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Reverse norms and L infinity exponential decay for a class of degenerate evolution systems arising in kinetic theory PDF

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Preview Reverse norms and L infinity exponential decay for a class of degenerate evolution systems arising in kinetic theory

REVERSE NORMS AND L∞ EXPONENTIAL DECAY FOR A CLASS OF DEGENERATE EVOLUTION SYSTEMS ARISING IN KINETIC THEORY ALINPOGAN AND KEVINZUMBRUN Abstract. Weconsiderthequestionofexponentialdecaytoequilibriumofsolutionsofanabstract 7 1 classofdegenerateevolutionequationsonaHilbertspacemodelingthesteadyBoltzmannandother 0 kineticequations. Specifically, we provideconditions suitable for construction of a stable manifold 2 in a particular “reverse L∞” norm and examine when these doand donot hold. n a J Miami University Indiana University 6 Department of Mathematics Department of Mathematics 1 301 S. Patterson Ave. 831 E. Third St. ] Oxford, OH 45056, USA Bloomington, IN 47405, USA P A . h Contents t a 1. Introduction 1 m 1.1. The reverse norm 2 [ 1.2. Results and counterexamples 3 1 2. Results 4 v 2.1. Invariance of H under the action of D 4 3 ∗ ∗ 4 2.2. Existence and uniqueness of an H∗ stable manifold of equation (1.1) 6 4 3. Counterexamples 7 4 3.1. Counterexample (i) 7 0 3.2. Counterexample (ii) 9 . 1 4. Discussion and open problems 10 0 7 References 11 1 : v i X 1. Introduction r a In this note, we study decay to zero for a class of degenerate evolution equations (1.1) Γu′+u= D(u,u) on a real Hilbert space H, where D(·,·) is a bounded bilinear map on H and Γ is a (fixed) bounded linear operator on H that is self-adjoint, one-to-one, but not invertible. As described in [8, In- troduction, Eqs. (1.5), (1.8)], this is equivalent to the study of decay to equilibria of shock and boundary layer solutions of a class of kinetic equations including the steady Boltzmann equation. This equation is most easily understood via spectral decomposition of Γ, converting (1.1) to a family of scalar equations (1.2) (γ ∂ +1)u = D (u,u), λ x λ λ A. P. research was partially supported undertheSummerResearch Grant program, Miami University. Research of K.Z. was partially supported underNSFgrant no. DMS-0300487. 1 indexed by λ ∈ Λ, where u is the coordinate of u associated with spectrum γ , real, in the λ λ eigendecomposition of Γ, with kuk2 = |u |2dµ . Here dµ denotes spectral measure associated H Λ λ λ with λ, and the set {γλ : λ ∈ Λ} is bounRded with an accumulation point at 0. The linearized equations about zero, (γ ∂ +1)u = 0, have a stable subspace consisting of λ x λ (1.3) u (x)= u (0)e−x/γλ, λ λ for any u(0) ∈ H with u (0) = 0 whenever γ < 0, satisfying the uniform exponential bound λ λ ku(x)kH ≤ Ce−x/supλ∈Λ|γλ|ku(0)kH. On the other hand, derivatives γ−1e−x/γλ of functions in the stable subspace, being multiplied by λ the unbounded factor γ−1, may not even lie in H. We denote as the H1 stable subspace the subset λ of solutions in the stable subspace that are contained in H1(R ,H), namely, those with + u(0) ∈ dom(Γ−1/2). See [8] for further details. In [8], it was shown that there exists an H1 stable manifold, consisting of all solutions of the full equationthataresufficientlysmallinH1(R ,H),lyingtangenttotheH1 stablesubspace,onwhich + solutionsdecay uniformlyexponentially inH1(R ,H), hencedecay pointwiseatratee−βx, forsome + β > 0. However, it was also shown that the linearized solution operator (Γ∂ +Id)−1, though a x bounded operator on all Lp(R ,H), 1 < p < ∞, is unbounded on L1(R ,H) and L∞(R ,H), as + + + a consequence of which the usual stable manifold construction fails in L∞(R ,H). It was cited + as an interesting open problem in [8] whether there exists a “full” stable manifold, tangent to the entire linear stable subspace. Here, we consider a specific approach to this problem based on the introduction of a nonstandard “reverse norm.” 1.1. The reverse norm. Let us first review the standard fixed point construction of the stable manifold (as carried out for finite dimensions in, e.g., [2, 3]) within the context considered in [8]. Define χ (x) to be the cutoff function returning 1 for x ≥ 0 and 0 otherwise, and let Π be + s projection onto the stable subspace of Γ and T (·) the semigroup induced by (1.1) restricted to the s stable subspace of −Γ−1. Then, solutions of (1.1) on R may be expressed as + (1.4) u= T (·)Π u(0)+χ (Γ∂ +Id)−1 χ D (u,u) , s s + x + ∗ (cid:0) (cid:1) whereD :L2(R ,H)×L2(R,H) → L1(R,H)isthebilinearmapdefinedbyD (u,v) = D(u(·),v(·)). ∗ + ∗ Equation (1.4) can be seen as a variant of the usual variation of constants formula, so long as the inverse (Γ∂ +Id)−1 is well-defined on functions of the form χ D (u,u). Indeed, with some x + ∗ elaborations, (1.4) is used in [5, 6, 8] effectively as the definition of a mild solution of (1.1) on R . + What we seek, then, is a Banach Space Z of functions on R that is continuously embedded in + L∞(R,H), closed under the action of (Γ∂ +Id)−1 and of D , in the sense that x ∗ (1.5) D (v,v) ∈ Z for any v ∈ Z and kD (v,v)k . kvk2 for any v ∈Z. ∗ ∗ Z Z Moreover, the space Z should be large enough to contain the subspace of trajectories {T (·)h : h∈ s RangeΠ }. When these properties hold, one readily sees that the fixed-point equation (1.4) is a s contraction, yieldingexistenceanduniquenessofthestablemanifold; fordetailsoftheconstruction, definition of mild solution, etc., see the similar analysis of [8]. To this end, we introduce the reverse norm 1/2 (1.6) kukH∗ := (cid:16)ZΛ(xs∈uRp+|uλ(x)|)2dµλ(cid:17) , 2 where dµ denotes spectral measure associated with λ, and the space H of functions on R with ∗ + finite k·kH∗ norm. We see readily that (Γ∂x +Id) is boundedly invertible on H∗, with resolvent kernel given in u coordinates by the scalar resolvent kernel λ (1.7) Rλ(x,y) = γλ−1e(x−y)/γλ−1 whenever (x−y)γλ < 0, and 0 otherwise. For, (1.7) is integrable with respect to x, hence bounded coordinate-by-coordinate with respect to L∞(R+), as therefore is the square integral k·kH∗ of all coordinates. The question thus reduces to: “under what conditions on D is the extension D closed with ∗ respect to H ?” When such conditions are met, we have existence of a unique stable manifold in ∗ H , tangent to the full stable subspace of the bi-semigroup generated by −Γ−1, answering the open ∗ question of [8]; see Theorem 1.5 and Corollary 1.6. 1.2. Results and counterexamples. We first look for a sharp abstract condition that charac- terizes condition (1.5). For any α ∈ L2(dµ ) we introduce the set λ (1.8) E = v ∈ H: |v |≤ |α |for any λ ∈ Λ . α λ λ (cid:8) (cid:9) We recall that here (v ) denote the spectral coordinates of v ∈ H. Next, for any α ∈ L2(dµ ) λ λ∈Λ λ we define S(α) by (1.9) S (α) = sup |D (v,v)|. λ λ v∈Eα The following three results are established in Section 2. Proposition 1.1. The reversed-norm space H is closed under the action of D if and only if ∗ ∗ (1.10) kS(α)kL2(dµλ) . kαk2L2(dµλ) for any α ∈ L2(dµλ). Let D be expressed in terms of a kernel D :Λ3 → R, via (1.11) D (v,w) = D(λ,ν,σ)v w dµ dµ . λ ν σ ν σ ZΛ2 Then, two sufficient conditions are as follows. Proposition 1.2 (Hilbert-Schmidt condition). The reversed-norm space H is closed under the ∗ action of D provided ∗ (1.12) |D(λ,ν,σ)|2dµ dµ dµ < +∞. λ ν σ ZΛ3 Proposition 1.3 (Absolute boundedness condition). The reversed-norm space H is closed under ∗ the action of D provided the bilinear map |D| with kernel |D(λ,γ,σ)| is bounded from H×H→ H. ∗ Remark 1.4. In [8], the form of equation (1.1) arose through linearization about u¯ ∈ H of (1.13) Γu′ = D˜(u,u), D˜ a bounded bilinear map; that is, the term u on the lefthand side of (1.1) corresponds to the relation−2Id = D(u¯,·). But,thiscontradicts(1.12),since(1.12)togetherwiththeCauchy-Schwarz inequality gives kDkL2(dµ3) ≤ ku¯k2HkD˜k2H→H < ∞. Thus, the Hilbert-Schmidt criterion, though appealing, is not relevant to the problem originally considered in [8], in particular not to the case of the steady Boltzmann equation. Using these results we can prove the following existence and uniqueness result. 3 Theorem 1.5. Assume that H is closed under the action of D (for example, that |D| is bounded ∗ ∗ from H×H to H, or (1.12) is satisfied). Then, for any integer r ≥ 1 there exists a Cr local stable manifold M near 0, expressible as a graph of Cr function J : H → H , that is locally invariant s s s u under the flow of equation Γu′+u= D(u,u) and uniquely determined by the property that u∈ H . ∗ Using the existence result above we obtain the following exponential decay result for solutions of equation (1.1): Corollary 1.6. Assume that H is closed under the action of D , and let u∗ ∈ H be a solution of ∗ ∗ ∗ equation Γu′ +u = D(u,u). Then, there exist β ∈ (0,kΓk−1) such that u∗ ∈ H . In particular, ∗,β we have that there exists β > 0 such that ku∗(x)k . e−β|x| for any x ∈ R . ± Itis straightforward toconstructkernels D originating fromlinearization of (1.13)andsatisfying the condition of Proposition 1.3 but not (1.12). Hence, Proposition 1.3 gives existence of a full stable manifold in some cases relevant to the scenario originally considered in [8]. However, this condition too is not sharp. In Section 3 we give two counterexamples showing that Propositions 1.2 and 1.3 provide only sufficient conditions guaranteeing that H is closed under the action of D . ∗ ∗ Finally,inSection4,wediscusspossiblegeneralizations, andopenproblems,inparticularasregards the important example of the steady Boltzmann equation, our main interest in [8]. 2. Results In this section we prove our results stated in the introduction. First, we give necessary and sufficient conditions that guarantee that the extension D is bounded. Then, we sketch the proof ∗ of existence and uniqueness of a stable manifold of equation (1.1) assuming boundedness of D . ∗ 2.1. Invariance of H under the action of D . ∗ ∗ Proof. ofProposition 1.1. First,weassumethatcondition(1.5)holdsforZ = H . Fixα∈ L2(dµ ). ∗ λ From the definition of S(α) in (1.9) we have that for any λ ∈ Λ there exists vλ ∈ E ⊂ H such α that 1 (2.1) S (α) ≤ |D (vλ,vλ)| for any λ ∈ Λ. λ λ 2 Since the linear operator Γ is self-adjoint, its spectral decomposition Λ is contained in R. It follows that for any λ ∈ Λ there exist x ∈ R such that x 6= x for any λ 6= ν. Next, for any λ ∈ Λ we λ + λ ν construct a function w : R → C such that w (x ) = vν for any ν ∈ Λ and 0 otherwise. From λ + λ ν λ (1.8) we obtain that (2.2) sup|w (x)| = sup|vν|≤ |α | for any λ ∈Λ. λ λ λ x≥0 ν∈Λ Since (vν) ∈ L2(dµ ) we have that the function λ → vν is µ-measurable for any ν ∈ Λ, which λ λ∈Λ λ λ implies that the function λ → sup |w (x)| is µ-measurable. Since α ∈ L2(dµ ) from (2.2) we x≥0 λ λ conclude that w = (wλ)λ∈Λ ∈H∗ and kwkH∗ ≤ kαkL2(dµλ). Moreover, we have that w(xλ) =vλ for any λ ∈ Λ. From (2.1) we obtain that 1 (2.3) sup|D (w(x),w(x)| ≥ |D (vλ,vλ)| ≥ S (α) for any λ ∈ Λ. λ λ λ 2 x≥0 Since H is closed under the action of the extension D , from (1.5) and (2.3) it follows that ∗ ∗ (2.4) ZΛ|Sλ(α)|2dµλ ≤ 2ZΛ(cid:0)sxu≥p0|Dλ(w(x),w(x)|(cid:1)2dµλ = 2kD∗(w,w)k2H∗ . kwk2H∗ ≤ kαk2L2(dµλ), 4 proving that S(α) ∈ L2(dµ ) and (1.10) holds true. Conversely, assume (1.10) and let w = λ (w ) ∈ H . From the definition of H we immediately infer that α = (α ) defined by α = λ ∗ ∗ λ λ∈Λ λ sup |w (x)| belongs to L2(dµ ). Moreover, from (1.8) and (1.9), respectively, we have that x≥0 λ λ w(x) ∈ E and |D (w(x),w(x))| ≤ S (α) for any x ≥ 0 and λ ∈ Λ. We conclude that α λ λ (2.5) ZΛ(cid:0)sxu≥p0|Dλ(w(x),w(x)|(cid:1)2dµλ ≤ ZΛ|Sλ(α)|2dµλ . kαk2L2(dµλ) = kwk2H∗, proving the proposition. (cid:3) Proof. of Proposition 1.2. The result follows from Proposition 1.1 by using a simple Cauchy- Schwartz argument. Indeed, for any α ∈ L2(dµλ) and v = (vλ)λ∈Λ ∈ Eα we have that kvkH ≤ kαk , thus L2(dµλ) 1 1 |D (v,v)| ≤ |D(λ,ν,σ)|2dµ dµ 2 |v |2|v |2dµ dµ 2 λ ν σ ν σ ν σ (cid:16)ZΛ2 (cid:17) (cid:16)ZΛ2 (cid:17) 1 1 ≤ (cid:16)ZΛ2|D(λ,ν,σ)|2dµνdµσ(cid:17)2kvk2H ≤ (cid:16)ZΛ2|D(λ,ν,σ)|2dµνdµσ(cid:17)2kαk2L2(dµλ). It follows that 1 S (α) ≤ |D(λ,ν,σ)|2dµ dµ 2kαk2 for any λ ∈Λ,α ∈ L2(dµ ). λ (cid:16)ZΛ2 ν σ(cid:17) L2(dµλ) λ Integrating this inequality with respect to λ ∈ Λ we obtain that |S (α)|2dµ ≤ |D(λ,ν,σ)|2dµ dµ dµ kαk4 for any α ∈ L2(dµ ), ZΛ λ λ ZΛ3 λ ν σ L2(dµλ) λ proving the proposition. (cid:3) Remark 2.1. We note that our Hilbert-Schmidt condition from (1.12) does depend on the spectral decomposition of the self-adjoint operator Γ. Moreover, for each λ ∈ Λ there exists T ∈ B(H) λ such that Dλ(u,v) = hu,Tλvi for any u,v ∈ H. From (1.11) we immediately infer that kTλkH→H = 1 |D(λ,ν,σ)|2dµ dµ 2 for any λ ∈ Λ. Therefore, (1.12) is equivalent to Λ2 ν σ (cid:0)R (cid:1) (2.6) kT k2 dµ < +∞. λ H→H λ Z Λ Replacing the k · kH→H norm in (2.6) by the Hilbert-Schmidt norm we obtain an even stronger Hilbert-Schmidt condition on the bilinear map D (2.7) kT k2 dµ < +∞, λ HS λ Z Λ that can be shown to be independent of the spectral decomposition of Γ or the choice of Hilbert bases on H. Proof. of Proposition 1.3. Fix v = (v ) ∈ H and let w ∈ H having spectral decomposition λ λ∈Λ ∗ (wλ)λ∈Λ where wλ = supx≥0|vλ(x)|. One can readily check that kvkH∗ = kwkH. Using that |D| 5 defines a bounded bilinear map on H we obtain that 2 2 sup|D (v(x),v(x))| dµ = sup D(λ,ν,σ)v (x)v (x)dµ dµ dµ λ λ ν σ ν σ λ ZΛ(cid:0)x≥0 (cid:1) ZΛx≥0(cid:12)ZΛ2 (cid:12) (cid:12) (cid:12) 2 ≤ sup |D(λ,ν,σ)||v (x)||v (x)| dµ dµ dµ ν σ ν σ λ ZΛ(cid:16)ZΛ2 x≥0(cid:0) (cid:1) (cid:17) 2 = |D(λ,ν,σ)|w w dµ dµ dµ ν σ ν σ λ ZΛ(cid:16)ZΛ2 (cid:17) = |D| (w,w) 2dµ = |D|(w,w) 2 . kwk4 = kvk4 Z λ λ H H H∗ Λ(cid:12) (cid:12) (cid:13) (cid:13) (2.8) (cid:12) (cid:12) (cid:13) (cid:13) From (2.8) we have that D (v,v) ∈ H and ∗ ∗ 1 (2.9) kD∗(v,v)kH∗ = (cid:16)ZΛ(cid:0)sxu≥p0|Dλ(v(x),v(x))|(cid:1)2dµλ(cid:17)2 . kvk2H∗, proving the proposition. (cid:3) 2.2. Existence and uniqueness of an H stable manifold of equation (1.1). Now we have ∗ all the ingredients needed to construct the stable manifold tangent at u= 0 to the stable subspace of the linearized equation u′ = −Γ−1u. Throughout this subsection we assume that the space H ∗ is closed under the action D in the sense of (1.5). ∗ Since Γ is similar to the operator of multiplication by (γ ) on L2(dµ ), we can immediately λ λ∈Λ λ infer that the stable/unstable subspace of equation u′ = −Γ−1u is given by H = {h ∈ H : s/u supp(h ) ⊆ Λ }, where Λ = {λ ∈ Λ : ±γ > 0}. Using that the linear operator Γ is self-adjoint, λ ± ± λ onecanreadily check that2πiωΓ+I isinvertible onHfor anyω ∈ Randsup k(2πiωΓ+I)−1k< ω∈R ∞. Thus, the Fourier multiplier K = (Γ∂ +I)−1 = F−1M F is bounded on L2(R,H), where x R M is the operator of multiplication on L2(R,H) by the operator valued function R : R → B(H) R defined by R(ω) = 2πiωΓ+I. Taking Fourier Transform in (1.1) and then solving for Fu we can see that its L2-solutions on R satisfy (1.4). + To solve (1.4) locally, we use a fixed point argument on a small closed ball centered at the origin in the weighted space H = {u : eβ|·|u ∈ H } for β ∈ [0,kΓk−1). The procedure requires the ∗,β ∗ − x following steps. First, using the representation of the stable semigroup (Ts(x)h)λ = e γλhλ for x ≥ 0, λ ∈ Λ and h∈ H , we prove that T (·)h ∈ H and s s ∗,β (2.10) kTs(·)hkH∗,β . khkHs for any h ∈ Hs. Next, we study the properties of the function Φ : H ×H → H defined by Φ(h,u) = T (·)h+ s ∗,β ∗,β s χ K(χ D (u,u)). Here we recall that χ is the characteristic function of R . To establish our + + ∗ + + results we need to prove that, provided H is closed under the action of D , there exist ε > 0 ∗ ∗ 1 and ε2 > 0 such that for any β ∈ [0,kΓk−1) the function Φ maps BHs(0,ε1) × BH∗,β(0,ε2) to BH∗,β(0,ε2)1 and 1 (2.11) kΦ(h,u)−Φ(h,v)kH∗,β ≤ 2ku−vkH∗,β for any h ∈BHs(0,ε1), u,v ∈ BH∗,β(0,ε2). We note that for any weight β ∈ [0,kΓk−1) the space H is closed under the action D provided ∗,β ∗ H is closed underthe action of D . Therefore, to prove(2.11) itis enough to show thattheFourier ∗ ∗ multiplier K can be extended to a bounded linear operator on H , for β ∈ [0,kΓk−1). This result ∗,β 1BX(0,ε) denotes theclosed ball in X of radius ε centered at the origin. 6 follows by a long but fairly simple computation using the following convolution representation of K. (2.12) (Kf) (x) = K (x−y,λ)f (y)dy for any x ≥ 0, λ ∈ Λ, f ∈ L2(R ,H)∩L∞(R ,H), λ λ + + ZR where the kernel K : R×Λ → Λ is given by (2.13) K (x,λ) =  −γ1λγe1λ−eγ−xλγxλ iiff xx <> 00 aanndd λλ ∈∈ ΛΛ−+ . 0 otherwise   Usingafixedpointargument,from(2.11)weobtainthatforanyh ∈ BHs(0,ε1)equationu= Φ(h,u) has a unique, local solution denoted u(·,h). Moreover, u(·,h) ∈ BH∗,β(0,ε2) depends smoothly on h ∈BHs(0,ε1) in the H∗,β norm. Using the representation (2.12) we have that (2.14) Πsu(0,h) = h for any h ∈ BHs(0,ε1). Next, we introduce the stable manifold of equation (1.1) by (2.15) Ms = {u(0,h) : h∈ BHs(0,ε1)}. From (2.14) we infer that Ms = Graph(Js), where Js : BHs(0,ε1) → Hu by Js(h) = Πuu(0,h). Moreover, since (Kf)(·+x ) = Kf(·+x ) for any x > 0 and f ∈ H , by using the uniqueness 0 0 0 ∗,β of solution of equation u = Φ(h,u), we conclude that the manifold M is invariant under the flow s of equation (1.1). Finally, by differentiating with respect to h in (1.4), we infer that J′(0) = 0 s proving that the manifold M is tangent at u= 0 to the stable subspace H . s s 3. Counterexamples In the remainder of the paper, we provide counterexamples showing that (i) boundedness of D does not imply boundedness of |D| (Subsection 3.1) and (ii) boundedness of D from H×H → H does not imply H is closed under the action of D . (Subsection 3.2). For simplicity, we work in ∗ ∗ the discrete case L2(dµ ) = ℓ2; however, the examples have obvious continuous counterparts. λ 3.1. Counterexample (i). Consider the matrix cosθ sinθ π (3.1) M (θ):= , 0 < θ < . 1 (cid:18)sinθ −cosθ(cid:19) 2 This has eigenvalues ±1, yet, separating positive and negative parts cosθ sinθ 0 0 (3.2) M+(θ):= , M+(θ):= , 1 (cid:18)sinθ 0 (cid:19) 1 (cid:18)0 cosθ(cid:19) we see that there exists θ ∈ (0, π) such that M+(θ ) has an eigenvalue ρ > 1, with an associ- 0 4 1 0 ated eigenvector q = (q ,q )T (by nonnegative version of Frobenius–Perron) having nonnegative 1 2 eigenvalues (indeed, calculation shows that the entries are strictly positive). Next, we recall the following decomposition: ℓ2 = ∞ R2j. We construct a linear operator j=1 M(θ) = ∞ M (θ) ∈ B( ∞ R2j), that is an infinLite matrix, recursively, defining its upper j=1 j j=1 lefthand 2Lj ×2j block M (θ)L, j ≥ 1, as follows j cosθM (θ) sinθM (θ) (3.3) M (θ):= j j , j+1 (cid:18)sinθMj(θ) −cosθMj(θ)(cid:19) 7 so that cosθM+(θ) sinθM+(θ) cosθM−(θ) sinθM−(θ) (3.4) M+ (θ):= j j , M− (θ):= j j . j+1 (cid:18)sinθM+(θ) −cosθM−(θ)(cid:19) j+1 (cid:18)sinθM−(θ) −cosθM+(θ)(cid:19) j j j j Lemma 3.1. For any j ≥ 1 and θ ∈ (0, π) the matrix M (θ) is an isometry on R2j, therefore the 2 j linear operator M(θ) = ∞ M (θ) is bounded on ℓ2. j=1 j L Proof. A direct computation shows that kM1(θ)vkR2 = kvkR2 for any v ∈ R2, showing that M1(θ) is an isometry on R2. Assume that M (θ) is an isometry on R2k. Let w = (w ,w ) ∈ R2k+1 = k 1 2 R2k ×R2k. From (3.3) we obtain that kM (θ)wk2 = kcosθM (θ)w +sinθM (θ)w k2 +ksinθM (θ)w −cosθM (θ)w k2 k+1 R2k+1 k 1 k 2 R2k k 1 k 2 R2k = cos2θkM (θ)w k2 +sin2θkM (θ)w k2 +2sinθ cosθhM (θ)w ,M (θ)w i k 1 R2k k 2 R2k k 1 k 2 R2k +sin2θkM (θ)w k2 +cos2θkM (θ)w k2 −2sinθ cosθhM (θ)w ,M (θ)w i k 1 R2k k 2 R2k k 1 k 2 R2k (3.5) = kM (θ)w k2 +kM (θ)w k2 = kw k2 +kw k2 = kwk2 , k 1 R2k k 2 R2k 1 R2k 2 R2k R2k+1 proving that M (θ) is an isometry on R2k+1. (cid:3) k+1 From Lemma 3.1 we can immediately infer that the map D : ℓ2×ℓ2 → ℓ2 defined by D(u,v) := hu,M(θ )vi e is a bounded bilinear map. Here {e } denotes the standard orthonormal Hilbert 0 ℓ2 1 n n≥ basis of ℓ2. However, |D| may, by recursion, be easily seen to be unbounded, by application to the nonnegative-entry (in fact, strictly positive entry) test vectors u ∈ R2j×2j, j ≥ 1, determined j recursively by q u (3.6) u = 1 j , for j ≥ 1 j+1 (cid:18)q2uj(cid:19) with u = q ∈ R2. We note that the following identity holds true: 1 (3.7) |D|(u,v) = hu,|M (θ )|vi for any u,v ∈ R2j×2j ⊂ ℓ2,j ≥ 1.2 j 0 R2j Using(3.7), we prove an estimate that will allows to immediately conclude that |D| is not bounded on H = ℓ2. Lemma 3.2. For M ∈ B(ℓ2) defined as in (3.3), the following inequalities hold true: (3.8) hu ,|M (θ )|u i ≥ ρjku k2 . j j 0 j R2j j R2j Proof. Estimate (3.8) holds with equality for j = 1, by construction. Assume that it holds for j ≤ k. Computing q u T cosθM+(θ ) sinθM+(θ ) q u hu ,|M (θ )|u i = 1 k k 0 k 0 1 k k+1 k+1 0 k+1 R2k+1 (cid:18)q2uk(cid:19) (cid:18)sinθMk+(θ0) cosθMk−(θ0)(cid:19)(cid:18)q2uk(cid:19) = huk,Mk+(θ0),ukiR2khq,M1+(θ0)qiR2 +huk,Mk−(θ0),ukiR2khq,M1−(θ0)qiR2, and using nonnegativity of entries to see that huk,Mk−(θ0),ukiR2khq,M1−(θ0)qiR2 ≥ 0, we obtain using the induction hypothesis that huk+1,|Mk+1(θ0)|uk+1iR2k+1 ≥ huk,Mk+(θ0),ukiR2khq,M1+(θ0)qiR2 ≥ ρkku k2 ρkqk2 = ρk+1ku k2 , k R2k R2 k+1 R2k+1 proving the claim. (cid:3) 2For a matrix A, we denote by|A|:=A++A−, where A± are thepositive/negative parts of A. 8 This shows it is not true that |D| is bounded when D is bounded, even restricted to a single coordinate. However, this property does not imply that H = {f = (f ) : sup |f (x)| ∈ ℓ2} ∗ n n≥1 x≥0 n is not closed in the sense of (1.5) under the action of the associated extension D . It is a necessary ∗ but not sufficient condition for a counterexample to that more primary question. And, indeed, it is easily seen that this cannot be violated by a bilinear map involving a single mode. 3.2. Counterexample(ii). Finally,wegivea(vectorial)counterexampleshowingthatDbounded from H×H → H does not imply that H is closed under the action of D . Similar to the previous ∗ ∗ counterexample wetake H = ℓ2 = ∞ R2j. Next, weintroducethematrix T := M (π) ∈R2j×2j, j=1 j j 4 j ≥ 1, where M (θ) is defined recuLrsively by (3.1) and (3.3). Let w = (w ) ∈ R2j be the j j j,k 1≤k≤2j vector defined by w = 2−j/2, 1 ≤ k ≤ 2j, j ≥ 1. A direct computation shows that j,k (3.9) kw k = 1 for any j ≥ 1. j R2j Moreover, from Lemma 3.1 we have that (3.10) kT vk = kvk for any v ∈ R2j, j ≥ 1. j R2j R2j We define D :R2j ×R2j → R2j be the bilinear form defined by D (u,v) = hu,w iT v. From (3.9) j j j j and (3.10) we obtain that (3.11) kD (u,v)k = |hu,w i|kT vk ≤ kuk kw k kvk = kuk kvk j R2j j j R2j R2j j R2j R2j R2j R2j foranyu,v ∈R2j andj ≥ 1. ItfollowsthatthebilinearmapD : ∞ R2j× ∞ R2j → ∞ R2j j=1 j=1 j=1 defined by L L L ∞ (3.12) D u,v = D (u ,v ) for u= (u ) ,v = (v ) ∈ R2j j j j j≥1 j j≥1 j j≥1 (cid:0) (cid:1) (cid:0) (cid:1) Mj=1 is well-defined and bounded. Indeed, from (3.11) and the Cauchy-Schwartz inequality we can immediately infer that kD(u,v)k ≤ kuk kvk , for any u,v ∈ ℓ2 = ∞ R2j. ℓ2 ℓ2 ℓ2 j=1 Below we use Proposition 1.1 to prove H = {f = (f ) : supL|f (x)| ∈ ℓ2} is not closed ∗ n n≥1 x≥0 n in the sense of (1.5) under the action of the associated extension D . We introduce the vectors ∗ α = (α ) ∈ R2j, j ≥ 1 by α = 1 . Thus, kα k = 1 for any j ≥ 1, which implies j j,k 1≤k≤2j j,k j2j/2 j R2j j that α = (α ) ∈ ℓ2 = ∞ R2j and kαk2 = ∞ kα k2 = ∞ 1 < ∞. In the case at j j≥1 j=1 ℓ2 j=1 j R2j j=1 j2 hand we have that L P P ∞ 1 (3.13) E = v = (v ) ∈ R2j :v = (v ) ∈ R2j, |v | ≤ , 1≤ k ≤ 2j,j ≥ 1 . α n j j≥1 Mj=1 j j,k 1≤k≤2j j,k j2j/2 o Next, we denote by D : R2j ×R2j → R, 1 ≤ k ≤ 2j, j ≥ 1 the components of the bilinear map j,k D , j ≥ 1. A crucial observation is that all the entries of T on the first row are equal to 2−j/2 for j j any j ≥ 1. It follows that 2j 2 (3.14) D (v,v) = 2−j v for any v ∈E . j,1 j,k α (cid:16)Xk=1 (cid:17) Similarly, for any k ∈ {2,...,2j} the k-th row of T consists of 2j−1 entries equal to 2−j/2 and j 2j−1 entries equal to −2−j/2. Thus, for each k ∈ {2,...,2j} there exits σ a permutation of the set k {1,...,2j} such that 2j−1 2j 2 2 (3.15) D (v,v) = 2−j v −2−j v for any v ∈ E . j,k j,σk(m) j,σk(m) α (cid:16)mX=1 (cid:17) (cid:16)m=X2j−1+1 (cid:17) 9 From (3.14) and (3.15) we obtain that 1 1 (3.16) sup |D (v,v)| = , sup |D (v,v)| = for any 2 ≤ k ≤ 2j, j ≥ 1. j,1 j2 j,k 4j2 v∈Eα v∈Eα Denoting by S (α) = (sup |D (v,v)|) ∈ R2j, we have that j v∈Eα j,k 1≤k≤2j 1 2j −1 2j (3.17) kS (α)k2 = + ≥ for any j ≥ 1, j R2j j4 16j4 16j4 which implies that S(α) = (S (α)) 6∈ ∞ R2j = ℓ2. From Proposition 1.1 we infer that H j j≥1 j=1 ∗ is not closed under the action of D∗. L 4. Discussion and open problems There are two important differences between our analysis here in Section 2.2 and the H1-stable manifold construction of [8]. First, at linear level, trajectories T (·)h ∈ H for any h ∈ H . s ∗,β s Therefore the condition u(0) ∈ dom(Γ−1/2) is not needed for membership in the stable subspace. Second, at nonlinear level, we may express the stable manifold simply as a graph over the stable manifold, requiring only Π u ∈ H , whereas in [8] we required the more complicated implicit s ∗ condition Π (u+D(u,u)) ∈ dom(Γ−1/2) ⇔ Π u′ ∈dom(Γ−1/2). s s This greatly simplifies the argument at the same time that it extends the results. Ontheotherhand,theanalysisof[8]appliedtotheimportantcaseofthesteadyBoltzmannequa- tion with hard-sphere collision kernel, the main example and motivation for our investigations. To the contrary, our Hilbert–Schmidt condition derived here does not hold for Boltzmann’s equation, and it is not at all clear how one would check that absolute boundednesscondition on the kernel. It might bethatone could show boundednessof D directly for Boltzmann’s equation, however, using ∗ the explicit structure of the collision operator (for example, the linearized collision operator may beexpressed [1, 10] as the sum of a positive real-valued multiplication operator boundedabove and below, and a compact operator Kˇ that is readily seen to satisfy the Hilbert-Schmidt condition). This would be a very interesting open problem to resolve. Whether or not one can verify the bounded-D condition, answerering the question of existence ∗ ofa“full”stablemanifold,thereremainsthesecondquestionwhethersolutionssmallinL∞(R ,H) + necessarilydecayexponentially. Asnotedin[8],averyinterestingobservationduetoFedjaNazarov [4] based on the indefinite Lyapunov functional relation hu,Γui′ = −kuk2 +o(kuk2) yields the L2- H H exponential decay result eβ|·|ku(·)k ∈ L2(R ) for some β > 0, hence (by interpolation) in any + Lp(R ), 2 ≤ p < ∞. However, it is not clear what happens in the critical norm p = ∞. This, too, + would be very interesting to resolve, either exhibiting a counterexample or proving decay. Another approach to construction of a “full” stable manifold for Boltzmann’s equation, as dis- cussed, e.g., in [7, 9], is to work in an appropriately weighted L∞(R ,B), whereB is some weighted + L∞ space in ξ (here ξ denotes the independent variable of velocity, as standard for Boltzmann’s equation [7, 8]), for which boundedness of D would follow immediately from boundedness of D, ∗ as would exponential decay of solutions merely small in L∞(R ), answering both questions posed + here. However, to date, it has not been shown that D is bounded in this setting, and it is not clear whether or not this is true; see [9]. This is another open problem that would be very interesting to resolve. 10

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