Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off
This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off and non mollified) hard potentials, we thus only deal with a moderate angular singularity. We prove an exponential in time convergence towards the equilibrium, improving results of Villani from  where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani et al. in . We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.
Mathematics Subject Classification (2010): 76P05 Rarefied gas flows, Boltzmann equation; 47H20 Semigroups of nonlinear operators; 35B40 Asymptotic behavior of solutions.
Keywords: Boltzmann equation without cut-off; hard potentials; spectral gap; dissipativity; exponential rate of convergence; long-time asymptotic.
1.1. The model
In the present paper, we investigate the asymptotic behavior of solutions to the spatially homogeneous Boltzmann equation without angular cut-off, that is, for long-range interactions. Previous works have shown that these solutions converge towards the Maxwellian equilibrium with a polynomial rate when time goes to infinity. Here, we are interested in improving the rate of convergence and we show an exponential decay to equilibrium.
We consider particles described by their space homogeneous distribution density . We hence study the so-called spatially homogeneous Boltzmann equation:
The Boltzmann collision operator is defined as
Here and below, we are using the shorthand notations , , and . In this expression, , and , are the velocities of a pair of particles before and after collision. We make a choice of parametrization of the set of solutions to the conservation of momentum and energy (physical law of elastic collisions):
so that the post-collisional velocities are given by:
The Boltzmann collision kernel only depends on the relative velocity and on the deviation angle through where and is the usual scalar product in . By a symmetry argument, one can always reduce to the case where is supported on i.e . So, without loss of generality, we make this assumption.
In this paper, we shall be concerned with the case when the kernel satisfies the following conditions:
it takes product form in its arguments as
the angular function is locally smooth, and has a nonintegrable singularity for , it satisfies for some and (moderate angular singularity)
the kinetic factor satisfies
this assumption could be relaxed to assuming only that satisfies for some .
Our main physical motivation comes from particles interacting according to a repulsive potential of the form
The assumptions made on throughout the paper include the case of potentials of the form (1.5) with . Indeed, for repulsive potentials of the form (1.5), the collision kernel cannot be computed explicitly but Maxwell  has shown that the collision kernel can be computed in terms of the interaction potential . More precisely, it satisfies the previous conditions (1.2), (1.3) and (1.4) in dimension (see [11, 12, 39]) with and .
One traditionally calls hard potentials the case (for which ), Maxwell molecules the case (for which ) and soft potentials the case (for which ). We can hence deduce that our assumptions made on include the case of hard potentials. The equation (1.1) preserves mass, momentum and energy. Indeed, at least formally, we have:
from which we deduce that a solution to the equation (1.1) is conservative, meaning that
We introduce the entropy and the entropy production . Boltzmann’s theorem asserts that
and states that any equilibrium (i.e any distribution which maximizes the entropy) is a Maxwellian distribution for some , and :
where , and are the mass, momentum and temperature of the gas:
Thanks to the conservation properties of the equation (1.6), the following equalities hold:
where is the initial datum.
Moreover, a solution of the Boltzmann equation is expected to converge towards the Maxwellian distribution when .
In this paper, we only consider the case of an initial datum satisfying
one can always reduce to this situation (see ). We then denote the Maxwellian with same mass, momentum and energy of : .
1.2. Function spaces and notations
We introduce some notations about weighted spaces. For some given Borel weight function on , we define the Lebesgue weighted space , , as the Lebesgue space associated to the norm
We also define the weighted Sobolev space , , , as the Sobolev space associated to the norm
Throughout this paper, we will use the same notation for positive constants that may change from line to line. Moreover, the notation will mean that there exist two constants , such that .
1.3. Main results and known results
Convergence to equilibrium
We first state our main result on exponential convergence to equilibrium.
where is defined in Theorem 1.4.
We improve a polynomial result of Villani  and generalize to our context similar exponential results known for simplified models. Mouhot in  proved such a result for the spatially homogeneous Boltzmann equation with hard potentials and Grad’s cut-off. Carrapatoso in  recently proved exponential decay to equilibrium for the homogeneous Landau equation with hard potentials which is the grazing collisions limit of the model we study in the present paper. Let us also mention the paper of Gualdani et al.  where an exponential decay to equilibrium is proved for the inhomogeneous Boltzmann equation for hard spheres (see also [27, 28, 26] for related works).
It is a known fact that our equation (1.1) admits solutions which are conservative and satisfy some suitable properties of regularity, we will call them smooth solutions. We here precise the meaning of this term and give an overview of results on the Cauchy theory of our equation.
Let be a nonnegative function defined on with finite mass, energy and entropy. We shall say that is a smooth solution to the equation (1.1) if the following conditions are fulfilled:
for any ,
where is the entropy production defined in (1.7);
for any and for any ,
where the last integral is define through the following formula
for any and for any ,
for any and for any , ,
Such a solution is known to exist. The problem of existence of solutions was first studied by Arkeryd in  where existence of solutions is proven for not too soft potentials, that is (Goudon  and Villani  then improved this result enlarging the class of considered). We mention that uniqueness of solution for hard potentials can be proved under some more restrictive conditions on the initial datum, see the paper of Desvillettes and Mouhot  where is supposed to be regular () and the paper of Fournier and Mouhot  where is supposed to be localized ( for some ) for hard potentials.
Concerning the moment production property, it was discovered by Elmroth  and Desvillettes  and improved by Wennberg , which justifies our point (1.9) in the definition of a smooth solution. We here point out the fact that this property is not anymore true for Maxwell molecules or soft potentials. As a consequence, our method, which relies partially on this property, works only for hard potentials.
Finally, we mention papers where regularization results are proven for “true” (that is non mollified) physical potentials:  by Alexandre et al. and  by Chen and He where the initial datum is supposed to have finite energy and entropy,  by Bally and Fournier where only the case is treated and  by Fournier under others conditions on the initial datum. Theorem 1.4 from  explains our point (1.10).
We now recall previous results on convergence to equilibrium for solutions to equation (1.1). It was first studied by Carlen and Carvalho [8, 9] and then by Toscani and Villani . Up to now, the best rate of convergence in our case was obtained by Villani in :
This result comes from [40, Theorem 4.1] which states that if is a function which satisfies the following lowerbound
then for any , there exists an explicit constant such that
It is a result from Mouhot [30, Theorem 1.2] that the lowerbound (1.11) holds for any smooth solution of our equation (1.1). Let us mention that lowerbounds of solutions were first studied by Carleman  (for hard spheres) and then by Pulvirenti and Wennberg  (for hard potentials with cut-off). Finally, Mouhot  extended these results to the spatially inhomogeneous case without cut-off. We here state Theorem 1.2 from  that we use: for any and for any exponent such that
a smooth solution to (1.1) satisfies
for some , .
Let us here emphasize that the method of Villani to prove the polynomial convergence towards equilibrium is purely nonlinear. Ours is based on the study of the linearized equation.
The linearized equation
We introduce the linearized operator. Considering the linearization , we obtain at first order the linearized equation around the equilibrium
for , . The null space of the operator is the -dimensional space
Our strategy is to combine the polynomial convergence to equilibrium and a spectral gap estimate on the linearized operator to show that if the solution enters some stability neighborhood of the equilibrium, then the convergence is exponential in time. Previous results on spectral gap estimates hold only in and the Cauchy theory for the nonlinear Boltzmann equation is constructed in -spaces with polynomial weight. In order to link the linear and the nonlinear theories, our approach consists in proving new spectral gap estimates for the linearized operator in spaces of type . To do that, we exhibit a convenient splitting of the linearized operator in such a way that we may use the abstract theorem from  which allows us to enlarge the space of spectral estimates of a given operator.
Here is the result we obtain on the linearized equation which provides a constructive spectral gap estimate for in and which is the cornerstone of the proof of Theorem 1.1.
Let and a collision kernel satisfying (1.2), (1.3) and (1.4). Consider the linearized Boltzmann operator defined in (1.13). Then for any positive (where is the spectral gap of in defined in Proposition 2.1 and is a constant depending on defined in Lemma 2.7), there exists an explicit constant , such that for any , we have the following estimate
where denotes the semigroup of and the projection onto .
Let us briefly review the existing results concerning spectral gap estimates for . Pao  studied spectral properties of the linearized operator for hard potentials by non-constructive and very technical means. This article was reviewed by Klaus . Then, Baranger and Mouhot gave the first explicit estimate on this spectral gap in  for hard potentials (). If we denote the Dirichlet form associated to :
and the orthogonal of defined in (1.14) and the projection onto , the Dirichlet form satisfies
for some constructive constant . This result was then improved by Mouhot  and later by Mouhot and Strain . In the last paper, it was conjectured that a spectral gap exists if and only if . This conjecture was finally proven by Gressman and Strain in .
Another question would be to obtain similar results in other spaces: spaces with and a polynomial weight or spaces with and a stretched exponential weight. Our computations do not allow to conclude in those cases, more precisely, we are not able to do the computations which allow to obtain the suitable splitting of the linear operator in order to apply the theorem of enlargement of the space of spectral estimates. As a consequence, we can not prove the existence of a spectral gap on those spaces. However, we believe that such results may hold.
We here point out that the knowledge of a spectral gap estimate in for the fractional Fokker-Planck equation (see ) is consistent with our result. Indeed, the behavior of the Boltzmann collision operator has been widely conjectured to be that of a fractional diffusion (see [14, 20, 38]).
Acknowledgments. We thank Stéphane Mischler for fruitful discussions and his encouragement.
2. The linearized equation
Here and below, we denote with . The aim of the present section is to prove Theorem 1.4. To do that, we exhibit a splitting of the linearized operator into two parts, one which is bounded and the second one which is dissipative. We can then apply the abstract theorem of enlargement of the functional space of the semigroup decay from Gualdani et al.  (see Subsection 2.4).
We now introduce notations about spectral theory of unbounded operators. 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 . 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  paragraph III-6.19) that , so that is indeed a projector.
When moreover the so-called “algebraic eigenspace” is finite dimensional we say that is a discrete eigenvalue, written as .
2.2. Spectral gap in
There is a constructive constant such that
2.3. Splitting of the linearized operator
We first split the linearized operator defined in (1.13) into two parts, separating the grazing collisions and the cut-off part, we define
for some to be chosen later, it induces the following splitting of :
In the rest of the paper, we shall use the notations
As far as the cut-off part is concerned, our strategy is similar as the one adopted in  for hard-spheres. For any , we consider bounded by one, which equals one on
and whose support is included in
We then denote the truncated operator
and the corresponding remainder operator
We also introduce
so that we have the following splitting: .
using the spherical coordinates to get the second equality and (1.3) to get the final one; and
We finally define
so that .
There exists a function depending on and tending to as tends to such that for any , the following estimate holds:
Let us first introduce a notation which is going to be useful in the sequel of the proof:
where the last equality comes from (1.3). We here underline the fact that considering a moderate singularity, meaning , is here needed to get the convergence of to as goes to .
We split into two parts in the following way:
this splitting corresponds to the splitting of as if denotes the collisional operator associated to the kernel .
We first deal with . Let us recall that we have . In the following computation, we denote :
where we used that for any , , to get the last inequality.
We here emphasize that this computation is particularly convenient in the case since . In the case, it is trickier and for now, we are not able to adapt it to get the wanted estimates.
We now use the classical pre-post collisional change of variables to pursue the computation:
We hence deduce that
We now estimate the difference :
Then, we use the fact
which implies that
for some constant depending on .
We here point out that this kind of estimate does not hold in the case of a stretched exponential weight. Indeed, taking the gradient of a stretched exponential function, there is not anymore a gain in the degree as in the case of a polynomial function.
We finally obtain
where we used spherical coordinates to obtain the second inequality and (2.2) to obtain the last one.
We now deal with . We split it into two parts: