9 Localization of Solutions to Stochastic Slow Diffusion Equations: Finite Speed of Propagation
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3.9 Localization of Solutions to Stochastic Slow Diffusion Equations: Finite. . .
79
2 O. As mentioned in the introduction, O is an open and bounded domain of Rd
with smooth boundary @O, d D 1; 2; 3. Everywhere below, X is the strong solution
to equation with initial data x.
Below, we are only concerned with small T > 0, so we may assume that T Ä 1.
Furthermore, for a function g W Œ0; 1 ! R, we define its ˛-Hölder norm, ˛ 2 .0; 1/,
by
0
jgj WD sup
s;t20;1
sÔt
jg.t/ g.s/j
jt sj
Let for 2 0; 12
˛
D f! 2 ˝ W jˇk .!/j˛ Ä R; 1 Ä k Ä Ng :
˝H;R
˛
Then, ˝H;R
% ˝ as R ! 1 P-a.s.
Now, we are ready to formulate the main result.
Theorem 3.9.1 Assume that d D 1; 2; 3 and 1 < m Ä 5, and that x 2 L1 .O/,
x 0, is such that
support fxg
where r0 > 0 and
0
Bcr0 . 0 /;
(3.98)
2 O. Fix ˛ 2 0; 12 and let for R > 0
0
N
Á1=2
X
1
ı.R/ WD @
c1 1
jrek j1
mC1 2
kD1
"
1
.1
exp
2
!
1
k
N
X
1
m/
c2 C
jek j1
2
kD1
!#!
^ 1;
k
where c1 ; c2 (depending on R) are as in Lemma 3.9.2 below and
Define for T 2 .0; 1
(
ı.R/
˝T
WD
as in (3.96).
)
sup jˇk .t/j Ä ı.R/ for all 1 Ä k Ä N :
t2Œ0;T
ı.R/
˛
Then, for ! 2 ˝T \ ˝H;R
, there is a decreasing function r. ; !/ W Œ0; T ! .0; r0 ;
and t.!/ 2 .0; T such that for all 0 Ä t Ä t.!/,
X.t; !/ D 0 on Br.t;!/ . 0 / Br.t.!/;!/ . 0 /; and
X.t; !/ 6Á 0 on Bcr.t;!/ Bcr.t.!/;!/ . 0 /:
(3.99)
80
3 Equations with Maximal Monotone Nonlinearities
ı.R/
Since ˝T
% ˝ as T ! 0 up to a P-zero set, and hence
P
[[
!
ı.m/
˝1=n
\
˛
˝H;m
D 1;
m2N n2N
it follows that we have finite speed of propagation of disturbances .“localization”/
for .Xt /t 0 P-a.s.
As explicitly follows from the proof, the function t ! r.t/ is a process adapted
to the filtration fFt g.
ı.R/
˛
Roughly speaking, Theorem 3.9.1 amounts to saying that, for ! 2 ˝T \ ˝H;R
and for a time interval Œ0; t.!/ sufficiently small, the stochastic flow X D X.t; ; !/
propagates with finite speed.
If we set rT .!/ D lim r.t; !/, we see by (3.99) that X.t; !/ D 0 on BrT .!/ ,
t!T
8t 2 .0; t.!// and X.t/ 6Á 0 on BcrT .!/ : It is not clear whether rT .!/ D 0 for some
T > 0, that is, whether the “hole filling” property holds in this case (see [89]).
It should be mentioned also that the assumption x
0 in O was made only to
give a physical meaning to the propagation process.
The conditions m Ä 5 and x 2 L1 might seem unnatural, but they are technical
assumptions required by the work [15] on which the present proof essentially relies.
3.9.1 Proof of Theorem 3.9.1
Without loss of generality we may take 0 D 0 2 O and set Br D Br .0/: The
method of the proof relies on some sharp integral energy type estimates of X D X.t/
on arbitrary balls Br O.
It is convenient to rewrite, as in previous situation, Eq. (3.93) as a deterministic
equation with random coefficients. To this aim we consider the transformation (3.45)
y.t/ D eW.t/ X.t/; t
where W.t/ D
N
X
0;
(3.100)
k ek ˇk .t/:
kD1
Then we have, see (3.46),
8
dy
1
ˆ
ˆ
eW .ym e mW / C
y D 0; t > 0; P-a.s.;
<
dt
2
y.0/ D x;
ˆ
ˆ
: ym 2 H 1 .O/; 8t > 0; P-a.s.;
0
(3.101)
3.9 Localization of Solutions to Stochastic Slow Diffusion Equations: Finite. . .
81
where
D
N
X
2 2
k ek :
(3.102)
kD1
By Theorem 3.6.1, we have P-a.s.
0; ym .t/e
y
mW.t/
2 H01 \ L
mC1
m
; a.e. t
0:
(3.103)
As a matter of fact, one has
Lemma 3.9.2 Assume that 1 Ä d Ä 3 and m 2 Œ1; 5. Then, if x 2 L1 , the solution
y to (3.101) satisfies P-a.s. for every T > 0
y 2 L1 ..0; T/
O/ \ C.Œ0; TI H
1
ym 2 L2 .0; TI H01 /;
dy
2 L2 .0; TI H
dt
1
/;
/:
(3.104)
(3.105)
Moreover, for every T 2 .0; 1, ˛ 2 .0; 12 /, R > 0, there exist constants c1 ; c2 > 0
depending on ˛; R; O; jxj1 , max .jek j1 ; jrek j1 ; j ek j1 /, but not on T such that
1ÄkÄN
˛
,
P-a.s. on ˝H;R
"
kykL1 ..0;T/
O/
#
Ä c1 exp c2 max
sup jˇk .t/j :
1ÄkÄN t2Œ0;T
(3.106)
The first part of Lemma 3.9.2 is just Theorem 3.6.1, while (3.101) follows by
Theorem 3.6.2.
Before we introduce our crucial energy functional
in (3.113) below and
explaining the idea of the proof subsequently, we need some preparations by a few
estimates on the solution y to (3.101). Everywhere in the following we fix ˛ 2 .0; 12 /,
˛ > 0 and assume that x 0 so that (3.103) holds and fix T 2 .0; 1.
By Green’s formula, it follows from (3.101) that
1
mC1
D
for all
Z
Z
O
ymC1 .t; / . /d C
1
mC1
2 C01 .O/.
1
C
2
Z
O
Z
t
r.ym e
ds
0
Z
/ r.eW ym /d
Z
t
ymC1
ds
0
mW
O
d
O
xmC1 . / . /d ; t 2 .0; T/;
(3.107)
82
3 Equations with Maximal Monotone Nonlinearities
Fix r > 0 and let " 2 C1 .RC / be a cut-off function such that
0 Ä s Ä r C ", " .s/ D 0 for s r C 2" and for " D 1ŒrC";rC2" ,
ˇ
ˇ
lim ˇ "0 .s/ C
"!0 ˇ
ˇ
1 ˇˇ
"ˇ
" .s/
" .s/
D 0;
D 1 for
(3.108)
uniformly in s 2 Œ0; 1/. Roughly speaking, this means that " is a smooth
approximation of the function " .s/ D 1 on Œ0; r C ", " .s/ D 0 on Œr C 2"; 1/,
1
.s r "/ C 1 on Œr C "; r C 2".
" .s/ D
"
If in (3.107) we take D " .j j/ (for " small enough), setting " . / D " .j j/;
2 O, we obtain that
1
mC1
Z
Z
.y.t; //
mC1
O
" .j
j/d C
r.ye
ds
0
C
Z
t
1
2
O
Z
t
/ r.eW ym
W m
Z
WymC1
ds
0
O
"d D
" /d
1
mC1
Z
xmC1
O
"d
:
(3.109)
On the other hand, we have
Z
Z
W m
W m
r.ye / r.e y " /d D
jr.ye
O
W m 2
O
1
C.m C 1/
2
Z
C .r.ye
O
/ j
Z
"e
.mC1/W
.r.ye
O
W m
d
/
rW/eW ym
W m
"d
/.s; / "0 .j j/.eW ym /.s; /d ;
/
(3.110)
(Since 2 C2 .O/, the above calculation is justified.)
j j
˛
\
Everywhere in the following, the estimates are taken P-a.s. on the set ˝H;R
where . / D
ı.R/
˝T .
We set B"r D BrC2" n BrC" . Then, by (3.109), (3.110), we see that
1
mC1
Ä
Z
Z
ymC1 .t; /d C
ds
BrC"
1
mC1
C
Z
BrC2"
Z t
Z
"x
mC1
Z
t
1
2
0Z
t
BZ
rC2"
"e
ds
0
.mC1/W
" Wy
jr.ye
mC1
W m 2
/ j d ds
d ds
BrC2"
d
.m C 1/
.r.ye W /m W/ " eW ym d ds
BrC2"
0
Z tZ
.r.ye W /m /.s; /.e ym /.s; / "0 .j j/d ds:
0
B"r
(3.111)
3.9 Localization of Solutions to Stochastic Slow Diffusion Equations: Finite. . .
83
On the other hand, we have
Z tZ
B"r
0
j.r.ye
Z tZ
Ä
0
" .j
j
B"r
0
Z tZ
/eW ym "0 .j j/jd ds
/
W m
e.1
j/j jr.ye
/ j e
!12
d ds
(3.112)
!12
m/W 2m
y j "0 .j j/jd ds :
B"r
0
W m 2 .mC1/W
We introduce the energy function
.t; r/ D
Z tZ
0
jr.ye
W m 2 .mC1/W
/ j e
d ds; t 2 Œ0; T; r
0:
In order to prove (3.99), our aim in the following is to show that
differential inequality of the form
@
.t; r/
@r
(3.113)
Br
CtÂ
1
ı.R/
˛
. .t; r//ı on ˝H;R
\ ˝T
satisfies a
for t 2 Œ0; T; r 2 Œ0; r0 ;
where 0 < Â < 1 and 0 < ı < 1 and from which (3.99) will follow.
Taking into account that function is absolutely continuous in r, we have by
(3.108), a.e. on .0; r0 /,
Z tZ
lim
"!0 0
B"r
j "0 .j j/jjr.ye
W m 2 .mC1/W
/ j e
d ds D
@
.t; r/:
@r
Then, letting " ! 0 in (3.111), (3.112), we obtain that
1
mC1
Z Z
1 t
WymC1 d ds
2 0 Br
Z tZ
.m C 1/
.r.ye W /m rW/eW ym d ds
Z
ymC1 .t; /d C .t; r/ C
Br
1
Ä
mC1
Z
xmC1 d
0
Br
Br
(3.114)
Â
C
@
.t; r/
@r
Ã 12 ÂZ
Z
t
y2m e.1
ds
0
ı.R/
m/W
Ã 12
d
˙r
˛
\ ˝T ; t 2 Œ0; T; r 2 Œ0; r0 :
on ˝H;R
;
84
3 Equations with Maximal Monotone Nonlinearities
In order to estimate the right-hand side of (3.114), we introduce the following
notations
Z Z
1 t
K.t; r/ D
WymC1 ds d
(3.115)
2 0 Br
H.t; r/ D sup
1
mC1
Z
ymC1 .s; /d ; 0 Ä s Ä t ;
(3.116)
Br
and note that by assumption (3.96) we have
Z tZ
1
K.t; r/
ymC1 d ds; 8t 2 Œ0; T; r 2 Œ0; r0 :
2
0
Br
(3.117)
Then (3.114) yields, for r 2 .0; r0 ,
Z tZ
H.t; r/ C .t; r/ C K.t; r/Ä .m C 1/
j.r.ye W /m rW/eW ym jd ds
0
Br
Ã 12 ÂZ t Z
Â
Ã 12
@
.t; r/
C
y2m e.1 m/W d ds
@r
0
˙r
(3.118)
because x Á 0 on Br . We note that, by the trace theorem, the surface integral arising
in the right-hand side of formula (3.118) is well defined because r.ye W /m 2
L2 .Œ0; T O/ and, by Lemma 3.9.2, y 2 L1 ..0; T/ O/ P-a.s.
Now, we are going to estimate the right-hand side of (3.118).
By Cauchy–Schwarz and (3.117), we have
Z tZ
0
j.r.ye
/
W m
rW/eW ym jd ds
Br
1=2
Ä kym 1 e.1 m/W jrWj2 kL1 ..0;T/ O/
ÂZ t Z
Ã 12 ÂZ t Z
Ã 12
m mW 2 .mC1/W
mC1
ds
jr.y e
/j e
d
ds
y
d
Ä .2
Ä
0
Br
1 1=2 m 1 .1 m/W
/
ky
e
1=2
jrWj2 kL1 ..0;T/
0
Br
1
1
2
2
O/ . .t; r// .K.t; r//
1
ı.R/
˛
. .t; r/ C K.t; r//; 8t 2 .0; T; r 2 .0; r0 ; on ˝H;R
\ ˝T ;
2.m C 1/
(3.119)
by the definition of ı.R/.
By (3.118), it follows that
H.t; r/ C .t; r/ C K.t; r/
Â
Ã 12 ÂZ t Z
Ã 12
@
2m .1 m/W
Ä
.t; r/
ds
y e
d
@r
˙r
0
ı.R/
˛
8t 2 Œ0; T; r 2 Œ0; r0 ; on ˝H;R
\ ˝T :
(3.120)
3.9 Localization of Solutions to Stochastic Slow Diffusion Equations: Finite. . .
85
In order to estimate the surface integral from the right-hand side of (3.120), we
invoke the following interpolation-trace inequality (see, e.g., Lemma 2.2 in [54])
jzjL2 .˙r / Ä C.jrzjL2 .Br / C jzjL
C1 .B
r/
/Â jzjL1
Â
C1 .B
r/
;
(3.121)
for all 2 Œ0; 1 and Â D .d.1
/ C C 1/=.d.1
/ C 2. C 1//: Clearly,
Â 2 12 ; 1 .
We shall apply this inequality for z D .ym e W /m and D m1 . We obtain, by
(3.23) that
ÂZ
2m .1 m/W
y e
Ã 12
Ä
d
˙r
1=2
Ä Cke.1Cm/W kL1 ..0;T/
jym e
Äe
C
1=2
ke.1Cm/W kL1 ..0;T/ O/
O/ .jr.ye
ÂZ
˙r
/ jL2 .Br / Cjym e
W m
W 2m
.ye
mW
/ d
j
L
mC1
m
Ã 12
.Br /
/Â
mW 1 Â
j
L
ÂZ
mC1
m
.Br /
jr.y e
m
mW
2 .mC1/W
/j e
Ã 12
d
!Â
CH
m
mC1
.t; r/
Br
m
.H mC1 .t; r//1
Â
ı.R/
˛
; on ˝H;R
\ ˝T ;
ı.R/
˛
\ ˝T ,
where, as will be the case below, e
C is a positive function of ! 2 ˝H;R
independent of t and r, which may change below from line to line.
1
Integrating over .0; t/ and applying first Minkowski’s (since Â
) and then
2
Hölder’s inequality yields
ÂZ
Z
t
2m .1 m/W
ds
0
Äe
C
y e
˙r
Z
Äe
CH
d
ÂZ
t
jr.y e
m
ds
0
Ã 12
m.1 Â /
mC1
mW
2 .mC1/W
/j e
d CH
2m
mC1
.s; r/
ÃÂ
H
2m.1 Â /
mC1
! 12
.s; r/
Br
.t; r/t
1 Â
2
1
m
ı.R/
˛
.. .t; r// 2 C H mC1 .t; r//Â ; on ˝H;R
\ ˝T :
Substituting the latter into (3.120), we obtain that
Ã 12
m.1 Â /
m
@
1
. 2 C H mC1 /Â H mC1
@r
Â Ã 12
ÁÂ
m.1 Â /
m
@
1 Â
1
2 H .mC1/Â C H .mC1/Â
Äe
Ct 2
;
@r
ı.R/
˛
\ ˝T :
8t 2 Œ0; T; r 2 Œ0; r0 ; on ˝H;R
CH Äe
Ct
1 Â
2
Â
(3.122)
86
3 Equations with Maximal Monotone Nonlinearities
On the other hand, for H0 D H.T; r0 /, we have the estimate
1
2
m.1 Â /
1
2
m
H .mC1/Â C H .mC1/Â Ä
m.1 Â /
m
H .mC1/Â C H0mC1
1
2
m.1 Â /
1
1
m.1 Â /
H .mC1/Â C 2 Ä e
C. C H/ 2 C .mC1/Â ;
m 1
where e
C WD 2 max.1; H02.mC1/ / and where we used that by Young’s inequality, for
all p; q 2 .0; 1/,
p
H q Ä . C H/pCq :
Substituting the latter into (3.122) yields
1 Â
CH Äe
Ct 2
Â
@
@r
Ã 12
Â
. C H/ 2 C
m.1 Â /
mC1
ı.R/
on .0; T/
.0; r0 /
˛
˝H;R
\ ˝T ;
on .0; T/
.0; r0 /
˛
˝H;R
\ ˝T :
and therefore
Â
@
.t; r/
@r
Ã 12
Â 1
e
Ct 2 . .t; r//
2 Â
2
m.1 Â /
mC1
ı.R/
(3.123)
Equivalently,
@'
.t; r/
@r
e
CtÂ
1
; on .0; T/
.r.t/; r0 /
ı.R/
˛
˝H;R
\ ˝T ;
(3.124)
where
'.t; r/ D . .t; r//Â C
2m.1 Â /
mC1
1
;
(3.125)
and
0 j .t; r/ > 0g ^ r0 :
r.t/ WD inffr
We note that, by continuity,
.t; r.t// D 0
and that, since .t; r/ is increasing in t and r, we have .t; r/ > 0; if r > r.t/,
and that t 7! r.t/ is decreasing in t. Furthermore, the same is true for ' defined in
Â/
1 > 0, because 0 < Â < 1 and m > 1.
(3.125), since Â C 2m.1
mC1
Moreover, by (3.23) and (3.122) we see that
X.t; / D 0
for
2 Br.t/ :
3.9 Localization of Solutions to Stochastic Slow Diffusion Equations: Finite. . .
˛
We recall that r.t/ D r.t; !/ depends on ! 2 ˝. Now, fix ! 2 ˝H;R
aim is to show that
9 t.!/ 2 .0; T such that r.t; !/ > 0; 8t 2 Œ0; t.!/:
87
ı.R/
˝T . Our
(3.126)
Since we already noted that .t; r/ > 0, if r > r.t/, by (3.101), (3.125) and (3.100),
we deduce the property in (3.99) from (3.126). To show (3.36), we first note that by
(3.124) for all t 2 .0; T/
e
CtÂ
'.t; r0 /.!/
1
.r0
r.t; !//;
hence
r.t; !/
.e
C.!// 1 t1
r0
Â
'.t; r0 /.!/:
So, because 0 < Â < 1, we can find t D t.!/ 2 .0; T/, small enough, so that
the right-hand side is strictly positive. Now, (3.36) follows, since, as noted earlier,
t 7! r.t; !/ is decreasing in t, which completes the proof of (3.99). By elementary
considerations for ı > 0, we have
r
P.˝Tı /
2
N
1
T
e
2 ı2
!N
ı 2 =.2T/
:
Hence ˝Tı % ˝ as T ! 0 up to a P-zero set and the last part of the assertion also
follows.
Remark 3.9.3 In the deterministic case, for O D Rd the finite speed propagation
property: support fxg
Br0 . 0 / H) support fX.t/g
Br.t/ .e0 / for some e0 2 Rd
e0 /, where
and r D r.t/; follows by the comparison principle X.t; / Ä U.t C ;
U D U.t; / is the Barenblatt source solution
U.t; / D t
d
.m 1/dC2
Ä
C
1
m 1
j j2
m 1
2
2m..m 1/d C 2/ t .m 1/dC2
(3.127)
C
2
(see [89]) and which has the support in f.t; /I j j2 Ä C1 t .m 1/dC2 g.
At least in the simpler case, where the noise is not function valued, i.e.
independent of , this is similar in the stochastic case. More precisely, for m D 2,
d D 1, O D R1 and W.t/ D ˇ.t/ Dstandard, real-valued Brownian motion, the
function
ÂZ t
Ã
1
Z.t; / D U
k.s/ds;
k.t/; k.t/ D eˇ.t/ 2 t
0
88
3 Equations with Maximal Monotone Nonlinearities
.t; /I j j2 Ä C1
is a solution to (3.93) and support Z
Rt
0 k.s/ds
2
3
(see [78]
for details). However, on bounded domains, it is not clear, whether this is applicable.
t
u
Remark 3.9.4 We refer to [3, 54, 89, 90] for corresponding localization results in
deterministic case. As a matter of fact the energy method used here was introduced
by S. N. Antonsev and developed in [3].
The finite dimensional structure of the Wiener process W.t/ was essential for
the present approach, which is based on sharp estimates on solutions to Eq. (3.101).
A direct application of the above energy method in L2 .˝I L2 .0; TI H 1 / failed for
general cylindrical Wiener processes W.t/.
t
u
3.10 The Logarithmic Diffusion Equation
We consider here the nonlinear diffusion equation with linear multiplicative noise
(logarithmic diffusion equation)
8
ˆ
dX.t/ D log.X.t//dt C .X.t//dW.t/;
ˆ
ˆ
ˆ
ˆ
<
X.0/ D x 2 H 1 ;
ˆ
ˆ
ˆ
ˆ
ˆ
: X.t/ D 1; on .0; T/ @O
in .0; T/
O
(3.128)
where
.X.t//dW.t/ D KX.t/. A/
DK
1
X
dW.t/
(3.129)
˛h .X.t/eh /dWh .t/:
hD1
with Dirichlet
Here K is a positive constant, f˛h g are the eigenvalues of A D
boundary conditions and > d3 (see Example 2.1.2). Moreover W is a cylindrical
Wiener process in H 1 .
Definition 3.10.1 The process X is called a distributional solution of (3.128) if the
following conditions hold.
(i) X 2 L2W .˝I C.Œ0; TI H 1 \ L1 ..0; T/
(ii) X > 0 a.e. in .0; T/ O ˝.
(iii) log X 2 L2W .0; TI L2 .˝I H01 //.
O
˝/.
3.10 The Logarithmic Diffusion Equation
Z
t
(iv)
0
89
log X.s/ds 2 L2W .˝I C.Œ0; TI H01 /.
(v) We have
Z
Z
t
X.t/ D x C
0
where
t
log.X.s//ds C K
0
.X.s//A
dW.s/; 8 t 2 Œ0; T; P-a.s.;
(3.130)
is considered in sense of distributions on O.
As mentioned earlier in Sect. 1.1, Eq. (3.128) can be viewed as a superfast
diffusion equation as the limit case m D 0 of Eq. (1.5) that is
Â
dX.t/ D div
rX.t/
X.t/
Ã
C KX.t/A
dW.t/
(3.131)
and it models the dynamics of plasma in a magnetic field perturbed by a multiplicative Gaussian noise. We also recall that the deterministic version of (3.130)
(equivalently (3.128)) arises in the Riemannian geometry, as a model for the
evolution of conformally flat metric driven by its Ricci curvature flow. Taking into
account that by the rescaling transformation X D eKY.t/A W.t/ , Eq. (3.128) reduces
formally to the deterministic equation
@Y
De
@t
W
.log Y/
e
W
WC
1
K2 X 2 2
˛ eh Y;
2 hD1 h
8 ! 2 ˝:
one might obtain a similar geometric interpretation to stochastic equation (3.128).
Here is the main result of the section.
Theorem 3.10.2 Let x 2 L2 such that x > 0; x log x 2 L2 . Then for each T > 0
there is a unique strong solution X to (3.128). Moreover Xj log Xj 2 L1 ..0; T/
˝ O/
Proof Arguing as in Sect. 3.3 we consider for any > 0 the approximating problem
8
< dX .t/ D
:
.ˇ .X .t// C X .t//dt C .X .t// dW.t/;
(3.132)
X .0/ D x;
where ˇ are the Yosida approximation of the maximal monotone function ˇ W
R!R
ˇ.r/ D
log r if r > 0
¿ if r Ä 0:
(3.133)