,Z)=\frac{2(Z+s+2K+1)}{s},$

$Z_{2}(s)={\mathcal{D}}^{2}\leq-2K-1-2s+Cs^{2}$ and $Z_{2}(0)\leq-2K-1<0$.
This contradiction shows that our assumption was wrong and ${\mathcal{D}}>0$
is bounded. $\Box$

###  2.3. Counting the number of periods for which the boundedness of density
is guaranteed

Based on estimates (13), (14) we can find the number of periods for which the
breaking is guaranteed not to occur.

Let, for definiteness, ${\mathcal{D}}_{0}$ be negative. Assume that the
initial data are such that the solution does not blow up and on the curve $L$
there exist points $(s=L_{1},{\mathcal{D}}=0)$ and
$(s=L_{2},{\mathcal{D}}=0)$, $L_{1}<L_{2}$. However, we can use $(L_{2},0)$ as
initial data and check whether the equation $Z_{1}(s)=0$ has the second root
($L_{3}$) at $s<0$ besides $L_{2}$. This would mean that there $L$ (and
${\mathcal{D}}(t,s)$) makes at least two revolutions and at the axis
${\mathcal{D}}=0$ there exist two points of intersection $s=L_{3}$ and
$s=L_{4}$, $L_{3}<L_{4}$, where $L_{3}$ and $l_{4}$ are the roots of equation
$Z_{2}(s)=0$. Then we can use $(L_{4},0)$ as initial data again and so on. We
can continue this procedure until at some step $n$ for the point $(L_{2n},0)$,
taken as initial data, we do not obtain a sufficient condition for boundedness
of ${\mathcal{D}}$. This means that we can guarantee $n$ revolutions of the
trajectory ${\mathcal{D}}(t,s)$. The iterative process of counting revolutions
is easy to implement numerically, if we know $Z_{1}$ and $Z_{2}$ (see the
example in Sec.6.3. Nevertheless, the estimate of numbers of revolutions from
below may be rough.

###  2.4. Estimates of the guaranteed time of existence of the bounded density

Let $\mbox{div}\,{\bf V}_{0}<0$ and for the characteristic starting from the
point $x_{0}$ we can guarantee $n$ revolutions of the trajectory
${\mathcal{D}}(t,s)$ around the origin. Then the guaranteed lifetime of a
solution with bounded density, denoted as $T(x_{0})$, can be estimated as

$T_{l}(x_{0})<T(x_{0})<T_{L}(x_{0}),$

where $T_{l}$ and $T_{L}$ is the time of passage of $n$ turns along a compound
spiral lines $l$ and $L$, respectively. Thus,

$\displaystyle
T_{l}=-\int\limits_{s_{0}}^{l^{1}}\frac{ds}{s\sqrt{Z_{2}(0;s)}}-\int\limits_{l^{1}}^{l^{2}}\frac{ds}{s\sqrt{Z_{1}(1;s)}}-\int\limits_{l^{2}}^{l_{3}}\frac{ds}{s\sqrt{Z_{2}(2;s)}}-\cdots-\int\limits_{l^{2n}}^{l_{2n+1}}\frac{ds}{s\sqrt{Z_{2}(n;s)}},$
$\displaystyle
T_{L}=-\int\limits_{s_{0}}^{L^{1}}\frac{ds}{s\sqrt{Z_{1}(0;s)}}-\int\limits_{L^{1}}^{L^{2}}\frac{ds}{s\sqrt{Z_{2}(1;s)}}-\int\limits_{L^{2}}^{L_{3}}\frac{ds}{s\sqrt{Z_{1}(2;s)}}-\cdots-\int\limits_{L^{2n}}^{s_{0}}\frac{ds}{s\sqrt{Z_{1}(n;s)}},$

where we denoted $Z_{i}(k;s)$, $i=1,2$, $k=0,1,\dots,n$, the functions
$Z_{1}(s)$ and $Z_{2}(s)$, based on initial data at points $(L_{2k},0)$ for
$Z_{1}(k,s)$ and $(L_{2k-1},0)$ for $Z_{2}(k,s)$.

For $\mbox{div}\,{\bf V}_{0}\geq 0$ the formulas are similar, except for the
sum $T_{l}$ begins from
$-\int\limits_{s_{0}}^{l^{2}}\frac{ds}{s\sqrt{Z_{2}(0;s)}}$ and the sum
$T_{L}$ begins from
$-\int\limits_{s_{0}}^{L^{2}}\frac{ds}{s\sqrt{Z_{1}(0;s)}}$.

The time $T_{*}$ of existence of the smooth solution with a bounded density
can be estimated as

$T_{*}>\inf\limits_{x_{0}\in\mathbb{R}}T_{l}(x_{0}),$

however $\sup\limits_{x_{0}\in\mathbb{R}}T_{L}(x_{0})$ can be less than
$T_{*}$.

######  Remark 2.1.

The technique of counting the number of oscillations before the blow-up in the
case of multidimensional electrostatic non-relativistic oscillations is
similar to the case of 1D relativistic oscillations [10].

##  3\. Particular cases

###  3.1. 1D oscillations, a criterion of a singularity formation

In this case $({\bf V},{\bf E})=({\bf V}(x_{1}),{\bf E}(x_{1}))$,
$V_{2}=E_{2}=0$, ${\bf\Xi}=0$, condition (5) evidently holds. It is easy to
see that here $J=0$ and system (7) turns into

$\displaystyle\dot{\lambda}={\mathcal{D}}(1-\lambda),\quad\dot{\mathcal{D}}=-{\mathcal{D}}^{2}-\lambda.$

Such a system has been considered in [9], where the following criterion for
the preservation of the global in time smoothness was obtained: at each point
$x_{0}\in{\mathbb{R}}$ (here $x=x_{1}$) the condition

(16) $\displaystyle{\Delta=(V_{10}^{\prime})^{2}+2(E_{10})^{\prime}-1<0.}$

holds.

###  3.2. Globally smooth affine solutions

######  Definition 3.

A solution ${\bf V},{\bf E}$ is called an affine solution if it has the form

(17) ${\bf V}={\mathcal{Q}}(t){\bf r},\quad{\bf E}={\mathcal{R}}(t){\bf r},$

with $(3\times 3)$ matrices ${\mathcal{Q}}=(q_{ij})$ and
${\mathcal{R}}=(r_{ij})$.

Condition (5) dictates $q_{ij}=q_{ji}$, $r_{ij}=r_{ji}$, so we get the matrix
equation

$\dot{\mathcal{Q}}+{\mathcal{Q}}^{2}+{\mathcal{R}}=0,\quad\dot{\mathcal{R}}=(1-{\rm
tr}{\mathcal{R}}){\mathcal{Q}}$

or quadratically nonlinear system of 12 ODEs.

For the plain oscillations $q_{3j}=q_{i3}=r_{3j}=r_{i3}=0$, $i,j=1,2$, so the
number of equations is 6.

######  Definition 4.

A solution ${\bf V},{\bf E}$ is called radially symmetric if

${\bf V}=F(t,r){\bf r},\,{\bf E}=G(t,r){\bf r},\,{\bf
r}=(x_{1},x_{2},x_{3}),\,r=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}.$

It is easy to check that for the affine solutions and radially symmetric
solutions $\mbox{curl}\,{\bf V}=0$ and condition (5) holds.

Let us consider solutions that are both affine and radially symmetric, i.e
$F=\alpha(t)$, $G=\beta(t)$. Here ${\mathcal{D}}=3\alpha$, $J=3\alpha^{2}$,
$\lambda=3\beta$, and the system (7) takes a closed form

$\displaystyle\dot{\alpha}=-{\alpha}^{2}-\beta,\quad\dot{\beta}=\alpha(1-3\beta),$

the first integral is

$\displaystyle\alpha^{2}=2\beta-1+K|1-3\beta|^{\frac{2}{3}},\quad
K=\frac{1-2\beta(0)+\alpha^{2}(0)}{|1-3\beta(0)|^{\frac{2}{3}}}.$

Recall that the positivity of density requires $1-3\beta>0,$ see (3). The
prevailing term of the right hand side as $\beta\to-\infty$ is $2\beta$, it
tends to $-\infty$ for any $K$. This means that $\alpha$ and $\beta$ are
bounded for any initial data, the density is bounded and the smooth in time
solution to the Cauchy problem exists for all radially symmetric initial data.

In the case of plane oscillations we have

${\bf V}=F(t,r){\bf r},\,{\bf E}=G(t,r){\bf r},\,{\bf
r}=(x_{1},x_{2},0),\,r=\sqrt{x_{1}^{2}+x_{2}^{2}}.$

For the corresponding affine solution ${\mathcal{D}}=2\alpha$, $J=\alpha^{2}$,
$\lambda=2\beta$, therefore system (7) takes a closed form

$\displaystyle\dot{\alpha}=-{\alpha}^{2}-\beta,\quad\dot{\beta}=\alpha(1-2\beta),$

the first integral is

$\displaystyle 2\alpha^{2}=-(1-2\beta)\ln|1-2\beta|+K(1-2\beta)+1,\quad
K=\frac{1-2\alpha^{2}(0)}{1-2\beta(0)}-\ln|1-2\beta(0)|.$

From the positivity of density we have $1-2\beta>0,$ therefore the prevailing
term of the right hand side as $\beta\to-\infty$ is the logarithmic one, it
tends to $-\infty$ for any $K$. This means that $\alpha$ and $\beta$ are
bounded for any initial data, the density is bounded and the smooth in time
solution to the Cauchy problem exists for all radially symmetric initial data
(see also [2], [11]).

These examples show, in particular, that there are three-dimensional initial
data that do not lead a finite time blow-up. However, as was proved in [11],
radially symmetric solutions of the cold plasma equation, which are not
affine, in the general case blow up even if they are arbitrarily small
perturbations of the zero stationary state.

##  4\. The plain and irrotational oscillations

In this section we specify two cases, where it is possible to obtain an
estimate giving a possibility to study the behavior of the trajectory in the
lower hyperplane.

###  4.1. Plain oscillations

For the case of plain oscillations ${\bf V}={\bf V}(x_{1},x_{2})$, ${\bf
E}={\bf E}(x_{1},x_{2})$, $V_{3}=E_{3}=0$, $\xi_{1}=\xi_{2}=0$, and we have

$\displaystyle\displaystyle{\partial\xi_{3}\over\partial t}$ $\displaystyle+$
$\displaystyle({\bf V}\cdot\nabla)\xi_{3}=-{\mathcal{D}}\xi_{3}.$

Along characteristics this equations can be written as
$\dot{\xi_{3}}=-{\mathcal{D}}{\xi_{3}}$, therefore (7) implies

(18) $\displaystyle\xi_{3}=C_{3}(\lambda-1),\quad C_{3}={\rm const}.$

Thus, from (15) we have

$\displaystyle 2J\leq{\mathcal{D}}^{2}+C^{2}_{3}s^{2},$

and (11) takes the form

$\displaystyle Q_{1}(s,Z)=\frac{2(2Z+s+C_{3}^{2}s^{2}+1)}{s}.$

Therefore

$\displaystyle Z_{1}(s)=A_{4}s^{4}-A_{2}s^{2}+A_{1}s+A_{0},$

where

$\displaystyle
A_{4}=\frac{Z_{0}+{\xi}_{30}^{2}+\frac{2}{3}s_{0}+\frac{1}{2}}{s_{0}^{4}},\quad
A_{2}=-\frac{|{\xi}_{30}|^{2}}{s_{0}^{2}},\quad A_{1}=-\frac{2}{3},\quad
A_{0}=-\frac{1}{2},$

the coefficients are substituted with the value of the constant $C_{3}$, found
from (18).

###  4.2. Irrotational oscillations

The case of irrotational oscillations is analogous. Since

$\displaystyle\displaystyle{\partial\xi_{i}\over\partial t}$ $\displaystyle+$
$\displaystyle({\bf
V}\cdot\nabla)\xi_{i}=-{\mathcal{D}}\xi_{i}-({\bf\Xi}\cdot\nabla)V_{i},\quad
i=1,2,3,$

then the uniqueness of the solution of the Cauchy problem implies that if
${\bf\Xi}_{0}=0$, then ${\bf\Xi}=0$ identically. Therefore, (9) gives

$\displaystyle 2J\leq{\mathcal{D}}^{2}$

and (11) is

(19) $\displaystyle Q_{1}(s,Z)=\frac{2(2Z+s+1)}{s}.$

Thus,

$\displaystyle Z_{1}(s)=A_{4}s^{4}+A_{1}s+A_{0},$

where

$\displaystyle
A_{4}=\frac{Z_{0}+\frac{2}{3}s_{0}+\frac{1}{2}}{s_{0}^{4}},\quad
A_{1}=-\frac{2}{3},\quad A_{0}=-\frac{1}{2}.$

For the both case the following lemma holds.

######  Lemma 4.1.

Let at a point $x_{0}\in{\mathbb{R}}^{3}$ the initial conditions (6) are such
that ${\bf V}_{0}=(V_{1}(x_{1},x_{2}),V_{2}(x_{1},x_{2}),0),$ ${\bf
E}_{0}=(E_{1}(x_{1},x_{2}),E_{2}(x_{1},x_{2}),0),$ or ${\rm curl}{\bf
V}_{0}=0$ and

(20) $\displaystyle{\rm div}{\bf V}_{0}<0,\quad{\Delta_{-}=({\rm div}{\bf
V}_{0})^{2}+|{\rm curl}{\bf V}_{0}|^{2}+\frac{2}{3}{\rm div}{\bf
E}_{0}-\frac{1}{6}<0,}$

or

(21) $\displaystyle{\rm div}{\bf V}_{0}=0,\quad{\rm div}{\bf
E}_{0}>0,\quad{\Delta_{-}=|{\rm curl}{\bf V}_{0}|^{2}+\frac{2}{3}{\rm div}{\bf
E}_{0}-\frac{1}{6}<0}.$

Then there exists a moment $t_{*}$ such that the trajectory $(s,\mathcal{D})$
turns out in the upper hyperplane ${\mathcal{D}}<0$ (a breaking of
oscillations does not occur in the first period).

###### Proof.

The sense of Lemma 2.1 is such that the trajectory ${\mathcal{D}}(s,t)$ does
not go to infinity during the first oscillation in the lower hyperplane
${\mathcal{D}}<0$. As follows from Sec.2, the trajectory in the lower
hyperplane ${\mathcal{D}}<0$ is bounded from below by the curve
$-\sqrt{Z_{2}(s)}$. In its turn, ${Z_{2}(s)}$ is bounded if and only if
$A_{4}<0$, where $A_{4}$ is defined in (4.1). The condition $A_{4}<0$ is
exactly (20). Further, since the motion along the phase trajectory is
clockwise, then (21) signifies that the trajectory will occur in the lower
hyperplane with the condition (20) at moment $t>0$. $\Box$

Thus, as a corollary of Lemmas 2.2 and 4.1, we get the following theorem.

######  Theorem 4.1.

Let at each point $x_{0}\in{\mathbb{R}}^{3}$ the initial conditions (6) are
such that ${\bf V}_{0}=(V_{1}(x_{1},x_{2}),V_{2}(x_{1},x_{2}),0),$ ${\bf
E}_{0}=(E_{1}(x_{1},x_{2}),E_{2}(x_{1},x_{2}),0),$ or ${\rm rot}{\bf V}_{0}=0$
and one of conditions (20) and (21) holds. Then the density $n$, found as (3)
from the solution of the problem (4), (6), is bounded during the first period
of oscillations.

###### Proof.

According to conditions (20) or (21), every trajectory
${\mathcal{D}}={\mathcal{D}}(s,t)$ starts from a point do not laying in the
upper hyperplane and turns out in the upper hyperplane (Lemma 4.1) and then
again turns out in the lower hyperplane (possibly, at the point, do not
satisfying to (20) or (21)) (Lemma 2.2). However, the trajectories starting
from each point $x_{0}\in{\mathbb{R}}^{3}$ make as least one full rotation
around the origin $\lambda={\mathcal{D}}=0$, therefore ${\mathcal{D}}$ is
bounded during the first period of oscillations, and (3) implies that $n$ is
also bounded. $\Box$

###  4.3. Affine solutions, an example

For the affine solutions (17) Theorem 4.1 gives the following sufficient
conditions for maintaining smoothness in the first period of oscillations:

${\rm tr}\,Q(0)<0,\quad\Delta_{-}=({\rm tr}\,Q(0))^{2}+\frac{2}{3}{\rm
tr}\,R(0)-\frac{1}{6}<0,$

or

(22) ${\rm tr}\,Q(0)=0,\quad{\rm tr}\,R(0)>0,\quad\Delta_{-}=4\,{\rm
tr}\,R(0)-1<0.$

For the axisymmetric 3D case, (22) implies $\beta<\frac{1}{12}$, whereas as is
shown in Sec.3.2 the solution is globally smooth for $\beta<\frac{1}{3}$.

For the axisymmetric 2D and 1D cases, (22) gives $\beta<\frac{1}{8}$ and
$\beta<\frac{1}{4}$, respectively. However, the solution is globally smooth
for $\beta<\frac{1}{2}$ for both these cases.

##  5\. Radially symmetric oscillations in any dimensions

For the radially symmetric case we can obtain a two-sided estimate (9) and
obtain the find the time for which we guarantee the existence of the smooth
solution to the problem (4), (6). This result can be considered as an addition
to the results of the recent paper [11], where we proved that any solution to
the Cauchy problem for ${\bf d}=2$ and ${\bf d}=3$ generally blows up, even
for arbitrary small perturbation to the zero steady state.

The calculations are similar in the space of any dimension ${\bf d}\geq 1$, we
assume

${\bf V}=F(t,r){\bf r},\,{\bf E}=G(t,r){\bf r},\,{\bf
r}=(x_{1},x_{2},\dots,x_{\bf d}),\,r=\sqrt{x_{1}^{2}+x_{2}^{2}+\dots+x_{\bf
d}^{2}}.$

However, we are interested mostly in physical dimension ${\bf d}=3$. For the
plain oscillations (${\bf d}=2$ )

(23) ${\bf V}=F(t,r){\bf r},\,{\bf E}=G(t,r){\bf r},\,{\bf
r}=(x_{1},x_{2},0),\,r=\sqrt{x_{1}^{2}+x_{2}^{2}}.$

It is easy to check that $\mbox{curl}\,{\bf V}=0$ and condition (5) holds.

Further,

$\displaystyle{\mathcal{D}}={\bf d}\,F+F_{r}\,r,\quad\lambda={\bf
d}\,G+G_{r}\,r,\quad J=({\bf d}-1)\,F\,F_{r}\,r+\frac{({\bf d}-1){\bf
d}}{2}\,F^{2},$

therefore

$\displaystyle 2J$ $\displaystyle=$ $\displaystyle 2\,({\bf
d}-1)\,{\mathcal{D}}\,F-({\bf d}-1)\,{\bf d}\,F^{2}.$

Further, if we substitute (23) to (4), we get

(24) $\displaystyle\dot{G}=F-{\bf d}FG,\quad\dot{F}=-F^{2}-G,$

where $\dot{f}=\displaystyle{\partial f\over\partial
t}+Fr\displaystyle{\partial f\over\partial r}$.

On the phase plane $(G,F)$ system (24) implies one equation

$\displaystyle\frac{1}{2}\displaystyle{dF^{2}\over dG}=-\frac{F^{2}+G}{1-{\bf
d}G},$

which is linear with respect to $Y=F^{2}$ and can be explicitly integrated.
Indeed, we have for ${\bf d}=2$

(25) $\displaystyle 2Y=(2G-1)\ln|1-2G|+C_{2}(2G-1)-1,$ $\displaystyle
C_{2}=\frac{1+2F^{2}(0,r_{0})}{2G(0,r_{0})-1}-\ln|1-2G(0,r_{0})|,$

for ${\bf d}=1$ and ${\bf d}\geq 3$

(26) $\displaystyle Y=\frac{2G-1}{{\bf d}-2}+C_{\bf d}|1-{\bf
d}G|^{\frac{2}{{\bf d}}},\quad C_{\bf d}=\frac{1-2G(0,r_{0})+({\bf
d}-2)F^{2}(0,r_{0})}{({\bf d}-2)|1-{\bf d}G(0,r_{0})|^{\frac{2}{{\bf d}}}}.$

If $G(0,r_{0})<\frac{1}{{\bf d}}$, $r_{0}\in\bar{\mathbb{R}}$, then the phase
trajectories of (24) are in the half-plane $G<\frac{1}{{\bf d}}$. In this
half-plane the leading term in the right hand side of (25) is
$(2\,G-1)\ln|1-2\,G|$, and the leading term in the right hand side of (26) is
$\frac{2\,G-1}{{\bf d}-2}$. Therefore $F$ and $G$ are bounded for any $t>0$.

Along every characteristic curve starting from $r_{0}\in\bar{\mathbb{R}}_{+}$
the functions $F(t)$ and $G(t)$, solution of (24), are periodic with the
period

$T=2\int\limits_{G_{-}}^{G_{+}}\frac{d\eta}{(1-{\bf d}\,\eta)F(\eta)},$

$F$ is given as (25) or (26), $G_{-}<0$ and $G_{+}>0$ are the lesser and
greater roots of the equation $F(G)=0$. Moreover,
$\int\limits_{0}^{T}F(\tau)\,d\tau=0$.

We refer to [11] for a detailed analysis of the phase portrait and the period
of oscillations.

Thud, we can obtain the evaluating functions $Q_{1}$ and $Q_{2}$ in (10).

Indeed, the following upper estimate holds:

(27) $\displaystyle 2J\leq 2({\bf d}-1)F\mathcal{D}\leq$
$\displaystyle-\left(\sigma_{1}\mathcal{D}-\frac{({\bf
d}-1)F}{\sigma_{1}}\right)^{2}+\sigma_{1}^{2}\mathcal{D}^{2}+\frac{({\bf
d}-1)^{2}}{\sigma_{1}^{2}}\,F^{2}\leq\sigma_{1}^{2}\mathcal{D}^{2}+\frac{({\bf
d}-1)^{2}}{\sigma_{1}^{2}}\,F_{+}^{2},\quad\sigma_{1}>0.$

The lower estimate is analogous:

(28) $\displaystyle 2J\geq 2({\bf d}-1)F\mathcal{D}-{\bf d}({\bf
d}-1)F_{+}^{2}\geq$ $\displaystyle\left(\sigma_{2}\mathcal{D}+\frac{({\bf
d}-1)F}{\sigma_{2}}\right)^{2}-\sigma_{2}^{2}\mathcal{D}^{2}-\frac{({\bf
d}-1)^{2}F^{2}}{\sigma_{2}^{2}}-{\bf d}({\bf d}-1)F_{+}^{2}\geq$
$\displaystyle-\sigma_{2}^{2}\mathcal{D}^{2}-\frac{({\bf d}-1)({\bf
d}(\sigma_{2}^{2}+1)-1)}{\sigma_{2}^{2}}\,F_{+}^{2},\quad\sigma_{2}>0.$

The parameters $\sigma_{i}$, $i=1,2$, in both estimates can be different and
are chosen at our convenience.

Thus, taking into account (27) and (28), we getestimating functions

$\displaystyle\bar{Q}_{1}(s,Z)=\frac{2((1+\sigma_{1}^{2})Z+s+\frac{({\bf
d}-1)^{2}F_{+}^{2}}{\sigma_{1}^{2}}+1)}{s},$
$\displaystyle\bar{Q}_{2}(s,Z)=\frac{2((1-\sigma_{2}^{2})Z+s-\frac{({\bf
d}-1)({\bf d}(\sigma_{2}^{2}+1)-1)F_{+}^{2}}{\sigma_{2}^{2}}+1)}{s}.$

The equations $\frac{dZ}{ds}=\bar{Q}_{1}(s,Z)$ and
$\frac{dZ}{ds}=\bar{Q}_{2}(s,Z)$ can be solved, their solutions are

(29) $\displaystyle\bar{Z}_{1}(s)=-\frac{2s}{1+2\sigma_{1}^{2}}-\frac{({\bf
d}-1)^{2}F_{+}^{2}+\sigma_{1}^{2}}{\sigma_{1}^{2}(1+\sigma_{1}^{2})}+C_{1}\,s^{2(1+\sigma_{1}^{2})},$
(30) $\displaystyle
C_{1}=s_{0}^{-2(1+\sigma_{1}^{2})}\left(Z(0)+\frac{2\sigma_{1}^{2}(1+\sigma_{1}^{2})s_{0}+(1+2\sigma_{1}^{2})(({\bf
d}-1)^{2}F_{+}^{2}+\sigma_{1}^{2})}{\sigma_{1}^{2}(1+2\sigma_{1}^{2})(1+\sigma_{1}^{2})}\right),$

and

(31)
$\displaystyle\bar{Z}_{2}(s)=-\frac{2s}{1-2\sigma_{2}^{2}}+\frac{K-1}{1-\sigma_{2}^{2}}+C_{2}\,s^{2(1-\sigma_{2}^{2})},\quad
K=\frac{({\bf d}-1)({\bf d}(\sigma_{2}^{2}+1)-1)F_{+}^{2}}{\sigma_{2}^{2}},$
(32) $\displaystyle
C_{2}=s_{0}^{-2(1-\sigma_{2}^{2})}\left(Z(0)+\frac{2(s_{0}-1)}{2\sigma_{2}^{2}-1}+\frac{K}{1-\sigma_{2}^{2}}\right).$

Thus, we get

$\bar{Q}_{1}(s,Z)\leq Q(s,Z,J)\leq\bar{Q}_{2}(s,Z),$

The new estimate $\bar{Q}_{1}(s,Z)\leq Q(s,Z,J)$ is not necessarily better
that the previous estimate $Q_{1}(s,Z)\leq Q(s,Z,J)$, see (19). Nevertheless,
for the initial data (6) sufficiently small in the uniform norm, the new
estimate improves the lower bound for the existence time of a smooth solution.

##  6\. A model example in 2D

We restrict ourselves to a special kind of initial data, the most interesting
from the point of view of physics.

Motivated by the form of a standard laser pulse [12] we choose as the initial
data (6)

$|{\bf
E}_{0}(\rho)|=\left(\displaystyle{a_{*}\over\rho_{*}}\right)^{2}\rho\exp\left(-\displaystyle{\rho^{2}\over\rho_{*}^{2}}\right),\quad{\bf
V}_{0}(\rho)=0,\quad\rho=\sqrt{x_{1}^{2}+x_{2}^{2}},$

$a_{*}$ and $\rho_{*}$ are parameters. For the sake of simplicity, we change
the space variable to $r=\rho/\rho_{*}$ and reduce the data to

(33) $|{\bf E}_{0}(r)|=K\,e^{-r^{2}}\,{\bf r},\quad{\bf V}_{0}(r)=0,\quad
r=\sqrt{x_{1}^{2}+x_{2}^{2}},\quad K=\displaystyle{a_{*}^{2}\over\rho_{*}}>0.$

Fig.1 presents the curves $C_{1}=0$ and $C_{2}=0$ on the plane
$(\sigma,\lambda)$ ($\sigma=\sigma_{1}$ or $\sigma_{2}$) for
$\mathcal{D}_{0}=0$ for different values of $F_{+}$. Below (above) $C_{i}$,
lie such values of $\lambda_{0}=s_{0}+1$ that $C_{i}<0$ ($C_{i}\geq 0$) and
$Z_{i}$ is bounded (unbounded), $i=1,2$. This conclusion follows from the
analysis of leading terms in (29), (31).

Figure 1. Graphs of $C_{1}(\lambda,\sigma)=0$ (1) and
$C_{2}(\lambda,\sigma)=0$ (2), $\lambda=s+1$, for $F_{+}=0.05$,
$F_{+}<\frac{\sigma^{4}}{2\sigma^{2}+1}$, left. Graph of
$C_{2}(\lambda,\sigma)=0$ (2), for $F_{+}=2.5$,
$F_{+}>\frac{\sigma^{4}}{2\sigma^{2}+1}$, right.

Since the trajectory on the phase plane can go to infinity only at
$\mathcal{D}<0$, we study the estimate function under this condition.

###  6.1. $\mathcal{D}<0$, above estimate $\bar{Z}_{1}$

Thus, we have to study the curve $C_{1}=0$, given as (30) for $Z(0)=0$. We
find $s_{0}$ from this equation and denote

$S_{1}=-\frac{1}{2}\frac{(1+2\sigma_{1}^{2})(F_{+}^{2}+\sigma_{1}^{2})}{\sigma_{1}^{2}(1+\sigma_{1}^{2})}.$

Condition (21) implies that that the solution keeps smoothness at the first
rotation provided $\lambda_{0}<\frac{1}{4}$ ($s_{0}<-\frac{3}{4}$) at the most
”dangerous” point $r=0$ ($G(r)$ has the maximum here).

Figure 2. Dependency $F_{+}(\lambda)$ (see (34)), left. Dependency
$\lambda^{*}_{1}(\sigma)$ (1) and $\lambda^{*}_{2}(\sigma)$ (2); the maximum
$(\bar{\sigma}_{1},\Lambda_{1})$ and the minimum
$(\bar{\sigma}_{2},\Lambda_{2})$, right.

We have to keep in mind that $F_{+}$ cannot be chosen arbitrary. Indeed,
according to (23), at the point $r=0$ we chose $\lambda_{0}$ together with
$G(0)=\frac{1}{2}\lambda_{0}$.

Then we can use (25), where the constant $C_{2}$ is found from $Y(0)=0$,
$G(0)=\frac{1}{2}\lambda_{0}$. Then

(34) $\bar{F}_{+}=F_{+}(\lambda_{0})=\frac{1}{2}\sqrt{4e^{-C-1}-2},\quad
C=\frac{1}{\lambda_{0}-1}-\ln\left(\frac{1-\lambda_{0}}{2}\right),$

where $F_{+}(\lambda_{0})=\sqrt{Y(G_{m})}$, $G_{m}=-e^{-C-1}$ is the maximum
point of $Y(G)$. Graph of $F_{+}(\lambda_{0})$ is presented in Fig.2, left.

Let us introduce the function
$\lambda_{1}(F_{+}(\lambda_{0}),\sigma_{1})=S_{1}(F_{+}(\lambda_{0}),\sigma_{1})+1\,\lambda_{0}\in(0,1)$.
Taking into account (34), we can write an explicit expression for this
function

$\lambda_{1}(F_{+}(\lambda_{0}),\sigma_{1})=\frac{1+4\sigma_{1}^{2}-(2\sigma_{1}^{2}+1)(1-\lambda_{0})e^{\frac{\lambda_{0}}{1-\lambda_{0}}}}{4\sigma_{1}^{2}(\sigma_{1}^{2}+1)}.$

We look for $\lambda_{0}\leq\lambda_{1}(\lambda_{0})$. If such
$\lambda_{0}>\frac{1}{4}$, we obtain a sufficient condition for the smoothness
of the solution on the first oscillation. Thus, we have to find the fixed
point $\lambda_{1}^{*}$ of the mapping $\lambda_{1}(F_{+}(\lambda_{0}))$. This
value can be expressed in terms of the Lambert W function [4]:

$\displaystyle\lambda^{*}_{1}(\sigma_{1})$ $\displaystyle=$
$\displaystyle\frac{(4\sigma_{1}^{4}-1)L_{W}(\sigma_{1})-4\sigma_{1}^{2}-1}{(4\sigma_{1}^{4}-1)\,L_{W}(\sigma_{1})-4\sigma_{1}^{2}(1+\sigma_{1}^{2})},$
$\displaystyle L_{W}$ $\displaystyle=$ $\displaystyle{\rm
LambertW}\left(k,\frac{1}{2\sigma_{1}^{2}-1}\,e^{\frac{4\sigma_{1}^{2}+1}{4\sigma_{1}^{2}-1}}\right),$

where $k=-1$ for $\sigma_{1}<\frac{1}{\sqrt{(}2)}$ and $k=0$ for
$\sigma_{1}>\frac{1}{\sqrt{(}2)}$.

Let us maximize $\lambda^{*}_{1}(\sigma_{1})$ (this can be done numerically).
The maximum point is $\bar{\sigma}_{1}=0.5032\dots$, the maximum value
$\Lambda_{1}\equiv\lambda^{*}_{1}(\bar{\sigma}_{1})=0.3058\dots$. We can see
that the sufficient condition (21) is improved, because now we can guarantee
at least one revolution of every trajectory on the phase plane for the data
(33) with $\lambda_{0}\in[0,\lambda_{*}^{1}]$, i.e.
$K<\frac{\lambda_{*}^{1}}{2}=0.1529\dots$ instead of $K<0.125$.

###  6.2. $\mathcal{D}<0$, below estimate $\bar{Z}_{2}$

The analysis is analogous to the previous subsection. We find $s_{0}$ from the
equation $C_{2}=0$, given as (32) for $Z(0)=0$, and denote

(35)
$S_{2}=\frac{1}{2}\frac{(2\sigma_{2}^{2}-1)(F_{+}^{2}(2\sigma_{2}^{2}+1)-\sigma_{2}^{4})}{\sigma_{2}^{2}(\sigma_{2}^{2}-1)}.$

Since the denominator of $S_{2}$ vanishes at $\sigma_{2}=1$, we restrict
ourselves by the interval $\sigma_{2}\in(0,1)$.

For $F_{+}<\bar{F}_{+}=\frac{\sigma_{2}^{4}}{2\sigma_{2}^{2}}$ the function
$S_{2}$ has a minimum of $S_{2}$ on the interval $\sigma_{2}\in(0,1)$, for
$F_{+}>\bar{F}_{+}$ the function $S_{2}$ decays with $\sigma_{2}$ (see Fig.1).

We consider the function
$\lambda_{2}(F_{+}(\lambda_{0}),\sigma_{2})=S_{2}(F_{+}(\lambda_{0}),\sigma_{2})+1\,\lambda_{0}\in(0,1)$,
its explicit expression is

$\lambda_{2}(F_{+}(\lambda_{0}),\sigma_{2})=\frac{(2\sigma_{2}^{2}-1)(\frac{1}{2}((1-\lambda_{0})e^{\frac{\lambda_{0}}{1-\lambda_{0}}}-1)(2\sigma_{2}^{2}+1)-\sigma_{2}^{4})}{2\sigma_{2}^{2}(\sigma_{2}^{2}-1)}.$

Now we need to find $\lambda_{0}\geq\lambda_{2}(\lambda_{0})$. If such
$\lambda_{0}<1$, we obtain a sufficient condition for the blow-up on the first
oscillation. The fixed point $\lambda_{2}^{*}$ of the mapping
$\lambda_{2}(F_{+}(\lambda_{0}))$ exists only for
$\sigma_{2}>\frac{1}{\sqrt{2}}$, it can be expressed in terms of the Lambert W
function:

$\displaystyle\lambda^{*}_{2}(\sigma_{2})$ $\displaystyle=$
$\displaystyle\frac{(2\sigma_{2}^{2}-1)\Sigma_{1}(\sigma_{2})L_{W}(\sigma_{2})+\Sigma_{2}(\sigma_{2})}{(2\sigma_{2}^{2}-1)\Sigma_{1}(\sigma_{2})L_{W}(\sigma_{2})-4\sigma_{2}^{2}(\sigma_{2}^{2}-1)},$
$\displaystyle\Sigma_{1}$ $\displaystyle=$ $\displaystyle
2\sigma_{4}+2\sigma_{2}+1,\quad\Sigma_{1}=4\sigma_{6}-2\sigma_{4}+4\sigma_{2}-1,$
$\displaystyle L_{W}$ $\displaystyle=$ $\displaystyle{\rm
LambertW}\left(-1,\frac{1+2\sigma_{2}^{2}}{\Sigma_{1}}\,e^{\frac{\Sigma_{2}}{(1-2\sigma_{2}^{2})\Sigma_{1}}}\right).$

We minimize $\lambda^{*}_{2}(\sigma_{2})$ with respect to $\sigma_{2}$. The
minimum point is $\bar{\sigma}_{2}=0.9423\dots$, the minimum value
$\Lambda_{2}\equiv\lambda^{*}_{2}(\bar{\sigma}_{2})=0.5754\dots$.

Fig.2, right, presents the functions $\lambda^{*}_{1}(\sigma_{1})$,
$\lambda^{*}_{2}(\sigma_{2})$ and their extrema.

We summarize our results.

######  Proposition 6.1.

Let us consider the solution to the Cauchy problem (4), (33).

  * •

If $K<\frac{1}{2}\Lambda_{1}=0.1529\dots$, then the solution keeps $C^{1}$ \-
smoothness during at least the first oscillation;

  * •

if $K>\frac{1}{2}\Lambda_{2}=0.2877\dots$, then the solution blows up within
the first oscillation.

###  6.3. Example of the estimate of the number of oscillations before the
blow up

Let us choose the data (33) with $K=0.1$. We use the technique, described in
Secs. 2.3 and 2.4 by means of functions $\bar{Z}_{1}$ and $\bar{Z}_{2}$.

Let us list its steps.

1\. Given $K=\frac{1}{2}\lambda_{0}$, $K<\frac{1}{2}\Lambda_{1}$, find
$F_{+}(\lambda_{0})$ by formula (34).

2\. For this value of $F_{+}$ we consider
$\bar{Z}_{1}=\bar{Z}_{1}(\lambda,\bar{\sigma}_{1})$ (see (29)). The graph of
$-\sqrt{\bar{Z}_{1}}$ is the first part of the curve $L$.

3\. We find $\lambda_{0}^{1}<0$, the second root of equation
$\bar{Z}_{1}(\lambda,\bar{\sigma}_{1})=0$, and use it as the new initial data.
Thus, we find new $F_{+}$ and construct the function
$\bar{Z}_{2}(\lambda,\bar{\sigma}_{2})$. The graph of $\sqrt{\bar{Z}_{2}}$ is
the next link of the curve $L$.

4\. We find $\lambda_{0}^{2}>0$, the second root of equation
$\bar{Z}_{2}(\lambda,\bar{\sigma}_{2})=0$, and use it as the new initial data,
and repeat Step 2. Thus, we get the third link of the curve $L$, and so on.

5\. To find the guaranteed number of revolutions ($n$) we construct the curve
$L$ as described in Sec.2.3. The iteration procedure stops if on the next step
$C_{1}\geq 0$.

6\. We construct the curve $l$ as described in Sec.2.3 for this $n$. The
constant $T_{l}$, computed by the formula from Sec.2.4, is the estimate of the
time of the existence of the smooth solution from below.

Figure 3. Bounds for the projection of the phase trajectory on the plane
$(\lambda,\mathcal{D})$. The starting point is $(0.1,0)$. Left: the above
bound $L$, the guaranteed number of oscillations is $3$. Right: the respective
below bound $l$.

Fig.3, left, presents the curve $L$, which bounds the projection of the phase
trajectory on the plane $(\lambda,\mathcal{D})$ from above. Fig.3, right,
presents the curve $l$, which bounds the projection of the phase trajectory on
the plane $(\lambda,\mathcal{D})$ from below. The method guarantees 3
oscillations before the blow-up. The guaranteed time of smoothness

$T_{*}>\inf\limits_{r_{0}\in\mathbb{R}}T_{l}(r_{0})=T_{l}(0)=18.8685...$

The estimate of this guaranteed time from above is $19.1298...$

######  Remark 6.1.

A series of computations with the data (33) was performed in [3], [2], so we
have an opportunity to compare our results with the results of sophisticated
numerics. In [3] computations are made for $a_{*}=0.365$, $\rho_{*}=0.6$, i.e.
$K\approx 0.222$, the breaking time is about $\theta_{*}=35$ (dimensionless
units). Thus, the solution keeps smoothness within the first oscillation.
However, Proposition 6.1 does not guarantee the preservation of smoothness
during the first oscillation. This means that the sufficient condition from
Proposition 6.1 is still too rough.

######  Remark 6.2.

The method for refining estimates outlined in this section can be easily
adapted to the 3D radially symmetric case.

##  7\. Discussion

In this paper, we constructed a method, that allows one to obtain a sufficient
condition that guarantees the boundedness of the density component on a given
time interval for the solution to the Cauchy problem of the system of PDE
describing general 3D electrostatic oscillations. For particular cases (plain
and irrotational oscollations) we find an explicit sufficient condition for
preventing a blow-up in the first period of oscillations. Next, we compare the
sufficient condition with the known criteria for the formation of
singularities, if such a possibility exists. Further, we consider the case of
2D axisymmetric oscillations with special initial data, where there is a
possibility of comparison with the results of a numerical study of the blow-up
process. For this case, we obtain a sufficient condition for the preservation
of smoothness, which is less rough than in the general case, as well as a
sufficient condition for a blow-up.

As shown in 3, in the 1D case, as well as for the affine and radially
symmetric cases in many dimensions, they exist globally in $t$ smooth
solutions of the Cauchy problem. It is very interesting to check whether there
are globally smooth solutions in the multidimensional case without these
restrictive assumptions. The numerics suggest that any other solution
necessarily blows up. Recently, a finite-time blow-up for general radially
symmetric initial data, even arbitrarily small, has been proved
analytically[11]. The only exception is the initial data in the form of so-
called simple waves, where the solution either tends to the affine one or
blows up. The present paper can be regarded as an addition to this result,
since it makes it possible to estimate the lifetime of a smooth solution. Now
it is not known where affine solutions without radial symmetry necessarily
blows up.

Another important open question is the description of the complete class of
solutions corresponding to electrostatic oscillations. The solutions
considered in this article belong to this class, and the hypothesis is that in
the multidimensional case, electrostatic oscillations must necessarily be
radially symmetric or affine. However, this fact has not yet been proven.

## Acknowledgment

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

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