On Measure Solutions of the Boltzmann Equation part II…

On Measure Solutions of the Boltzmann Equation part II: Rate of Convergence to Equilibrium

Xuguang Lu and Clément Mouhot
Abstract.

The paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. We prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy. The proof is based on the regularizing property of the iterated collision operators, exponential moment production estimates, and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data. We also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the Maxwellian equilibrium. The constants in these convergence rates depend only on the collision kernel and conserved quantities (mass, momentum, and energy). We finally use these convergence rates in order to deduce global-in-time strong stability of measure solutions.


Mathematics Subject Classification (2000): 35Q Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05], 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05].


Keywords: Boltzmann equation; spatially homogeneous; hard potentials; measure solutions; equilibrium; exponential rate of convergence; eternal solution; global-in-time stability.

1. Introduction

The Boltzmann equation describes evolution of a dilute gas. Investigations of the spatially homogeneous Boltzmann equation have made a lot of progresses in the last decades and it is hoped to provide useful clues for the understanding of the complete (spatially inhomogeneous) Boltzmann equation. The complete equation is more realistic and interesting to physics and mathematics but remains still largely out of reach mathematically and will most likely need long term preparations and efforts. For review and references of these areas, the reader may consult for instance [30, 13, 22].

The present paper is a follow-up to our previous work [22] on measure-valued solutions111As in our previous work [22], the “measure-valued solutions” will be also called “measure solutions”. to the spatially homogeneous Boltzmann equation for hard potentials. In this second part, we prove that, under some angular cutoff assumptions (which include the hard sphere model), solutions with measure-valued initial data having finite mass and energy converge strongly to equilibrium in the exponential rate , where is the spectral gap of the corresponding linearized collision operator. This sharp exponential rate was first proved in [25] for initial data with bounded energy, and belonging to (for the hard sphere model) or to (for all hard potentials with cutoff). The core idea underlying our improvement of this result to measure solutions is that instead of considering a one-step iteration of the collision integral which produces the integrability for the hard sphere model (as first observed by Abrahamsson [1], elaborating upon an idea in [23]), we consider a multi-steps iteration which produces the integrability for all hard potentials with angular cutoff. This, together with approximation by solutions through the Mehler transform, and the property of the exponential moment production, enables us to apply the results of [25] and obtain the same convergence rate for measure solutions. We also obtain a lower bound of the convergence rate and establish the global in time strong stability estimate. As a consequence we prove that, for any hard potentials with cutoff, there are no eternal measure solutions with finite and non-zero temperature, apart from the Maxwellians.

1.1. The spatially homogeneous Boltzmann equation

The spatially homogeneous Boltzmann equation takes the form

(1.1)

with some given initial data , where is the collision integral defined by

(1.2)

In the latter expression, and stand for velocities of two particles after and before their collision, and the microscopic conservation laws of an elastic collision

(1.3)

induce the following relations:

(1.4)

for some unit vector .

The collision kernel under consideration is assumed to have the following product form

(1.5)

where is a nonnegative Borel function on . This corresponds to the so-called inverse power-law interaction potentials between particles, and the condition corresponds to the so-called hard potentials. Throughout this paper we assume that the function satisfies Grad’s angular cutoff:

(1.6)

and it is always assumed that , where denotes the Lebesgue measure of the -dimensional sphere (recall that in the case we have and ). This enables us to split the collision integral as

with the two bilinear operators

(1.7)
(1.8)

which are nonnegative when applied to nonnegative functions.

The bilinear operators are bounded from to for , where is a subspace of defined by

(1.9)

where we have used the standard notation

Since in the equation (1.1), , by replacing

one can assume without loss of generality that the function is even: for all . This in turn implies that the polar form of satisfies

(1.10)

1.2. The definition of the solutions

The equation (1.1) is usually solved as an integral equation as follows. Given any , we say that a nonnegative Lebesgue measurable function on is a mild solution to (1.1) if for every , belongs to , , and there is a Lebesgue null set (which is independent of ) such that

(1.11)

The bilinear operators can now be extended to measures. For every , let with the norm be the Banach space of real Borel measures on defined by

(1.12)

where the positive Borel measure is the total variation of . This norm can also be defined by duality:

(1.13)

The latter form is convenient when dealing with the difference of two positive measures. The norms and are related by

(1.14)

For any (), we define the Borel measures and

through Riesz’s representation theorem by

(1.15)
(1.16)

for all bounded Borel functions , where

(1.17)

and in case we define to be a fixed unit vector . It is easily shown (see Proposition 2.3 of [22]) that the extended bilinear operators are also bounded from to for : if then and

(1.18)
(1.19)

Let us finally define the cone of positive distributions with moments bounded:

We can now define the notion of solutions that we shall use in this paper. We note that the condition as assumed in the following definition is mainly used for ensuring the existence of solutions.

Definition 1.1 (Measure strong solutions).

Let be given by (1.5) with and with satisfysing the condition (1.6). Let . We say that , or simply , is a measure strong solution of equation (1.1) if it satisfies the following:

  • and

    (1.20)

Furthermore is called a conservative solution if conserves the mass, momentum and energy, i.e.

Observe that (1.18) and (1.19) imply the strong continuity of and therefore the strong continuity of . Hence the differential equation (1.20) is equivalent to the integral equation

(1.21)

where the integral is taken in the sense of the Riemann integration or more generally in the sense of the Bochner integration. Recall also that here the derivative and integral are defined by

for all Borel sets .

1.3. Recall of the main results of the first part

The following results concerning moment production and uniqueness of conservative solutions which will be used in the present paper are extracted from our previous paper [22]. The following properties (a) and (b) are a kind of “gain of decay” property of the flow stating and quantifying how moments of the solutions become bounded for any positive time even they are not bounded at initial time; the following properties (c)-(d)-(e) concern the stability of the flow.

Theorem 1.2 ([22]).

Let be defined in (1.5) with and with the condition (1.6). Then for any with , there exists a unique conservative measure strong solution of equation (1.1) satisfying . Moreover this solution satisfies:

  • satisfies the moment production estimate:

    (1.22)

    where ,

    (1.23)
    (1.24)
  • If or if

    (1.25)

    then satisfies the exponential moment production estimate:

    (1.26)

    where

    (1.27)

    and depends only on the function and .

  • Let be a conservative measure strong solutions of equation (1.1) on the time-interval with an initial datum for some . Then:

    • If , then

      (1.28)

      where

      (1.29)

      and is an explicit positive continuous function on .

    • If , then

      (1.30)

      where

      and is defined by (1.23) with .

  • If is absolutely continuous with respect to the Lebesgue measure, i.e.

    then is also absolutely continuous with respect to the Lebesgue measure: for all , and is the unique conservative mild solution of equation (1.1) with the initial datum .

  • If is not a single Dirac distribution, then there is a sequence , , of conservative mild solutions of equation (1.1) with initial data satisfying

    (1.31)

    such that

    (1.32)

    Besides, the initial data can be chosen of the form where is a subsequence of the Mehler transforms of .

Remarks 1.3.
  1. In the physical case, and , the moment estimates (1.22) and (1.26) also hold for conservative weak measure solutions of equation (1.1) without angular cutoff (see [22]).

  2. The Mehler transform

    of a measure (which is not a single Dirac distribution) will be studied in Section 4 (after introducing other notations) where we shall show that has a further convenient property:

    and thus it is a useful tool in order to reduce the study of properties of measure solutions to that of solutions. Here is the Maxwellian (equilibrium) having the same mass, momentum, and energy as , see (1.42)-(1.43) below.

1.4. Normalization

In most of the estimates in this paper, we shall try as much as possible to make explicit the dependence on the basic constants in the assumptions. But first let us study the reduction that can be obtained by scaling arguments.

Under the assumption (1.6), it is easily seen that is a measure solution of equation (1.1) with the angular function if and only if is a measure solution of equation (1.1) with the scaled angular function . Therefore without loss of generality we can assume the normalization

(1.33)

Next given any , and , we define the bounded positive linear operator on as follows: for any , there is a unique such that (thanks to Riesz representation theorem),

We call the normalization operator associated with . The inverse of is given by , i.e.

It is easily seen that for every

(1.34)
(1.35)
(1.36)

where

(1.37)
(1.38)

We then introduce the subclass of by

(1.39)

In other words, means that has the mass , mean-velocity , and the kinetic temperature . It is obvious that conserves mass, momentum, and energy is equivalent to that conserves mass, mean-velocity , and kinetic temperature.

When restricting on , it is easily seen that

Similarly we define by

(1.40)

In this case, the normalization operator is written directly as

(1.41)

Recall that the Maxwellian is given by

(1.42)

For notational convenience we shall do not distinguish between a Maxwellian distribution and its density function : we write without risk of confusion that

(1.43)

Due to the homogeneity of , we have

and then by Fubini theorem we get (denoting simply when no ambiguity is possible)

Since is linear and bounded, this implies that if is a measure strong solution of equation (1.1) and , then

This together with (1.34)-(1.36) leads to the following statement:

Proposition 1.4 (Normalization).

Let be defined by (1.5) with and with the condition (1.33). Let with , and , and let be the unique conservative measure strong solution of equation (1.1) with the initial datum . Let be the Maxwellian defined by (1.42), let be the normalization operator, and let . Then:

  • The normalization is the unique conservative measure strong solution of equation (1.1) with the initial datum .

  • For all

    where and are given in (1.37)-(1.38).

1.5. Linearized collision operator and spectral gap

For any nonnegative Borel function on we define the weighted Lebesgue space with by

Let as defined in (1.5) with and with satisfying (1.33). Let be the Maxwellian with mass , mean velocity and temperature defined in (1.42), and let

be the linearized collision operator associated with and , i.e.

(1.44)

It is well-known that the spectrum of is contained in and has a positive spectral gap , i.e.

Moreover by simple calculations, one has the following scaling property on this spectral gap

In the spatially homogeneous case, the study of the linearized collision operator goes back to Hilbert [17, 18] who computed the collisional invariant, the linearized operator and its kernel in the hard spheres case, and showed the boundedness and “complete continuity” of its non-local part. Carleman [6] then proved the existence of a spectral gap by using Weyl’s theorem and the compactness of the non-local part proved by Hilbert. Grad [14, 15] then extended these results to the case of hard potentials with cutoff. All these results are based on non-constructive arguments. The first constructive estimates in the hard spheres case were obtained only recently in [2] (see also [24] for more general interactions, and [26] for a review). Let us also mention the works [32, 3, 4] for the different setting of Maxwell molecules where the eigenbasis and eigenvalues can be explicitly computed by Fourier transform methods. Although these techniques do not apply here, the explicit formula computed are an important source of inspiration for dealing with more general physical models.

1.6. Main results

In order to use the results obtained in [25] (see also [31, 27]) for solutions, we shall need the following additional assumptions for some of our main results:

(1.45)
(1.46)

Recall that for the hard sphere model, i.e. and , the conditions (1.45)-(1.46) are satisfied.

The first main result of this paper is concerned with the upper bound of the rate of convergence to equilibrium when the dimension is greater or equal to .

Theorem 1.5 (Sharp exponential relaxation rate).

Suppose and let be given by (1.5) with and with satisfying (1.33), (1.45), and (1.46). Let , and , and let

be the spectral gap for the linearized collision operator (1.44) associated with and the Maxwellian . Then for any conservative measure strong solution of the equation (1.1) with we have:

where

with and given in (1.35) and (1.36), and with some constant which depends only on , , and the function (through the bounds (1.45), (1.46)).

Remark 1.6.
  1. It should be noted that, in addition to the exponential rate, Theorem 1.5 also shows that for the hard potentials considered here, the convergence to equilibrium is grossly determined, i.e. the speed of the convergence only depends only on the collision kernel and the conserved macroscopic quantities (mass, momentum, energy). This is essentially different from those for non-hard potentials (i.e. ), see for instance [11].

  2. Applying Theorem 1.5 to the normal initial data and the Maxwellian , and using we have

    where depends only on , , and the function . Then by normalization (using Proposition 1.4) and the relation , we conclude that if , then for the same constant we have

    (1.47)

    This estimate will be used in proving our next results Corollary 1.9 and Theorem 1.10.

  3. In general, in this paper we say that a constant depends only on some parameters , if is an explicit continuous function of where