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Construction of Green's functions for the Boltzmann equations
Shih-Hsien YuDepartment of Mathematics
National University of Singapore
Motivation to investigate Green’s function for Boltzmann equation before 2003
• Nonlinear time-asymptotic stability of a Boltzmann shock profile
Zero total macroscopic perturbations
• Nonlinear time-asymptotic stability of a Knudsen layer for the Boltzmann EquationMach number <-1
Green’s function of linearized equation around a global Maxwellian,
• Fourier transformation
• The inverse transformation
Lfff xt
tLie )(
.0|| around Analytic :)( detxGR
Litix 3
)(),(
detxG
detxG
txGtxGtxG
LitixS
LitixL
SL
||
)(
||
)(
),(
),(
),(),(),(
t
LL
tS eOG
x
122
)1()(
)1()( 2
OGLH
tL n
x
• Initial value problem
• Particle-like wave-like decomposition
KffffLfff xtxt )(
)()()0,,(
*33
xxf
Lfff xt
0)0,,(
,
0)0,,(
1 ,
0)0,,(
)()()0,,(
2
1
1
01111
*33
0
00000
xR
KfLRRR
xf
kKffff
xf
fKfff
xxf
fKfff
n
nnnxnt
k
kkkxkt
xt
xt
n
knk Rff
2
0
t
LHn eOf
LemmaMixture
nx
)1(
)(2 2
)1(
)(2 2 ORLHn n
x
)1( )(2 2 OR
LHn nx
n
knk RftxG
2
0
),(
t
LL
tS
LH
tL
eOG
OG
x
nx
122
2
)1(
)1(
)(
)(
),(),(),( txGtxGtxG SL
ntL
n
kk
tS RGfG
2
0
t
LL
n
kk
tS eOfG
x
)1()(
2
0 2
Pointwise of structure of the Green’s function
• Space dimension=3
• Space dimension=1
Macroscopic wave structure of 1-D Green’s function
Application: Pointwise time-asymptotic stability of a global Maxwellian state in 1-D.
Green’s function of linearized equation around a global Maxwellian M, , in a half-space problem x>0. 1 Lfff xt
0|0|
1
0|
0
1
111 )(),0(
0)0,(
0 ,)data shif(
0),0(
)()0,(
0 ,
tbtg
xg
xLgggt
tf
xfxf
xLfff xtxt
Green identity:
t
dssgstxGtxg0
1 )],0()[,(),(
.),,0( ,1M 2
00
11
dtdetgthen)Mach(If t
Boundary value estimates ( a priori estimate):
Approximate boundary data for case |Mach(M)|<1
0|0|
1
11 )(),0(
0)0,(
0 ,
tbtg
xg
xLggg xt
Upwind damping approximation to the boundary data
00
0
1
1
0
0|0|
3322
1
11
11
|)),(),0((|),0(
)],0()[,(),(
)),0(),,0((
)(),0(
0)0,(
0 ),),(),((
txhtbtq
dsswstxGtxh
dsswswe
tbtw
xw
xEEwEEw
Lwww
t
t
xt
An approximation to the full boundary data.
0101
0
11
0|0|
3322
1
11
11
|)),(),0((|),0(
)],0()[,(),(
)(),0(
0)0,(
0 ),),(),((
txgtbtb
dsswstxGtxg
tbtw
xw
xEEwEEw
Lwww
t
xt
001
0
1
0|0|
3322
1
11
11
|)),(),0((|),0(
)],0()[,(),(
)(),0(
0)0,(
0 ),),(),((
txgtbtb
dsswstxGtxg
tbtw
xw
xEEwEEw
Lwww
kkk
t
k
k
xt
0
),(),(k
k txgtxg
Green’s function of linearized equation around a stationary shock profile . , 1 fLff xt
Separation of wave structures
Transversal wave
Compressive wave
ct0
10
ionDecomposit Hyperbolic
fff
fff
1. Shift data
)()0,(
01
xhxf
fLff xt
2)()(
L
xxhy
)]()[(
)]()[(),(
xhGx
xhGxtxgt
t
dyyhtyxGxhG t )(),()]([
gLtx xt )(),( 1
0)0,(
),(),(),(1
xq
qLqq
txgtxftxq
xt
2)(),(
L
xtxgy
2. Hyperbolic Decomposition
ct0
10
ionDecomposit Hyperbolic
fff
fff
0)0,(
1
xq
qLqq xt
Transversal wave
Compressive wave
3. Transverse Operator and Local Wave Front tracing
||
1
t1
)1(
),(][)(
,0),(][)(
:),]([
3,
x
Lxt
xt
eO
txTL
txTL
txT
0
),(])[)((
)(')(),]([
:),]([
10
dxtxDTLP
xtltxD
txD
xt
4. Coupling of T and D operators
),(])[)((),]([ 1 txDTLtxR xt
5. Respond to Coupling
0][P
0),(][)(
0
t1
dxR
txTLxt
0)0,(
),]([),()( 1
xy
txRtxyLxt
6. Approximation to Respond, Compressive Operator
).,(),](C[
0)0,(
),]([
),()( t1
txYtx
xY
txR
YtxYLxt
.),]([)1(),(2||
0
2||
0
2
2
2
2
dxdttxReOdxdttxYeL
tx
L
tx
6. T-C scheme for
0)0,(
1
xq
qLqq xt
0)0,(
0 ,),(),(
0)0,(
),](C[
]C)[(),(
0)0,(
,),](C[
, ]C)[(),(
1
1
1
111
t1
1
1
111
t1
1
xq
qLtxqtxq
xq
qL
tx(x,t)
(x,t)DTtxq
xq
qL
tx(x,t)
(x,t)DTtxq
xt
kk
k
kkkxt
kk
kk
xt
An estimates
.0)(),( s.t.
]1,1[)supp()()0,(
0
00
1
lim
xctxfc
hxhxf
fLff
t
xt
A Diagram for '0cf
A Diagram for general pattern
+ extra time decaying rate in microscopic component nonlinear stability of Boltzmann shock profile
Applications of the Green’s functions
• Nonlinear invariant manifolds for steady Boltzmann flow
. , )),,((),( 31 RRxxgQxgx
.point Fixed :M),( ],,[ uxg
M/)M()( ,MM
),(1
fQffg
fLffx
Applications of the Green’s functions
• Milne’s problme
],,[
0|0|
31
M),(lim
given :)(),0(
,0 )),,((),(
11
ux
x
xg
bg
RxxgQxg
Sone’s Diagram for Condensation-Evaporation