Framework for linear kinetic equations

h P H¹|ť

T^NˆR^NhM pvq dxdv “ 0 ) is the orthogonal of all constant functions@h₈D

in@

¨ , ¨D

, and C “ ∇x. Conditions (i)-(iii) are triv-ially satisfied since (i) A only acts on velocities, whereas C only acts on space, (ii) rA, A^˚s “ Id2N

and (iii) rB, Cs “ 0. Thanks to the Poincaré²⁹inequality

@h P H¹ s.t.

ij

T^NˆR^N

hM pvq dxdv “ 0 : ||∇xh||²` ||∇vh||²ě K2||h||²,

the operator A^˚A ` C^˚C is coercive on H¹{K. Further,@@h , hDD “ Grhs, and so Theorem6.1tells us that there exists a constant λ ą 0 such that

d

dtGrhs ď ´2λGrhs . (6.36)

Generally, it is not possible to show ||hptq||H¹ ď ||hp0q||H¹e^´λtas explained above. However, it follows from norm-equivalence between Gr¨s and || ¨ ||²H¹and from (6.36) that

||hptq||H¹ ď c0||hp0q||H¹e^´λt

on H¹{K for some λ ą 0 and c⁰ą 1. This is exactly what we mean by saying that L is hypocoercive on H¹{K.

6.2 Framework for linear kinetic equations

In Chapter5, we focus on a specific example of a linear kinetic equation conserving mass, a class of equations for which the general hypocoercivity theory simplifies greatly [135]. For a detailed account of the general method, see [298,296] and the references therein. Consider the abstract ODE

d

dtf ` Tf “ Qf (6.37)

29Jules Henri Poincaré (1854-1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science, and often described as a ’polymath’. He was proponent of the view, known as conventionalism, that it is not an objective question which model of geometry best fits physical space, but is rather a matter of which model we find most convenient.

governing the evolution of a density f pt, x, vq, where x and v denote the space and velocity vari-ables respectively, and f pt, ¨q P H for all t ě 0 for some Hilbert space H. Here, T denotes a skew-symmetric transport operator and Q is a collision operator that is assumed to be negative semi-definite. Both operators are possibly unbounded. Further, let us assume that we have exis-tence of a unique equilibrium distribution F P H of unit mass satisfying TF “ QF . The goal is to show convergence to F in the norm } ¨ } corresponding to the Hilbert space H for initial data finP H of unit mass.

Hypocoercivity as a method has been developed for equations where the collision part of the operator only acts on the velocity variable. In particular, denoting by Π the projection onto velocity-independent densities, Πf :“ ρ^fF {ρF with ρ^f :“ ş fdv, we have ΠQ “ QΠ “ 0. Since the mixing only occurs in the velocity variable, it is not directly obvious why one would expect to observe convergence to equilibrium both in space and in velocity. However, with the good as-sumptions on T, the mixing in v can be transferred to x via transport effects.

Under the assumptions that T is skew-symmetric and Q is negative semi-definite, one obtains the H -theorem

d

dt}f }²“@Qf, fD ď 0 , (6.38)

In other words, } ¨ }²is a Lyapunov functional for equation (6.37). However, this does not give us any information about the kernel of T. Further, since Q is only negative semi-definite and not coercive, we cannot directly derive convergence to equilibrium from identity (6.38) as the decay in } ¨ }²pauses as soon as the solution f ptq reaches the kernel of Q without necessarily being in the kernel of T. As described in the previous section, this can be remedied by adding a suitable mixed term as an equivalent norm, for which the operator is coercive. In this section, we describe how to formulate the framework of Theorem6.1for linear kinetic equations conserving mass without recourse to commutators, following the functional setting in [135]. The main difference of the approach taken in [135] compared to [298] is to work in an L²-framework instead of H¹, giving important physical information on the behaviour of solutions. For example, one can obtain ex-ponential decay even if the initial datum finoscillates wildly, meaning that the hypocoercivity method is not sensitive to the regularity of fin. Even though hypoellipticity may provide H¹ -regularity, there are two advantages to showing convergence in L²: firstly, the approach in [135]

also applies to equations that are not hypoelliptic, and secondly, an L²-framework is preferable if one is interested in physical applications and dependence on the initial data. We also point out that H¹-regularisation with global estimates in weighted norms has not been done yet for equa-tion (6.37). In order to work in L², definition (6.35) is replaced with a different generalised entropy using a suitable auxiliary operator.

6.2.1 Generalised entropy

The main idea of the convergence proof for hypocoercive operators is to find a Lyapunov func-tional, a generalised entropy, that is better than the ’natural’ entropy } ¨ }², by adding carefully cho-sen lower-order terms. This approach is motivated by [184] in the context of commutator theory for hypoelliptic operators, see Section6.1. In the case of a linear kinetic equation of type (6.37), a suitable generalised entropy G : H Ñ R`is given by

Grf s :“ 1

2}f }²` ε@Af, fD , ε ą 0 with

A :“ p1 ` pTΠq^˚TΠq^´1pTΠq^˚. (6.39)

The ^˚-notation refers to the adjoint in the inner product @

¨, ¨D

corresponding to H. Note that pTΠq^˚TΠ is an elliptic operator. The operator A is bounded and regularises the solution to (6.37) in the space variable (and it is not the same as the operator A in Section6.1). The idea of choosing this generalised entropy is due to [135] and allows to use the projection Π instead of having to deal with ∇v. Here, pTΠq^˚plays the role of the mixed term@∇_xh , ∇vhD

in (6.35), and choosing A “ pTΠq^˚would be enough to build a hypocoercivity theory along the lines of Theorem6.1. The main idea of choosing A as in (6.39) is borrowed from Hérau [184]: replacing the H¹-norm plus a mixed term with a mixed term only, but which is divided by a second order operator to obtain an operator of order zero (i.e. no derivatives). Here, the operator A is of order ´1, but allows to show that solutions to (6.37) decay exponentially fast in L², i.e. the aim is to find an explicit λ ą 0 such thatdt^dG ď ´λG and show that G is norm-equivalent to || ¨ ||².

6.2.2 Microscopic and macroscopic coercivity

Let us differentiate G along trajectories of the system, d

dtGrf s “@Qf, fD ´ ε@ATΠf, fD ´ ε@ATp1 ´ Πqf, fD ` ε@TAf, fD ` ε@AQf, fD , (6.40) using the fact that T is skew-symmetric, and so@Tf, fD “ 0, as well as QA “ 0 which follows since g :“ Af satisfies g “ ´ΠTf ` ΠT²Πg and so it is in the kernel of Q. The first term can be conrolled by the following microscopic coercivity assumption: there exists λmą 0 such that

´@Qf, fD ě λm}p1 ´ Πqf }². (6.41)

In other words, this means that we require the collision operator Q to be coercive on the comple-ment of its kernel. In order to control the second term in (6.40), we need that the elliptic operator ATΠ satisfies a Poincaré inequality, which corresponds to a spectral gap on the macroscopic level.

This can be formulated as the following macroscopic coercivity assumption: there exists λ^M ą 0

such that

}TΠf }²ě λM}Πf }²ùñ@ATΠf, fD ě λM

1 ` λM

}Πf }². (6.42)

In other words, the restriction of T to Ker Q is coercive.

6.2.3 Diffusive macroscopic limit

Take a change of variables pt, x, vq ÞÑ pt{ε², x{ε, vq in equation (6.37) depending on 0 ă ε ! 1 such that the rescaled density f^εpt, x, vq “ f`t{ε², x{ε, v˘

satisfies ε²d

dtf^ε` εTf^ε“ Qf^ε. (6.43)

Consider fluctuations around the set of velocity-independent densities, that is f^ε“ Πf^ε` εR^εfor some R^ε P H. Substituting this ansatz into (6.43) and projecting onto the kernel of Q, we obtain the conservation law

εd

dtpΠf^εq ` ΠTΠf<sup>ε</sup>` εΠTR^ε“ 0 , (6.44) since Π²“ Π, ΠR^ε“ Πp1 ´ Πqf^ε{ε “ 0 and ΠQ “ 0. Assuming that f^εÑ f⁰and R^εÑ R⁰in the limit ε Ñ 0, we obtain the identity

ΠTΠ “ 0 . (6.45)

It follows from (6.43) that Qf⁰ “ 0, and since f⁰is in Ker Q, we conclude that f⁰“ Πf⁰. Further, dividing (6.43) by ε and using that QΠ “ 0, we have

εd

dtf^ε` Tf^ε“ Qf^ε{ε “ Qp1 ´ Πqf^ε{ε “ QR^ε.

Therefore, we obtain in the limit that Tf⁰“ QR⁰. Recalling that f⁰“ Πf⁰, we have R⁰“ ˆQ^´1Tf⁰“ ˆQ^´1TΠf⁰, Q :“ Q|ˆ _p1´ΠqH.

Finally, dividing (6.44) by ε and using (6.45), we obtain in the limit ε Ñ 0 the macroscopic equation BtΠf⁰´ pTΠq^˚Qˆ^´1TΠf⁰“ 0 ,

where we used that T is skew-symmetric T^˚ “ ´T, and Π^˚ “ Π. In other words, assuming ΠTΠ “ 0 corresponds to a diffusive macroscopic limit of equation (6.37).

6.2.4 Exponential convergence

The price to pay by using the generalised entropy G is that one needs to be able to control the last three terms in (6.40) also. The assumption ΠTΠ “ 0 yields [135, Lemma 1]

}Af } ď 1

2}p1 ´ Πqf } , }TAf } ď }p1 ´ Πqf } .

It follows from the first estimate that G is norm-equivalent to the Hilbert space norm } ¨ }²if ε ă 1.

Finally, it remains to show that the following auxiliary operators are bounded:

@ATp1 ´ Πqf, fD ` @AQf, fD ď CM}p1 ´ Πqf }² (6.46)

for some constant C^M ą 0. Putting all the bounds together, we obtain exponential decay of }f ptq}, i.e. hypocoercivity, with an explicitly computable rate depending on λ^m, λ^M, C^M, assuming that (6.41), (6.42), (6.46) and ΠTΠ “ 0 hold. For the detailed proof of this statement, see [135].

Applications of the hypocoercivity approach in the linear kinetic setting include equations con-taining confinement terms and different types of collision operators with mass conservation, such as the Fokker–Planck equation, scattering models and the linearised BGK equations, see [135] and the references therein. Further recent applications include the fibre-lay down process (5.33) for a stationary conveyor belt [134], a velocity-jump model for bacterial chemotaxis [69], and particles interacting with a vibrating medium [1].

7 Part II: Results

In Chapter5, we apply the hypocoercivity method described above to the linear kinetic equation modelling the fibre lay-down in the production process of non-woven textiles as formulated in (5.33). The full hypocoercivity analysis of the long-time behaviour of solutions to this kinetic model in the case of a stationary conveyor belt κ “ 0 is completed in [134]. In the case κ “ 0, there exists a unique global normalised equilibrium distribution

F0pxq “ e^{´V pxq} ş

R²e^{V pxq}dx.

For technical applications in the production process of non-wovens, one is interested in a model including the movement of the conveyor belt, and in Chapter5, we extend the results in [134] to the case κ ą 0. This is not a trivial task for several reasons. First of all, for a moving conveyor belt, we are not able to find a stationary state for equation (5.33) explicitly. The hypocoercivity method however is used to find estimates about rates of convergence after the existence and uniqueness of a steady state have been established.

Secondly, adding the movement of the belt breaks the symmetry of the problem, and the operator assumptions required for the hypocoercivity strategy to work do not hold in the ’natural’ func-tional framework. However, the hypocoercivity theory is based on a priori estimates [135], and is therefore stable under perturbation. We will show in Chapter5how the hypocoercivity technique can be adapted to this context under the assumption that the conveyor belt moves slow enough.