A unified treatment of ODEs under Osgood and Sobolev type conditions

0.15 MB·English

by  Huaiqian Li

A unified treatment of ODEs under Osgood and
Sobolev type conditions
Huaiqian Li1, Dejun Luo2∗
1 1School of Mathematical Sciences, Beijing Normal University,Beijing, China
1
0 2Key Lab of Random Complex Structuresand Data Science, Academy of Mathematics and Systems Science,
2 Chinese Academy of Sciences, Beijing 100190, China

l
u
J

3
Abstract
1
In this paper we prove the existence, uniqueness and regularity of the DiPerna–Lions

] flow generated by a vector field which is “almost everywhere Osgood continuous”, following
A
Crippa and de Lellis’s direct method. As an application, we show the well-posedness of
C
transport equations in the space of nonnegative integrable functions.
.
h
t Keywords: DiPerna–Lionstheory,Sobolevregularity,Osgoodcondition,regularLagrangian
a
flow, transport equation
m
MSC 2010: 37C10, 35B65
[

1
v 1 Introduction
6
9
Intheseminalpaper[7],DiPernaandLionsestablishedtheexistenceanduniquenessofthequasi-
4
2 invariant flow of measurable maps generated by a Sobolev vector field with boundeddivergence.
.
7 Their method is quite indirect in the sense that they first established the well-posedness of
0 the corresponding transport equation, from which they deduced the results on ODE. Their
1
methodology is now called the DiPerna–Lions theory and can be seen as a generalization of the
1
: classical method of characteristics. It has subsequently been extended to the case of BV vector
v
i fields by Ambrosio [1, 2], via the well-posedness of the continuity equation. For the development
X
of this theory in the infinite dimensional Wiener space, see [3, 9]. Recently, Crippa and de Lellis
r
a [6] obtained some a-priori estimates on the flow (called regular Lagrangian flow there) which
enable them to give a direct construction of the flow (see the extension to the case of stochastic
differential equations in [20, 10]).
To introduce the setting of the present work, we recall the key ingredient in Crippa–de
Lellis’sdirectmethod,namely, aSobolevvector fieldb ∈ W1,p(Rd)(p ≥ 1)is“almosteverywhere
loc
Lipschitz continuous” (it holds even for BV vector fields, see [6, Lemma A.3]). More precisely,
there are a negligible subset N ⊂ Rd and a constant C depending only on the dimension d,
d
such that for all x,y ∈/ N and |x−y|≤ R, one has
|b(x)−b(y)| ≤ C |x−y| M |∇b|(x)+M |∇b|(y) , (1.1)
d R R
(cid:0) (cid:1)
where M f is the local maximal function of f ∈ L1 (Rd):
R loc
1
M f(x):= sup |f(y)|dy.
R Ld(B(x,r)) Z
0<r≤R B(x,r)
∗Email: [email protected]
1
Here Ld is the Lebesgue measure on Rd and B(x,r) is the ball centered at x with radius r. If
x = 0 is the origin, we will simply write B(r) instead of B(0,r). For a proof of the inequality
(1.1), see [10, Appendix]; one can also find a more complete discussion in [4] for higher order
Sobolev spaces. Using the inequality (1.1), Crippa and de Lellis estimated the following type of
quantity
|X (x)−X˜ (x)|
t t
log +1 dx (1.2)
Z (cid:18) δ (cid:19)
B(R)
in terms of R,δ and the Lp-norms of ∇b,∇˜b on some ball, where X and X˜ are respectively the
t t
flows associated to the Sobolev vector fields b and˜b. We refer the readers to [5] for an extension
of some of the results.
On the other hand, the study of stochastic differential equations with non-Lipschitz coeffi-
cients has attracted intensive attentions in the past decade, see for instance [17, 11, 19, 16]. In
particular, S. Fang and T. Zhang considered in [11] the general Osgood condition:
|b(x)−b(y)| ≤ C|x−y|r(|x−y|2), |x−y|≤ c , (1.3)
0
where r : (0,c ] → [1,∞) is a continuous function defined on a neighborhood of 0 and satisfies
0
c0 ds = ∞. Under this condition and assuming that the ODE
0 sr(s)
R
dX
t
= b(X ), X = x
t 0
dt
has no explosion, they proved that the solution X is a flow of homeomorphisms on Rd (see
t
[11, Theorem 2.7]). If in addition r(s) = log 1 and the generalized divergence of b is bounded,
s
then it is proved in [8, Theorem 1.8] that the Lebesgue measure Ld is also quasi-invariant under
the flow X . In a recent paper [15], the second named author generalized this result to the
t
Stratonovich SDE with smooth diffusion coefficients, using Kunita’s expression for the Radon–
Nikodym derivative of the stochastic flow (see [13, Lemma 4.3.1]).
Inspired by these two types of conditions (1.1) and (1.3), we consider in this work the
following assumption on the time dependent measurable vector field b : [0,T]×Rd → Rd:
(H) there are g ∈L1 [0,T],L1 (Rd) and negligible subsets N , such that for all t ∈ [0,T] and
loc t
x,y ∈/ N , one ha(cid:0)s (cid:1)
t
|b (x)−b (y)| ≤ g (x)+g (y) ρ(|x−y|), (1.4)
t t t t
(cid:0) (cid:1)
where ρ∈ C(R ,R ) is strictly increasing, ρ(0) = 0 and ds = ∞.
+ + 0+ ρ(s)
R
The typical examples of the function ρ are ρ(s) = s, slog 1, s log 1 loglog 1 ,···. Notice
s s s
that the two latter functions are only well defined on some sm(cid:0)all int(cid:1)er(cid:0)val (0,c (cid:1)], but we can
0
extend their domain of definition by piecing them together with radials. In this paper we fix
such a function ρ. Similarly we may call a function satisfying (1.4) “almost everywhere Osgood
continuous”. It is clear that if we take g = C M |∇b | and ρ(s) = s for all s ≥ 0, then the
t d R t
inequality (1.4) is reduced to (1.1). On the other hand, if g is essentially bounded, then (1.4)
becomes the general Osgood condition (1.3), except on the negligible set N (we can redefine
t
b on this null set to get a continuous vector field). Therefore, this paper can be seen as a
t
unified treatment of the two different types of conditions. We would like to mention that the
assumptions like (H) were considered in [18], but the function ρ was always taken as ρ(s) = s
for all s ≥ 0.
The paper is organized as follows. In Section 2, we construct the flow of measurable maps
under the condition (H) and the boundedness of the divergence of b, following Crippa and de
2
Lellis’s directmethod. Weapplythisresulttoshowthewell-posedness ofthetransportequation
in the space of nonnegative integrable solutions. Then in Section 3, we prove a regularity
property of the flow, which is weaker than the approximate differentiability discussed in [6]. We
also prove a compactness result on the flow. To avoid technical complexities, we assume that
the vector fields are bounded throughout this paper.
2 Preparations
Wefirstgivethedefinitionoftheflowassociatedtoavectorfieldb(alsocalledregularLagrangian
flow in [1, 6]).
Definition2.1(RegularLagrangianflow). Letb ∈ L1 ([0,T]×Rd,Rd). AmapX :[0,T]×Rd →
loc
Rd is called the regular Lagrangian flow associated to the vector field b if
(i) for a.e. x ∈ Rd, the function t → X (x) is absolutely continuous and satisfies
t
t
X (x) = x+ b (X (x))ds, for all t ∈ [0,T];
t s s
Z
0
(ii) there exists a constant L > 0 independent of t ∈ [0,T] such that (X ) Ld ≤ LLd.
t #
Recall that Ld is the Lebesgue measure on Rd and (X ) Ld is the push-forward of Ld by the
t #
flow X . L will be called the compressibility constant of the flow X.
t
Next we introduce some notations and results that will be used in the subsequent sections.
Denote by Γ = C([0,T],Rd), i.e. the space of continuous paths in Rd. For γ ∈ Γ , we write
T T
kγk for its supremum norm. Let δ > 0, we define an auxiliary function by (cf. [11, (2.7)])
∞,T
ξ ds
ψ (ξ) = , ξ ≥ 0. (2.1)
δ
Z ρ(s)+δ
0
Note that if ρ(s)= s for all s ≥ 0, then
ξ ds ξ
ψ (ξ) = = log +1
δ
Z s+δ (cid:18)δ (cid:19)
0
which is the functional used in (1.2). Here are some properties of ψ .
δ
Lemma 2.2. (1) lim ψ (ξ) =+∞ for all ξ > 0;
δ↓0 δ
(2) for any δ > 0, the function ψ is concave.
δ
Proof. Property (1) follows from the fact that ds = ∞. To prove (2), we notice that
0+ ρ(s)
ψ′(s) = 1 . Since s 7→ ρ(s) is increasing, wRe see that the derivative ψ′(s) is monotone
δ ρ(s)+δ δ
decreasing, hence ψ is concave. (cid:3)
δ
Theconcavity of ψ will play an importantrolein thearguments of Sections 3 and4. Finally
δ
we give an inequality concerning the local maximal function (see [6, Lemma A.2]).
Lemma 2.3. Let R, λ and α be positive constants. Then there is C depending only on the
d
dimension d, such that
C
Ld{x ∈ B(R): M f(x) > α} ≤ d |f(y)|dy.
λ
α Z
B(R+λ)
3
3 Existence and uniqueness of the regular Lagrangian flow
Inorder to prove the existence and uniquenessof the flow generated by avector field b satisfying
the assumption (H), we first establish an a-priori estimate.
Theorem 3.1. Let b and ˜b be time dependent bounded vector fields satisfying (H) with g and
g˜ respectively. Let X and X˜ be the regular Lagrangian flows associated to b and ˜b, with the
compressibility constants L and L˜ respectively. Then for any R > 0 and t ≤ T,
L˜
Z ψδ kX·(x)−X˜·(x)k∞,T dx ≤ (L+L˜)kgkL1([0,T]×B(R¯))+ δkb−˜bkL1([0,T]×B(R¯)),
B(R) (cid:0) (cid:1)
where ψδ isdefined in (2.1), k·k∞,T isthe supremumnorm inΓT andR¯ = R+T kbkL∞∨k˜bkL∞ .
(cid:0) (cid:1)
Proof. By Definition 2.1(1), for a.e. x ∈ Rd, the function t 7→ |X (x)− X˜ (x)| is Lipschitz
t t
continuous, hence
d d
ψ |X (x)−X˜ (x)| = ψ′ |X (x)−X˜ (x)| |X (x)−X˜ (x)|
dt δ t t δ t t dt t t
(cid:2) (cid:0) (cid:1)(cid:3) (cid:0) (cid:1)
b (X (x))−˜b (X˜ (x))
t t t t
≤ .
(cid:12)ρ |X (x)−X˜ (x)| +δ(cid:12)
(cid:12) t t (cid:12)
(cid:0) (cid:1)
Integrating from 0 to t and noticing that ψ (0) = 0, we get
δ
t b (X (x))−˜b (X˜ (x))
ψ |X (x)−X˜ (x)| ≤ s s s s ds.
δ t t Z (cid:12)ρ |X (x)−X˜ (x)| +δ(cid:12)
(cid:0) (cid:1) 0 (cid:12) s s (cid:12)
(cid:0) (cid:1)
As a result,
T b (X˜ (x))−˜b (X˜ (x))
sup ψ |X (x)−X˜ (x)| ≤ t t t t dt.
0≤t≤T δ(cid:0) t t (cid:1) Z0 (cid:12)(cid:12)ρ |Xt(x)−X˜t(x)| +δ(cid:12)(cid:12)
(cid:0) (cid:1)
Since the function ψ is continuous, we arrive at
δ
T b (X˜ (x))−˜b (X˜ (x))
ψ kX(x)−X˜ (x)k ≤ t t t t dt.
δ · · ∞,T Z (cid:12)ρ |X (x)−X˜ (x)| +δ(cid:12)
(cid:0) (cid:1) 0 (cid:12) t t (cid:12)
(cid:0) (cid:1)
Therefore
T b (X˜ (x))−˜b (X˜ (x))
ψ kX(x)−X˜ (x)k dx ≤ t t t t dxdt. (3.1)
Z δ · · ∞,T Z Z (cid:12)ρ |X (x)−X˜ (x)| +δ(cid:12)
B(R) (cid:0) (cid:1) 0 B(R) (cid:12) t t (cid:12)
(cid:0) (cid:1)
Denote by I the integral on the right hand side of (3.1). Using the triangular inequality, we
obtain
T b (X (x))−b (X˜ (x)) T b (X˜ (x))−˜b (X˜ (x))
t t t t t t t t
I ≤ dxdt+ dxdt. (3.2)
Z Z (cid:12)ρ |X (x)−X˜ (x)| +δ(cid:12) Z Z (cid:12)ρ |X (x)−X˜ (x)| +δ(cid:12)
0 B(R) (cid:12) t t (cid:12) 0 B(R) (cid:12) t t (cid:12)
(cid:0) (cid:1) (cid:0) (cid:1)
Since the flows X and X˜ leave the Lebesgue measure absolutely continuous, we can apply the
t t
condition (H) for b and obtain that for a.e. x ∈B(R),
t
b (X (x))−b (X˜ (x)) ≤ g (X (x))+g (X˜ (x)) ρ |X (x)−X˜ (x)| .
t t t t t t t t t t
(cid:12) (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)
(cid:12) (cid:12)
4
NextitisclearthatfromDefinition2.1(i), onehaskX·(x)k∞,T ≤ R+TkbkL∞ andkX˜·(x)k∞,T ≤
R+Tk˜bkL∞ for a.e. x ∈ B(R). Therefore, by the definition of the compressibility constants L
and L˜, the first term on the right hand side of (3.2) can be estimated as follows:
T b (X (x))−b (X˜ (x)) T
t t t t dxdt ≤ g (X (x))+g (X˜ (x)) dxdt
Z Z (cid:12)ρ |X (x)−X˜ (x)| +δ(cid:12) Z Z t t t t
0 B(R) (cid:12) t t (cid:12) 0 B(R)(cid:0) (cid:1)
(cid:0) (cid:1) T T
≤L g (y)dydt+L˜ g (y)dydt
t t
Z0 ZB(R¯) Z0 ZB(R¯)
=(L+L˜)kgkL1([0,T]×B(R¯)). (3.3)
Moreover, the second integral in (3.2) is dominated by
1 T L˜ T
b (X˜ (x))−˜b (X˜ (x)) dxdt ≤ |b (y)−˜b (y)|dydt
t t t t t t
δ Z0 ZB(R)(cid:12) (cid:12) δ Z0 ZB(R¯)
(cid:12) (cid:12) L˜
= δkb−˜bkL1([0,T]×B(R¯)).
Combining this with (3.2) and (3.3), we obtain
L˜
I ≤ (L+L˜)kgkL1([0,T]×B(R¯))+ δkb−˜bkL1([0,T]×B(R¯)).
Substituting I into (3.1), we complete the proof. (cid:3)
Now we can prove the main result of this section.
Theorem 3.2 (Existence and uniqueness of the regular Lagrangian flow). Assume that b :
[0,T]×Rd → Rd is a bounded vector field satisfying (H) with g ∈ L1 [0,T],L1 (Rd) . Moreover,
loc
the distributional divergence div(b) of b exists and [div(b)]− ∈ L1 (cid:0)[0,T],L∞(Rd) .(cid:1) Then there
exists a unique regular Lagrangian flow generated by b. (cid:0) (cid:1)
Proof. Step 1: Uniqueness. Suppose there are two regular Lagrangian flows X and Xˆ as-
t t
sociated to b with compressibility constants L and Lˆ respectively. Applying Theorem 3.1, we
have
ψδ kX·(x)−Xˆ·(x)k∞,T dx ≤ (L+Lˆ)kgkL1([0,T]×B(R¯)), (3.4)
Z
B(R) (cid:0) (cid:1)
where R¯ = R+TkbkL∞. If Ld{x ∈ B(R) : X·(x) 6= Xˆ·(x)} > 0, then there is ε0 > 0 such that
K := {x ∈ B(R): kX(x)−Xˆ (x)k ≥ ε } has positive measure. Thus by (3.4),
ε0 · · ∞,T 0
(L+Lˆ)kgkL1([0,T]×B(R¯)) ≥ ψδ kX·(x)−Xˆ·(x)k∞,T dx ≥ ψδ(ε0)Ld(Kε0).
Z
Kε0 (cid:0) (cid:1)
Letting δ ↓ 0, we deduce from Lemma 2.2(1) that
∞ > (L+Lˆ)kgkL1([0,T]×B(R¯)) ≥ ∞,
which is a contradiction. Hence N = {x ∈ B(R):X(x) 6= Xˆ (x)} is Ld-negligible. We conclude
· ·
that the two flows X and Xˆ coincide with each other on the interval [0,T].
t t
Step 2: Existence. Let {χ : n ≥ 1} be a sequence of standard convolution kernels. For
n
t ∈ [0,T], define bn = b ∗χ , i.e. the convolution of b and χ . Then for every n ≥ 1, bn is a
t t n t n
time dependent smooth vector field, and
kbntkL∞ ≤ kbtkL∞, [div(bnt)]− L∞ ≤ [div(bt)]− L∞, t ∈ [0,T].
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
5
Let Xn be the smooth flow generated by bn, then it is easy to know that (Xn) Ld ≤ L Ld,
t t t # n
where
T T
L = exp [div(bn)]− dt ≤ exp [div(b )]− dt =:L.
n (cid:18)Z t L∞ (cid:19) (cid:18)Z t L∞ (cid:19)
0 (cid:13) (cid:13) 0 (cid:13) (cid:13)
Now we show that bn s(cid:13)atisfies (H)(cid:13)with gn = g ∗χ for(cid:13)all n ≥ 1(cid:13). To this end, we fix any
t t t n
two points x,y ∈ Rd. We have by the definition of convolution,
|bn(x)−bn(y)| ≤ |b (x−z)−b (y−z)|χ (z)dz.
t t t t n
ZRd
Now we shall make use of the condition (H). Note that (x−N )∪(y−N ) is a negligible subset.
t t
When z ∈/ (x−N )∪(y−N ), one has x−z ∈/ N and y−z ∈/ N , hence by (H),
t t t t
|b (x−z)−b (y−z)| ≤ g (x−z)+g (y−z) ρ(|x−y|).
t t t t
(cid:0) (cid:1)
As a result,
|bn(x)−bn(y)| ≤ g (x−z)+g (y−z) ρ(|x−y|)χ (z)dz
t t t t n
ZRd
(cid:0) (cid:1)
= gn(x)+gn(y) ρ(|x−y|). (3.5)
t t
(cid:0) (cid:1)
Thus bn satisfies (H) with the function gn. Notice that (3.5) holds for all x,y ∈ Rd.
t t
From the above arguments, we can apply Theorem 3.1 to the sequence of smooth flows
{Xn :n ≥ 1} and get
t
ψ kXn(x)−Xm(x)k dx
δ · · ∞,T
Z
B(R) (cid:0) (cid:1)
L
≤ (Ln+Lm)kgnkL1([0,T]×B(R¯)) + δmkbn −bmkL1([0,T]×B(R¯))
L
≤ 2LkgkL1([0,T]×B(R¯+1))+ δkbn −bmkL1([0,T]×B(R¯)). (3.6)
Set
δ = δn,m := kbn−bmkL1([0,T]×B(R¯))
which tends to 0 as n,m → ∞, since bn converges to b in L1 [0,T],L1 (Rd) . Then we obtain
loc
(cid:0) (cid:1)
Z ψδn,m kX·n(x)−X·m(x)k∞,T dx ≤ 2LkgkL1([0,T]×B(R¯+1)) +L =:C < ∞. (3.7)
B(R) (cid:0) (cid:1)
We will show that {Xn : n ≥ 1} is a Cauchy sequence in L1(B(R),Γ ). For any η > 0, let
· T
K = {x ∈ B(R):kXn(x)−Xm(x)k ≤ η}
n,m · · ∞,T
= x ∈ B(R): ψ kXn(x)−Xm(x)k ≤ ψ (η) .
δn,m · · ∞,T δn,m
(cid:8) (cid:0) (cid:1) (cid:9)
By Chebyshev’s inequality and (3.7),
1 C
Ld(B(R)\K ) ≤ ψ kXn(x)−Xm(x)k dx ≤ .
n,m ψ (η) Z δn,m · · ∞,T ψ (η)
δn,m B(R) (cid:0) (cid:1) δn,m
Therefore
kXn(x)−Xm(x)k dx = + kXn(x)−Xm(x)k dx
Z · · ∞,T (cid:18)Z Z (cid:19) · · ∞,T
B(R) Kn,m B(R)\Kn,m
≤ ηLd(Kn,m)+2(R+TkbkL∞)Ld(B(R)\Kn,m)
C
≤ ηLd(B(R))+2R¯ .
ψ (η)
δn,m
6
Note that as n,m → ∞, δ → 0, thus by Lemma 2.2(1), ψ (η) → ∞ for any η > 0.
n,m δn,m
Consequently,
lim kXn(x)−Xm(x)k dx ≤ ηLd(B(R)).
· · ∞,T
n,m→∞Z
B(R)
By the arbitrariness of η > 0, we conclude that {Xn : n ≥ 1} is a Cauchy sequence in
·
L1(B(R),Γ ) for any R > 0. Therefore, there exists a measurable map X : Rd → Γ which
T T
is the limit in L1 (Rd,Γ ) of Xn. We can find a subsequence {n : k ≥ 1}, such that for a.e.
loc T k
x ∈ Rd, Xnk(x) converges to X (x) uniformly in t ∈ [0,T]. Hence we still have
t t
kX·(x)k∞,T ≤ R+TkbkL∞ = R¯, for a.e. x ∈ B(R). (3.8)
Now we prove that X is a regular Lagrangian flow generated by b. Firstly, for any φ ∈
t
C (Rd,R ), we have by the Fatou lemma,
c +
φ(X (x))dx ≤ lim φ(Xnk(x))dx ≤ L φ(y)dy.
t t
ZRd k→∞ZRd ZRd
This implies
(X ) Ld ≤ LLd, for all t ∈ [0,T]; (3.9)
t #
thus Definition 2.1(ii) is satisfied. Secondly, we show that for Ld-a.e. x ∈ Rd, t → X (x) is an
t
integral curve of the vector field b . To this end, we estimate the quantity
t
t t
Jn := sup bn(Xn(x))ds− b (X (x))ds dx.
Z (cid:12)Z s s Z s s (cid:12)
B(R)0≤t≤T(cid:12) 0 0 (cid:12)
(cid:12) (cid:12)
By the triangular inequality, Jn is do(cid:12)minated by the sum of (cid:12)
T
Jn := bn(Xn(x))−b (Xn(x)) dsdx
1 s s s s
Z Z
B(R) 0 (cid:12) (cid:12)
(cid:12) (cid:12)
and
T
Jn := b (Xn(x))−b (X (x)) dsdx.
2 s s s s
Z Z
B(R) 0 (cid:12) (cid:12)
(cid:12) (cid:12)
For the first term, we have
T T
Jn = bn(Xn(x))−b (Xn(x)) dxds ≤ L bn(y)−b (y) dyds.
1 s s s s s s
Z0 ZB(R)(cid:12) (cid:12) Z0 ZB(R¯)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Hence
lim Jn = 0. (3.10)
1
n→∞
For any ε > 0, we take a vector field ˆb ∈ C1([0,T]×B(R¯),Rd) such that
T
ˆb (x)−b (x) dxds < ε.
s s
Z0 ZB(R¯)(cid:12) (cid:12)
(cid:12) (cid:12)
Again by the triangular inequality,
T T
Jn ≤ b (Xn(x))−ˆb (Xn(x)) dxds+ ˆb (Xn(x))−ˆb (X (x)) dxds
2 s s s s s s s s
Z Z Z Z
0 B(R)(cid:12) (cid:12) 0 B(R)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
T
+ ˆb (X (x))−b (X (x)) dxds
s s s s
Z Z
0 B(R)(cid:12) (cid:12)
=:Jn +Jn +(cid:12) Jn . (cid:12)
2,1 2,2 2,3
7
Since (Xn) Ld ≤ LLd for all s ∈ [0,T], we have
s #
T
Jn ≤ L b (y)−ˆb (y) dyds < Lε.
2,1 s s
Z0 ZB(R¯)(cid:12) (cid:12)
(cid:12) (cid:12)
By (3.8) and (3.9), the same argument leads to
Jn < Lε.
2,3
Moreover, bythechoice ofˆb,thereisC > 0suchthatsup ∇ˆb ≤ C . Therefore
1 0≤s≤T s L∞(B(R¯)) 1
(cid:13) (cid:13)
(cid:13) (cid:13)
T
Jn ≤ C |Xn(x)−X (x)|dxds ≤ C T kXn(x)−X(x)k dx → 0
2,2 1 s s 1 · · ∞,T
Z Z Z
0 B(R) B(R)
as n goes to ∞. Summing up the above arguments, we get
lim Jn ≤ 2Lε.
2
n→∞
Sinceε> 0isarbitrary, weobtainlim Jn = 0. Combiningthiswith(3.10), wefinallyobtain
n→∞ 2
lim Jn = 0. Therefore, letting n → ∞ in the equality
n→∞
t
Xn(x) = x+ bn(Xn(x))ds for all t ≤ T,
t s s
Z
0
we see that both sides converge in L1 (Rd,Γ ) to X(x) and x+ ·b (X (x))ds respectively.
loc T · 0 s s
Hence for Ld-a.e. x ∈ Rd, it holds R
t
X (x) = x+ b (X (x))ds for all t ∈ [0,T];
t s s
Z
0
that is, t → X (x) is an integral curve of the vector field b . To sum up, X is a regular
t t t
Lagrangian flow generated by b. (cid:3)
Remark 3.3. We can relax the condition [div(b)]− ∈ L1 [0,T],L∞(Rd) to be [div(b)]− ∈
L1 [0,T],L∞(Rd) , since we have good estimate on the grow(cid:0)th of the flow X(cid:1) .
loc t
(cid:0) (cid:1)
Asabyproductoftheaboveresult,wehavethewell-posednessofthecorrespondingtransport
equation
∂
u +b ·∇u +div(b )u = 0, u| = u . (3.11)
t t t t t t=0 0
∂t
Corollary 3.4 (Well-posedness of the transport equation). Assume the conditions of Theo-
rem 3.2. Then for any integrable function u ≥ 0, the transport equation (3.11) has a unique
0
nonnegative solution in L∞([0,T],L1(Rd)).
Proof. The proof of the existence part is quite standard, see for instance the case p = 1 in [7,
Proposition II.1] (it is easy to see that the nonnegative property of solutions is preserved in the
limit process).
For the uniqueness of solutions, noticing that the equation (3.11) is equivalent to the conti-
nuity equation
∂
u +D ·(b u ) = 0, u| = u ,
t x t t t=0 0
∂t
8
where D · is the generalized divergence operator. Next since u ∈ L∞([0,T],L1(Rd,R )), one
x +
has
T |b (x)|
t
Z0 ZRd 1+|x|ut(x)dxdt ≤ kbkL∞([0,T]×Rd)kukL∞([0,T],L1(Rd)) < ∞.
By [2, Theorem 3.2], u is a superposition solution, that is, there exists a measure η ∈M (Rd×
+
Γ ) concentrated on the set of pairs (x,γ) such that γ is an absolutely continuous integral curve
T
of b with γ(0) = x, and
t
ϕu dx = ϕ(γ(t))dη(x,γ), for all ϕ ∈ C (Rd).
t b
ZRd ZRd×ΓT
Recall that Γ = C([0,T],Rd). Let {η : x ∈ Rd} be the disintegration of η with respect to the
T x
measure µ (dx) := u (x)dx. Then for µ -a.e. x ∈ Rd, η concentrates on the integral curve of
0 0 0 x
b starting from x. By the uniqueness of the regular Lagrangian flow X proved in Theorem 3.2,
t t
we have η {X(x)} = 1 for µ -a.e. x ∈ Rd. Therefore
x · 0
ϕu dx = ϕ(γ(t))dη (γ) u (x)dx
t x 0
ZRd ZRd(cid:18)ZΓT (cid:19)
= ϕ(X (x))u (x)dx.
t 0
ZRd
This gives the uniqueness of nonnegative integrable solutions to (3.11). (cid:3)
Remark 3.5. If one wants to prove the uniqueness of solutions for any initial condition u ∈
0
L1(Rd), then one has to extend the superposition principle proved in [2, Theorem 3.2] to the case
of solutions of the continuity equation consisting of signed measures with finite total variation.
Underthecondition(H),itseemstotheauthorsthatoneisunabletoprovethewellposedness
of the transport equation (3.11) by following DiPerna-Lions’s original approach, that is, by
proving an estimate on the commutator
r (b ,u )= (b ·∇u )∗χ −b ·∇(u ∗χ ),
n t t t t n t t n
where χ is the standard convolution kernel. This can be seen from the proof of [7, Lemma II.1]
n
(or[2, Proposition4.1]), whichessentially relies onthe“almost everywhereLipschitzcontinuity”
of Sobolev vector fields.
4 Regularity of the flow
Inthis section, wefirstprove aregularity resulton theregular Lagrangian flow, apropertymuch
weaker than the approximate differentiability discussed in [6]. We need the following notation:
for a bounded measurable subset U with positive measure, define the average of f ∈ L1 (Rd)
loc
on U by
1
− fdx = f dx.
Z Ld(U) Z
U U
Then the local maximal function
M f(x)= sup − |f(y)|dy.
R
Z
0<r≤R B(x,r)
Now we can prove
9
Theorem4.1. Letbbeabounded vectorfieldsatisfying (H), and [div(b)]− ∈ L1 [0,T],L∞(Rd) .
Let X be the unique regular Lagrangian flow associated to b. Then for any R > 0(cid:0) and sufficient(cid:1)ly
small ε, there are a measurable subset E ⊂ B(R) and some constant C depending on R,d and
g, such that Ld(B(R)\E)≤ ε and for all x,y ∈ E, one has
|X (x)−X (y)| ≤ ψ−1 (C/ε).
t t |x−y|
Hereψ−1 istheinversefunctionofψ . NotethatbyLemma2.2(1), wehavelim ψ−1(ξ) = 0
r r r↓0 r
for all ξ > 0. Therefore this theorem implies that X is uniformly continuous in E, since when
t
y → x in the subset E, the quantity ψ−1 (C/ε) decreases to 0. Unfortunately, the function
|x−y|
ψ−1 does not have an explicit expression, unless ρ(s) =s for all s ≥ 0 (see Remark 4.2).
r
Proof of Theorem 4.1. We follow the ideas of [6, Remark 2.4] (see also [14, Proposition 5.2]).
For 0≤ t ≤ T, 0< r ≤ 2R and x ∈B(R), define
Q(t,x,r) = − ψ (|X (x)−X (y)|)dy.
r t t
Z
B(x,r)
Then
Q(0,x,r) = − ψ (|x−y|)dy ≤ − ψ (r)dy ≤ 1.
r r
Z Z
B(x,r) B(x,r)
By Definition 2.1(i), we see that t → Q(t,x,r) is Lipschitz and
d d
Q(t,x,r) = − ψ′(|X (x)−X (y)|) |X (x)−X (y)|dy
dt Z r t t dt t t
B(x,r)
b (X (x))−b (X (y))
t t t t
≤ − dy.
Z (cid:12)ρ(|X (x)−X (y)|)+r(cid:12)
B(x,r) (cid:12) t t (cid:12)
Using the assumption (H) on b, we have
d
Q(t,x,r) ≤ − g (X (x))+g (X (y)) dy = g (X (x))+− g (X (y))dy.
t t t t t t t t
dt Z Z
B(x,r)(cid:0) (cid:1) B(x,r)
Integrating both sides with respect to time from 0 to t, we arrive at
t t
Q(t,x,r) ≤ Q(0,x,r)+ g (X (x))ds+ − g (X (y))dyds
s s s s
Z Z Z
0 0 B(x,r)
T T
≤ 1+ g (X (x))ds+ − g (X (y))dyds. (4.1)
s s s s
Z Z Z
0 0 B(x,r)
Denote by Φ(x)= T g (X (x))ds for a.e. x ∈ Rd. Then for all t ≤ T,
0 s s
R
Q(t,x,r) ≤ 1+Φ(x)+− Φ(y)dy.
Z
B(x,r)
Therefore
sup sup Q(t,x,r) ≤ 1+Φ(x)+M Φ(x). (4.2)
2R
0≤t≤T0<r≤2R
For η > 0 sufficiently small, we have
I := Ld{x ∈ B(R):1+Φ(x)+M Φ(x)> 1/η}
2R
≤ Ld{x ∈ B(R):Φ(x) > 1/(3η)}+Ld{x ∈B(R) :M Φ(x) > 1/(3η)}.
2R
10