Symmetries and global solvability of the isothermal gas dynamics equations

0.36 MB·English

by  Philippe G. LeFloch

SYMMETRIES AND GLOBAL SOLVABILITY
OF THE ISOTHERMAL GAS DYNAMICS EQUATIONS
PHILIPPE G.LEFLOCH1 ANDVLADIMIR SHELUKHIN2
Abstract. WestudytheCauchyproblemassociatedwiththesystemoftwoconservationlaws
7 arising in isothermal gas dynamics, in which the pressure and the density are related by the
0 γ-law equation p(ρ)∼ ργ with γ =1. Our results complete those obtained earlier for γ > 1.
0 Weprovetheglobalexistenceandcompactnessofentropysolutionsgeneratedbythevanishing
2 viscosity method. The proof relies on compensated compactness arguments and symmetry
group analysis. Interestingly, we make use here of the fact that the isothermal gas dynamics
n
system is invariantmodulo a linear scalingof the density. This property enables us to reduce
a
J ourproblemtothatwithasmallinitialdensity.
Onesymmetrygroupassociatedwiththelinearhyperbolicequationsdescribingallentropies
3
oftheEulerequationsgivesrisetoafundamentalsolutionwithinitialdataimposedtotheline

ρ=1. Thisisincontrast tothecommonapproach (when γ>1)whichprescribesinitialdata
]
P onthevacuumlineρ=0. Theentropiesweconstructhereareweakentropies,i.e. theyvanish
when the density vanishes. Another feature of our proof lies in the reduction theorem which
A
makesuseofthefamilyofweakentropiestoshowthataYoungmeasuremustreducetoaDirac
h. mass. Thisstep isbasedon new convergence resultsforregularizedproducts of measuresand
t functionsofbounded variation.
a
m
[
1. Introduction
1
v We consider the Euler equations for compressible fluids
0
0 (1.1) ∂tρ+∂x(ρu)=0,
1 (1.2) ∂ (ρu)+∂ (ρu2+p(ρ))=0,
1 t x
0 where ρ 0 denotes the density, u the velocity, and p(ρ) 0 the pressure. We assume that the
7 fluid is g≥overnedby the isothermal equation of state ≥
0
/ (1.3) p(ρ)=k2ρ,
h
t where k > 0 is a constant. Observe that the scaling u ku, t t/k allows one to reduce the
a → →
system (1.1)–(1.3) to the same system with k =1.
m
The existence of weak solutions (containing jump discontinuities) for the Cauchy problem
:
v associated with (1.1)–(1.3) was first established by Nishida [24] (in the Lagrangianformulation).
i The solutions obtained by Nishida have bounded variation and remain bounded away from the
X
vacuum. For background on the BV theory we refer to [6, 16].
r
a By contrast, we are interested here in solutions in a much weaker functional class and in solu-
tions possibly reachingthe vacuum ρ=0. Near the vacuum, the system (1.1)–(1.3)is degenerate
and,inparticular,thevelocityucannotbedefineduniquely. Indeed,thepresentpaperisdevoted
to developing the existence theory in a framework covering solutions satisfying
ρ L (Π), ρu C(ρ+ρ logρ), Π=R (0,T),
∞
∈ | |≤ | | ×
with a constantC >0 depending solelyon initialdata. The time interval(0,T)is arbitrary. Our
proof extends DiPerna’s pioneering work [10] concerned with the pressure law p(ρ) ργ.
∼
1LaboratoireJacques-LouisLions&CentreNationaldelaRechercheScientifique,UniversityofParis6,4place
Jussieu,75252Paris,France. E-mail: lefl[email protected].
2 Lavrentyev Institute of Hydrodynamics, Prospect Lavrentyeva 15, Novosibirsk, 630090, Russia. E-mail :
[email protected]
2000 AMSSubject Classification: 35L,35L65, 76N,76L05
Key Words: Euler equations, isothermal compressible fluids, mathematical entropy, compensated compactness,
existencetheory.
1
2 LEFLOCHANDSHELUKHIN
2. Main result
Introducing the momentum variable m := ρu, one can reformulate the Cauchy problem asso-
ciated with (1.1)–(1.3) as follows:
∂ ρ+∂ m=0,
t x
(2.1) m2
∂ m+∂ ( +ρ)=0,
t x
ρ
with initial condition
(2.2) ρt=0 =ρ0, mt=0 =m0 :=ρ0u0.
| |
where ρ ,u are prescribed. Let us first recall the following terminology. A pair of (smooth)
0 0
functions η = η(m,ρ), q = q(m,ρ) is called an entropy pair if, for any smooth solution (m,ρ) of
(2.1), one also has
∂ η(m,ρ)+∂ q(m,ρ)=0.
t x
More precisely, we consider entropies η,q C2(Ω) C1(Ω¯) in any domain of the form
∈ ∩
Ω:= 0<ρ<ρ , m <c ρ(1+ lnρ) , c >0, ρ >0.
∗ | | ∗ | | ∗ ∗
It is easily checked tha(cid:8)t η,q must solve the equations (cid:9)
m m2
(2.3) q =2 η +η , q =η η ,
m ρ m ρ ρ m− ρ2 m
which implies that
p(ρ) 1
′
(2.4) η = η = η .
ρρ ρ2 uu ρ2 uu
A pair (η,q) is said to be a weak entropy if η(0,0) = q(0,0) = 0. It is said to be convex if in
addition, η is convex with respect to the conservative variables (ρ,m).
Given an initial data m , ρ L (R) obeying the inequalities
0 0 ∞
∈
(2.5) ρ (x) 0, m (x) c ρ (x)(1+ lnρ (x)), x R
0 0 0 0 0
≥ | |≤ | | ∈
for some constant c > 0, an entropy solution to the Cauchy problem (2.1)-(2.2) on the time
0
interval (0,T) is, by definition, a pair of functions (m,ρ) L (Π) satisfying the inequalities
∞
∈
(2.6) ρ(x,t) 0, m(x,t) cρ(x,t)(1+ lnρ(x,t)), (x,t) Π
≥ | |≤ | | ∈
for some positive constant c, together with the inequality
(2.7) η(m,ρ)∂ ϕ+q(m,ρ)∂ ϕ dxdt+ η(m ,ρ )ϕ(,0)dx 0
t x 0 0
ZZΠ(cid:16) (cid:17) ZR · ≥
for every convex, weak entropy pair (η,q) and every non-negative function ϕ D(R [0,T))
∈ ×
(smooth functions with compact support).
The main results established in the present paper are summarized in Theorems 2.1–2.3below.
Theorem 2.1. (Cauchy problem in momentum-density variables.) Given an arbitrary time in-
terval (0,T) and an initial data (m ,ρ ) L (R) satisfying the condition (2.5), there exists an
0 0 ∞
∈
entropy solution (m,ρ) of the Cauchy problem (2.1)-(2.2) satisfying the inequalities (2.6), with a
constant c depending on c only.
0
To prove this theorem it will be convenient to introduce the Riemann invariants W and Z by
W :=ρeu, Z :=ρe−u,
or equivalently
ρ=f (W,Z):=(WZ)1/2, ρu=f (W,Z):=(WZ)1/2ln(W/Z)1/2.
1 2
One can then reformulate the Cauchy problem (2.1)-(2.2) in terms of W,Z, as follows
∂ f (W,Z)+∂ f (W,Z)=0,
t 1 x 2
(2.8)
∂ f (W,Z)+∂ (f (W,Z)+f (W,Z))=0, f :=(WZ)1/2 ln(W/Z)1/2 2,
t 2 x 3 1 3
(cid:0) (cid:1)
ISOTHERMAL COMPRESSIBLE FLUIDS 3
(2.9) W t=0 =W0 :=ρ0eu0, Z t=0 =Z0 :=ρ0e−u0.
| |
a pair of non-negative functions W,Z L (Π) if then called an entropy solution to the problem
∞
∈
(2.8)-(2.9) if
η(W,Z) ∂ ϕ+q(W,Z)∂ ϕ dxdt+ η(W ,Z )ϕ(,0)dx 0
t x 0 0
ZZΠ(cid:16) (cid:17) (cid:17) ZR · ≥
for any non-negative function ϕ D(R [0,T)), where
e ∈ e × e
η˜(W,Z):=η(f (W,Z),f (W,Z)), q˜(W,Z):=q(f (W,Z),f (W,Z)),
2 1 2 1
and (η,q) is any convex, weak entropy pair in the sense introduced above.
Theorem 2.1 above will be obtained as a corollary of the following result.
Theorem 2.2. (Cauchy problem in Riemann invariant variables.) Given non-negative functions
W ,Z L (R), the Cauchy problem (2.8)-(2.9) has an entropy solution on any time interval
0 0 ∞
∈
(0,T).
It is checked immediately that, if (W,Z) is an entropy solution given by Theorem 2.2, then
the functions m := f (W,Z) and ρ := f (W,Z) determine an entropy solution of the problem
2 1
(2.1)-(2.2).
OnemoreconsequenceofTheorem2.2concernstheoriginalproblem(1.1)–(1.3)inthedensity-
velocity variables. Defining the density and velocity from the Riemann variables by
u:=ln(W/Z)1/2, ρ:=(WZ)1/2,
we deduce also the following result from Theorem 2.2.
Theorem 2.3. (Cauchy problem in velocity-density variables.) Let (0,T) be a time interval.
Given any measurable functions u and ρ satisfying the conditions
0 0
0 ρ L (R), u (x) c (1+ lnρ (x)), x R
0 ∞ 0 0 0
≤ ∈ | |≤ | | ∈
for some positive constant c , there exist measurable functions u = u(x,t) and ρ = ρ(x,t) such
0
that
0 ρ L∞(Π), u(x,t) c(1+ lnρ(x,t)), (x,t) Π
≤ ∈ | |≤ | | ∈
(where c > 0 is a constant depending on c ) and (u,ρ) is an entropy solution of the problem
0
(1.1)-(1.3) in the sense that the entropy inequality
η(ρ,ρu)∂ ϕ+q(ρ,ρu)∂ ϕ dxdt+ η(ρ ,ρ u )ϕ(,0)dx 0
t x 0 0 0
ZZΠ(cid:16) (cid:17) ZR · ≥
holds for any convex, weak entropy pair (η,q) and any function ϕ as in Theorem 2.1.
The novel features of our proof of the above results are :
the use of symmetry and scaling properties of both the isothermal Euler equations and
•
the entropy-waveequation,
ananalysisofnew nonconservativeproductsoffunctions withboundedvariationbymea-
•
sures.
We rely on two classical ingredients. The first tool is the compensated compactness method
introduced by Tartar in [32, 33]. (See also Murat [22].) This method allows to show that a
weakly convergent sequence (of approximate solutions given by the viscosity method) is actually
strongly convergent: such a result is achieved by a “reduction lemma” (to point mass measures)
forYoungmeasuresrepresentingthelimitingbehaviorofthesequence. Tartarmethodwasapplied
to systems of conservation laws by DiPerna [9, 10]. For a completely different approach to the
vanishing viscosity method, we refer to Bianchini and Bressan [2]. Still another perspective is
introduced in LeFloch [17].
The second main tool is the symmetry group analysis of differential equations which goes
back to Lie’s classical works. The first symmetry property we use concerns the system (1.1)–
(1.3) itself: we observe that it is invariant with respect to the scaling ρ λρ (λ being an
→
4 LEFLOCHANDSHELUKHIN
arbitrary parameter). This property allows us to assume that the density is sufficiently small
when performing the reduction of the Young measures.
To generate the class of weak entropies, we calculate all the Lie groups associated with the
entropy equation (2.4) for the function η. By using one of them we construct the fundamental
solution with initial data prescribed on the line ρ = 1. This is in contrast with the standard
approach which prescribes initial data on the vacuum line ρ=0.
TheneedofalargefamilyofweakentropiesfortheYoungmeasurereductionwasdemonstrated
by DiPernafor the isentropicgas dynamics equationswith the pressurelaw p=ργ, γ >1. When
γ = 2n+3, with n being integer, DiPerna used weak entropies which are progressive waves given
2n+1
byLax. ThemethodofTartarandDiPernawasthenextendedbySerre[29]tostrictlyhyperbolic
systems of two conservation laws, by Chen, et al. [3, 8] to fluid equations with γ (1,5/3] and
∈
by Lions, Perthame, Souganidis, and Tadmor [18, 19] to the full range γ > 1. The theory was
extended to real fluid equations by Chen and LeFloch [4, 15, 5]. We also mention the important
work by Perthame and Tzavarason the kinetic formulationfor systems of two conservationlaws;
see [26, 27]. The success of these works relies on a detailed analysis of the fundamental solution
of the entropy wave equation (2.4), which is a degenerate, linear wave equation.
When γ = 1 the analysis developed in [4, 5] for the construction of entropies does not work
because the equation (2.4) degenerates at a higher degree and the Cauchy problem at the line
ρ = 0 becomes highly singular. One novelty of the present paper is to rely on symmetry group
argument to identify the entropy kernel.
For the convenience of the reader we summarize now the main steps of the proof of Theorems
2.1–2.3.
Step1. Werelyonthevanishingviscositymethodandfirstconstructasequenceofapproximate
solutions (uǫ,ρǫ), ǫ 0, defined on the strip Π, and such that
↓
2ǫr ρǫ ρ <1
2
≤ ≤
for some r > 1. The constant ρ can be chosen to be arbitrary small by introducing a rescaled,
2
initial density λρ . We will thus establish first Theorems 2.1 to 2.3 in the case when the initial
0
density is small. Then we will treat the general case by observing that the system (1.1)–(1.3) is
invariant via the symmetry (u,ρ) (u,λρ). More precisely, given an entropy solution (u,ρ) of
→
the problem (1.1)–(1.3) with initial data (u ,ρ ), the pair (u,ρ) := (u,λρ) is also an entropy
0 0 ′ ′
solution with the initial data (u ,ρ )=(u ,λρ ).
′0 ′0 0 0
Step 2. Next, we prove that there is a sequence ǫ 0 such that
↓
(2.10) Wǫ :=ρǫeuǫ ⇀W, Zǫ :=ρǫe uǫ ⇀Z weakly ⋆ in L (Π)
− ∞loc
and there exist Young measures ν , associated with the sequence ǫ 0 and defined on the
x,t
↓
(W,Z) plane for each point (x,t) Π, such that
− ∈
limF(Wǫ(x,t),Zǫ(x,t))= F(α,β)dν = ν ,F =: F in D(R2)
x,t x,t ′
ǫ 0 h i h i
→ ZZ
for any F(α,β) C (R2). The crucial point in the compensated compactness argument is to
loc
∈
prove that ν is a point mass measure. In that case the convergence in (2.10) becomes strong in
any Lr (Π), 1 r < .
loc ≤ ∞
Step 3. Given two entropy pairs (η ,q ), obeying the conditions of Theorem 2.1, we check that
i i
Tartar’s commutation relations
(2.11) ν ,η q η q = ν ,η ν ,q ν ,η ν ,q
x,t 1 2 2 1 x,t 1 x,t 2 x,t 2 x,t 1
h − i h ih i−h ih i
hold. Here,weapplytheso-calleddiv-curllemmaofMurat[22]andTartar[32,33]. Theobjective
is to prove that the measure ν is a point mass measure by using a “sufficiently large” class of
entropy pairs in (2.11).
ISOTHERMAL COMPRESSIBLE FLUIDS 5
Step4. Toproducealargefamilyofentropypairs,wehavetoconstructafundamentalsolution
χ(R,u s) (where R := lnρ) of the entropy equation (2.4). To this end, we rely on symmetry
−
group arguments for the equation (2.4). We find that it has an invariant solution
1
η(u,ρ)=√ρf(u2 ln2ρ), where ξf′′(ξ)+f′(ξ)+ f(ξ)=0.
− 16
Then we define
χ=eR/2f(u s2 R2)1 .
u s<R
| − | − | − | | |
The function f(ξ) can be represented by a Bessel function of zero index.
Step 5. Then we search for the entropy pairs in the form
(2.12) η = χ(R,u s)ψ(s)ds, q = σ(R,u,s)ds,
−
Z Z
whereψ L1(R)isarbitraryandwedescribepropertiesofthekernelsχ,σ. Inparticular,wefind
∈
that σ = uχ(R,u s)+h(R,u s), where the function h is given by an explicit formula. We
− −
also will show that
Pχ:=∂ χ=eR/2 δ δ +Gχ(R,u s)1 ,
s s=u R s=u+R u s<R
−| |− | | − | − | | |
(cid:16) (cid:17)
Ph:=∂ h=eR/2 δ +δ +Gh(R,u s)1 ,
s s=u R s=u+R u s<R
−| | | | − | − | | |
where Gχ(R,v) and Gh(R,v) are(cid:16)bounded, continuous(cid:17)functions.
Step 6. Finally, we plug the entropy pairs (2.12) in Tartar’s commutation relations,but in the
formderivedby Chen and LeFloch[4]. We arriveafter cancellationof ψ atthe following equality
in D(R)3
′
χ P h h P χ P χ + h P χ χ P h P χ = P h P χ P χ P h χ ,
1 2 2 1 2 2 3 3 1 3 3 1 3 3 2 2 3 3 2 2 3 3 2 2 1
h − ih i h − ih i −h − ih i
wherethenotationsg :=g(R,u,s )andP g :=∂ g(R,u,s )areused. Thenwetestthisequality
i i i i si i
with the function
1 s s s s
1 2 1 3
ψ(s )ϕ ( − )ϕ ( − ),
δ2 1 2 δ 3 δ
where ψ D(R) and ϕ are molifiers such that
j
∈
s2
∞
Y := ϕ (s )ϕ (s ) ϕ (s )ϕ (s ) ds ds =0.
2 2 3 3 3 2 2 3 2 3
− 6
Z−∞Z−∞(cid:0) (cid:1)
This identity involves products of measures by functions of bounded variation. Such products
were earlier discussed by Dal Maso, LeFloch, and Murat [7].
By letting δ go to zero we obtain the equalities
(2.13) Y D(ρ)ρ ρ′dν(W ,Z )dν(W,Z)=0,
′ ′
ZZW,Z ZZ W′<W ∩ Z′<1/W p
(cid:8) (cid:9) (cid:8) (cid:9)
(2.14) Y D(ρ)ρ ρ′dν(W′,Z′)dν(W,Z)=0,
ZZW,Z ZZ W′<1/Z ∩ Z′<Z p
where (cid:8) (cid:9) (cid:8) (cid:9)
1 15 1
ρ=(WZ)1/2, D(ρ)=√ρ( + ln ), ρ′ =(W′Z′)1/2,
−2 8 ρ
and the measure dν(W ,Z ) is a copy of dν on the (W ,Z )-plane. At this point we choose
′ ′ ′ ′
the constant ρ2 (see Step 1) small enough to ensure the inequality D(ρ) √ρ/2. Hence, it
≥
follows from (2.13),(2.14) that dν = αδ +µ and α(1 α) = 0, where P(x,t) is a point
x,t P x,t
−
on the (W,Z) plane and the support of the measure supp µ lies in the set ρ = 0 . This
x,t
− { }
representationformula for the measureν enables us to justify the passageto the limit as ǫ 0.
x,t
↓
We summarize Step 6 in the following key result.
6 LEFLOCHANDSHELUKHIN
Theorem 2.4. Let (m ,ρ ) be a bounded in L (Π) sequence of entropy solutions of the problem
n n ∞
(2.1) and such that
0 ρ , m cρ (1+ lnρ )
n n n n
≤ | |≤ | |
uniformlyinn. Then,passingtoasubsequenceifnecessary,(m ,ρ )convergesalmosteverywhere
n n
in Π to an entropy solution (m,ρ) of (2.1).
3. Vanishing viscosity method
Given parameters ǫ,ǫ >0 we consider the Cauchy problem
1
(3.1) ρ +(ρu) =ǫρ +2ǫ u ,
t x xx 1 x
(3.2) (ρu) +(ρu2) +ρ =ǫ(ρu) +ǫ (u2) +2ǫ (lnρ) ,
t x x xx 1 x 1 x
with initial condition
(3.3) ρ =ρǫ +2ǫ , u =uǫ.
|t=0 0 1 |t=0 0
Inthissectionweestablishtheexistenceofsmoothsolutionstothisproblem. Laterinthissection
we will assume that ǫ =ǫr for some r >1. The positivity of the density will be obtained by the
1
following argument.
Lemma 3.1. (Positivity for convection-diffusion equations.) If v = v(x,t) is a smooth bounded
solution of the Cauchy problem
(3.4) v +(uv) =ǫv , v =v (x),
t x xx t=0 0
|
where u=u(x,t) L (Π) and u L (R), then v 0 provided v 0.
∞ 0 ∞ 0
∈ ∈ ≥ ≥
Proof. Given R>0, let ψ :R R be a non-increasing function of class C2 such that ψ(x)=1
+
→
for x [0,R], ψ(x) = e x for x 2R, and ψ(x) is a non-negative polynomial for R x 2R.
−
Denot∈e Ψ(x)=ψ(x) for x R.≥Clearly, ≤ ≤
| | ∈
′ c1 ′′ c1
(3.5) Ψ (x) Ψ(x), Ψ (x) Ψ(x)
| |≤ R | |≤ R2
for some constant c >0. The map
1
v2+µ2 µ, v 0,
Uµ(v)= 0, − v ≤>0,
(cid:26) p
is a regularizationof the mapping v v :=max v,0 .
7→ − {− }
Using (3.4) and (3.5) we can compute the t-derivative of the integral ΨU (v)dx:
µ
d ΨU dx+ǫ Ψv2∂2Uµ dx R
dt µ x ∂v2
Z Z
∂2U ∂U
(3.6) = µvv (ǫΨ +uΨ)dx+ v µ(ǫΨ +uΨ )dx
∂v2 x x ∂v xx x
Z Z
∂2U ∂U
µv v Ψ(ǫc /R+ u)dx+ v µ Ψ(ǫc /R2+ uc /R)dx.
≤ ∂v2 | x| 1 | | | ∂v | 1 | | 1
Z Z
Observe that
c u c u
ǫv2 v v (ǫc /R+ u)=ǫ(v v( 1 + | |))2 v2( 1 + | |)2,
x− | x| 1 | | | x|− 2R 2ǫ − 2R 2ǫ
∂2U µ2v2
v2 µ ,
∂v2 ≤ (v2+µ2)3/2
and
∂U
µ
v v as µ 0.
∂v → − →
Weintegrate(3.6)withrespecttotandletµtendtozero,bytakingintoaccountthatU (v )=0:
µ 0
t
ǫc u c
1 1
Ψv dx Ψv ( + | | )dxdτ.
− ≤ − R2 R
Z Z Z
0
ISOTHERMAL COMPRESSIBLE FLUIDS 7
By Gronwall’s lemma, Ψv dx=0. We thus conclude that v 0. (cid:3)
− ≥
As a consequence ofRLemma 3.1, we deduce that any bounded solution (u,ρ) of the problem
(3.1)–(3.3) has the following property:
(3.7) ρ 2ǫ uniformly in ǫ.
1
≥
Namely, this is clear since the function v =ρ 2ǫ solves the problem
1
−
v +(uv) =ǫv , v 0.
t x xx t=0
| ≥
Fromnow on, we assume that the initial data ρǫ anduǫ belong to the Sobolev space H2+β(R)
0 0
for some 0<β <1 and satisfy
0 ρǫ M, uǫ u ,
≤ 0 ≤ k 0k∞ ≤ 1
and
uǫ u , ρǫ ρ in L1 (R),
0 → 0 0 → 0 loc
where u := u and M := ρ .
1 0 0
k k∞ k k∞
Lemma 3.2. Let (u,ρ) be a smooth bounded solution of the Cauchy problem (3.1)–(3.3). Then
there exist positive constants c , ρ , W , and Z such that
1 1 1 1
2ǫ ρ ρ , m :=ρ u c ρ(1+ lnρ) m , ρ :=(2ǫ +M)eu1,
1 1 1 1 1 1
≤ ≤ | | | |≤ | | ≤
m :=c sup ρ(1+ lnρ),
1 1
| |
(3.8) 0 ρ ρ1
≤ ≤
0 W :=ρeu W , 0 Z :=ρe u Z ,
1 − 1
≤ ≤ ≤ ≤
uniformly in ǫ.
Proof. Passing to the Riemann invariant variables
w :=u+lnρ, z :=u lnρ,
−
we can rewrite the system (3.1)-(3.2) as
2ǫ ǫz 3ǫw ǫz2
w +w u+1 1 + x x =ǫw x,
t x xx
− ρ 2 − 4 − 4
(cid:16) (cid:17)
2ǫ ǫw 3ǫz ǫw2
z +z (u 1+ 1 x + x)=ǫz + x.
t x xx
− ρ − 2 4 4
By the maximum principle,
w maxw (x), z minz (x).
0 0
≤ ≥
Now, the estimates (3.8) is a simple consequence of these inequalities. (cid:3)
By the estimates (3.8)there existsequencesWǫn, Zǫn, ρǫn, andmǫn := ρǫnuǫn anda family of
non-negativeprobabilitymeasuresν ,calledYoungmeasures,definedonthe (W,Z)-plane,such
x,t
that
(3.9) Wǫn ⇀W, Zǫn ⇀Z, ρǫn ⇀ρ, ρǫnuǫn ⇀m weakly ⋆ in L∞loc(Π),
and
F(Wǫn(x,t),Zǫn(x,t)) F ϕ(x,t)dtdx 0,
−h i →
ZZΠ(cid:16) (cid:17)
where we have set F := F(W,Z)dν for any test function ϕ D(R2) and any continuous
h i W,Z x,t ∈
function F(W,Z) C (R2). Moreover,
∈ loc R
suppν (W,Z):0 W W , 0 Z Z .
x,t 1 1
⊂{ ≤ ≤ ≤ ≤ }
For a proof that to each bounded sequence v (x,t) one can associate a Young measure µ , we
n x,t
refer to Tartar [32] and Ball [1]; see also [30].
8 LEFLOCHANDSHELUKHIN
Lemma 3.3. (Entropy dissipation estimate.) The smooth solution (u,ρ) of the Cauchy problem
(3.1)-(3.3) satisfies the estimate
ǫρ2
(3.10) k ρx +ǫρu2xkL1loc(Π) ≤c
uniformly in ǫ.
Proof. The identity
∂ ρu2 ǫρ2
+(1+ρlnρ ρ) + x +ǫρu2
∂t 2 − ρ x
(3.11)
(cid:16) (cid:17)
∂ ρu3 ρu2 ǫ u3
1
= +uρlnρ ǫρ lnρ 2ǫ ulnρ ǫ( ) =: J
x 1 x x
−∂x{ 2 − − − 2 − 3 } −
follows immediately from (3.1) and (3.2). Multiplying this identity by the function Ψ(x) intro-
duced in the proof of Lemma 3.1 and integrating with respect to x we deduce, in view of the
estimates (3.7) and (3.8),
1 ǫρ2 ρu2
JΨ dx Ψ( x +ǫρu2)dx+c Ψ(1+ )dx.
x ≤ 2 ρ x 2
Z Z Z
Hence, we have
T ǫρ2
Ψ( x +ǫρu2)dxdt c,
ρ x ≤
Z0 Z
which yields the desired estimate. (cid:3)
We rewrite the equations (3.1)-(3.2) as a quasi-linear parabolic system:
(3.12) u +a (u,ρ,u ρ )=ǫu , ρ +a (u,ρ,u ρ )=ǫρ ,
t 1 x x xx t 2 x x xx
where we have set
ρ 2ǫρ u 2ǫ ρ
x x x 1 x
a :=uu , a :=(ρu) 2ǫ u .
1 x− ρ − ρ − ρ2 2 x− 1 x
In view of (3.7) and (3.8), we obtain the global a priori estimates
2ǫ ρ ρ , u c(u ,ρ ,ǫ ).
1 1 1 1 1
≤ ≤ | |≤
With these estimates at hand, it is a standard matter to derive estimates in Ho¨lder’s norms,
depending on ǫ, by standard techniques of the theory quasi-linear parabolic equations [13]. We
will only sketch the derivation. Let ζ(x,t) be a smooth function such that ζ = 0 only if x ω,
6 ∈
where ω is an interval [x σ,x +σ]. Denote
0 0
−
u(n) :=max u n,0 .
{ − }
Multiplyingthesecondequationin(3.12)byζ2ρ(n) andintegratingwithrespecttox,oneobtains
d
ζ2 ρ(n) 2dx+ǫ ζ2 ρ(n) 2dx γ (ζ2+ζ ζ )ρ(n) 2dx+γ ζ1 dx.
dt | | | x | ≤ x | t| | | ρ≥n
Z Z Z Z
Similarly, for the velocity variable one gets
d
ζ2 u(n) 2dx+ǫ ζ2 u(n) 2dx γ (ζ2+ζ ζ )u(n) 2+ζ1 +ǫζ2 ρ(n) 2dx.
dt | | | x | ≤ x | t| | | ρ≥n | x |
Z Z Z
These inequalities imply that u and ρ belong to a class B (Q,M,γ,r,δ,n) [13] (Chapter II, 7,
2
§
formula (7.5)), for some parameters Q,M,γ,r,δ, and n. Then it follows that the estimate
ku,ρkHα,α/2(ω×[0,T]) ≤c
holds for some α (0,1).
∈
In the same manner, one can estimate the Ho¨lder norm of the derivatives u , u , u , ρ , ρ ,
x xx t x xx
and ρ , in the same way as done in [11] for a general class of parabolic systems.
t
We now arrive at the main existence result, concerning the viscous approximation (3.1)-(3.3).
ISOTHERMAL COMPRESSIBLE FLUIDS 9
Lemma 3.4. (Existence of smooth solution of the regularized system.) Let uǫ, ρǫ L Hβ ,
0 0 ∈ ∞∩ loc
0<β <1. Then the Cauchy problem (3.1)-(3.3) has a unique solution such that
u,ρ L (Π) H2+β,1+β/2(Π).
∈ ∞ ∩ loc
Now,wesetǫ =ǫr,r>1,andstudycompactnessoftheviscoussolutions(uǫ,ρǫ)whenǫ 0.
1
→
Lemma 3.5. Given an entropy entropy-flux pair (η(m,ρ),q(m,ρ)), m=ρu, the sequence
∂ηǫ ∂qǫ
θǫ := +
∂t ∂x
is compact in Wl−oc1,2(Π), where ηǫ =η(mǫ,ρǫ), qǫ =q(mǫ,ρǫ).
Proof. We use the following lemma due to Murat’s lemma [23].
Let Q R2 be a bounded domain, Q C1,1. Let A bea compact set in W 1,2(Q), B be a bounded
−
⊂ ∈
set in the space of bounded Radon measures M(Q), and C be a bounded set in W 1,p(Q) for some
−
p (2, ]. Further, let D D(Q) be such that
′
∈ ∞ ⊂
D (A+B) C.
⊂ ∩
Then there exists E, a compact set in W 1,2(Q) such that D E.
−
⊂
By definition, the functions η(m,ρ) and q(m,ρ) solve the system
2m m2
q = η +η , q =η η .
m ρ m ρ ρ m− ρ2 m
Hence, calculations show that
ηǫ mηǫ mηǫ m2ηǫ ηǫ
(3.13) θǫ =2ǫ m ( ρ + m)+2ǫ ( ρ m + m)+ǫηǫρ +ǫηǫ m =
1 x ρ ρ2 1 − ρ2 − ρ3 ρ ρ xx m xx
ρ ηǫ
ǫ u (qǫ +ηǫ) 2ǫ x m +ǫηǫ ǫ[ηǫ ρ2 +ηǫ m2 +2ηǫ ρ m ].
1 x m ρ − 1 ρ xx− ρρ x mm x ρm x x
We check the conditions of Murat’s lemma. By Lemma 3.2, the sequence θǫ is bounded in
Wl−oc1,∞(Π). Hence, it is enough to show that ǫηxǫ → 0 in L2loc(Π) and the residual sequence
θǫ ǫηǫ is bounded in L1 (Π).
− xx loc
We have
qǫ +ηǫ
ǫηǫ =ǫρu ηǫ +ǫρ m ρ.
x x m x 2
Thus, by estimates (3.8) and (3.10), ǫηǫ 0 in L2 .
x → loc
Consider the sequence θǫ ǫηǫ . We have
− xx
ρ ηǫ
θǫ ǫηǫ = ǫ[ηǫ ρ2 +ηǫ m2 +2ηǫ ρ m ]+ǫ u (qǫ +ηǫ) 2ǫ x m.
− xx − ρρ x mm x ρm x x 1 x m ρ − 1 ρ
Each term on the right hand-side is bounded in L1 provided ǫ =ǫ. Indeed, by (3.7),
loc 1
2ǫ ρ1/2 u 2ǫ ρ √2ǫρ
2ǫ u = 1 | x| √2ǫρ1/2 u , 1| x| | x|.
1| x| ρ1/2 ≤ | x| ρ ≤ ρ1/2
Moreover,if ǫ =0(ǫ),
1
ρ ηǫ
(3.14) ǫ u (qǫ +ηǫ) 2ǫ x m 0 in L2 (Π).
1 x m ρ − 1 ρ → loc
The other terms are treated similarly. This completes the proof. (cid:3)
Given two entropy pairs (η (m,ρ),q (m,ρ)), (i=1,2), from Lemma 3.5, we define
i i
η˜(W,Z)=η (f (W,Z),f (W,Z)), q˜(W,Z)=q (f (W,Z),f (W,Z)).
i i 2 1 i i 2 1
Clearly, the functions
∂ η˜ǫ+∂ q˜ǫ
t i x i
are compact in Wl−oc1,2(Π). Hence, by the div-curl lemma [32], Tartar’s commutation relation
(3.15) η˜ q˜ η˜ q˜ = η˜ q˜ η˜ q˜
1 2 2 1 1 2 2 1
h − i h ih i−h ih i
10 LEFLOCHANDSHELUKHIN
is valid.
For reader’s convenience, we remind that the div-curl lemma states the following.
Let Q R2 be a bounded domain, Q C1,1. Let
⊂ ∈
wk ⇀w, wk ⇀w , vk ⇀v , vk ⇀v ,
1 2 2 1 1 2 2
weakly in L2(Q), as k . With curl(w ,w ) denoting ∂w /∂x ∂w /∂x , suppose that
1 2 2 1 1 2
→ ∞ −
the sequences div(vk,vk) and curl(wk,wk) lie in a compact subset E of W 1,2(Q). Then, for a
1 2 1 2 −
subsequence,
vkwk+vkwk v w +v w in D(Q) as k .
1 1 2 2 → 1 1 2 2 ′ →∞
The further claim is due to the fact that system (1.1)-(1.3) is invariant with respect to the
scaling ρ λρ.
→
Lemma 3.6. If (m,ρ) is an entropy solution with initial data (m ,ρ ) then (cm,cρ) is also the
0 0
entropy solution with the initial data (cm ,cρ ), where c is an arbitrary positive constant.
0 0
Proof. The claim follows easily from the fact that the pair (η(cm,cρ),q(cm,cρ)) is an entropy-
entropy flux pair as soon as the pair (η(m,ρ),q(m,ρ)) is an entropy-entropy flux pair.
Given λ>0, let us consider the auxiliary problem
(3.16) ρ +(ρu) =ǫρ +2ǫ u ,
t x xx 2 x
(3.17) (ρu) +(ρu2) +ρ =ǫ(ρu) +ǫ (u2) +2ǫ (lnρ) ,
t x x xx 2 x 2 x
(3.18) ρ =λρǫ(x)+2ǫ , u =uǫ(x),
|t=0 0 2 |t=0 0
where ǫ =λǫ =λǫr.
2 1
The main feature of the auxiliary problem is the following. If the functions (u ,ρ ) solve the
ǫ ǫ
′ ′
problem (3.1)-(3.3) then the functions (u ,ρ ) solve the problem (3.16)-(3.18) with ρ = λρ .
ǫ ǫ ǫ ǫ
The solution (u ,ρ ) of problem (3.16)-(3.18) obeys the estimates
ǫ ǫ
(3.19) 2ǫ ρ (2ǫ +λ ρ )e u0 ∞ =:ρ , u ρ cρ (1+ lnρ )
2 ǫ 2 0 k k 2 ǫ ǫ ǫ ǫ
≤ ≤ k k∞ | |≤ | |
uniformly in ǫ. Lemmas 3.3-3.5 are also valid for (u ,ρ ). The correspondingYoung measureν
ǫ ǫ x,t
has a finite support:
(3.20) supp ν (W,Z):0 W W , 0 Z Z :=K.
x,t 2 2
⊂{ ≤ ≤ ≤ ≤ }
We impose the following smallness conditions for λ:
1 8
(3.21) ρ <1, ln .
2
ρ ≥ 15
2
Assumethatthesolution(u ,ρ )ofthe auxiliaryproblemconvergestoanentropysolution(m,ρ)
ǫ ǫ
of the problem (2.1):
(u ρ ,ρ ) (m,ρ) almost everywhere in Π.
ǫ ǫ ǫ
→
The initial data for (m,ρ) are
ρ =λρ , m =λm .
t=0 0 t=0 0
| |
′ ′
By Lemma 3.6, the functions (m ,ρ ) = (m/λ,ρ/λ) is an entropy solution of the same problem
with the initial data
′ ′
ρ =ρ , m =m .
t=0 0 t=0 0
| |
Thus, it is enough to study convergence of the solutions to the auxiliary problem.
With the condition (3.21) at hand, the function
1 15R
D(R):=( + | |)eR/2, R:=lnρ,
−2 8
from Section 5 admits the estimate D(R) 1eR/2. Hence, D(R) vanishes only at the vacuum
≥ 2
points ρ=0.