波茲曼方程式柔和解的存在性 - 政大學術集成

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(2) ‧ 國. 學. 政治大 Existence Result of Mild Solution 立 of Boltzmann Equation ‧. !"#$ n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. .

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(4) Contents. . i. Abstract. ii. . iii. 1 Introduction. 立. 政治大. 1. 2 Basic Kinetic Theory of Gases and the Boltzmann Equation. 學. ‧ 國. 4. Binary elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.2. The distribution functions and macroscopic fluid variables . . . . .. 7. 2.3. The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . .. 11. ‧. 2.1. n. Ch. er. io. al. sit. y. Nat. 3 Element Properties of the Collision Operator. i n U. v. 15. 3.1. Collision invariants . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.2. The H-Theorem and the Maxwellian distribution. . . . . . . . . . .. 17. 3.3. Grad’s angular cutoff potential . . . . . . . . . . . . . . . . . . . . .. 20. engchi. 4 Hydrodynamical Limits:Expansions and Decomposition. 22. 4.1. A formal discussion on conservation laws . . . . . . . . . . . . . . .. 22. 4.2. Hilbert expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 4.3. Chapman-Enskog expansion . . . . . . . . . . . . . . . . . . . . . .. 25. 4.4. Micro-macro decomposition . . . . . . . . . . . . . . . . . . . . . .. 29.

(5) 5 Solutions in a Weighted Lebesgue Space. 35. 5.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 5.2. Existence of the mild solution . . . . . . . . . . . . . . . . . . . . .. 41. 5.3. Uniform stability of solutions . . . . . . . . . . . . . . . . . . . . .. 42. References. 44. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

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(8) Abstract In this thesis, we consider the initial-value problem for the Boltzmann equation of the form   f + ξ · ∇ f = Q(f, f ), (t, x, ξ) ∈ R × R3 × R3 , t x +  f (0, x, ξ) = f (x, ξ), 3 (x, ξ) ∈ R × R3 . 0 We prove that the existence of mild solution in the weighted Lebesgue space ½ L1ω (R3x. ×. R3ξ ). ≡. 立. 政治¯¯¯ 大 f ∈ L (R × R ) ¯ f ω ∈ L (R 1. 3 x. 3 ξ. 1. ¾ 3 x. ×. R3ξ ). ‧ 國. 學. by using Banach’s fixed point theorem, and that the uniform stability of solution with respect to the weighted norm. Here ω ≡ M(ξ). , and. ‧. M(ξ) is any Maxwellian.. −1 2. Nat. sit. y. Key words: Boltzmann equation, Maxwellian, existence of mild solution-. n. al. er. io. s, uniform stability.. Ch. engchi. ii. i n U. v.

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(10) ÷Ý   f + ξ · ∇ f = Q(f, f ), (t, x, ξ) ∈ R × R3 × R3 , t x +  f (0, x, ξ) = f (x, ξ), (x, ξ) ∈ R3 × R3 . 0. 5 Banach %k !hp ½. L1ω (R3x. ×. R3ξ ). ≡. 1. f ∈L. (R3x. ×. ¯ ¯. R3ξ ) ¯¯ f ω. ¾. 1. ∈L. (R3x. ×. R3ξ ). 政治大 D "#n%í$0 %÷&'(# 立 ω ≡ M(ξ) ) M(ξ) ?*w +,-./ −1 2. ‧ 國. 學. ‧. :

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(12) 1. Introduction The kinetic theory, introduced by Boltzmann [17] at the end of the nineteenth. century, provides a description of gases at an intermediate level between the hydrodynamic description which does not allow to take into account phenomena far from thermodynamic equilibrium, and the atomistic description which is often too complex. Nevertheless, a gas flow may be modelled at either the macroscopic or the microscopic level. The macroscopic model regards the gas as a continuous medium. 治政 flow properties such as the velocity, density, pressure,大 and temperature. The Euler 立provide the conventional mathematical model of a gas and Navier-Stokes equations and the description is in terms of the spatial and temporal variations of the familiar. ‧ 國. 學. as a continuum.. On the other hand, the microscopic or molecular model recognizes the partic-. ‧. ulate structure of the gas as a myriad of discrete molecules and ideally provides information on the position, velocity, and state of every molecule at all times. The. Nat. sit. y. mathematical model at this level is the celebrated Boltzmann equation, which is. er. io. a mathematical model of the phenomenological kinetic theory of gases which describes the evolution, in time and space, of the one-particle distribution function. n. al. Ch. i n U. v. for a simple monatomic gas of a large number of identical particles.. engchi. Furthermore, the degree of rarefaction of a gas is generally expressed through the Knudsen number, denoted by κ (or Kn), which is the ratio of the mean free path λ to some characteristic length L of the domain containing the gas, i.e., κ=. λ . L. (1.1). It plays an essential role in the analysis of asymptotic relations between the Boltzmann equation and macroscopic fluid equations. In this paper, we study the existence theory and stability of solutions to the Boltzmann equation. The first existence theory of the solutions to the Boltzmann equation goes back to 1932 when T.S.Carleman [18] proved the existence of global (in time) solutions to the Cauchy problem for the spatially homogeneous case.. 1.

(13) It should be stressed that this is 2 years before the incompressible Navier-Stokes equation was solved by J.Leray on the existence of global weak solutions. On the other hand, the research on the spatially inhomogeneous Boltzmann equation started much later. It is only in 1965 when Grad [11] constructed the first local solutions near the Maxwellian, and it is in 1974 when the first author of this note constructed global solutions that are also near the Maxwellian, extending Grad’s mathematical framework, [19]. However, the process made afterward was remarkable. Up to the present, three. 治政 The difference is due to difference of function spaces used 大for solving the Boltzmann 立the methods of proof employed are also different. The equation, and consequently,. different methods have been developed for establishing the global existence theory.. ‧ 國. 學. following is a short summary of the solutions established so far :. ‧. 1. Solutions in L∞ framework: Grad’s scheme was extended to construct global solutions in the L∞ space for various initial and initial boundary value. sit. y. Nat. problems. See, [19], etc.. er. io. 2. Solutions in L1 framework: DiPerna-Lions [5] constructed global L1 solutions. al. iv n C renormalized solutions because are the weak h ethey i U solutions to the Boltzmann h n c g equation in the renormalized form. n. without smallness assumption on initial data. These solutions are called the. 3. Solutions in L2 framework: Recently, the L2 energy method, which is familiar in the theory of nonlinear PDE’s, because available for the Boltzmann equation by introducing a new decomposition of the equation and solutions, called the macro-micro decomposition. This was developed by Liu-Yang-Yu [16], Liu-Yu [12], Yang-Zhao [22] and independently by Guo [6, 7, 8, 9, 10]. Also, the green function for the Cauchy problem was constructed by Liu-Yu [13]. 4. Solutions in L∞ β framework: The first global existence theorem was established 5 3 k 3 in the space L∞ β (Rξ ; H (Rx )), β > 2 , k ≥ 2, by using the spectral analysis. 2.

(14) by Nishida-Imai [14], where 3 L∞ β (Rξ ). ½ ¾ β ∞ 3 ≡ f : (1+ | ξ |) f ∈ L (Rξ ) .. The same result was obtained by Shizuta [15] for the torus case with the 5 3 k 3 + space L∞ β (Rξ ; C (Tx )), β > 2 , k ∈ Z . Recently, Yang-Ukai [20] present3 ∞ 3 ed a function space L2 (R3x × R3ξ ) ∩ L∞ β (Rξ ; L (Rx )), β >. 3 , 2. in which the. Cauchy problem is globally well-posed in a mild sense without any regularity conditions.. 治政 This thesis contains five sections. In section 1, 大 we give an introduction. In 立 elastic collisions and the well-known Boltzmann esection 2, we review the binary ‧ 國. 學. quation. In section 3, we introduce collision invariants, which aid in obtaining fluid dynamic conservation equations and the H-Theorem. This theorem in turn leads to. ‧. the well-known Maxwellian distribution. We end section 3 by introducing the Grad’s angular cutoff potential, which plays a crucial role of the existence theory. Further-. Nat. sit. y. more, thanks to [16, 21, 22], we study a presentation of the few rigorous results. er. io. on the fluid dynamical limit available in section 4. These are Hilbert expansion, Chapman-Enskog expansion, and macro-micro decomposition. They explain how. n. al. Ch. i n U. v. the compressible Euler equations and compressible Navier-Stokes equations arise in. engchi. suitable limits from the Boltzmann equation. Finally, in section 5, we study the existence of mild solution to the Boltzmann equation in the weighted Lebesgue space L1ω (R3x × R3ξ ) by using the Banach’s fixed point theorem, and the uniform L1ω -type stability of solution.. 3.

(15) 2. Basic Kinetic Theory of Gases and the Boltzmann Equation This section begins with a review of scattering concepts important to derivation. of the Boltzmann equation. More details can be found in [1, 2, 3].. 2.1. Binary elastic collisions. 政治大. Intermolecular collisions in dilute gases are overwhelmingly likely to be binary. 立. collisions involving just two molecules. An elastic collision is defined as one in which. ‧ 國. 學. there is no interchange of translational and internal energy. We may denote the pre-collision velocities of the two collision partners in a typical binary collision by. ‧. ξ and ξ∗ . Given the physical properties of the molecules and the orientation of the trajectories, our task is to determine the post-collision velocities ξ 0 and ξ∗0 .. sit. y. Nat. Linear momentum and energy must be conserved in the collision. This requires. n. al. er. io. m1 ξ + m2 ξ∗ = m1 ξ 0 + m2 ξ∗0 = (m1 + m2 )ξm. and. Ch. engchi. i n U. (2.1). v. m1 | ξ |2 +m2 | ξ∗ |2 = m1 | ξ 0 |2 +m2 | ξ∗0 |2 ,. (2.2). where m1 and m2 are the masses of the two molecules and ξm is the velocity of the center of mass of the pair of molecules. Equation (2.1) shows that this center of mass velocity is not affected by the collision. The pre-collision and post-collision values of the relative velocity between the molecules may be defined by    ξ = ξ − ξ∗ ,   r     ξ0 = ξ0 − ξ0 . ∗ r. 4. (2.3).

(16) Equations (2.1) and (2.3) give    ξ = ξm +  . m2 ξ, m1 +m2 r.     ξ =ξ − ∗ m. (2.4) m2 ξ. m1 +m2 r. The pre-collision velocities relative to the center of mass are ξ − ξm and ξ∗ − ξm . Equation (2.4) shows that these velocities are antiparallel in this frame of reference and, if the molecules are point centers of force, the force between them remains in the plane containing the two velocities. The collision is therefore planer in the. 政治大. center of mass frame. Similarly, equations (2.1) and (2.3) imply   2  ξ0 , ξ 0 = ξm + m1m+m  2 r . 立. ‧ 國. (2.5). 學.     ξ0 = ξ − m ∗. m2 ξ0 . m1 +m2 r. This shows that the post-collision velocities are also anti-parallel in the center of. ‧. mass frame.. er. io. sit. y. Nat. Furthermore, equations (2.4) and (2.5) show that    m1 | ξ |2 +m2 | ξ∗ |2 = (m1 + m2 ) | ξm |2 +mr | ξr |2 ,  .     m | ξ 0 |2 +m | ξ 0 |2 = (m + m ) | ξ |2 +m | ξ 0 |2 . 1 2 1 2 m r ∗ r. n. al. where mr :=. m1 m2 m1 +m2. Ch. engchi. i n U. (2.6). v. is called the reduced mass. A comparison of equation (2.6). with the energy equation (2.2) shows that the magnitude of the relative velocity is unchanged by collision, i.e., ξr0 = ξr .. (2.7). Since both ξm and ξr may be calculated from the pre-collision velocities, the determination of the post-collision velocities reduces to the calculation of the change in direction n of the relative velocity vector. Now we rewrite (2.1) and (2.2) as    ξ + ξ∗ = ξ 0 + ξ∗0 ,       | ξ |2 + | ξ |2 =| ξ 0 |2 + | ξ 0 |2 . ∗ ∗ 5. (2.8).

(17) Introduce a unit vector n directed along ξ − ξ 0 ; this direction bisects the directions of ξ and −ξ 0 . Because of our definition of n we have ξ 0 = ξ + nC,. (2.9). where C is a scalar to be determined. The first of equations (2.8) gives ξ∗0 = ξ∗ − nC.. (2.10). Substituting (2.9) and (2.10) into (2.8) we obtain. 政治大. | ξ |2 +2n · ξ + C 2 + | ξ∗ |2 −2n · ξ∗0 C + C 2 =| ξ |2 + | ξ∗ |2 ,. 立. which implies that. ‧ 國. 學. n · (ξ∗ − ξ∗ )C + C 2 = 0.. Hence, dismissing the case C = 0 which corresponds to a trivial solution of the. ‧. conservation equations, we have. Nat. n. al. Ch. engchi. y. sit.    ξ 0 = ξ − n[n · (ξ − ξ∗ )],  . (2.11). er. io. Thus, we have. C = −n · (ξ − ξ∗ ).. i n U. v.     ξ 0 = ξ + n[n · (ξ − ξ )]. ∗ ∗ ∗. (2.12). On the other hand, if we consider the case on a hard sphere ¯ ½ ¾ ¯ 2 2¯ S+ ≡ Ω ∈ S ¯ (ξ − ξ∗ ) · Ω ≥ 0 , then (2.12) becomes.    ξ 0 = ξ − [(ξ − ξ∗ ) · Ω]Ω,       ξ 0 = ξ + [(ξ − ξ ) · Ω]Ω. ∗ ∗ ∗. (2.13). In this case, we use B(|ξ − ξ∗ |, ϑ) to denote the collision cross section, where ¶ · ¸ µ π (ξ − ξ∗ ) · Ω ∈ 0, (2.14) ϑ = arccos |ξ − ξ∗ | 2 6.

(18) is the scattering angle between ξ − ξ∗ and Ω. The definition of B depends on the physics of collision. In fact, for the inverse power interaction potential, B takes the form of B(|ξ − ξ∗ |, ϑ) = bγ (ϑ)|ξ − ξ∗ |γ , −2 < γ ≤ 1,. (2.15). where bγ (ϑ) is the collision kernel. However, different values of γ corresponds to different kinds of interactions, namely, γ ≥ 0 corresponds to the hard intersection potential, and γ < 0 corresponds to the soft intersection potential. Moreover, the hard sphere model satisfies. 政治大. B(|ξ − ξ∗ |, ϑ) = σ|ξ − ξ∗ | cos ϑ. 立. with σ being the radius of the hard sphere.. ‧ 國. 學. The distribution functions and macroscopic fluid variables. ‧. 2.2. sit. y. Nat. io. er. A gas flow would be completely described, in the classical sense, by listings of the position, velocity, and internal state of every molecule at a particular instant.. n. al. Ch. i n U. v. The number of molecules in a real gas is so large that such a description is un-. engchi. thinkable, and we must resort to a statistical description in terms of probability distributions. To see this, consider a sample of gas which is homogeneous in physical space and contains N identical molecules. A typical molecule has a velocity ξ with component ξ 1 , ξ 2 , ξ 3 in the direction of the Cartesian axes x1 , x2 , and x3 . Note that x1 , x2 , and x3 define a space called physical space, and a volume element in physical space may be denoted by dx. Similarly, ξ 1 , ξ 2 , and ξ 3 define velocity space, and a volume element may be denoted by dξ. The product dxdξ then denotes a volume element in phase space, which is the 6-dimensional (in general, 6N -dimensional) space formed by the combination of physical space and velocity space. The state of the gas is therefore modelled by a distribution function in phase space. In the case. 7.

(19) of a monatomic gas, the function f (t, x, ξ), (t, x, ξ) ∈ R+ × R3 × R3 is termed the velocity distribution function. Its definition involves probability concepts; any result in which it appears will be a result as to the probable, or average, behavior of the gas; in addition, f (t, x, ξ)dxdξ. (2.16). 治政 Notice that f can never be negative and must either have 大 finite bounds in velocity space or tend to zero as |ξ|立 −→ ∞.. represents the number of particles, which at time t, have position x and velocity ξ.. ‧ 國. 學. Furthermore, in order to relate the macroscopic properties to this distribution function, we must determine the relationship between the function and the average. ‧. value of any molecule quantity q. This quantity is either a constant or a function. y. R3. al. (2.17). er. io. sit. Nat. of the molecule velocity. The mean value principle gives Z q= qf dξ.. n. Now, we introduce some macroscopic fluid variables, and f will be assumed to. Ch. i n U. v. be an expected mass density in phase space. Define the mass density Z ρ(t, x) ≡ f (t, x, ξ)dξ.. engchi R3. (2.18). The fluid velocity (or bulk velocity of gas) u(t, x) is the average of the molecular velocities ξ at a certain point x and time instant t; since f is proportional to the probability for a molecular to have a given velocity, u is given by Z Z ξf (t, x, ξ)dξ ξf (t, x, ξ)dξ 3 3 R R Z . = u(t, x) ≡ ρ(t, x) f (t, x, ξ)dξ. (2.19). R3. Equation(2.19) can also be written as follows: Z ρ(t, x)u(t, x) = ξf (t, x, ξ)dξ =: m(t, x),. R3. 8. (2.20).

(20) or, using components: Z i. i. m (t, x) ≡ ρ(t, x)u (t, x) ≡. R. ξ i f (t, x, ξ)dξ, i = 1, 2, 3.. (2.21). 3. The fluid velocity u is what we can directly perceive of the molecular motion by means of macroscopic observation; it is zero for a gas in equilibrium in a box at rest. Every molecule has its own velocity ξ, which can be decomposed into the sum of u and other velocity c≡ξ−u. (2.22). 政治大. called the random or peculiar velocity. By (2.17), we have Z Z Z Z c= cf dξ = ξf dξ − uf dξ = ξf dξ − uρ = 0,. 立R. R3. R3. 3. R3. ‧ 國. 學. i.e., the average of c is zero.. The quantity mi that appears in equation (2.21) is the ith component of the. ‧. mass flow or of the momentum density of the gas. Other quantities of similar nature. y. Nat. are the momentum flow:. mij ≡. (i, j = 1, 2, 3);. 3. n. al. (2.23). er. io. R. ξ i ξ j f dξ. sit. Z. the energy density per unit volume:. i n C h1 Z i U | fhdξ; w ≡ e n |gξ c. v. 2. 2 R3. (2.24). and the energy flow: Z 1 ξ i | ξ |2 f dξ r ≡ 2 R3 i. (i = 1, 2, 3).. (2.25). Equation (2.25) shows that the momentum flow is described by the components of a symmetric tensor of second order, because we need to describe the flow in the ith direction of the momentum in the jth direction. Note also that Z Z i j c c f dξ = (ξ i − ui )(ξ j − uj )f dξ. R3. R3. = mij − ρui uj , 9.

(21) which implies that Z i j. mij = ρu u + where. R3. ci cj f dξ = ρui uj + pij ,. (2.26). Z pij ≡. R. ci cj f dξ. (i, j = 1, 2, 3). (2.27). 3. plays the role of stress tensor. Furthermore, we define the internal energy E per unit mass (associated with random motions) by. Z 1 | c |2 f dξ. ρE = 2 R3 Z Z 1 1 2 | c | f dξ = | ξ − u |2 f dξ 2 R3 2 R3. 政治大. 立. 學. ‧ 國. Note that. (2.28). = w − 21 ρ | u |2 .. ‧. er. io. Similarly, we define. Z a 1 = l. n. v 1, 2, 3), ni Cch| c | f dξ, (i = U engchi called the components of the so-called heat-flow vector, and we can obtain q. i. µ i. i. r = ρu. (2.29). sit. y. Nat. Thus the energy density per unit volume can also be written as follows: µ ¶ 1 1 2 2 | u | +E . w = ρ | u | +ρE = ρ 2 2. i. 2. 2 R3. 1 | u |2 +E 2. ¶ +. 3 X. uj pij + q i. (i = 1, 2, 3).. (2.30). (2.31). i=1. The decomposition in equation (2.31) shows that the microscopic energy flow is a sum of a macroscopic flow of energy (both kinetic and internal), of the work (per unit area and unit time) done by stresses, and of the heat flow. We end this subsection with the definition of pressure p in terms of f ; p is nothing other than 1/3 of the spur or trace (i.e. the sum of the three diagonal terms) of pij and thus is given by Z 1 | c |2 f dξ. p= 3 R3 10. (2.32).

(22) If we compare this with the definition of the specific internal energy E, given in equation (2.28), we obtain the relation: 2 p = ρE. 3. (2.33). In order to complete the connection, as a simple mathematical consequence of the Boltzmann equation, we can derive five differential relations satisfied by the macroscopic quantities introduced above; these relations describe the balance of mass, momentum, and energy and have the same form as in continuum mechanics,. 政治大. and we will introduce these in the following sections.. The Boltzmann equation. 學. ‧ 國. 2.3. 立. Now, we can drive the well-known Boltzmann equation. At a particular instant,. ‧. the number of molecules in the phase space element dxdξ is given by equation (2.16).. y. Nat. The rate of change of the number of molecules in the element is. al. sit. (2.34). er. io. ∂ f dxdξ. ∂t. v. n. The processes that contribute to the change in the number of molecules within dxdξ, which are. Ch. engchi. i n U. (i) The convection of molecules across the face of dx by the molecular velocity ξ. The representation of the phase space element as separate volume elements in physical and velocity space emphasizes the fact that x and ξ are treated as independent variables. (ii) The ’convection’ of molecules across the surface of dξ as a result of the external force per unit mass F. (iii) The scattering of molecules into and out of dxdξ as a result of intermolecular collisions. The gas is assumed to be dilute. One consequence of this assumption is that a collision may be assumed to be an instantaneous event at a. 11.

(23) fixed location in physical space. A second major consequence of the dilute gas assumption is that all collisions may be assumed to be binary collisions. First consider process (i) which is a conservative process across the surface dx. The number of molecules in the phase space element is f dxdξ, so the number density of class ξ molecules within dx is f dξ. Note that the flux of the quantity q across the element per unit area per unit time in the direction of e is 4qξ · e. So, the total flux is obtained by summing over all velocity classes and can be written. 政qξ · e.治大. 立. (2.35). Hence, equations (2.35) and (2.17) then enable the net inflow of molecules of this. ‧ 國. 學. class ξ across the surface of dx to be written as Z − f ξ · ex dSx dξ,. (2.36). ‧. Sx. where Sx is the total area of the surface of dx, dSx is the element of this surface,. sit. y. Nat. and ex is the unit normal vector of this unit element. By the Gauss-divergence. n. al. er. io. Theorem, we have Z Z − f ξ · ex dSx dξ = − Sx. v ni. ∇ · (f ξ)d(dx)dξ = −∇ · (f ξ)dxdξ. Ch dx. engchi U. (2.37). because f and ξ are constants within dx.. Also, since we are considering only molecules of class ξ, the velocity ξ may be taken outside the divergence in physical space. Therefore, the inflow of molecules of class ξ across the surface of dx due to the velocity ξ is −ξ · ∇x f dxdξ,. (2.38). and the inflow of molecules across the surface of dξ due to the external force F per unit mass is −F · ∇ξ f dxdξ.. (2.39). Furthermore, the volume swept out in physical space by the collision cross section B(|ξ − ξ∗ |, ϑ) in the hard sphere is B(|ξ − ξ∗ |, ϑ)dΩ, and the number of class 12.

(24) ξ∗ molecules per unit volume in physical space is f∗ dξ∗ , where f∗ denotes the value of f at ξ∗ . The number of collisions in the hard sphere per unit time is f∗ B(|ξ − ξ∗ |, ϑ)dΩdξ∗ .. (2.40). Since the number of class ξ molecules in the phase space element is f dξdx, the number of class ξ, ξ∗ −→ ξ 0 , ξ∗0 collisions per unit time in the element is f f∗ B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξdx.. (2.41). Just as f denotes the value of the velocity distribution function f at ξ, f∗. 政治大. denotes the value of f at ξ∗ . Similarly, f 0 and f∗0 denote the values of f at ξ 0 and. 立. ξ∗0 , respectively.. ‧ 國. 學. The existence of inverse collisions means that an analysis, exactly similar to that leading to equation (2.40) may be made for the collisions of class ξ, ξ∗ ←− ξ 0 , ξ∗0. ‧. that scatter molecules into class ξ. This yields. y. (2.42). sit. Nat. f 0 f∗0 B(|ξ 0 − ξ∗0 |, ϑ)dΩ0 dξ∗0 dξ 0 dx. io. al. er. for the collision rate in the phase space element dξ 0 dx. By equation (2.7), we have. n. | B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξ |=| B(|ξ 0 − ξ∗0 |, ϑ)dΩ0 dξ∗0 dξ 0 | .. Ch. So, equation (2.42) may be written. engchi U. v ni. f 0 f∗0 B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξdx.. (2.43). (2.44). The rate of increase of molecules of class ξ in the phase space element dξdx as a result of the combined direct and inverse collisions of class ξ, ξ∗ ←→ ξ 0 , ξ∗0 is obtained by the subtracting the loss rate (equation (2.40)) from the rate of gain(equation (2.43)). This gives (f 0 f∗0 − f f∗ )B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξdx.. (2.45). The total rate of increase of molecules of class ξ in the element as a result of collisions is given by. Z. Z. R3 S2+. (f 0 f∗0 − f f∗ )B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξdx.. 13. (2.46).

(25) Hence, equations (2.34),(2.38),(2.39), and (2.46) give ∂ f dξdx = −ξ · ∇x f dξdx − F · ∇ξ f dξdx ∂t Z Z + (f 0 f∗0 − f f∗ )B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξdx.. (2.47). R3 S2+. If the latter terms are transferred to the left-hand side and the complete equation is divided by dξdx, we have the Boltzmann equation for a sample dilute gas: Z Z ∂ (f 0 f∗0 − f f∗ )B(|ξ − ξ∗ |, ϑ)dΩdξ∗ . (2.48) f + ξ · ∇x f + F · ∇ξ f = ∂t R3 S2+. 政治大 operator or simply the collision term, and is denoted by Q(f, f ). 立. The term on the right-hand side of the Boltzmann equation is called the collision. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 14. i n U. v.

(26) 3. Element Properties of the Collision Operator In this section, we review some element properties of the collision operator. and introduce the celebrated H-Theorem. Finally, we introduce the Grad’s angular cutoff potential, which plays a role of the existence theory.. 3.1. Collision invariants. 政治大. In this subsection, we study some elementary properties of the collision integral on a hard sphere. 立 Z ½ S2+. ≡. (f 0 f∗0 − f f∗ )B(|ξ − ξ∗ |, ϑ)dΩdξ∗. ¾ Ω ∈ S : (ξ − ξ∗ ) · Ω ≥ 0 .. (3.1). ‧. ‧ 國. where. R S. 3× 2 +. 學. Q(f, f ) =. 2. sit. y. Nat. Actually it turns out that it is more convenient to study the slightly more general. n. al. er. io. bilinear expression associated with Q(f, f ), i.e., Z 1 (f 0 g∗0 + g 0 f∗0 − f g∗ − gf∗ )B(|ξ − ξ∗ |, ϑ)dΩdξ∗ . Q(f, g)(ξ) = 2 R3 ×S2+. Ch. engchi U. v ni. (3.2). It is clear that when g ≡ f , equation (3.2) reduces to equation (3.1) and Q(f, g) = Q(g, f ).. (3.3). Our first aim is to study the eightfold integral: Z Z 1 (f 0 g∗0 + g 0 f∗0 − f g∗ − gf∗ )ψ(ξ)B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξ, Q(f, g)ψ(ξ)dξ = 2 2 3 3 3 R ×R ×S+ R (3.4) where f , g, and ψ are functions such that the indicated integrals exist and the order of the integration does not matter. Furthermore, by the Fubini’s Theorem and the. 15.

(27) ∂(ξ,ξ∗ ) ∂(ξ 0 ,ξ∗0 ). fact that the Jacobian | Z. |= 1, we have. Z 1 (f 0 g∗0 + g 0 f∗0 − f g∗ − gf∗ )ψ(ξ)B(|ξ − ξ∗ |, ϑ)dΩdξdξ∗ Q(f, g)ψ(ξ)dξ = 2 R3 ×R3 ×S2+ R3 (3.5). Z. =. 1 (f g∗ + gf∗ − f 0 g∗0 − g 0 f∗0 )ψ(ξ 0 )B(|ξ 0 − ξ∗0 |, ϑ)dΩdξ 0 dξ∗0 2 R3 ×R3 ×S2+ (3.6). Z. =. 1 (f g∗ + gf∗ − f 0 g∗0 − g 0 f∗0 )ψ(ξ 0 )B(|ξ − ξ∗ |, ϑ)dΩdξdξ∗ 2 R3 ×R3 ×S2+ Z. 1 2 R3 ×R3 ×S2+. 立. ∗. ∗. 0 0 ∗. 0 ∗. 0 0 ∗. ∗. 學. ‧ 國. =. (3.7) 政治大 (f g + gf − f g − g f )ψ(ξ )B(|ξ − ξ |, ϑ)dΩdξdξ. ∗. (3.8). Taking the sum of equations (3.4), (3.5), (3.7), and (3.8) and dividing by four we. ‧. obtain Z Z 1 (f 0 g∗0 +g 0 f∗0 −f g∗ −gf∗ )(ψ+ψ∗ −ψ 0 −ψ∗0 )B(|ξ−ξ∗ |, ϑ)dΩdξdξ∗ . Q(f, g)ψ(ξ)dξ = 2 8 3 3 3 R ×R ×S+ R (3.9). er. io. sit. y. Nat. This relation expresses a basic property of the collision term, which is frequently. n. al. Ch. i n U. v. used. In particular, when g ≡ f , equation (3.9) becomes Z Z 1 (f 0 f∗0 −f f∗ )(ψ +ψ∗ −ψ 0 −ψ∗0 )B(|ξ −ξ∗ |, ϑ)dΩdξdξ∗ . Q(f, f )ψ(ξ)dξ = 2 4 3 3 3 R ×R ×S+ R (3.10). engchi. Observe that the integral in (3.9) is zero independent of the particular functions f and g, if ψ + ψ∗ = ψ 0 + ψ∗0. (3.11). is valid almost everywhere in velocity space. Since the integral appearing in the left-hand side of equation (3.10) is the rate of change of the average value of the function ψ due to collisions, the functions satisfying (3.11) is called collision invariants.. 16.

(28) Definition 3.1. A function ψ(ξ) is called the collision invariant if it satisfies equation (3.11), or, equivalently, if it satisfies Z Q(f, g)ψ(ξ)dξ = 0.. (3.12). R3. for all f, g ∈ C0∞ (R3ξ , R+ ). Remark 3.1.. 0. (i) In fact, Q has five collision invariants in R3 , which are. 政治大. 1 ψ0 (ξ) = 1, ψi (ξ) = ξ i (i = 1, 2, 3), ψ4 (ξ) = |ξ|2 . 2. 立. (3.13). This leads to the conservation laws of the Boltzmann equation (2.48) introduced. ‧ 國. 學. latter.. (ii) The first discussion of equation (3.11) is due to Boltzmann [17], who assumed. ‧. φ to be differentiable twice and arrived at the result that the most general solution. y. n. al. where A, B, and C are constants.. 3.2. sit. io. ψ(ξ) = A + B · ξ + C | ξ |2 ,. (3.14). er. Nat. of equation (3.11) is given by. Ch. engchi. i n U. v. The H-Theorem and the Maxwellian distribution. Now consider the case that the external force F = 0, and let f be a nonnegative solution to (2.48) with rapid decay properties in (x, ξ). Then the Boltzmann equation (2.48) becomes ∂f = Q(f, f ). ∂t. (3.15). Furthermore, if we let f be a density function of a gas, and since it is nonnegative, we may define the H-function by Z H(t) =. R3 ×R3. f log(f )dxdξ.. 17. (3.16).

(29) Theorem 3.1 (The Boltzmann’s H-Theorem). If f is a nonnegative solution to the equation (3.15), i.e., the Boltzmann equation (2.48) is under the assumptions that f is a nonnegative solution to (2.48) with rapid decay properties in (x, ξ) and F = 0, then dH ≤ 0. dt. (3.17). Proof. Note that µ ¶ Z ∂ dH = f log (f ) dxdξ dt 3 ×R3 ∂t R Z ∂f = (1 + log (f ))dxdξ 3 ×R3 ∂t R µ ¶ Z = Q(f, f )(ξ) 1 + log (f ) dxdξ 3 ×R3 R Z = Q(f, f )(ξ) log (f )dxdξ 3 ×R3 R Z = Q(f, f )(ξ∗ ) log (f∗ )dxdξ∗ 3 ×R3 R Z = −Q(f, f )(ξ 0 ) log (f 0 )dxdξ 3 3 ZR × R = −Q(f, f )(ξ∗0 ) log (f∗0 )dxdξ∗. 政治大. 立. n. i n U. engchi. Since the logarithm function is increasing on (0, ∞), then ¶ µ f f∗ 0 0 ≤ 0. (f∗ f − f f∗ ) log f 0 f∗0 Thus, from (3.25) and (3.26), we conclude that. In above proof, we also obtain. 18. dH ≤ 0. dt. y. sit. (3.21). (3.23) (3.24). v. because ψ ≡ 1 is a collision invariant. From (3.21) to (3.24), we obtain ¶ µ Z 1 f f∗ dH 0 0 = (f f − f f∗ ) log B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξdx. dt 4 R3 ×R3 ×R3 ×S2+ ∗ f 0 f∗0. Ch. (3.20). (3.22). er. io. al. (3.19). ‧. ‧ 國. 學. Nat. R3 ×R3. (3.18). (3.25). (3.26).

(30) Corollary 3.1. If f is a nonnegative function such that log (f )Q(f, f ) is integrable and the manipulations of the previous section when ψ ≡ log (f ), then the Boltzmann inequality. Z. R3. log (f )Q(f, f )dξ ≤ 0.. (3.27). holds. In particular, the equality holds if and only if log (f ) is a collision invariant by equation (3.26), or equivalently, f = exp (a + b · ξ + c | ξ |2 ), where a, b, and c are constants. Remark 3.2.. 0. 立. 政治大. ‧ 國. µ. Z. − f f∗ ) log. ¶ f f∗ B(|ξ − ξ∗ |, ϑ)dΩdξ∗ dξ (3.29) f 0 f∗0. ‧. R3 ×R3 ×S2+. (f∗0 f 0. 學. (i) The integral. D(t, x) := −. (3.28). in the equation (3.25) is called the entropy dissipation integral. It is non-. Nat. sit. y. negative as seen (3.26).. al. µ i v¶ | ξ −n u| ρ C − , f≡ h e n exp (2πRθ) 2Rθ gchi U. n. (3.28) as. er. io. (ii) In the equation (3.28) c must be negative, since f ∈ L1 (R3 ). If we rewrite 2. 3/2. (3.30). where θ > 0 is the temperature and R is the gas constant. The function appearing in equation (3.30) is the celebrated Maxwell distribution or Maxwellian, and is always denoted by M(ξ) or M[ρ,u,θ] (ξ). (iii) The Maxwellian is known to describe the velocity distribution of a gas in an equilibrium state. Here (ρ, u, θ) are taken to be parameters, and if they are constants, M is called a global Maxwellian while if they are functions of (t, x), it is called a local Maxwellian. (iv) Without loss of generality, we may take a global Maxwellian as µ ¶ |ξ|2 1 exp − M ≡ M[1,0,1] (ξ) = (2π)3/2 2 19. (3.31).

(31) in R3 . Evidently, the global Maxwellian is a stationary solution of the Boltzmann equation (2.48) if the external force F is absent. (v) By the Corollary 3.1, we conclude that Z Q(f, f ) = 0 ⇐⇒ log (f )Q(f, f )dξ = 0 ⇐⇒ f ≡ M(ξ).. R3. Remark 3.3. The H-Theorem provides many physical implications as follows: (i) It says that the entropy is increasing with time.. 政治大. (ii) As far as the total entropy dissipation integral Z Ddx. 立. ‧ 國. 學. R3. is bounded in t, the H-function may play a role of the Lyapounov function,. ‧. to prove that the solution of the Boltzmann equation converges to a limit. Corollary 3.1 then asserts that this limit should be a Maxwellian; that is, the. Nat. sit. y. Maxwellian is the only possible asyptotically stable stationary solution of the. io. er. Boltzmann equation.. al. n. iv n C described by the Maxwellian,hnot by any other e n g c h i Udistribution functions. Thus,. (iii) Physically, this can be rephrased as the equilibrium state of the gas is uniquely. one can say that the Maxwellian is built-in in the Boltzmann equation.. 3.3. Grad’s angular cutoff potential Note that the properties of the collision operator Q stated in the previous. section are valid only when the relevant integrals converge. However, there are two different singular properties of the collision kernel, which are the strong singularity at ϑ =. π 2. in (2.15) due to the grazing collision and the unboundedness as |ξ| −→ ∞.. The first singularity does not guarantee the convergence of integral over S2 in (3.1) under a mild assumption on f such that it is bounded. To avoid this difficulty,. 20.

(32) π 2. Grad [11] introduced an idea to cutoff the singularity at ϑ =. assuming that bγ (ϑ). vanishes near ϑ = π2 . The assumption was highly successful for existence theory of the Boltzmann equation in the sense that almost all progresses made after Grad owe to his idea. This is known as the Grad’s angular cutoff assumption. In the following sections, we assume that bγ (ϑ) is a nonnegative measurable function satisfying. Z. S2+. bγ (ϑ)dΩ ≥ b0 , bγ (ϑ) ≤ b1 | cos ϑ|. (3.32). for some constants b0 , b1 > 0.. 政治大 this, let M = M(ξ) be any Maxwellian and introduce the function 立Z. Under this assumption, the collision operator Q becomes well-defined. To see. ‧ 國. R3 ×S2+. B(|ξ − ξ∗ |, ϑ)M(ξ∗ )dΩdξ∗ ,. 學. νM (ξ) =. (3.33). which satisfies under the assumption (3.32),. ‧. ν0 (1 + |ξ|)γ ≤ νM (ξ) ≤ ν1 (1 + |ξ|)γ ,. Nat. sit. y. (3.34). for some positive constants ν0 and ν1 . Notice that (3.34) only holds in the case on. io. n. al. er. hard potentials, i.e., γ ∈ [0, 1].. Ch. engchi. 21. i n U. v.

(33) 4. Hydrodynamical Limits:Expansions and Decomposition In this section, we will use the decomposition of the solution into the macroscopic. and microscopic components to reformulate the Boltzmann equation ft + ξ · ∇ x f =. 1 Q(f, f ), (t, x, ξ) ∈ R+ × R3 × R3 , κ. (4.1). where κ is the Knudsen number introduced in section 1, as a system of conservation. 政治大 components. This kind of thinking 立 is similar to the Hilbert and Champman-Enskog laws for the macroscopic components coupled with an equation for the microscopic. ‧ 國. 學. expansions where the leading term is a local Maxwellian with its macroscopic com-. ponents governed by the conservation laws, either the Euler equations or NavierStokes equations.. ‧ sit. y. Nat. n. al. er. A formal discussion on conservation laws. io. 4.1. Ch. i n U. v. Note that if we multiply both sides of (4.1) by one of the element collision. engchi. invariants ψα (α = 0, 1, 2, 3, 4) which are given by (3.13), and integrate with respect to ξ, we have. Z. R3 or. Z ∂t. R3. ψα f dξ +. 3 X. ψα (ft + ξ · ∇x f )dξ = 0 Z ∂xj. j=1. R3. ξ j ψα f dξ = 0 (α = 0, 1, 2, 3, 4).. If α = 0, then equation (4.2) becomes ρt +. 3 X j=1. Z ∂xj. R. ξ j f dξ = 0, 3. which implies ρt + divx m = 0. 22. (4.2).

(34) If α = i, i = 1, 2, 3, then equation (4.2) becomes mit. +. Z. 3 X. ∂xj. R. ξ j ξ i f dξ = 0.. 3. j=1. By (2.23) and (2.26), we get mit. +. 3 X. ∂xj (mi uj + pij ) = 0.. j=1. Similarly, if α = 4 we have µ ∂t. 1 ρ | u |2 +ρE 2. ¶ +. 3 X. 立. j=1. ¶ X 治政 ρu + up 大 ·. µ. j. ∂xj. 1 | u |2 +E 2. ¸. 3. i. ji. + qj = 0.. i=1. ‧. ‧ 國. 學. Hence, form above discussion, we have the system of conservation laws:   ρt + divx m = 0,         Ã 3 !    X   mi + (mi uj + pij ) = 0, i = 1, 2, 3, t j j=1 x        ( 3 · µ  ¶ X ¶¸ · µ ¸) 3  2 X  2 | u |  |u|  +E + ui pji + q j + ρuj = 0.   ρ 2 +E 2. n. al. er. io. sit. y. Nat t. (4.3). j=1. Ch. i=1. engchi U. v ni. xj. These equations have the so-called conservation form because they express the circumstance that a certain quantity neither created nor destroyed in a certain region D ⊆ R3 because something is flowing through the boundary ∂D.. 4.2. Hilbert expansion. In this subsection, we assume that κ is a small constant and use it as the parameter for the expansion. In 1912, Hilbert [4] introduced the following famous expansion of the solution to the Boltzmann equation: f=. ∞ X n=0. 23. κn fn .. (4.4).

(35) By putting this expansion into the Boltzmann equation (4.1) and comparing the terms by the order of κ, we have the following equations for fn :   Q = 0, 0  (f ) + ξ · ∇ f = Q , n ≥ 1, n−1 t. x n−1. (4.5). n. where Q0 = Q(f0 , f0 ), Qn = 2Q(f0 , fn ) +. n−1 X. Q(fk , fn−k ), n ≥ 1.. (4.6). k=1. Hence, by the Remark 3.2 (iv), the first equation in (4.6) implies that f0 is a local. 政治 µ大 |ξ−u ρ. Maxwellian, i.e.,. 0. 立. (2πRθ0 )3/2. exp. −. |. ¶. 2Rθ0. where ρ0 , u0 , and θ0 are functions of (t, x), and f0 satisfies. ,. (4.7). 學. ‧ 國. f0 = M0 ≡ M[ρ0 ,u0 ,θ0 ] =. 0 2. ‧. (f0 )t + ξ · ∇x f0 = Q1 .. (4.8). y. Nat. Here, Q1 is a microscopic component which is orthogonal to the five collision in-. io. Z. er. of conservation laws. sit. variants ψα (ξ), α = 0, 1, · · · , 4. The solvability condition for (4.8) gives the system. n. a l ψ ((f ) + ξ · ∇ f )dξ = 0, i v R n C U which are exactly the compressiblehEuler i e nequations gch α. 0 t. x 0. (4.9). 3.    ρ0t + ∇x · (ρ0 u0 ) = 0,         (ρ0 u0 )t + ∇x · (ρ0 u0 ⊗ u0 ) + ∇x p0 = 0,           £ρ0 (E 0 + 1 |u0 |2 )¤ + ∇ · ©[ρ0 (E 0 + 1 |u0 |2 ) + p0 ]uª = 0, x 2 2 t. (4.10). where the pressure function is given by p0 = Rρ0 θ0 and the internal energy E 0 = 3 Rθ0 . 2. For n ≥ 1, let Sn =. n−1 X. Q(fk , fn−k ),. k=1. 24. (4.11).

(36) and LM0 h = 2Q(h, f0 ),. (4.12). which is the linearized collision operator with respect to the local Maxwellian M0 . Then under the solvability condition for the second equation in (4.5) Z ¡ ¢ ψα (fn−1 )t + ξ · ∇x fn−1 dξ = 0,. R3. (4.13). fn can be represented in terms of fk for k = 0, 1, · · · , n − 1 by fn =. 4 X. ¡ ¢ cα ψα M0 + L−1 (fn−1 )t + ξ · ∇x fn−1 − Sn .. α=0. 立. 政治大. ‧ 國. 學. Thus, the conservation laws Z ¡ ¢ ψα (fn )t + ξ · ∇x fn dξ = 0, α = 0, 1, · · · , 4,. R3. (4.14). (4.15). are the systems of linearized Euler equations around the fluid variables (ρ0 , u0 , θ0 ). ‧. for the macroscopic components in fn .. y. Nat. Since to determine the value of fn in the Hilbert expansion involves the dif-. sit. ferentiation of fn−1 , by induction, the convergence of this expansion can only be. er. io. expected when the solution is infinitely differentiable and bounded with respect to. al. n. iv n C vergence, especially in the presenthof initial layer, shock e n g c h i U layer and boundary layer the Knudsen number κ. Therefore, in general, the Hilbert expansion does not con-. where the value of the differentiation grows when κ decreases.. 4.3. Chapman-Enskog expansion. The Chapman-Enskog expansion was introduced by Chapman and Enskog in 1916 and 1917 independently. The advantage of this expansion is that the first order correction yields the Navier-Stokes equations for the macroscopic components so that the viscosity and heat conductivity are correctly represented. Formally, we can write. ∞. X ∂ (n) fn ∂f = κn . ∂t ∂t n=0 25. (4.16).

(37) Since the conserved quantities are unexpanded, the consistency requires that for n ≥ 1,. Z. R3. ψα fn dξ = 0, α = 0, 1, 2, 3, 4,. (4.17). which implies that all the function fn for n ≥ 1 are microscopic. Substituting (4.4) and (4.16) into the Boltzmann equation (4.1), we have    Q0 = Q(f0 , f0 ) = 0,     n−1 (k)  X  ∂ fn−k−1   + ξ · ∇x fn−1 = 2Q(f0 , fn ) + Sn , n ≥ 1,   ∂t k=0. 立. (4.18). 政治大. where the notation has the same meaning as in the last subsection. However, notice. ‧ 國. 學. that here each fn is a functional of the conserved quantities which are not expanded.. ‧. Again, the first equation in (4.18) implies that f0 must be a local Maxwellian, i.e., µ ¶ | ξ − u |2 ρ exp − . (4.19) f0 = M ≡ M[ρ,u,θ] = (2πRθ)3/2 2Rθ. n. al. er. io. ∂ (0) f0 + ξ · ∇x f0 = LM f1 . ∂t. sit. y. Nat. Furthermore, for n = 1, the second equation in (4.18) can be written as. i n U. (4.20). v. The solvability condition for (4.20) immediately gives the following Euler equations  (0)   ∂ ρ = − ∂ (ρui ),   ∂t ∂xi         (0) i ∂ u ∂ui 1 ∂p (4.21) = −uj j − , i, j = 1, 2, 3, i  ∂t ∂x ρ ∂x         i (0)    ∂ θ = −ui ∂θ − 2 θ ∂u , ∂t ∂xi 3 ∂xi. Ch. engchi. where p = Rρθ, here and in what follows, the summation is over any repeated indices. By plugging the expression of the local Maxwellian of f0 into the equation (4.20), we have µ ¶ 1 |c|2 3 1 j 0 j 1 0 MB ρ + − MB0 θ + c B u = L M f1 , ρ θ 2Rθ 2 Rθ 26. (4.22).

(38) where c = ξ − u is the random velocity introduced in (2.22) and B 0 is the following linear operator B0 ≡. ∂ (0) + ξ · ∇x . ∂t. Now we can substitute the time derivative. ∂ (0) ∂t. (4.23). of (4.21) into the equation (4.22) to. obtain µ. ¶ µ ¶ 5 ci ∂θ 1 2 ∂ui |c|2 1 i j − M c c − |c| δij M j = LM f1 . + 2Rθ 2 θ ∂xi Rθ 3 ∂x. (4.24). By using the Burnett functions defined by  2   Aj (ξ) = |ξ| 2−5 ξ j , j = 1, 2, 3,  . 政治大. 立. (4.25). f1 =. L−1 M. µ ³ c ´ ∂θ ³ c ´ ∂ui ¶ √ M i + Bij √ M j . RAi √ ∂x ∂x Rθ Rθ. ‧. ‧ 國. we have. 學.     B (ξ) = ξ i ξ j − 1 δ | ξ |2 , i, j = 1, 2, 3, ij 3 ij. (4.26). y. Nat. Notice that we have used the fact that the operator LM is invertible in the micro-. sit. scopic space which is the space orthogonal to the null space of LM .. n. al. er. io. Before going further, we review the properties of the Burnett functions. Proposition 4.1. Let. Ch. i n U. v. e n gB c=hLi B.. A0 = L−1 M A,. 0. M. (4.27). Then there exist positive functions a(r) and b(r) defined on [0, ∞) such that A0 (ξ) = −a(|ξ|)A(ξ),. B 0 (ξ) = −b(|ξ|)B(ξ).. (4.28). And the following properties hold, where h·, ·i denotes the inner product of L2 (R3 ). (1) h−Ai , A0i i is positive and independent of i. (2) hAi , A0j i = 0 for any i 6= j. 0 (3) hAi , Bjk i = 0 for any i, j, k.. 27.

(39) 0 0 (4) hBij , Bkl i = hBkl , Bij0 i = hBji , Bkl i holds and is independent of i, j for any. fixed k, l. (5) −hBij , Bij0 i is positive and independent of i, j when i 6= j. 0 (6) −hBii , Bjj i is positive and independent of i, j when i 6= j.. (7) −hBii , Bii0 i is positive and independent of i. 0 (8) hBij , Bkl i = 0 unless either (i, j) = (k, l) or (l, k), or i = j and k = l.. 政治大. 0 (9) hBii , Bii0 i − hBii , Bjj i = 2hBij , Bij0 i holds for any i 6= j.. 立. These proofs are quite technical and can be found in [23].. ‧ 國. 學. With the Burnett functions, the viscosity µ(θ) and heat condutivity coefficient. ‧. ℵ(θ) can be represented by  ¶ µ µ ¶ ¶ µ Z c c  −1  LM Bij √ M dξ > 0, i 6= j,  µ(θ) = −κRθ Bij √   Rθ Rθ R3 . y. Nat. sit. (4.29). er. io.  ¶ µ µ ¶ ¶ µ Z   c c  −1 2  LM Al √ M dξ > 0,  ℵ(θ) = −κR θ Al √ Rθ Rθ R3. al. n. iv n C U the first order approxih e n glaws into the conservation c htoi include. Note that these coefficients are independent of the density function ρ. Now, if we put f1. mation, then the conservation laws take the form Z ¡ ¢ ψα (f0 )t + ξ · ∇x (f0 + κf1 ) dξ = 0.. R3. (4.30). Since f1 is microscopic, its contribution to the conservation of mass is zero. And its contribution to the equations of conservation of momentum and energy is represented by the stress tensor and heat flux: Z Z κ (1) (1) i j ci |c|2 f1 dξ. pij = κ c c f1 dξ, qi = 2 3 3 R R. 28. (4.31).

(40) With the proposition 4.1, it is straightforward to calculate the stress tensor and heat flux in terms of the fluid variables: µ i ¶  ∂uk ∂u ∂uj 2 (1)   µ(θ) p = −µ(θ) + + δij ,  ij  ∂xj ∂xi 3 ∂xk  (4.32).      q (1) = −ℵ(θ) ∂θ . i ∂xi. In summary, the first order approximation in the Chapman-Enskog expansion is the compressible Navier-Stokes equations:     ρt + divx m = 0,         ¢¤ £ ¡ i  j 2 i i    mt + divx (m u) + pxi = µ(θ) uxj + uxi − 3 δij divx u xj , i, j = 1, 2, 3,. 學. ‧ 國. 立. 政治大. (4.33). ‧.   ¶¸ ½· µ ¶ ¸ ¾ · µ    1 1 2 2  + divx ρ E + 2 |u| + p u ρ E + 2 |u|     t ¶ µ   ¡ ¢  j 2 i i   = µ(θ)u uxj + uxi − 3 δij divx u + ℵ(θ)θxi xj .. y. Nat. sit. Again, similar but tedious calculation can be used to find the next terms, f2 , f3 , · · · ,. n. al. er. io. in the Chapman-Enskog expansion, however, without good mathematical theory.. 4.4. Ch. engchi Micro-macro decomposition. i n U. v. In this subsection, we study a micro-macro decomposition of the Boltzmann equation (4.1), which is based on the decomposition of the solution into its macroscopic (fluid dynamic) and microscopic (kinetic) component, we can reformulate the Boltzmann equation into a system of conservation laws for the time evolution of the macroscopic variables and an equation for the time evolution of the microscopic variable. The main idea is not to have any approximation, but a complete description of the solutions to the Boltzmann equation so that the analytic techniques from the theory of conservation laws can be applied in the study of the Boltzmann equation.. 29.

(41) To see this, let f (t, x, ξ) be the solution to the Boltzmann equation (4.1), we will decompose it into the macroscopic fluid part, the local Maxwellian M = M(t, x, ξ) = M[ρ,u,θ] (ξ), and the microscopic, non-fluid part G = G(t, x, ξ), f (t, x, ξ) = M(t, x, ξ) + G(t, x, ξ).. (4.34). The local Maxwellian M(t, x, ξ) is constructed from the fluid variables, the five conserved quantities, the mass density ρ(t, x), momentum m(t, x) and energy E(t, x) + | u(t, x) |2 of the Boltzmann equation: Z    ρ(t, x) ≡ f (t, x, ξ)dξ,   3  R       Z  i m (t, x) ≡ ψi f (t, x, ξ)dξ, (i = 1, 2, 3),  3 R       Z   ¢ ¡   1 2  ρ E + 2 | u | (t, x) ≡ ψ4 f (t, x, ξ)dξ,. 立. (4.35). ‧ 國. 學. R3. ¶ | ξ − u |2 − . 2Rθ. y. µ. Nat. ρ exp M ≡ M[ρ,u,θ] (ξ) ≡ (2πRθ)3/2. ‧. and. 政治大. (4.36). sit. 1 2. n. al. er. io. With respect to the local Maxwellian, we define an inner product in ξ ∈ R3 as Z 1 h(ξ)g(ξ)dξ (4.37) hh, giM ≡ R3 M. Ch. engchi. i n U. v. for functions h, g of ξ such that the integral is well defined. Using this inner product with respect to this Maxwellian, the subspace spanned by the collision invariants has the following set of orthogonal basis:  1   χ 0 ≡ χ0 (ξ; ρ, u, θ) ≡ √ M,   ρ           ξ i − ui   √ χ ≡ χ (ξ; ρ, u, θ) ≡ M, f or i = 1, 2, 3,  i i   Rθρ   µ ¶  2  | ξ − u | 1   − 3 M, χ4 ≡ χ4 (ξ; ρ, u, θ) ≡ √    Rθ 6ρ           hχ , χ i = δ f or α, β = 0, 1, 2, 3, 4. α β αβ 30. (4.38).

(42) With this basis, we can define the macroscopic projection P0 and microscopic projection P1 as follows: P0 h ≡. 4 X. hh, χα iM χα ,. (4.39). α=0. P1 h = h − P0 h.. (4.40). Note that the operators P0 and P1 are projections, that is, P0 P0 = P0 ,. P1 P1 = P1 ,. P0 P1 = P1 P0 = 0.. 治政 the local Maxwellian states so that the linear collision大 operator L 立. We view the above decomposition of Boltzmann equation as the linearization around M. is. ‧ 國. (4.41). 學. L ≡ L[ρ,u,θ] g = Q(M + g, M + g) − Q(g, g).. Definition 4.1. A function h(ξ) is called non-fluid if it gives raise to zero conserved. ‧. quantities, that is,. Z h(ξ)ψα dξ = 0,. α = 0, 1, 2, 3, 4.. (4.42). sit. y. Nat. R3. er. io. Note that functions in the range of the microscopic projection P1 are nonfluid. It is clear that for the solution f (t, x, ξ) of the form (4.34) of the Boltzmann. al. n. equation (4.1), we have. Ch. i n U. v. e n gPcf h= iG.. P0 f = M,. 1. From the decomposition of the solution f = M + G, the Boltzmann equation (4.1) becomes (M + G)t + ξ · ∇x (M + G) =. ¢ 1¡ 2Q(G, M) + Q(G, G) . κ. (4.43). We now decompose the Boltzmann equation. Using the same method as above, if we multiply equation (4.43) by ψα. (α = 0, 1, 2, 3, 4) and integrate with respect. 31.

(43) to ξ we obtain the system of conservation laws:   ρt + divx m = 0,         Ã 3 !  Z   X   mi + i j + pxi + mu ψi (ξ · ∇x G)dξ = 0, i = 1, 2, 3, t 3 R j=1 xj         ¶ ¸¾ ¶¸ · µ Z 3 ½ · µ  X  | u |2  |u|2 j  +E +p + ψ4 (ξ · ∇x G)dξ = 0. + u ρ   ρ 2 +E 2 3 t. R. xj. j=1. (4.44). 政治大 The microscopic equation is obtained by applying the microscopic projection 立 to the Boltzmann equation (4.43). Note that M , as a function of ξ, is in the. Here p is the pressure for the monatomic gases same as Equation(2.33).. t. 學. ‧ 國. P1. space spanned by χα , α = 0, 1, 2, 3, 4. Hence. ‧. P0 Mt = Mt .. R3. α = 0, 1, 2, 3, 4.. y. hψα dξ = 0,. io. sit. Nat. Note also that P0 h = 0 if Z. n. al. er. Thus the projection of collision terms under P0 is zero. We also have Z Z Gt ψα dξ = ∂t Gψα dξ = 0.. R3. Therefore, we have. Ch. e n gR c h i 3. i n U. v. P0 Gt = 0, and so P1 (Mt + Gt ) = (Mt + Gt ) − P0 (Mt + Gt ) = Gt . With these, the microscopic equation is ¡ ¢ 1 1 Gt + P1 ξ · ∇x G + ξ · ∇x M = LG + Q(G, G). κ κ From (4.45), we have µ ¶ µ ¶ ¡ ¢ −1 −1 G = κL P1 ξ · ∇ x M + L κ ∂t G + P1 ξ · ∇x G − Q(G, G) µ ¶ = κL−1 P1 ξ · ∇x M + Θ,. 32. (4.45). (4.46).

(44) where. µ −1. Θ := L. ¶ ¡ ¢ κ ∂t G + P1 ξ · ∇x G − Q(G, G) ,. and substitute this into (4.44) yields the following fluid-type system for the macroscopic components:   ρt + divx m = 0,         Ã 3 !  Z   X   i j i  mt + mu + pxi + κ ψi (ξ · ∇x L−1 P1 ξ · ∇x M)dξ    R3  xj¶  Z j=1µ   + ψi ξ · ∇x Θ = 0, i = 1, 2, 3, R3         ¶ ¸¾ ¶¸ · µ Z 3 ½ · µ  X  | u |2  |u|2 j  +E +p + κ ψ4 (ξ · ∇x L−1 P1 ξ · ∇x M)dξ + u ρ ρ 2 +E    2  R3 xj  µ t j=1 ¶ Z      + ψ4 ξ · ∇x Θ = 0.. 立. ‧. ‧ 國. 學. R3. 政治大. (4.47). sit. y. n. al. R3. er. io. R3. Nat. In the above system, the terms Z Z −1 −κ ψi (ξ · ∇x L P1 ξ · ∇x M)dξ = −κ ψi (ξ · ∇x L−1 [ρ,u,θ] P1 ξ · ∇x M[ρ,u,θ] )dξ. i C h = −κ ψ (ξ · U ∇n L e n g cR h i Z. i. 3. x. v. −1 [1,u,θ] P1 ξ. · ∇x M[1,u,θ] )dξ. and Z Z −1 −κ ψ4 (ξ · ∇x L P1 ξ · ∇x M)dξ = −κ ψ4 (ξ · ∇x L−1 [ρ,u,θ] P1 ξ · ∇x M[ρ,u,θ] )dξ. R3. R3. Z = −κ. R3. ψ4 (ξ · ∇x L−1 [1,u,θ] P1 ξ · ∇x M[1,u,θ] )dξ. are the viscosity and heat conductivity terms which are the same as those in the compressible Navier-Stokes equations; and they are independent of the density gradient ∇x ρ. In fact, with the Burnett functions defined by (4.25), and the viscosity coefficient µ(θ) and heat conductivity coefficient ℵ(θ) defined by (4.29), the fluid-. 33.

(45) type system (4.47) can be written as    ρt + divx m = 0,         Ã 3 !   X    mit + m i uj + pxi     j j=1  x  ¶¸ µ ¶ µ Z  3 ·  X  2 j  = i − ψi ξ · ∇x Θ dξ, i = 1, 2, 3, µ(θ) uxj + uxi − δij divx u 3 3 j R x j=1         ¶ ¸¾ ¶¸ · µ 3 ½ · µ  X  | u |2  |u|2 j  +E +p + u ρ ρ 2 +E    2  xj t j=1   ¶¸ µ ¶ µ · µ ¶  Z 3 3  X X  2 j  i  + ℵ(θ)θxj − ψ4 ξ · ∇x Θ dξ. µ(θ) uxj + uxi − δij divx u   = 3 R3 xj xj j=1 j=1. 立. 政治大. ‧ 國. 學. (4.48). From this fluid-type system, we can easily deduce the structure of the compress-. ‧. ible Euler and the compressible Navier-Stokes equations. For example, when the Knudsen number κ is set to be zero, the system (4.48) becomes the compressible. y. Nat. sit. Euler equations; and when Θ is set to be zero in (4.48), it becomes the compressible. al. er. io. Navier-Stokes equations. These fluid equations as derived through the Hilbert and. n. Chapman-Enskog expansions are approximations to the Boltzmann equation. How-. Ch. i n U. v. ever, the above system is part of Boltzmann equation. Nevertheless, our approach. engchi. is consistent in spirit with these expansions in that the higher order terms beyond first order in the expansions must satisfy a solvability condition, which means that these terms are microscopic. Furthermore, the above analysis also indicates that if we deduce the compressible Navier-Stokes equations from the Boltzmann equation, the viscosity coefficient µ(θ) > 0 and the heat conductivity coefficient ℵ(θ) > 0 are smooth functions of the temperature θ.. 34.

(46) 5. Solutions in a Weighted Lebesgue Space This section we are going to discuss the initial-value problem for the Boltzmann. equation for the hard-sphere monatomic gas in the whole space R3 :   f + ξ · ∇ f = Q(f, f ), (t, x, ξ) ∈ R × R3 × R3 t x +  f (0, x, ξ) = f (x, ξ), (x, ξ) ∈ R3 × R3 ,. (5.1). 0. where Z 1 (f 0 g∗0 + g 0 f∗0 − f g∗ − gf∗ )B(|ξ − ξ∗ |, ϑ)dΩdξ∗ , Q(f, g)(ξ) ≡ 2 R3 ×S2+ ½ ¾ 2 2 S+ = Ω ∈ S : (ξ − ξ∗ ) · Ω ≥ 0 ,. 政治大. 立. ‧ 國. 學. and. £ ¤ £ ¤ ξ 0 = ξ − (ξ − ξ∗ ) · Ω Ω, ξ∗0 = ξ∗ + (ξ − ξ∗ ) · Ω Ω. ‧. as before.. sit. n. al. er. io. Preliminaries. y. Nat. 5.1. Ch. engchi. i n U. v. In this subsection, we give the definition of the mild solution, and present some lemmas which will be needed in the proofs of our main results.. Motivation for definition of mild solutions. We first consider the case for the collision term Q(f, f ) ≡ 0 in (5.1) so that the Boltzmann equation in (5.1) becomes the free transport equation with initial data:   f + ξ · ∇ f = 0, (t, x, ξ) ∈ R+ × R3 × R3 , t x  f (0, x, ξ) = f (x, ξ), (x, ξ) ∈ R3 × R3 .. (5.2). 0. Fix any point (t, x, ξ) ∈ R+ × R3 × R3 and define z(s) := f (t + s, x + sξ, ξ) for s ∈ R.. 35. (5.3).

(47) Then d z(s) = ∇x f (t + s, x + sξ, ξ) · ξ + ft (t + s, x + sξ, ξ) = 0. ds Hence, z is a constant function of s, and therefore for each point (t, x, ξ), f is constant on the line through (t, x, ξ) with the direction (1, ξ, 0) ∈ R5 . Furthermore, since f (0, x − tξ, ξ) = f0 (x − tξ, ξ), we deduce f (t, x, ξ) = f0 (x − tξ, ξ), (t, x, ξ) ∈ R+ × R3 × R3 .. (5.4). Hence, if (5.2) has a sufficiently regular solution f , it must certainly be given by. 政治大. (5.4). And conversely, we can check directly that if f0 is C 1 , then f defined by (5.4) is indeed a solution of (5.2).. 立. Now, we return to the I.V.P. (5.1). As above, fix (t, x, ξ) ∈ R+ × R3 × R3 and,. ‧ 國. 學. inspired by the calculation above, set z(s) := f (t + s, x + sξ, ξ) for s ∈ R. Then d z(s) = ∇x f (t+s, x+sξ, ξ)·ξ +ft (t+s, x+sξ, ξ) = Q(f, f )(t+s, x+sξ, ξ). (5.5) ds. ‧. Thus,. Nat. n. al. =. Ch. d z(s)ds −t ds. e nZ g c h i 0. =. sit. 0. er. io. Z. y. f (t, x, ξ) − f0 (x − tξ, ξ) = z(0) − z(−t). i n U. v. Q(f, f )(t + s, x + sξ, ξ)ds. −t. Z. t. =. Q(f, f )(s, x + (s − t)ξ, ξ)ds, 0. which implies that Z. t. f (t, x, ξ) = f0 (x − tξ, ξ) +. Q(f, f )(s, x + (s − t)ξ, ξ)ds.. (5.6). 0. The definition of mild solutions can be stated as follows: Definition 5.1. A nonnegative function f ∈ C([0, T ); L1+ (R3x × R3ξ )) is a mild solution to (5.1) with a nonnegative initial data f0 if and only if for all t ∈ [0, T ) and a.e. (x, ξ) ∈ R3 × R3 , f satisfies the integral equation (5.6).. 36.

(48) Next, we define a weighted Banach space as follows: Let M ≡ M(ξ) be any −1 2. . Define ¯ ½ ¾ ¯ 1 3 3 1 3 3 ¯ Lω (Rx × Rξ ) = f ∈ L (Rx × Rξ ) ¯ kf kL1ω < +∞ ,. Maxwellian, and let ω := M. where. (5.7). Z kf kL1ω :=. R3 ×R3. | f | ω dξdx.. (5.8). To prove our main results, we now state the assumptions on cross section and the collision frequency as follows:. 政治大 (A ) The cross section B(|ξ − ξ |, ϑ) satisfies (2.15). 立 1. ∗. ‧ 國. 學. (A2 ) bγ (ϑ) is a nonnegative measurable function satisfying (3.32) for γ ∈ [0, 1] and inequality (3.34) holds.. ‧. Under these assumptions, we have the following estimates for Q.. y. Nat. sit. Lemma 5.1. Under the assumptions (A1 ) and (A2 ), for any p ∈ [1, ∞], k ∈ [0, 1],. al. n. ° ¢° ° −k ¡ ° °νM ω Q(f, f ) − Q(g, g) °. er. io. and f, g ∈ Lp (R3x × R3ξ ), there exists a constant C > 0 such that Lp. ½µ ≤C. Ch. ¶. 1−k kνM ωf kLp. +. 1−k kνM ωgkLp. engchi. i n U. v. µ. kω(f − g)kLp + kωf kLp + kωf kLp. ¶ 1−k kνM ω(f. (5.9) where k · kLp ≡ k · kLp (R3x ×R3ξ ) . Proof. Write 1 Q(f, g) = [Q+ (f, g) + Q+ (g, f ) − Q− (f, g) − Q− (g, f )], 2 where. (5.10). Z +. Q (f, g) =. R3 ×S2+. f 0 g∗0 B(|ξ − ξ∗ |, ϑ)dΩdξ∗ ,. 37. (5.11). ¾ − g)kLp ,.

(49) and. Z −. Q (f, g) =. f g∗ B(|ξ − ξ∗ |, ϑ)dΩdξ∗ .. R3 ×S2+. (5.12). Then we can write Q(f, f ) − Q(g, g) = [Q+ (f, f ) − Q+ (g, g)] + [Q− (g, g) − Q− (f, f )].. (5.13) 1. 1. First, we prove this lemma for Q+ . To do this, put f = M 2 h1 and g = M 2 h2 . By the conservation laws of collision (2.8) and the definition of Maxwellian (3.30), we have. 政治大. M(ξ 0 )M(ξ∗0 ) = M(ξ)M(ξ∗ ).. 立. Hence, the Hölder inequality gives. (5.14). ‧ 國. Z. 1. 1. io. R3 ×S2+. M(ξ) 2 M(ξ∗ ) 2 |h1 (ξ 0 )h1 (ξ∗0 ) − h2 (ξ 0 )h2 (ξ∗0 )|B(|ξ − ξ∗ |, ϑ) dΩdξ∗. al. n. µZ ≤. sit. =. y. Nat. Z. R3 ×S2+. q 2. q 2. Ch. ¶1/q µ Z q. er. R3 ×S2+. |f 0 f∗0 − g 0 g∗0 |B(|ξ − ξ∗ |, ϑ) dΩdξ∗. ‧. ≤. 學. |Q+ (f, f ) − Q+ (g, g)|. v ni. e n g c h i UR. M(ξ) M(ξ∗ ) B(|ξ − ξ∗ |, ϑ) dΩdξ∗. S. 3× 2 +. |h1 (ξ. µZ ≤ CνM (ξ)M(ξ). ¡. 0. ≤ C νM (ξ ) +. 1 2. R3 ×S2+ ¢. Z. R3 ×S2+. 1 p. )h1 (ξ∗0 ). − h2 (ξ. ¶1/p. νM (ξ∗0 ). where p ∈ [1, ∞),. ¶1/p 0. +. 1 q. |h1 (ξ. 0. )h1 (ξ∗0 ). − h2 (ξ. 0. )h2 (ξ∗0 )|p. dΩdξ∗. µZ M(ξ). 1 2. ¶1/p. R3 ×S2+. |h1 (ξ. 0. )h1 (ξ∗0 ). − h2 (ξ. 0. )h2 (ξ∗0 )|p. dΩdξ∗. = 1,. ¡ ¢γq q M(ξ∗ ) 2 B(|ξ − ξ∗ |, ϑ)q dΩdξ∗ ≤ C 1 + |ξ| ≤ CνM (ξ)q. 38. ,. 0. )h2 (ξ∗0 )|p. dΩdξ∗.

(50) by assumption (A2 ), and ¡ ¢γ νM (ξ) ≤ C 1 + |ξ| ¡ ¢γ = C 1 + |ξ 0 − [(ξ 0 − ξ∗0 ) · Ω]Ω| ¡ ¢γ ≤ C 2 + |ξ 0 | + |ξ∗0 | ¡ ¢ ≤ C νM (ξ 0 ) + νM (ξ∗0 ). 政治大 Since the Jacobian of the change of variable (ξ, ξ , x, Ω) ←→ (ξ , ξ , x, Ω) is unity, 立. by (2.13) and (3.34). Here C is a positive constant depending on b∗ , ν0 , ν1 , ρ, u, θ. 0. ∗. ‧ 國 ·. ≤C. νM (ξ∗0 )1−k. n. al. +. Ch. ½µ. ¶ 1−k kνM ωf kLp. +. − h2 (ξ. 0. ¶1/p. )h2 (ξ∗0 )|p. dΩdξ∗0 dξ 0 dx. y 0 1−k. νM (ξ ). io. R3 ×R3 ×R3 ×S2+. ¡. ¢(1−k)p νM (ξ∗0 ) |h1 (ξ 0 )h1 (ξ∗0 ). 1−k kνM ωgkLp. sit. ½Z ≤C. νM (ξ ) +. Nat. R3 ×R3 ×R3 ×S2+. 0. ¸ ¢¡ ¢ p 0 0 0 0 0 0 |h1 (ξ )||h1 (ξ∗ ) − h2 (ξ∗ )| + |h1 (ξ ) − h2 (ξ )||h2 (ξ∗ )| ¾1/p 0 0 dΩdξ∗ dξ dx. engchi. er. ≤C. ¡. Lp. ‧. µZ. 學. then by the Minkowski’s inequality, we can obtain ° ¢° ° −k ¡ + ° °νM ω Q (f, f ) − Q+ (g, g) °. 0 ∗. i n U. v. µ. kω(f − g)kLp + kωf kLp + kωf kLp. ¶ 1−k kνM ω(f. This proves (5.9) for Q+ for the case p ∈ [1, ∞). The case p = ∞ can be proved similarly, and the proof for Q− is also similar. Now, the proof of this lemma is complete.. Lemma 5.2. Under the assumptions (A1 ) and (A2 ), for any f, g ∈ L1ω (R3x × R3ξ ), there exists a constant C > 0 such that ¡ ¢ kQ(f, f ) − Q(g, g)kL1ω ≤ CkνM kL∞ kf kL1ω + kgkL1ω kf − gkL1ω .. 39. (5.15). ¾ − g)kLp ..

(51) Proof. Setting k = 0 and p = 1 in (5.9), and using the Hölder inequality, we obtain kQ(f, f ) − Q(g, g)kL1ω ½µ ≤C. ¶ kνM ωf kL1 + kνM ωgkL1 kf − gkL1ω. µ ¶ ¾ + kf kL1ω + kf kL1ω kνM ω(f − g)kL1. ¡ ¢ ≤ 2CkνM kL∞ kf kL1ω + kgkL1ω kf − gkL1ω , as desired.. 立. 政治大. Furthermore, we recall that a mapping F : X −→ X, where X is a Banach. ‧ 國. 學. space with norm k·kX , is said to be a contraction of X if there is a positive constant α < 1 such that. ‧. k F (u) − F (v) kX ≤ α k u − v kX , ∀u, v ∈ X.. Nat. y. (5.16). sit. Lemma 5.3 (Banach’s Fixed Point Theorem). A contraction F of a Banach. er. io. space X has a unique fixed point.. al. n. iv n C U Finally, in order to derive ourhmain of mild solutions, we e n result g c hfori stability. also need a fundamental inequality which is the well-known Gronwall’s inequality :. Lemma 5.4 (Gronwall’s Inequality). Let u : [a, b] −→ [0, ∞) and v : [a, b] −→ R be two continuous functions, and let C be a constant. If Z t v(t) ≤ C + v(s)u(s) ds. (5.17). a. for t ∈ [a, b], then we have µZ v(t) ≤ C exp. u(s) ds a. for t ∈ [a, b].. 40. ¶. t. (5.18).

(52) 5.2. Existence of the mild solution. Now, we give the crucial estimate for the local existence of the mild solution to (5.1) by the Banach’s fixed point theorem. To do this, given T > 0, let us denote the closed subset X0 of L1ω (R3x × R3ξ ) by ¯ ½ ¾ ¯ ¯ X0 = f ∈ X ¯ kf kL1ω ≤ 2M0 for some M0 > 0 , and. ¯ ¾ ½ ¯ C([0, T ]; X0 ) = f (t, x, ξ) ¯¯ k f k< ∞ ,. 政治大. where. 立 k f k:= max k f k t∈[0,T ]. (5.19). (5.20). R3x ×R3ξ ) .. (5.21). L1ω (. ‧ 國. 學. Theorem 5.1. Under the assumptions (A1 ) and (A2 ), if f0 ≥ 0, f0 ∈ L1ω (R3x × R3ξ ) ∩ C(R3x × R3ξ ) and kf0 kL1ω ≤ M0 , then there exists a unique mild solution. ‧. f ∈ C([0, T ]; X0 ) to the initial-value problem (5.1).. y. Nat. sit. Proof. Let X := C([0, T ]; X0 ). We will apply Banach’s fixed point theorem in the. er. io. space X. From the integral equation (5.6), we may define a mapping F on X by Z t F (f ) = f0 (x − tξ, ξ) + Q(f, f )(s, x + (s − t)ξ, ξ)ds, t ∈ [0, T ]. (5.22). n. al. 0. Ch. engchi. i n U. v. Assume kf0 kL1ω ≤ M0 , then by lemma 5.2, we see that F (f ) ∈ X. Furthermore, for f, g ∈ X, we have Z. t. F (f ) − F (g) =. (Q(f, f ) − Q(g, g))(s, x + (s − t)ξ, ξ)ds, t ∈ [0, T ].. (5.23). 0. Apply lemma 5.2 again, we obtain Z tµZ k F (f ) − F (g) kL1ω ≤ 0. R3 ×R3. ¶ | (Q(f, f ) − Q(g, g))(s, x + (s − t)ξ, ξ) | ω dξdx ds. ≤ 4M0 CT kνM kL∞ kf − gk = α k f − g k,. 41.