Existence and sharp localization in velocity of small-amplitude Boltzmann shocks

Using a weighted $H^s$-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman--Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere potential, recovering and slightly sharpening results obtained by Caflisch and Nicolaenko using different techniques. A key technical point in both analyses is that the linearized collision operator $L$ is negative definite on its range, not only in the standard square-root Maxwellian weighted norm for which it is self-adjoint, but also in norms with nearby weights. Exploring this issue further, we show that $L$ is negative definite on its range in a much wider class of norms including norms with weights asymptotic nearly to a full Maxwellian rather than its square root. This yields sharp localization in velocity at near-Maxwellian rate, rather than the square-root rate obtained in previous analyses


Introduction
In this paper, we study existence and structure of small-amplitude shock profiles x, t ∈ R, where f (x, t, ξ) ∈ R denotes the distribution of velocities ξ ∈ R 3 at point x, t, τ > 0 is the Knudsen number, and and various collision kernels C. Our main example is the hard sphere case, for which (1.5) C(Ω, ξ) = Ω · ξ .
See, e.g., [Gl] for further details. Note that Q is in this case not symmetric.
Other standard examples we have in mind are associated with the class of hard cutoff potentials defined by Grad [G], as considered in [CN]. By small-amplitude, we mean that the density ρ(x, t) := 1 f (x, t) := R 3 f (x, t, ξ)dξ is confined within an ε 0 -neighborhood of some fixed reference density ρ 0 > 0 for all x, t, for ε 0 > 0 sufficiently small, where, throughout our analysis, we have fixed τ ≡ 1.
Substituting (1.1) into (1.2), we seek, equivalently, stationary solutions of the traveling-wave equation By frame-indifference, we may without loss of generality take s = 0.
Associated with these invariants are the macroscopic fluid-dynamical variables (1.8) u := Rf =: (ρ, ρv 1 , ρv 2 , ρv 3 , ρE) T , where ρ is density, v = (v 1 , v 2 , v 3 ) is velocity, E = e + 1 2 |v| 2 is total energy density, and e is internal energy density. Here, we are assuming that f (x, t, ·) is confined to a space H to be specified later such that the integral converges for f ∈ H.
The set of equilibrium states Q(f, f ) = 0 are exactly (see, e.g., [Gl]) the Maxwellians Making the equilibrium assumption f = M u , we obtain a closed system of equations for the fluid-dynamical variables consisting of the one-dimensional Euler equations (1.9) with pressure p = p(ρ, E) given by the monatomic ideal gas equation of state (1.11) p = (2/3)ρE.
This corresponds to the zeroth-order approximation obtained by formal Chapman-Enskog expansion about a Maxwellian state [G, KMN], where the expansion can be taken equivalently in powers of τ , or, as pointed out in [L,MaZ1], in powers of k, where k is the frequency in x, t of perturbations. In the present context, it is the latter derivation that is relevant, since (as we shall see better in a moment) we seek slowly varying solutions near a constant, Maxwellian, state. The next-, and presumably more accurate, first-order Chapman-Enskog approximation yields the one-dimensional Navier-Stokes equations (1.12) where temperature T is related to internal energy by e = 3 2 RT , R the universal gas constant, and (1.13) µ = µ(T ) > 0 and κ = κ(T ) > 0 are coefficients of viscosity and heat conduction. In the hard sphere case, these may be computed explicitly as µ(T ) = (RT ) 1/2 µ(1/R), κ(T ) = (RT ) 1/2 κ(1/R) (Chapman's formulae). For derivations, see, e.g., [KMN], Section 3. By (1.9), the fluid-dynamical variables associated with a traveling wave (1.1) must satisfy (1.14) −s∂ x ρ + ∂ x (ρv 1 ) = 0 −s∂ x (ρv) + ∂ x (v 1 ρv + pe 1 ) = 0 −s∂ x (ρE) + ∂ x (v 1 (ρE + p)) = 0, hence, integrating from x = −∞ to x = +∞, the Rankine- Noting that endstates f ± of (1.1) by (1.6) necessarily satisfy Q(f, f ) ± = 0, we find that they are Maxwellians f ± = M u ± , and so the associated pressures p ± = p(f ± ) are given by the ideal gas formula (1.11), recovering the standard fact that endstates of a Boltzmann shock (1.1) are Maxwellians with fluid-dynamical variables corresponding to fluid-dynamical shock waves of the Euler equations with monatomic ideal gas equation of state [G, CN].
This gives a rigorous if straightforward connection between Boltzmann shocks and their zeroth order Chapman-Enskog approximation. The following, main result of this paper gives a rigorous connection to the first-order Chapman-Enskog approximation given by the Navier-Stokes equations (1.12) in the limit as shock amplitude goes to zero.
Recall [Gi], for an ideal-gas equation of state (1.11) under assumptions (1.13), that for each pair of end-states u ± satisfying the Rankine-Hugoniot conditions (1.15), the Navier-Stokes equations (1.12) admit a unique up to translation smooth traveling-wave solution or Navier-Stokes shock. Moreover, denoting shock amplitude by ε := |u + − u − |, we have for ε > 0 sufficiently small the asymptotic description [Pe] (1.16) Up to this point in the discussion, we have made essentially no assumption on the nature of the collision kernel C(Ω, ξ). For the analysis of exact profiles, we require specific properties of C. For simplicity of exposition, we specialize hereafter to the hard-sphere case (1.5). As discussed in Section 11, the arguments extend to a more general class of kernels including the hard cutoff potentials of Grad [G]. Then, our main result is as follows.
Theorem 1.1. In the hard-sphere case (1.5), for any given fluid-dynamical reference state u 0 and η > 0, there exist ε 0 > 0, δ k > 0, and C k > 0 such that for |u + − u 0 | ≤ ε 0 and ε = |u + − u − | ≤ ε 0 , the standing-wave equation (1.6) has a solutionf satisfying for all whereū := Rf is the associated fluid-dynamical profile. Moreover, up to translation, this solution is unique among functions satisfying for 0 ≤ k ≤ 2, c k sufficiently small, the weaker estimate Existence of small-amplitude Boltzmann profiles was established some time ago in [CN] for the full class of hard cutoff potentials, viewing them as bifurcations from the constant Maxwellian solution f ≡ M u − , with the somewhat weaker existence result 0 ≤ β ≤ 1, but also the somewhat stronger result of uniqueness among solutions satisfying for C > 0 bounded and ε > 0 sufficiently small. For the hard sphere potential, positivity of profiles, and the improved estimate (1.18) were shown by Liu and Yu [LY] by a "macromicro decomposition" method in which fluid (macroscopic, or equilibrium) and transient (microscopic) effects are separated and estimated by different techniques. This was used in [LY] to establish time-evolutionary stability of profiles with respect to perturbations of zero fluid-dynamical mass, u(x)dx = 0, and thus, assuming the existence result of [CN], to establish positivity of Boltzmann profiles by the positive maximum principle for the Boltzmann equation (1.2) together with convergence to the Boltzmann profile of its own Maxwellian approximation: by definition, a perturbation of zero relative mass in fluiddynamical variables. The purpose of the present paper is to obtain existence from first principles by an elementary argument in the spirit of [LY], based on approximate Chapman-Enskog expansion combined with Kawashima type energy estimates [K] (the macro-micro decomposition of the reference), but carried out for the stationary (traveling-wave) rather than the timeevolutionary equations, and estimating the finite-dimensional fluid part using sharp ODE estimates in place of the sophisticated energy estimates of [LY]. 1 In this latter part, we are much aided by the more favorable properties of the stationary fluid equations, a rather standard boundary value ODE system, as compared to the time-evolutionary equations, a hyperbolic-parabolic system of PDE. This in a sense completes the analysis of [LY], providing by a common set of techniques both existence (through the present argument) and (through the argument of [LY]) positivity. At the same time it gives a truly elementary proof of existence of Boltzmann profiles.
For similar results in the general finite-dimensional relaxation case, see [MeZ1,MTZ]. The key new technical observations needed for the infinite-dimensional case are a way of choosing Kawashima compensators of finite rank (see Remark 4.3), and the fact that the linearized collision operator remains negative definite on its range not only in norms of square-root Maxwellian weight where it is self-adjoint, but also in norms with nearby weights; this allows coordinatization with respect to a single global Maxwellian, avoiding unbounded commutators associated with a changing local Maxwellian frame.
In passing, we obtain also the new result of sharp localization in velocity at near-Maxwellian rate (1.17), which comes from improved estimates on the linearized collision operator independent of the basic argument. A key technical point in all three analyses- [CN], [LY], and the present one-is that the linearized collision operator L is negative definite on its range, not only in the standard square-root Maxwellian weighted norm, but also in norms with nearby weights. Exploring this issue further, we show that L is negative definite on its range in a much wider class of norms including norms with weights asymptotic nearly to a full Maxwellian rather than its square root. This observation, of interest in its own right, yields through the same existence argument sharp localization in velocity at near-Maxwellian rate, rather than the square-root rate obtained in previous analyses.
Finally, we note that stability of small-amplitude Boltzmann shocks has been shown in [LY] with respect to small H s perturbations with zero mass in fluid variables. It would be very interesting to continue along the same lines to obtain a complete nonlinear stability result as in [SX] or [MaZ1,MaZ2], with respect to general, not necessarily zero mass, perturbations.

The nonlinear collision operator
We begin by a careful study of the collision operator. Related results may be found, for example, in [C, GPS].

Splitting of the collision operator
In view of definition (1.3), we split into gain and loss parts [G, Gl], where, for Ω defined as in (1.4),
Lemma 2.1. In the hard-sphere case (1.5), for h ≥ 0 with ξ h ∈ L 1 , ν h is positive, continuous and This implies the upper bound. Next, which implies the lower bound for |ξ| > (B+1)/2A. For ξ bounded, the integral is continuous and bounded from below.
Remark 2.8. The estimates above were proved for convenience for the Gaussian weights ω = e |ξ| 2 and ω s . They immediately extend to any Maxwellian weight M u and M s u .

The linearized collision operator
We next study the linearized collision operator about a Maxwellian or nearby velocity distribution. Fix a reference state u. The associated Maxwellian M u is denoted by M . For s ∈]0, 1], let H s denote the space of functions f on R 3 such that Note that M ∈ H s for all s < 1. The space H 1 2 plays a particular role as it will be clear below.
where R is the operator (1.7) defining the thermodynamical variables. .
Proof. The action from ξ − 1 2 H s to ξ 1 2 H s is a consequence of Corollary 2.4 and Remark 2.8. That the image is contained in ker R follows from the known properties of the collision operator: Given a function a, the linearized collision operator at a is (3.3) L a g = Q(a, g) + Q(g, a).
In particular, we consider first the linearized operator at a = M : We denote by P U and P V the orthogonal projection from H 1 2 to U and V respectively. In the language of [LY], U is the macroscopic part of f and V is the microscopic part.
Note that U ⊂ ξ − 1 2 H 1 2 and U ⊂ H s for all s < 1. Therefore Remark 3.4. The projections P U and P V do not commute with the operator of multiplication by ξ 1 . They are not orthogonal in H s for s > 1 2 , but still produce a continuous decomposition H s = U ⊕ V s .

Proposition 3.5. L is (formally) self adjoint and nonpositive in H
ii) there is δ > 0 such that for all f ∈ ξ −1 H: Notes on the proof. This is a classical result in the theory of Boltzmann equation in the hard sphere case and more generally in the case of hard cut off potentials (see e.g. [C, G, Gl, CN]) 1. The analysis of section 2 splits L into L = −ν(ξ) + K, with for j = 1, 2, 3, implying the symmetry of L in H 1 2 . 2. One can also argue as follows. By Boltzmann's H-theorem, for all f withe enough decay at infinity. Hence, Taylor expanding about the Maxwellian M , a minimizer of Q(f, f ) log f dξ, we obtain symmetry and nonnegativity of the Hessian, giving nonnegativity of L on H 1 2 and also formal self-adjointness. 3. It is known that is K compact in H 1 2 and that ker L = U (this can be proved using the formulas above). By self-adjointness of L on H 1 2 , to establish strict negativity on V, it is sufficient to establish a spectral gap between the eigenvalue zero and the essential spectrum of L. But, this follows from Weyl's Lemma by comparison of L = −ν + K with the multiplication operator by −ν(ξ) ≤ −c 0 < 0.
In a more explicit form, the inequality (3.9) reads We also point out the following properties which are freely used below and which follow from the symmetry of L in H 1 2 :

Coercivity on H s
With λ ≥ 0 to be determined later on, introduce the equivalent norms: Proposition 3.6. For 1 2 ≤ s < 1, the operator L is continuous from ξ − 1 2 H s to ( ξ 1 2 H s )∩V and from ξ −1 H s to V s , and formally coercive on V s for the norm (3.14). More precisely, there are λ ≥ 0 and δ > 0 such that for all f ∈ ξ −1 V s : Proof. We want to prove that Following the analysis of Section 2, Moreover, by (3.12), there is δ 0 such that Hence: We choose λ such that for all ξ implying the inequality (3.16).
Remark 3.7. Included in the bound (3.15) is the observation that both the first-order Chapman-Enskog approximationŪ N S and the entire hierarchy of higher-order Chapman-Enskog correctors lie in H s , any 0 < s < 1, something that is not immediately obvious. Indeed, looking closely at the inversion of L a , we see that they in fact decay at successively higher polynomial multipliers of the full Maxwellian rate.

Comparison
We consider the linearized operator L a g = Q(a, g) + Q(g, a) at a, not necessarily nonnegative, close to M . In (3.19) the H s scalar product has to be understood as the integral Since L a f and Lf belong to V and thanks to the definition of the modified scalar product H s , With (3.17), this implies (3.18) with a new constant C. With Proposition 3.6, this implies (3.19).
Remark 3.9. Since U is finite dimensional, one can use any norm for P U f in the estimates (3.17) (3.18) and (3.19) above.
Remark 3.10. We have in mind that ε(a) can be taken arbitrarily small. This holds if a = M u and u is close to u since , when s < 1, as |u − u| → 0 by Lebesgue's dominated convergence theorem.

Abstract formulation
We now rephrase the problem in a general framework, for the square-root Maxwellian norm H = H 1 2 in which we carry out the main analysis. We treat general weights in Section 10.2, by a bootstrap argument. Taking the shock speed equal to 0 by frame-indifference, we consider (1.6) as the abstract standing-wave ODE independent of U (semilinearity of the Boltzmann equation), and Q as in (1.3), (1.5).

Bounds on the transport operator
The collision operator has been studied acting in spaces H s associated to our reference Maxwellian M . We have the following evident facts regarding the transport operator A.

Kawashima multiplier
We next construct a Kawashima compensator as in [K,MeZ1,MTZ], but taking special care that the operator remain bounded in this infinite-dimensional setting.
Proposition 4.2. There are C, δ > 0, λ ≥ 0 and there is a finite rank operator K ∈ L (H −1 , H 1 ) such that such K is skew symmetric in H s and satisfies meaning that We first check that the genuine coupling condition is satisfied, i.e. that there is no eigenvector of A in ker L = U. Indeed, using the basis φ j of U given in (3.6), an eigenvector of A with eigenvalue τ in U is a linear combination α j φ j such that the polynomial is identically zero. Equating to zero the term of degree 3 implies that α 4 = 0. Equating to zero the coefficient of the terms of degree 2 implies that α j = 0 for j = 1, 2, 3, and finally α 0 = 0. Thus the property is satisfied. b) We look for K as (4.5) with θ > 0 a parameter to be chosen and Here * means the adjoint with respect to the scalar product in H. We have used Proposition 4.1.
Thus, with A 11 := P U AP U , The condition a) means that A 11 (restricted to U) has no eigenvector in ker A 21 = ker A * 21 A 21 , with A * 21 A 21 symmetric positive semidefinite and A 11 symmetric. Since dim U is finite (equal to 5), this implies by the standard, finite-dimensional construction of Kawashima et al [K] that one can choose K 11 Moreover, since dim U is finite, there is another c 1 > 0 such that Thus, using Proposition 3.9 for a = M : where the norms are taken in L (H 1 ; H −1 ). All these operators have finite rank ≤ n and are bounded. Thus if θ is small enough, this shows that Re (KA − L) is definite positive in the sense of (4.4). Using the perturbation Lemma 3.8 implies that the estimate remains true for a satisfying (3.20).
Remark 4.3. The construction above, by reduction to the equlibrium manifold, is essentially different from the original proof of [K] in the finite-dimensional case, which would yield a symmetrizer of infinite rank. The advantage of finite rank is that we need not worry about boundedness of the operator. We note that this is related to methods in the Boltzmann literature in which the Kawashima compensator is replaced by estimates on a reduced Chapman-Enskog approximation such as the Grad 13 moments model or the Navier-Stokes approximation, again to avoid possible boundedness issues; see, e.g., [G, LY]. See also the related construction of [GMWZ] in the case that u is scalar, for which K 11 may be taken equal to zero. We note that we could apply the same reduction argument to the reduced problem and proceed by iteration to this scalar case, thus obtaining an alternative proof in the finite-dimensional case as well.

Reduction to bounded operators
In the hard-sphere case (1.5), we may rescale the equations to obtain a problem involving only bounded operators. We have H 1 ⊂ H ⊂ H −1 , bounded operators from H 1 to H −1 and we work with the scalar product of H. We can multiply the equations on the left by ξ −1 : By Corollary 2.4, the corresponding collision operatorŝ

The framework
At this point, we have reduced to the following abstract problem, with semilinear structure quite similar to that treated in the finite-dimensional analysis of [MeZ1]. Working in H with operatorsÃ andQ and dropping tildes, we study the standing-wave ODE (4.10) with U taking its values in an infinite dimensional space H.

Assumptions on the full system
We make the following assumptions, verified above for the Boltzmann equation in the hardsphere case with A, Q replaced byÃ,Q.
For U ∈ H, we denote by L U the bounded operator V → 2Q(U, V ), that is the differential of Q(U, U ). We denote by P U and P V the orthogonal projectors on U and V respectively. We use the notations U = u + v, with u = P U U and v = P V U .
Assumption 4.5. We are given a reference state U (in a smaller space M ⊂ H) such that L = L U is self adjoint with kernel U and L is definite negative on V.
Lemma 4.6. There are δ > 0, ε 0 and C ≥ such that for a ∈ M and U ∈ H: Proof. By continuity of Q, (4.9), there is C such that Lemma 4.7. In an H-neighborhood of U , the zero set of Q is given by a smooth (indeed Proof. Assumption 4.5 and the Implicit Function Theorem, together with the observation that Q as a continuous biinear form (in sense (4.9)) is C ∞ in the Frechet sense.
We further assume the Kawashima condition established in Proposition 4.3.
Assumption 4.8. There is a skew symmetric bounded operator K ∈ L (H) and a constant γ > 0 such that Using (4.13), this implies Lemma 4.9. There are γ > 0 and ε 0 > 0 such that for a ∈ H satisfying (4.12) and U ∈ H:

Assumptions on the reduced system
Finally, the equilibria are parametrized by u: where v * is the smooth mapping from a neighborhood of u to a neighborhood of v = v * (u) in V, as described in Lemma 4.7.
Recall from [Y] that the reduced, Navier-Stokes type equations obtained by Chapman-Enskog expansions are Note also, by the Implicit Function Theorem, that dv * (u) = −∂ v q −1 ∂ u q(u, v * (u)).
An important property of the Chapman-Enskog approximation, following either by direct computation or by coordinate-independence of the physical derivation, is that the form (4.21)-(4.23) of the equations is coordinate invariant, changing tensorially with respect to constant linear coordinate changes; moreover, the change in functions h * , b * due to a constant linear coordinate change may be computed directly from (4.20) using the coordinate change in u alone. From this we find in the Boltzmann case that (4.20) is equivalent through a constant linear coordinate change to the Navier-Stokes equations (1.12) with monatomic ideal gas equation of state and viscosity and heat conduction coefficients satisfying (1.13). We make the following assumptions on the reduced system, verified for the Navier-Stokes equations (hence satisfied for the Boltzmann equation) in [MaZ3]. Finally, we assume that the classical theory of weak shocks can be applied to (4.20), requiring that the flux f * have a genuinely nonlinear eigenvalue near 0.
Assumption 4.11. In a neighborhood U * of a given base state u 0 , dh * has a simple eigenvalue α near zero, with α(u 0 ) = 0, and such that the associated hyperbolic characteristic field is genuinely nonlinear, i.e., after a choice of orientation, ∇α · r(u 0 ) < 0, where r denotes the eigendirection associated with α.

The basic estimate
With these preparations, we can establish existence by an argument almost identical to that used in [MeZ1] to treat the finite-dimensional case: indeed, somewhat simpler. The single difference is that in carrying out the basic symmetric energy estimates controlling microscopic variables we do not attempt to exactly symmetrize L a at each x value as was done in [MeZ1], but only use the fact that each L a is approximately symmetric by construction. This is important in the infinite-dimensional case, since exact symmetrization can (and does in the Boltzmann case) introduce unbounded commutator terms that wreck the argument. To isolate this important technical point, we carry out the key estimate here, before describing the rest of the argument.
We consider the equation We assume that (4.26) Lemma 4.13. There is a constant C such that for ε sufficiently small, one has Here, the norms L 2 , H 1 etc denote the norms in L 2 (R; H), H 1 (R; H) etc.
Proof. Introduce the symmetrizer (4.28) where Re T = 1 2 (T + T * ) and the adjoint is taken in L 2 (R; H). We have used that [∂ x , L a ] = L ∂xa by linearity of L with respect to a. Thus Therefore, for U ∈ H 2 (R), (4.11), (4.15) and the continuity of K and Q imply that We note that Taking λ large enough and using (4.25) yields In the opposite direction, The estimate (4.27) follows provided that ε is small enough. This proves the lemma under the additional assumption that U ∈ H 2 . When U ∈ H 1 , the estimate follows using Friedrichs mollifiers.

Basic L 2 result
We now describe a simpler version of our main result, carried out in the L 2 norm H. For clarity of exposition, we carry out the entire argument in this more transparent context, indicating afterward in Section 10 how to extend to the general (pointwise, higher weight) norms described in Theorem 1.1.

Chapman-Enskog approximation
Integrating the first equation of (4.10) and noticing that the end states (u ± , v ± ) must be equilibria and thus satisfy v ± = v * (u ± ), we obtain Because f is linear, the first equation reads The idea of Chapman-Enskog approximation is that v − v * (u) is small (compared to the fluctuations u − u ± ). Taylor expanding the second equation, we obtain The derivative of (5.2) implies that Therefore, (5.3) can be replaced by where c * is defined at (4.23). Substituting in (5.2), we thus obtain the approximate viscous profile ODE where b * is as defined in (4.22). Motivated by (5.3)-(5.5), we define an approximate solution (ū N S ,v N S ) of (5.1) by choosingū N S as a solution of andv N S as the first approximation given by (5.3) Small amplitude shock profiles solutions of (5.6) are constructed using the center manifold analysis of [Pe] under conditions (i)-(iv) of Assumption 4.10; see discussion in [MaZ5].
Proposition 5.1 ( [MaZ5]). Under Assumptions 4.10 and 4.11, in a neighborhood of (u 0 , u 0 ) in R n × R n , there is a smooth manifold S of dimension n passing through (u 0 , u 0 ), such that for (u − , u + ) ∈ S with amplitude ε := |u + − u − | > 0 sufficiently small, and direction (u + − u − )/ε sufficiently close to r(u 0 ), the zero speed shock profile equation (5.6) has a unique (up to translation) solutionū N S in U * . The shock profile is necessarily of Lax type: i.e., with dimensions of the unstable subspace of dh * (u − ) and the stable subspace of dh * (u + ) summing to one plus the dimension of u, that is n + 1.
Moreover, there is θ > 0 and for all k there is C k independent of (u − , u + ) and ε, such that x ≷ 0. We denote by S + the set of (u − , u + ) ∈ S with amplitude ε := |u + − u − | > 0 sufficiently small and direction (u + − u − )/ε sufficiently close to r(u 0 ) such that the profileū N S exists. Given (u − , u + ) ∈ S + with associated profileū N S , we definev N S by (5.7) and (5.9)Ū N S := (ū N S ,v N S ).
It is an approximate solution of (5.1) in the following sense: Proof. Given the choice ofv N S , the first equation is a rewriting of the profile equation (5.6).
Next, note that , where here O(·) denote smooth functions ofū N S and its derivatives, which vanish as indicated. With similar notations, the Taylor expansion of q and the definition ofv N S show that . satisfies the estimates stated in (5.11).
Remark 5.3. One may check that if we did not include the correction from equilibrium on the righthand side of (5.7), taking instead the simpler prescriptionv N S = v * (ū N S ) as in [LY], then the residual error that would result in (5.10) would be too large for our later iteration scheme to close. This is a crucial difference between our analysis and the analysis of [LY]. The prescriptionŪ N S corresponds to the first-order Chapman-Enskog approximation in both variables, u and v together.

Basic L 2 result
We are now ready to state the basic L 2 version of our main result. Define a base state U 0 = (u 0 , v * (u 0 )) and a neighborhood U = U * × V.

Outline of the proof
We describe now the main steps in the proof of Proposition 5.4, exactly following the finitedimensional analysis of [MeZ1].

Nonlinear perturbation equations
Defining the perturbation variable U :=Ū −Ū N S , and expanding aboutŪ N S , we obtain from (5.1) the nonlinear perturbation equations where the remainder N (U ) is a smooth function of U N S and U , vanishing at second order at U = 0: We push the reduction a little further, using that Therefore the equation reads Differentiating the first line, it implies that The linearized operator A∂ x − dQ(Ū ) about an exact solutionŪ of the profile equations has kernelŪ ′ , by translation invariance, so is not invertible. Thus, the linear operators L ε * and L ε * are not expected to be invertible, and we shall see later that they are not. Nonetheless, one can check that L ε * is surjective in Sobolev spaces and define a right inverse L ε * (L ε * ) † ≡ I, or solution operator (L ε * ) † of the equation as recorded by Proposition 6.2 below. Note that L ε * is not surjective because the first equation requires a zero mass condition on the source term. This is why we solve the integrated equation (6.5) and not (6.7).
To define the partial inverse (L ε * ) † , we specify one solution of (6.8) by adding the codimension one internal condition: where ℓ ε is a certain unit vector to be specified below.
Remark 6.1. There is a large flexibility in the choice of ℓ ε . Conditions like (6.9) are known to fix the indeterminacy in the resolution of the linearized profile equation from (5.6) and it remains well adapted in the present context, see section 8 below. A possible choice, would be to choose ℓ ε independent of ε and parallel to the left eigenvector of dh * (u 0 ) for the eigenvalue 0 (see Assumption 4.11), which, together with the asymptotics of Proposition 5.1, gives (6.10) ℓ ε ·Ū ′ N S (0) ∼ ε 2 = 0.

Fixed-point iteration scheme
The coefficients and the error term R v are smooth functions ofū N S and its derivative, thus behave like smooth functions of εx. Thus, it is natural to solve the equations in spaces which reflect this scaling. We do not introduce explicitly the change of variablesx = εx, but introduce norms which correspond to the usual H s norms in thex variable : We also introduce weighted spaces and norms, which encounter for the exponential decay of the source and solution: introduce the notations.

Proof of the basic result
Proof of Theorem 5.4. The profileū N S exists if ε is small enough. The estimates (5.8) imply that with C s independent of ε and δ, provided that δ ≤ θ/2. Similarly, (5.11) implies that and (6.4) implies that Moreover, with the choice of norms (6.11), the Sobolev inequality reads Moreover, for smooth functions Φ, there are nonlinear estimates which also extend to weighted spaces, for δ ≤ 1: In particular, this implies that for s ≥ 1, δ ≤ min{1, θ/2} and ε small enough: where the first constant C is independent of s. Similarly, Combining these estimates, we find that provided that ε ≤ ε 0 , δ ≤ min{1, θ/2} and U L ∞ ≤ 1.
Consider first the case s = 2. Then, T maps the ball provided that U L ∞ ≤ 1 and V L ∞ ≤ 1, from which we readily find that, for ε > 0 sufficiently small, T is contractive on B ε,δ , whence, by the Contraction-Mapping Theorem, there exists a unique solution U ε of (6.17) in B ε,δ for ε sufficiently small. Moreover, from the contraction property with c < 1, we obtain as usual that U ε,δ by (6.26). In particular, e εδ x U ε = O(ε 2 ) in H 2 ε and by the Sobolev embedding For s ≥ 3, the estimates (6.26) show that for ε ≤ ε 1 independent of s, the iterates T n (0) are bounded in H s ε,δ , and similarly that T n (0) − T (0) = O(ε 2 ) in H s ε,δ , implying that the limit U belongs to H s ε,δ with norm O(ε 2 ). Together with the Sobolev inequality (6.21), this implies the pointwise estimates (1.17).
Finally, the assertion about uniqueness follows by uniqueness in B cε,δ under the additional phase condition (6.9) for the choice δ = 0 and c > 0 sufficiently small (noting by our argument that also B cε,δ is mapped to itself for ε sufficiently small, for any c > 0), together with the observation that phase condition (6.9) may be achieved for any solution 0) and so (by the Implicit Function Theorem applied to h(a) := ε −2 ℓ ε · u a ), together with ℓ ε · u 0 = o(ε) and the assumed property that ℓ a ·ū ′ N S (0) ∼ ε 2 coming from our choice of ℓ ε ; see (6.10), Remark 6.1) the inner product ℓ a ·ū ′ N S (0) may be set to zero by appropriate choice of a = o(ε −1 ) leaving U a in the same o(ε) neighborhood, by the computation U a − U 0 ∼ ∂ a U · a ∼ o(ε −1 )ε 2 .
It remains to prove existence of the linearized solution operator and the linearized bounds (6.15), which tasks will be the work of most of the rest of the paper. We concentrate first on estimates, and prove the existence next, using a viscosity method combined with (the single new step in treating the infinite-dimensional case) discretization in velocity.

Internal and high frequency estimates
We begin by establishing a priori estimates on solutions of the equation (6.8) This will be done in two stages. In the first stage, carried out in this section, we establish energy estimates showing that "microscopic", or "internal", variables consisting of v and derivatives of (u, v) are controlled by and small with respect to the "macroscopic", or "fluid" variable, u. As discussed in Section 4.5, this is the main new aspect in the infinite-dimensional case.
In the second stage, carried out in Section 8, we estimate the macroscopic variable u by Chapman-Enskog approximation combined with finite-dimensional ODE techniques such as have been used in the study of fluid-dynamical shocks [MaZ4,MaZ5,Z1,Z2,GMWZ], exactly as in the finite-dimensional analysis of [MeZ1].

The basic H 1 estimate
We consider the equation and its differentiated form: The internal variables are U ′ = (u ′ , v ′ ) andṽ where is the linearization about (ū N S ,v N S ) of the key variable v − v * (u) arising in the Chapman-Enskog expansion of Section 5.1. Noting that pu = 0 at the reference point U by Assumption 4.5, we have the important fact that on the set of U we consider (ε 2 close toŪ N S , so ε close to U ), so that v andṽ are nearly equivalent.

The approximate equations
It remains only to estimate u L 2 ε,δ in order to close the estimates and establish (7.5). To this end, we work with the first equation in (7.1) and estimate it by comparison with the Chapman-Enskog approximation (see the computations Section 5.1), exactly as in the finite-dimensional case [MeZ1].
From the second equation where we use the notationsṽ of Proposition 7.1, we find Introducingṽ in the first equation, yields . Therefore, (8.1) can be modified to This implies that u satisfies the linearized profile equation

L 2 estimates and proof of the main estimates
The following estimate was established in [MeZ1] using standard finite-dimensional ODE techniques; for completeness, we recall the proof here as well, in Section 8.3 below.
uniquely specified by the property that the solution u = (b * ∂ x − dh * ) † h satisfies for a certain unit vector ℓ ε .
Taking this proposition for granted, we finish the proof of the main estimates in Proposition 6.2.
Knowing a bound for u L 2 ε,δ , Proposition 7.2 immediately implies Proposition 8.3. There are constants C, ε 0 > 0 and δ 0 > 0 and for s ≥ 3 there is a constant C s such that for ε ∈]0, ε 0 ], δ ∈ [0, δ 0 ], f ∈ H s+1 ε,δ , g ∈ H s ε,δ and U ∈ H s ε,δ satisfying (6.8) and (6.9), one has Remark 8.4. The estimate of Proposition 8.1 may be recognized as somewhat similar to the estimates of Goodman [Go] obtained by energy methods in the time-evolutionary case, the same ones used by Liu and Yu [LY] to control the macroscopic variable u. More precisely, the argument is a simplified version of the one used by Plaza and Zumbrun [PZ] to show time-evolutionary stability of general small-amplitude waves.

Proof of Proposition 8.1
By Assumption 4.10(i), we may assume that there are linear coordinates u = (u 1 , u 2 ) ∈ R n 1 × R n 2 and h = (h 1 , h 2 ) ∈ R n 1 × R n 2 , with n 2 = rank b * (ū) such that the equationb * u ′ −dh * u = h has the form: Assumption 4.10(ii) implies that the left upper corner blockā 11 is uniformly invertible. Solving the first equation for u 1 , we obtain the reduced nondegenerate ordinary differential equationb Note that detdh * = detā 11 detǎ by standard block determinant identities, so that detǎ ∼ detdh * by Assumption 4.10(ii). Moreover, as established in [MaZ4], by Assumption 4.11 and the construction of the profileū N S we find that m := (b) −1ǎ has the following properties: i) with m ± denoting the end points values of m, there is θ > 0 such that for all k : ii) m(x) has a single simple eigenvalue of order ε, dented by εµ(x), and there is c > 0 such that for all x and ε the other eigenvalues λ satisfy |Re λ| ≥ c; iii) the end point values µ ± of µ satisfy (8.20) µ − ≥ α µ + ≤ −α for some α > 0 independent of ε.
In the strictly parabolic case det b * = 0, this follows by a lemma of Majda and Pego [MP].
At this point, we have reduced to the case with m having the properties listed above. The important feature is that m ′ = O(ε 2 ) << ε, the spectral gap between stable, unstable, and ε-order subspaces of m. The conditions above imply that there is a matrix ω such that where the spectrum of p ± lies in ±Re λ ≥ c. Moreover, ω and p satisfies estimates similar to (8.19). The change of variables u 2 = ωz reduces (8.21) to The equations (z + ) ′ − p + z + = h + and (z − ) ′ − p − z − = h − either by standard linear theory [He] or by symmetrizer estimates as in [GMWZ], admit unique solutions in weighted L 2 spaces, satisfying e δ|x| z ± L 2 ≤ C e δ|x| h ± L 2 , provided that δ remains small, typically δ < |Re p ± |.
The equation z ′ 0 −εµz 0 = h 0 may be converted by the change of coordinates x →x := εx to (8.20). This equation is underdetermined with index one, reflecting the translation-invariance of the underlying equations. However, the operator ∂x −μ has a bounded L 2 right inverse (∂x −μ) −1 , as may be seen by adjoining an additional artificial constraint (8.24)z 0 (0) = 0 fixing the phase. This can be seen by solving explicitly the equation or applying the gap lemma of [MeZ2] to reduce the problem to two constant-coefficient equations onx ≷ 0, with boundary conditions at z = 0. We obtain as a result that e δ|x|z 0 L 2 ≤ C e δ|x|h 0 L 2 if δ < min{α, θ}, which yields by rescaling the estimate Together with the (better) previous estimates, this gives existence and uniqueness for the equation , this implies that for ε small enough, the equation (8.22) with z 0 (0) = 0 has a unique solution. Tracing back to the original variables u, the condition z 0 (0) = 0 translates into a condition of the form ℓ ε · u(0) = 0. Therefore, the equationb * u ′ −df * u = h has a unique solution such u that ℓ ε · u(0) = 0, which satisfies e εδ|x| u L 2 ≤ Cε −1 e εδ|x| h L 2 for δ and ε small enough, finishing the proof of Proposition 8.1.

Existence for the linearized problem
The desired estimates (6.14) and (6.15) are given by Propositions 8.2 and 8.3. It remains to prove existence for the linearized problem with phase condition u(0) · r(ε) = 0. This we carry out using a vanishing viscosity argument. Fixing ε, consider in place of L ε * U = F the family of modified equations Differentiating the first equation yields where dQ(x) denotes here the matrix dQ(ū N S , v * (ū N S )).

Uniform estimates
We first prove uniform a-priori estimates. We denote by S the Schwartz space and for δ ≥ 0, by S εδ the space of functions u such that e εδ x u ∈ S , with x = √ 1 + x 2 as in (6.12).
Proposition 9.1. There are constants ε 0 > 0, δ 0 > 0 and η 0 > 0, and for all s ≥ 2 a constant C s , such that for ε ∈]0, ε 0 ], δ ∈ [0, δ 0 ], η ∈]0, η 0 ], and U and F in S εδ (R), satisfying (9.1) Proof. The argument of Proposition 7.1 goes through essentially unchanged, with new η terms providing additional favorable higher-derivative terms sufficient to absorb new higherderivative errors coming from the Kawashima part. Thus we are led to equations of the form (7.2) with the additional term −ηU ′′ in the left hand side. Using the symmetrizer S (4.28), one gains η U ′′ 2 L 2 +λ U ′ 2 L 2 in the minorization of Re (SF, U ) and loses commutator terms which are dominated by which can be absorbed by the left hand side yielding uniform estimates Going back to (9.2), this implies uniform estimates of the form for δ = 0, and next for δ ∈ [0, δ 0 ] with δ 0 > 0 small, as in the proof of Proposition 7.1.
When commuting derivatives to the equation, the additional term η∂ 2 x brings no new term and the proof of Proposition 7.2 can be repeated without changes, yielding estimates of the form Next, applying the Chapman-Enskog argument of Section 8 to the viscous system, we obtain in place of (8.3) the equation where the final η term coming from artificial viscosity is treated as a source. One applies Proposition 8.1 to estimate ε u L 2 ε,δ by the L 2 ε,δ -norm of the right hand side, and continuing as in the proof of Proposition 8.2, the estimate (8.13) is now replaced by Therefore, for η small, the new O(η) terms can be absorbed, and (9.3) for s = 2 follows as before. The higher order estimates follow from (9.6).

Existence
We now prove existence and uniqueness for (9.1). First, recast the the problem as a firstorder system (9.9) and (9.10) Next, consider this as a transmission problem or a doubled boundary value problem on x ≷ 0, with boundary condtitions given by the n + 2r matching conditions U(0 − ) = U(0 + ) at x = 0 together with the phase condition ℓ ε · u(0) = 0, that is n + 2r + 1 conditions in all: Note that the operator-valued coefficient matrix A converges exponentially to its endstates at ±∞, by exponential convergence ofŪ N S and boundedness of A, Q.
Lemma 9.2. There is θ 1 > 0 such that for ε small enough , the limiting coefficient matrices A ± have no eigenvalue in the strip |Re z| ≤ εδ 0 .
Remark 9.3. The same reasoning can be applied to prove that A actually has a simple eigenvalue such that |z − εµ| ≤ 1 2 ε|µ|.

Finite-dimensional case
We first review the case that U is finite-dimensional, recalling for completeness the analysis of [MeZ1].
Proof. Noting that the coefficient matrix A converges exponentially to A ± at ±∞, we may apply the conjugation lemma of [MeZ1] to convert the equation (9.9) by an asymptotically trivial change of coordinates U = T (x)Z to a constant-coefficient problems on {±x ≥ 0}, with n + 2r + 1 modified boundary conditions determined by the value of the transformation T at x = 0, where A ± := A(±∞), and Z ± (x) := Z(x) for ±x > 0. By standard boundary-value theory (see, e.g., [He]), to prove existence and uniqueness in the Schwartz space for the problem (9.9) on {x < 0} and {x > 0} with transmission conditions (9.11), it is sufficient to show that (i) the limiting coefficient matrices A ± are hyperbolic, i.e., have no pure imaginary eigenvalues, (ii) the number of boundary conditions is equal to the number of stable (i.e., negative real part) eigenvalues of A + plus the number of unstable eigenvalues (i.e., positive real part) of A − , and (iii) there exists no nontrivial solution of the homogeneous equation f = 0, g = 0. Moreover, since the eigenvalues of A ± are located in {|Re z| ≥ θ 1 ε, the conjugated form (9.16) of the equation show that if the source term f has an exponential decay e −εδ x at infinity, then the bounded solution also has the same exponential decay, provided that δ < θ 1 . Therefore, the three conditions above are also sufficient to prove existence and uniqueness in S εδ if ε and δ are small.
Note that (i) is a consequence of Lemma 9.2, while (iii) follows from the estimate (9.3). To verify (ii), it is enough to establish the formulae where A * ± 11 = dh * (u ± ) = A 11 + A 12 dv * (u ± ) and S(M ) and U(M ) denote the stable and unstable subspaces of a matrix M . We note that A * ± 11 = dh * (u ± ) are invertible, with dimensions of the stable subspace of A * + 11 and the unstable subspace of A * − 11 summing to n + 1, by Proposition 5.1. Thus, (9.17) implies that dim S(A + ) + dim U(A − ) = 2r + dim S(A * + 11 ) + dim U(A * − 11 ) = 2r + n + 1 as claimed.
To establish (9.17), introduce the variableṽ = v + Q −1 22 Q 21 u, and the variable corresponding toṽ ′ scaled by a factor η 1 2 , that isw = η 1 2 w + η − 1 2 Q −1 22 Q 21 (A 11 u + A 12 v). After this change of variables, the matrix A it conjugated to A with (9.18) η From (i), the matrix η 1 2 A has no eigenvelue on the imaginary axis, and the number of eigenvalues in {Re λ > 0} is independent of η, and thus can be determined taking η to infinity. The limiting matrix has r eigenvalues in {Re λ > 0}, r eigenvalues in {Re λ < 0} and the eigenvalue 0 with multiplicity n, since −Q 22 has its spectrum in {Re λ > 0}. The classical perturbation theory as in [MaZ1] shows that for η − 1 2 small, η 1 2 A has n eigenvalues of order η − 1 2 , close to the spectrum of A * 11 with error O(η −1 ). Thus, for η > 0 large, η 1 2 A has r + dim S(A * 11 ) eigenvalue in {Re λ < 0}, proving (9.17). The proof of the Proposition is now complete.

Finite dimensional approximations
To treat the infinite-dimensional case, we proceed by finite-dimensional approximations. Let Q = Q M and K = K M denote the operators Q U and K U evaluated at the equilibrium M = M (u), so that Moreover, the Kawashima multiplier has the form (9.21) Thanks to (9.20) the condition (4.14) is satisfied for θ small enough as soon as (9.22) Re K 11 A 11 + K 12 A 21 ≥ cId, c > 0.
Consider an increasing sequence of finite dimensional subspaces Similarly, let H r = U ⊕ V r . Let Π r (U ) denote the orthogonal projector onto H r , so that (9.24) Π r = Π * r , and define is uniformly satisfied for U in a neighborhood of M and r large enough.
Proof. (i) follows by (9.24) and symmetry of A on H.
On {0} ⊕ V r , Q r (U ) is a perturbation of π r Q 22 π r where π r is the (usual) orthogonal projection onto V r , with To prove (9.26), is is sufficient to prove the property at U = M . Restricting to V r by π r , one has A r = A 11 A 12 π r π r A 21 π r A 22 π r , K r = θ K 11 K 12 π r π r K 21 0 .
Note that K 11 A 11 + K 12 π r A 21 is an n-dimensional perturbation of K 11 A 11 + K 12 A 21 whose real part is definite positive. Therefore, for r large enough, Re K r A r 11 = Re K 11 A 11 + K 12 π r A 21 ≥ cId with c independent of r. Since π r Q 22 π r ≥ c 1 Id on V r uniformly in r, and since the other blocks of K r A r are uniformly O(θ), the condition (9.26) is satisfied for r large enough and θ sufficiently small.
Corollary 9.6. On H r the equation is well posed, and there are uniform estimates in r, for r sufficiently large.
Corollary 9.7. On H the equation is well posed.

Other norms
We now briefly discuss the modifications needed to obtain the full result of Theorem 1.1.

Pointwise velocity estimates
Proof of Theorem 1.1 (H 1/2 ). To obtain pointwise bounds with respect to velocity, we carry out the same argument as in the proof of Proposition 5.4, substituting in place of the L 2 norm | · | in ξ, the weighted H s (Sobolev) norm |f | s := s k=0 C −k |∂ k ξ f |2, C > 0 sufficiently large, similarly as we did for the x-variable in order to get pointwise bounds in x.
We have only to observe that differentiating the linearized equations in ξ gives the same principal part applied to the ξ-derivative of U , plus commuator terms. Since commutator terms, both for the linearized collision operator L and the transport operator A are of one lower derivative in ξ and also one lower factor in ξ (straightforward computation differentiating |ξ − ξ ′ |, ξ 1 , respectively) for the hard-sphere case, we easily find that commutator terms are absorbable for C > 0 sufficiently large by lower order estimates already carried out.
Thus, we obtain all the same estimates as before and the argument closes to give the same result in the stronger norm | · | s . (Note: a detail is to observe that truncation errors of the approximate solution are of the same order in the Sobolev norm, which follows by the corresponding property of the Maxwellian.) Applying the Sobolev embedding estimate in ξ, we obtain (1.17) for η = 1/2, which evidently implies the same estimate for η ≥ 1/2.

Higher weights
Proof of Theorem 1.1 (H s ). To extend our results from H 1 2 to H s , we use a simple bootstrap argument together with the key observation that the H s norm of P U f is controlled (by equivalence of finite-dimensional norms) by the H 1 2 estimates already obtained. Namely, starting similarly as in (4.24) with the equation A∂ x − L a U = F , P U F = f , P V F = g, we find, taking the H s -inner product of U against this equation and applying the result of Proposition 3.6 and recalling that A is formally self-adjoint in H s , we obtain the estimate (10.1) P V U L 2 ≤ C f L 2 + g L 2 + P U U L 2 .
Differentiating the equations k times and taking the inner product with ∂ k x U , we find, similarly, the higher-derivative estimate Specializing now to the case (6.8), (6.9), and bounding the H s norm of P U U by a constant times the H 1 2 bound obtained already in our previous analysis, we recover the key bounds (6.14)-(6.15) of Proposition 6.2 in the general space H s . With this bound, the entire contraction mapping argument goes through in H s , since this relies only on boundedness estimates on A, Q already obtained, the estimate (5.2) (still valid in H s ), and the linearized estimates (6.14)-(6.15), yielding (1.17)(i) and (ii) as claimed, for any η > 0.
The estimate (1.17)(iii) then follows by Remark 3.7 estimating decay in velocity ξ of the approximating profilef N S .
Remark 10.1. We emphasize that L is not approximately self-adjoint with respect to H s , s >> 1/2, and, likewise, the splitting H s = U ⊕ V s using projectors P U and P V is not orthogonal in this norm. For this reason, we obtain term P U U H k ε,δ in the righthand side of (10.2) and not ε P U U H k ε,δ as in the H 1/2 computations. The missing ε factor was crucial in closing the argument in H 1/2 and estimating P U U . However, with P U U already bounded it is no longer needed, since our final estimates make no distinction between P U U and P V U components; that is, the lost ε factor is needed only to close the loop between microscopic and macroscopic estimates, and not to bound P V U in terms of P U U .

Other potentials
Finally, we briefly indicate the changes needed to accomodate general hard cutoff potentials. Recall [CN, GPS] that these give structure L = −ν(ξ) + K, where ν ∼ ξ β , 0 < β < 1, and K is compact, and similarly for Q. Dividing by ν ∼ ξ β as before, we can thus obtain Q, L bounded, but this leaves A unbounded. Nonetheless, a closer look shows that the Kawashima compensator K as constructed is still bounded, the key point. For, examining A 12 , we see that it decays as a polynomial in ξ times full Maxwellian, so is clearly bounded in H s for s < 1.
Since the norm of A does not enter except through the good term KA, our basic microestimates therefore still survive. Of course, the macro-estimates, since finite-dimensional, survive as well. (This follows by the same estimate that shows that K as constructed is bounded; that is, one has only to check that A 12 and A 21 entries remain bounded, thanks to Maxwellian rate decay.) Thus, the argument goes through as before, also for these more general potentials.