Mixed Value Problem for a One-Dimensional Nonlinear Nonstationary Twelve moment BoltzmannS System Equations with the Maxwell-Auzhan Boundary Conditions

Sh. Akimzhanova

Sh. Akimzhanova

I. INTRODUCTION

Many problems of the rarefied gas dynamics require the solution of one or another problem for the Boltzmann equation. The prediction of aerodynamic characteristics of aircraft at very high speeds and at high altitudes is an important problem in aerospace engineering. It is impossible to determine the aerodynamic characteristics of aircraft at high altitudes. The interaction of gas molecules with the surfaces of real bodies has been little studied. The aerodynamic characteristics of aircraft at very high speeds and at high altitudes can be determined by the methods of the theory of a rarefied gas [1]. To analyze the aerodynamic characteristics of aircraft in the transient regime, the complete integro-differential Boltzmann equation is used with appropriate boundary conditions. The determination of the boundary conditions on the surfaces that are streamlined with a rarefied gas is one of the most important questions in the kinetic theory of gases. In high-altitude aerodynamics, the interaction of gas with the surface of a streamlined body plays an important role [2]. The aerothermodynamic characteristics of bodies to the gas flow are determined by the transfer of momentum and energy to the surface of the body, that is, the connection between the velocities and the energies of the molecules incident on the surface and the molecules reflected from it, which is the essence of the kinetic boundary conditions on the surface. Maxwell's boundary condition for solving specific problems more accurately describes the interaction of gas molecules with the surface. One of the approximate methods for solving the initial-boundary value problem for the Boltzmann equation is the moment method. With the help of the moment method, it is possible to determine the aerodynamic characteristics of aircraft, such as atmospheric parameters, flight speed, geometric parameters, and the like. We note that in work [3] two new models of boundary conditions were proposed: diffusive-moment and mirror-moment, generalizing the known boundary conditions of Cherchinyani, and in [4] aerodynamic characteristics of space vehicles were studied by the method of direct static modeling (Monte Carlo method) and various model of the interaction of gas molecules with the surface and their effect on aerodynamic characteristic. Moment methods are different from each other as sets of various systems of basic functions. For example, Grad in works [5] and [6] received moment system through decomposition of particles distribution function by Hermitte polynomials near the local Maxwell's distributions. Grad used Cartesian coordinates of velocities and Grad's moment system contained unknown hydrodynamic characteristics as density, temperature, average speed, etc. In [7] obtained the moment system which differs from the system of equations of Grad. In this case was used the spherical coordinates of velocity and distribution function is decomposed into a series by the eigenfunctions of the linearized collision operator [1], [8], which is the product of the Sonin polynomials and spherical functions. The expansion coefficients, the moments of the distribution function are defined differently from Grad. The resulting system of equations corresponding to the partial sum of the series, which was call the Boltzmann's moment system equations, is nonlinear hyperbolic system relative to the moments of the particles distribution function. Differential part of the resulting system is linear and quadratic nonlinearity is shaped as moments of the distribution function. Quadratic forms – the moments of the nonlinear collision integrals – are calculated in [9] and are expressed in terms of coefficients Talmi [10] and Klebsh-Gordon [11].

In [12] - [13], moment systems for the spatially homogeneous Boltzmann equation and the conditions for the representability of the solution of the spatially homogeneous Boltzmann equation in the form of the Poincare series were obtained. The method proposed in [12](application of the Fourier transform with respect to the velocity variable in the isotropic case) greatly simplified the collision integral and, hence, the calculation of the moments from the collision integral. In [13] the result of [12] is generalized for the case of anisotropic scattering.

In work [14] presented a systematic nonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical theory. This paper is fundamental work where closed systems of moment equations describe a transition regime.

The Boltzmann equation is equivalent to an infinite system of differential equations relative to the moments of the particle distribution function in the complete system of eigenfunctions of linearized operator. As a rule, limited study to finite moment system equations as solving the infinite system of equations does not seem to be possible.

Finite system of moment equations for a specific task with a certain degree of accuracy replaces the Boltzmann equation. It's necessary, also roughly, to replace the boundary conditions for the particle distribution function by a number of macroscopic conditions for the moments, i.e. there arises the problem of boundary conditions for a finite system of equations that approximate the microscopic boundary conditions for the Boltzmann equation. The question of boundary conditions for a finite system of moment equations can be divided into two parts: how many conditions must be imposed and how they should be prepared. From microscopic boundary conditions for the Boltzmann equation there can be obtained an infinite set of boundary conditions for each type of decomposition. However, the number of boundary conditions is determined not by the number of moment equations, i.e. it is impossible, for example, take as much boundary conditions as equations, although the number of moment equations affects the number of boundary conditions. In addition, the boundary conditions must be consistent with the moment equations and the resulting problem must be correct.

Grad in [5] described the construction of an infinite sequence of boundary conditions without the consent of the order of approximation for the decomposition of the boundary conditions and the expansion of the Boltzmann equation. Boundary conditions, even a one-dimensional Grad's moment system equations is a very difficult task, because Grad's moment system of equations is a hyperbolic system and this system of equations contains as coefficients unknown parameters like density, temperature, average speed, etc. In such case the characteristic equation is also dependent on the unknown parameters and thus appears to be very difficult to formulate the boundary conditions for the moment system. In work [15] had been discussed the boundary conditions for the 13-moment Grad system.

In work [7] had been shown approximation of the homogeneous boundary condition for the particle distribution function and proved the correctness of the initial-boundary value problem for nonstationary nonlinear Boltzmann's moment system equations in a three-dimensional space. More precisely, was proved the existence of a unique generalized solution of the initial-boundary value problem for the Boltzmann's moment system equations in the space of functions that are continuous in time and square summable in the space of variables. In addition, the same study shows the approximation of the microscopic boundary conditions for the Boltzmann equation. The boundary condition can be formulated as follows: determine the mirrored half of the distribution function from the known half, corresponding to the falling particles. The boundary condition is specified as an integral relation between particles falling to the boundary and particles reflected from the boundary (assuming that we know the probability of an event that a particle falling to the boundary with velocity $\nu_{i}$ reflects with velocity $\nu_{r}$ ).

In this article it is proved existence and uniqueness of solution of the problem with initial and boundary conditions of Maxwell-Auzhan (we consider pure specular reflection from the boundary) for the nonstationary nonlinear one-dimensional Boltzmann's twelve-moment system equations in space of functions continuous in time and summable in square by spatial variable.

Existence and uniqueness of the solutions of initial and boundary value problem for twelve-moment one-dimensional Boltzmann's system of equations with boundary conditions of Maxwell-Auzhan In this section we prove existence and uniqueness of solutions of the initial and boundary value problem for twelve-moment one-dimensional Boltzmann's system of equations with boundary conditions of Maxwell-Auzhan in space of functions continuous in time and summable in square by spatial variable. Theorem of existence of global in time solution of the initial and boundary value problem for 3-dimensional nonlinear Boltzmann equation with boundary conditions of Maxwell proved in work [16].

Statement of the problem: Find the solution of initial-boundary value problem for a homogeneous one-dimensional Boltzmann equation

where $f \equiv f(t, x, v)$ is a particle distribution function in space of velocity and time; $f^0(x, v)$ is distribution of the particles at the initial time (fixed function);

$J(f, f) \equiv \int [f(v') f(w') - f(v) f(w)] \sigma (\cos x) dw dv$ nonlinear collision operator, recorded for Maxwell molecules, n is external unit normal vector of the boundary. Condition (3) is a natural boundary condition for the Boltzmann equation, which makes it possible to determine the reflected half of the distribution function f, if we know the half corresponding to the falling particles. According to (3), some part of falling particles reflected specularly, and other particles are absorbed into the wall and emitted with the Maxwell distribution with corresponding wall temperature $T_0$ .

Formula (3) refers to the case of a wall at rest; otherwise v must be replaced by $v - u_{0}$ , $u_{0}$ being the velocity of the wall. $\beta$ , $T_{0}$ , $u_{0}$ may vary from point to point and with time [8].

For one-dimensional problems of the Eigen functions of linearized operator are [1], [8]:

where $S_{n}^{l+\frac{1}{2}}\left(\frac{\alpha^{2}|v|^{2}}{2}\right)$ is Sonin polynomials, $P_{l}(\cos\theta)$ is Legendre polynomials, $\Gamma$ is Gamma function.

To find an approximate solution of the problem (1)-(3) we apply the Galerkin method. We define the approximate solution of one-dimensional problem (1)-(3) as follows:

where $n = (0,0,1)$ with $x = a$ and $n = (0,0 - 1)$ with $x = -a$ ;

$f_{0}\big(\alpha |v|\big)$ is the global Maxwell distribution;

In the general case, the approximation of the boundary condition (3) depends on the parity or oddness of the approximation of the Boltzmann's moment system of equations [17]. When approximating a microscopic boundary condition, we took into account the approximation of the Boltzmann equation by moment equations corresponding to the fifth approximation (the twelve moment system of equations).

Thus, the approximation orders for the expansion of the boundary condition and the expansion of the Boltzmann equation are consistent. Macroscopic conditions(7)were called Maxwell-Auzhan boundary conditions[17].

Boltzmann's system of moment equations (5), corresponding to the decomposition (4) can be written in expanded form

where the moments of the collision integral can be expressed in terms of coefficients Talmi and Klebsh-Gordon as follows [6]

$\left\langle N_3L_3n_3l_3:l|n_1l_1n_2l_2:l\right\rangle$ is generalized Talmi coefficients, $(l_{1}0l_{2}0 / l0)$ is Klebsh-Gordon coefficients.

If in (9) $2n + l$ is from 0 to 5, we get the system of equations, corresponding fifth approximation of Boltzmann's moment system equations or twelve-moment Boltzmann's system equations.

The initial and boundary value problem for twelve-moment Boltzmann's system equations with boundary conditions of Maxwell-Auzhan (7) can be written in vector-matrix form [18](we consider pure specular reflection from the boundary $\beta = 1$ ):

where $A'$ is transpose matrix; $u_{0}(x)=\left(f_{00}^{0}(x),\quad f_{02}^{0}(x),\quad f_{10}^{0}(x),\quad f_{04}^{0}(x),\quad f_{12}^{0}(x),\quad f_{20}^{0}(x)\right)^{\prime}$ , $w_{0}(x)=\left(f_{01}^{0}(x),\quad f_{03}^{0}(x),\quad f_{05}^{0}(x),\quad f_{11}^{0}(x),\quad f_{13}^{0}(x),\quad f_{21}^{0}(x)\right)^{\prime}$ are given initial vector functions,

J _ {1} (u, v) = ig (0, I _ {0 4}, 0, I _ {1 2}, I _ {2 0} ig), J _ {2} (u, v) = ig (0, I _ {0 3}, I _ {0 5}, I _ {1 1}, I _ {1 3}, I _ {2 1} ig),

The matrices $A$ and $B$ are degenerated. Required to find a solution to the system of equations (10) satisfying the initial condition (11) and boundary conditions (12) and (13). Before we prove the existence and uniqueness of the solution of the problem (10)-(13).

For the problem (10)-(13) following theorem takes place.

Theorem: If $U_0 = (u_0(x), w_0(x)) \in L^2[-a, a]$ , then problem (10)-(13) has unique solution in domain $[-a, a] \times [0, T]$ , belonging to the space $C([0, T]; L^2[-a, a])$ , moreover

where $C_1$ is constant independent from $U = (u,w)^{\prime}$ and $T\sim 0$ $\left(\left\| U_0\right\|_{L^2[-a,a]}^{-1}\right)$ .

Proof: Let $U_0 \in L^2[-a, a]$ . Let's prove estimation (14). We multiple first equation of system (10) by $u$ and second equation by $w$ , and integrate from $-a$ to $a$ :

After integration by parts we receive

Taking into account boundary conditions (12)-(13) we rewrite equality (15) in following form

Let's use spherical representation [19] of vector $U(t,x) = r(t)\omega (t,x)$ , where $\omega (t,x) = (\omega_{1}(t,x),\omega_{2}(t,x))^{\prime},r(t) = \| U(t,.)\|_{L^{2}[-a,a]},\| \omega \|_{L^{2}[-a,a]} = 1$ .

Substituting the values $u = r(t)\omega_1(t,x)$ , $w = r(t)\omega_2(t,x)_{\text{into (17)}}$ we have that

where

Let's study equation (17) with initial condition

Solution of the problem (17) -(18) has following form

If $R(t) \equiv \int_{0}^{t} Q(\tau) \exp \left( -\int_{0}^{\tau} P(\xi) d\xi \right) d\tau \leq 0, \forall t_{\text{then}} r(t)_{\text{is bounded for}} \forall t \in [0, +\infty)$ .

Let $R(t) > 0$ . We denote by $T_{1}$ the moment of time at which

Then $r(t)$ is bounded for $\forall t \in [0, T]$ , where $T < T_1$ , moreover $T_1 \sim 0\left(\|U_0\|^-\right)$ , since integrand $Q(\tau)\exp\left(-\int_{0}^{t} P(\xi)d\xi\right)$ is bounded. Hence $\forall t \in [0, T]$ takes place a priori estimation (14).

Now we prove the existence of a solution for (10)-(13) with help of Galerkin method. Let us $\{v_l(x)\}_{l=1}^{\infty}$ be a basis in space $L_2[-a,a]$ , where dimension of vector $v_l(x)$ is equal to dimension of vector $U$ . For each $m$ we define an approximate solution $U_{m}$ of (10)-(13) as follows:

where $U_{0m}$ is the orthogonal projection in $L^{2}$ of function $U_{0}$ on the subspace, spanned by $v_{1},\ldots,v_{m}$ .

We represent $v_{j}(x)$ in the form $v_{j}^{(1)} = \left(v_{j}^{1}, v_{j}^{2}\right)'$ , where

The coefficients $c_{jm}(t)$ are determined from the equations

where $d_{im}$ is $i$ -th component of $U_{0m}$ .

We multiply (21) by $c_{im}(t)$ and sum over $i$ from 1 to $m$ :

With help of above shown arguments now we prove that $r_m(t)$ is bounded in some time interval $[0, T_m]$ , where

where $C_2$ is constant and independent from $m$ . Then solvability of system equations (19) - (22) or (23)-(24) follows from estimation (25).

Thus, the sequence $\{U_m\}$ of approximate solutions of problem (10)-(13) is uniformly bounded in function space $C([0,T];L^2 [-a,a])$ . Moreover, homogeneous system of equations $rE + \frac{1}{\alpha} A\xi$ with respect to $\tau,\xi$ has only trivial solution. Then it follows from results in [20] that $U_{m}\rightarrow U$ is week in $C([0,T];L^2 [-a,a])$ and $J(U_{m})\to J(U)$ is week in $C([0,T];L^2 [-a,a])$ as $m\to \infty$ . Further, it can be shown by standard methods that limit element is a weak solution of the problem (10)-(13).

The theorem is proved.

II. CONCLUSION

We prove the theorem of existence and uniqueness of the local solution of the initial and boundary value problem for twelve-moment one-dimensional Boltzmann's system of equations with boundary conditions of Maxwell-Auzhan in space of functions continuous in time and summable in square by spatial variable, because the solution existence time depends on the norm of the initial vector function at the power minus one. Therefore, the smaller the norm of the initial vector function, the longer the solution existence time of the initial and boundary value problem for six-moment one-dimensional Boltzmann's system of equations and vice versa.