Factorization of Non-Symmetric Operators and Exponential -Theorem
We present an abstract method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker-Planck equation in the torus, and to the linearized Boltzmann equation in the torus.
We finally use this information on the linearized Boltzmann semigroup to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in , , with sharp rate of decay in time.
As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in  where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the -theorem.
Preliminary version of July 5, 2019
Mathematics Subject Classification (2000): 47D06 One-parameter semigroups and linear evolution equations [See also 34G10, 34K30], 35P15 Estimation of eigenvalues, upper and lower bounds, 47H20 Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07], 35Q84 Fokker-Planck equations, 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05].
Keywords: spectral gap; semigroup; spectral mapping theorem; quantitative; Plancherel theorem; coercivity; hypocoercivity; dissipativity; hypodissipativity; Fokker-Planck equation; Boltzmann equation; -theorem; exponential rate; stretched exponential weight; Povzner estimate; averaging lemma; thermalization; entropy.
- 1 Introduction
- 2 Factorization and quantitative spectral mapping theorems
- 3 The Fokker-Planck equation
4 The linearized Boltzmann equation
- 4.1 Review of the decay results on the semigroup
- 4.2 The main hypodissipativity results
- 4.3 Strategy of the proof
- 4.4 Integral estimates with polynomial weight on the remainder
- 4.5 Pointwise estimates on the remainder
- 4.6 Dissipativity estimate on the coercive part
- 4.7 Regularization estimates in the velocity variable
- 4.8 Iterated averaging lemma
- 4.9 Proof of the main hypodissipativity result
- 4.10 Structure of singularities for the linearized flow
- 5 The nonlinear Boltzmann equation
1.1. The problem at hand
This paper deals with (1) the study of resolvent estimates and decay properties for a class of linear operators and semigroups, and (2) the study of relaxation to equilibrium for some kinetic evolution equations, which makes use of the previous abstract tools.
Let us give a brief sketch of the first problem. Consider two Banach spaces , and two unbounded closed linear operators and respectively on and with spectrum . They generate two -semigroups and respectively in and . Further assume that , and is dense in . The theoretical question we address in this work is the following:
Can one deduce quantitative informations on and in terms of informations on and ?
We provide here an answer for a class of operators which split as , where the spectrum of is well localized and the iterated convolution maps to with proper time-decay control for some . We then prove that (1) inherits most of the spectral gap properties of , (2) explicit estimates on the rate of decay of the semigroup can be computed from the ones on . The core of the proposed method is a quantitative and robust factorization argument on the resolvents and semigroups, reminiscent of the Dyson series.
In a second part of this paper, we then show that the kinetic Fokker-Planck operator and the linearized Boltzmann operator for hard sphere interactions satisfy the above abstract assumptions, and we thus extend their spectral-gap properties from the linearization space (a space with Gaussian weight prescribed by the equilibrium) to larger Banach spaces (for example with polynomial decay). It is worth mentioning that the proposed method provides optimal rate of decay and there is no loss of accuracy in the extension process from to (as would be the case in, say, interpolation approaches).
Proving the abstract assumption requires significant technical efforts for the Boltzmann equation and leads to the introduction of new tools: some specific estimates on the collision operator, some iterated averaging lemma and a nonlinear non-symmetric energy method. As a conclusion we obtain a set of new stability results for the Boltzmann equation for hard spheres interactions in the torus as discussed in the next section.
The motivation for the abstract part of this paper, i.e. enlarging the functional space where spectral properties are known to hold for a linear operator, comes from nonlinear PDE analysis.
The first motivation is when the linearized stability theory of a nonlinear PDE is not compatible with the nonlinear theory. More precisely the natural function space where the linearized equation is well-posed and stable, with nice symmetric or skew-symmetric properties for instance, is “too small” for the nonlinear PDE in the sense that no well-posedness theorem is known even locally in time (or even conjectured to be false) in such a small space. This is the case for the classical Boltzmann equation and therefore it is a key obstacle in obtaining perturbative result in natural physical spaces and connecting the nonlinear results to the perturbative theory.
This is related to the famous -theorem of Boltzmann. The natural question of understanding mathematically the -theorem was emphasized by Truesdell and Muncaster [91, pp 560-561] thirty years ago: “Much effort has been spent toward proof that place-dependent solutions exist for all time. […] The main problem is really to discover and specify the circumstances that give rise to solutions which persist forever. Only after having done that can we expect to construct proofs that such solutions exist, are unique, and are regular.”
The precise issue of the rate of convergence in the -theorem was then put forward by Cercignani  (see also ) when he conjectured a linear relationship between the entropy production functional and the relative entropy functional, in the spatially homogeneous case. While this conjecture has been shown to be false in general , it gave a formidable impulse to the works on the Boltzmann equation in the last two decades [28, 27, 89, 17, 95]. It has been shown to be almost true in , in the sense that polynomial inequalities relating the relative entropy and the entropy production hold for powers close to , and it was an important inspiration for the work  in the spatially inhomogeneous case.
However, due to the fact that Cercignani’s conjecture is false for physical models , these important progresses in the far from equilibrium regime were unable to answer the natural conjecture about the correct timescale in the -theorem, and to prove the exponential decay in time of the relative entropy. Proving this exponential rate of relaxation was thus pointed out as a key open problem in the lecture notes [86, Subsection 1.8, page 62]. This has motivated the work  which answers this question, but only in the spatially homogeneous case.
In the present paper we answer this question for the full Boltzmann equation for hard spheres in the torus. We work in the same setting as in , that is under some a priori regularity assumptions (Sobolev norms and polynomial moments bounds). We are able to connect the nonlinear theory in  with the perturbative stability theory first discovered in  and then revisited with quantitative energy estimates in several works including  and . This connexion relies on the development of a perturbative stability theory in natural physical spaces thanks to the abstract extension method. Let us mention here the important papers [8, 9, 99, 100] which proved for instance nonlinear stability in spaces of the form with and large enough, by non-constructive methods.
We emphasize the dramatic gap between the spatially homogeneous situation and the spatially inhomogeneous one. In the first case the linearized equation is coercive and the linearized semigroup is self-adjoint or sectorial, whereas in the second case the equation is hypocoercive and the linearized semigroup is neither sectorial, nor even hypoelliptic.
The second main motivation for the abstract method developed here is considered in other papers [67, 10]. It concerns the existence, uniqueness and stability of stationary solutions for degenerate perturbations of a known reference equation, when the perturbation makes the steady solutions leave the natural linearization space of the reference equation. Further works concerning spatially inhomogeneous granular gases are in progress.
1.3. Main results
We can summarize the main results established in this paper as follows:
Section 2. We prove an abstract theory for enlarging (Theorem 2.1) the space where the spectral gap and the discrete part of the spectrum is known for a certain class of unbounded closed operators. We then prove a corresponding abstract theory for enlarging (Theorem 2.13) the space where explicit decay semigroup estimates are known, for this class of operators. This can also be seen as a theory for obtaining quantitative spectral mapping theorems in this setting, and it works in Banach spaces.
Section 3. We prove a set of results concerning Fokker-Planck equations. The main outcome is the proof of an explicit spectral gap estimate on the semigroup in , as small as wanted, for the kinetic Fokker-Planck equation in the torus with super-harmonic potential (see Theorems 3.1 and 3.12).
Section 4. We prove a set of results concerning the linearized Boltzmann equation. The main outcome is the proof of explicit spectral gap estimates on the linearized semigroup in and with polynomial moments (see Theorem 4.2). More generally we prove explicit spectral gap estimates in any space of the form , , with polynomial or stretched exponential weight , including the borderline cases and . We also make use of the factorization method in order to study the structure of singularities of the linearized flow (see Subsection 4.10).
Section 5. We finally prove a set of results concerning the nonlinear Boltzmann equation in perturbative setting. The main outcomes of this section are: (1) The construction of perturbative solutions close to the equilibrium or close to the spatially homogeneous case in , with polynomial or stretched exponential weight , including the borderline cases and without assumption on the derivatives: see Theorem 5.3 in a close-to-equilibrium setting, and Theorem 5.5 in a close-to-spatially-homogeneous setting. (2) We give a proof of the exponential -theorem: we show exponential decay in time of the relative entropy of solutions to the fully nonlinear Boltzmann equation, conditionnally to some regularity and moment bounds. Such rate is proven to be sharp. This answers the conjecture in [32, 86] (see Theorem 5.7). We also finally apply the factorization method and the Duhamel principle to study the structure of singularities of the nonlinear flow in perturbative regime (see Subsection 5.7).
Below we give a precise statement of what seems to us the main result established in this paper.
The Boltzmann equation
with hard spheres collision kernel and periodic boundary conditions is globally well-posed for non-negative initial data close enough to the Maxwellian equilibrium or to a spatially homogeneous profile in , .
The corresponding solutions decay exponentially fast in time with constructive estimates and with the same rate as the linearized flow in the space . For large enough (with explicit threshold) this rate is the sharp rate given by the spectral gap of the linearized flow in .
Moreover any solution that is a priori bounded uniformly in time in with some large satisfies the exponential decay in time with sharp rate in norm, as well as in relative entropy.
We thank Claude Bardos, José Alfrédo Cañizo, Miguel Escobedo, Bertrand Lods, Mustapha Mokhtar-Kharroubi, Robert Strain for fruitful comments and discussions. The third author also wishes to thank Thierry Gallay for numerous stimulating discussions about the spectral theory of non-self-adjoint operators, and also for pointing out the recent preprint . The authors wish to thank the funding of the ANR project MADCOF for the visit of MPG in Université Paris-Dauphine in spring 2009 where this work was started. The third author’s work is supported by the ERC starting grant MATKIT. The first author is supported by NSF-DMS 1109682. Support from IPAM (University of California Los Angeles) and ICES (The University of Texas at Austin) is also gratefully acknowledged.
2. Factorization and quantitative spectral mapping theorems
2.1. Notation and definitions
For a given real number , we define the half complex plane
For some given Banach spaces and we denote by the space of bounded linear operators from to and we denote by or the associated norm operator. We write when . We denote by the space of closed unbounded linear operators from to with dense domain, and in the case .
For a Banach space and we denote by , , its semigroup, by its domain, by its null space and by its range. We also denote by its spectrum, so that for any belonging to the resolvent set the operator is invertible and the resolvent operator
is well-defined, belongs to and has range equal to . We recall that is said to be an eigenvalue if . Moreover an eigenvalue is said to be isolated if
In the case when is an isolated eigenvalue we may define the associated spectral projector by
with . Note that this definition is independent of the value of as the application , is holomorphic. For any isolated, it is well-known (see [59, III-(6.19)]) that , so that is indeed a projector, and that the associated projected semigroup
When moreover the algebraic eigenspace is finite dimensional we say that is a discrete eigenvalue, written as . In that case, is a meromorphic function on a neighborhood of , with non-removable finite-order pole , and there exists such that
On the other hand, for any we may also define the “classical algebraic eigenspace”
We have then if is an eigenvalue and if is an isolated eigenvalue.
Finally for any such that
where are distinct discrete eigenvalues, we define without any risk of ambiguity
2.2. Factorization and spectral analysis
The main abstract factorization and enlargement result is:
Theorem 2.1 (Enlargement of the functional space).
Consider two Banach spaces and such that with continuous embedding and is dense in . Consider an operator such that . Finally consider a set as defined above.
Localization of the spectrum in . There are some distinct complex numbers , (with the convention if ) such that
Decomposition. There exist some operators defined on such that and
is such that is bounded in uniformly on and as , in particular
is a bounded operator on ;
There is such that the operator is bounded in uniformly on .
Then we have in :
The spectrum satisfies: .
For any the resolvent satisfies:
For any , , we have
and at the level of the spectral projectors
In words, assumption (H1) is a weak formulation of a spectral gap in the initial functional space . The assumption (H2) is better understood in the simplest case , where it means that one may decompose into a regularizing part (in the generalized sense of the “change of space” ) and another part whose spectrum is “well localized” in : for instance when is dissipative with then the assumption (H2)-(i) is satisfied.
There are many variants of sets of hypothesis for the decomposition assumption. In particular, assumptions (H2)-(i) and (H2)-(iii) could be weakened. However, (1) these assumptions are always fulfilled by the operators we have in mind, (2) when we weaken (H2)-(i) and/or (H2)-(iii) we have to compensate them by making other structure assumptions. We present below after the proof a possible variant of Theorem 2.1.
One may relax (H2)-(i) into and the bound in (H2)-(iii) could be asked merely locally uniformly in .
One may replace by any nonempty open connected set .
This theorem and the next ones in this section can also be extended to the case where is not necessarily included in . This will be studied and applied to some PDE problems in future works.
Proof of Theorem 2.1.
Let us denote and let us define for
Observe that thanks to the assumption (H2), the operator is well-defined and bounded on .
Step 1. is a right-inverse of on . For any , we compute
Step 2. is invertible on . First we observe that there exists such that is invertible in . Indeed, we write
with for , large enough, thanks to assumption (H2)-(i). As a consequence is invertible and so is as the product of two invertible operators.
Since we assume that is invertible in for some , we have . And if
for some , then is invertible on the disc with
and then again, arguing as before, on since is a left-inverse of for any . Then in order to prove that is invertible for any , we argue as follows. For a given we consider a continuous path from to included in , i.e. a continuous function such that , . Because of assumption (H2) we know that , , and are locally uniformly bounded in on , which implies
Since is invertible we deduce that is invertible with locally bounded around with a bound which is uniform along (and a similar series expansion as in (2.4)). By a continuation argument we hence obtain that is invertible in all along the path with
Hence we conclude that is invertible with .
This completes the proof of this step and proves together with the point (ii) of the conclusion.
Step 3. Spectrum, eigenspaces and spectral projectors. On the one hand, we have
so that . The other inclusion was proved in the previous step, so that these two sets are equals. We have proved
Now, we consider a given eigenvalue of in . We know (see [59, paragraph I.3]) that in the following Laurent series holds
for close to and for some bounded operators , . The operators satisfy the range inclusions
This Laurent series is convergent on . The Cauchy formula for meromorphic functions applied to the circle with small enough thus implies that
since is a discrete eigenvalue.
Using the definition of the spectral projection operator (2.1), the above expansions and the Cauchy theorem we get for any small
where the first integral has range included in and the second integral vanishes in the limit . We deduce that
we conclude that and for any and .
Finally, the proof of is straightfoward from the equality
and the integral formula (2.1) defining the projection operator. ∎
Let us shortly present a variant of the latter result where the assumption (H2) is replaced by a more algebraic one. The proof is then purely based on the factorization method and somehow simpler. The drawback is that it requires some additional assumption on at the level of the small space (which however is not so restrictive for a PDE’s application perspective but can be painful to check).
Theorem 2.3 (Enlargement of the functional space, purely algebraic version).
Consider the same setting as in Theorem 2.1, assumption (H1), and where assumption (H2) is replaced by
Decomposition. There exist operators on such that (with corresponding extensions on ) and
and are closed unbounded operators on and (with domain containing and ) and
is a bounded operator on .
There is such that the operator is bounded from to for any .
Then the same conclusions as in Theorem 2.1 hold.
Actually there is no need in the proof that for is a bounded operator, and therefore assumption (H2’) could be further relaxed to assuming only (bijectivity is already known in from the invertibility of ). However these subtleties are not used at the level of the applications we have in mind.
Proof of Theorem 2.3.
The Step 1 is unchanged, only the proofs of Steps 2 and 3 are modified:
Step 2. is invertible on . Consider . First observe that if the operator is bijective, then composing to the left the equation
by yields and we deduce that the inverse map is bounded (i.e. is an invertible operator in ) together with the desired formula for the resolvent. Since has a right-inverse it is surjective.
Let us prove that it is injective. Consider :
We denote and obtain
and therefore, from assumption (H2’), we deduce that . Finally . Since is injective we conclude that .
This completes the proof of this step and proves together with the point (ii) of the conclusion.
Step 3. Spectrum, eigenspaces and spectral projectors. On the one hand,
so that . Since the other inclusion was proved in the previous step, we conclude that
On the other hand, let us consider an eigenvalue , for , some integer and some :
Using the decomposition of (H2) and denoting we deduce
By expanding this identity we obtain
where is a finite sum of powers of (with terms and exponents between and ). By iterating this equality ( and commute), we get
This implies, arguing as in the previous step, that and finally . This proves that
and since the eigenvalues are discrete, it straightforwardly completes the proof of the conclusions (i) and (ii). Finally, the fact that is a straightforward consequence of when and of the formula (2.1) for the projector operator. ∎
Let us first introduce the notion of hypodissipative operators and discuss its relation with the classical notions of dissipative operators and coercive operators as well as its relation with the recent terminology of hypocoercive operators (see mainly  and then [77, 53, 35] for related references).
Definition 2.5 (Hypodissipativity).
Consider a Banach space and some operator . We say that is hypodissipative on if there exists some norm on equivalent to the initial norm such that
where is the duality bracket for the duality in and and is the dual set of defined by
An hypodissipative operator such that in the above definition is nothing but a dissipative operator, or in other words, is an accretive operator.
When is an Hilbert norm on , we have and (2.5) writes
where is the scalar product associated to . In this Hilbert setting such a hypodissipative operator shall be called equivalently hypocoercive.
When is an Hilbert norm on , the above definition corresponds to the classical definition of a coercive operator.
In other words, in a Banach space (resp. an Hilbert space) , an operator is hypodissipative (resp. hypocoercive) on if is dissipative (resp. coercive) on endowed with a norm (resp. an Hilbert norm) equivalent to the initial one. Therefore the notions of hypodissipativity and hypocoercivity are invariant under change of equivalent norm.
The concept of hypodissipativity seems to us interesting since it clarifies the terminology and draws a bridge between works in the PDE community, in the semigroup community and in the spectral analysis community. For convenience such links are summarized in the theorem below. This theorem is a non standard formulation of the classical Hille-Yosida theorem on -dissipative operators and semigroups, and therefore we omit the proof.
Consider a Banach space and the generator of a -semigroup . We denote by its resolvent. For given constants , the following assertions are equivalent:
the semigroup satisfies the growth estimate
and there exists some norm on equivalent to the norm :
We recall that is maximal if
This further condition leads to the notion of -hypodissipative, -dissipative, -hypocoercive, -coercive operators.
The Hille-Yosida theorem is classically presented as the necessary and sufficient conditions for an operator to be the generator of a semigroup. Then one assumes, additionally to the above conditions, that is maximal for some given . Here in our statement, the existence of the semigroup being assumed, the maximality condition is automatic, and Theorem 2.7 details how the operator’s, resolvent’s and the associated semigroup’s estimates are linked.
In other words, the notion of hypodissipativity is just another formulation of the minimal assumption for estimating the growth of a semigroup. Its advantage is that it is arguably more natural from a PDE viewpoint.
Let us now give a synthetic statement adapted to our purpose. We omit the proof which is a straightforward consequence of the Lumer-Philipps or Hille-Yosida theorems together with basic matrix linear algebra on the finite-dimensional eigenspaces. The classical reference for this topic is .
Consider a Banach space , a generator of a -semigroup , and distinct , . The following assertions are equivalent:
There exist linearly independent vectors so that the subspace is invariant under the action of , and
Moreover there exist linearly independent vectors so that the subspace is invariant under the action of . These two families satisfy the orthogonality conditions and the operator is hypodissipative on :
There exists a decomposition where (1) and are invariant by the action of , (2) for any is a finite-dimensional space included in , and (3) is hypodissipative on :
There exist some finite-dimensional projection operators which commute with and such that if , and some operators with , nilpotent, so that the following estimate holds
for some constant .
The spectrum of satisfies
and is hypodissipative on .
Moreover, if one (and then all) of these assertions is true, we have
As a consequence, we may write
where is holomorphic and bounded on for any and
When is a Hilbert space and is a self-adjoint operator, the assumption (i) is satisfied with , , as soon as there exist normalized such that if , for all , and
2.4. Factorization and quantitative spectral mapping theorems
The goal of this subsection is to establish quantitative decay estimates on the semigroup in the larger space . Let us r