\documentstyle{SibMatJ} % \TestXML %\Rus \topmatter \Author Skubachevskii \Initial A. \Initial L. \Gender he \Email alskubachevskii\@yandex.ru \AffilRef 1 \AffilRef 2 \endAuthor \Affil 1 \Division S.~M. Nikolskii Mathematical Institute \Organization RUDN University \City Moscow \Country Russia \endAffil \Affil 2 \Organization Moscow Center for Fundamental and Applied Mathematics \City Moscow \Country Russia \endAffil \datesubmitted March 29, 2026\enddatesubmitted %\daterevised April 2, 2026\enddaterevised \dateaccepted April 2, 2026\enddateaccepted \UDclass 517.955+517.958 \endUDclass \thanks The work was supported by the Ministry of Science and Higher Education of the Russian Federation within the state assignment (project FSSF--2026--0013). \endthanks \dedication Devoted to the 70th anniversary of my friend, professor Gennady Demidenko, whose textbooks and monographs helped me in my research. \enddedication \title The Cauchy Problem for the Vlasov--Poisson System with External Magnetic Field \endtitle \abstract We consider the Vlasov--Poisson system with external homogeneous magnetic field describing kinetics of two-component rarefied plasma in the three-dimensional case. For any initial density distribution functions with compact supports, we obtain sufficient conditions for external magnetic field that provide global existence of density distribution functions with arbitrary small growth of their supports with respect to space variables. \endabstract \keywords Vlasov--Poisson system, Cauchy problem, plasma confinement \endkeywords \endtopmatter \head 1. Introduction \endhead The Vlasov equations were derived in~[1]. Global classical solutions to the Cauchy problem for the Vlasov--Poisson system were studied in~[2--6]. The Vlasov--Poisson equations have many important applications: to the theory of high temperature plasma, to stellar dynamics, etc. One of the most exciting applications is connected with controlled thermonuclear fusion. Since plasma in a fusion reactor has a very high temperature, collisions of charged particles with the wall of the reactor may lead to a destruction of the wall. Therefore one of the most important problems in plasma physics is confinement of high temperature plasma at some distance from the wall. In order to provide plasma confinement physicists are using external magnetic field of a special form~[7]. In this paper we consider a simplest model of plasma confinement: the Cauchy problem for the Vlasov--Poisson system with external magnetic field. \Sec*{Section~2} deals with the statement of problem, notation and formulation of the main result. \Sec*{Section~3} is devoted to the existence of global classical solution. In \Sec*{Section~4}, we obtain a priori estimate for the norm of electric field strength. \Sec*{Section~5} deals with the properties of characteristics for the Vlasov equations. In \Sec*{Section~6}, we obtain sufficient conditions for external magnetic field, which provide existence of density distribution functions with arbitrary small growth of their supports with respect to space variables as the time changes on a finite interval. Conditions for existence of classical solutions to mixed problems for the Vlasov--Poisson system with external magnetic field providing confinement of two-component plasma in the cases of half-space, infinite cylinder (``mirror trap'') and torus (``tokamak'') were studied in~[8--15]. \head 2. Statement of Problem. Main Result \endhead We consider the Vlasov--Poisson system of equations \iftex $$ \align &-\Delta \varphi(x,t) = 4\pi\int\limits_{\Bbb{R}^3}\sum_{\beta}\beta f^\beta(x,v,t)dv,\quad x\in \Bbb{R}^3,\ t \in (0,T),\ \beta=\pm 1, \tag2.1 \\ &\frac{\partial f^\beta}{\partial t}+(v,\nabla_x f^\beta)+\beta(-\nabla_x\varphi+[v,B(x)], \nabla_v f^\beta)=0, \quad x,v\in\Bbb{R}^3,\ t \in (0,T),\ \beta=\pm 1, \tag2.2 \endalign $$ \else $$ -\Delta \varphi(x,t) = 4\pi\int\limits_{\Bbb{R}^3}\sum_{\beta}\beta f^\beta(x,v,t)dv,\quad x\in \Bbb{R}^3,\ t \in (0,T),\ \beta=\pm 1, \tag2.1 $$ $$ \frac{\partial f^\beta}{\partial t}+(v,\nabla_x f^\beta)+\beta(-\nabla_x\varphi+[v,B(x)], \nabla_v f^\beta)=0, \quad x,v\in\Bbb{R}^3,\ t \in (0,T),\ \beta=\pm 1, \tag2.2 $$ \fi with the initial conditions $$ f^\beta(x,v,t)\big|_{t=0}= f_0^\beta(x,v),\quad x,v\in\Bbb{R}^3,\ \beta=\pm 1. \tag2.3 $$ Here $ f^\beta=f^\beta(x,v,t)\geq 0 $ is the density distribution function of positively charged ions if $ \beta=+1$ and negatively charged electrons if $ \beta=-1 $, at the point $ x=(x_1,x_2,x_3)$ with velocity $ v=(v_1,v_2,v_3) $ at the moment $ t $; $ \varphi=\varphi(x,t) $ is a potential of self-consistent electric field; $ \nabla_x $ and $ \nabla_v $ are gradients with respect to $ x $ and $v$, respectively; $ B=B(x) $ is the induction of external magnetic field; $(.,.)$ is the scalar product in $ \Bbb{R}^3 $; $[.,.] $ is the vector product in $ \Bbb{R}^3 $. Suppose that the potential $ \varphi $ satisfies the decreasing condition at infinity $$ \lim_{|x|\to\infty}\varphi(x,t)=0,\quad 0\le t\le T. \tag2.4 $$ Denote by $ {C^s}(\Bbb{R}^n) $, $s\geq 0 $, $n\in\Bbb{N}$, the H\"{o}lder space of continuous functions on $\Bbb{R}^n$ having all continuous derivatives on $ \Bbb{R}^n $ up to the $k$th order, $k=[s]$, with the finite norm \iftex $$ \align & \|u\|_{s}=\max_{|\alpha|\leq k}\,\sup_{x}|D^{\alpha}u(x)| \quad\text{if } s=k\in\Bbb{Z},\ 0\leq k, \tag2.5 \\ & \|u\|_{s}=\|u\|_{k}+\|u\|_{\sigma} \quad\text{if } s=k+\sigma,\ 0\leq k\in\Bbb{Z},\ 0<\sigma<1, \tag2.6 \endalign $$ \else $$ \|u\|_{s}=\max_{|\alpha|\leq k}\,\sup_{x}|D^{\alpha}u(x)| \quad\text{if } s=k\in\Bbb{Z},\ 0\leq k, \tag2.5 $$ $$ \|u\|_{s}=\|u\|_{k}+\|u\|_{\sigma} \quad\text{if } s=k+\sigma,\ 0\leq k\in\Bbb{Z},\ 0<\sigma<1, \tag2.6 $$ \fi where $$ \align & |u|_{\sigma}=\max_{|\alpha|=k}\,\sup_{x\neq y}{|x-y|^{-\sigma}| D^{\alpha}u(x)-D^{\alpha}u(y)|}, \\ & D^{\alpha}=\Bigl(\frac{\partial}{\partial x_1}\Bigr)^{\alpha_1}\dotsm \Bigl(\frac{\partial}{\partial x_n}\Bigr)^{\alpha_n}, \quad\alpha=(\alpha_1,\dots,\alpha_n),\ |\alpha|=\alpha_1+\dots+\alpha_n. \endalign $$ Let $C(\Bbb{R}^n)=C^0(\Bbb{R}^n)$, and let $\overset{\circ}\to{C}^k{(\Bbb{R}^n)} $, $ k,n\in\Bbb{N} $, be a set of continuously differentiable functions in~$ \Bbb{R}^n $ with compact supports. We introduce the Banach space $ C([0,T], C^s(\Bbb{R}^3)) $, $ s\geq 0 $, consisting of continuous functions $ [0,T]\ni t\mapsto\varphi(.,t)\in C^s(\Bbb{R}^3) $ with the norm $$ \|\varphi\|_{s;T}=\sup_{0\leq t\leq T}\|\varphi(.,t)\|_{s}. \tag2.7 $$ Let $ B_\rho(x^0)=\{x\in\Bbb{R}^3 : |x-x^0|<\rho\} $, and let $ B_\rho=B_\rho(0)$. Denote by $|B_\rho|=\frac{4\pi\rho^3}{3}$ the volume of the ball $ B_\rho$. Further, we denote by $ k_i $ and $ c_j $ positive constants in inequalities, which do not depend on functions contained in these inequalities. Denote by $\widehat{C}^s(\Bbb{R}^3) $ the space of vector-functions $ Y=(Y_1,Y_2,Y_3)^ T $, having coordinates $ Y_i\in C^s(\Bbb{R}^3) $, with the norm $$ \langle Y\rangle _s=\Biggl\{\sum_{i=1}^{3}\|Y_i\|_s^2\Biggr\}^{1/2}. \tag2.8 $$ Assume that the following conditions hold. \proclaim{Condition 2.1} Let $ B \in \widehat{C}^1(\Bbb{R}^3) $. \endproclaim \proclaim{Condition 2.2} Let $ f_0^\beta \in \overset{\circ}\to{C}^1(\Bbb{R}^6) $, and let $$ \operatorname{supp}f_0^\beta \subset D_0 := B_\lambda \times B_\rho, $$ where $ \lambda, \rho > 0 $. \endproclaim \proclaim{Condition 2.3} Let $ B(x) = (0, 0, b) $ for $ x \in \{ x \in \Bbb{R}^3 : |x'| < 2\lambda \}$, where $ x = (x', x_3) $. \endproclaim The main result of this paper can be formulated as following. \proclaim{Theorem 2.1} Let \Par{Condition 2.1}{Conditions~{\rm2.1}}--\Par{Condition 2.3}{\rm2.3} hold. Then for any $ \delta, 0 < \delta < \lambda$, there is $b > 0$ such that there exists a solution of problem~\Tag(2.1)--\Tag(2.4) $\{\varphi,f^\beta\}$, $ \varphi \in C([0, T], C^{2+\sigma}(\Bbb{R}^3))$, $ f^\beta \in C^1(\Bbb{R}^3 \times \Bbb{R}^3 \times [0, T]) $ having the following property: $\operatorname{supp} f^\beta \subset ( \{ x \in \Bbb{R}^3 : |x'| < \lambda + \delta \} \cap B_{\lambda_1}) \times B_{\rho_1}$, where $\lambda < \lambda_1 < \infty, \, \rho < \rho_1 < \infty$. \endproclaim Further we obtain \Par*{Theorem~2.1} as a corollary from \Par*{Theorem~6.1}, in which we describe explicit condition for a constant $b$; see \Tag(5.9). A proof of \Par*{Theorem~6.1} consists of three steps. First, we will formulate lemma on the existence of global classical solution to the problem~\Tag(2.1)--\Tag(2.4). Second, we will obtain a priori estimate for the norm of electric field strength through the norm of density distribution function for charged particles. At the third step we will estimate deviations of trajectories of particles from the Larmor trajectories. From these estimates we derive the conclusion of \Par*{Theorem~2.1}. \head 3. Existence of Global Classical Solutions \endhead \proclaim{Lemma 3.1} Let \Par{Condition 2.1}{Conditions~{\rm2.1}} and \Par{Condition 2.2}{\rm2.2} hold. Then there exists a solution of problem \Tag(2.1)--\Tag(2.4) $\{\varphi, f^\beta\}$ such that $ \varphi \in C([0, T], C^{2+\sigma}(\Bbb{R}^3)) $ and $ f^\beta \in C^1(\Bbb{R}^3 \times \Bbb{R}^3 \times [0, T])$. Moreover, $ f^\beta$ have compact supports, $\beta = \pm 1 $. \endproclaim \demo{Proof} By virtue of Theorem 1 from~[2] and the theorem from~[4], there exists a solution of the problem \Tag(2.1)--\Tag(2.4) $\varphi \in C([0, T], C^{2+\sigma}(\Bbb{R}^3))$, $f^\beta \in C^1(\Bbb{R}^3 \times \Bbb{R}^3 \times [0, T])$, $\beta = \pm 1 $. Moreover, $$ \aligned R(T): =\biggl\{1+\max\limits_{\beta}\sup\limits_{v\in \Bbb{R}^3 } |v|:\ &\text{there exist } x\in \Bbb{R}^3 \text{ and } t \in [0, T] \\ &\text{such that }f^\beta(x, v, t) \neq 0 \biggr\} < \infty. \endaligned \tag3.1 $$ We note that unlike papers~[2] and~[4] we have the additional term $\beta([v,B(x)], \nabla_vf^\beta)$ in the Vlasov equations, which provides plasma confinement. However, we still can use a standard proof for the existence of global solution, since it is based on the invariance of the Lebesgue measure with respect to transformations generated by vector-field %\? $$ \{V^\beta_\varphi(\tau), -\beta \nabla_x\varphi(X^\beta_\varphi(\tau),\tau) + \beta [V^\beta_\varphi(\tau),B(X^\beta_\varphi(\tau))]\} $$ in characteristic equations and a priori estimates of velocities following from the equality $(V^\beta_\varphi(\tau), [V^\beta_\varphi(\tau),B(X^\beta_\varphi(\tau))])=0$. \qed\enddemo \demo{Remark 3.1} Generally speaking $R(T)$ depends also on $f^\beta_0( \beta = \pm 1 )$. \enddemo \head 4. A Priori Estimate for the Norm of Electric Field Strength \endhead We define the space $C([0, T], C_\Omega^s(\Bbb{R}^3))$, where $ C_\Omega^s(\Bbb{R}^3)=\{w\in C^s(\Bbb{R}^3):\operatorname{supp} w\subset\Omega\}$, $\Omega \subset \Bbb{R}^3 $ is a~bounded domain. Let $C_0(\Bbb{R}^3) = \{ g \in C(\Bbb{R}^3); g(x) \rightarrow 0\quad \text{as } |x| \rightarrow \infty \}$. We consider the equation $$ -\Delta u(x) = F(x), \quad x \in \Bbb{R}^3, \tag4.1 $$ with the decreasing condition at infinity $$ u(x) \to 0 \quad \text{as} \quad |x| \to \infty. \tag4.2 $$ \proclaim{Lemma 4.1} Let $\Omega \subset \Bbb{R}^3$ be a bounded domain. Then, for any function $F \in C_\Omega^\sigma(\Bbb{R}^3)$, $0 < \sigma < 1$, there exists a unique solution of equation \Tag(4.1) with condition \Tag(4.2) $u \in C^{2+\sigma}(\Bbb{R}^3) \cap C_0(\Bbb{R}^3)$. \endproclaim Moreover, $$ \||\nabla u|\|_0 \leqslant c_1 \|F\|_0, \tag4.3 $$ where $$ 0 0$ does not depend on $x$, i.e., $w \in C_0(\Bbb{R}^3)$. Therefore, problem \Tag(4.1), \Tag(4.2) has the classical solution $u = w\in C^{2+\sigma}(\Bbb{R}^3) \cap C_0(\Bbb{R}^3)$. Differentiating the right hand side of \Tag(4.4), we obtain the expression for constant $c_1$. In order to prove a uniqueness of solution to problem \Tag(4.1), \Tag(4.2), we put $F(x)=0$. Then, by virtue of Liouville theorem, $u(x)$ is a polynomial. However, a nontrivial polynomial does not satisfy the decreasing condition \Tag(4.2). \qed\enddemo Let $\{\varphi,f^\beta\}$, $\varphi\in C([0,T], C^{2+\sigma}(\Bbb{R}^3))$, $ f^\beta\in C^1(\Bbb{R}^3\times\Bbb{R}^3\times[0,T])$, $\beta=\pm 1$, be a solution of problem \Tag(2.1)--\Tag(2.4). For the present function $\varphi\in C([0,T], C^{2+\sigma}(\Bbb{R}^3))$, the Vlasov equations \Tag(2.2) with the initial conditions \Tag(2.3) can be solved with the help of characteristics method. For this purpose, we consider the following system of ordinary differential equations: \iftex $$ \align &\frac{d X_{\varphi}^{\beta}(\tau)}{d\tau} = V_{\varphi}^\beta (\tau),\quad 0<\tau0$ is a constant from inequality \Tag(4.3). \endproclaim \demo{Proof} Clearly, $$ f^{\beta}(x,v,t)= f_0^{\beta}(S_\varphi^{\beta}(0,x,v,t)), \quad x\in\Bbb{R}^3,\ v\in\Bbb{R}^3,\ t\in(0,T). \tag4.11 $$ From \Par*{Lemma 4.1} and inequality \Tag(3.1) we obtain $$ \align \||\nabla\varphi |\|_{0,\tau} &\leq c_{1}{4\pi}\sup\limits_{x,t}\int\limits_{\Bbb{R}^3}| \sum\limits_\beta\beta f^\beta(x,v,t)|dv \\ &\leq c_1 4\pi\sup\limits_{x,t}\int\limits_{\Bbb{R}^3}\max\limits_{\beta} f^{\beta}(x,v,t) \,dv \\ &=c_1 4\pi\sup\limits_{x,t}\int\limits_{\Bbb{R}^3}\max\limits_{\beta} f_0^{\beta}(S_{\varphi}^{\beta}(0,x,v,t)),dv \\ &\leq c_1|B_{R(T)}|{4\pi}\max\limits_{\beta}\sup\limits_{(y,w)\in B_{{R}_0}}|f_0^\beta(y,w)|, \quad x \in \Bbb{R}^3,\ t\in[0,T], \endalign $$ where $R_0=(\lambda^2+\rho^2)^{1/2}$. From this it follows inequality \Tag(4.10). \qed\enddemo \head 5. Some Properties of Characteristics \endhead We again consider the system of ordinary differential equations \Tag(4.6) and \Tag(4.7) with initial conditions \Tag(4.8) and \Tag(4.9). \proclaim{Lemma 5.1} Let \Par{Condition 2.1}{Conditions~{\rm2.1}}--\Par{Condition 2.3}{\rm2.3} hold. Then for every $(x, v)\in D_{0}$ and $0\leq t\leq T$ the following estimates take place \iftex $$ \align &|V_{\varphi}^{\beta}(t,x,v,0)|<\rho_1, \tag5.1 \\ &|X_{\varphi}^{\beta}(t,x,v,0)|<\lambda_1, \tag5.2 \endalign $$ \else $$ |V_{\varphi}^{\beta}(t,x,v,0)|<\rho_1, \tag5.1 $$ $$ |X_{\varphi}^{\beta}(t,x,v,0)|<\lambda_1, \tag5.2 $$ \fi where $\rho_1=\rho+T c_1|B_{R(T)}|{4\pi} \max\nolimits_{\beta} \|f_0^\beta\|_0, \lambda_1=\lambda+\rho_1T$. \endproclaim \demo{Proof} Multiplying the left and the right sides %\? of equation \Tag(4.7) by $V_\varphi^\beta$, we have $$ \frac{1}{2}\frac{d}{d\tau}| V_{\varphi}^{\beta}(\tau,x,v,0)|^2 =-\beta(\nabla_{x}{\varphi}(X_\varphi^{\beta}(\tau,x,v,0), V_{\varphi}^{\beta}(\tau,x,v,0)),\quad 0 \leq \tau \leq T. $$ From this equality and from the Cauchy--Bunyakovsky inequality we obtain $$ \frac{1}{2}\frac{d}{d\tau}|V_{\varphi}^{\beta}(\tau,x,v,0)|^2 \leq|\nabla_x\varphi(X_\varphi^{\beta}(\tau,x,v,0) |\cdot|V_{\varphi}^{\beta}(\tau,x,v,0)|, \quad 0 \leq \tau \leq T. $$ Hence, $$ \frac{d}{d\tau}|V_{\varphi}^{\beta}(\tau,x,v,0)|\leq |\nabla_x\varphi(X_\varphi^{\beta}(\tau,x,v,0))|,\quad 0\leq \tau \leq T. \tag5.3 $$ Integrating \Tag(5.3) by $\tau$ from $0$ to $t$, by virtue of \Par*{Lemma~4.2} and \Par*{Condition~2.2}, we have $$ \aligned &|V_{\varphi}^{\beta}(t,x,v,0)|\leq |v|+ \int\limits_{0}\limits^ {t}|\nabla_x\varphi(X_\varphi^{\beta}(\tau,x,v,0))|d\tau \\ &<\rho + Tc_1|B_{R(T)}|{4\pi}\max\limits_{\beta} \|f_0^\beta\|_0. \endaligned \tag5.4 $$ Integrating equation \Tag(4.6) by $\tau$ from $0$ to $t$ and using inequality \Tag(5.4), we obtain \Tag(5.2). \qed\enddemo We introduce the matrix $R(\theta)$ by the formula: $$ R(\theta)=\pmatrix \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \endpmatrix,\quad \theta\in\Bbb{R}. $$ As it is known, multiplication by the matrix $R(\theta)$ corresponds to rotation by the angle $\theta$ on the plane. We now formulate some properties of this operator, which allow us to study the behavior of characteristics of Vlasov equations. \proclaim{Lemma 5.2 \rm (see [11])}\Label{L5.2} \Item (a) $R(\theta_{1})R(\theta_{2})=R(\theta_{1}+\theta_{2})$, $\theta_{1},\theta_{2}\in\Bbb{R}$. \Item (b) $R(\theta)^{m}=R(m\theta)$, $\theta\in\Bbb{R}$, $m\in\Bbb{Z}$. \Item (c) $\frac{d}{d\theta}R(\theta)=R(\frac{\pi}{2})R(\theta)=R(\theta+\frac{\pi}{2})$, $\theta\in\Bbb{R}$. \Item (d) $|R(\theta)x|=|x|$, $\theta\in\Bbb{R}$, $x\in\Bbb{R}^{2}$. \Item (e) $\exp(tR(\theta))=\exp(t\cos\theta)R(t\sin\theta)$. \endproclaim \demo{Proof} Properties \Par{L5.2}{(a)}--\Par{L5.2}{(d)} are obvious. We prove property \Par{L5.2}{(e)}. From property \Par{L5.2}{(b)} and the definition of the matrix $R(\theta)$, it follows that $$ \exp(tR(\theta)) = \sum_{n=0}^{\infty} \frac{t^n}{n!} R(\theta)^n = \sum_{n=0}^{\infty} \frac{t^n}{n!} R(n\theta) = \sum_{n=0}^{\infty} \frac{t^n}{n!} \pmatrix \cos (n\theta) & -\sin (n\theta) \\ \sin (n\theta) & \cos (n\theta) \endpmatrix . \tag5.5 $$ From the Euler formula we have $$ \exp(t \exp(i\theta)) = \exp(t (\cos \theta + i \sin \theta)) = \exp(t \cos \theta)[\cos(t \sin \theta) + i \sin(t \sin \theta)]. \tag5.6 $$ It easy to see that $$ \exp(t \exp(i\theta)) = \sum_{n=0}^{\infty} \frac{t^n}{n!} \exp(in\theta) = \sum_{n=0}^{\infty} \frac{t^n}{n!}\cos (n\theta) + i \sum_{n=0}^{\infty} \frac{t^n}{n!}\sin (n\theta). \tag5.7 $$ From equations \Tag(5.6) and \Tag(5.7), we obtain $$ \aligned \ \sum\limits_{n=0}^\infty \frac{t^n}{n!} \cos(n\theta) = \exp(t\cos\theta)\cos(t\sin\theta), \\ \sum\limits_{n=0}^\infty \frac{t^n}{n!} \sin(n\theta) = \exp(t\cos\theta)\sin(t\sin\theta). \endaligned \tag5.8 $$ Equalities \Tag(5.5) and \Tag(5.8) imply that $$ \align \exp(tR(\theta)) &= \exp(t\cos\theta) \pmatrix \cos(t\sin\theta) & -\sin(t\sin\theta) \\ \sin(t\sin\theta) & \cos(t\sin\theta) \endpmatrix \\ &= \exp(t\cos\theta)R(t\sin\theta). \qed \endalign $$ \enddemo Let $x'=(x_1,x_2)$, and let $X_\lambda^{\beta'}(\tau, x, v, 0):= (X^{\beta}_{\lambda,1}(\tau, x, v, 0), X^{\beta}_{\lambda,2}(\tau, x, v, 0))$. Assume that the following condition is fulfilled. \proclaim{Condition 5.1} Let a constant $b$ in \Par*{Condition~2.3} satisfy the inequality $$ \frac{2}{\delta}(\rho+Tc_{1}|B_{R(T)}|{4\pi}\max\limits_{\beta}\|f_0^\beta\|_0)< b, \tag5.9 $$ where $c_1>0$ is a constant from \Par*{Lemma~4.1}. \endproclaim The next lemma is a generalization of Lemma 3.3 from [11]. \proclaim{Lemma 5.3} Let \Par{Condition 2.1}{Conditions~{\rm2.1}}--\Par{Condition 2.3}{\rm2.3} and \Par*{Condition {\rm5.1}} hold. Then the solution of problem \Tag(4.6)--\Tag(4.9) given by $(X^{\beta}_{\varphi}(\tau, x, v, 0), V^{\beta}_{\varphi}(\tau, x, v, 0)) := S^{\beta}_{\varphi}(\tau, x, v, 0)$ has the following properties: for all $(x,v)\in D_0$ and $\tau\in[0,T]$, $$ |X^{\beta'}_{\varphi}(\tau, x, v, 0)-x'|<\delta, \quad X^{\beta}_{\varphi}(\tau, x, v, 0)\in B_{\lambda_1}, \quad V^{\beta}_{\varphi}(\tau, x, v, 0)\in B_{\rho_1}, $$ where $0<\delta<\lambda$, $\lambda_1=\lambda+\rho_1T$, $\rho_1=\rho+T{c_1}|B_{R(T)}|{4\pi}\max\nolimits_{\beta} \|f_0^\beta\|_0$. \endproclaim \demo{Proof} We prove that $$ |X^{\beta'}_{\varphi}(\tau, x, v, 0)-x'|<\delta \quad\text {for any } \tau\in[0,T]. \tag5.10 $$ Assume to the contrary that there is $\tau_0\in[0,T]$ such that $$ |X^{\beta'}_{\varphi}(\tau, x, v, 0)-x'|\ge \delta. $$ Since $X^{\beta'}_{\varphi}(0, x, v, 0)=x'$, %then for some $0<\tau_1\leqslant\tau_0\leqslant T$, we have \iftex $$ \align & |X^{\beta'}_{\varphi}(\tau_1, x, v, 0)-x'|=\delta, \tag5.11 \\ & |X^{\beta'}_{\varphi}(\tau, x, v, 0)-x'|<\delta,\; \tau \in [0,\tau_1). \tag5.12 \endalign $$ \else $$ |X^{\beta'}_{\varphi}(\tau_1, x, v, 0)-x'|=\delta, \tag5.11 $$ $$ |X^{\beta'}_{\varphi}(\tau, x, v, 0)-x'|<\delta,\; \tau \in [0,\tau_1). \tag5.12 $$ \fi Since $0<\delta<\lambda$, \Par*{Condition~2.3} allows us to rewrite characteristic equation \Tag(4.7) in the form $$ \frac{dV_\varphi^\beta(\tau)}{d\tau} =-\beta\nabla_x \varphi(X_\varphi^{\beta},\tau)+ \beta\pmatrix 0 & b & 0 \\ -b & 0 & 0 \\ 0 & 0 & 0\endpmatrix V_\varphi^{\beta}(\tau),\quad\tau\in(0,\tau_1). $$ Therefore, we have $$ \frac{d}{d\tau} \pmatrix V_{\varphi,1}^{\beta}(\tau) \\ V_{\varphi,2}^{\beta}(\tau) \endpmatrix + \beta b R\Bigl( \frac{\pi}{2} \Bigr) \pmatrix V_{\varphi,1}^{\beta}(\tau) \\ V_{\varphi,2}^{\beta}(\tau) \endpmatrix = -\beta\nabla_{(x_1,x_2)}\varphi(X_\varphi^\beta,\tau), \quad \tau\in(0,\tau_1). $$ If we multiply the last equation by $\exp(\tau\beta b R(\frac{\pi}{2}))$, we get $$ \frac{d}{d\tau} \biggl[ \exp\Bigl( \tau \beta b R\Bigl(\frac{\pi}{2}\Bigr)\Bigr) \pmatrix V_{\varphi,1}^{\beta}(\tau) \\ V_{\varphi,2}^{\beta}(\tau) \endpmatrix \biggr] = -\beta\exp\Bigl(\tau\beta b R\Bigl(\frac{\pi}{2}\Bigr)\Bigr) \nabla_{(x_1,x_2)}\varphi(X_\varphi^\beta,\tau), \quad \tau\in(0,\tau_1). \tag5.13 $$ Integrating equation \Tag(5.13) from $0$ to $t$, $t\in(0,\tau_1]$, we obtain $$ \exp\Bigl( t \beta b R\Bigl(\frac{\pi}{2}\Bigr)\Bigr) \pmatrix V_{\varphi,1}^{\beta}(t) \\ V_{\varphi,2}^{\beta}(t) \endpmatrix - \pmatrix v_1\\ v_2\endpmatrix =- \beta\int\limits_{0}\limits^ {t}\exp\Bigl(\tau\beta b R\Bigl(\frac{\pi}{2}\Bigr)\Bigr)\nabla_{(x_1,x_2)} \varphi(X_{\varphi}^{\beta},\tau)\, d\tau. \tag5.14 $$ \Par*{Lemma~5.2}\itm(e) implies that $$ % \label{5.15} \exp\Bigl(t\beta b R\Bigl(\frac{\pi}{2}\Bigr)\Bigr)=R(t\beta b). \tag5.15 $$ From \Tag(5.14) and \Tag(5.15) it follows that $$ \pmatrix V_{\varphi,1}^{\beta}(t) \\ V_{\varphi,2}^{\beta}(t) \endpmatrix =R(-t\beta b) \pmatrix v_1\\ v_2 \endpmatrix - \beta\int\limits_{0}\limits^ {t}R((\tau-t) \beta b) \nabla_{(x_1,x_2)}\varphi(X_{\varphi}^{\beta},\tau)\, d\tau, \quad t\in(0,\tau_1]. \tag5.16 $$ From \Tag(4.6) and \Tag(5.16) it follows that $$ \pmatrix X_{\varphi,1}^{\beta}(\tau_1) \\ X_{\varphi,2}^{\beta}(\tau_1)\endpmatrix =\pmatrix x_1\\ x_2\endpmatrix +I_1+I_2, \tag5.17 $$ where $$ \align & I_1=\int\limits_{0}\limits^ {\tau_1}R(-t\beta b)\pmatrix v_1\\ v_2\endpmatrix \, dt, \\ & I_2=-\beta \int\limits_{0}^{\tau_1}dt\int\limits_{0}^{t} R((\tau-t)\beta b)\nabla_{(x_1,x_2)}\varphi(X_{\varphi}^{\beta},\tau)\,d\tau. \endalign $$ We now estimate the integrals $I_1$ and $I_2$. \Par*{Lemma~5.2}\itm(c) implies that $$ R (-t \beta b ) = -\frac{1}{\beta b} \frac{d}{dt} \Bigl(R \Bigl(-t \beta b - \frac{\pi}{2}\Bigr)\Bigr). $$ Therefore, by virtue of \Par*{Lemma~5.2}\itm(a), we have $$ \align I_1 &= \frac{1}{\beta b} \{-R(-\tau_1 \beta b)+E\}R \Bigl(-\frac{\pi}{2}\Bigr) \pmatrix v_1 \\ v_2\endpmatrix \\ &=\frac{1}{\beta b}\pmatrix 1-\cos(\tau_1 \beta b)&-\sin(\tau_1 \beta b) \\ \sin(\tau_1 \beta b)&1-\cos(\tau_1 \beta b)\endpmatrix \pmatrix 0 & 1\\ -1 & 0\endpmatrix \pmatrix v_1 \\ v_2\endpmatrix \\ &=\frac{1}{\beta b}\pmatrix \beta\sin(\tau_1 b)v_1+(1-\cos(\tau_1 b))v_2 \\ -(1-\cos(\tau_2 b))v_1+\beta\sin(\tau_1 b)v_2\endpmatrix , \endalign $$ where $ E = \pmatrix 1 & 0\\ 0 & 1\endpmatrix$. Thus we have $$ \aligned |I_1| &=\frac{1}{b}\bigl\{\{\beta\sin(\tau_1 b)v_1+(1-\cos(\tau_1 b))v_2\}^2 \\ &\qquad+ (-(1-\cos(\tau_1 b))v_1+\beta\sin(\tau_1 b)v_2)^2\bigr\}^{1/2} \\ &=\frac{1}{b}\bigl\{(v_{1}^{2}+v_{2}^{2})( %\?лишняя ((1-\cos(\tau_{1}b))^{2} +\sin^{2}(\tau_1 b))^{\frac{1}{2}}\bigr\} \\ &=\frac{1}{b}|v|{\sqrt{2}\sqrt{1-\cos(\tau_1 b)}}\leq\frac{2}{b}|v|. \endaligned \tag5.18 $$ Changing the order of integration and using again \Par*{Lemma~5.2}\itm(c), we obtain $$ \align I_2&=-\beta\int\limits_{0}^{\tau_1} \Biggl\{\int\limits_{\tau}^{\tau_1}R((\tau-t)\beta b)\,dt\Biggr\} \nabla_{(x_1,x_2)}\varphi(X_\varphi^\beta,\tau)\,d\tau \\ &=\frac{1}{b}\int\limits_{0}^{\tau_1}\{R((\tau-\tau_1)\beta b)-E\} R(-\tfrac{\pi}{2})\nabla_{(x_1,x_2)}\varphi(X_\varphi^{\beta},\tau) \, d \tau \\ &= \frac{1}{b} \int\limits_{0}^{\tau_1} \pmatrix \cos((\tau-\tau_1)b)-1&-\beta\sin((\tau-\tau_1) b) \\ \beta \sin((\tau-\tau_1) b)&\cos((\tau-\tau_1) b)-1\endpmatrix \pmatrix 0 & 1\\ -1 & 0\endpmatrix \pmatrix \frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_1} \\ \frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_2} \endpmatrix d\tau \\ &=\frac{1}{b} \int\limits_0^{\tau_1} \pmatrix \beta\sin((\tau-\tau_1)b)\ \frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_1} + \cos((\tau-\tau_1)b)-1) \ \frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_2} \\ (1-\cos((\tau-\tau_1)b)) \frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_1} +\beta\sin((\tau-\tau_1)b) \frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_2} \endpmatrix d\tau. \endalign $$ Finally, we have $$ \aligned |I_{2}| &\leq \frac{1}{b}\int\limits_{0}^{\tau_1}\Biggl\{((1-\cos((\tau-\tau_{1})b))^2 + \sin^2((\tau-\tau_1)b) \\ &\qquad\times\biggl(\Bigl(\frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_1}\Bigr)^2 +\Bigl(\frac{\partial \varphi(X_\varphi^\beta,\tau)}{\partial x_2}\Bigr)^2\biggr) \Biggr\}^{1/2}d\tau \\ &=\frac{1}{b} \int\limits_{0}^{\tau_1}((1-\cos((\tau-\tau_1)b))^2+ \sin^2((\tau-\tau_1)b))^{1/2}|\nabla_{(x_1,x_2)}\varphi(X^{\beta}_{\varphi},\tau)| d\tau \\ &\leq\frac{2}{b}\int\limits_{0}^{\tau_1}|\nabla_{(x_1,x_2)} \varphi(X^{\beta}_{\varphi},\tau)|d\tau \leq \frac{2T}{b}\||\nabla\varphi|\|_{0,{T}}. \endaligned \tag5.19 $$ From \Tag(5.11), \Tag(5.17)--\Tag(5.19) and \Par*{Lemma~4.2} it follows that $$ \aligned \delta &= |X^{\beta'}_{\varphi}(\tau_1,x,v,0)-x'| \leq |I_{1}|+|I_{2}| \\ &\leq\frac{2}{b}\bigl(\rho+T\||\nabla\varphi|\|_{0,T}\bigr) \leq\frac{2}{b}\bigl(\rho + Tc_1 |B_{R(T)}| 4\pi\max_{\beta}\|f^{\beta}_0\|\bigr). \endaligned \tag5.20 $$ By virtue of \Tag(5.9), we obtain $$ \frac{2}{b}\bigl(\rho + Tc_1 |B_{R(T)}| 4\pi\max_{\beta}\|f^{\beta}_0\|_0\bigr)<\delta. $$ We have contradiction with \Tag(5.20). This implies \Tag(5.10). From \Par*{Lemma~5.1} it follows that $X_\varphi^\beta(\tau,x,v;0)\in B_{\lambda_1}$ and $ V_\varphi^\beta(\tau,x,v;0)\in B_{\rho_1} $ for $((x,v)\in D_0 $ and $\tau\in[0,T]$. \qed\enddemo \head 6. Proof of \Par*{Theorem 2.1} \endhead First we will prove the following auxiliary statement. \proclaim{Lemma 6.1} Let the assumptions of \Par*{Lemma~{\rm5.3}} hold. Then $\operatorname{supp} f_0^\beta(S_\varphi^\beta(0,.,.,t)) %\? \subset D_1:=( B_{\lambda_1}\cap\{ x:|x'|<\lambda+\delta\} )\times B_{\rho_1}$ for $t\in[0,T]$. \endproclaim \demo{Proof} By virtue of \Par*{Condition~2.2} it is sufficient to prove that $S_\varphi^\beta(t, x, v, 0)\in D_1$ for $(x,v) \in D_0$, $t \in [0,T]$. This statement follows from \Par*{Lemma~5.3}. \qed\enddemo We define the function $f^\beta(x, v, t)$ by the formula $$ f^{\beta}(x, v, t)= \cases f_{0}^{\beta} (S_{\varphi}^{\beta}(0, x, v, t)), & (x, v) \in D_{1},\ t\in[0, T], \\ 0, &(x,v)\in(\Bbb{R}^3\times\Bbb{R}^3)\setminus D_{1},\ t\in[0, T]. \endcases \tag6.1 $$ Let assumptions of \Par*{Lemma~5.3} hold. Then, by virtue of \Par*{Lemma~3.1}, there exists a global classical solution of problem \Tag(2.1)--\Tag(2.4) $\{\varphi,f^\beta\}$, $\varphi \in C([0,T],C^{2+\sigma}(\Bbb{R}^3)) $, $ f^\beta \in C^1(\Bbb{R}^3\times\Bbb{R}^3\times[0,T]) $. From \Par*{Lemma~6.1} and representation \Tag(6.1) it follows that $\operatorname{supp} f^\beta = \operatorname{supp} f_0^\beta(S_\varphi^\beta (0,.,., t)) %\? \subset D_1$. Thus we have proved the following result. \proclaim{Theorem 6.1} Let \Par{Condition 2.1}{Conditions~{\rm2.1}}--\Par{Condition 2.3}{\rm2.3} be fulfilled. Then, for any $\delta$, $0<\delta<\lambda$, and $b$ satisfying \Par*{Condition~{\rm5.1}}, there exists a~global classical solution of problem \Tag(2.1)--\Tag(2.4) $ \{ \varphi,f^\beta\}$, $ \varphi\in C ([0,T],C^{2+\sigma}(\Bbb{R}^3))$, $f^\beta \in C^1(\Bbb{R}^3 \times\Bbb{R}^3\times[0,T])$ such that $\operatorname{supp} f^\beta \subset D_1$, $\beta = \pm 1$. \endproclaim \Par*{Theorem~2.1} follows from \Par*{Theorem~6.1}. Moreover, \Par*{Condition~5.1} allows us to define the external magnetic field, which provides plasma confinement. \acknowledgment I express my gratitude to the reviewer for valuable comments contributing to the improvement of the article. \Refs \ref\no 1 \by Vlasov~L.A. \paper Vibrational properties of the electronic gas \jour Zh. Eksp. Teoret. Fiz. \yr 1938 \vol 8 \pages 291--318 \endref \ref\no 2 \by Batt~J. \paper Global symmetric solutions of the initial value problem of stellar dynamics \jour J.~Differ. Equ. \yr 1977 \vol 25 \pages 342--364 \endref \ref\no 3 \by Pfaffelmoser~K. \paper Global classical solutions of the Vlasov--Poisson system in three dimensions for general initial data \jour J.~Differ. Equ. \yr 1992 \vol 95 \pages 281--303 \endref \ref\no 4 \by Sch\"{a}ffer~J. \paper Global existence of smooth solutions to the Vlasov--Poisson system in three dimensions \jour Comm. Part. Diff. Equat. \yr 1991 \vol 16 \pages 1313--1335 \endref \ref\no 5 \by Horst~E. \paper On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation. 1. General theory \jour Math. Methods Appl. Sci. \yr 1981 \vol 3 \pages 229--248 \endref \ref\no 6 \by Horst~E. \paper On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation. 2. Special cases \jour Math. Methods Appl. Sci. \yr 1982 \vol 4 \pages 19--32 \endref \ref\no 7 \by Miyamoto~K. \book Fundamentals of Plasma Physics and Controlled Fusion \publaddr Iwanami %Book Service Centre \publ Tokyo \yr 1997 \endref \ref\no 8 \by Skubachevskii~L.L. \paper On the unique solvability of mixed problems for the system of Vlasov--Poisson equations in a~half-space \jour Dokl. Math. \yr 2012 \vol 85 \pages 255--258 \endref \ref\no 9 \by Skubachevskii~L.L. \paper Initial-boundary value problems for the Vlasov--Poisson equations in a~half-space \jour Proc. Steklov Inst. Math. \yr 2013 \vol 283 \pages 197--225 \endref \ref\no 10 \by Skubachevskii~L.L. \paper Vlasov--Poisson equations for a two-component plasma in a~homogeneous magnetic field \jour Russian Math. Surveys \yr 2014 \vol 69 \pages 291--330 \endref \ref\no 11 \by Skubachevskii~L.L. and Tsuzuki~Y. \paper Classical solutions of the Vlasov--Poisson equations with external magnetic field in a~half-space \jour Comput. Math. Math. Phys. \yr 2017 \vol 57 \pages 541--557 \endref \ref\no 12 \by Belyaeva~Yu.O., Gebhard~B., and Skubachevskii~L.L. \paper A~general way to confined stationary Vlasov--Poisson plasma configurations \jour Kinet. Rel. Models \yr 2021 \vol 14 \pages 257--282 \endref \ref\no 13 \by Skubachevskii~L.L. \paper On the existence of global solutions for the Vlasov--Poisson System in a~half-space and plasma confinement \jour Lobachevskii~J. Math. \yr 2024 \vol 45 \pages 851--863 \endref \ref\no 14 \by Skubachevskii~L.L. \paper A~priori estimate of solutions to the first mixed problem for Vlasov--Poisson system and plasma confinement \jour Sb.: Math. \yr 2025 \vol 216 \pages 1178--1192 \endref \ref\no 15 \by Knopf~P. \paper Confined steady states of Vlasov--Poisson plasma in an infinitely long cylinder \jour Math. Methods Appl. Sci. \yr 2019 \vol 42 \pages 6369--6384 \endref \ref\no 16 \by Ladyzhenskaya~O.A. and Ural'tseva~N.N. \book Linear and Quasilinear Elliptic Equations \publaddr New York \publ Academic \yr 1968 \endref \ref\no 17 \by Gilbarg~D. and Trudinger~N. \book Elliptic Partial Differential Equations of Second Order \publaddr Berlin \publ Springer \yr 1983 \endref \endRefs \enddocument