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On the properties of multidimensional electrostatic oscillations of an electron plasma

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.


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]) ∂n ∂t + div (nV) = 0, ∂V ∂t n and V = (V 1 , V 2 , V 3 ) are the density and velocity of electrons, E = (E 1 , E 2 , E 3 ) and B = (B 1 , B 2 , B 3 ) are vectors of electric and magnetic fields. All components of solution depends on t ∈ R + and x ∈ 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. rot E and the magnetic field B does not change with time. For simplicity, we set 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 thus, n can be removed from the system. Thus, we get ∂V ∂t together with (5) curl E = 0, curl ((1 − div E)V) = 0.
Consider the initial data 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 ∈ 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).

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 (λ 0 , D 0 ) = (divE 0 (x 0 ), divV 0 (x 0 )), x 0 ∈ R 3 on the plane (λ, D), intersects the semi-axis D = 0, λ < 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 (λ, D).
In Sec.2 we show that if a two-sided estimate 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 (λ, 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.

Method for obtaining a sufficient conditions for boundedness of density
We denote for convenience s = λ − 1 ≤ 0 (the inequality follows from (3), since n ≥ 0). Thus, we get Let us assume that the following inequality is known: Then, taking into account the sign of s, we obtain the estimate: where 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 and for s < s 0 the inverse inequality where Z k (s) are the solutions to problems dZ ds = Q k (s, Z), Z(s 0 ) = Z 0 , k = 1, 2. Since Z ≥ 0, then the phase trajectory outgoing from the point (s 0 , Z 0 ), is bounded, if the solution Z 1 (s) → −∞ as s → −∞. If Z 2 (s) → +∞ as s → −∞, 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 dZ ds = Q 1 (s, Z) and dZ ds = Q 2 (s, Z). First of all, note that on the plane (s, D) the motion is clockwise (for positive 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 − ≤ S + , or one root S + .
If the equation Z 2 (s) = 0 has two roots on the semiaxis s < 0, we denote them s − and s + , s − ≤ s + .
The trajectory D(s, t) lies between the curves Z 2 (s) and Z 1 (s). Denote by s k * the point at which D(s, t) intersects the axis D = 0 for the k-th time. If D > 0, then Let us compose the curve L as follows: it consists of the part of trajectory Z 2 (s) if D > 0 and switches to the part of trajectory − Z 1 (s) if D < 0 at the axis D = 0. If − √ Z 1 (s) = 0 has two roots at s < 0, then L switches once again the the axis D = 0 to the trajectory Z 2 (s), and so on. Denote by L k the point at which L intersects the axis D = 0 for the k-th time for D 0 < 0 and in the k − 1-th time for D 0 ≥ 0. Thus, the trajectory D(s, t) makes at least one revolution if the number of points L k is more then 2 (L 2 , L 3 , . . . ) for D 0 ≥ 0 and more then 1 In its turn, we compose the curve l as follows: it consists of the part of trajectory Z 1 (s) if D > 0, switches to the part of trajectory − Z 2 (s) if D < 0 at the axis D = 0 and switches once again the the axis D = 0 to the trajectory Z 1 (s), and so on. Denote by l k the point at which l intersects the axis D = 0 for the k-th time for D 0 < 0 and in the k − 1-th time for D 0 ≥ 0.
We note that s k * for all k ∈ N lies between l k and L k .
2.1. General estimates of J.
whereΞ andD are defined as Ξ and D with the change of all derivatives Proof. The estimate (15) is a corollary of the elementary inequality −(a − b) 2 ≤ 2ab ≤ (a + b) 2 , for every a, b ∈ R. Estimate (15) does not yet provide an opportunity to find explicit bounds, but we will use it for particular cases.  Proof. As follows from the second equation of (7), the motion along the projection D = D(s, t) onto the plane (s, D) is clockwise for s < 0. Therefore, in the upper hyperplane D > 0 there are two possibilities for a trajectory starting from any point (s 0 , D 0 > 0): 1. to cross the axis D = 0 and fall into the lower hyperplane, which means that the projection of the phase trajectory D(s, t) was bounded for D > 0, or 2. to tend to +∞ as s → −0 and t → t * ≤ ∞. (This would mean that a vacuum point is formed along the characteristic).
Assume that the latter possibility realizes, i.e. D(t) → +∞ and s(t) → −0 as t → t * ≤ ∞. We see from (7) This contradiction shows that our assumption was wrong and D > 0 is bounded.

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, 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 , D = 0) and (s = L 2 , 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 D(t, s)) makes at least two revolutions and at the axis 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 D. This means that we can guarantee n revolutions of the trajectory 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 div V 0 < 0 and for the characteristic starting from the point x 0 we can guarantee n revolutions of the trajectory 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 , where T l and T L is the time of passage of n turns along a compound spiral lines l and L, respectively. Thus, where we denoted Z i (k; s), i = 1, 2, k = 0, 1, . . . , 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). The time T * of existence of the smooth solution with a bounded density can be estimated as Remark 2.1. The technique of counting the number of oscillations before the blowup in the case of multidimensional electrostatic non-relativistic oscillations is similar to the case of 1D relativistic oscillations [10].

Particular cases
3.1. 1D oscillations, a criterion of a singularity formation. In this case (5) evidently holds. It is easy to see that here J = 0 and system (7) turns intȯ 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 holds.
Condition (5) dictates q ij = q ji , r ij = r ji , so we get the matrix equatioṅ 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.
It is easy to check that for the affine solutions and radially symmetric solutions curl V = 0 and condition (5) holds.
Let us consider solutions that are both affine and radially symmetric, i.e F = α(t), G = β(t). Here D = 3α, J = 3α 2 , λ = 3β, and the system (7) takes a closed formα Recall that the positivity of density requires 1 − 3β > 0, see (3). The prevailing term of the right hand side as β → −∞ is 2β, it tends to −∞ for any K. This means that α and β 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 For the corresponding affine solution D = 2α, J = α 2 , λ = 2β, therefore system (7) takes a closed formα From the positivity of density we have 1 − 2β > 0, therefore the prevailing term of the right hand side as β → −∞ is the logarithmic one, it tends to −∞ for any K. This means that α and β 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.

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.

Plain oscillations. For the case of plain oscillations
Along characteristics this equations can be written asξ 3 = −Dξ 3 , therefore (7) implies Thus, from (15) we have 2J ≤ D 2 + C 2 3 s 2 , and (11) takes the form 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 then the uniqueness of the solution of the Cauchy problem implies that if Ξ 0 = 0, then Ξ = 0 identically. Therefore, (9) gives 2J ≤ D 2 and (11) is Thus, For the both case the following lemma holds.
Then there exists a moment t * such that the trajectory (s, D) turns out in the upper hyperplane 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 D(s, t) does not go to infinity during the first oscillation in the lower hyperplane D < 0. As follows from Sec.2, the trajectory in the lower hyperplane D < 0 is bounded from below by the curve − Z 2 (s). In its turn, Z 2 (s) is bounded if and only if A 4 < 0, where A 4 is defined in (19). 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.
Thus, as a corollary of Lemmas 2.2 and 4.1, we get the following theorem.

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 d = 2 and 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 d ≥ 1, we assume However, we are interested mostly in physical dimension d = 3. For the plain oscillations (d = 2 ) It is easy to check that curl V = 0 and condition (5) holds. Further, Further, if we substitute (23) to (4), we geṫ On the phase plane (G, F ) system (24) implies one equation which is linear with respect to Y = F 2 and can be explicitly integrated. Indeed, we have for d = 2 If G(0, r 0 ) < 1 d , r 0 ∈R, then the phase trajectories of (24) are in the halfplane G < 1 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 2 G−1 d−2 . Therefore F and G are bounded for any t > 0.
Along every characteristic curve starting from r 0 ∈R + the functions F (t) and G(t), solution of (24), are periodic with the period 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: The lower estimate is analogous: The parameters σ 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 s .
The equations dZ ds =Q 1 (s, Z) and dZ ds =Q 2 (s, Z) can be solved, their solutions areZ Thus, we getQ 1 (s, Z) ≤ Q(s, Z, J) ≤Q 2 (s, Z), The new estimateQ 1 (s, Z) ≤ Q(s, Z, J) is not necessarily better that the previous estimate Q 1 (s, Z) ≤ 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.

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.
Since the trajectory on the phase plane can go to infinity only at D < 0, we study the estimate function under this condition.
Condition (21) implies that that the solution keeps smoothness at the first rotation provided λ 0 < 1 4 (s 0 < − 3 4 ) at the most "dangerous" point r = 0 (G(r) has the maximum here).
3. We find λ 1 0 < 0, the second root of equationZ 1 (λ,σ 1 ) = 0, and use it as the new initial data. Thus, we find new F + and construct the functionZ 2 (λ,σ 2 ). The graph of Z 2 is the next link of the curve L. 4. We find λ 2 0 > 0, the second root of equationZ 2 (λ,σ 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 ≥ 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. Fig.3, left, presents the curve L, which bounds the projection of the phase trajectory on the plane (λ, D) from above. Fig.3, right, presents the curve l, which bounds the projection of the phase trajectory on the plane (λ, D) from below. The method guarantees 3 oscillations before the blow-up. The guaranteed time of smoothness 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, ρ * = 0.6, i.e. K ≈ 0.222, the breaking time is about θ * = 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.

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.