4 The Cauchy problem
In Corollary 3.1, we established an almost sure hydrodynamic limit for initial measures correspond- ing to the Riemann problem with R-valued initial densities. In this section we prove that this implies
Theorem 2.1, that is the almost sure hydrodynamic limit for any initial sequence associated with any measurable initial density profile thus we add in this Section the hypothesis p. finite range, which
was not necessary for Riemann a.s. hydrodynamics. This passage is inspired by Glimm’s scheme, a well-known procedure in the theory of hyperbolic conservation laws, by which one constructs gen-
eral entropy solutions using only Riemann solutions see e.g. [39, Chapter 5]. In [8, Section 5], we undertook such a derivation for convergence in probability. In the present context of almost sure
convergence, new error analysis is necessary. In particular, we have to do an explicit time discretiza- tion vs. the “instantaneous limit” of [7, Section 3, Theorem 3.2] or [8, Section 5] for the analogue
of 105 below, we need estimates uniform in time Lemma 4.2, and each approximation step requires a control with exponential bounds Proposition 4.2 and Lemma 4.3.
4.1 Preliminary results
For two measures α, β ∈ M
+
R with compact support, we define ∆α, β := sup
x∈R
α−∞, x] − β−∞, x] 95
When α or β is of the form u.d x for u. ∈ L
∞
R with compact support, we simply write u in 95 instead of u.d x. A connection between ∆ and vague convergence is given by the following
technical lemma, whose proof is left to the reader.
Lemma 4.1. i Let α
N N
be a M
+
R-valued sequence supported on a common compact subset of R
, and u. ∈ L
∞
R. The following statements are equivalent: a α
N
→ u.d x as N → ∞; b ∆α
N
, u. → 0 as N → ∞. ii Let
α
N
.
N
be a sequence of M
+
R-valued functions α
N
: T → M
+
R, where T is an arbitrary set, such that the measures
α
N
t are supported on a common compact subset of R. Assume that, for some
α : [0, +∞ → M
+
R, ∆α
N
t, αt converges to 0 uniformly on T . Then α
N
. converges to
α. uniformly on T . The following proposition is a collection of results on entropy solutions. We first recall two defini-
tions. A sequence u
n
, n ∈ N of Borel measurable functions on R is said to converge to u in L
1 loc
R if and only if
lim
n→∞
Z
I
u
n
x − ux d x = 0
for every bounded interval I ⊂ R. The variation of a function u. on an interval I ⊂ R is defined by TV
I
[u.] = sup
n−1
X
i=0
ux
i+1
− ux
i
: n ∈ N, x , . . . , x
n
∈ I, x · · · x
n
We shall simply write TV for TV
R
. We say that u = u., . defined on R × R
+∗
has locally bounded space variation if for every bounded space interval I ⊂ R and bounded time interval J ⊂ R
+∗
sup
t∈J
TV
I
[u., t] +∞
26
Proposition 4.1.
o Let u., . be the entropy solution to 10 with Cauchy datum u ∈ L
∞
R. Then the map-
ping t 7→ u., t lies in C [0, +∞, L
1 loc
R. i If u
. is a.e. R-valued, then so is the corresponding entropy solution u., t to 10 at later times.
ii If u
i
. has finite variation, that is TVu
i
. +∞, then so does u
i
., t for every t 0, and TVu
i
., t ≤ TVu
i
.. iii Finite propagation property: Assume u
i
., . i ∈ {1, 2} is the entropy solution to 10 with Cauchy data u
i
.. Let V = ||G
′
||
∞
:= sup
ρ
G
′
ρ . Then: a for every x y and 0 ≤ t y − x2V ,
Z
y−V t x+V t
u
1
z, t − u
2
z, t dz ≤
Z
y x
u
1
z − u
2
z dz
96 In particular, if u
1
is supported resp. coincides with u
2
in [−R, R] for some R 0, u
1
., t is supported resp. coincides with u
2
., t in [−R − V t, R + V t]. b If R
R
u
i
zdz +∞, ∆u
1
., t, u
2
., t ≤ ∆u
1
., u
2
. 97
iv Let x = −∞ x
1
· · · x
n
x
n+1
= +∞ and ǫ := min
0≤k≤n
x
k+1
− x
k
. Denote by u . the
piecewise constant profile with value r
k
on I
k
:= x
k
, x
k+1
. Then, for t ǫ2V , the entropy solution u., t to 10 with Cauchy datum u
. is given by ux, t = R
r
k−1
,r
k
x − x
k
, t, ∀x ∈ x
k−1
+ V t, x
k+1
− V t Properties o, ii and iii are standard, see [28; 39; 29]. Properties i and iv are respectively
[8, Lemma 5.3] and [7, Lemma 3.4]. The latter states that the entropy solution starting from a piecewise constant profile can be constructed at small times as a superposition of successive
non-interacting Riemann waves. This is a consequence of iii. Note that the whole space is indeed covered by the definition of ux, t in iv, since we have x
k+1
− V t ≥ x
k
+ V t for t ≤ ǫ2V . The next lemma improves [8, Lemma 5.5] by deriving an approximation uniform in time.
Lemma 4.2. Assume u . is a.e. R-valued, has bounded support and finite variation, and let x, t 7→
ux, t be the entropy solution to 10 with Cauchy datum u .. For every ǫ 0, let P
ǫ
be the set of piecewise constant R-valued functions on R with compact support and step lengths at least
ǫ, and set δ
ǫ
t := ǫ
−1
inf{∆u., u., t : u. ∈ P
ǫ
} Then there is a sequence
ǫ
n
↓ 0 as n → ∞ such that δ
ǫ
n
converges to 0 uniformly on any bounded subset of R
+
. Proof of Lemma 4.2. We first argue that, for every
ǫ 0, δ
ǫ
is a continuous function. Indeed, by 27
Proposition 4.1, iii,a for every T 0, there exists a bounded set K
T
⊂ R such that the support of u., t is contained in K
T
for every t ∈ [0, T ]. Since ∆v, w ≤
Z
R
|vx − wx| d x 98
for v, w ∈ L
∞
R with compact support, it follows by Proposition 4.1, o and Lemma 4.1, i that lim
s→t
∆u., s, u., t = 0 99
for every t ≥ 0. This and the inequality δ
ǫ
t − δ
ǫ
s ≤ ǫ
−1
∆u., s, u., t imply continuity of
δ
ǫ
. By Proposition 4.1, i and ii, u., t has bounded, finite space variation, and is R-valued. Hence, by [8, Lemma 5.5], for any given
δ 0, for ǫ 0 small enough, there exists an approximation u
ǫ,δ
., t ∈ P
ǫ
of u., t with ∆
u
ǫ,δ
., t, u., t
≤ ǫδ. This implies δ
ǫ
t → 0 as ǫ → 0 for every t 0. Let T be some countable dense subset of [0, +∞. By the diagonal extraction
procedure, we can find a sequence ǫ
n
↓ 0 such that δ
ǫ
n
t ↓ 0 for each t ∈ T . By continuity of δ
ǫ
we also have that
δ
ǫ
n
t ↓ 0 for every t ∈ [0, +∞. Dini’s theorem implies that δ
ǫ
n
converges uniformly to 0 on every bounded subset of [0, +∞.
We now quote [10, Proposition 3.1], which yields that ∆ is an “almost” nonincreasing func-
tional for two coupled particle systems:
Proposition 4.2. Assume p. is finite range. Then there exist constants C 0 and c 0, depending
only on b., . and p., such that the following holds. For every N ∈ N, η
, ξ
∈ X
2
with η
+ ξ
:= P
x∈Z
[η x + ξ
x] +∞, and every γ 0, the event ∀t 0 : ∆α
N
η
t
η ,
ω, α
N
η
t
ξ ,
ω ≤ ∆α
N
η , α
N
ξ + γ
100 has IP-probability at least 1 − C
η +
ξ e
−cN γ
. We finally recall the finite propagation property at particle level see [8, Lemma 5.2], which is a
microscopic analogue of Proposition 4.1, iii.
Lemma 4.3. There exist constants v and C, depending only on b., . and p., such that the following holds. For any x, y ∈ Z, any
η ,
ξ ∈ X
2
, and any 0 t y − x2v: if η
and ξ
coincide on the site interval [x, y], then with IP-probability at least 1 − e
−C t
, η
s
η ,
ω and η
s
ξ ,
ω coincide on the site interval [x + v t, y − v t] ∩ Z for every s ∈ [0, t].
Remark 4.1. The time uniformity in Proposition 4.2 and Lemma 4.3 does not appear in the original statements repectively, [10, Proposition 3.1] and [8, Lemma 5.2], but follows in each case from the
proof.
4.2 Proof of Theorem 2.1
