From gas dynamics with large friction to gradient flows describing diffusion theories

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by  Corrado Lattanzio

FROM GAS DYNAMICS WITH LARGE FRICTION TO GRADIENT FLOWS
DESCRIBING DIFFUSION THEORIES
CORRADOLATTANZIOANDATHANASIOSE.TZAVARAS
6 Abstract. We study the emergence of gradient flows in Wasserstein distance as high friction limits of
1 an abstract Euler flow generated by an energy functional. We develop a relative energy calculation that
0 connects theEulerflow tothegradientflowinthe diffusivelimitregime. Weapplythisapproach toprove
2 convergence fromthe Euler-Poissonsystem with frictionto the Keller-Segel system in the regimethat the
latterhassmoothsolutions. ThesamemethodologyisusedtoestablishconvergencefromtheEuler-Korteweg
p
theorywithmonotone pressurelawstotheCahn-Hilliardequation.
e
S

1
2
Contents
]
P
1. Introduction 1
A
2. A large friction theory converging toward gradient flows 3
. 2.1. The energy equation 4
h
t 2.2. Relative energy identity for the relaxation theory 5
a
2.3. Confinement potentials 6
m
2.4. The analysis of the diffusive limit 7
[
2.5. Relative energy estimate for the gradient flow 8

2 3. From the Euler-Poissonsystem with friction to the Keller-Segel system 8
v 3.1. Preliminaries 9
6
3.2. Relative energy estimate 10
6
3.3. Stability estimate and convergence of the relaxation limit 14
9
5 4. From the Euler-Korteweg system with friction to the Cahn–Hilliard equation 17
0 4.1. Relative energy estimate 18
. 4.2. Convergence to Cahn-Hilliard in the large friction limit 22
1
0 References 23
6
1
:
v
Xi 1. Introduction
r Following the works of Jordan-Kinderlehrer-Otto [18] and Otto [25] a large interest was generated for
a
diffusive equations induced as gradient flows of functionals in the form:
δE(ρ)
ρ −div ρ∇ =0. (1.1)
t x x
δρ
(cid:18) (cid:19)
A key novelty of the approach introduced in these papers is the use of the Wasserstein space of probability
measures as a framework where the gradient flow is considered; for a complete theory we refer to the
monograph [1].
The objective of this work is to explore the induction of such diffusion problems as high friction limits of
abstract Euler flows of the form
∂ρ
+div (ρu)=0
x
∂t
 (1.2)
ρ∂u +ρu·∇ u=−ρ∇ δE −ζρu,
x x
∂t δρ
 1
where ζ > 0 is a (large) friction coefficient ζ > 0 and E(ρ) is a functional on the density that generates
the evolution. This problem is introduced in [13] (the Hamiltonian flow case ζ = 0) with the objective to
put in a common framework several commonly used systems in applications, like the Euler equations, the
Euler-Poisson system and the Euler-Korteweg theory. In this work we study the emergence of the system
(1.1) from the system (1.2) in the high friction regime ζ →∞.
This type ofproblembelongsto the generalrealmofdiffusive limits, whichhas beenaddressedinvarious
contexts with several techniques; we refer to [11] for a survey. The simplest example of a high friction limit
that fits within the present functional framework (from (1.2) to (1.1)) is the limit from the Euler system
with friction to the porous media equation. This has been addressed again with various methodologies, see
e.g. [23, 16, 17] and in particular [21] using the relative energy method adopted here.
We develop a general methodology for treating the diffusive limit from (1.2) to (1.1) and apply it to two
examples: First, we consider generalized Keller-Segel type models
ρ =div ∇ p(ρ)−C ρ∇ c
t x x x x
(1.3)
(−△xc+β(cid:0)c=ρ−<ρ>. (cid:1)
as high friction imits of the Euler-Poissonsystem with attractive potentials (C >0) and friction:
x
ρ +div m=0
t x
m +div m⊗m +∇ p(ρ)=−ζm+C ρ∇ c (1.4)
 t x ρ x x x
−△ c+βc=ρ−<ρ>.
x
This example corresponds tothe choice of the functional
E(ρ)= h(ρ)− 1C ρc dx,
2 x
Z
where h and p are linked by the thermodynamic(cid:0)consistency rel(cid:1)ations ρh′′(ρ)=p′(ρ), ρh′(ρ)=p(ρ)+h(ρ),
while c is the solution of the Poisson equation
−△ c+βc=ρ−<ρ>, <ρ>= ρdx, β ≥0,
x
Z
normalized by requiring < c >= 0 for β = 0. For alternative methodologies on this problem see [9, 22];
related models in the context of semiconductors devices with repulsive potentials C < 0 are analyzed in
x
[24, 20, 19]. For a study of the limiting Keller-Segel model (1.3) as a gradient flows we refer to [6].
As a secondparadigmentering into this framework,we consider the Euler-Kortewegsystem with friction
ρ +div m=0
t x
 m⊗m (1.5)
mt+divx ρ =−ζm−ρ∇x h′(ρ)−Cκ△xρ
(cid:0) (cid:1)
converging in the high-friction regime to the Cahn-Hilliard equation
ρ =div ρ∇ h′(ρ)−C △ ρ =div ∇ p(ρ)−C ρ∇ △ ρ , (1.6)
t x x κ x x x κ x x
which corresponds to the choic(cid:0)e of fu(cid:0)nctional (cid:1)(cid:1) (cid:0) (cid:1)
E(ρ)= h(ρ)+ 1C |∇ ρ|2 dx, C >0.
2 κ x κ
Z
(cid:0) (cid:1)
The technical tool consists of a functional form of the relative energy identity introduced in [13], and
inspiredby [21, 22]andthe relativeenergy calculationsofDafermos [7, 8]. The relativeenergymonitors the
distance between solutions in appropriate norms pertinent to the aforementioned equations (1.2) to (1.1).
It provides a very efficient tool to carry out the limiting process, as it is precisely adapted to the functional
framework of both problems (1.1) and (1.2).
The outline of this work is as follows. In Section 2 we introduce the relative kinetic energy
1
K(ρ,m|ρ¯,m¯):= ρ|u−u¯|2dx,
2
Z
2
and the relative potential energy
δE
E(ρ|ρ¯):=E(ρ)−E(ρ¯)− (ρ¯),ρ−ρ¯ ,
δρ
and use them to derive an identity for the distance betwee(cid:10)n two solution(cid:11)s of (1.2); see (2.17) in Section 2.2.
The same tool is used in order to measure the distance between solutions of (1.2) and (1.1) in Section 2.4.
It provides a yardstick to measure the distance in the relaxation limit. The identity carries seamlessly to
the limit and yields an identity between two solutions of (1.1) in terms of the relative potential energy; see
(2.29) in Section 2.5. After this formal calculation, we study the relaxation limits from weak solutions of
the hyperbolic relaxing model toward strong solutions of the diffusive equations. This is carried out in two
cases: fromtheEuler-PoissonsystemswithattractivepotentialstowardsKeller-SegeltypemodelsinSection
3, and from the Euler-Kortewegsystem with friction toward the Cahn-Hilliard equation in Section 4.
2. A large friction theory converging toward gradient flows
We startour analysis by presenting a relaxationtheory of large friction convergingtowardsgradient flow
dynamics. Thisformalismwillunifyinacommonframeworktheresultsoncovergencefromthecompressible
Euler system with friction to the porous media equation obtained in [21], with convergence results towards
Keller-Segel type systems (see [22] for preliminary results in this direction), or towards the Cahn-Hilliard
equation, obtained in the following sections. The specific cases will be obtained as particular examples of
the general framework via an appropriate choice of the entropy functional defining the flow of the limiting
equation.
To this aim, let us consider the following system of equations consisting of a conservation of mass and a
functional momentum equation
∂ρ
+div (ρu)=0
x
∂t
 (2.1)
ρ∂u +ρu·∇ u=−ρ∇ δE −ζρu,
x x
∂t δρ
where ρ≥0 is the density and u isthe velocity. Moreover, δE stands for the generatorof the first variation
δρ
of the functional E(ρ) (see the discussion in [13]), and the term −ζρu accounts for a damping force with
frictionalcoefficientζ >0. Forlargefrictionsζ = 1,afteraproperscalingoftime∂ 7→ε∂ ,(2.1)isrewritten
ε t t
as
∂ρ 1
+ div (ρu)=0
x
∂t ε
 (2.2)
ρ∂∂ut + 1ερu·∇xu=−ε12ρu− 1ερ∇xδδEρ ,
or, in terms of (ρ,m=ρu) 
1
ρ + div m=0
t x
ε
 (2.3)
mt+ 1ε divx m⊗ρ m =−ε12m− 1ερ∇xδEδ(ρρ).
Note that (2.3) is in conservationform except for the termρ∇xδδEρ. Nevertheless,for all examples treated in
this paper, we have
δE
−ρ∇ =∇ ·S, (2.4)
x x
δρ
whereS =S(ρ)willbeatensor-valuedfunctionalonρthatplaystheroleofastresstensorwithcomponents
S (ρ) with i,j =1,...,d. We refer to [13] for a discussion of the ramifications of that property.
ij
As ε↓0 in (2.3), we formally obtain the gradient flow dynamic
δE(ρ)
ρ −div ρ∇ =0. (2.5)
t x x
δρ
(cid:18) (cid:19)
3
Theobjectiveofthissectionistodescribethislargefrictionlimitusingtherelativeenergyidentitiesinduced
by the functional framework. Particularexamples will include variousinteresting systems (see the examples
in [13]) and in particular:
(1) the porous medium equation as limit of the Euler equation with friction [21] corresponds to the
choice
E(ρ)= h(ρ)dx;
Z
(2) the Keller-Segel system as limit of the Euler-Poisson system with friction (in the case of attractive
potentials), considered in Section 3, is given by the functional
E(ρ)= h(ρ)− 1C ρc dx,
2 x
Z
(cid:0) (cid:1)
where C >0 and c is viewed as a constraint in terms of the relation
x
−△ c+βc=ρ−<ρ>, <ρ>= ρdx, β ≥0;
x
Z
(3) the Cahn-Hilliard equation as limit of the the Euler-Korteweg system with friction corresponds to
the choice
E(ρ)= h(ρ)+ 1C |∇ ρ|2 dx, C >0
2 κ x κ
Z
(cid:0) (cid:1)
and is investigated in Section 4.
2.1. Theenergyequation. Westartbyreviewingandadaptingtotherelaxationframeworkcertainresults
from [13]. First, we derive the energy estimate for (2.2) or (2.3) in the functional setting. We assume that
the directional derivative (Gateaux derivative) of the functional E defined by
E(ρ+τψ)−E(ρ) d
dE(ρ;ψ)= lim = E(ρ+τψ)
τ→0 τ dτ τ=0
(cid:12)
(cid:12)
is linear in ψ and can be represented via a duality bracket (cid:12)
d δE
dE(ρ;ψ)= E(ρ+τψ) = (ρ),ψ , (2.6)
dτ τ=0 δρ
(cid:12)
(cid:12) (cid:10) (cid:11)
with δE(ρ) standing for the generator of the bracket. Th(cid:12)is property is always satisfied for Frechet differen-
δρ
tiable functionals. Using (2.4), the potential energy is computed via
d δE 1 δE 1
E(ρ)=h (ρ),ρ i=− h (ρ),div (ρu)i= S :∇ udx. (2.7)
t x x
dt δρ ε δρ ε
Z
Now, using again (2.4) and the momentum equation (2.3) with the standard multiplier u, we obtain the
2
usual kinetic energy relation
1 d 1 1 δE(ρ)
ρ|u|2dx=− ρ|u|2dx− ρu·∇ dx
2dt ε2 ε x δρ
Z Z Z
1 1
=− ρ|u|2dx− S :∇ udx,
ε2 ε x
Z Z
which, added to (2.7), finally leads to the standard energy relation
d 1 1
E(ρ)+ ρ|u|2dx + ρ|u|2dx=0. (2.8)
dt 2 ε2
(cid:18) Z (cid:19) Z
4
2.2. Relative energy identity for the relaxation theory. Next, we compare two different solutions
(ρ,m), (ρ¯,m¯) of (2.3) using the relativeentropyframework. To this end, we define also the secondvariation
of the functional E(ρ) via
δE(ρ+τϕ),ψ − δE(ρ),ψ
d2E(ρ;ψ,ϕ)= lim δρ δρ
τ→0(cid:10) τ(cid:11) (cid:10) (cid:11)
(whenever the limit exists), and we assume that this can be representedas a bilinear functional in the form
δE(ρ+τϕ),ψ − δE(ρ),ψ δ2E
d2E(ρ;ψ,ϕ)= lim δρ δρ = (ρ),(ψ,ϕ) . (2.9)
τ→0(cid:10) τ(cid:11) (cid:10) (cid:11) (cid:28)(cid:28)δρ2 (cid:29)(cid:29)
Moreover, in analogy to (2.6), we assume that the directional derivative of S(ρ) is expressed as a linear
functional via a duality bracket,
d δS
dS(ρ;ψ)= S(ρ+τψ) = (ρ),ψ , (2.10)
dτ τ=0 δρ
(cid:12)
in terms of the generator δS(ρ). (cid:12) (cid:10) (cid:11)
δρ (cid:12)
2.2.1. The relative potential energy. Define the relative potential energy,
δE
E(ρ|ρ¯):=E(ρ)−E(ρ¯)− (ρ¯),ρ−ρ¯ , (2.11)
δρ
as the quadratic part of the Taylor series expansion of the(cid:10)functional E(cid:11)with respect to a reference solution
ρ¯(x,t). IfE(ρ) isconvex,this quantitycanserveasameasureofdistancebetweenthetwosolutionsρ andρ¯.
Consider next the weak form of (2.4),
δE ∂ ∂ϕ
i
(ρ), (ρϕ ) =− S (ρ) dx.
j ij
δρ ∂x ∂x
j j
Z
(cid:10) (cid:11)
This relation is viewed as a functional in ρ; talking its directional derivative along a direction ψ, with ψ a
smooth test function, we obtain
δ2E ∂ δE ∂
(ρ), ψ, (ρϕ ) + (ρ), (ψϕ )
δρ2 ∂x j δρ ∂x j
(cid:28)(cid:28) j (cid:29)(cid:29) (cid:28) j (cid:29)
(cid:0) (cid:1) δS ∂ϕ
ij i
=− (ρ),ψ dx.
δρ ∂x
Z (cid:28) (cid:29) j
The two relations lead to (see [13, Section 2.1] for the details of this computation):
d 1 ∂u¯ 1 δE δE
i
E(ρ|ρ¯)= S (ρ|ρ¯) dx− (ρ)− (ρ¯),div ρ(u−u¯) , (2.12)
ij x
dt ε ∂x ε δρ δρ
j
Z
(cid:10) (cid:0) (cid:1)(cid:11)
where S(ρ|ρ¯) stands for the relative stress tensor:
δS
S(ρ|ρ¯):=S(ρ)−S(ρ¯)− (ρ¯),ρ−ρ¯ . (2.13)
δρ
(cid:10) (cid:11)
2.2.2. The relative kinetic energy. Next consider the kinetic energy
1|m|2
K(ρ,m)= dx (2.14)
2 ρ
Z
viewed as a (not strictly) convex functional on the density ρ and the momentum m = ρu. The relative
kinetic energy is expressed in the form
K(ρ,m|ρ¯,m¯):= k(ρ,m)−k(ρ¯,m¯)−∇k(ρ¯,m¯)·(ρ−ρ¯,m−m¯)dx
Z
1|m|2 1|m¯|2 1|m¯|2 m¯
= − − − , ·(ρ−ρ¯,m−m¯)dx (2.15)
2 ρ 2 ρ¯ 2 ρ¯2 ρ¯
Z
1 (cid:0) (cid:1)
= ρ|u−u¯|2dx,
2
Z
5
To compute its evolution, consider the difference of the two equations satisfied by (ρ,u) and (ρ¯,u¯), that is
1 1
∂ (u−u¯)+ (u·∇ )(u−u¯)+ (u−u¯)·∇ u¯
t x x
ε ε
1 1 (cid:0)δE(ρ) δE(ρ(cid:1)¯)
=− (u−u¯)− ∇ − .
ε2 ε x δρ δρ
(cid:18) (cid:19)
Multiplying this relation by u−u¯ we end up with
1 1 1
∂ |u−u¯|2+ (u·∇ )|u−u¯|2+ ∇ u¯:(u−u¯)⊗(u−u¯)
t x x
2 2ε ε
1 1 δE(ρ) δE(ρ¯)
=− |u−u¯|2− (u−u¯)·∇ − ,
ε2 ε x δρ δρ
(cid:18) (cid:19)
which, using (2.3) and integrating over space leads to the balance of the relative kinetic energy
1
1 d 1
ρ|u−u¯|2dx+ ρ|u−u¯|2dx=
2dt ε2
Z Z (2.16)
1 1 δE δE
− ρ∇ u¯:(u−u¯)⊗(u−u¯)dx+ (ρ)− (ρ¯),div ρ(u−u¯) .
x x
ε ε δρ δρ
Z
(cid:10) (cid:0) (cid:1)(cid:11)
2.2.3. The functional form of the relative energy formula. Summing (2.12) to (2.16) we obtain the relative
energy identity
d 1 1
E(ρ|ρ¯)+ ρ|u−u¯|2dx + ρ|u−u¯|2dx
dt 2 ε2
(cid:18) Z (cid:19) Z (2.17)
1 1
= ∇ u¯:S(ρ|ρ¯)dx− ρ∇ u¯:(u−u¯)⊗(u−u¯)dx,
x x
ε ε
Z Z
where E(ρ|ρ¯) and S(ρ|ρ¯) stand for the relative potential energy and relative stress functionals defined in
(2.11) and (2.13), respectively. The main property which leads to the above relation is the fact that the
contributions of the term
1 δE δE
D = (ρ)− (ρ¯),div ρ(u−u¯)
x
ε δρ δρ
in (2.12) and (2.16) offset each other, as(cid:10)for the terms involvin(cid:0)g the stre(cid:1)s(cid:11)s tensor S in the derivation of the
energy relation (2.8).
2.3. Confinement potentials. Itisexpedienttogiveanextensionofthe calculationforsystemsdrivenby
a confinement potential V =V(x),
∂ρ
+ 1div (ρu)=0
∂t ε x
 (2.18)
ρ∂∂ut + 1ερu·∇xu=−ε12ρu− 1ερ∇xδδEρ − 1ερ∇xV .
The potential energy is nowgiven by the functional
F(ρ)=E(ρ)+ ρV(x)dx (2.19)
Z
and we require that E(ρ) satisfies (2.4) for some stress functional S(ρ). Note, that the potential energy
functional splits into the potential energy of the contact forces E(ρ) and the potential energy of the body
forces ρV. The latter is not expected to be associated to a stress, and also is not invariant under space
translations (which is connected to the hypothesis (2.4)). Under the hypothesis (2.4) for E, we may write
R
the weak form of (2.18) for (ρ,m=ρu)
ρ + 1div m=0
t ε x
(2.20)
 m⊗m
mt+ 1εdivx ρ =−ε12m− 1ε∇x·S(ρ)− 1ερ∇xV .
6

Proceeding along the lines of the calculations in Section 2.1 we see that
d 1 1
F(ρ)= S :∇ udx+ ρ∇ V ·udx, (2.21)
x x
dt ε ε
Z Z
1 d 1 1 1
ρ|u|2dx=− ρ|u|2dx− S :∇ udx− ρ∇ V ·udx, (2.22)
2dt ε2 ε x ε x
Z Z Z Z
and the total energy for (2.18) reads
d 1 1
F(ρ)+ ρ|u|2dx + ρ|u|2dx=0. (2.23)
dt 2 ε2
(cid:18) Z (cid:19) Z
On the other hand, due to the formulas
F(ρ|ρ¯)=E(ρ|ρ¯),
δF δF δE δE
(ρ)− (ρ¯)= (ρ)− (ρ¯),
δρ δρ δρ δρ
thecalculationsinSections2.2.1and2.2.2remainessentiallyunaffected,andthefinalrelativeenergyformula,
comparing two solutions (ρ,u) and ρ¯,u¯) of the system (2.18) with confinement potential, takes exactly the
same form as (2.17):
d 1 1
E(ρ|ρ¯)+ ρ|u−u¯|2dx + ρ|u−u¯|2dx
dt 2 ε2
(cid:18) Z (cid:19) Z (2.24)
1 1
= ∇ u¯:S(ρ|ρ¯)dx− ρ∇ u¯:(u−u¯)⊗(u−u¯)dx.
x x
ε ε
Z Z
2.4. The analysis of the diffusive limit. We returnnow to the system(2.2) without confinementpoten-
tial. In this section we aim to compare a solution (ρ,ρu) of (2.3) with a smooth solution ρ¯of (2.5). To this
end, we define
δE(ρ¯)
m¯ =ρ¯u¯=−ερ¯∇ (2.25)
x
δρ
and visualize the pair (ρ¯,m¯ =ρ¯u¯) as an approximate solution of (2.3), that is
1
ρ¯ + div m¯ =0
t x
ε
 (2.26)
m¯ + 1div m¯ ⊗m¯ =− 1 m¯ − 1ρ¯∇ δE(ρ¯) +e¯,
t ε x ρ¯ ε2 ε x δρ
where 
1 m¯ ⊗m¯
e¯=m¯ + div .
t x
ε ρ¯
Using (2.25) and the smoothness of ρ¯ we see that e¯ is a O(ε) error term. The only difference from the
calculations of section 2.2 lies in the relative kinetic energy (2.16). Presently, u¯ satisfies the approximate
equation
1 1 1 δE(ρ¯) e¯
u¯ + (u¯·∇ )u¯=− u¯− ∇ + ,
t ε x ε2 ε x δρ ρ¯
and, following the analysis in section 2.2.2, we obtain
1 1 1
∂ |u−u¯|2+ (u·∇ )|u−u¯|2+ ∇ u¯:(u−u¯)⊗(u−u¯)
t x x
2 2ε ε
1 1 δE(ρ) δE(ρ¯) e¯
=− |u−u¯|2− (u−u¯)·∇ − −(u−u¯)· , (2.27)
2ε2 ε x δρ δρ ρ¯
(cid:18) (cid:19)
and the relative energy relation
d 1 1
E(ρ|ρ¯)+ ρ|u−u¯|2dx + ρ|u−u¯|2dx
dt 2 ε2
(cid:18) Z (cid:19) Z
1 1 e¯
= ∇ u¯:S(ρ|ρ¯)dx− ρ∇ u¯:(u−u¯)⊗(u−u¯)dx− ρ(u−u¯)· dx. (2.28)
x x
ε ε ρ¯
Z Z Z
7
Sinceu¯=O(ε), forsmoothsolutionsthecoefficientsofthequadratictermsareO(1)inε. Moreover,thelast
(error)term at the right hand side of (2.28) is controlled in terms of the distance w.r.t. the equilibrium, i.e.
by 1ε−2ρ|u−u¯|2,andanO(ε4)termdepending ontotalmassofρ andthe smooth,strictlypositive solution
2
ρ¯of (2.5). This relation is therefore instrumental to control the relaxation limit.
2.5. Relative energy estimate for the gradient flow. The above calculations induce a relative energy
estimate for comparing two solutions ρ and ρ¯of the limiting gradientflow (2.5). Indeed, using in (2.17) the
expressions (2.25) for the two velocities u and u¯ at equilibrium we obtain
d δE(ρ) δE(ρ¯) 2 ∂2 δE(ρ¯)
E(ρ|ρ¯)+ ρ ∇ − dx=− S (ρ|ρ¯) dx. (2.29)
x ij
dt δρ δρ ∂x ∂x δρ
Z (cid:12)(cid:12) (cid:16) (cid:17)(cid:12)(cid:12) Z j i
Note thatinthis calculationthe(cid:12)effectofthe kinetic en(cid:12)ergydropsout,andthe derivationof (2.29)usesonly
(cid:12) (cid:12)
(2.12) and the transport equations for ρ and ρ¯; the term D becomes dissipative with the velociy choices
δE(ρ) δE(ρ¯)
u=−ε∇ and u¯=−ε∇
x x
δρ δρ
leading to the gradient flow.
3. From the Euler-Poisson system with friction to the Keller-Segel system
Inthissection,weshallmakeprecisethefunctionalsettingfortheEuler–Poissonsystemwithanattractive
potential and friction converging to Keller–Segel type models. The Euler-Poissonsystem reads
1
ρ + div m=0
t x
ε

mt+ 1ε divx m⊗ρ m + 1ε∇xp(ρ)=−ε12m+ Cεxρ∇xc (3.1)
where t ∈ R, x ∈ Tn, n =2−,3△txhce+phβycs=icaρll−y<relρev>an,t dimensions, ρ ≥ 0, c ∈ R, m = ρu ∈ Rn. We assume
thattheinternalenergyh(ρ)andthepressurep(ρ)areconnectedthroughtheusualthermodynamicrelation
ρh′′(ρ)=p′(ρ), ρh′(ρ)=p(ρ)+h(ρ) with p′(ρ)>0. (H)
Moreover,we impose the conditions on the pressure that for some constants k >0 and A>0,
k
h(ρ)= ργ +o(ργ), as ρ→+∞ (A )
1
γ−1
p′(ρ)
|p′′(ρ)|≤A ∀ρ>0. (A )
2
ρ
Theseconditionsaresatisfiedbytheusualγ–law: p(ρ)=kργ withγ >1. Fortheconstantsγ >1,β ≥0and
C >0 (chemosensitive coefficient) appearing in (3.1) we will impose various smallness/largenessconditions
x
that are precised later. Finally, the elliptic equation in (3.1) is provided with periodic boundary conditions
and it shall be intended for zero mean solutions c in the case β = 0; in that equation, as specified in the
previous section, <ρ> stand for the mean of ρ.
Formally,after an appropriatescaling of the moment, at the limit ε↓0 we obtain m=C ρ∇ c−∇ p(ρ)
x x x
and therefore the formal limit of (3.1) is given by the Keller-Segel type model:
ρ +div C ρ∇ c−∇ p(ρ) =0
t x x x x
(3.2)
(−△xc+β(cid:0)c=ρ−<ρ>. (cid:1)
8
3.1. Preliminaries. Standard hyperbolic theory suggests to employ for (3.1) the usual entropy–entropy
flux pair
1|m|2 1 |m|2
η(ρ,m)= +h(ρ), , q(ρ,m)= m +mh′(ρ), (3.3)
2 ρ 2 ρ2
depictingthemechanicalenergyanditsflux. Werecallthatfortheparticularcaseofγ–lawgases,p(ρ)=kργ,
h takes the form
k 1
ργ = p(ρ) for γ >1;
h(ρ)= γ−1 γ−1

kρlogρ for γ =1.

An entropy weak solution of (3.1) satisfies in the sense of distribution the entropy inequality

1 1 1 |m|2 C
x
η(ρ,m) + div q(ρ,m)≤− ∇ η(ρ,m)·m=− + m·∇ c (3.4)
t ε x ε2 m ε2 ρ ε x
On the other hand, smooth solutions of (3.2) satisfy the entropy identity
|∇ p(ρ)|2
h(ρ) +div h′(ρ)(C ρ∇ c−∇ p(ρ)) =− x +C ∇ p(ρ)·∇ c. (3.5)
t x x x x x x x
ρ
Thisformofthe energyequati(cid:0)onsisinadequateto carry(cid:1)outthe relativeentropyanalysisofthe forthcoming
section.
Next, we present a variant of the energy equation, inspired by the formal analysis of Section 2. To start,
the solution of the elliptic equation (3.1) can be expressed via convolution with the Green’s function,
3
c(x)=(K∗ρ)(x)= K(x−y)ρ(y)dy,
Z
where K is a symmetric function. The energy of the Euler-Poissonsystem takes the form
E(ρ)= h(ρ)− 1C ρc dx
2 x
Z (3.6)
(cid:0) (cid:1)
= h(ρ)dx− 1C ρ(x)K(x−y)ρ(y)dxdy,
2 x
Z ZZ
and the symmetry of K implies
δE
−ρ∇ (ρ)=−ρ∇ (h′(ρ)−C c). (3.7)
x x x
δρ
Next,weshowthatfortheEuler-Poissonsystemthereisastressassocatedwith(3.7). Indeed,multiplying
(3.1) by ∇ c we end up with
3 x
ρ∇ c=∇ 1|∇ c|2+ βc2+<ρ>c −div (∇ c⊗∇ c),
x x 2 x 2 x x x
(cid:16) (cid:17)
so that
−ρ∇ (h′(ρ)−C c) (3.8)
x x
=div − p(ρ)− 1C (βc2+|∇ c|2)−C <ρ>c I−C ∇ c⊗∇ c .
x 2 x x x x x x
This determines the stress(cid:16)S in(cid:2)(2.4) as (cid:3) (cid:17)
S =− p(ρ)− 1C (βc2+|∇ c|2)−C <ρ>c I−C ∇ c⊗∇ c, (3.9)
2 x x x x x x
where I is the identity ma(cid:2)trix. Note that the pressure has a co(cid:3)ntribution coming from the mean-field
interactionterm. Theinductionofapressurefromthemeanfiledinteractioncanalsobeseefromtheenergy
identity
ρcdx= βc2+|∇ c|2 dx, (3.10)
x
Z Z
obtained directly from the elliptic equation (3.1) . (cid:0) (cid:1)
3
Following the general framework of Section 2.1, the potential energy satisfies
d 1
h(ρ)− 1C ρc dx=− h′(ρ)−C c div (ρu)dx
dt 2 x ε x x
Z Z
(cid:0) (cid:1) 9 (cid:0) (cid:1)
1
= S :∇ udx,
x
ε
Z
the kinetic energy is
1 d |m|2 1 |m|2 1
dx=− dx− ρu·∇ (h′(ρ)−C c)dx
2dt ρ ε2 ρ ε x x
Z Z Z
1 |m|2 1
=− dx− S :∇ udx,
ε2 ρ ε x
Z Z
and the total energy reads
d 1 |m|2
η(ρ,m)− 1C ρc dx+ dx=0.
dt 2 x ε2 ρ
Z Z
In accordance to the usual prac(cid:0)tice in conservat(cid:1)ion laws, we will define entropy dissipative solutions
(ρ,m,c) of (3.1) as weak solutions satifysing the weak form of the energy inequality
d 1 |m|2
η(ρ,m)− 1C ρc dx+ dx≤0, (3.11)
dt 2 x ε2 ρ
Z Z
inthesenseofdistributions. Clearly(cid:0)(3.11)and(3.4)(cid:1)areequivalent(forsmoothsolutions),howevertheform
(3.11) together with (3.10) suggest that to control the potential energy is tantamount to obtaining control
of the H1 norm of c. The above estimate is the starting point to obtain the stability estimate in terms of
relative entropy and the corresponding analysis of the relaxation limit in the next section.
3.2. Relative energy estimate. Inthis sectionweperformarelativeenergycomputationbetweenaweak
solution(ρ,m,c)of (3.1)andastrongsolution(ρ¯,c¯)of (3.2). Thefinalformula(3.20)turnsouttobeaspecial
case of formula (2.28) derived in Section 2 for smooth solutions, and of the relative energy computation for
the Euler–Poissonsystemdiscussedin [13, Section 2.5]. Nevertheless,we shallprovide here a directproofof
this identity. The reason is twofold: (i) to justify the relative energy estimate among a weak and a strong
solution, (ii) to account for the effect of error terms appropriate for the relaxationlimit problem. We recall
the framework of weak solutions we shall refer to.
Definition 3.1. A function (ρ,m,c) with ρ ∈ C([0,∞);L1(Tn)∩Lγ(Tn)), m ∈ C ([0,∞); L1(Tn) n ,
c ∈ C([0,∞);H1(Tn)) ρ ≥ 0 and m⊗m ∈ L1 (((0,∞)×Tn))n×nis a dissipative weak periodic solution of
ρ loc (cid:0) (cid:0) (cid:1) (cid:1)
(3.1) with finite total energy if
• (ρ,m,c) satisfies the weak form of (3.1);
• (ρ,m,c) satisfies the following integrated form of the energy inequality (3.11):
1 |m|2
− η(ρ,m)− 1C ρc θ˙(t)dxdt+ θ(t)dxdt
2 x ε2 ρ
ZZ (cid:16) (cid:17) ZZ
(3.12)
≤ η(ρ,m)− 1C ρc θ(0)dx,
2 x
t=0
Z (cid:16) (cid:17)(cid:12)
for any non-negative θ ∈W1,∞[0,∞) compactly supported(cid:12) on [0,∞),
(cid:12)
• (ρ,m) satisfies the following bounds, natural within the given framework:
sup ρdx=M <∞,
t∈(0,T)Z
sup η(ρ,m)− 1C ρc dx<∞. (3.13)
2 x
t∈(0,T)Z
(cid:0) (cid:1)
Remark 3.2. The regularity requested in the above definition is the one needed to rewrite the equation in
terms of the divergence of the stress tensor S in (3.9), and it is implied by the finite energy condition. This
relies on the Lγ integrability of ρ and elliptic regularity estimates for c, and it is proved in Section 3.3; see
Lemma 3.6. Besides the appropriate smallness condition on the chemosensitive coefficient C > 0 (cfr.
x
(H )), we shall require that γ lies in the relevant range for which (3.2) has regular solutions, that is (H ).
c exp
The existence theory for the Euler-Poisson system (3.1) with attractive potential C > 0 (treated here)
x
is largely an open problem. Repulsive potentials, C < 0, offer a subtle stabilizing mechanism leading to
x
10