Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier-Stokes equations

0.12 MB·English

by  Xiangdi Huang

EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS OF THE
COMPRESSIBLE SPHERICALLY SYMMETRIC NAVIER-STOKES
EQUATIONS
XIANGDIHUANG
5
1
0
2
Abstract. OneofthemostinfluentialfundamentaltoolsinharmonicanalysisisRiesz
n
transform. It maps Lp functionsto Lp functionsfor any p ∈ (1,∞) which plays an
a
J importantrole in singular operators. As an application in fluid dynamics, the norm
equivalencebetweenk∇uk andkdivuk +kcurluk iswellestablishedforp∈(1,∞).
1 Lp Lp Lp
However,sinceRieszoperatorssentboundedfunctionsonlytoBMOfunctions,there

] is no hope to bound k∇uk in terms of kdivuk + kcurluk . As pointed out by
P L∞ L∞ L∞
Hoff[SIAMJ.Math. Anal. 37(2006),No. 6,1742-1760],thisisthemainobstacleto
A
obtainuniquenessofweaksolutionsforisentropiccompressibleflows.
.
h Fortunately,basedonnewobservations,see Lemma2.2, we derivean exactesti-
t matefork∇uk ≤(2+1/N)kdivuk foranyN-dimensionalradiallysymmetricvector
a L∞ L∞
m functionsu. Asadirectapplication,wegiveanaffirmativeanswertotheopenprob-
lemofuniquenessofsomeweaksolutionstothecompressiblesphericallysymmetric
[
flowsinaboundedball.

1
v
2
6 1. Introductionand main results
2
0
0 We are concerned with the isentropic system of compressible Navier-Stokes equa-
1. tionswhichreads as
0 ρ +div(ρU) = 0,
5 t
(1.1)
1 (ρU) +div(ρU ⊗U)+∇P = µ△U +(µ+λ)∇(divU),
:  t
v
i where t ≥ 0,x∈ Ω ⊂ RN(N = 2,3), ρ = ρ(t,x) and U = U(t,x) are the density and
X
fluid velocityrespectively,and P = P(ρ)isthepressuregivenby astateequation
r
a (1.2) P(ρ) = aργ
with the adiabatic constant γ > 1 and a positiveconstant a. The shear viscosity µ and
thebulkoneλare constantssatisfyingthephysicalhypothesis
N
(1.3) µ > 0, µ+ λ ≥ 0.
2
ThedomainΩisaboundedball witharadiusR, namely,
(1.4) Ω = B = {x ∈ RN; |x| ≤ R < ∞}.
R
Westudyan initialboundaryvalueproblemfor(1.1)withtheinitialcondition
(1.5) (ρ,U)(0,x) = (ρ ,U )(x), x ∈ Ω,
0 0
and theboundarycondition
(1.6) U(t,x) = 0, t ≥ 0, x ∈ ∂Ω,
2010MathematicsSubjectClassification. 35Q30,76N10
1
2 XIANGDIHUANG
andwearelookingforthesmoothsphericallysymmetricsolution(ρ,U)oftheproblem
(1.1), (1.5),(1.6)which enjoystheform
x
(1.7) ρ(t,x) = ρ(t,|x|), U(t,x) = u(t,|x|) .
|x|
Then,fortheinitialdatatobeconsistentwiththeform(1.7),weassumetheinitialdata
(ρ ,U )alsotakes theform
0 0
x
(1.8) ρ = ρ (|x|), U = u (|x|) .
0 0 0 0 |x|
In thispaper, wefurtherassumetheinitialdensityisuniformlypositive,that is,
(1.9) ρ = ρ (|x|) ≥ ρ > 0, x ∈ Ω
0 0
for a positive constant ρ. Then it is noted that as long as the classical solution of
(1.1),(1.5),(1.6) exists the density ρ is positive, that is, the vacuum never occurs. It is
also notedthatsincetheassumption(1.7)implies
(1.10) U(t,x)+U(t,−x) = 0, x ∈ Ω,
wenecessarily haveU(t,0) = 0 (alsoU (0) = 0).
0
There are many results about the existence of local and global strong solutions in
timeoftheisentropicsystemofcompressibleNavier-Stokesequationswhentheinitial
densityisuniformlypositive(referto[1,13,15,16,23–25,28,29]andtheirgeneraliza-
tion [20–22,27] to the full system including the conservation law of energy). On the
otherhand, for theinitialdensityallowingvacuum,thelocal well-posednessofstrong
solutions of the isentropic system was established by Kim [17]. For strong solutions
with spatial symmetries, the authors in [18] proved the global existence of radially
symmetricstrongsolutionsoftheisentropicsysteminan annulardomain,evenallow-
ing vacuum initially. However, it still remains open whether there exist global strong
solutionswhicharesphericallysymmetricinaball. Themaindifficultieslieonthelack
ofestimatesof thedensityand velocitynearthecenter. In the casevacuumappears, it
isworthnotingthatXin[30]establishedablow-upresultwhichshowsthatiftheinitial
densityhasacompactsupport,thenanysmoothsolutiontotheCauchyproblemofthe
fullsystemofcompressibleNavier-Stokesequationswithoutheatconductionblowsup
in a finite time. The same blowup phenomenon occurs also for the isentropic system.
Indeed,Zhang-Fang([31],Theorem1.8)showedthatif(ρ,U) ∈ C1([0,T];Hk)(k > 3)
isasphericallysymmetricsolutiontotheCauchyproblemwiththecompactsupported
initial density, then the upper limit of T must be finite. On the other hand, it’s unclear
whether the strong (classical) solutions lose their regularity in a finite time when the
initialdensityis uniformlyaway fromvacuum.
On the otherhand, there are amount ofliteratures investigatingtheglobal existence
of weak solutions to the compressibleNavier-Stokes equations, such as ”finite energy
solutions” proposed and developed by Lions [19], Hoff [7] and Feireisl [4], etc. One
remarkable result is due to Jiang-Zhang [14], where they prove a global existence for
three-dimensional compressible spherically symmetric flows with γ > 1. However,
whether their weak solution is unique remains a long-standing open problem. Mean-
while, Desjardins [3] built a more regular weak solution for three-dimensional torus.
Inspiredby hiswork and anewobservationin Lemma2.2, wewillgivesomepositive
answerto thesphericallysymmetricflowin thispaper.
SPHERICALLYSYMMETRICFLOWS 3
In the spherical coordinates, the original system (1.1) under the assumption (1.7)
takes theform
ρu
ρ +(ρu) +(N −1) = 0,
t r
r
(1.11) 
(ρu)t + ρu2 +P(ρ) +(N −1)ρu2 = κ ur +(N −1)u
whereκ = 2µ+λ. Now,(cid:16)weconside(cid:17)rr thefollowinrgLagr(cid:18)angiantransfror(cid:19)mr ation:
r
(1.12) t = t, y = ρ(t,s)sN−1ds.
Z
0
Then, itfollowsfrom (1.10)that
(1.13) y = −ρurN−1, r = u, r = (ρrN−1)−1,
t t y
and thesystem(1.11)can befurtherreduced to
ρ +ρ2(rN−1u) = 0,
t y
(1.14)

r1−Nut +Py = κ ρ(rN−1u)y y
(cid:16) (cid:17)
wheret ≥ 0,y ∈ [0,M ]andM is defined by
0 0
R R
(1.15) M = ρ (r)rN−1dr = ρ(t,r)rN−1dr,
0 0
Z Z
0 0
according totheconservationofmass. Notethat
(1.16) r(t,0) = 0, r(t,M ) = R.
0
Wedenoteby E theinitialenergy
0
R u2 aργ
(1.17) E = ρ 0 + 0 rN−1dr,
0 Z 0 2 γ−1!
0
and defineacuboid Q , fort ≥ 0 and y ∈ [0,M ], as
t,y 0
(1.18) Q = [0,t]×[y,M ].
t,y 0
From nowon,wedenotefor p ∈ [1,∞)anda radiallysymmetricfunction f = f(|x|)
1
R p
(1.19) kfkLp(Ω) = ( fpdx)1p = ωN fprN−1dr
ZΩ Z0 !
where
(1.20) ω = N|B |, |B | standsforthevolumeofN-dimensionalunitball.
N 1 1
WefirstprovealocalexistenceofweaksolutionstothecompressibleNavier-Stokes
equationsinTheorem 1.1.
Theinitialdataare supposedtosatisfy(1.8-1.9)and
ρ ∈ L∞(Ω)
0
(1.21)
U ∈ H1(Ω)N.
 0

4 XIANGDIHUANG
Theorem 1.1. Assume Ω = B is a bounded ball in RN for N = 2,3 and γ > 1, then
R
there exists T ∈ (0,+∞] and a weak solution (ρ,U) to the Navier-Stokes equations
0
(1.1)in(0,T )suchthatforallT < T ,
0 0
ρ ∈ L∞((0,T)×Ω),
ρU˙ = ρ(U +U ·∇U),∇U˙ ∈ L2((0,T)×Ω)N,
(1.22) divU ∈ L2t(0,T;L∞(Ω)),
Remark1.1. Thekey∇idUea∈tLo∞e(s0ta,bTl;isLh2(lΩoc)a)Nle∩xiLst2e(n0c,eTo;fLw∞e(Ωak))solutionwithregularity
(1.22) is to derive uniform upper bound of the density, which is analogous to Des-
jardins [3]. However, there are two obstacles in bounded domain. First of all, due
to the lack of commutator estimates, whether there is local weak solution with higher
regularity (1.22) for system (1.1) remains unknown for initial boundary value prob-
lem. We rewrite it in spherically coordinate and derive some new estimates to play a
critical role as commutator estimates frequently used by [3,19] for Cauchy problem
and torus. On the other hand, general global finite energy weak solution with γ > 1
was proved by Jiang-Zhang [14]. However, their weak solution is much less regular
than (1.22) and leave a challenging problem on uniqueness of such weak solutions.
The main value in Theorem 1.1 is to weak the assumption from γ > 3 in [3] to γ > 1
for three-dimensional spherically symmetric flow, and further more, give an affirma-
tive answer on uniqueness of weak solutions stemming from regularity class (1.22),
which is our main issue in Theorem 1.2. The technical part lies on the combination
ofCaffarelli-Kohn-Nirenberg[2]inequalitieswithweightsandpointwiseestimatesfor
radiallysymmetricfunctions,seeLemmas 2.1-2.2.
Remark 1.2. From many early works on the blowup criterion [10–12,26] of strong
solutions to the compressible Navier-Stokes equations, the uniform bound of the den-
sity induces the regularity of ρU˙, ∇U˙, divU and curlU, as indicated by the first three
lines in Theorem 1.1. Besides, one of the most important observations is Lemma 2.2,
which gives desired bound for k∇Uk in terms of kdivUk . This is a key ingredient
L∞ L∞
in provingtheuniquenessof weaksolutionsillustratedin thefollowingtheorem.
Theorem1.2. Let(ρi,Ui)betwoweaksolutionsof(1)obtainedbyTheorem1.1. Then
(1.23) ρ1 = ρ2, U1 = U2 a.eon(0,T)×Ω.
Remark 1.3. There are many weak-strong uniqueness results [5,6, 8] concerning
compressible Navier-Stokes equations. However, weak-weak uniqueness incorporates
more difficulties and need special attention. Up to now, the most far reaching result
appears in [8]. As pointed out by Hoff [8], the main obstacle to prevent us from es-
tablishing uniqueness is whether we can prove ∇U ∈ L1L∞ instead of ∇U ∈ L1BMO.
Such a fact was first verified by Hoff [9], where theLipschitzregularityof thevelocity
U was establishedwith piecewiseCα density. This is a onlyresultconcerning unique-
ness of weak solutions. Withthe help of Lemma 2.2 and Theorem 1.1, we also give an
affirmativeanswerto thesphericallysymmetriccase.
2. Proof ofTheorem 1.1
FirstwerecallthefollowingfamousCaffarelli-Kohn-Nirenberg[2]inequalitieswith
weights.
SPHERICALLYSYMMETRICFLOWS 5
Lemma2.1(Caffarelli-Kohn-Nirenberg). Thereexistsapositiveconstantsuchthatthe
followinginequalityholdsforallu ∈ C∞(Rn)
0
(2.1) |x|γu ≤ C |x|αDua |x|βu1−a
Lr Lp Lq
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
ifand onlyif thefollowingr(cid:12)elatio(cid:12)nshold(cid:12): (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
1 γ 1 α−1 1 β
(2.2) + = a + +(1−a) +
r n p n ! q n!
(2.3) 0 ≤ α−σ if a > 0,
and
1 α−1 1 γ
(2.4) α−σ ≤ 1 ifa > 0 and + = + .
p n r n
satisfying
p,q ≥ 1, r > 0, 0 ≤ a ≤ 1
1 α−1 1 β 1 γ
(2.5) + , + , + > 0,
p n q n r n
γ = aσ+(1−a)β
Furthermore, on anycompact set in which (2.5) and 0 ≤ α−σ ≤ 1 hold,the constant
C isbounded.
ThefollowinglemmaisessentialinprovingTheorems1.1and 1.2.
Lemma 2.2. Assume Ω is either a bounded ball B with radius R or the whole space
R
RN, then for any N-dimensional radially symmetric vector functions U(x) = u(|x|) x
|x|
for x ∈ Ω,we havethefollowingestimates
(1) Wehave
1 1
(2.6) kdivUk ≤ k∇Uk ≤ (2+ )kdivUk .
L∞(Ω) L∞(Ω) L∞(Ω)
N N
(2) For p ∈ [1,∞), wehave
|ur| ≤ (N1)1−1pr−Npω−N1pkdivUkLp(Ω)
(2.7) |ur| ≤ 1+(N −1)(N1)1−1pr−Npω−N1p!kdivUkLp(Ω)
where 
(2.8) ω = N|B |.
N,r 1
Proof. Set r = |x|, obviously,
u
(2.9) divU = u +(N −1) , curlU = 0.
r
r
Denoteby
u
(2.10) u +(N −1) = F.
r
r
It followsfrom (2.10)that
(2.11) (rN−1u) = rN−1F,
r
6 XIANGDIHUANG
which gives
r r rN
(2.12) rN−1u = sN−1Fds ≤ kFk ( sN−1ds) = kFk .
L∞ L∞
Z Z N
0 0
Consequently,
u 1
(2.13) | | ≤ kFk .
L∞
r N
And
u 2N −1
(2.14) ku k ≤ kFk +(N −1)k k ≤ kFk .
r L∞ L∞ L∞ L∞
r N
On theotherhand, asr = xi,we have
xi r
x x x x
∂ (u j) = u i j +u( j)
xi r r r2 r xi
(2.15)
x x x x u
= i ju +(δ − i j) .
r2 r ij r2 r
It immediatelyimplies
u 1
(2.16) k∇Uk ≤ ku k +2k k ≤ (2+ )kFk .
L∞(Ω) r L∞(Ω) L∞(Ω) L∞(Ω)
r N
For p ∈ (1,∞),r ∈ (0,∞),oneobtains
r
rN−1u = sN−1Fds
Z
0
r 1 r 1−1
p p
(2.17) ≤ FpsN−1ds sN−1ds
Z ! Z !
0 0
≤ (rNN)1−1pω−N1pkFkLp(Ω)
Hence,
(2.18) |u| ≤ (N1)1−1pr1−Npω−N1pkFkLp(Ω)
Therefore, (2.7)followsfrom(2.18)and (2.9).
ThisfinishestheproofofLemma. (cid:3)
Weonlyprovethecasewhen N = 3 sincethecase N = 2is evensimpler. Through-
out of this section, we assume that (ρ,U) with the form (1.7) is the solution to the
initial boundary value problem (1.1),(1.5),(1.6) in [0,T] × Ω, and we denote by C
genericpositiveconstantsonlydependingon theinitialdataand timeT.
WegiveasketchofproofofTheorem1.1,sincethemainideacanbeborrowedfrom
Desjardins[3]. Denote
(2.19) Φ(t) = 1+kρk +kPk2 +k∇Uk2 .
L∞(Ω) L2(Ω) L2(Ω)
Ourmainprocedureis toderivethefollowingestimatefor t < 1,
t
(2.20) Φ ≤ C +Cexp Cexp( χ(Φ)λ(s)ds) .
Z !
0
for some positive increasing smooth funtion χ(x) and integrable function λ(s). The
existenceofT < 1followsimmediatelyfrom(2.20).
0
SPHERICALLYSYMMETRICFLOWS 7
Say denotingbyζ(t)therighthandsideof(2.20), weconcludethat
d
(2.21) ζ(t) ≤ Cπ(ζ(t))λ(t),
dt
forsomesmoothfunctionπ. Thus,wehave
ζ(t) du t
(2.22) ≤ λ(s)ds,
Z π(u) Z
0 0
so thatthereexistsT such thatforall T < T < 1,
0 0
(2.23) Φ ≤ C .
T
We first have the following basic energy estimate. Since the proof is standard, we
omitit.
Lemma 2.3. It holdsforany0 ≤ t ≤ T,
|U|2 aργ t |U |2 aργ
(2.24) ρ + dx+κ |∇U|2dxdτ ≤ ρ 0 + 0 dx,
ZΩ 2 γ−1! Z0 ZΩ ZΩ 0 2 γ−1!
or equivalently,
R u2 aργ t R u2
ρ + rN−1dr+κ u2 + rN−1drdτ
Z 2 γ−1! Z Z r r2!
0 0 0
(2.25)
R u2 aργ
≤ ρ 0 + 0 rN−1dr = E .
Z 0 2 γ−1! 0
0
Denotedby
(2.26) G = κdivU −P
as effectiveflux. Themomentumequations(1.1) can berewrittenas
2
(2.27) ρU˙ = ∇G.
The main difficulty lies in the bound of the density. We will work it in spherical
coordinateassystem(1.14)as follows.
Lemma2.4. Thereexistsasmoothpositiveincreasingfunctionχ(x)andanintegrable
functionλ(t)suchthat
(2.28)
t
ρ(t,y) ≤ Cexp Cβ(t)+Cexp Cβ(t)+C χ(Φ)λ(s)ds , (t,y) ∈ [0,1]×[0,M ].
0
Z !!
0
where
(2.29) β(t) = kPk2 +k∇Uk2
L2(Ω) L2(Ω)
Remark 2.1. Fromtheworkof[3],refer to (103),we can prove
t
(2.30) β(t) ≤ Cexp C χ(Φ)λ(s)ds .
Z !
0
Therefore, theleftwork isconcentratedon Lemma 2.4.
8 XIANGDIHUANG
Proof. Step 1. Inviewof(1.14), itholds
ρ
κ(logρ) = κ( t) = −κ(ρ(rN−1u) ) = −r1−Nu − p
ty y y y t y
ρ
(2.31)
u2
= −(r1−Nu) − p −(N −1) .
t y rN
Thus,integrating(2.31)over(0,t)×(0,y),wededuce that
ρ(t,y) ρ (y) y
κlog = κlog 0 + (r1−Nu)(0,z)−(r1−Nu)(t,z) dz
ρ(t,0) ρ (0) Z
(2.32) 0 0 (cid:16) (cid:17)
t t y u2(s,z)
+ (p(s,0)− p(s,y))ds− (N −1) dzds,
Z Z Z rN
0 0 0
which isequivalentto
ρ(t,y) ρ (y) y
= 0 exp κ−1 ((r1−Nu)(0,z)−(r1−Nu)(t,z))dz
ρ(t,0) ρ (0) Z !
0 0
t
·exp κ−1 (p(s,0)− p(s,y))ds
Z !
(2.33) 0
t y u2(s,z)
·exp −κ−1 (N −1) dzds .
Z Z rN !
0 0
ρ (y)
= 0 Π3 Ψ.
ρ (0) i=1 i
0
Wefirst deal withΨ andΨ .
1 3
Step 2. It follows from Lemma 2.2 with N = 3, energy equality (2.25) and γ > 1
that
y r r
r1−N|u|dy = ρ|u|dr ≤ Ck∇UkL2(Ω) ρs−21ds
Z Z Z
0 0 0
≤ Ck∇UkL2(Ω) rρ6γs2ds 61γ r s−36γγ+−21ds 1−61γ
Z ! Z !
0 0
(2.34)
3(γ−1) 1
≤ Cr 6γ kPkLγ6(Ω)k∇UkL2(Ω)
≤ C +CkPk2 +Ck∇Uk2
L6(Ω) L2(Ω)
t
≤ C +Cβ(t)+C kPk12dt,
∞
Z
0
wherewe usedthefollowingfact.
Firstrecall that
(2.35) (P6) +div(P6u)+(6γ−1)Pdivu = 0,
t
oneimmediatelyhas
t t
(2.36) kPk6 ≤ kP k6 +C |P6divu|dxds ≤ C +C kPk12ds.
L6(Ω) 0 L6(Ω) Z0 ZΩ Z0 ∞
Hence,
t
(2.37) Ψ ,Ψ−1 ≤ Cexp Cβ(t)+C kPk12ds .
1 1 Z ∞ !
0
SPHERICALLYSYMMETRICFLOWS 9
Step3. Similarly,recall Lemma2.2andG = κdivU −P = κF −P
r
r2u = s2Fds
Z
0
r 1 r 5
6 6
(2.38) ≤ F6s4ds s58ds
Z ! Z !
0 0
r
≤ Cr163( F6s4ds)61
Z
0
Hence,
r r
(2.39) |u|2/r ≤ Cr−23( F6s4ds)31 ≤ Cr−23( G6s4ds)31 +Crkρk6γ.
∞
Z Z
0 0
Also,wewillusethefollowingCKN inequalityeasilyfrom Lemma2.1
(2.40) ks32GkL6(0,r) ≤ Cks∇GkL322(0,r)ksGkL312(0,r)
orequivalently,
r
(2.41) G6s4ds ≤ Ck∇Gk4 kGk2
Z L2(B(0,r)) L2(B(0,r))
0
Indeed,inviewof(2.1),taken = 1,γ = 2,r = 6,α = β = 1 > σ = 1,p = q = 2,a = 2.
3 2 3
Nowweareready togiveestimatesforΨ .
3
(2.42)
t y |u|2 t r ρ|u|2
r1−N dydτ = dsdτ, r ∈ (0,R)
Z Z r Z Z s
0 0 0 0
t r r
≤ C ( F6s4ds)31 ρs−32dsdτ
Z Z Z
0 0 0
t r
≤ C r13kρk∞( G6s4ds)13 +r13kρk2∞γ+1dτ
Z Z
0 0
t
≤ C kρk kGk32 kGk34 +kρk2γ+1 dτ
Z0 (cid:18) ∞ L2(B(0,r)) H1(B(0,r)) ∞ (cid:19)
t
= C kρk kGk32 kGk34 +kρk2γ+1 dτ
Z0 (cid:18) ∞ L2(B(0,r)) H1(B(0,r)) ∞ (cid:19)
t
≤ C 1+kρk k∇Uk2 +kρk k∇Uk32 k∇Gk34 +kρk2γ+1 dτ
Z0 (cid:18) ∞ L2(B(0,r)) ∞ L2 L2(B(0,r)) ∞ (cid:19)
t
≤ C 1+kρk−1k∇Gk2 +k∇Uk2 (kρk5 +1)+kρk6γ+1 dτ
Z ∞ L2(B(0,r)) L2(B(0,r)) ∞ ∞
0 (cid:16) (cid:17)
t
≤ C 1+kρ21U˙k2 +C(kρk5 +1)k∇Uk2 +kρk6γ+1 dτ
Z L2(B(0,r)) ∞ L2 ∞
0 (cid:16) (cid:17)
The estimates of kρ1U˙k then follows similarly as (103) in Desjardins [3], one
2 L2
concludesthat
t
(2.43) Ψ ≤ 1 ≤ Ψ−1 ≤ Cexp C χ(Φ)λ(s)ds .
3 3 Z !
0
10 XIANGDIHUANG
Step4. Wecan rewrite(2.33)as
t
(2.44) ρ(t,y) = P(t)U(t,y)exp −κ−1 p(s,y)ds
Z !
0
where
ρ(t,0) t
(2.45) P(t) = exp κ−1 p(s,0)ds
ρ (0) Z !
0 0
and
(2.46) U(t,y) = ρ (y)Ψ Ψ
0 1 3
Ontheotherhand,itfollowsfrom(2.44)that
d γ t aγ γ t
exp p(s,y)ds = ρ(t,y)γexp p(s,y)ds
dt κ Z ! κ κ Z !
(2.47) 0 0
aγ
= (P(t)U(t,y))γ,
κ
which implies
1 t aγ t 1/γ
(2.48) exp p(s,y)ds = 1+ (P(s)U(s,y))γds .
κ Z ! κ Z !
0 0
Next,weare inapositionto estimateP(t). First,observethat
M0 dy R RN
(2.49) = rN−1dr = .
Z ρ(t,y) Z N
0 0
In viewof(2.44)and(2.48), wehave
P(t)U(t,y)
(2.50) ρ(t,y) = .
1+ aγ t(P(s)U(s,y))γds 1/γ
κ 0
(cid:16) R (cid:17)
Then, P(t)can beestimatedas
RN M0 P(t)
P(t) = dy
N Z ρ(t,y)
0
M0 1+ aγ t(P(s)U(s,y))γds 1/γ
= κ 0 dy
(cid:16) R (cid:17)
Z U(t,y)
0
M0 1
≤ C dy
Z U(t,y)
0
(2.51)
aγ M0 t 1/γ
+C( )1/γ supU(t,y) supU−1(t,y) P(s)γds dy
κ   Z Z !
QT,0 QT,0  0 0
≤ CM supU−1(t,y)
0
 
QT,0  t 1/γ
+CM supU(t,y) supU−1(t,y) P(s)γds .
0
   Z !
QT,0 QT,0  0