Classical flows of vector fields with exponential or sub-exponential summability
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Classical lows o ec o ields wi h exponen ial o sub-exponen ial summabili y
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Published e sion
Amb osio, Luigi; Nicolussi Golo, Sebas iano; Se a Cassano, F ancesco
Amb osio, L., Nicolussi Golo, S., & Se a Cassano, F. (2023). Classical lows o ec o ields wi h
exponen ial o sub-exponen ial summabili y. Jou nal o Di e en ial Equa ions, 372(5), 458-504.
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Jou nal o Di e en ial Equa ions 372 (2023) 458–504
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Classical lows o ec o ields wi h exponen ial o
sub-exponen ial summabili y
Luigi Amb osio a, Sebas iano Nicolussi Golo b,∗,
F ancesco Se a Cassano c
aScuola No male Supe io e, Pisa, I aly
bDepa men o Ma hema ics and S a is ics, Uni e si y o Jy äskylä, Finland
cDipa imen o di Ma ema ica, Uni e si à di T en o, I aly
Recei ed 2 Augus 2022; e ised 23 May 2023; accep ed 3 July 2023
Abs ac
We show ha ec o ields bwhose spa ial de i a i e Dxbsa is ies a O licz summabili y condi ion ha e
a spa ially con inuous ep esen a i e and a e well-posed. Fo he case o sub-exponen ial summabili y, hei
lows sa is y a Lusin (N) condi ion in a quan i a i e o m, oo. Fu he mo e, we p o e ha i Dxbsa is ies
a sui able exponen ial summabili y condi ion hen he low associa ed o bhas Sobole egula i y, wi hou
assuming boundedness o di xb. We hen apply hese esul s o he ep esen a ion and Sobole egula i y
o weak solu ions o he Cauchy p oblem o he anspo and con inui y equa ions.
©2023 The Au ho s. Published by Else ie Inc. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons .o g /licenses /by /4 .0/).
MSC: 35F10; 35A01; 35A02
Keywo ds: Vec o ields; Flow; Sobole –O licz spaces; T anspo equa ion; Con inui y equa ion
*Co esponding au ho .
E-mail add esses: [email p o ec ed] (L. Amb osio), [email p o ec ed] (S. Nicolussi Golo),
[email p o ec ed] (F. Se a Cassano).
h ps://doi.o g/10.1016/j.jde.2023.07.005
0022-0396/©2023 The Au ho s. Published by Else ie Inc. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons .o g /licenses /by /4 .0/).
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
Con en s
1. In oduc ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
1.1. Well-posedness ................................................. 461
1.2. Regula i y..................................................... 462
1.3. T anspo and con inui y equa ions .................................... 465
1.4. S uc u e o he pape ............................................. 466
Acknowledgmen s .................................................... 466
2. P elimina ies on homeomo phisms......................................... 466
2.1. Weak de i a i es o homeomo phisms.................................. 466
2.2. Mappings o ini e dis o ion ........................................ 467
3. P elimina ies on lows o ec o ields....................................... 469
3.1. Well-posedness o ec o ields....................................... 469
3.2. The low o a well-posed ec o ield................................... 470
4. Well-posedness wi h O licz condi ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
4.1. A class o subexponen ial summabili y ypes ............................. 479
5. Regula i y o he low wi h subexponen ial summabili y . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
6. Regula i y o he low wi h exponen ial summabili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
7. Applica ions o PDEs .................................................. 495
8. Examples .......................................................... 500
8.1. Well-posedness does no imply absolu e con inui y o he low ................. 500
8.2. Sub-exponen ial condi ion does no imply high Sobole egula i y............... 501
8.3. Exponen ial summabili y does no imply he di e gence in BMO and i is only su icien
o he Sobole egula i y .......................................... 502
Da a a ailabili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Re e ences.............................................................. 503
1. In oduc ion
In his pape we a e conce ned wi h he s udy o he exis ence and uniqueness o classical
solu ions o he Cauchy p oblem o he ODE sys em
˙γ( )=b( , γ ( ))
γ(s)=x, (1)
wi h x∈, an open domain in Rn, s∈I, an open in e al in R, and b:I× →Rna con inu-
ous, possibly non-au onomous ec o ield E en hough we will mos ly deal wi h he case when
bis con inuous, we will poin ou which p oo s can easily be adap ed o he case when bis only
measu able wi h espec o . I solu ions o (1)exis and a e unique o e e y sand x, we say
ha he ec o ield bis well-posed in I×(o in I×, see De ini ion 3.1 o a mo e p ecise
s a emen ). Fo e e y well posed ec o ield b:I× →Rnwe ha e a low, ha is, a map
X:I×I× →, de ined as X( , s, x) :=γ( ) whe e γis he unique absolu ely con inuous
solu ion o (1). Mo e p ecisely, o each , s∈Iwe deno e by ( ,s) ⊂ he open se o all
x∈such ha he pa h s a ing a xa ime scan be ex ended un il ime (see Sec ion 3.2 and
Rema k 3.3). Then X( , s, ·)is a well de ined homeomo phism ( ,s) →(s, ) (see Rema k 3.3).
Di Pe na–Lions [17] ca ied ou a pionee ing and a - eaching heo y by in oducing a gene al-
ized no ion o low o ec o ields b∈L1
loc((0, T); W1,1
loc (Rn, Rn)) wi h impo an applica ions
459
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
o he exis ence and uniqueness o weak solu ions o he Cauchy p oblem o he anspo equa-
ion associa ed o a weakly di e en iable ec o ield b, ha is,
∂ u+b·Dxu=0in(0,T)×Rn
u(0,·)=¯u. (2)
The heo y was la e ema kably ex ended by he i s au ho [2] o ec o ields b( , ·)wi h
BV egula i y. In hese wo ks, he egula i y o bis pai ed wi h he boundedness o i s spa ial
di e gence, ha is
di xb∈L1((0,T);L∞(Rn)), (3)
which ensu es he exis ence and uniqueness o he gene alized low o b. I (3) does no hold,
hen uniqueness o he low may ail, as i was al eady shown in [17, Sec ion IV.1]. The exis ence
and uniqueness o a gene alized low associa ed o a weakly egula ec o ield bhas been he
objec o an in ensi e s udy wi h applica ions o he Cauchy p oblem o he anspo equa ion
as well as o he con inui y equa ion associa ed o b, ha is,
∂ ρ+di x(bρ) =0in(0,T)×Rn
ρ(0,·)=¯ρin Rn.(4)
Exis ence, uniqueness and egula i y o solu ions o hese h ee p oblems (1), (2) and (4)a e
connec ed wi h each o he . In pa icula , he exis ence o a unique low Xwi h enough egula i y
implies exis ence and uniqueness o solu ions o bo h he anspo equa ion and he con inui y
equa ion. A ai ly comple e accoun o he de elopmen in his opic can be ound in [5] and
e e ences he ein. A sample o he li e a u e on his subjec is [4,8–10,13,11,12,14–16,27,29,28].
Ou con ibu ion ocuses on wo p oblems. Fi s , we wan o weaken he boundedness as-
sump ion on he di e gence (3). We will show in Theo em A ha sub-exponen ial summabili y
o Dxbgua an ees he exis ence o a unique classical low (in he Di Pe na–Lions–Amb osio
heo y, lows ha e a weake de ini ion).
Second, we wan o ind condi ions on b o he low o ha e Sobole egula i y, ins ead o
jus Lpin eg abili y. I is well-known ha high Lpin eg abili y o ma ix Jacobian Dxb, e en
coupled wi h (3), is no enough in o de o p o ide Sobole egula i y o he low X(see, o
ins ance, [26]). A s a egy used in he ecen pape s [13,8]was o s eng hen he hypo heses by
equi ing exponen ial summabili y o Dxb. We e e in pa icula o he ecen pape [8], whe e
i has been shown ha bhas a unique low wi h Sobole egula i y unde he condi ion
sup
∈Rˆ
Tn
exp(βDxb( ,x))dx<∞and di xb∈L∞
loc(R×Tn), (5)
o some β>0, whe e Tnis he n-dimensional o us. We p o e analogous esul s wi hou con-
di ions on he di e gence o bin Theo em D, see also Rema k 6.3.
Ou esul s a e o h ee ypes. We i s p o ide in eg al condi ions o sub-exponen ial ype on
Db ha ensu e well-posedness. Then, we s udy he Sobole egula i y o he homeomo phisms
460
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
X( , s, ·). Finally, we apply hese esul s o bo h he anspo equa ion and he con inui y equa-
ion.
1.1. Well-posedness
Le us ocus, i s , on he well-posedness. I Dxbsa is ies an exponen ial summabili y, ha
is, condi ions o he o m
ˆ
I
ˆ
exp (βDxb( ,x))dxd <+∞ (6)
o some β>0, hen i is well-known ha bis well-posed. Indeed, in his case, he ec o ield
b( , ·)sa is ies a so-called Log-Lipschi z condi ion; see, o ins ance, [8,32]. Howe e , e o mu-
la ing he condi ion o exponen ial summabili y in a O licz-like o m, we ex end he esul o
some sub-exponen ial cases.
Theo em A. Le :[0, +∞) →(0, +∞)be a non dec easing locally Lipschi z unc ion. As-
sume ha
(A.I) i n >1, he e exis s α∈(1, n
n−1)such ha α−1
αis con ex, while, i n =1, is con ex;
(A.II) he e exis s C≥1such ha :[C, +∞) →[(Cθ), +∞)is bijec i e and
(s1)(s2)≤(Cs1s2) o all s1,s
2≥C;(7)
(A.III)
∞
ˆ
1
(s)
s(s) ds=+∞.
Le b∈L1
loc(I; W1,1
loc (; Rn)) and assume ha o e e y o∈ he e exis c>0, R>0such ha
B(o, 2R) ⊂and he unc ion
→ψ( ):= ˆ
B(o,2R)
(cDxb( ,z))dz(8)
belongs o L1
loc(I). Then b( , ·) : →Rnhas a con inuous ep esen a i e ˜
b( , ·) o a.e. ∈I
and ˜
bis well-posed in I×. Mo eo e , i he e exis s m ∈L1(I) such ha
|b( ,x)|≤m( ) o a.e. ∈I, o a.e. x∈B(o,R), (9)
hen ˜
bis also well-posed in I×.
No ice ha a byp oduc o he p oo o Theo em Ais ha he Sobole -O licz space W1L()
embeds in C0(), wi h modulus o con inui y ha depends only on . See [7, Sec ion 2.6] o
he de ini ion o W1L().
461
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
The p oo is inspi ed om [32]. In ac , we shall p o e ha a con inuous ep esen a i e ˜
b( , ·)
o b( , ·)sa is ies he Osgood’s c i e ion (see [23, Chap. III, Co olla y 6.2] o P oposi ion 3.2
below). The p oo o his esul is gi en in Sec ion 4.
We no e in P oposi ion 4.4 ha i an inc easing unc ion sa is ies condi ion (A.III) o
Theo em A, hen i does no ha e polynomial g ow h. Examples o unc ions sa is ying he
p ope ies (A.I)-(A.III) a e o he o m
Ek,β(s) =exp ⎛
⎜
⎜
⎜
⎝
s
log(s) log log(s) . . . (log ...log
k- imes
s)β⎞
⎟
⎟
⎟
⎠
o s≥¯s, (10)
wi h Ek,β(s) =Ek,β(¯s) o s<¯s, whe e ¯sis la ge enough, k≥1is an in ege and 0 ≤β≤1, see
P oposi ion 4.6. No ice ha he asymp o ic beha io o Ek,β as s→∞is almos sha p in o de
ha assump ion (A.III) holds, see Rema k 4.7. I Ek,1(c Dxb) ∈L1
loc() o some c>0, we
say ha Dxbsa is ies a subexponen ial summabili y o o de k. The e o e, Theo em Ashows
ha , i Dxbhas subexponen ial summabili y, hen bhas a classical unique low. Howe e , we
s ess ha , unde he hypo hesis o Theo em A, Dxbdoes no need o be in Lp
loc(I ×) o
each p>1(see Rema k 4.5).
1.2. Regula i y
Mo ing on o he egula i y o he low, we can p o e ha , i Dxbsa is ies a subexponen-
ial summabili y o o de 1, he associa ed low X( , s, ·)sa is ies a weak egula i y p ope y,
namely i maps he Lebesgue measu e in o absolu ely con inuous measu es. No ice ha , in his
case, Dxb( ,·)does belong o Lp
loc() o e e y p>1, o almos e e y ∈I. A quan i a i e
e sion, ha we ob ain adap ing [11], is he ollowing.
Theo em B. Le b∈L1
loc(I; W1,1
loc (Rn; Rn)). Suppose ha
|b( ,x)|
1+|x|log+|x|∈L1(I;L∞(Rn)) , (11)
and
ˆ
I
ˆ
Rn
exp Dxb
log+Dxb( , x) dγn(x)d < +∞,(12)
whe e γnis he Gaussian measu e on Rn, namely
γn:= 1
(2π)n/2exp −|x|2
2Ln
wi h Ln he Lebesgue measu e in Rn.
462
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
Then he space con inuous ep esen a i e ˜
b, g an ed by he Sobole embedding, is well-posed
in I×and he associa ed low Xo ˜
bis globally de ined, ha is, X:I×I×Rn→Rn. Mo e-
o e , o e e y , s∈I, he image measu e X( , s, ·)#Lnis absolu ely con inuous wi h espec o
Lnand he e exis s a posi i e cons an α0( , s) >0such ha
d
dLn(X( , s, ·)#Ln)∈Lα
loc (Rn)(13)
o each 0 <α<α
0(s, ), whe e Lα
loc (Rn)is he O licz space wi h
α:[0,+∞)→[0,+∞), α(w) :=wexp((log+w)α),
and α0:I×I→Ris con inuous wi h α0( , ) =1 o e e y ∈I.
See Sec ion 5 o he p oo . By i ial conside a ions, in he one-dimensional case, we can
imp o e Theo em B o absolu ely con inui y o he low.
Theo em C. I n =1and bsa is ies he condi ions o Theo em B, hen, o e e y , s∈I, he
map X( , s, ·)is an absolu ely con inuous homeomo phism be ween in e als o R.
Sobole egula i y s a ed in Theo em Cis sha p, as we show in Example 8.2. We don’ know
whe he Theo em Ccan be ex ended o he case o sub-exponen ial summabili y o o de k>1.
In highe dimensions, o subexponen ial summabili y, we ha e a pa ial nega i e esul wi h
an example in Sec ion 8.2, o he example cons uc ed in [8], see also Rema k 7.2: he e a e
ec o ields sa is ying a subexponen ial summabili y o o de 1whose low is no in W1,p
loc o any
p>n. Howe e , i emains open whe he ec o ields sa is ying a subexponen ial summabili y
o o de 1 can ail o ha e he low in W1,1
loc .
On he o he hand, in highe dimensions, o exponen ial summabili y we ha e a posi i e
esul :
Theo em D. Le I⊂Rand ⊂Rnbe bounded open se s and le b∈L1
loc(I; W1,1
loc (; Rn)) be
bounded. Assume ha o some p>2n he ec o ield bsa is ies he global geome ic condi ion
p:=ˆ
ˆ
I
max
n
n−p,(dis (x, ∂))
n
n−p
(sup |b|)
n
n−pexp p 2
p−nDxb(s,x)dsdx<+∞,(14)
wi h equal o he leng h o I. Then he space-con inuous ep esen a i e ˜
bis well-posed in I×
hanks o Theo em Aand, in addi ion, o e e y ∈Iand o almos e e y s∈I, one has
o a.e. s∈I, X( ,s,·)∈W1,p(( ,s);Rn)and X(s, ,·)∈W1,p((s, );Rn), (15)
wi h
ˆ
I
ˆ
( ,s)
DxX( ,s,x)pdxds≤
n
p−np.(16)
463
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
Wi h ega ds o (15), a lowe Sobole egula i y o he low can be p o ed o all pai s o
imes s, ∈I, see Co olla y 6.9. Fo b∈C0(I; C1(; Rn)), we ha e a mo e de ailed s a emen
in Theo em 6.1. We may also conside he case when b:I×Rn→Rnand he suppo o b( , ·)
is con ained in a compac se independen o . See also Rema ks 6.3 and 6.2.
Theo em E. Le I⊂Rbe a bounded open in e al and le b∈L1
loc(I; W1,1
loc (Rn; Rn)) be a
bounded ec o ield. Assume ha he e exis a bounded open se ⊂Rnand p>nsuch ha
sp (b( , ·)) ⊂ o each ∈I(17)
and (6)holds wi h β=p2/(p −n) and equal o he leng h o I.
Then he space-con inuous ep esen a i e ˜
b, unde s ood as ec o ield in I×, is well-posed
in I× hanks o Theo em A, ( ,s) =and o e e y , s∈Ione has
X( ,s,·)∈W1,p(;Rn)wi h
ˆ
DxX( ,s,x)pdx≤1
ˆ
I
ˆ
exp p 2
p−nDxb( , y)dyd . (18)
No ice also ha , in he p e ious heo em, we ob iously ha e ha he low map X:I×I×
Rn→Rno bis iden ically equal o he iden i y on I×I×(Rn ).
Rema k 1.1. (1) Since dis (x, ∂) is bounded, (14) can be s a ed in he equi alen o m
ˆ
ˆ
I
(dis (x, ∂))n/(n−p) exp p2
p−nDxb(s,x)dsdx<+∞.
We used ha speci ic o m o he pu pose o he es ima e (15).
(2) I α=n/(p −n) is smalle han 1 (i.e., p>2n) and ∂ is egula , hen he geome ic
condi ion (14), hanks o he Hölde inequali y, is implied by he simple condi ion
ˆ
ˆ
I
exp (cDxb(s,x))dsdx<+∞ (19)
p o ided c>(1/α)p2/(p −n), whe e (1/α)=(1 −α)−1is he dual exponen .
We also poin ou ha he Sobole exponen p, ela ed o he cons an in he exponen ial
in eg abili y condi ion in (14), may no be sha p. In o he wo ds, we can ha e p=∞bu
X( , s, ·) ∈W1,p(( ,s); Rn), see Example 8.3.
Se e al egula i y p ope ies o X ollow om Theo em D, see Co olla ies 6.7,6.8 and 6.10.
In pa icula we show ha , as in Theo em B, X( , s, ·)#LnLn, o each , s∈I.
Rema k 1.2. Suppose ha b∈L1
loc(I; W1,1
loc (Rn; Rn)) has compac suppo and (19) holds o
some c>0. Fo e e y p>2n, we ge (18)on subin e als o Icon aining sand o leng h
such ha p2
p−n=c. Since p2
p−n>4n o p>2n, hen <c
4n.
464
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
1.3. T anspo and con inui y equa ions
Finally, we apply Theo em Band Theo em D o he anspo equa ion (2) and he con inui y
equa ion (4), espec i ely. By s anda d me hods, we p o ide exis ence, uniqueness, ep esen a-
ion and egula i y o solu ions.
In [12, Theo em 1], uniqueness o weak solu ions u ∈L∞((0, T); L∞(Rn)) has been p o ed,
p o ided ha he spa ial de i a i es o bsa is y a sub-exponen ial summabili y and ¯u∈L∞(Rn).
Ou nex esul p o ides e en in his con ex he classical ep esen a ion o his unique solu ion
in e ms o he low map, g an ed by Theo em B.
Theo em F (Rep esen a ion o he solu ions o he anspo equa ion). Le bbe a ec o ield
as in Theo em B, and le Xbe he low associa ed o he con inuous ep esen a i e b. Then, o
each ¯u∈L∞(Rn), o each ∈[0, T], he unc ion
( ,x) := ¯uX( ,0,·)−1(x) o a.e. x∈Rn,(20)
is he unique weak solu ion in L∞((0, T); L∞(Rn)) o he Cauchy p oblem o he anspo
equa ion (2), unde s ood in he sense o dis ibu ions.
By means o he heo y o maps wi h ini e dis o ion, see o ins ance [25], we can also show
Sobole egula i y o he solu ion o he anspo equa ion, see Co olla y 7.1.
By applying he well-posedness and ep esen a ion esul s p o ed in [3,14] (see Theo em 7.4
and Rema k 7.5), we make mo e explici he ep esen a ion o he weak solu ions o (4), unde
he assump ions o Theo em Band Theo em E. I is use ul o deal wi h he case whe e ρ( , ·)
belongs o he space o signed Bo el measu es on Rnwi h ini e o al a ia ion, which we will
deno e by M(Rn).
Theo em G (Rep esen a ion o he solu ions o he con inui y equa ion). Le I=(0, T), le
b:I×Rn→Rnbe a ec o ield wi h b( , ·)con inuous o a.e. ∈I.
(i) Suppose ha bsa is ies he assump ions o Theo em Band le Xbe he low associa ed o b.
Then, o each signed measu e ¯ρ∈M(Rn),
ρ =ρ( ,·)=X( ,0,·)#¯ρ ∈[0,T](21)
is he unique weak solu ion o he Cauchy p oblem (4)in L∞((0, T); M(Rn)).
Mo eo e , i ¯ρ∈L1(Rn), ρ can be ep esen ed as
ρ :=(¯ρJX, )◦X(0, ,·), (22)
whe e JX, (y) =dX( ,0,·)#Ln
dLn(X( , 0, y)). In pa icula we ha e ρ∈L∞((0, T); L1(Rn)).
(ii) I bsa is ies he s onge hypo heses o Theo em Eand ¯ρ∈L1(Rn), hen he unique weak
solu ion o he Cauchy p oblem (4)can also be ep esen ed as
ρ =¯ρ
JX( ,0,·)◦X(0, ,·), (23)
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
P oo . Le us conside he compac cu e ={( , X∞( , s∞, x∞)) : ∈J}and le Ube he
open se J×, wi h JJIand chosen in such a way ha ⊂U. By assump ion,
´Jsup|bh( , ·) −b( , ·)| d is in ini esimal as h →∞. Le us conside he maximal solu ions
γh( ) o he ODE ela i e o bh, s a ing om xha shand wi h g aph emaining in U(in pa ic-
ula , es ic ions o Xh(·, sh, xh)), and le Jh⊂Ih
(sh,xh) hei maximal exis ence in e als. Since
he cu es γha e equicon inuous, i is clea ha limi poin s o he g aph o hese cu es exis , and
ha any limi poin is he g aph o a solu ion γ∞ o he ODE ela i e o b∞, wi h γ∞(s∞) =x∞,
de ined on an in e al J∞and wi h he p ope y ha , as ends o one, i any, o he ex eme
poin s o J∞di e en om ∂I, ( , γ∞( )) ends o ∂U. Since UI×, he well-posedness
o b∞yields ha he cu e γ∞is no only a es ic ion o he maximal cu e X∞( , s∞, x∞),
∈I∞
(s∞,x∞), bu also con ains he cu e X∞( , s∞, x∞), ∈J, since his es ic ed cu e does
no ouch he bounda y o U.
The p e ious lemma g an s, in pa icula , he lowe semicon inui y o (s, x) →leng h(I(s,x))
in I×. Mo eo e , a simple con adic ion a gumen gi es
o any compac in e al J⊂I∞
(s∞,x∞)and any open se A, one has
∃¯
hsuch ha J⊂Ih
(s,x) o each h> ¯
hand (s, x) ∈J×A;(34)
A∞
( ,s) implies Ah
( ,s) o hla ge enough (35)
and he uni o m con e gence o Xh( , s, ·) o X∞( , s, ·)on A.
4. Well-posedness wi h O licz condi ion
In his sec ion we a e going o p o e Theo em A. We ix some dimensional cons an s o
n ≥1. Le ωn=|B(0, 1)|be he olume o he uni Euclidean ball in Rnand σn−1=nωn he
pe ime e o B(0, 1); le τn=|B(0, 1) ∩B(q, 1)| o any q∈∂B(0, 1); inally, Cnis he cons an
om Lemma 4.1 and κn=2τ−1
nωnCnσn−1will appea in Lemma 4.3.
Lemma 4.1. Le b∈W1,1(B(x, ); Rn)and assume ha xis a Lebesgue poin o b. Then, o
some dimensional cons an Cn, one has
−
ˆ
B(x, )
|b(x) −b(y)|dy≤Cnˆ
B(x, )
Db(y)
|x−y|n−1dy. (36)
P oo . We assume o simplici y n ≥2. F om he same a gumen o [18, Lemma 1, Sec. 4.5.2],
based on a adial in eg a ion, o all x∈Rn, >0, ∈(0, 1)and ∈C1(B(x, )) we ha e
−
ˆ
B(x, ) B(x, )
| (x)− (y)|dy≤
ˆ
−
ˆ
B(x,s)
|y−x||D (y )|dyds
≤
ˆ
s−
ˆ
B(x,s)
|D (y )|dyds.
(37)
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
Assume now ha ∈L1
loc(), xis a Lebesgue poin o , B(x, ) and ∈W1,1(B(x, )).
Le δ:= dis (B(x, ), Rn ), δ:= {x∈ :dis (x, Rn ) >δ/2}and (ρh)hbe a sequence
o molli ie s in Rn. Then i is well-de ined he sequence o egula ized unc ions h:= ∗ρh:
δ→Ri h >1/δ and i sa is ies he ollowing p ope ies: h∈C∞(δ),
h(z) → (z), o each z∈δLebegue poin o ,ash→∞,(38)
h→ in W1,1(B(x, )) . (39)
Applying (37) wi h ≡ hand le h →∞, by (38) and (39), we ge he same inequali y when
∈W1,1(B(x, )) and xis a Lebesgue poin o . Now we can le →0 and use Fubini’s
heo em in he igh hand side o ge
−
ˆ
B(x, )
| (x)− (y)|dy≤1
nˆ
B(x, )
|D (y )|
|x−y|n−1dy. (40)
Finally, no ice ha he e is a cons an cnsuch ha , i Mis a n ×nma ix wi h ows Mj, hen
j|Mj| ≤cnM. The e o e,
−
ˆ
B(x, )
|b(x) −b(y)|dy≤
n
j=1−
ˆ
B(x, )
|bj(x) −bj(y)|dy
≤1
n
n
j=1ˆ
B(x, )
|Dbj(y)|
|x−y|n−1dy≤cn
nˆ
B(x, )
Db(y)
|x−y|n−1dy,
which p o es (36) wi h Cn=cn/n.
Recall Jensen’s inequali y
⎛
⎝ˆ
X
χ(x)dμ(x)⎞
⎠≤ˆ
X
(χ(x)) dμ(x), (41)
whe e :[0, +∞) →[0, +∞)is a con ex unc ion, μis a p obabili y measu e on Xand χ:
X→Ris a μ-measu able non-nega i e unc ion. The ollowing lemma is a di ec applica ion o
Jensen’s inequali y.
Lemma 4.2. Le :[0, +∞) →[0, +∞)be a con ex unc ion. I n ≥1, x∈Rn, >0, χ∈
L1
loc(Rn)is non-nega i e, hen
⎛
⎜
⎝
1
ˆ
B(x, )
χ(z)
|z−x|n−1dz⎞
⎟
⎠≤1
σn−1ˆ
B(x, )
(σn−1χ(z))
|x−z|n−1dz. (42)
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
P oo . Using pola coo dina es, one easily checks ha
ˆ
B(0, )
1
|z|n−1dz= σn−1.
The e o e, he measu e dμ := χB(x, )(z)
σn−1
dz
|x−z|n−1is a p obabili y measu e on Rnand Jensen’s
inequali y (41) applies.
Lemma 4.3. Le :[0, +∞) →[0, +∞)be a con ex non-dec easing unc ion. Le o∈Rn,
R>0and b∈W1,1(B(o, 2R); Rn). Then, o all α∈(1, n
n−1), and all x, y∈B(o, R) dis inc
Lebesgue poin s o b, one has
|b(x) −b(y)|
|x−y|≤1
σn−1⎛
⎜
⎝ˆ
B(o,2R)
1
|z|α(n−1)dz⎞
⎟
⎠
1/α
×
×1
|x−y|⎛
⎜
⎝ˆ
B(o,2R)
(κnDb(z))α
α−1dz⎞
⎟
⎠
α−1
α
.
(43)
In he case n =1, he es ima e (43)is unde s ood o hold o α∈(1, +∞)and we also ha e
|b(x) −b(y)|
|x−y|≤1
2|x−y|ˆ
B(o,2R)
(κnDb(z))dz. (44)
P oo . Se :=|x−y| >0 and W=B(x, ) ∩B(y, ). No ice ha |W| =τn n. Then
|b(x) −b(y)|≤−
ˆ
W
|b(x) −b(z)|dz+−
ˆ
W
|b(y) −b(z)|dz
≤ωn
τn⎛
⎜
⎝−
ˆ
B(x, )
|b(x) −b(z)|dz+−
ˆ
B(y, )
|b(y) −b(z)|dz⎞
⎟
⎠
≤2ωn
τn
sup
w∈B(o,R) −
ˆ
B(w, )
|b(w) −b(z)|dz.
Since is non-dec easing, we ha e:
|b(x) −b(y)|
|x−y|≤sup
w∈B(o,R)
⎛
⎜
⎝
2ωn
τn−
ˆ
B(w, )
|b(w) −b(z)|dz⎞
⎟
⎠
474
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
by (36)
≤sup
w∈B(o,R)
⎛
⎜
⎝
2ωnCn
τnˆ
B(w, )
Db(z)
|w−z|n−1dz⎞
⎟
⎠
by (42)
≤1
σn−1
sup
w∈B(w, ) ˆ
B(w, )
2ωnCnσn−1
τnDb(z)1
|w−z|n−1dz.
I n =1, we ha e al eady ob ained (44). Howe e , i α>1, we apply he Hölde inequali y o
ob ain
ˆ
B(w, )
(κnDb(z))1
|w−z|n−1dz
≤⎛
⎜
⎝ˆ
B(w, )
(κnDb(z))α
α−1dz⎞
⎟
⎠
α−1
α
×⎛
⎜
⎝ˆ
B(w, )
1
|w−z|α(n−1)dz⎞
⎟
⎠
1/α
≤⎛
⎜
⎝ˆ
B(o,2R)
(κnDb(z))α
α−1dz⎞
⎟
⎠
α−1
α
×⎛
⎜
⎝ˆ
B(0,2R)
1
|z|α(n−1)dz⎞
⎟
⎠
1/α
.
No ice ha ´B(0,2R)
1
|z|α(n−1)dz<∞i and only i α(n −1) <n, i.e., α< n
n−1when n >1o
α<∞when n =1. Applying his es ima e o he o me inequali y, we ob ain (43).
P oo o Theo em A.No ice ha assump ion (A.I) implies ha is also con ex, and condi-
ion (7)is equi alen o
−1(s1s2)≤C−1(s1)−1(s2)∀s1,s
2≥(C). (45)
Le n >1, ix α∈(1, n
n−1)as in he assump ion (A.I), so ha (s) :=(s/κn)α−1
αis con ex,
whe e κnis he cons an de ined a he beginning o he sec ion.
Fix o∈and he co esponding R>0 and c>0as in (8). We claim ha he space con inuous
ep esen a i e ˜
bis well posed in I×B(o, R). Since b:I×B(o, R) →Rnis well posed and
sa is ies (8)i and only i cb does, in he p oo o he claim we can assume wi h no loss o
gene ali y c=1. Applying Lemma 4.3 wi h he abo e α∈(1, n
n−1)and , we ha e, o almos
e e y and all x, y∈B(o, R) dis inc Lebegue poin s o b( , ·), one has
|b( ,x) −b( ,y)|
κn|x−y|α−1
α
≤C(n,R,α) 1
|x−y|⎛
⎜
⎝ˆ
B(o,2R)
(Db( ,z))dz⎞
⎟
⎠
α−1
α
.
We w i e s1∨s2 o max{s1, s2}and s1∧s2 o min{s1, s2}. Then, by applying (45) wice we
ob ain ha
475
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
|b( ,x) −b( ,y)|
κn|x−y|≤C2
−1((C)∨C(n,R,α) α
α−1)×
×−1⎛
⎜
⎝(C)∨ˆ
B(o,2R)
(Db( ,z))dz⎞
⎟
⎠×
×−1(C)∨1
|x−y|α
α−1.
I we se
ω(δ) :=δ−1(C)∨1
δα
α−1(46)
and
ϕ( ) :=κnC2
−1((C)∨C(n,R,α) α
α−1)
−1⎛
⎜
⎝(C)∨ˆ
B(o,2R)
(Db( ,z))dz⎞
⎟
⎠,
(47)
we ob ain ha ω∈C0((0, ∞)) and bsa is ies
|b( ,x) −b( ,y)|≤ϕ( )ω(|x−y|)(48)
o each x, y∈B(o, R) Lebesgue poin s o b( , ·). F om (48), i ollows ha b( , ·)is uni o mly
con inuous on he se o Lebesgue poin s o b( , ·)con ained in B(o, R). Since his se is dense
in B(o, R), i ollows ha he e exis s a unique con inuous ex ension ˜
b( , ·) :B(o, R) →Rn
s ill sa is ying (48)on he whole B(o, R). The e o e we can conclude ha ˜
b( , ·) :Rn→Rnis
con inuous, condi ion holds (48) on B(o, R) and
˜
b=ba.e. on I×B(o,R). (49)
Mo eo e , i is clea ha ϕ∈L1
loc(I; R), because −1is conca e and he unc ion in (8) belongs
o L1
loc(I; R)by assump ion.
We claim ha condi ion (A.III) implies ´1
0
1
ω(δ) dδ=∞and limδ→0ω(δ) =0. Indeed, on he
one hand, using he mono onici y o −1and he change o a iables s=−1((1/δ) α
α−1), we
ob ain, i ¯
δ:=(C)1−α
α∧1 and ¯s=−1((1/¯
δ) α
α−1),
1
ˆ
0
1
ω(δ) dδ≥
¯
δ
ˆ
0
1
δ−1((1/δ) α
α−1)
dδ=α−1
α
∞
ˆ
¯s
(s)
s(s) ds=+∞.
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
On he o he hand, o p o e lim
δ→0ω(δ) =lim
δ→0δ−1((1/δ) α
α−1) =0, we only need o show ha he
unc ion δ→δ−1((1/δ) α
α−1)is mono one o δsmall enough, because he abo e in eg al is no
bounded. Again wi h he change o a iables s=−1((1/δ) α
α−1), we see ha his mono onici y
is equi alen o he mono onici y o he unc ion φ:s→s−1(s)−α−1
α o sla ge. Inspec ing he
de i a i e o φ, we see ha
φ(s) =(s)−α−1
α
s2−α−1
α
(s)
s(s) −1<0,
and so φis mono one o ssu icien ly la ge. We conclude ha , o each o∈and such ha
ϕ( ) <+∞ he e exis s R>0 such ha B(o, R) ⊂and ˜
b( , ·) :B(o, R) →Rnsa is ying
Osgood’s condi ion
|˜
b( ,x) −˜
b( ,y)|≤ϕ( )ω(|x−y|) o each x,y ∈B(o,R). (50)
|˜
b( ,x)|≤|˜
b( ,x) −˜
b( ,x0)|+|˜
b( ,x0)|≤ϕ( )ω(|x−x0|)+|˜
b( ,x0)|
≤ω(2R)ϕ( ) +|˜
b( ,x0)| o a.e. ∈Iand o each x, x0∈B(o,R).
(51)
Now obse e ha , since b∈L1
loc(I ×; Rn), by (49), ˜
b∈L1
loc(I ×B(o, R); Rn). Thus he e
exis s x0∈B(o, R) such ha |˜
b( , x0)| ∈L1
loc(I). By (50), (51) and P oposi ion 3.2 (i), we ob ain
ha ˜
b:I×B(o, R) →Rnis well-posed. As a consequence also he ec o ield ˜
b:I× →Rn
is well-posed.
Mo eo e , i (9) holds, i also ollows ha
|˜
b( ,x)|≤m( ) o a.e. ∈I, o a.e. x∈B(o,R).
Thus, by applying now P oposi ion 3.2 (i ), we can conclude ha he ec o ield ˜
b:I×
B(o, R) →Rnis well-posed, which implies ha ˜
b:I× →Rnis well-posed, oo.
When n =1, le (s) :=(s/κn). Then we can epea he same a gumen s o he p e ious
case, showing ha he space con inuous ep esen a i e o bsa is ies Osgood’s condi ion.
The nex p oposi ion shows why we could no use a s anda d Sobole embedding in he p oo
o Theo em A o ob ain a s onge Sobole egula i y o b. See also Rema k 4.5 below.
P oposi ion 4.4. Condi ion (A.III) in Theo em Aimplies ha canno ha e polynomial g ow h,
ha is,
∀m∈Nlim sup
s→∞
(s)
sm=+∞,(52)
bu i does no imply ha has mo e han polynomial g ow h.
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
P oo . On he one hand, we ha e
∞=
∞
ˆ
1
(s)
s(s) ds=∞
k=1
k+1
ˆ
k
(s)
s(s) ds
≤∞
k=1
1
k
k+1
ˆ
k
(s)
(s) ds=∞
k=1
1
k(log((k +1)) −log((k)))
=−log((1)) +∞
k=2
1
k2−klog((k)) +lim sup
k→∞
log((k))
k
I he se ies in he las ow is in ini e, hen o e e y 0 <α<1 he e exis s a sequence kj→∞
such ha
1
k2
j−kj
log((kj)) ≥1
k1+α
j
,
ha is, o all jwi h kj≥2,
(kj)≥exp 1
2k1−α
j.
I ins ead he se ies in he las ow is ini e, hen lim supk→∞
log((k))
k=∞, ha is, he e exis s a
sequence kj→∞such ha
(kj)≥exp kj.
On he o he hand, we canno imp o e he lim sup in (52) wi h a lim in (o a lim). Indeed,
wi h a simila es ima e as abo e, we ob ain
∞
ˆ
1
(s)
s(s) ds≥−1
2log((1)) +∞
k=2
1
k2+klog((k)).
Thus, aking a sequence {kj}jspa se enough, one can cons uc a piece-wise linea inc easing
con ex unc ion :[0, +∞) →[0, +∞)such ha (kj) =exp(k2
j+kj)( hus ´∞
1
(s)
s(s) ds=
∞) and (xj) =x2
j o some kj<x
j<k
j+1(hence lim in s→∞ (s)
s2<∞).
Le us gi e mo e de ails abou such cons uc ion. As a s a , de ine k1=1, (1) =exp(2)
and α1=exp(2). Then de ine ( ) =α1 o ∈[0, x1], whe e x1is so ha (x1) =x2
1, ha
is, x1=α1. I e a i ely, gi en xjand αj, and he unc ion de ined in [0, xj]wi h (xj) =x2
j,
de ine
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
kj+1=2xj
αj+1=exp(k2
j+1+kj+1)−x2
j
kj+1−xj
( ) =(xj)+αj+1( −xj) o ∈[xj,x
j+1]
whe e xj+1is so ha (xj)+αj+1(xj+1−xj)=x2
j+1
By cons uc ion, we ha e (kj+1) =exp(k2
j+1+kj+1)and (xj+1) =x2
j+1. Mo eo e , since
exp(x2+x) >x
2 o x≥0, we ha e αj+1>α
jand hus he esul ing unc ion is con ex; in
ac , is he sup o he linea unc ions →(xj) +αj+1( −xj), j=1, ..., and hus i is
con ex.
Rema k 4.5. Using he unc ion cons uc ed in he p e ious P oposi ion 4.4, we can gi e an
example o a con inuous ec o ield b:R →R ha is well posed, e en hough b/∈W1,p
loc (R) o
all p>2.
Indeed, de ine b( ) =´
0χ(s) ds, whe e χ∈L1(R)is he unc ion
χ(s)=∞
j=1
χIj(s) ·xjwhe e Ij:=⎛
⎝
j−1
k=1
1
k2x2
k
,
j
k=1
1
k2x2
k⎞
⎠.
No ice ha , by cons uc ion,
exp(4x2
j+xj)≤exp(k2
j+kj)=(kj+1)<(x
j+1)=x2
j+1,
and hus xj+1>exp(2x2
j+xj) >exp(xj) >exp(j) o all j≥1. I ollows ha ∞
j=1Ijis
bounded and ha
ˆ
R
χ(s)pds=∞
j=1
xp
j
j2x2
j=∞
j=1
xp−2
j
j2
is ini e i and only i p≤2.
4.1. A class o subexponen ial summabili y ypes
Examples o unc ions sa is ying he p ope ies lis ed in Theo em Aa e o he o m Ek,β(s)
as in (10), as we will show in his sec ion. I is clea ha he subexponen ial summabili y o ype
Ek,β implies he subexponen ial summabili y o ype Ek,β o all β≤β≤1 and k≥k.
I Dxbsa is ies an exponen ial summabili y, hen i is well-known ha bis well-posed.
Indeed, in his case, he ec o ield bsa is ies a so-called Log-Lipschi z condi ion (see, o
ins ance, [8,32]). Theo em Aex ends he well-posedness o subexponen ial summabili y o de
k≥1. We will show in Sec ion 8.2 ha he uppe bound on β o he subexponen ial summabili y
o de 1is in ac necessa y; see also [12, Sec ion 6], [11, p. 1240].
479
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
Le us now show ha Ek,β sa is ies he assump ions o Theo em A. Le us in oduce some no-
a ion in o de o be e ep esen Ek,β. Le us deno e by Ek:R →R he k h-i e a ed exponen ial
unc ion, ha is, by induc ion on k,
E1(s) :=exp(s), Ek+1(s) :=exp (Ek(s))i s∈R,k≥1.
Since lims→−∞E1(s) =0, no ice ha lims→−∞Ek(s) =Ek−1(0) =:sk o all k>1, and ha
limk→∞sk=∞. Deno e by Lk:Ek(R) →R he k h-i e a ed loga i hm unc ion as he in e se
o Ek, ha is
Lk(s) :=E−1
k(s) i s∈Ek(R)=(sk,+∞).
De ine hen Pk(s) :=k
j=1Lj(s), so ha one can easily check ha
L
k+1(s) =1
sPk(s),so ha P
k(s) =Pk(s)
s
k
j=1
1
Pj(s).
Wi h his no a ion, we ha e
Ek,β(s) =exp s
Pk−1(s)Lβ
k(s).
A di ec compu a ion shows ha
E
k,β =Ek,βHk,β whe e Hk,β =1
Pk−1Lβ
k⎛
⎝1−
k−1
j=1
1
Pj−β
Pk⎞
⎠,(53)
and ha
H
k,β =1
sPk−1Lβ
k−k−1
j=1
1
Pj+β
Pk1−
k−1
j=1
1
Pj−β
Pk+
k−1
j=1
1
Pjj
i=1
1
Pi+β
Pk.(54)
P oposi ion 4.6. The unc ion Ek,β :R →Rsa is ies he assump ions o Theo em A o each
in ege k≥1and 0 ≤β≤1.
P oo . Mono onici y o Ek,β:F om (53)i is e iden ha E
k,β is posi i e o sla ge enough and
hus Ek,β is s ic ly inc easing in [α, +∞) o αla ge enough.
Ve i ica ion o (A.I): I γ>0, hen
d2
ds2Ek,β(s)γ=γEk,β(s)γγH2
k,β +H
k,β.
Because o he p esence o he ac o 1/s in (54), one can see ha lims→∞
H
k,β
H2
k,β =0 and hus
d2
ds2Ek,β(s)γis posi i e o sla ge enough. Thus we ob ain ha Ek,β(s)γis con ex o sla ge.
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
Ve i ica ion o (A.II): Fi s ly, no ice ha log(s ) =log(s) +log( ) ≤log(s) log( ) o all
s, >e. The e o e, by induc ion on k, we ha e Lk(s ) ≤Lk(s)Lk( ) o all s, >E
n(1). Hence,
Pk(s ) =k
j=1Lj(s ) ≤k
j=1Lj(s)Lj( ) =Pk(s)Pk( ), o all s, >E
n(1).
Secondly, condi ion (7), i.e., Ek,β( s) ≤Ek,β(s)Ek,β( ) o s, la ge enough, is equi alen o
s
Pk−1(s)Lk(s)β+
Pk−1( )Lk( )β≤s
Pk−1(s )Lk(s )β,
ha is
Pk−1( )Lk( )β
+Pk−1(s)Lk(s)β
s≤Pk−1(s)Lk(s)βPk−1( )Lk( )β
Pk−1(s )Lk(s )β.(55)
Thi dly, on he one hand we ha e lim →∞ Pk−1( )Lk( )β
=0, and hus he le -hand side o (55)
smalle han 1 o sand la ge. On he o he hand, by he ini ial obse a ion,
Pk−1(s)Lk(s)βPk−1( )Lk( )β
Pk−1(s )Lk(s )β≥Pk−1(s)Lk(s)βPk−1( )Lk( )β
Pk−1(s)Pk−1( )Lk(s)βLk( )β=1.
So, inequali y (55) holds ue and so condi ion (7).
Ve i ica ion o (A.III): No ice ha , by (53), o s>E
k(1),
E
k,β(s)
sEk,β(s) ∼s→∞
1
sPk−1(s)Lk(s)β.
Thus, since ´∞
Ek(1)
1
sPk−1(s)Lk(s)βds=∞i and only i β≤1, we also ha e ´∞
1
E
k,β(s)
sEk,β(s) ds=+∞
i and only i β≤1.
Rema k 4.7. Obse e ha , i (A.III) holds, hen
lim sup
s→∞
Pk−1(s)Lk(s)1+α(s)
(s) =∞ o e e y α>0 and k≥1.(56)
5. Regula i y o he low wi h subexponen ial summabili y
This sec ion is de o ed o he p oo o Theo em Band i s consequence in dimension 1 as
w i en in Theo em C. Le us ecall ha , i :[0, +∞) →[0, +∞)is an inc easing homeo-
mo phism, so ha (0) =0 and lim →+∞ ( ) =+∞, he O licz space L(Rn)is he space o
measu able unc ions :Rn→R o which he Luxembou g no m
L:= in ⎧
⎨
⎩
λ>0:ˆ
Rn
(x)
λdx≤1⎫
⎬
⎭
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
To p o e (65), we need o ake a limi in (75) along a sequence o se s A ha ills . To
his aim, we de ine
A:={x∈:dis (x, ∂) > }
o >0, plug Ain (75) and ake →0.
Fo he le -hand side o (75), we obse e ha A1
( ,s) ⊂A2
( ,s) whene e 1>
2. Thus, we
ha e
lim
→0ˆ
I
ˆ
A
( ,s)
DxX( ,s,x)pdxds=ˆ
I
ˆ
( ,s)
DxX( ,s,x)pdxds.
The igh -hand side o (75)is mo e icky. De ine ( , y) :=exp pq Dxb( ,y), δ(x) :=
dis (x, ∂), and α=n
n−p. No ice ha , since p≥2n, we ha e α∈(−1, 0).
We claim ha
lim in
→0ˆ
A
ˆ
I
α
A( , y) ( , y) d dy≤ˆ
ˆ
I
α
( , y) ( , y) d dy. (76)
I K, he mono one con e gence heo em implies ha
lim
→0ˆ
K
ˆ
I
α
A( , y) ( , y) d dy=ˆ
K
ˆ
I
α
( , y) ( , y) d dy.
Thus, (76)is shown i we can p o e ha
in
Klim in
→0ˆ
A K
ˆ
I
α
A( , y) ( , y) d dy=0.(77)
Indeed, hanks o (66), o e e y Kwe ha e
######
lim in
→0ˆ
A
ˆ
I
α
A( , y) ( , y) d dy−ˆ
ˆ
I
α
( , y) ( , y) d dy######
≤lim in
→0ˆ
A K
ˆ
I
α
A( , y) ( , y) d dy+ˆ
K
ˆ
I
α
( , y) ( , y) d dy.
I he e is a sequence Kjsuch ha
lim
j→∞lim in
→0ˆ
A Kj
ˆ
I
α
A( , y) ( , y) d dy=0,
hen limj→∞ | Kj| =0, and limj→∞´ Kj´Iα
( , y) ( , y) d dy=0, and so (76) holds.
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
To p o e (77)we use he inequali y (62). I Kis la ge enough, so ha dis (y, ∂A) < o all
y∈A K, he inequali y (62) simpli ies o
A( , y)bL∞≥dis (y, ∂A).
We use he la e inequali y o compu e he ollowing a e aged in eg al
1
η
η
ˆ
0
ˆ
A K
ˆ
I
α
A( , y) ( , y) d dyd
≤1
η
η
ˆ
0
ˆ
A K
ˆ
I
dis (y, ∂A)α
bα
L∞
( ,y)d dyd
=1
bα
L∞ˆ
K
ˆ
I
⎛
⎝1
η
η
ˆ
0
χA(y)dis (y, ∂A)αd⎞
⎠ ( ,y)d dy.
Since δ(y) ≤dis (y, ∂A) +, i.e., dis (y, ∂A) ≥δ(y) −, we compu e
η
ˆ
0
χA(y)dis (y, ∂A)αd≤
η
ˆ
0
χA(y)(δ(y) −)αd
=
min{η,δ(y)}
ˆ
0
(δ(y) −)αd
(∗)
≤min 2−α,2
α+1$ηδ(y)α.
In (∗), we conside ed wo cases: i s , when η<δ(y)/2, he in eg al is bounded by 2−αδ(y)αη;
second, when η≥δ(y)/2, using he ac ha α>−1, he in eg al is bounded by 2
α+1ηδ(y)α.
The e o e, he e exis s η0>0 such ha o all η∈(0, η0), we can es ima e he a e aged
in eg al wi h
1
η
η
ˆ
0
ˆ
A K
ˆ
I
dis (y, ∂A)α ( ,y)d dyd≤2−αˆ
K
ˆ
I
δ(y)α ( ,y)d dy.
I ollows ha
lim in
→0ˆ
A K
ˆ
I
dis (y, ∂A)α ( ,y)d dy≤2−αˆ
K
ˆ
I
δ(y)α ( ,y)d dy.
Then, since we assumed ´´Iδ(y)α ( , y) d dy<∞in (63), we ob ain he es ima e (77) and
hus ou claim (76).
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
We ha e hus comple ed he p oo o (65), and hus (64). Nex we wo k on he second pa o
he heo em. I (67) holds, i.e., ˆ
≤( , x) ≤ o all ∈Iand x∈, hen
p≤
p≤
ˆ
n
p−n
p,(78)
since p≥2n. We can epea he s eps in (71) and (72) wi hou he in eg al in s, and hen ap-
ply (65) and (78) o ob ain
ˆ
( ,s)
DxX( ,s,x)pdx
≤⎛
⎝ˆ
I
ˆ
χ(s, )(y)
(s,X(s, ,y))
qexp p q Dxb( ,y)dyd ⎞
⎠
1/q
×
×⎛
⎝ˆ
I
ˆ
DyX(s, ,y)nq dyd ⎞
⎠
1/q
≤1
ˆ
ˆ
n2
p(p−n)
p,
and his p o es (68).
I (69) holds, hen he iple in eg al in (73)is ini e also o A =and p>n. In his case,
we can ob ain di ec ly (75) o A =wi hou passing h ough he app oxima ion A, whe e we
used he s onge hypo hesis p>2n.
The ex ension o Theo em 6.1 o he case o Sobole spa ial egula i y equi es a global
app oxima ion o bby mo e egula ec o ields ha seems o be no i ial, also because o
he weigh unc ion (s, x) depending on he ec o ield i sel . In addi ion, he non-doubling
p ope y o he exponen ial unc ion is sou ce o ex a di icul ies, when pe o ming s anda d
con olu ion a gumen s. In he case o a weigh independen o , we add essed his p oblem in
he no e [7], om which we ex ac he ollowing esul .
Theo em 6.6 ([7, Theo em 3 and Rema k 25]). Le be a bounded open se and le w: →
(0, +∞), wi h w+w−1∈L∞
loc().
(i) I b∈L1(I ; W1,1
loc (; Rn)) sa is ies
ˆ
I
ˆ
w(x)exp(cDxb(s,x))dxds<+∞,(79)
o some c>0, hen he e exis bh∈C∞(I ×; Rn)sa is ying, whene e ,
bhcon e ge o ˜
bin L1(I;C(;Rn)) as h→∞,(80)
whe e ˜
bis he space con inuous ep esen a i e o b,
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
Dxbh→Dxbin L1(I;L1(;Rn)), as h→∞,(81)
and
lim
h→∞ˆ
I
ˆ
w(x)exp(cDxbh(s, x))dxds=ˆ
I
ˆ
w(x)exp(cDxb(s,x))dxds. (82)
(ii) I b∈L1(I ; W1,1
loc (Rn; Rn)) and he e exis s a bounded open se such ha
sp (b( , ·)) ⊂ o each ∈I, (83)
and bsa is ies (79)on , hen he e exis bh∈C∞(I ×Rn; Rn)s ill sa is ying (80), (81),
(82)and also
sp (bh( , ·)) ⊂ o each ∈Iand h∈N.(84)
P oo o Theo em D.Le
w(x) :=max n/(n−p),(dis (x, ∂))n/(n−p)
(sup |b|)n/(n−p) ,
so ha , wi h c=p2/(p −n), one has
ˆ
I
ˆ
w(x)exp(cDb(s,x))dxds<+∞
and we may apply Theo em 6.6 (i) o b. Since bis bounded, we can also assume, by a unca ion
a gumen , ha sup|bh| ≤sup |b|. In addi ion, he quan i y
pin (66)is ini e and sa is ies
p≤
p, using he inequali y (62).
I ollows om hese conside a ions ha
lim sup
h→∞
p,h ≤lim sup
h→∞
p,h ≤p<+∞,
whe e
p,h :=ˆ
I
ˆ
(h(s, x))n/(n−p) expp 2
p−nDxbh(s, x)dxds, (85)
and simila ly p,h is de ined wi h bh.
Now, in o de o apply (65) om Theo em 6.1 o bhand he pass o he limi h →∞, i
su ices by Fa ou’s lemma o p o e ha
ˆ
( ,s)
DxX( ,s,x)pdx≤lim in
h→∞ ˆ
h
( ,s)
DxXh( ,s,x)pdx, (86)
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
whe e h
( ,s) and Xha e ela i e o he ec o ields bh. Now, (35), de i ed om Lemma 3.5,
yields ha o any open domain A ( ,s) one has A h
( ,s) o hla ge enough, and ha
Xh( , s, ·)con e ge o X( , s, ·)uni o mly on A. Hence he lowe semicon inui y o w→
´A|Dxw|pdxyields
ˆ
A
DxX( ,s,x)pdx≤lim in
h→∞ ˆ
A
DxXh( ,s,x)pdx≤lim in
h→∞ ˆ
h
( ,s)
DxXh( ,s,x)pdx.
Le ing A ↑( ,s) we ob ain he claimed semicon inui y p ope y (86).
P oo o Theo em E.Fi s assume ha bis unde s ood as ec o ield b:I× →Rn. Le
w≡1so ha , wi h c= p2/(p −n), one has
ˆ
I
ˆ
exp(cDb(s,x))dxds<+∞.
F om Theo em 6.6 (ii), le bh:I× →Rnbe he egula sequence o ec o ields sa is ying
(80), (81), (82) and (84).
I ollows om (83) ha h( , x) =and hus
lim sup
h→∞
p,h ≤1
ˆ
I
ˆ
exp(cDb(s,x))dxds<+∞,
whe e
p,h is he quan i y in (85).
Now, in o de o apply (68) wi h p>n om Theo em 6.1 o bhand hen pass o he limi
h →∞, we only need o use (86)again.
Assume now ha bis unde s ood as ec o ield b:I×Rn→Rn. I is clea ha he low
map X:I×I×Rn→Rnis iden ically equal o he iden i y on I×I×(Rn ). Since (18)
holds o any bounded open se , hen X( , s, ·) ∈W1,p
loc (Rn).
Co olla y 6.7. Unde he assump ions o Theo em D, we ha e, o e e y ∈I,
o almos e e y s∈I
X( ,s,·)#Ln( ,s) =JX(s, , ·)Ln(s, ) =1
JX( ,s,X(s, ,·))
Ln(s, ) ,(87)
whe e JX( , s, ·) =de (DxX( , s, ·)) ∈Lp/n(( ,s))is non-ze o almos e e ywhe e. Unde he
assump ions o Theo em E, hen (87)holds o e e y s, ∈I eplacing bo h ( ,s) and (s, ) by
Rn.
In ac , JXis s ic ly posi i e, as we will show in he nex co olla y.
P oo . Since X( , s, ·)−1=X(s, , ·)is also in W1,p
loc ((s, ); Rn)wi h p>2n, hen bo h maps
X( , s, ·)and X( , s, ·)−1a e di e en iable almos e e ywhe e by Lemma 2.4. The e o e, we
492
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
ha e JX( , s, x) = 0 by Lemma 2.1. By Lemma 2.4, X( , s, ·)sa is ies Lusin’s (N) condi ion,
and hus, by Lemma 2.2, he a ea o mula holds, ha is, (87) holds. Mo eo e , by he Laplace
expansion o he de e minan , one can easily p o e, by induc ion on he o de ma ix and Hölde
inequali y, ha JX( , s, ·) =de (DxX( , s, ·)) ∈Lp/n(( ,s)). The same a gumen applies unde
he assump ions o Theo em E.
Co olla y 6.8. Unde he assump ions o Theo em D, o e e y (s, x) ∈I×, i we se
y( ) :=DxX( ,s,x), B( ) :=(Dxb)( , X( , s, x)),
J( ):=de (DxX( , s, x)), β( ) :=di xb( ,X( ,s,x)) = ace(B( )),
hen yand Ja e absolu ely con inuous solu ions o he ini ial alue p oblems
˙y( ) =B( )y( ),
y(s) =Id.(88)
˙
J( )=β( )J( ),
J(s)=1.(89)
Mo eo e , o almos e e y ( , s, x) ∈Dbwe ha e
DxX( ,s,x) =exp ⎛
⎝
ˆ
s
Dxb( ,X( ,s,x))d ⎞
⎠,(90)
JX( ,s,x)=exp ⎛
⎝
ˆ
s
di xb( ,X( ,s,x))d ⎞
⎠.(91)
In pa icula , JX>0almos e e ywhe e.
P oo . Fi s , we claim ha , gi en s, o almos e e y x he ma ix Bbelongs o L1(I(s,x); Rn2),
whe e I(s,x) is de ined in Sec ion 3.2. Indeed, he change o a iables z=X( , s, x) and he
iden i y JX(s, , X( , s, x)) =1/JX( , s, x) gi e
ˆ
I
ˆ
(s, )
(Dxb)( , X( , s, x))dxd =ˆ
I
ˆ
( ,s)
(Dxb)( , z)JX(s, ,z)dzd .
The la e in eg al is ini e because Dxb∈Lq o all qand JX∈Lp/n. The e o e, he claim is
ue.
Second, using he same app oxima ion o bas in he p oo o Theo em D, we know ha
DxXh→DxXweakly in Lp(Db), since DxXha e uni o mly bounded in Lp(Db).
Thi d, we see ha he dis ibu ional de i a i e ∂ DxXhas he ollowing o m: o e e y φ∈
C∞
c(Db; Rn),
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
∂ DxX[φ]=−ˆ
Db
DxX∂ φd dsdx
=−lim
→0ˆ
Db
DxX∂ φd dsdx
=lim
→0ˆ
Db
Dxb( , X( , s, x))DxX( ,s,x)φd dsdx.
Now, X→Xuni o mly on compac se s and Dxb→Dxbin Lq
loc(Db) o all q. So, using
Hölde inequali y and he Lebesgue domina ed con e gence heo em, we ob ain ha he limi
abo e is
∂ DxX[φ]= ˆ
R2+n
Dxb( ,X( ,s,x))DxX( ,s,x)φ d dsdx.
In o he wo ds, ∂ DxX=Dxb( , X( , s, x))DxX( , s, x). This shows ha yis solu ion o he
Cauchy sys em (88). Since Bis in eg able, hen we ge (90)by in eg a ing his Cauchy sys em.
Finally, he alidi y o (89) ollows in a s anda d way om (88) and (89) implies (91).
Co olla y 6.9. Unde he assump ions o Theo em D, o e e y 1 < <p−nand o e e y
s, ∈I, we ha e ha
X( ,s,·)∈W1, (( ,s);Rn). (92)
P oo . The p oo is an imp o emen o (15) h ough an applica ion o Co olla y 6.8, Lemma 2.7
and P oposi ion 2.8. Indeed, o e e y 1 < <p−n he e exis s 0 <q<p
2−p2
p−nsuch ha
=p2(p −n)
q(p −n) +p2, ha is,
p− =q
p+n
p−n−1
.
So, gi en s, ∈I,(15) implies ha he e exis s u ∈Isuch ha
X(s, ,·)=X(s,u,X(u, ,·))
and bo h maps X(s, u, ·)and X(u, , ·)belong o W1,p on hei domains, wi h non-ze o Jacobian
by Co olla y 6.8. We hen apply Lemma 2.7 and P oposi ion 2.8 o p o e ha hei composi ion
is o class W1, on i s domain.
Co olla y 6.10. Unde he assump ions o Theo em D, we ha e
X∈W1,p(Db;Rn).
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L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
P oo . We know ha Xis con inuous and, om (16), we ha e ha DxX∈Lp(Db; Rn2). Mo e-
o e , by he iden i y X( , s, x) =x+´
sb( , X( , s, x)) d , we ha e
∂ X( ,s,x) =b( ,X( ,s,x)),
and hus ∂ Xis con inuous. Finally, by di e en ia ing wi h espec o s he semig oup iden-
i y (33)in he o m X( , , x) =X( , s, X(s, , x)), we ge
∂sX( ,s,X(s, ,x)) +DxX( , s, X(s, , x))b(s, X(s, , x)) =0,
o all , s, ∈Rand x∈Rn o which he exp ession makes sense. The e o e, ∂sX( , s, y) =
−DxX( , s, y)b(s, y) and so ∂sX∈Lp(Db; Rn).
Rema k 6.11. Co olla y 6.10 imp o es [13, Theo em 4] and [8, Co olla y 1.8], by d opping
assump ion (3), ha is di xb∈L1
loc(R, L∞(Rn)). No ice ha , i n =1, assump ion (3) educes
o he classical Lipschi z condi ion o bwi h espec o x, uni o mly in . Co olla y 6.10 also
applies o a non Lipschi z one-dimensional ec o ield b(see Example 8.3 below).
Rema k 6.12. Co olla y 6.10 looks almos sha p. Indeed, i was p o ed in [26] ha no Sobole
egula i y can be expec ed o he low, when assuming only ha b∈L1(I; W1,p(Rn; Rn)) o
all ini e p∈[1, +∞), e en when bis compac ly suppo ed and di e gence- ee. See also Re-
ma k 5.1.
7. Applica ions o PDEs
In his sec ion we apply he Sobole egula i y o lows o ge ing he ep esen a ion o weak
solu ions o he Cauchy p oblems bo h o he anspo and con inui y equa ions.
P oo o Theo em F.By [12, Theo em 1], we can in e he uniqueness o weak solu ions u ∈
L∞((0, T); L∞(Rn)) o (2), p o ided ha he spa ial de i a i e o bsa is ies a sub-exponen ial
summabili y and ¯u∈L∞(Rn). The e o e we ha e only o show ha he unc ion in (20)is a
weak solu ion o (2), ha is,
T
ˆ
0
ˆ
Rn
(∂ ϕ+di (b ϕ) )d dx =−ˆ
Rn
¯uϕ(0,·)dx(93)
o each ϕ∈C∞
c([0, T) ×Rn). We di ide he p oo in wo s eps.
1s s ep. Le us assume ha ¯u∈C∞(Rn) ∩L∞(Rn). Le b:I×Rn→Rnbe he amily o
ec o ields de ined as
b( , x) := (˜
b( , ·)∗ρ)(x) i ( , x) ∈I×Rn,
whe e
˜
b( , x) := (˜
b,1( ,x),..., ˜
b,n( , x)) ,
495
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
˜
b,i( , x) := max{{min{bi( , x), 1/},−1/}i ( , x) ∈I×Rn,i=1,...,n,
and (ρ)deno es a amily o molli ie s depending on he space a iable x. Then, s anda d p op-
e ies o con olu ions yield
|b( , x)|≤ √n
o each ( , x) ∈I×Rn,(94)
b( , ·)∈C∞(Rn;Rn)and Dxb( , x) =ˆ
Rn
Dxρ(x −y) ˜
b( , y) dy ∀( , x) ∈I×Rn,
(95)
b( , ·):Rn→Rnis Lipschi z con inuous, uni o mly wi h espec o ∈I, (96)
b→bin L1(I;L1(;Rn)) as →0, whene e Rn.(97)
Then, om (96)we ge ha bis well-posed and he low maps Xassocia ed o bis globally
de ined, ha is, X:I×Rn→Rnand is Lipschi z egula . De ine
( , x) =¯u(X(0, ,x)).
By (97), Lemma 3.5 and he subsequen ema k, X→Xuni o mly on compac subse s o
I×Rn. Since ¯uis assumed o be con inuous, we ob ain ha → uni o mly on compac se s
o I×Rn. Mo eo e , since ¯u∈L∞(Rn), by he domina ed con e gence heo em, we can also
assume ha → in L2on each compac se o I×Rn.
By he classical Cauchy-Lipschi z heo y (see, o ins ance, [5, Sec ion 2]), i is well-known
ha is a classical solu ion o (2) wi h bin place o band, in pa icula , a weak solu ion, i.e.,
T
ˆ
0
ˆ
Rn
(∂ ϕ+di (bϕ))dxd =−ˆ
Rn
¯u(x)ϕ(0,x)dx,
o each ϕ∈C∞
c([0, T) ×Rn). Since ϕhas compac suppo in [0, T) ×Rn, i is easy o check
ha di (bϕ) →di (bϕ) in L2. Hence
lim
→0+
T
ˆ
0
ˆ
Rn
(∂ ϕ+di (bϕ))dxd =
T
ˆ
0
ˆ
Rn
(∂ ϕ+di (bϕ))dxd ,
ha is, is a weak solu ion o (2).
2nd s ep. Le ¯u∈L∞(Rn)and, by molli ica ion in Rn, le ¯uj∈L∞(Rn) ∩C∞(Rn), wi h
j∈N, be a sequence o unc ions ha con e ges o ¯ualmos e e ywhe e and such ha ¯ujL∞≤
¯uL∞ o e e y j.
De ine
j( , x) =¯uj(X(0, ,x)).
496
L. Amb osio, S. Nicolussi Golo and F. Se a Cassano Jou nal o Di e en ial Equa ions 372 (2023) 458–504
By Theo em B, o e e y he homeomo phism X(0, , ·)sa is ies he Lusin (N) condi ion, and
he e o e j( , ·) → ( , ·)almos e e ywhe e in Rn. Since his is ue o e e y , we ge ha
j→ almos e e ywhe e in [0, T] ×Rn. Mo eo e , i is clea ha jL∞≤¯uL∞ o all j.
By he p e ious s ep, we ha e, o all j,
T
ˆ
0
ˆ
Rn
j(∂ ϕ+di (bϕ))dxd =−ˆ
Rn
¯uj(x)ϕ(0,x)dx,
o each ϕ∈C∞
c([0, T) ×Rn). We can now apply he Domina ed Con e gence Theo em and
pass o he limi j→∞ o ob ain ha is a weak solu ion o (2).
Co olla y 7.1 (Sobole egula i y o he solu ions o he anspo equa ion). Le b∈C0(I ×
Rn; Rn)as in Theo em Eand le Xbe he low o b. Le p>2nand 1 ≤˜q≤q<∞be such ha
˜q
q−˜q=q
p+n
p−n−1
,i.e., ˜q=pq(p −n)
q(p −n) +p2.(98)
I ¯u∈L∞(Rn) ∩W1,q
loc (Rn), hen he unc ion in (20)sa is ies
∈L∞([0,T];W1,˜q
loc (Rn)) .
P oo . By Theo em E, he unc ion :=X(0, , ·)and i s in e se a e in W1,p
loc wi h p>2n. By
Lemma 2.7 and Co olla y 6.8, is o ini e dis o ion and K
q∈L
loc o =q
p+n
p−n−1
. By
P oposi ion 2.8, he composi ion ope a o Tis con inuous om W1,q
loc (2) o W1,˜q
loc (1). Since
( , ·) =T(¯u), he p oo is concluded.
Rema k 7.2. No ice ha he p opaga ion o egula i y, in he spi i o Co olla y 7.1, may ail be-
low he exponen ial summabili y o Dxb, e en hough ¯u∈C∞
c(Rn). Indeed, in [8, Theo em 2.1],
he au ho s cons uc ed a di e gence- ee ec o ield b:R ×Rn→Rn(n ≥2) sa is ying he
subexponen ial summabili y condi ion (12), and a weak bounded, compac ly suppo ed solu-
ion u( , x) o (2) such ha ¯u:= u(0, ·) ∈C∞
c(Rn)bu u( , ·) /∈˙
Ws,p(Rn) o all >0, s>0
and p≥1, whe e ˙
Ws,p(Rn)deno es he so-called homogeneous Sobole space. The example is
based on he wo k [1]. Le us ecall ha when s=1 and 1 <p<∞, hen ˙
W1,p(Rn) ∩Lp(Rn)
coincides wi h he classical Sobole space W1,p(Rn)(see [1, Sec ion 2]).
Le us poin ou ha , al hough bsa is ies he hypo hesis o Theo em F, u( , ·) /∈W1,p(Rn) o
each ∈(0, ∞)and p∈(1, ∞). This implies ha he low o bhas no Sobole egula i y W1,p
o some p>n, o he wise he same p oo o Co olla y 7.1 could be epea ed. Ou Example 8.2
shows he same phenomenon. No ice ha we don’ know whe he u( , ·) /∈W1,1(Rn).
Rema k 7.3. Co olla y 7.1 shows ha he Sobole egula i y o he low Ximplies Sobole
egula i y o solu ions o he anspo equa ion in he o m (20). We no ice ha he con e se
implica ion is almos ue. Indeed, suppose ha o e e y 1 ≤˜q≤q<∞sa is ying (98), and o
e e y ¯u∈L∞(Rn) ∩W1,q
loc (Rn), he unc ion in (20) sa is ies ∈L∞((−T, T); W1,˜q
loc (Rn)).
497
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