Linear Instability of Three-Dimensional Dynamic Equilibrium States for Two-Component Vlasov–Poisson Plasma

Abstract

The problem of controlling plasma is one of the most essential challenges in the creation of experimental facilities for thermonuclear fusion. In this study, a mathematical model of a two-component Vlasov–Poisson plasma is used to study the stability of spatial dynamic equilibria in this plasma. Applying the direct Lyapunov method, we obtain results that demonstrate that three-dimensional (3D) dynamic equilibrium states of the Vlasov–Poisson plasma are absolutely unstable with respect to small spatial perturbations. The sufficient conditions for linear practical instability are obtained for the 3D dynamic equilibria of a two-component Vlasov–Poisson plasma. An a priori exponential lower estimate is constructed, and initial data are found for small spatial perturbations that grow with time. Finally, analytical examples are presented for exact stationary solutions to the mathematical model of Vlasov–Poisson plasma and the growing small 3D perturbations superimposed on these solutions.

1. Introduction

One of the most known partial differential equations in mathematical physics for the kinetic theory of plasma is the Vlasov–Poisson system. The Vlasov and Poisson equations jointly describe the dynamics of charged plasma particles while taking the long-range Coulomb force into consideration through a self-consistent electric field in the electrostatic approximation [1]. It is impossible to decide whether thermonuclear fusion can be controlled without first solving the problem of stability for plasma equilibria [2]. Therefore, developing a mathematical theory for stability is of great significance for exploring plasma and its properties.

It is worth noting that the Vlasov–Poisson system of equations can be linearized if perturbations of the distribution function and the potential of the self-consistent electric field are small with respect to plasma equilibrium states. The linearization procedure results in a system that characterizes plasma dynamics through both an electrostatic approximation and a linear statement [2,3]. The linearized Vlasov–Poisson equations are used to describe the collisionless movement of electrons that interact with each other through Coulomb repulsive forces against the background of a uniform distribution of resting ions inside the physical space.

We reviewed mathematical results on the stability and instability of dynamic equilibria of the Vlasov–Poisson system circumscribing a two-component plasma and found that these equilibrium states have been extensively studied. For instance, some fundamental results have been reported in Refs. [3,4,5,6,7,8,9]. For example, in Refs. [4,5], it was shown that the sufficient stability condition prohibits small perturbations in the form of normal waves that grow over time. Furthermore, this sufficient stability condition was generalized to the case of finite perturbations in Ref. [3], and is termed the so-called sufficient stability condition for dynamic equilibria of a two-component Vlasov–Poisson plasma [6]. Furthermore, in Ref. [3], the stability of these equilibrium states was studied throughout the entire continuum, and a constraint on the $L_2$ norm of perturbations was provided by estimating the required convexity. In 1993, Jürgen Batt and Gerhard Rein described that the spatially homogeneous stationary solutions with non-positive energy gradient and compact support are (nonlinearly) stable in the $L_1$ norm with respect to weak solutions to the initial value problem [7]. Yan Guo and Walter Strauss explored a system characterized by the Vlasov–Poisson equations in a three-dimensional (3D) cube with specular boundary conditions [8] and proved that an equilibrium that satisfies the Penrose linear instability condition and decays like $\mathcal{O}\left(\right|v{|}^{-3})$ (with the velocity v of electrons) is nonlinearly unstable. In addition, in 2021, Daniel Han-Kwan, Toan Nguyen, and Frederic Rousset studied the Vlasov–Poisson system linearized around suitably stable homogeneous equilibrium states and established dispersive $L_{\infty }$ decay estimates in the physical space [9]. Other interesting results on this topic have been published recently; for example, in Refs. [10,11,12,13,14].

However, in his earlier papers [15,16], one of the authors of this paper has reversed the sufficient linear stability condition [4,5,6] for 1D dynamic equilibria of a two-component Vlasov–Poisson plasma and proved that growing small 1D perturbations in the form of normal waves exist despite this sufficient condition holding. In this paper, those results are extended to the problem on linear stability for 3D dynamic equilibrium states of a two-component Vlasov–Poisson plasma containing electrons and ions in the case where stationary distribution functions of electrons are variable over the physical continuum and in the velocity space, whereas the ions are isotropically distributed across the physical continuum and do not move within the velocity space. In what follows, it is shown that the stability results from Refs. [15,16] are also applicable in this case. Just put, as mentioned above, the results of others are not valid under any small 3D perturbations, while the results presented in Refs. [15,16] hold for all small spatial perturbations.

Another significant aspect regarding the stability of plasma equilibria is Landau damping [17,18,19,20]. Lev Landau considered a two-component Vlasov–Poisson plasma [17] and assumed that the frequency of electron plasma vibrations is sufficiently high such that the collisions between electrons and ions are negligible and the collision integral can be ignored in the Vlasov equation. In this scenario, the distribution function of ions can be treated as an invariant, and the distribution of electrons only vibrates. Then, Landau showed that vibrations of the electric field in plasma are always damped, and the dependence of frequency and the damping decrement on the wave vector was determined for small and large enough values of the latter. Many other researchers have continued to expand upon and generalize Landau’s results [18,19,20]. However, Landau did not consider all small perturbations, i.e., he did not explore small perturbations of the self-consistent electric field potential as functions of time exclusively. For such perturbations, the gradient (2) and the Laplacian (3) in Ref. [17] of the electric field potential will degenerate, and the formula (9) of Ref. [17] cannot be derived; therefore, there is no Landau damping. In this paper, analytical examples are given to prove that the subclass of small perturbations for self-consistent electric field potential that are described by time functions only is non-empty.

2. Kinetic Case

2.1. Exact Stationary Solutions

In the 3D setting, the mathematical model for a boundless electrically neutral collisionless plasma in the electrostatic approximation has the following index form [1,21,22]:

$$\begin{array}{c}{\displaystyle \frac{\partial f}{\partial t}}+v_i{\displaystyle \frac{\partial f}{\partial x_i}}-{\displaystyle \frac{\partial \phi }{\partial x_i}}{\displaystyle \frac{\partial f}{\partial v_i}}=0,\\ {\displaystyle \frac{{\partial }^2\phi }{\partial x_i^2}}=4\pi \left(1-{\int }_{{\mathcal{R}}^3}f(x_i,v_i,t)d{v}_{1}d{v}_{2}d{v}_3\right);\\ f=f(x_i,v_i,t)\ge 0;f(x_i,v_i,0)=f_0(x_i,v_i),\end{array}$$

(1)

where f is the electron distribution function, $x_i$ and $v_i$ (i = 1, 2, 3) denote the electron coordinates and velocities, t is the time, $\phi$ denotes the potential of a self-consistent electric field, $f_0$ is the initial data, and $\pi$ denotes a known constant. It is supposed that for the argument $v_i$, the function f decreases when $|\overrightarrow{v}|$ tends to infinity and, for the argument $x_i$, this function along with the functions $\partial \phi /\partial x_i$ are either periodic or decreasing when $|\overrightarrow{x}|$ tends to infinity.

In the following, E denotes the functional of full energy and C denotes the law of electron number conservation [3]. Note that these quantities are conserved along exact evolutionary solutions to the mixed problem (1):

$$\begin{array}{c}E\equiv {\displaystyle \frac{1}{2}}\int {\int }_{{\mathcal{R}}^3}f{v}_i^{2}d\overrightarrow{x}d\overrightarrow{v}+{\displaystyle \frac{1}{2}}{\int }_{{\mathcal{R}}^3}\phi ({\int }_{{\mathcal{R}}^3}fd\overrightarrow{v}-1)d\overrightarrow{x}=\mathrm{const},\\ C\equiv \int {\int }_{{\mathcal{R}}^3}\mathsf{\Phi }\left(f\right)d\overrightarrow{x}d\overrightarrow{v}=\mathrm{const}.\end{array}$$

(2)

Here, $d\overrightarrow{x}=d{x}_{1}d{x}_{2}d{x}_3$ is a volume element in the physical continuum ${\mathcal{R}}^3$, $d\overrightarrow{v}=d{v}_{1}d{v}_{2}d{v}_3$ is a volume element in the velocity space ${\mathcal{R}}^3$, and $\mathsf{\Phi }$ denotes an arbitrary function of the argument f.

Then, we assume that the initial boundary value problem (1) has the following exact stationary solutions:

$$\begin{array}{c}\hfill f=f^0(x_i,v_i),\phi ={\phi }^0\left(x_i\right)\end{array}$$

(3)

which satisfy the following system of stationary equations:

$$\begin{array}{c}{v}_i{\displaystyle \frac{\partial f^0}{\partial x_i}}={\displaystyle \frac{\partial {\phi }^0}{\partial x_i}}{\displaystyle \frac{\partial f^0}{\partial v_i}},\\ {\displaystyle \frac{{\partial }^2{\phi }^0}{\partial x_i^2}}=4\pi \left(1-{\int }_{{\mathcal{R}}^3}{f}^0(x_i,v_i)d\overrightarrow{v}\right),\end{array}$$

(4)

where we deem that $f^0$ is the stationary electron distribution function, and ${\phi }^0$ is the stationary potential of a self-consistent electric field.

2.2. Newcomb–Gardner–Rosenbluth Sufficient Condition

Furthermore, we study whether the exact stationary solutions (3) and (4) are stable in relation to small 3D perturbations. According to the linearization procedure, one has the following form for the exact non-stationary solutions (1): $f=f^0(x_i,v_i)+f^{\prime }(x_i,v_i,t)$ and $\phi ={\phi }^0\left(x_i\right)+{\phi }^{\prime }(x_i,t)$. (As the perturbations $f^{\prime }$ and ${\phi }^{\prime }$ are sufficiently small, the quadratic terms associated with these perturbations are negligible in the Vlasov equation.) Then, we linearize the system (1) in the vicinity of its exact stationary solutions (3) and (4) to obtain the following equations:

$$\begin{array}{c}{\displaystyle \frac{\partial f^{\prime }}{\partial t}}+v_i{\displaystyle \frac{\partial f^{\prime }}{\partial x_i}}-{\displaystyle \frac{\partial {\phi }^0}{\partial x_i}}{\displaystyle \frac{\partial f^{\prime }}{\partial v_i}}-{\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}}{\displaystyle \frac{\partial f^0}{\partial v_i}}=0,\\ {\displaystyle \frac{{\partial }^2{\phi }^{\prime }}{\partial x_i^2}}=-4\pi {\int }_{{\mathcal{R}}^3}{f}^{\prime }(x_i,v_i,t)d\overrightarrow{v};\\ f^{\prime }(x_i,v_i,0)=f_0^{\prime }(x_i,v_i).\end{array}$$

(5)

Here, the boundary conditions satisfy the following: the perturbation $f^{\prime }$ (5) vanishes when $|\overrightarrow{v}|$ tends to infinity, while when $|\overrightarrow{x}|$ tends to infinity, this perturbation together with the functions $\partial {\phi }^{\prime }/\partial x_i$ are either periodic or vanishing.

Next, we compose the functional $I=E+C$ (2), which ensures preservation over time in accordance with the mixed problem (1). When the ratio

$$\begin{array}{c}\hfill {\displaystyle \frac{v_i^2}{2}}+{\phi }^0=-{\displaystyle \frac{d\mathsf{\Phi }}{df}}\left(f^0\right)\end{array}$$

(6)

is satisfied, one has that the first variation in integral I is equal to zero, and the functional I reaches its extreme values on the exact stationary solutions (3) and (4). Then, one has the second variation in the integral I:

$$\begin{array}{c}\hfill {\delta }^{2}I={\displaystyle \frac{1}{2}}\int {\int }_{{\mathcal{R}}^3}{\displaystyle \frac{d^2\mathsf{\Phi }}{d{f}^2}}\left(f^0\right){\left(\delta f\right)}^{2}d\overrightarrow{x}d\overrightarrow{v}+{\displaystyle \frac{1}{8\pi }}{\int }_{{\mathcal{R}}^3}{\left({\displaystyle \frac{\partial \delta \phi }{\partial x_i}}\right)}^{2}d\overrightarrow{x}.\end{array}$$

Taking the first variations $\delta f$ and $\delta \phi$ as small spatial perturbations $f^{\prime }$ and ${\phi }^{\prime }$, respectively, one can obtain the linear analog of the full-energy functional $E_1$ for evolutionary solutions to the linearized initial boundary value problem (5) in the following form:

$$\begin{array}{c}\hfill E_1\equiv {\displaystyle \frac{1}{2}}\int {\int }_{{\mathcal{R}}^3}{\displaystyle \frac{d^2\mathsf{\Phi }}{d{f}^2}}\left(f^0\right)f^{\prime 2}d\overrightarrow{x}d\overrightarrow{v}+{\displaystyle \frac{1}{8\pi }}{\int }_{{\mathcal{R}}^3}{\left({\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}}\right)}^{2}d\overrightarrow{x}=\mathrm{const}.\end{array}$$

(7)

It is straightforward to check that the integral $E_1$ (7) is a non-negative quantity when it satisfies the following inequality:

$$\begin{array}{c}{\displaystyle \frac{d^2\mathsf{\Phi }}{d{f}^2}}\left(f^0\right)\ge 0\\ \left\{-{\displaystyle \frac{d}{d{f}^0}}\left({\displaystyle \frac{v_i^2}{2}}+{\phi }^0\right)\ge 0\Rightarrow {\displaystyle \frac{d\left(\frac{v_i^2}{2}+{\phi }^0\right)}{d{f}^0}}\le 0\right\},\end{array}$$

(8)

where using expression (6), we can rewrite the first inequality (8) as shown in the curly brackets.

Now, it can be said that the exact stationary solutions (3) and (4) are stable with respect to small 3D perturbations, as the functional $E_1$ has a defined sign. Based on the variational method [3,23], the derived inequality (8) is a sufficient condition for linear stability of the exact stationary solutions in relation to small spatial perturbations (5). The inequality (8) is consistent with the known Newcomb–Gardner–Rosenbluth result [3,4,5,6], which states that monotonically decreasing stationary distribution functions $f^0$ are stable. It is worth mentioning that the relation (8) is also known as the Newcomb–Gardner–Rosenbluth sufficient condition for stability.

According to the single evolutionary equation—that is, the Vlasov equation—the integrals E and C (2) are interdependent along the solutions to the mixed problem (1). This signifies that the Newcomb–Gardner–Rosenbluth sufficient condition (8) for linear stability of dynamic equilibrium states (3) and (4) in a two-component Vlasov–Poisson plasma is valid only when the considered perturbations (5) belong to some incomplete unclosed subclass. An analytical example is given in Section 2.3 just below to verify the correctness of this conclusion.

2.3. Analytical Example

As it is complicated to find solutions to the boundary value problem (4) and the initial boundary value problem (5), we further explore simpler problems, namely, suppose that the mathematical model (1) of a two-component Vlasov–Poisson plasma has the following exact stationary solutions:

$$\begin{array}{c}\hfill f=f^0\left(v_i\right),\phi ={\phi }^0\equiv \mathrm{const}\end{array}$$

(9)

which satisfy the integral equation

$$\begin{array}{c}\hfill {\int }_{{\mathcal{R}}^3}{f}^0\left(v_i\right)d\overrightarrow{v}=1.\end{array}$$

(10)

Similarly to that in Section 2.2, we linearize the system (1) in the vicinity of the exact stationary solutions (9) and (10) and obtain the following equations:

$$\begin{array}{c}{\displaystyle \frac{\partial f^{\prime }}{\partial t}}+v_i{\displaystyle \frac{\partial f^{\prime }}{\partial x_i}}={\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}}{\displaystyle \frac{\partial f^0}{\partial v_i}},\\ {\displaystyle \frac{{\partial }^2{\phi }^{\prime }}{\partial x_i^2}}=-4\pi {\int }_{{\mathcal{R}}^3}{f}^{\prime }(x_i,v_i,t)d\overrightarrow{v}.\end{array}$$

(11)

It is not complicated to calculate the exact stationary solution $f^0\left(v_i\right)$ (9), as follows:

$$\begin{array}{c}\hfill f^0(v_1,v_2,v_3)={\pi }^{-{\displaystyle \frac{3}{2}}}\exp (-v_1^2-v_2^2-v_3^2),\end{array}$$

(12)

such that the integral Equation (10) turns into an identity. We choose small perturbations for the potential of a self-consistent electric field in the following form:

$$\begin{array}{c}\hfill {\phi }^{\prime }=C_1\exp \left(\alpha t\right)(C_1\equiv \mathrm{const},\alpha \equiv \mathrm{const}>0).\end{array}$$

(13)

Substituting the expressions (12) and (13) into the system (11), one gets the following equation for perturbations $f^{\prime }$ of the electron distribution function and its general solution:

$$\begin{array}{c}\hfill f_t^{\prime }+v_1{\displaystyle \frac{\partial f^{\prime }}{\partial x_1}}+v_2{\displaystyle \frac{\partial f^{\prime }}{\partial x_2}}+v_3{\displaystyle \frac{\partial f^{\prime }}{\partial x_3}}=0;\\ \hfill F\left(f^{\prime },t-{\displaystyle \frac{x_1}{v_1}},t-{\displaystyle \frac{x_2}{v_2}},t-{\displaystyle \frac{x_3}{v_3}}\right)=0.\end{array}$$

(14)

In accordance with the relations in Equation (14), one can set three different solutions for the function $f^{\prime }$, which satisfy the linearized Poisson Equation (11) and the boundary conditions:

$$\begin{array}{c}{f}^{\prime }=v_1\exp \left[{\alpha }_1\left(t-{\displaystyle \frac{x_1}{v_1}}\right)-v_1^2\right]h_1\left(t-{\displaystyle \frac{x_2}{v_2}},t-{\displaystyle \frac{x_3}{v_3}}\right)\mathrm{for}{\displaystyle \frac{x_1}{v_1}}>0,\\ f^{\prime }=v_2\exp \left[{\alpha }_2\left(t-{\displaystyle \frac{x_2}{v_2}}\right)-v_2^2\right]h_2\left(t-{\displaystyle \frac{x_1}{v_1}},t-{\displaystyle \frac{x_3}{v_3}}\right)\mathrm{for}{\displaystyle \frac{x_2}{v_2}}>0,\\ f^{\prime }=v_3\exp \left[{\alpha }_3\left(t-{\displaystyle \frac{x_3}{v_3}}\right)-v_3^2\right]h_3\left(t-{\displaystyle \frac{x_1}{v_1}},t-{\displaystyle \frac{x_2}{v_2}}\right)\mathrm{for}{\displaystyle \frac{x_3}{v_3}}>0.\end{array}$$

(15)

Here, ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ serve as positive constants; $v_1\exp \left[{\alpha }_1\left(t-x_1/v_1\right)-v_1^2\right]$, $v_2\exp [{\alpha }_2(t-x_2/v_2)-v_2^2]$, and $v_3\exp \left[{\alpha }_3\left(t-x_3/v_3\right)-v_3^2\right]$ are odd functions in the independent variables $v_1$, $v_2$, and $v_3$, respectively; and $h_1\left(t-x_2/v_2,t-x_3/v_3\right)$, $h_2\left(t-x_1/v_1,\right.$$\left.t-x_3/v_3\right)$, and $h_3\left(t-x_1/v_1,t-x_2/v_2\right)$ serve as functions for which the volume integral in the second equation of system (11) exists. Alongside this, the corresponding boundary conditions for perturbations $f^{\prime }$ in Equations (11) and (15) are also satisfied. Finally, to ensure that all three functions $f^{\prime }$ (15) do not contain singular points, we redefine these functions by continuity at such points: ${\left.\left({\left.f^{\prime }\right|}_{t=0,x_1=0}\right)\right|}_{v_1=0}=0$, ${\left.\left({\left.f^{\prime }\right|}_{t=0,x_2=0}\right)\right|}_{v_2=0}=0$, and ${\left.\left({\left.f^{\prime }\right|}_{t=0,x_3=0}\right)\right|}_{v_3=0}=0$. These equalities can be considered as initial conditions for perturbed electron distribution functions $f^{\prime }$ at the origin coordinates.

Thus, an analytical example of small 3D perturbations (11) in the form of normal modes (13) and (15) that are superimposed on a decreasing stationary distribution function $f^0$ (9) and (12) but which; nevertheless, grow with time has been constructed.

This example has a natural physical meaning: it characterizes a change in the distribution function due to the spread of electrons from the vicinity of the reference point to infinity such that the value of the distribution function remains unchanged at the reference point, decreases in its vicinity, and increases at infinity.

On the grounds that the differential operators in the Vlasov–Poisson equation (11) degenerate partially for small spatial perturbations (13) and (15), the linear analog (7) of the full-energy functional—obtained via the variational method [3,16] for small 3D perturbations (5) in the general case—does not appear in this situation.

As a consequence, one can see that small spatial perturbations (13) and (15) are not applicable for the linear analog (7) of full-energy integral, and that these perturbations grow over time. The Newcomb–Gardner–Rosenbluth stability condition (8) does not affect these perturbations, and is formal in nature. Moreover, Equations (13) and (15) serve as a counterexample for three known spectral results: the Newcomb–Gardner theorem [2,4,5], the Penrose criterion [2,24], and Landau damping [17,18,19,20]. Therefore, these spectral results are formal too.

Let us now prove the instability for exact stationary solutions (3) and (4), regardless of the condition (8).

3. Gas-Dynamic Case

3.1. Vlasov–Poisson Equations as a Gas-Dynamic System

Now, we apply the technique of instability proof developed for hydrodynamic equations [15,16]. To this end, we use a non-degenerate change in independent variables [25,26,27] in the form [15,16]; that is, a hydrodynamic substitution is carried out:

$$\begin{array}{c}{\overrightarrow{v}}_0:\overrightarrow{v}=\overrightarrow{u}(x_i,v_{0i},t),{\displaystyle \frac{d{\overrightarrow{v}}_0}{dt}}=0;\\ f(x_i,v_i,t)=\rho (x_i,v_{0i},t)J^{-1},J\equiv {\displaystyle \frac{D(u_1,u_2,u_3)}{D(v_{01},v_{02},v_{03})}}\ne 0,\end{array}$$

(16)

where ${\overrightarrow{v}}_0=(v_{01},v_{02},v_{03})$ denotes the Lagrangian coordinates, $\overrightarrow{u}=(u_1,u_2,u_3)$ is the velocity field, $\rho$ denotes the field of electron density, and J is the Jacobian.

As a result, according to the replacement (16), the mixed problem (1) transforms into the following gas-dynamic form:

$$\begin{array}{c}{\displaystyle \frac{\partial u_i}{\partial t}}+u_k{\displaystyle \frac{\partial u_i}{\partial x_k}}=-{\displaystyle \frac{\partial \phi }{\partial x_i}},{\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i\right)=0,\\ {\displaystyle \frac{{\partial }^2\phi }{\partial x_i^2}}=4\pi \left(1-{\int }_{{\mathcal{R}}^3}\rho (x_i,v_{0i},t)d{\overrightarrow{v}}_0\right);\\ \overrightarrow{u}(x_i,v_{0i},0)={\overrightarrow{u}}_0(x_i,v_{0i}),\rho (x_i,v_{0i},0)={\rho }_0(x_i,v_{0i}).\end{array}$$

(17)

The boundary conditions are as follows: the density field $\rho$ decays as $|{\overrightarrow{v}}_0|\to \infty$ and, as $|\overrightarrow{x}|\to \infty$, this field, along with the functions $\partial \phi /\partial x_i$, approach zero or are periodic in the argument $x_i$.

Using the transformed system (17), one obtains the functional of full energy and the integral of motion, which are conserved over time:

$$\begin{array}{c}{E}_2\equiv {\displaystyle \frac{1}{2}}\underset{{\mathcal{R}}^3}{\int \int }\rho u_i^{2}d\overrightarrow{x}d{\overrightarrow{v}}_0+{\displaystyle \frac{1}{2}}{\int }_{{\mathcal{R}}^3}\phi \left({\int }_{{\mathcal{R}}^3}\rho d{\overrightarrow{v}}_0-1\right)d\overrightarrow{x}=\mathrm{const},\\ C_2\equiv \underset{{\mathcal{R}}^3}{\int \int }{\mathsf{\Phi }}_1\left(\kappa \right)Jd\overrightarrow{x}d{\overrightarrow{v}}_0=\mathrm{const}.\end{array}$$

(18)

Here, $\kappa \equiv \rho /J$ denotes the reverse vorticity, ${\mathsf{\Phi }}_1$ is an arbitrary function of its argument, and $d{\overrightarrow{v}}_0=d{v}_{01}d{v}_{02}d{v}_{03}$ denotes a volume element in the Lagrangian coordinates continuum ${\mathcal{R}}^3$.

Let us assume that the initial boundary value problem (17) has the following exact stationary solutions:

$$u_i=u_i^0(x_i,v_{0i}),\rho ={\rho }^0(x_i,v_{0i}),\phi ={\phi }^0\left(x_i\right)$$

(19)

which satisfy the following system of stationary equations:

$$\begin{array}{c}{u}_k^0{\displaystyle \frac{\partial u_i^0}{\partial x_k}}=-{\displaystyle \frac{\partial {\phi }^0}{\partial x_i}},\\ {\displaystyle \frac{{\partial }^2{\phi }^0}{\partial x_i^2}}=4\pi \left(1-{\int }_{{\mathcal{R}}^3}{\rho }^0(x_i,v_{0i})d{\overrightarrow{v}}_0\right),\\ {\displaystyle \frac{\partial }{\partial x_i}}\left({\rho }^0{u}_i\right)=0.\end{array}$$

(20)

Furthermore, let us now find out whether the exact stationary solutions (19) and (20) are stable with respect to small 3D perturbations ${\overrightarrow{u}}^{\prime }$, ${\rho }^{\prime }$, and ${\phi }^{\prime }$. To this end, we linearize the mixed problem (17) near these exact stationary solutions, which allows us to arrive at the following initial boundary value problem:

$$\begin{array}{c}{\displaystyle \frac{\partial u_i^{\prime }}{\partial t}}+u_k^0{\displaystyle \frac{\partial u_i^{\prime }}{\partial x_k}}+u_k^{\prime }{\displaystyle \frac{\partial u_i^0}{\partial x_k}}=-{\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}},\\ {\displaystyle \frac{\partial {\rho }^{\prime }}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}({\rho }^0{u}_i^{\prime }+{\rho }^{\prime }{u}_i^0)=0,\\ {\displaystyle \frac{{\partial }^2{\phi }^{\prime }}{\partial x_i^2}}=-4\pi {\int }_{{\mathcal{R}}^3}{\rho }^{\prime }(x_i,v_{0i},t)d{\overrightarrow{v}}_0;\\ {\overrightarrow{u}}^{\prime }(x_i,v_{0i},0)={\overrightarrow{u}}_0^{\prime }(x_i,v_{0i}),{\rho }^{\prime }(x_i,v_{0i},0)={\rho }_0^{\prime }(x_i,v_{0i})\end{array}$$

(21)

with the followng auxiliary equations:

$$\begin{array}{c}{\displaystyle \frac{\partial {\kappa }^{\prime }}{\partial t}}+u_i^0{\displaystyle \frac{\partial {\kappa }^{\prime }}{\partial x_i}}+u_i^{\prime }{\displaystyle \frac{\partial {\kappa }^0}{\partial x_i}}=0,\\ {\displaystyle \frac{\partial J^{\prime }}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}(J^0{u}_i^{\prime }+J^{\prime }{u}_i^0)=0,\end{array}$$

where ${\kappa }^{\prime }$ is the perturbed reverse vorticity, ${\kappa }^0$ denotes the stationary reverse vorticity, $J^{\prime }$ is the perturbed Jacobian, and $J^0$ denotes the stationary Jacobian.

3.2. Lagrange Displacements Field

Similarly to the kinetic case and by applying functional (18), the following linear analog of the full-energy integral is obtained:

$$\begin{array}{c}{E}_3\equiv {\displaystyle \frac{1}{2}}\underset{{\mathcal{R}}^3}{\int \int }\left[{\rho }^0{\left(u_i^{\prime }\right)}^2+2u_i^0{u}_i^{\prime }{\rho }^{\prime }+{\displaystyle \frac{d^2{\mathsf{\Phi }}_1}{d{\kappa }^2}}\left({\kappa }^0\right)J^0{\left({\kappa }^{\prime }\right)}^2\right]d\overrightarrow{x}d{\overrightarrow{v}}_0+{\displaystyle \frac{1}{8\pi }}\\ \times {\int }_{{\mathcal{R}}^3}{\left({\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}}\right)}^{2}d\overrightarrow{x}+\underset{{\mathcal{R}}^3}{\int \int }\left[{\mathsf{\Phi }}_1\left({\kappa }^0\right)+\left({\displaystyle \frac{{\left(u_1^0\right)}^2+{\left(u_2^0\right)}^2+{\left(u_3^0\right)}^2}{2}}+{\phi }^0\right){\kappa }^0\right]J_{1}d\overrightarrow{x}d{\overrightarrow{v}}_0=\mathrm{const};\\ {\displaystyle \frac{d{\mathsf{\Phi }}_1}{d\kappa }}\left({\kappa }^0\right)=-{\phi }^0-{\displaystyle \frac{{\left(u_1^0\right)}^2+{\left(u_2^0\right)}^2+{\left(u_3^0\right)}^2}{2}}.\end{array}$$

(22)

Here,

$$\begin{array}{c}{J}_1\equiv {\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_3^0}{\partial v_{03}}}+{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_2^0}{\partial v_{02}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{03}}}+{\displaystyle \frac{\partial u_1^0}{\partial v_{01}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{03}}}+{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_2^0}{\partial v_{03}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{01}}}\\ +{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_3^0}{\partial v_{01}}}+{\displaystyle \frac{\partial u_1^0}{\partial v_{02}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{01}}}+{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_2^0}{\partial v_{01}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{02}}}+{\displaystyle \frac{\partial u_1^0}{\partial v_{03}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{02}}}\\ +{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_3^0}{\partial v_{02}}}-{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_2^0}{\partial v_{02}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{01}}}-{\displaystyle \frac{\partial u_1^0}{\partial v_{03}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{01}}}-{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_3^0}{\partial v_{01}}}\\ -{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_3^0}{\partial v_{02}}}-{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_2^0}{\partial v_{03}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{02}}}-{\displaystyle \frac{\partial u_1^0}{\partial v_{01}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{03}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{02}}}-{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_2^0}{\partial v_{01}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{03}}}\\ -{\displaystyle \frac{\partial u_1^{\prime }}{\partial v_{02}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_3^0}{\partial v_{03}}}-{\displaystyle \frac{\partial u_1^0}{\partial v_{02}}}{\displaystyle \frac{\partial u_2^{\prime }}{\partial v_{01}}}{\displaystyle \frac{\partial u_3^{\prime }}{\partial v_{03}}}.\end{array}$$

The exact stationary solutions (19) and (20) to the mixed problem (17) are stable only if the functional $E_3$ (22) has a defined sign. In the case under study, based on the Sylvester criterion [28], the integral $E_3$ has no constant or definite sign; thus, the Newcomb–Gardner–Rosenbluth sufficient condition (8) for linear stability remains valid only for some incomplete, unclosed partial class of small spatial perturbations (21). Next, to show the instability for any stationary solutions (19) and (20) to initial boundary value problem (17) under study in relation to small 3D perturbations $u^{\prime }$, ${\rho }^{\prime }$, and ${\phi }^{\prime }$ (21), it is necessary to be able to select at least one perturbation which grows exponentially (or faster) with time.

Using the direct Lyapunov method [29,30] (this method in the Gubarev modification has been described in detail in Refs. [16,31,32,33]), one can obtain sufficient conditions for the linear practical instability for exact stationary solutions (19) and (20) to the mixed problem (17) with respect to small spatial perturbations (21). When these conditions for practical instability hold, one can then construct an a priori exponential lower estimate demonstrating that these small perturbations grow with time no slower than an exponential function [32].

Such perturbations can be described by means of the Lagrangian displacements field [34], which is defined by an equation of the form

$${\xi }_i={\xi }_i(x_i,v_{0i},t):{\displaystyle \frac{\partial {\xi }_i}{\partial t}}=u_i^{\prime }+{\xi }_k{\displaystyle \frac{\partial u_i^0}{\partial x_k}}-u_k^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_k}}.$$

(23)

In accordance with the definition of Lagrangian displacements field (23), the initial boundary value problem (21) can transform to the following mixed problem:

$$\begin{array}{c}{\displaystyle \frac{{\partial }^2{\xi }_i}{\partial t^2}}+2u_k^0{\displaystyle \frac{{\partial }^2{\xi }_i}{\partial x_k\partial t}}-{\xi }_k{\displaystyle \frac{\partial }{\partial x_k}}\left(u_m^0{\displaystyle \frac{\partial u_i^0}{\partial x_m}}\right)+u_k^0{\displaystyle \frac{\partial }{\partial x_k}}\left(u_m^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_m}}\right)=-{\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}},\\ {\rho }^{\prime }=-{\displaystyle \frac{\partial }{\partial x_i}}\left({\rho }^0{\xi }_i\right),\\ {\displaystyle \frac{{\partial }^2{\phi }^{\prime }}{\partial x_i^2}}=4\pi {\int }_{{\mathcal{R}}^3}{\displaystyle \frac{\partial }{\partial x_i}}\left({\rho }^0{\xi }_i\right)d{\overrightarrow{v}}_0;\\ {\xi }_i(x_i,v_{0i},0)={\left({\xi }_i\right)}_0(x_i,v_{0i}),{\displaystyle \frac{\partial {\xi }_i}{\partial t}}(x_i,v_{0i},0)={\left({\displaystyle \frac{\partial {\xi }_i}{\partial t}}\right)}_0(x_i,v_{0i}),\end{array}$$

(24)

with auxiliary equations

$$\begin{array}{c}{J}^{\prime }=-{\displaystyle \frac{\partial }{\partial x_i}}\left(J^0{\xi }_i\right),{\kappa }^{\prime }=-{\xi }_i{\displaystyle \frac{\partial {\kappa }^0}{\partial x_i}}.\end{array}$$

3.3. A Priori Exponential Lower Estimate

Now, let us introduce the Lyapunov functional [15,16,32] in the form

$$\begin{array}{c}\hfill M\equiv \int {\int }_{{\mathcal{R}}^3}{\rho }^0{\left({\xi }_i\right)}^{2}d{\overrightarrow{v}}_{0}d\overrightarrow{x}\ge 0,\end{array}$$

(25)

with its first and second derivatives with respect to t calculated according to the initial boundary value problem (24):

$$\begin{array}{c}\hfill {\displaystyle \frac{dM}{dt}}=2\int {\int }_{{\mathcal{R}}^3}{\rho }^0{\xi }_i({\displaystyle \frac{\partial {\xi }_i}{\partial t}}+u_k^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_k}})d{\overrightarrow{v}}_{0}d\overrightarrow{x},\\ \hfill {\displaystyle \frac{d^{2}M}{d{t}^2}}=2\int {\int }_{{\mathcal{R}}^3}{\rho }^0[{({\displaystyle \frac{\partial {\xi }_i}{\partial t}}+u_k^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_k}})}^2-{\xi }_i{\xi }_k{\displaystyle \frac{{\partial }^2{\phi }^0}{\partial x_i\partial x_k}}]d{\overrightarrow{v}}_{0}d\overrightarrow{x}-{\displaystyle \frac{1}{2\pi }}{\int }_{{\mathcal{R}}^3}{\left({\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}}\right)}^{2}d\overrightarrow{x}.\end{array}$$

Applying the third relation from the mixed problem (24), we acquire the expression for ${\phi }_x^{\prime }$ such that the constant of integration is equal to zero; without loss of generality, we consider the estimate for ${\left({\phi }_x^{\prime }\right)}^2$. Analogously, we evaluate the product ${\xi }_i{\xi }_k{\phi }_{x_i{x}_k}^0$ in the formula for $d^{2}M/d{t}^2$. As a result, we construct an original differential inequality [16,32] for the functional M (25), namely, the Gubarev inequality:

$${\displaystyle \frac{d^{2}M}{d{t}^2}}-2\lambda {\displaystyle \frac{dM}{dt}}+2({\lambda }^2+\gamma )M\ge 0,$$

(26)

where $\lambda$ is a constant and $\gamma$ denotes a known positive constant value.

If the parameter $\lambda$ is positive, the inequality (26) can be supplemented with a countable set of the following conditions [16,32]:

$$\begin{array}{c}M\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2\gamma }}}\right)>0;n=0,1,2,\dots ;\\ {\displaystyle \frac{dM}{dt}}\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2\gamma }}}\right)\ge 2\left(\lambda +{\displaystyle \frac{\gamma }{\lambda }}\right)M\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2\gamma }}}\right);\\ M\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2\gamma }}}\right)\equiv M\left(0\right)\exp \left({\displaystyle \frac{\lambda \pi n}{2\sqrt{{\lambda }^2+2\gamma }}}\right),\\ {\displaystyle \frac{dM}{dt}}\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2\gamma }}}\right)\equiv {\displaystyle \frac{dM}{dt}}\left(0\right)\exp \left({\displaystyle \frac{\lambda \pi n}{2\sqrt{{\lambda }^2+2\gamma }}}\right);\\ M\left(0\right)>0,{\displaystyle \frac{dM}{dt}}\left(0\right)\ge 2\left(\lambda +{\displaystyle \frac{\gamma }{\lambda }}\right)M\left(0\right).\end{array}$$

(27)

The first two inequalities in the additional conditions (27) are the sufficient conditions for linear practical instability [16,32].

In accordance with inequality (26) and conditions (27), an a priori exponential estimate from below for small 3D perturbations $\overrightarrow{\xi }$ (23) and (24) can be obtained:

$$M\left(t\right)\ge C_3{e}^{\lambda t}.$$

(28)

Here, $C_3$ is a known positive constant.

An indefiniteness of the integral $E_3$ (22) in sign allows us to solve initial boundary value problem (24) in relation to small spatial perturbations which are in the form of normal waves [32,35]. Furthermore, as the positive constant $\lambda$ is arbitrary, supposing that the mixed problem (24) has even at least one solution which grows in time and corresponds to a small 3D perturbation in the form of a normal wave, this solution will automatically satisfy the Gubarev inequality (26), the countable set of conditions (27), and the a priori exponential lower estimate (28). It can be inferred that the solutions to the initial boundary value problem (24) with additional conditions (27) make up a non-empty set, in which these solutions serve as small spatial perturbations in the form of normal waves. According to the definition of Lyapunov instability for solutions to systems of differential equations [36], the a priori exponential lower estimate (28) indicates that there is at least one small 3D perturbation ${\xi }_i(x_i,v_{0i},t)$ (23) and (24) for the exact stationary solutions (19) and (20) to the mixed problem (17), which grows over time and has a growth rate which is no less than that of an exponential function.

References [37,38] adhered to the point of view that if theoretical instability takes place on a semi-infinite time interval, then practical instability may or may not exist on a finite time interval at the same time. However, we demonstrated that sufficient conditions for linear practical instability hold if and only if linear theoretical stability conditions are not met. It is worth noting that the sufficient conditions for linear practical instability are constructive; this means that these conditions can be used in experiments, numerical calculations, and so on.

It should be noted that there is a quite simple physical explanation for an absolute linear theoretical instability of the exact stationary solutions (19) and (20) to the initial boundary value problem (17) with respect to small spatial perturbations ${\xi }_i(x_i,v_{0i},t)$ (23) and (24): as the stationary potential of a self-consistent electric field ${\phi }^0\left(x_i\right)$ is constant in the velocity continuum, there are no forces in the two-component Vlasov–Poisson plasma that can prevent the stationary distribution function ${\rho }^0(x_i,v_{0i})$ of electrons from developing in the phase space and the evolution of small 3D perturbations (23) and (24) that grow with time.

Subsequently, in Section 3.4, we suggest another example to confirm these conclusions.

3.4. Another Analytical Example

By analogy with the instance for the kinetic case, we create an example with particular solutions in this Section. At first, let us assume that the mixed problem (17) has the following exact stationary solutions:

$$u_i=u_i^0\left(v_{0i}\right),\rho ={\rho }^0\left(v_{0i}\right),\phi ={\phi }^0\equiv \mathrm{const}$$

(29)

satisfying the stationary integral equation

$${\int }_{{\mathcal{R}}^3}{\rho }^0(x_i,v_{0i})d{\overrightarrow{v}}_0=1.$$

(30)

Next, let us check whether the exact stationary solutions (29) and (30) are stable in relation to small spatial perturbations ${\overrightarrow{u}}^{\prime }$, ${\rho }^{\prime }$, and ${\phi }^{\prime }$. To this end, let us linearize the initial boundary value problem (17) in the vicinity of these stationary solutions, which allows us to obtain a mixed problem

$$\begin{array}{c}{\displaystyle \frac{\partial u_i^{\prime }}{\partial t}}+u_k^0{\displaystyle \frac{\partial u_i^{\prime }}{\partial x_k}}=-{\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}},\\ {\displaystyle \frac{\partial {\rho }^{\prime }}{\partial t}}+{\rho }^0{\displaystyle \frac{\partial u_i^{\prime }}{\partial x_i}}+u_i^0{\displaystyle \frac{\partial {\rho }^{\prime }}{\partial x_i}}=0,\\ {\displaystyle \frac{{\partial }^2{\phi }^{\prime }}{\partial x_i^2}}=-4\pi {\int }_{{\mathcal{R}}^3}{\rho }^{\prime }(x_i,v_{0i},t)d{\overrightarrow{v}}_0;\\ {\overrightarrow{u}}^{\prime }(x_i,v_{0i},0)={\overrightarrow{u}}_0^{\prime }(x_i,v_{0i}),{\rho }^{\prime }(x_i,v_{0i},0)={\rho }_0^{\prime }(x_i,v_{0i})\end{array}$$

(31)

and the following auxiliary equations:

$$\begin{array}{c}{\displaystyle \frac{\partial {\kappa }^{\prime }}{\partial t}}+u_i^0{\displaystyle \frac{\partial {\kappa }^{\prime }}{\partial x_i}}=0,\\ {\displaystyle \frac{\partial J^{\prime }}{\partial t}}+J^0{\displaystyle \frac{\partial u_i^{\prime }}{\partial x_i}}+u_i^0{\displaystyle \frac{\partial J^{\prime }}{\partial x_i}}=0.\end{array}$$

As in Section 3.2 above, the linear analog of the full-energy functional has the following ad hoc form:

$$\begin{array}{c}{E}_4\equiv {\displaystyle \frac{1}{2}}\underset{{\mathcal{R}}^3}{\int \int }\left[{\rho }^0{\left(u_i^{\prime }\right)}^2+2u_i^0{u}_i^{\prime }{\rho }^{\prime }+{\displaystyle \frac{d^2{\mathsf{\Phi }}_1}{d{\kappa }^2}}\left({\kappa }^0\right)J^0{\left({\kappa }^{\prime }\right)}^2\right]d\overrightarrow{x}d{\overrightarrow{v}}_0\\ +{\displaystyle \frac{1}{8\pi }}{\int }_{{\mathcal{R}}^3}{\left({\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}}\right)}^{2}d\overrightarrow{x}\\ +\underset{{\mathcal{R}}^3}{\int \int }\left[{\mathsf{\Phi }}_1\left({\kappa }^0\right)+{\kappa }^0{\displaystyle \frac{{\left(u_1^0\right)}^2+{\left(u_2^0\right)}^2+{\left(u_3^0\right)}^2+{\phi }^0}{2}}\right]J_{1}d\overrightarrow{x}d{\overrightarrow{v}}_0=\mathrm{const};\end{array}$$

(32)

$${\displaystyle \frac{d{\mathsf{\Phi }}_1}{d\kappa }}\left({\kappa }^0\right)=-{\displaystyle \frac{{\left(u_1^0\right)}^2+{\left(u_2^0\right)}^2+{\left(u_3^0\right)}^2+{\phi }^0}{2}}.$$

Now, we redefine the Lagrangian displacements field [34]

$$\begin{array}{c}\hfill {\xi }_i={\xi }_i(x_i,v_{0i},t);{\displaystyle \frac{\partial {\xi }_i}{\partial t}}=u_i^{\prime }-u_k^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_k}}\end{array}$$

(33)

for small 3D perturbations (31) and (32). In accordance with Equation (33), the mixed problem (31) and auxiliary equations have the following form:

$$\begin{array}{c}{\displaystyle \frac{{\partial }^2{\xi }_i}{\partial t^2}}+2u_k^0{\displaystyle \frac{{\partial }^2{\xi }_i}{\partial x_k\partial t}}+u_k^0{u}_m^0{\displaystyle \frac{{\partial }^2{\xi }_i}{\partial x_k\partial x_m}}=-{\displaystyle \frac{\partial {\phi }^{\prime }}{\partial x_i}},\\ {\rho }^{\prime }=-{\rho }^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_i}},\\ {\displaystyle \frac{{\partial }^2{\phi }^{\prime }}{\partial x_i^2}}=4\pi {\int }_{{\mathcal{R}}^3}{\rho }^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_i}}d{\overrightarrow{v}}_0;\\ {\xi }_i(x_i,v_{0i},0)={\left({\xi }_i\right)}_0(x_i,v_{0i}),{\displaystyle \frac{\partial {\xi }_i}{\partial t}}(x_i,v_{0i},0)={\left({\displaystyle \frac{\partial {\xi }_i}{\partial t}}\right)}_0(x_i,v_{0i});\end{array}$$

(34)

$$\begin{array}{c}{J}^{\prime }=-J^0{\displaystyle \frac{\partial {\xi }_i}{\partial x_i}},{\kappa }^{\prime }=0.\end{array}$$

Furthermore, we re-introduce the Lyapunov functional [15,16,32]:

$$\begin{array}{c}\hfill M\equiv \int {\int }_{{\mathcal{R}}^3}{\rho }^0{\left({\xi }_i\right)}^{2}d{\overrightarrow{v}}_{0}d\overrightarrow{x}\ge 0,\end{array}$$

(35)

and calculate its first and second derivatives with respect to t. Then, analogous to the previous narration, we can obtain an original differential inequality [16,32] with an arbitrary parameter $\lambda$ for the functional M (35), namely, the Gubarev inequality

$${\displaystyle \frac{d^{2}M}{d{t}^2}}-2\lambda {\displaystyle \frac{dM}{dt}}+2({\lambda }^2+{\alpha }_4)M\ge 0,$$

(36)

where ${\alpha }_4\equiv 4\pi {\int }_{{\mathcal{R}}^3}{\rho }^{0}d{\overrightarrow{v}}_0>0$ denotes a constant value.

If the parameter $\lambda >0$, we, similar to that in Section 3.3, obtain a countable set of conditions [16,32]:

$$\begin{array}{c}M\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2{\alpha }_4}}}\right)>0;n=0,1,2,\dots ;\\ {\displaystyle \frac{dM}{dt}}\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2{\alpha }_4}}}\right)\ge 2\left(\lambda +{\displaystyle \frac{{\alpha }_4}{\lambda }}\right)M\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2{\alpha }_4}}}\right);\\ M\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2{\alpha }_4}}}\right)\equiv M\left(0\right)\exp \left({\displaystyle \frac{\lambda \pi n}{2\sqrt{{\lambda }^2+2{\alpha }_4}}}\right);M\left(0\right)>0;\\ {\displaystyle \frac{dM}{dt}}\left({\displaystyle \frac{\pi n}{2\sqrt{{\lambda }^2+2{\alpha }_4}}}\right)\equiv {\displaystyle \frac{dM}{dt}}\left(0\right)\exp \left({\displaystyle \frac{\lambda \pi n}{2\sqrt{{\lambda }^2+2{\alpha }_4}}}\right);\\ {\displaystyle \frac{dM}{dt}}\left(0\right)\ge 2\left(\lambda +{\displaystyle \frac{{\alpha }_4}{\lambda }}\right)M\left(0\right).\end{array}$$

(37)

As mentioned in this Section above, the first two inequalities in the system (37) can be interpreted as sufficient conditions for linear practical instability [16,32].

Then, applying the relation (36) and the countable set (37), one receives an a priori exponential estimate from below for small spatial perturbations $\overrightarrow{\xi }$ (33) and (34):

$$M\left(t\right)\ge C_4{e}^{\lambda t};C_4\equiv \mathrm{const}>0.$$

(38)

Here, $C_4$ is a known constant. As the a priori exponential lower estimate (38) is constructed without any constraints, we proved that exact stationary solutions (29) and (30) to the system of Equation (17) are absolutely linearly unstable with respect to the studied small spatial perturbations.

According to the Euler–Poisson integral [39], exact stationary solutions (29) and (30) to the initial boundary value problem (17) have the following form:

$${\rho }^0={\pi }^{-{\displaystyle \frac{3}{2}}}{e}^{-v_{01}^2-v_{02}^2-v_{03}^2},u_i^0=v_{0i},{\phi }^0\equiv \mathrm{const}$$

(39)

and satisfy the relation (30).

Next, let us choose the following form for small 3D perturbations ${\phi }^{\prime }$:

$${\phi }^{\prime }=C_5{e}^{{\alpha }_{5}t}(C_5\equiv \mathrm{const},{\alpha }_5\equiv \mathrm{const}>0).$$

(40)

To obtain the expressions for ${\xi }_i$, we substitute the functions (39) and (40) into the system (34) and obtain the following formulas:

$$\begin{array}{cc}\hfill {\xi }_1=& x_3{v}_{03}{\left(v_{01}\right)}^2\exp \left\{{\beta }_1\left(t-{\displaystyle \frac{x_3}{v_{03}}}\right)\right\}{f}_1\left({\displaystyle \frac{x_1}{v_{01}}}-{\displaystyle \frac{x_3}{v_{03}}},{\displaystyle \frac{x_2}{v_{02}}}-{\displaystyle \frac{x_3}{v_{03}}}\right)\hfill \\ \hfill & +v_{01}{v}_{02}{\left(v_{03}\right)}^2\exp \left\{{\gamma }_1\left(t-{\displaystyle \frac{x_3}{v_{03}}}\right)\right\}{g}_1\left({\displaystyle \frac{x_1}{v_{01}}}-{\displaystyle \frac{x_3}{v_{03}}},{\displaystyle \frac{x_2}{v_{02}}}-{\displaystyle \frac{x_3}{v_{03}}}\right),\hfill \\ \hfill {\xi }_2=& x_3{v}_{03}{v}_{02}{v}_{01}\exp \left\{{\beta }_2\left(t-{\displaystyle \frac{x_3}{v_{03}}}\right)\right\}{f}_2\left({\displaystyle \frac{x_1}{v_{01}}}-{\displaystyle \frac{x_3}{v_{03}}},{\displaystyle \frac{x_2}{v_{02}}}-{\displaystyle \frac{x_3}{v_{03}}}\right)\hfill \\ \hfill & +{\left(v_{02}\right)}^2{\left(v_{03}\right)}^2\exp \left\{{\gamma }_2\left(t-{\displaystyle \frac{x_3}{v_{03}}}\right)\right\}{g}_2\left({\displaystyle \frac{x_1}{v_{01}}}-{\displaystyle \frac{x_3}{v_{03}}},{\displaystyle \frac{x_2}{v_{02}}}-{\displaystyle \frac{x_3}{v_{03}}}\right),\hfill \\ \hfill {\xi }_3=& x_3{v}_{01}{\left(v_{03}\right)}^2\exp \left\{{\beta }_3\left(t-{\displaystyle \frac{x_3}{v_{03}}}\right)\right\}{f}_3\left({\displaystyle \frac{x_1}{v_{01}}}-{\displaystyle \frac{x_3}{v_{03}}},{\displaystyle \frac{x_2}{v_{02}}}-{\displaystyle \frac{x_3}{v_{03}}}\right)\hfill \\ \hfill & +v_{02}{\left(v_{03}\right)}^3\exp \left\{{\gamma }_3\left(t-{\displaystyle \frac{x_3}{v_{03}}}\right)\right\}{g}_3\left({\displaystyle \frac{x_1}{v_{01}}}-{\displaystyle \frac{x_3}{v_{03}}},{\displaystyle \frac{x_2}{v_{02}}}-{\displaystyle \frac{x_3}{v_{03}}}\right),\hfill \end{array}$$

(41)

where ${\beta }_i$ and ${\gamma }_i$ are positive constants; $i=1,2,3$; and $x_i/v_{0i}>0$. Furthermore, the volumetric integral of the functions $f_i$ and $g_i$ in the third equation of system (34) exists and, due to the oddness of its integrand, this integral is equal to zero. Finally, redefining the functions ${\xi }_i(x_i,v_{0i},t)$ (41) separately by continuity, one has the following equalities:

$$\left({\xi }_1{|}_{t=0,x_1=0}\right){|}_{v_{01}=0}=0,\left({\xi }_2{|}_{t=0,x_2=0}\right){{|}_{v_{02}=0}=0,\left({\xi }_3{|}_{t=0,x_3=0}\right)|}_{v_{03}=0}=0.$$

From the example considered, one can see that although the functions ${\rho }^0$ (29), (30), and (39) are decreasing, we still constructed a perturbation which is growing. Therefore, we can infer that the exact stationary solutions under consideration are absolutely unstable. As in the kinetic case, this analytical example is a counterexample to both the Newcomb–Gardner–Rosenbluth linear stability sufficient condition [3,4,5] and the Penrose criterion [2,24].

In short, we can use the sufficient conditions (37) (see the first two inequalities) to determine linear practical instability at finite time intervals. If these conditions are met, the dynamic equilibrium states of plasma is unstable; however, if the sufficient conditions are not met, then the plasma dynamic equilibrium states is stable in a practical sense—at least in relation to small spatial perturbations in the form of normal waves. This signifies that new, potentially desirable plasma confinement devices can be created, which may be managed through real-time feedback based on these practical instability conditions.

4. Conclusions

In summary, we have studied the linear stability of dynamic equilibria for a two-component Vlasov–Poisson plasma containing electrons and ions in 3D space. Using the direct Lyapunov method with the Gubarev modification, we have shown that when the stationary distribution functions of ions and electrons are isotropic in the physical continuum, the exact stationary solutions (19) and (20) are absolutely unstable with respect to small spatial perturbations ${\xi }_i(x_i,v_{0i},t)$ (23) and (24). We have also constructed counterexamples to the Newcomb–Gardner–Rosenbluth sufficient conditions (8) for linear stability, the Penrose criterion, and Landau damping, pointing out that these classical results hold if and only if the explored small 3D perturbations come from some incomplete unclosed subclasses. It is worth noting that when studying stability problems, it is essential to determine whether the class of perturbations is complete and closed. Additionally, we have found sufficient conditions (see the first two inequalities from countable set (27)) for linear practical instability of dynamic equilibrium states for the considered two-component Vlasov–Poisson plasma. For small spatial perturbations ${\xi }_i(x_i,v_{0i},t)$ (23) and (24), an a priori exponential estimate (28) from below is established, and the initial data are specified for small perturbations that grow over time.

Concisely, the main objective of this study are achieved. Concretely, it is proven that the dynamic equilibria of a two-component Vlasov–Poisson plasma are absolutely unstable regarding small 3D perturbations; that is, there are always small spatial perturbations that grow in time. It is worth emphasizing that all of the results given here were obtained via the direct Lyapunov method with the Gubarev modification [16,31,32,33] in a general manner, without simplifying the linearized equations or imposing any additional restrictions.

Furthermore, the Gubarev algorithm for constructing Lyapunov functionals that grow with time is believed to be helpful for various types of linear theoretical and practical stability problems of hydrodynamic or kinetic type. Furthermore, when a functional of the type M (25) increases along the solutions to the linearized initial boundary value problems of hydrodynamic type, it can be applied to solve the instability for plasmic dynamic equilibrium states with respect to small 3D perturbations at semi-infinite (theoretical instability) or finite (practical instability) time intervals. In addition, one can readily choose a constant $\lambda$ in the exponent indicator from the lower estimate (28), which allows us to conceive any growing solution to the initial boundary value problem (23), (24), and (27) as an analog of the solution to an ill-posed problem in the sense of Hadamard [40].

In summary, we can collect experimental data on the time evolution of electrons for a two-component Vlasov–Poisson plasma using recording equipment and apply the sufficient conditions (27) for linear practical instability to clarify whether small spatial perturbations ${\xi }_i(x_i,v_{0i},t)$ (23) and (24) have a tendency to grow exponentially and in an unbounded manner over time, thereby destroying the plasmic dynamic equilibria (19) and (20) (or, equivalently, (3) and (4)). It can be concluded that the results of this research are important for controlling plasma, and we believe that the obtained results have great significance for the development of controlled thermonuclear fusion experimental devices.