Aspects of dissipation for compressible fluids and kinetic theory Tai-Ping Liu1,2 rsta.royalsocietypublishing.org

Research Cite this article: Liu T-P. 2013 Aspects of dissipation for compressible fluids and kinetic theory. Phil Trans R Soc A 371: 20120347. http://dx.doi.org/10.1098/rsta.2012.0347

One contribution of 11 to a Theme Issue ‘Entropy and convexity for nonlinear partial differential equations’.

1 Institute of Mathematics, Academia Sinica, Taipei 115, Taiwan,

Republic of China 2 Department of Mathematics, Stanford University, Stanford, CA 94305, USA There are several types of dissipation—the viscosity and heat conductivity, the nonlinearity and the coupling of distinct characteristics—that can occur in the system of conservation laws. In addition to these corresponding types of dissipation, there are other consequences of the H-theorem and the boundary dissipative effects for the kinetic theory. We discuss these issues and raise questions.

1. Introduction Subject Areas: differential equations

Consider the system of hyperbolic and viscous conservation laws ut + ∇x · f (u) = 0

(1.1)

ut + ∇x · f (u) = ∇x · (B(u)∇x u).

(1.2)

Keywords: dissipation, nonlinearity, shock waves

and

Author for correspondence: Tai-Ping Liu e-mail: [email protected]

The viscous conservation laws (1.2) are dissipative owing to the presence of the viscosity matrix B(u). There is the basic entropy condition for the hyperbolic conservation laws (1.1) in the form of entropy inequality  η(u)t + ∇x · q(u) ≤ 0

(1.3)

to ensure that the shock waves are the zero dissipation limit, B → 0, of solutions for the viscous conservation laws (1.2). The entropy inequality is an equality for smooth solutions, and so there is the compatibility condition for the entropy η(u) and the entropy flux q(u) ∇u η(u) · ∇u f (u) = ∇u q(u).

(1.4)

The entropy function is usually assumed to be convex η (u) > 0.

(1.5)

0

{−si [η(ui− ) − η(ui+ )] + f (ui+ ) − f (ui+ )}(t) dt ≤ ∞,

(1.6)

i

where si , i = 1, 2, . . . , are the corresponding shock speeds. The entropy condition is fundamental to the functional analytical study of the conservation laws. The nonlinearity, through the entropy condition, induces the inviscid dissipation. In the next section, we will comment on the fact that the strength of the dissipation depends on the nonlinearity of the flux function f (u), that is, it depends on the constitutive laws of the material under study. For the viscous conservation laws (1.2), there are then two aspects of dissipation: those induced by the nonlinearity of the flux f (u) and those induced by the viscosity matrix B(u). We illustrate this through simple examples in §3. The varying degrees of nonlinearity give rise to varying wave forms. These are the basic shock waves and expansion waves. The combination of these waves gives rise to rich wave behaviours, as illustrated in §4. In §5, we consider the system of conservation laws and indicate the complexity of wave behaviour, in particular the combination of compression and expansion waves pertaining to distinct characteristic fields. We turn to the kinetic theory from §6 on. In the kinetic theory, besides the space variables x = (x1 , x2 , x3 ) and the time variable t, there is the microscopic velocity ξ = (ξ 1 , ξ 2 , ξ 3 ) upon which the density distribution f depends, f = f (x, ξ , t). The macroscopic variables are computed from the microscopic density function f  ρ(x, , t) ≡  ρv(x, , t) ≡  ρe(x, , t) ≡

and

R3 R3

f (x, ξ , t) dξ , density, ξ f (x, ξ , t) dξ , momentum, |ξ − v|2 f (x, ξ , t) dξ , internal energy, 2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪  ⎪ ⎪ ⎪ i i j j 3 ⎪ pij (x, , t) ≡ (ξ − v )(ξ − v )f (x, ξ , t) dξ , P = {pij }i,j=1 , stress tensor⎪ ⎪ ⎪ 3 ⎪ R ⎪ ⎪ ⎪  ⎪ 2 ⎪ |ξ − v| ⎪ i i ⎪ ⎭ f (x, ξ , t) dξ , heat flux. qi (x, , t) ≡ (ξ − v ) 3 2 R R3

(1.7)

Here, v = (v 1 , v 2 , v 3 ) is the fluid velocity. The main equation in the kinetic theory is the Boltzmann equation 1 ft + ξ · ∇x f = Q(f , f ). k

(1.8)

The simplest equation, the free molecular flow equation ft + ξ · ∇x f = 0,

(1.9)

can be used to focus on the dissipative effect of the boundary condition. In §6, we present the basic notions of conservation laws and H-theorem, and comment on the relation of the kinetic theory to fluid dynamics in §7. Perhaps the most significant aspect of the kinetic theory in relation to the fluid dynamics is the behaviour near the boundary. There is a variety of boundary conditions for the kinetic theory. We will consider the most dissipative one, the diffuse reflection boundary condition, in §8. Finally, in §9, we make some concluding remarks.

......................................................

∞ 

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The usefulness of the entropy inequality is that it gives an a priori estimate. For instance, for the one-dimensional case, if a solution contains shock waves (ui− , ui+ )(t), i = 1, 2, . . ., at time t, then the entropy inequality yields

2. Dissipation and nonlinearity

3

(2.1)

where the flux function f (u) encodes the physical property in a material. To understand the dissipative property of the solution operator, one considers the entropy and entropy flux functions η(u) =

u2 , 2

q(u) =

u

vf  (v) dv.

For convex flux f  (u) > 0, the entropy dissipation across a shock wave (u− , u+ ) is − s[η(u+ ) − η(u− )] + [q(u+ ) − q(u− )] − |u+ − u− |3 .

(2.2)

Here, s is the shock speed s=

f (u+ ) − f (u− ) . u + − u−

For the simplest convex conservation law, i.e. the Hopf equation,   u2 ut + = 0, 2

(2.3)

x

we have (η(u), q(u)) = (u2 /2, u3 /3) and − s[η(u+ ) − η(u− )] + [q(u+ ) − q(u− )] = −

1 |u+ − u− |3 . 12

(2.4)

The Hopf equation (2.3) models the compression waves in gases. Thus for a convex conservation law, the entropy inequality (1.6) yields the a priori estimate of the total strength of shock waves (ui− , ui+ ), i = 1, 2, . . . , in a solution, say with compact support in x, ∞ |ui+ − ui− |3 (t) dt = O(1). (2.5) 0

This indicates that each shock wave should decay at least with the rate of t−α for some α > 13 in order for the above integral to converge, and thereby be bounded. It is a celebrated result of K. O. Friedrichs that the solution should converge to the N-wave. The idea of a generalized characteristic, as proposed by Glimm, gives an elegant proof that, in fact, the decay of the first and last shock waves is t−1/2 , with the ones in between decaying at the rate of t−1 [2–4]. Take another example of weaker nonlinearity of the flux function   u3 = 0. (2.6) ut + 3 x

In this case, the odd nonlinearity (2.6) models the shear waves in the elasticity. The entropy pair is (η(u), q(u)) = (u2 /2, u4 /4) and the entropy inequality (1.6) yields ∞ |ui+ − ui− |4 (t) dt = O(1). (2.7) 0

Again, this indicates the decay of the shock strength at the rate of t−α for some α > 14 . Precise computations  show that the strength of the shock waves decays at the rate of t−1/3 . Remark 2.1. The entropy inequality yields a global estimate. The advantage is its relative simplicity and easy usage. To yield a precise estimate, other methods, here the generalized characteristic method, are needed to take the full measure of the degree of the dissipation as the

......................................................

ut + f (u)x = 0,

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Consider scalar hyperbolic conservation law in one space dimension

−∞

x

x

On the other hand, the cubic nonlinearity possesses only one time invariant, ∞ p+q= u(y, t) dy. −∞

(2.9)

This subtle and basic difference can only be detected by a more quantitative method than the entropy method.

3. Nonlinearity and viscosity Consider the viscous version of the hyperbolic conservation laws (2.3) and (2.6),   u2 = μuxx ut + 2

(3.1)

x



and ut +

u3 3

 = μuxx .

(3.2)

x

The Burgers equation (3.1) is important for various reasons. It can be solved explicitly by the Hopf–Cole transformation [2,7]. The transformation yields several important pieces of information on the nonlinear waves (e.g. ). Here, we point out another basic property of the balance of nonlinearity and dissipation in the Burgers equation: consider the Burgers kernel B(x, t) = B(x, t; c, μ), the solution of the Burgers equation with initial value a multiple of the delta function ⎫ Bt + BBx = μBxx , ⎪ ⎪ ⎪ ⎪ ⎬ B(x, 0) = cδ(x) (3.3) ⎪ ⎪ x 1 ⎪ ⎪ and B(x, t) = √ φ √ .⎭ t t Here, we have noted that there is a scaling property similar to the heat kernel so that there exists a √ function φ(ξ ) depending on the scalar ξ ≡ x/ t. The Hopf–Cole transformation, or direct scaling computations, yields 2 2 √ c/2μ √ c/2μ μ(e − 1) e−x /4μt μ(e − 1) e−ξ /4μ 1 , φ(ξ ) = . (3.4) B(x, t) = √ √ ∞  √ 2 2 ∞ t π + x/√4μt (ec/2μ − 1) e−y dy π + ξ/√4μ (ec/2μ − 1) e−y dy Consider the corresponding multiple of the heat kernel

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

Ht + HHx = μHxx , H(x, 0) = cδ(x) 1 1 2 e−x /4μt ≡ √ φ0 H(x, t) = c √ 4π μt t

and

x √

t

⎪ ⎪ ⎪ ⎭ .⎪

Thus, the Burgers kernel and the heart kernel are different in the basic L1 norm, ∞ ∞ (H − B)(x, t) dx = (φ0 − φ)(ξ ) dξ = constant = 0, t > 0. −∞

−∞

(3.5)

(3.6)

The general solution u of the Burgers equation, with L1 initial data, approaches, timeasymptotically, the Burgers kernel with ∞ u(x, 0) dx = c, −∞

......................................................

x

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consequence of the strength of the nonlinearity. There is another important difference between the above two simple examples: the N-wave for the Hopf equation has two time invariants [2,3,6] x ∞ u(y, t) dy, q = sup u(y, t) dy. (2.8) p = inf

and so does the heat equation (see ). As the L1 norm is fundamental to the conservation laws, we see that the Burgers and heat equations are different in a basic way.

ux =t ˜ −1

and uxx =t ˜ −3/2 .

(3.7)

Thus, the nonlinear term uux and the viscous term μuxx have the same rate of time-asymptotic dissipation. Therefore, the nonlinear term and the viscous term balance out to yield a distinct time-asymptotic behaviour distinct from the heat equation. On the other hand, for the equation with cubic nonlinearity (3.2), the nonlinear term decays faster than the viscous term ˜ −2 μuxx =t ˜ −3/2 , u2 ux =t

as t → ∞.

Consequently, the time-asymptotic dissipation for the solutions of (3.2) is governed by a multiple of the heat kernel. Note here that, for the Burgers equation, the two limits μ → 0 and t → ∞ do not commute. In the limit μ → 0, there is the N-wave with two time invariants (2.8). On the other hand, as we have just seen, in the limit t → ∞, there is only one time invariant (2.9). For multi-dimensional waves, x ∈ Rm , m > 1, the heat kernel decays time-asymptotically as u=t ˜ −m/2 ,

∇x u=t ˜ −(m+1)/2

and ∇x2 u=t ˜ −(m+2)/2 .

(3.8)

So, a quadratic nonlinear convection term decays at a rate of t−m/2 t−(m+1)/2 = t−(2m+1)/2 faster than the rate of t−(m+2)/2 for the viscous term, as 2m + 1 > m + 2 for the multi-dimensional case m > 1. The reason for this is that in the higher space dimension, m > 1, there is stronger dispersion that makes the nonlinearity play a weaker role. The role of dispersion is well known for the nonlinear wave equations. This is also reflected in the study of the physical models such as the compressible Navier–Stokes equations. The analysis for the one-dimensional case  needs to be more nonlinear in this aspect than the multi-dimensional case .

4. Compression and expansion waves The nonlinearity in the hyperbolic conservation laws (2.1) gives rise to the compressive shock waves and the expansive rarefaction waves. These two types of waves behave differently. Shock waves are dissipative and irreversible. Thus they are highly stable. A compact-supported perturbation is attracted to the shock and vanishes in finite time, yielding a shift of the shock  wave. The shift can be computed by the L1 norm, the conservation law, udx = constant in time. The rarefaction waves are also nonlinearly stable, but only in Lp , p > 1 norms and not in L1 , the conservation norm. This is because the parts of the rarefaction wave can be translated away from each other and remain stable. There is the Oleinik entropy condition for a shock (u− , u+ ) for (2.1), f (u) − f (u− ) f (u+ ) − f (u− ) ≤ , for all u between u− and u+ . u − u− u + − u−

(4.1)

When f  (u) changes sign, there are wave patterns for a combination of expansion and compression waves. For the viscous conservation laws ut + f (u)x = μuxx ,

(4.2)

......................................................

u=t ˜ −1/2 ,

rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120347

Remark 3.1. The above analysis says that the way a Burgers solution dissipates is determined not only by the viscous term, but also by the nonlinearity term uux . This can also be seen easily by the scaling analysis as follows: the rates of the time-asymptotic dissipation of solutions of both heat and Burgers equations, in the sup norm, is

5

Consider a system of hyperbolic conservation laws in one space dimension ut + f (u)x = 0,

u ∈ Rn .

(5.1)

The characteristics λi (u), i = 1, 2, . . . , n, are defined by f  (u)ri (u) = λi (u)ri (u),

li f = λi li ,

li rj = δij , i, j = 1, 2, . . . , n.

(5.2)

For the system of hyperbolic conservation laws (5.1), the additional richness beyond that for the scalar laws is the coupling of waves pertaining to distinct characteristic families, λi (u) and λj (u), i = j. There are two theories, the pointwise estimates of Glimm , through a nonlinear functional in the total variation, and the compactness theory using the entropy inequality (e.g. [15,16]; see  and references therein). The study of nonlinear dissipation was initiated by the difficult work of Glimm & Lax  for two conservation laws and followed by others (see  and references therein). The study of nonlinear waves for genuinely nonlinear systems is better understood (see  and references therein). For systems which are not genuinely nonlinear, there are complicated wave interaction phenomena (e.g. ; see ). However, there remains the definitive quantitative study of wave interactions for such systems; see §3. Consider next the viscous conservation laws ut + f (u)x = (B(u)ux )x .

(5.3)

When the viscosity matrix B(u) is a multiple of the identity matrix, it is called the artificial viscosity, (5.4) ut + f (u)x = μuxx . The study of the system of viscous conservation laws is mostly confined to systems whose inviscid part is either genuinely nonlinear or linear degenerate. Even for such a system, the study of the wave propagation over shock waves was understood only recently, for artificial viscosity  and for physical viscosity . The study of rarefaction waves was also initiated recently . The differences between the shock wave and rarefaction waves for the viscous conservation laws, besides those already present for the scalar laws as mentioned in §4, are the following: the perturbation of the shock waves gives rise to waves of other characteristic families, the viscous version of the Glimm theory for hyperbolic conservation laws . The perturbation of the viscous rarefaction waves, on the other hand, gives rise to stronger coupling owing to the combined effect of nonlinear expansion and the viscosity . The study of rarefaction waves involves deeper understanding of the coupling of nonlinearity and viscosity. This is so already for artificial viscosity and can be seen easily as follows: an inviscid rarefaction wave pertaining to the expansion of the characteristic field λ = λi exactly satisfies the Hopf equation λt + λλx = 0. For viscous system (5.4), the propagation in the primary ith characteristic field is governed mainly by the Burgers equation λt + λλx = μλxx .

......................................................

5. System of conservation laws

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there is definitive understanding of wave behaviour only for the case of convex flux (see  and references therein). There are interesting qualitative studies of wave behaviour for the non-convex cases [10,11]. Quantitative study of the coupling of expansion and compression waves remains to be done. The effect of the boundary is another important element in the study of nonlinear waves. There is the interesting case of the boundary effect of the subscale propagation speed of the shock waves, done only for the Burgers equation [12,13]. The aforementioned coupling of expansion and compression waves for non-convex flux is similar to the problem with the boundary, as each of the wave acts as the boundary to the wave bordering it.

This, however, gives rise to an error contributing to the coupling to the other characteristic families ut + f (u)x = μuxx + error and error = μ(λx )2

n 

(lj ∇ri · ri )rj .

j=1

Note that the error is proportional to the viscosity μ. As the coupling coefficients lj ∇ri · ri , j = i, are in general non-zero, there is coupling and it is due to the viscosity μ. The coupling vanishes for the inviscid system, μ = 0. The coupling has the effect that, in a solution to the Riemann problem consisting of shock and rarefaction waves, the shock location is affected by the rarefaction wave by the order of μ log t . Such a sublinear scale is one of the subtle phenomena that vanishes in the zero viscosity limit. The study of nonlinear waves for conservation laws forms the basis for an important fluid dynamics aspect of the kinetic theory. The kinetic theory possesses other forms of dissipation as we will see in the following sections.

6. Kinetic theory: preliminaries The integration of the collision term Q(f , f ) times 1, ξ and |ξ |2 /2 is zero and represents the conservation of mass, momentum and energy. Integrate the Boltzmann equation (1.8) to obtain the conservation laws ⎫ ρt + ∇x · (ρv) = 0, mass, ⎪ ⎪ ⎬ (ρv)t + ∇x · [ρv ⊗ v + P] = 0, momentum (6.1) ⎪ ⎪ ⎭ and [ρe + 12 |v|2 ]t + ∇x · [(ρe + 12 |v|2 )v + v · P + q] = 0, energy. An important measure of dissipation is the H-theorem, the integration of the Boltzmann equation times log f ⎫  f f  ⎪ ⎪

=1 Ht + ∇ · H Q(f , f ) log ∗ dξ ≡ −J ≤ 0 ⎪ ⎬ 4 R3 ff∗ (6.2)   ⎪ ⎪

= H(x,

t) ≡ f log f dξ , H ξ f log f dξ .⎪ and H = H(x, t) ≡ ⎭ R3

R3

When the collision operator Q(f , f ) is zero, we say that the distribution f is in thermo-equilibrium. The ‘entropy production’ J is zero if and only if the distribution f is in thermo-equilibrium. Boltzmann also showed that this is so if and only if f is Gaussian in the microscopic velocity ξ f (x, t; ξ ) = Mρ(x,t),v(x,t),θ(x,t) =

ρ(x, t) 2 e−|ξ −v(x,t)| /2Rθ(x,t) . 3/2 (2π Rθ(x, t))

(6.3)

There has been deep analysis of using basic conservation laws and the H-theorem for the existence theory , and the asymptotic convergence to incompressible Navier–Stokes solutions (see  and references therein). There have also been substantial studies of J in terms of the distance of the Boltzmann solution f from its associated Maxwellian Mf , for the study of convergence to the Maxwellian . The combined effect of the transport and collision parts has been studied for its regularity property [30–33]. As we shall see, there are other equally important measures of dissipation for the Boltzmann equation coming from these as well as from the boundary effects.

......................................................

Here, the genuine nonlinearity of the i-characteristic field λi is assumed and the i-characteristic direction ri is normalized ∇u λi · ri = 1.

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The approximate rarefaction wave u is constructed using the Burgers solution λ to lie along the characteristic direction  ∇x,t u = (∇x,t λ)ri (u).

7. Kinetic theory: interior waves

8

the solution approaches the Maxwellian exponentially fast. The only dissipation mechanism is the H-theorem in the form of the positivity of J as a measure of the solution away from the Maxwellian . The situation for solutions periodic in space has been treated as a perturbation of the spatial homogeneous case. This is a good idea time-asymptotically, but not for short and intermediate times. The reason is that the positivity of J captures only part of the whole equilibrating mechanism of the Boltzmann operator. For the spatial inhomogeneous case, the left-hand side, the transport part of the Boltzmann equation (1.8), ft + ξ · ∇x f , is of equal weight to the right-hand side, the collision operator Q(f, f ). A partial understanding of this is to relate the Boltzmann equation to the fluid dynamics equations. There are the Chapmann– Enskog and the Hilbert expansions that give hints to the combined role of the transport and collision parts of the Boltzmann equation. These expansions are under the hypothesis of certain smoothness of the Boltzmann solution under study. Thus, they do not hold for the initial and shock layers. In those layers, the Boltzmann solution has a steep gradient. For instance, the compressible Navier–Stokes equations are derived by the Chapmann–Enskog expansion in the following way. First, when the distribution function is in local thermo-equilibrium, f = Mf Gaussian in the microscopic velocity ξ , the stress tensor is reduced to the pressure P = pI and the heat flux q = 0. In this case, the conservation laws (6.1) are reduced to the compressible Euler equations ⎫ ρt + ∇x · (ρv) = 0, ⎪ ⎪ ⎬ (ρv)t + ∇x · [ρv ⊗ v + pI] = 0 (7.1) ⎪ ⎪ ⎭ 2 2 1 1 and [ρe + 2 |v| ]t + ∇x · [ρ(e + 2 |v| )v + pv] = 0. The local Maxwellian distribution f = Mf (6.3) is not a solution to the Boltzmann equation in general, as the collision term Q(f , f ) = 0 but the transport part is not zero in general. A higher order approximation of compressible Navier–Stokes equations by the Chapman–Enskog expansion is to assume that G ≡ f − Mf , the part carries no mass, momentum or energy and is much smaller than the Maxwellian part, G Mf , and that the gradients in ∇(x,t) of both Mf and G are much smaller than Mf and G, respectively (e.g. ). In other words, the Boltzmann solution f is close to the local Maxwellian and it is very smooth. The compressible Navier–Stokes equations have more explicit dissipation parameters, the viscosity and heat conductivity. These parameters are the combined effect of the transport and collision parts of the Boltzmann equation and are proportional to the Knudsen number k. The relation of the Boltzmann equation to the fluid dynamics is useful for the study of the nonlinear waves for the Boltzmann equation. However, there is the caution that either the incompressible or the compressible Navier–Stokes equations fail to accurately approximate the Boltzmann equation in their respective physical situation. Much remains to be understood on this subtle and complex relationship. In fact, the study of the fluid dynamics from the point of view of the kinetic theory is so vital that there is now a modern fluid dynamics resulting from such studies .

8. Kinetic theory: boundary behaviour The kinetic theory is necessary for the the understanding of several physical phenomena induced by the solid boundary. The experiment with a radiometer highlights the fact that the classical

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1 ft = Q(f , f ), k

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For the spatial homogeneous Boltzmann equation

where n is the unit normal vector at the boundary, pointing to the gas region D. The gas reflected off the solid boundary is a multiple of MT(y) , the Maxwellian with the given boundary temperature T(y) on the solid and with zero fluid velocity, assuming here that the solid is not moving, 2

e−ξ /2RT . MT (ξ ) = (2π RT)3/2

(8.2)

The multiple of the Maxwellian in the first equation of (8.1),

1/2 2π j(y, t), RT(y) involves the flux j(y, t) on the solid to balance out so that the total flux on the boundary is zero  ξ ∗ · nf (y, ξ ∗ , t) dξ ∗ = 0. R3

Thus, the diffuse reflection boundary condition registers the solid temperature on the gas around it. The boundary condition has both the geometric and thermal effects of the solid. There are several standard boundary conditions. For example, the complete condensation boundary condition is conventionally used when evaporation or condensation takes place on the boundary. It prescribes the outgoing flow to be a given multiple c(y) of the Maxwellian, f (y, ξ , t) = c(y)MT(y) (ξ ),

y ∈ ∂D, ξ · n > 0.

(8.3)

The diffuse reflection boundary condition has a strong equilibrating effect. There is a common feature to the solution near the boundary, whatever the boundary condition prescribed; namely, that the distribution function is discontinuous in the microscopic velocity ξ at the boundary and that the macroscopic variables are singular near the boundary (see  for the analytical study of an example for thermal transpiration, and  for comprehensive asymptotic and computational studies). The diffuse reflection boundary condition has a strong equilibrating effect. We know that the collision operator also has an equilibrating effect through the H-theorem. In a bounded domain, it is common to consider these two effects together (see [28,37,38] and references therein). However, they play different roles in the physical modelling. Thus to highlight the boundary effect, one can consider the diffuse reflection boundary condition (8.1) for the free molecular flow (1.9). This problem has been studied recently. The result (see [39,40] and references therein) is that the equilibrating effect of the diffuse reflection boundary condition causes the gas to approach the stationary flow f → g, ∇x g = 0 at the rate of (t + 1)−d , d the spatial dimension. The reason for the slow, algebraic rate is that there are slow particles, |ξ | 1, that exist either in the initial distribution or in the reflected Maxwellian. These particles are slow to hit the boundary, and therefore the boundary condition has a weak effect on them. With the collision operator, the convergence rate is exponential. However, for large Knudsen number k the solution to the Boltzmann equation (1.8) with the diffuse boundary condition tends

......................................................

ξ ∗ ·n 0⎪ f (y, ξ , t) = ⎪ ⎬ RT(y) (8.1)  ⎪ ⎪ ⎪ ⎭ and j(y, t) = −ξ · nf (y, ξ , t) dξ ,

There are several versions and aspects of dissipation in the kinetic theory and gas dynamics. The analytical manifestations therefore take several forms. The entropy inequality in both the gas dynamics and the kinetic theory is simple and elegant in its formulation, and has generality in its applications. The functional analytical approaches have had great success, with some spectacular examples [15,16,26,29]. There is another more quantitative approach based on concrete constructions [14,18,21,42–44] on hyperbolic conservation laws (see  and references therein). Green’s function approach has been useful for the study of nonlinear waves for viscous conservation laws [4,22,23,45,46]. Green’s function approach [32,34,47,48] for the Boltzmann equation has yielded quantitative understanding of nonlinear waves and the boundary behaviour [41,49–51]. The relation between the gas dynamics and the kinetic theory is a rich field. This article only samples some of the analytical results around the entropy method. There is much room for future progress for the entropy method, Green’s function approach and other possibilities. Funding statement. Supported in part by an Investigator Award of Academia Sinica, National NSC grant no. 96-2628-M-001-011 and NSF grant no. DMS-0709248.

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9. Concluding remarks

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to the stationary solution exponentially fast, but with the exponent inversely proportional to the Knudsen number k . It would be interesting to study the situation of a small Knudsen number, k 1, for which the fluid-like behaviour, such as the Navier–Stokes flow for the momentum and the Fourier law for the temperature, would hold away from the boundary. However, as mentioned above, there would be discontinuity on the boundary and singularity near the boundary. This remains to be understood analytically. There have been substantial asymptotic and computational studies on these important issues .

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