Convergence rate to equilibrium for conservative scattering models on the torus: a new tauberian approach
Authors: Lods B. , Mokhtar-Kharroubi M. 
Published: November 2023

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of

C

0

C_{0}

-semigroups

(

V

(
t
)
)

t
⩾
0

\left (\mathcal {V}(t)\right )_{t \geqslant 0}

in

L

1

(

T

d

×

R

d

)

L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})

governing conservative linear kinetic equations on the torus with general scattering kernel

k
(
v
,

v
′

)

\boldsymbol {k}(v,v’)

and degenerate (i.e. not bounded away from zero) collision frequency
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis v right-parenthesis equals integral Underscript double-struck upper R Superscript d Baseline Endscripts bold-italic k left-parenthesis v prime comma v right-parenthesis bold-italic m left-parenthesis normal d v prime right-parenthesis">

σ
(
v
)
=

∫

R

d

k
(

v
′

,
v
)
m
(

d

v
′

)

\sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v’,v)\boldsymbol {m}(\mathrm {d}v’)

, (with

m
(

d

v
)

\boldsymbol {m}(\mathrm {d}v)

being absolutely continuous with respect to the Lebesgue measure). We show in particular that if

N

0

N_{0}

is the maximal integer

s
⩾
0

s \geqslant 0

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over sigma left-parenthesis dot right-parenthesis EndFraction integral Underscript double-struck upper R Superscript d Baseline Endscripts bold-italic k left-parenthesis dot comma v right-parenthesis sigma Superscript negative s Baseline left-parenthesis v right-parenthesis bold-italic m left-parenthesis normal d v right-parenthesis element-of upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis comma">

1

σ
(
⋅
)

∫

R

d

k
(
⋅
,
v
)

σ

−
s

(
v
)
m
(

d

v
)
∈

L

∞

(

R

d

)
,

\begin{equation*} \frac {1}{\sigma (\cdot )}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot ,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*}

then, for initial datum

f
f

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript double-struck upper T Superscript d Baseline times double-struck upper R Superscript d Baseline Endscripts StartAbsoluteValue f left-parenthesis x comma v right-parenthesis EndAbsoluteValue sigma Superscript minus upper N 0 Baseline left-parenthesis v right-parenthesis normal d x bold-italic m left-parenthesis normal d v right-parenthesis greater-than normal infinity">

∫

T

d

×

R

d

|

f
(
x
,
v
)

|

σ

−

N

0

(
v
)

d

x
m
(

d

v
)
>
∞

\displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) >\infty

it holds
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar script upper V left-parenthesis t right-parenthesis f minus rho Subscript f Baseline normal upper Psi double-vertical-bar Subscript upper L Sub Superscript 1 Subscript left-parenthesis double-struck upper T Sub Superscript d Subscript times double-struck upper R Sub Superscript d Subscript right-parenthesis Baseline equals StartFraction epsilon Subscript f Baseline left-parenthesis t right-parenthesis Over left-parenthesis 1 plus t right-parenthesis Superscript upper N 0 minus 1 Baseline EndFraction comma rho Subscript f Baseline colon-equal integral Underscript double-struck upper R Superscript d Baseline Endscripts f left-parenthesis x comma v right-parenthesis normal d x bold-italic m left-parenthesis normal d v right-parenthesis comma">

‖

V

(
t
)
f
−

ϱ

f

Ψ
‖

L

1

(

T

d

×

R

d

)

=

ε

f

(
t
)

(
1
+
t

)

N

0

−
1

,

ϱ

f

≔

∫

R

d

f
(
x
,
v
)

d

x
m
(

d

v
)
,

\begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*}

where

Ψ
\Psi

is the unique invariant density of

(

V

(
t
)
)

t
⩾
0

\left (\mathcal {V}(t)\right )_{t \geqslant 0}

and

lim

t
→
∞

ε

f

(
t
)
=
0

\lim _{t\to \infty }{\varepsilon }_{f}(t)=0

. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of

(

V

(
t
)
)

t
⩾
0

\left (\mathcal {V}(t)\right )_{t \geqslant 0}

and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp “subgeometric” convergence rate for Markov semigroups associated to general transition kernels.