Advances in Mathematical
Sciences and Applications
Vol. 25, No. 1 (2016), pp. 181-205

GAKKOTOSHOTOKYO JAPANMILD SOLUTIONS ARE WEAK SOLUTIONS IN A CLASSOF (NON)LINEAR MEASURE-VALUED EVOLUTIONEQUATIONS ON A BOUNDED DOMAIN
Joep H.M. Evers
Department of Mathematics, Simon Fraser University, Burnaby, Canada, andDepartment of Mathematics and Statistics, Dalhousie University, Halifax, Canada
8888 University Drive, Burnaby BC V5A 1S6, Canada(E-mail: jevers@sfu.ca)

Abstract. We study the connection between mild and weak solutions for a class ofmeasure-valued evolution equations on the bounded domain [0, 1]. Mass moves, driven
by a velocity field that is either a function of the spatial variable only, v = v(x), or
depends on the solution µ itself: v = v[µ](x). The flow is stopped at the boundaries of[0, 1], while mass is gated away by a certain right-hand side. In previous works [16, 18],
we showed the existence and uniqueness of appropriately defined mild solutions forv = v(x) and v = v[µ](x), respectively. In the current paper we define weak solutions(by specifying the weak formulation and the space of test functions). The main result
is that the aforementioned mild solutions are weak solutions, both when v = v(x) andwhen v = v[µ](x).

————————————————————
Communicated by A. Muntean; Received June 2, 2016.
AMS Subject Classification: 28A33, 34A12, 45D05, 35F16.
Keyworks: Measure-valued equations, nonlinearities, time discretization, flux boundary condition, mild
solutions, weak solutions, particle systems.

181

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1

Introduction

Measure-valued evolution equations have been used in a large number of recent mathematical publications to model for instance animal aggregations [6, 9], structured
populations [1, 7, 13, 21], pedestrian dynamics [12], and defects in metallic crystals[28]. The majority of works that study well-posedness of measure-valued equations andproperties of their solutions treat these equations in the full space –see for instance also
[2, 8, 10, 11, 27]– although many relevant problems involve boundaries and boundeddomains. Examples of such problems –apart from the ones mentioned above– areintracellular transport processes, cf. [17, Section 1], and manufacturing chains [20].
Defining mathematically and physically ‘correct’ boundary conditions is a challenge,
however. The present paper is a continuation of the author’s work (in collaboration with
Hille and Muntean) that focuses explicitly on bounded domains and boundary conditions.
Our first step is to consider a one-dimensional measure-valued transport equation
restricted to the unit interval [0, 1] where mass moves according to a prescribed velocity
field v and stops when reaching the boundary. A short-hand notation for this equation
is:
∂
∂
µt +
(v µt ) = Ff (µt ).
(1.1)
∂t
∂x
Here, the perturbation map Ff : M([0, 1]) → M([0, 1]) is given by Ff (µ) := f · µ.
In [16], we proved the well-posedness of this equation, in the sense of mild solutions, andthe convergence of solutions corresponding to a sequence (fn )n∈N in the right-hand side.
Some specific choices for (fn )n∈N represent for instance effects in a boundary layer thatapproximate, as n → ∞, sink or source effects localized on the boundary (flux boundary
conditions). The boundary layer corresponds to exactly those regions in [0, 1] where thefunctions fn are nonzero.
Next, we want to consider (1.1) for velocity fields that are no longer fixed elements of BL([0, 1]). Instead of v, we write v[µ] for the velocity field that depends
functionally on the measure µ. An example isZ
K(x − y) dµ(y) = (K ∗ µ)(x),v[µ](x) :=

(1.2)

[0,1]

where the convolution encodes nonlocal interactions due to a kernel K in a population
with distribution µ. This is a widely used choice of v, e.g. in interacting particle systems
or biological aggregation models. The example (1.2) is a special case of the class ofvelocity fields (see Assumption 4.1) that are admissible in the framework of the current
paper.
For such solution-dependent v = v[µ], the transport equation on [0, 1] becomes∂∂µt +(v[µt ] µt ) = Ff (µt ).
∂t
∂x

(1.3)

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The well-posedness of (1.3) for f ∈ BL([0, 1]) was proved in [18] in the sense of mild
solutions. The analysis turns out to build on the analysis for (1.1), hence it was useful toconsider (1.1) before the more general (1.3). For v = v[µ], mild solutions are defined as
the limit of a sequence (µk )k∈N of so-called Euler approximations. Such µk is constructedon each subinterval (tkj , tkj+1 ] as a mild solution to (1.1) for velocity v[µktk ]. Within ajsubinterval, v[µktk ] is a fixed element of BL([0, 1]) that is the same for all time t ∈ (tkj , tkj+1 ].
j

For further details on how µk is defined, see Section 4.
The works [5] and [21] treat comparable models, however posed on infinite domains. They consider weak solutions, and in fact, they construct those weak solutions
that –roughly speaking– correspond to ‘our’ mild solutions.
The aim of the current paper is:
to investigate how and in which sense the mild solutions from [16, 18]
correspond to weak solutions like the ones in e.g. [5, 21].
Note that the essence of weak solutions lies in the specific choice of the weak formulationand of the space of test functions that appear in the definition; cf. e.g. Definition 3.3
with weak formulation (3.2) and space of test functions (3.3).
Compared to e.g. [5, 21], our case is more complicated due to the bounded domain; the material flow is induced by the velocity v in the interior, but it is stopped once
characteristics reach any of the boundary points. The stopped flow introduces subtleties
when trying to find the appropriate definition of weak solutions. The domain in [21] is infact [0, ∞). Their velocity is required to point inward at x = 0, though, which is sufficient
to make sure that no mass escapes the domain. For us, a demand on the sign of the velocity at x = 0 or x = 1 is too restrictive; cf. the remark we make about this in [18, Section 1].
In the current work, we overcome these difficulties and give the appropriate definition of weak solutions. We believe this indeed is the appropriate definition because of
the following main result of this paper, that consists of two parts. Formulated in plain
words in a pseudo-theorem, the first part of this result reads:
Theorem. Mild solutions to (1.1) are weak solutions (in an appropriate sense).
A more precise formulation follows in Theorem 3.5. Next, we use this property oneach of the subintervals in an Euler approximation and show that in the limit as themesh size goes to zero, we obtain a weak solution to (1.3). In other words:
Theorem. Mild solutions to (1.3) are weak solutions.
This result is stated in full detail in Theorem 4.8.
Our justification for speaking about appropriate definition, is exactly the fact that

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we show the relation between these weak solutions and mild solutions in this paper (moreabout this in Section 4, directly after Definition 4.4). Mild solutions have a considerable
advantage over weak solutions in the sense that it is directly clear how they should beinterpreted, whereas defining weak solutions involves some seemingly arbitrary choices.
Which choices to make is not directly evident from modelling considerations.
On the other hand, as was argued in [18, Section 1], the mild formulation in terms of thevariation of constants formula (3.1) follows directly from a probabilistic interpretation.
For more details, see [16, Section 6]. Moreover, the exact form of the variation ofcontsants formula is unambiguous, provided the system that is to be modelled.
Subsequently, mild solutions for v = v[µ] follow in a straight-forward manner, using thevariation of constants formula as a building block; cf. (4.2) and Definition 4.5.
The structure of this paper is as follows. Section 2 provides preliminaries on the
stopped flow on the interval [0, 1] induced by the velocity field v : [0, 1] → R. In Section
3, we recall the results from [16] regarding the existence and uniqueness of mild solutions
to (1.1). We introduce the concept of weak solutions, and show in Theorem 3.5 thatthe mild solutions from [16] are weak solutions. Section 4 briefly recalls the main ideas
from [18]: the construction of Euler approximations using solutions to the variationof constants formula as building blocks. Theorem 4.5 repeats the result that Euler
approximations converge as the mesh size goes to zero; this result is an alternative way ofsaying that mild solutions to (1.3) exist and are unique. We show in Section 4 that thesemild solutions are weak solutions (in an appropriate sense). The paper is concluded bya section (Section 5) in which we discuss the wider context of our results and the open
issues that are subject for follow-up work.

2

Preliminaries

This section contains the preliminaries that are needed for the arguments in this paper.
These preliminaries were presented before in [16, 18]. We assume that the reader isfamiliar with elementary measure-theoretical concepts, such as finite Borel measures, thetotal variation norm k · kTV , and the dual bounded Lipschitz norm k · k∗BL . An overview
of the basic concepts used in this paper can be found in Appendix A.
The rest of this section is devoted to properties of the flow induced on [0, 1] bysome fixed v ∈ BL([0, 1]), a bounded Lipschitz velocity field. This flow is a fundamental
mechanism in the model considered in this paper.
We assume that a single particle (‘individual’) is moving in the domain [0, 1] deterministically, described by the differential equation for its position x(t) at timet:
ẋ(t) = v(x(t)),(2.1)
x(0) = x0 .
A solution to (2.1) is unique, it exists for time up to reaching the boundary 0 or 1 anddepends continuously on initial conditions. Let x( · ; x0 ) be this solution and Ix0 be its

185
maximal interval of existence. Defineτ∂ (x0 ) := sup Ix0 ∈ [0, ∞],i.e. τ∂ (x0 ) is the time at which the solution starting at x0 reaches the boundary (if it
happens) when x0 is an interior point. Note that τ∂ (x0 ) = 0 when x0 is a boundary point
where v points outwards, while τ∂ (x0 ) > 0 when x0 is a boundary point where v vanishes
or points inwards.
The stopped flow on [0, 1] associated to v is the family of maps Φt : [0, 1] → [0, 1],
t > 0, defined by(x(t; x0 ),if t ∈ Ix0 ,Φt (x0 ) :=(2.2)
x(τ∂ (x0 ); x0 ),otherwise.
To lift the dynamics to the space of measures, we define Pt : M([0, 1]) → M([0, 1]) bymeans of the push-forward under Φt : for all µ ∈ M([0, 1]),
Pt µ := Φt #µ = µ ◦ Φ−1
t ;

(2.3)

see (A.2). Clearly, Pt maps positive measures to positive measures and Pt is mass preserving on positive measures. Since the family of maps (Φt )t>0 forms a semigroup, so do
the maps Pt in the space M([0, 1]). That is, (Pt )t>0 is a Markov semigroup on M[0, 1](cf. [24]). The basic estimate
kPt µkTV 6 kµkTV(2.4)
holds for µ ∈ M([0, 1]). In [18, Section 2.2] a number of other properties (bounds andLipschitz-like estimates) of (Pt )t>0 are given.

3

Mild and weak solutions for prescribed velocity

Mild solutions to (1.1) are defined in the following sense:
Definition 3.1 (See [16, Definition 2.4]). A measure-valued mild solution to the Cauchyproblem associated to (1.1) on [0, T ] with initial value ν ∈ M([0, 1]) is a continuous mapµ : [0, T ] → M([0, 1])BL that is k·kTV -bounded and that satisfies the variation of constants
formulaZtµ t = Pt ν +Pt−s Ff (µs ) ds
for all t ∈ [0, T ].
(3.1)
0

Amongst others, we showed in [16] that mild solutions in the sense of Definition 3.1 exist
and are unique. We repeat those results in the following theorem.
Theorem 3.2 (Existence and uniqueness of mild solutions to (1.1)). Let f : [0, 1] → R bea piecewise bounded Lipschitz function such that v(x) 6= 0 at any point x of discontinuityof f . Then for each T > 0 and µ0 ∈ M([0, 1]) there exists a unique continuous and locallyk · kTV -bounded solution to (3.1).

186
Proof. See [16, Propositions 3.1 and 3.3] for details.
In the rest of this section we will compare mild solutions provided by Theorem 3.2 tosolutions in a different sense: weak solutions. Recall that h·, ·i denotes the duality paring
between finite Borel measures on [0, 1] and bounded measurable functions on [0, 1]; see(A.1).
Definition 3.3 (Weak solution to (1.1)). Fix T > 0, let v ∈ BL([0, 1]) and let f : [0, 1] →R be piecewise bounded Lipschitz. Then µ : [0, T ] → M([0, 1]) is a weak solution to (1.1)
corresponding to initial condition ν0 ∈ M([0, 1]), if
ZThµT , ψ(·, T )i − hν0 , ψ(·, 0)i =ZThµt , ∂t ψ(·, t) + ∂x ψ(·, t) · vi dt +0

hFf (µt ), ψ(·, t)i dt
0(3.2)
is satisfied for all1ψ ∈ Λ := ψ ∈ C ([0, 1] × [0, T ]) : ∂x ψ(0, t) = ∂x ψ(1, t) = 0 for all 0 6 t 6 T . (3.3)
T

Remark 3.4. Note that the boundary conditions imposed on the test functions are closely
related to the behaviour of the stopped flow at the boundary points x = 0 and x = 1:
no flux. This is a general phenomenon. See for instance [26, pp. 63–64 and 140], where
several spaces of test functions are given, depending on which behaviour at the boundary
is to be modelled. For the sake of completeness, we note that [26] treats Brownian motion(diffusion). The used operator A is the corresponding infinitesimal generator acting –asis common in probabilistic literature– on the test functions, not on the solution itself.
The main result of this section is the following theorem.
Theorem 3.5 (Mild solutions to (1.1) are weak solutions). Let µ : [0, T ] → M([0, 1]) be
the mild solution provided by Theorem 3.2, corresponding to initial value ν0 ∈ M([0, 1]).
Then, µ is a weak solution of (1.1).
For the proof of Theorem 3.5 we were inspired by the proof of [22, Proposition 3.7].

Proof. Let ψ be an arbitrary element from the set of test functions given in (3.3). Recall
that Iy is the maximal interval of existence of a solution to (2.1) –i.e. restricted to [0, 1]–with initial condition y. Recall moreover that τ∂ (y) = sup Iy , i.e. τ∂ (y) is the time atwhich the solution starting at y reaches the boundary (if it happens) when y is an interior
point. Note that τ∂ (y) = 0 when y is a boundary point where v points outwards, whileτ∂ (y) > 0 when y is a boundary point where v vanishes or points inwards.

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ConsiderZT
hPt ν0 , ∂x ψ(·, t) · v + ∂t ψ(·, t)i dt0ZThν0 , ∂x ψ(Φt (·), t) · v(Φt (·)) + ∂t ψ(Φt (·), t)i dt=0
Z ZT =∂x ψ(Φt (y), t) · v(Φt (y)) + ∂t ψ(Φt (y), t) dt dν0 (y)[0,1] 0

Z

τ∂Z
(y)∧T

[0,1]

0

=

Z

d
ψ(Φt (y), t) dt dν0 (y)
dt

ZT




∂x ψ(Φt (y), t) · v(Φt (y)) + ∂t ψ(Φt (y), t) dt dν0 (y),

+

(3.4)

[0,1] τ∂ (y)∧Twhere the truncation is defined as τ∂ (y)∧T := min(τ∂ (y), T ); this is a continuous function
in y. Interchanging the order of integration is allowed by Fubini’s theorem, because
the integrand is bounded. The subdivision of the domain [0, T ] with respect to τ∂ isnecessary, since the semigroup Φt represents the stopped flow and therefore the identitydΦ (y) = v(Φt (y)) is only valid if t ∈ Iy . Hence, only in the first integral on the right-handdt tside of (3.4), the chain ruledψ(Φt (y), t) = ∂x ψ(Φt (y), t) · v(Φt (y)) + ∂t ψ(Φt (y), t)
dt

(3.5)

can be used. Note that at time τ∂ (y) ∧ T this identity at least holds one-sidedly ast % (τ∂ (y)∧T ), which is sufficient for the first integral on the right-hand side to be correct.
Define, for z ∈ {0, 1}, the setsΩTz := {y ∈ [0, 1] : τ∂ (y) < T and Φτ∂ (y) (y) = z}.

(3.6)

These are connected subsets of [0, 1].
T
T
T
If y ∈ Ω := [0, 1] \ Ω0 ∪ Ω1 , then τ∂ (y) > T , and obviouslyZT∂x ψ(Φt (y), t) · v(Φt (y)) + ∂t ψ(Φt (y), t) dt = 0(3.7)

τ∂ (y)∧Tin the second integral on the right-hand side of (3.4), since the domain of integration is anullset (in fact, a single point).
If y ∈ ΩT0 ∪ ΩT1 , then τ∂ (y) < T and Φt (y) ∈ {0, 1} for all (τ∂ (y) ∧ T ) 6 t 6 T . Therefore,

188
taking into account the test functions’ boundary conditions ∂x ψ(0, t) = ∂x ψ(1, t) = 0, we
have that∂x ψ(Φt (y), t) = 0(3.8)
for all y ∈ ΩT0 ∪ ΩT1 and (τ∂ (y) ∧ T ) 6 t 6 T .
Due to (3.7) and (3.8), the second term on the right-hand side of (3.4) can be
written as
Z

ZT



∂x ψ(Φt (y), t) · v(Φt (y)) + ∂t ψ(Φt (y), t) dt dν0 (y)[0,1] τ∂ (y)∧T

Z ZT
=

Z ZT
∂t ψ(0, t) dt dν0 (y) +

τ (y)
ΩT
0 ∂

∂t ψ(1, t) dt dν0 (y)
τ (y)
ΩT
1 ∂


Z 

ψ( |{z}
0 , T ) − ψ 0, τ∂ (y) dν0 (y)=ΩT0=ΦT (y)Z ψ( |{z}
1 , T ) − ψ 1, τ∂ (y) dν0 (y).
+
ΩT
1

(3.9)

=ΦT (y)The first term on the right-hand side of (3.4) we treat as follows:
Z

τ∂Z
(y)∧T

[0,1]

0

d
ψ(Φt (y), t) dt dν0 (y)
dt
Z ZT
=

d
ψ(Φt (y), t) dt dν0 (y)
dt

ΩT 0

Z τZ∂ (y)
+
ΩT

0

d
ψ(Φt (y), t) dt dν0 (y) +dt

Z τZ∂ (y)
ΩT

1
0

Z 

=
ψ ΦT (y), T − ψ(y, 0) dν0 (y)dψ(Φt (y), t) dt dν0 (y)
dt

0

ΩT



Z 
Z 
+
ψ(0, τ∂ (y)) − ψ(y, 0) dν0 (y) +ψ(1, τ∂ (y)) − ψ(y, 0) dν0 (y).
ΩT
0

ΩT
1

(3.10)
Note that, for all y ∈ [0, 1], the function t 7→ ψ(Φt (y), t) is differentiable for all t > 0. If
t = τ∂ (y), then the differentiability follows from the boundary conditions on ψ.

189
Combining (3.4) with (3.9) and (3.10), we obtain that
ZT


Z 

hPt ν0 , ∂x ψ(·, t) · v + ∂t ψ(·, t)i dt =ψ ΦT (y), T − ψ(y, 0) dν0 (y)

0

[0,1]

= hPT ν0 , ψ(·, T )i − hν0 , ψ(·, 0)i .

(3.11)

Next, consider
ZT *Z t+Pt−s Ff (µs ) ds, ∂x ψ(·, t) · v + ∂t ψ(·, t)

0

dt

0

ZT Z thFf (µs ), ∂x ψ(Φt−s (·), t) · v(Φt−s (·)) + ∂t ψ(Φt−s (·), t)i ds dt

=
0

0

 +
ZT *
ZT 
Ff (µs ),∂x ψ(Φt−s (·), t) · v(Φt−s (·)) + ∂t ψ(Φt−s (·), t) dt ds (3.12)=
0s

We subdivide the domain of the spatial integration into ΩT0 −s , ΩT1 −s and ΩT −s := [0, 1] \(ΩT0 −s ∪ ΩT1 −s ), with the sets ΩTz −s defined analogous to (3.6).
For each z ∈ {0, 1}, we have
ZT Z ZT ∂x ψ(Φt−s (y), t) · v(Φt−s (y)) + ∂t ψ(Φt−s (y), t) dt dFf (µs )(y) ds

−s s
0 ΩT
z

ZT Z

τ∂Z
(y)+s

=
−s
0 ΩT
z

ZT Z

d
ψ(Φt−s (y), t) dt dFf (µs )(y) ds
dt

s

ZT

+
−s τ (y)+s
0 ΩT
∂
z




∂ ψ(z, t) ·v(z) + ∂t ψ(z, t)| {z }| x {z }=0

dt dFf (µs )(y) dsd= dt
ψ(z,t)ZT Z 
=ψ z, τ∂ (y) + s − ψ(y, s) dFf (µs )(y) ds
−s
0 ΩT
z


ZT Z 
ψ(z, T ) − ψ z, τ∂ (y) + s dFf (µs )(y) ds
+
−s
0 ΩT
z


ZT Z =ψ ΦT −s (y), T − ψ(y, s) dFf (µs )(y) ds.
−s
0 ΩT
z

(3.13)

190
Considering the spatial domain of integration ΩT −s , we find
ZT Z ZT ∂x ψ(Φt−s (y), t) · v(Φt−s (y)) + ∂t ψ(Φt−s (y), t) dt dFf (µs )(y) ds
0 ΩT −s s

ZT Z ZT=dψ(Φt−s (y), t) dt dFf (µs )(y) ds
dt

0 ΩT −s s
ZT Z ψ ΦT −s (y), T − ψ(y, s) dFf (µs )(y) ds.

=(3.14)

0 ΩT −s

Together, (3.12), (3.13) and (3.14) yield
ZT *Z tPt−s Ff (µs ) ds, ∂x ψ(·, t) · v + ∂t ψ(·, t)
0

ZT

+

Ff (µs ), ψ ΦT −s (·), T

dt =

0



ds

0

ZT
hFf (µs ), ψ(·, s)i ds.

−(3.15)

0

It follows from (3.11) and (3.15), and from the variation of constants formula (3.1) thatZT
hµt , ∂x ψ(·, t) · v + ∂t ψ(·, t)i dt = hPT ν0 , ψ(·, T )i − hν0 , ψ(·, 0)i0

ZTFf (µs ), ψ ΦT −s (·), T

+



ZT
ds −

0

hFf (µs ), ψ(·, s)i ds. (3.16)0Note that
ZT
Ff (µs ), ψ ΦT −s (·), TZThPT −s Ff (µs ), ψ(·, T )i ds

ds =

0

0

*ZT
=

+
PT −s Ff (µs ) ds, ψ(·, T ) ,0and hence
ZT
hPT ν0 , ψ(·, T )i +Ff (µs ), ψ ΦT −s (·), T

ds = hµT , ψ(·, T )i .

0

Equation (3.16) can thus be written as
ZT

ZT
hµt , ∂x ψ(·, t) · v + ∂t ψ(·, t)i dt = hµT , ψ(·, T )i − hν0 , ψ(·, 0)i −0hFf (µs ), ψ(·, s)i ds,0191
which shows that µ is a weak solution of (1.1).

4

Mild and weak solutions for measure-dependent
velocityIn this section we summarize the results of [18] and compare the concept of mild solutions
from [18] to weak solutions of (1.3). In [18], we generalized the assumptions on v from[16] in the following way to measure-dependent velocity fields:
Assumption 4.1 (Assumptions on the measure-dependent velocity field). Assume thatv : M([0, 1]) × [0, 1] → R is a mapping such that:(i) v[µ] ∈ BL([0, 1]), for each µ ∈ M([0, 1]).
Furthermore, assume that for any R > 0 there are constants KR , LR , MR such that forall µ, ν ∈ M([0, 1]) satisfying kµkTV 6 R and kνkTV 6 R, the following estimates hold:(ii) kv[µ]k∞ 6 KR ,(iii) | v[µ] |L 6 LR ,and(iv) kv[µ] − v[ν]k∞ 6 MR kµ − νk∗BL .
In [18], we proved well-posedness of (1.3):∂∂µt +(v[µt ] µt ) = Ff (µt )∂t∂x
on [0, 1], in the sense of mild solutions; cf. Definition 4.4 and Theorem 4.5. Like in [18],in this paper we restrict ourselves to f that is bounded Lipschitz on [0, 1]. See Section 5
for further discussion on this assumption.
Mild solutions for measure-dependent v = v[µ] are defined using mild solutions for
fixed v ∈ BL([0, 1]) as a building block via an Euler-like approach. We first summarize
the required notation used in [18].
Let v ∈ BL([0, 1]) and f ∈ BL([0, 1]) be arbitrary. For all t > 0, we define
Qt : M([0, 1]) → M([0, 1]) to be the operator that maps the initial condition to thesolution in the sense of Definition 3.1. Theorem 3.2 guarantees that this operator iswell-defined and continuous for k · k∗BL . Moreover, Q preserves positivity, due to [16,Corollary 3.4]. The set of operators (Qt )t>0 constitutes a semigroup and has useful otherproperties, like certain Lipschitz estimates. The exact results and their proofs can befound in [18, Section 2.3].
0
In the sequel we will write e.g. Qv and Qv to distinguish between the semigroups Q on
M([0, 1]) associated to v ∈ BL([0, 1]) and v 0 ∈ BL([0, 1]), respectively.

192
We now introduce the aforementioned forward-Euler-like approach to construct approximate solutions. Let T > 0 be given. Let N > 1 be fixed and define a set α ⊂ [0, T ]as follows:
α := tj ∈ [0, T ] : 0 6 j 6 N, t0 = 0, tN = T, tj < tj+1 ,(4.1)
which we call a partition of the interval [0, T ]. Here, N denotes the number of subintervals
in α.
Let µ0 ∈ M([0, 1]) be fixed. For a given partition α := {t0 , . . . , tN } ⊂ [0, T ],define a measure-valued trajectory µ ∈ C([0, T ]; M([0, 1])) byvj
µ := Qt−tµtj , if t ∈ (tj , tj+1 ];

j
 t
(4.2)
vj := v[µtj ]; µ =µ ,t=00

for all j ∈ {0, . . . , N − 1}. Here, (Qvt )t>0 denotes the semigroup introduced above. Notethat by Assumption 4.1, Part (i), vj = v[µtj ] ∈ BL([0, 1]) for each j.
We call this a forward-Euler-like approach, because it is the analogon of the forward Euler method for ODEs (cf. e.g. [4, Chapter 2]). See [18, Section 3] for further
explanation.
The conditions in Parts (ii)–(iv) of Assumption 4.1 are only required to hold formeasures in a TV-norm bounded set, in view of the following lemma:
Lemma 4.2. Let µ0 ∈ M([0, 1]) be given and let v : M([0, 1])×[0, 1] → R satisfy Assumption 4.1(i). For a given partition α := {t0 , . . . , tN } ⊂ [0, T ], let µ ∈ C([0, T ]; M([0, 1]))be defined by (4.2). Then for all t ∈ [0, T ]kµt k∗BL 6 kµt kTV 6 kµ0 kTV exp(kf k∞ T ).
This bound is in particular independent of t, N and the distribution of points within α.
Proof. See the proof of [18, Lemma 3.4] for details.
We construct sequences of Euler approximations, each following from a sequence of partitions (αk )k∈N that satisfies the following assumption:
Assumption 4.3 (Assumptions on the sequence of partitions). Let (αk )k∈N be a sequenceof partitions of [0, T ] and let (Nk )k∈N ⊂ N be the corresponding sequence such that eachαk is of the form
αk := tkj ∈ [0, T ] : 0 6 j 6 Nk , tk0 = 0, tkNk = T, tkj < tkj+1 .
(4.3)
DefineM (k) :=

max

j∈{0,...,Nk −1}

tkj+1 − tkj(4.4)

for all k ∈ N. Assume that the sequence (M (k) )k∈N is nonincreasing and M (k) → 0 as
k → ∞.

193
A mild solution is defined as follows:
Definition 4.4 (See [18, Definition 3.8]). Let the space of continuous maps from [0, T ]to M([0, 1]) be endowed with the metric defined for all µ, ν ∈ C([0, T ]; M([0, 1])) by
sup kµt − νt k∗BL .

(4.5)

t∈[0,T ]Let (αk )k∈N be a sequence of partitions satisfying Assumption 4.3. For each k ∈ N,
let µk ∈ C([0, T ]; M([0, 1])) be defined by (4.2) with partition αk . Then, for any suchsequence of partitions (αk )k∈N , any limit of a subsequence of (µk )k∈N is called a (measurevalued) mild solution of (1.3).
The name mild solutions is appropriate, first of all because they are constructed frompiecewise mild solutions in the sense of Definition 3.1.
Moreover, as their name suggests, weak solutions in general constitute a weaker solution
concept than mild solutions. That is, a mild solution (meaning: a solution of the variationof constants formula) is in general necessarily a weak solution. See e.g. [15, p. 4–5],[22, Proposition 3.7], Theorem 3.5 in this paper, and the way in which mild and weaksolutions are connected on [25, pp. 258–259]. In all of these references, the equations
treated are simpler than (1.3) that is considered in this section. Here, mild solutions are
the ones constructed in [18], being defined as the limit of Euler approximations. They
are not solutions of the variation of constants formula. However, they are still elementsof the set of weak solutions, as we will show in Theorem 4.8. This implication is an extra
justification for the name mild solutions.
In the rest of this paper we focus on positive measure-valued solutions, because
these are the only physically relevant solutions in many applications.
Theorem 4.5 (Existence and uniqueness of mild solutions to (1.3)). Let µ0 ∈ M+ ([0, 1])
be given and let v : M([0, 1]) × [0, 1] → R satisfy Assumption 4.1. Endow the spaceC([0, T ]; M([0, 1])) with the metric defined by (4.5). Then, there is a unique elementof C([0, T ]; M+ ([0, 1])) with initial condition µ0 , that is a mild solution in the sense of
Definition 4.4. That is, for each sequence of partitions (αk )k∈N satisfying Assumption 4.3,
the corresponding sequence (µk )k∈N defined by (4.2) is a sequence in C([0, T ]; M+ ([0, 1]))and has a unique limit as k → ∞.
Moreover, this limit is independent of the choice of (αk )k∈N .
Proof. See [18, Theorem 3.10] for details.
Definition 4.6 (Weak solution to (1.3)). Fix T > 0, let f ∈ BL([0, 1]) and let v :M([0, 1]) × [0, 1] → R satisfy Assumption 4.1. Then µ : [0, T ] → M([0, 1]) is a weaksolution to (1.3) corresponding to initial condition ν0 ∈ M([0, 1]), if
ZThµT , ψ(·, T )i − hν0 , ψ(·, 0)i =ZThµt , ∂t ψ(·, t) + ∂x ψ(·, t) · v[µt ]i dt +0

hFf (µt ), ψ(·, t)i dt
0

(4.6)
T

T

is satisfied for all ψ ∈ Λ , with Λ as defined in (3.3).

194
To show that mild solutions in the sense of Definition 4.4 are weak solutions in the senseof Definition 4.6, the following result is useful.
Lemma 4.7 (Cf. portmanteau theorem). Let (ν k )k∈N ⊂ M([0, 1]) and assume there isan R > 0 such that ν k ([0, 1]) 6 R for all k ∈ N. Let ν ∈ M([0, 1]). The following are
equivalent:(a) ν k , φ → hν, φi as k → ∞ for all φ ∈ BL([0, 1]);(b) ν k , φ → hν, φi as k → ∞ for all φ ∈ Cb ([0, 1]).
Proof. Apply [23, Theorem 13.16] to the sequence (ν k /R)k∈N ⊂ M61 ([0, 1]), and use theequivalence “(ii)⇔(iii)” therein, which implies the equivalence with (b) above.
The main result of this paper is the following theorem.
Theorem 4.8 (Mild solutions to (1.3) are weak solutions). Let µ : [0, T ] → M+ ([0, 1]) be
the mild solution provided by Theorem 4.5, corresponding to initial value ν0 ∈ M+ ([0, 1]).
Then, µ is a weak solution of (1.3).
The proof of Theorem 4.8 uses ideas from the proof of [22, Proposition 4.9].
Proof. Let (αk )k∈N be a sequence of partitions satisfying Assumption 4.3, and for eachk ∈ N, let µk ∈ C([0, T ]; M([0, 1])) be the corresponding sequence of Euler approximations defined by (4.2). Since µ is the unique mild solution, µ = limk→∞ µk holds withconvergence in the metric given by (4.5).
Fix a ψ ∈ ΛT . For each j ∈ {0, . . . , Nk − 1} we apply Theorem 3.5 to subinterval[tkj , tkj+1 ] in the Euler approximation. Note that the restriction ψ|[tkj ,tkj+1 ] is an appropriate
test function on the corresponding domain [0, 1] × [tkj , tkj+1 ]. The individual test functions
on these subdomains are therefore all derived from the same ψ on [0, 1]×[0, T ]. Moreover,
the required regularity per subdomain and the spatial boundary conditions are simply
inherited from ψ.
For each j ∈ {0, . . . , Nk − 1} we thus have
k

D

E

D

E

µktk , ψ(·, tkj+1 ) − µktk , ψ(·, tkj ) =
j+1

tj+1
Z
D

E
µkt , ∂t ψ(·, t) + ∂x ψ(·, t) · v[µktk ] dt
j

j

tkj
k

tj+1
Z

+
tkj

Ff (µkt ), ψ(·, t) dt,

(4.7)

195
while µktk = µk0 = ν0 . Note that within the integrals we do not need to write ψ|[tkj ,tkj+1 ] , but0
we can simply use ψ. Summation over j ∈ {0, . . . , Nk − 1} yieldsµkT , ψ(·, T )− hν0 , ψ(·, 0)i =tkj+1
Nk −1 ZDX
j=0Eµkt , ∂t ψ(·, t) + ∂x ψ(·, t) · v[µktk ]
j

dt

tkj

ZT
+

Ff (µkt ), ψ(·, t) dt

0ZTµkt , ∂t ψ(·, t) + ∂x ψ(·, t) · v̄ k (t, ·) dt

=
0

ZT
+

Ff (µkt ), ψ(·, t) dt.

(4.8)

0

Here, v̄ k : [0, T ] → BL([0, 1]) is defined by v̄ k (t, ·) := v[µktk ] whenever t ∈ (tkj , tkj+1 ], whilejv̄ k (0, ·) = v[ν0 ].
Note that, for each t ∈ [0, T ] fixed, ψ(·, t) ∈ BL([0, 1]) due to the assumed regularity on the test functions. Since µk → µ with respect to the metric in (4.5) as k → ∞,we have in particular thatk→∞µkT , ψ(·, T ) −→ hµT , ψ(·, T )i .

(4.9)

The second term on the right-hand side of (4.8) we treat as follows:
ZT

Ff (µkt ), ψ(·, t)

0

ZT

ZT
dt −

µkt − µt , ψ(·, t) · f (·)

hFf (µt ), ψ(·, t)i dt 6
0

dt

0

ZT
6

kψ(·, t)kBL · kf kBL · kµkt − µt k∗BL dt,

0(4.10)
where we used that for each t ∈ [0, T ] the product ψ(·, t) · f (·) is bounded Lipschitz andkψ(·, t) · f (·)kBL 6 kψ(·, t)kBL · kf kBL ; cf. (A.4).
Since ψ ∈ C 1 ([0, 1] × [0, T ]), we know thatsup|ψ| < ∞,

[0,1]×[0,T ]

and

sup
[0,1]×[0,T ]and hence,
sup kψ(·, t)kBL < ∞.
t∈[0,T ]|∂x ψ| < ∞,

196Consequently, we haveZT

Ff (µkt ), ψ(·, t)

0

ZT
dt −

hFf (µt ), ψ(·, t)i dt
0

6 sup kψ(·, t)kBL · kf kBL · sup kµkt − µt k∗BL · T,t∈[0,T ]t∈[0,T ]|{z}(4.11)→0 as k→∞and thus

ZTk→∞

Ff (µkt ), ψ(·, t) dt −→ZT
hFf (µt ), ψ(·, t)i dt.

(4.12)

0

0

Finally, we consider the first term on the right-hand side of (4.8).
For each t ∈ [0, T ], it holds that kµkt − µt k∗BL → 0 as k → ∞ since the convergence µk → µ
is in the metric (4.5). Then in particular, µkt , φ → hµt , φi for all φ ∈ BL([0, 1]). Sinceµkt ([0, 1]) = kµkt kTV , Lemma 4.2 provides a bound on µkt ([0, 1]) that is independent of kand t. We can therefore apply Lemma 4.7, and conclude that
µkt , φ → hµt , φi ,as k → ∞,(4.13)

for all φ ∈ Cb ([0, 1]).
For each t ∈ [0, T ], we have that ∂t ψ(·, t), ∂x ψ(·, t) ∈ Cb ([0, 1]) by the assumption thatψ ∈ Cb1 ([0, 1] × [0, T ]). Moreover, v[µt ] ∈ BL([0, 1]) and thus ∂x ψ(·, t) · v[µt ] ∈ Cb ([0, 1]).
Hence, (4.13) implies that
µkt , ∂t ψ(·, t) → hµt , ∂t ψ(·, t)i , and(4.14)µkt , ∂x ψ(·, t) · v[µt ] → hµt , ∂x ψ(·, t) · v[µt ]i(4.15)

is in Cb ([0, 1]) andas k → ∞.
For each t ∈ [0, T ], the function ∂x ψ(·, t) · v̄ k (t, ·) − v[µt ]thereforeµkt , ∂x ψ(·, t) · v̄ k (t, ·) − µkt , ∂x ψ(·, t) · v[µt ]Z=∂x ψ(x, t) · v̄ k (t, x) − v[µt ](x) dµkt (x)[0,1]

6 sup k∂x ψ(·, τ )k∞ · kv̄ k (t, ·) − v[µt ]k∞ · kµkt kTV .

(4.16)

τ ∈[0,T ]Due to Lemma 4.2, it holds for each t ∈ [0, T ] that kµkt kTV 6 R := kν0 kTV exp(kf k∞ T ).
The measure µt is positive for all t ∈ [0, T ], and thus
kµt kTV = kµt k∗BL 6 kµkt k∗BL + kµkt − µt k∗BL ,| {z } |
{z
}
6R

→0

(4.17)

197
whence kµt kTV 6 R. We can now use Assumption 4.1(iv) to estimate the term kv̄ k (t, ·) −v[µt ]k∞ .
Without loss of generality, assume that t > 0. Let j ∈ {0, . . . , Nk } be such that tkj < t 6
tkj+1 . Thenkv̄ k (t, ·) − v[µt ]k∞ = kv[µktk ] − v[µt ]k∞ 6 kv[µktk ] − v[µtkj ]k∞ + kv[µtkj ] − v[µt ]k∞jj

6 MR kµktk − µtkj k∗BL + MR kµtkj − µt k∗BLj

6 MR sup kµkτ − µτ k∗BL + MR kµtkj − µt k∗BL .
τ ∈[0,T ](4.18)
The first term on the right-hand side goes to zero, because of the uniform convergence ofµk to µ. The second term on the right-hand side goes to zero because t 7→ µt is continuousand t − tkj 6 M (k) → 0 as k → ∞. Consequently, the left-hand side of (4.18) must vanish
as k → ∞.
It follows from (4.16) and the fact that the left-hand side of (4.18) goes to zero, that
µkt , ∂x ψ(·, t) · v̄ k (t, ·) − µkt , ∂x ψ(·, t) · v[µt ]→ 0, as k → ∞.

(4.19)

This results yields, together with (4.15), that for all t ∈ [0, T ]µkt , ∂x ψ(·, t) · v̄ k (t, ·) → hµt , ∂x ψ(·, t) · v[µt ]i , as k → ∞,(4.20)

due to the triangle inequality.
Combining (4.14) and (4.20), we obtain that for all t ∈ [0, T ]µkt , ∂t ψ(·, t) + ∂x ψ(·, t) · v̄ k (t, ·) → hµt , ∂t ψ(·, t) + ∂x ψ(·, t) · v[µt ]i , as k → ∞. (4.21)
Since for each t ∈ [0, T ] and k ∈ N, the function ∂t ψ(·, t) + ∂x φ(·, t) · v̄ k (t, ·) is an elementof Cb ([0, 1]), it holds that!
µkt , ∂t ψ(·, t) + ∂x φ(·, t) · v̄ k (t, ·)6

sup k∂t ψ(·, τ )k∞ + sup k∂x ψ(·, τ )k∞ · KR · R,τ ∈[0,T ]τ ∈[0,T ]with the same R = kν0 kTV exp(kf k∞ T ) as before.
Assumption 4.1(ii).

(4.22)
Here we used Lemma 4.2 andBy the dominated convergence theorem, cf. e.g. [3, Theorem 4.2], we now obtainfrom (4.21) and (4.22) in particular that
ZT
0µkt , ∂t ψ(·, t) + ∂x ψ(·, t) · v̄ k (t, ·)

ZT
dt →

hµt , ∂t ψ(·, t) + ∂x ψ(·, t) · v[µt ]i dt, as k → ∞.
0

(4.23)

198
By taking the limit k → ∞ in (4.8), while taking (4.9), (4.12) and (4.23) into account,
we obtain
ZT
hµT , ψ(·, T )i − hν0 , ψ(·, 0)i =ZThµt , ∂t ψ(·, t) + ∂x ψ(·, t) · v[µt ]i dt +0

hFf (µt ), ψ(·, t)i dt.
0

Since ψ ∈ ΛT is chosen arbitrarily, this proves the statement of the theorem.

5

Context of the results and open issues

In the Discussion section of [18], we explained that in fact we would like to consider mild solutions (with measure-dependent velocity) corresponding to a sequence(fn )n∈N ⊂ BL([0, 1]), such that fn → f pointwise, and f is piecewise bounded Lipschitz.
Let (µfn )n∈N denote the corresponding sequence of measure-valued mild solutions, withmeasure-dependent velocity v = v[µfn ].
In [16] we specifically focussed on such sequence (fn )n∈N that describes a vanishingboundary layer in which mass is gated away from the domain. Assume there are regions
around 0 and 1 in which mass decays, and that these regions shrink to zero width. Thatis, fn is nonzero only in a region around x = 0 and x = 1, respectively. Moreover, thisregion shrinks to zero as n → ∞ and fn → f , where f satisfies f (x) = 0 if x ∈ (0, 1) andf (0) = f (1) = −1.
We want to know whether the sequence (µfn )n∈N converges, and whether the limit
coincides with the mild solution corresponding to f (if this mild solution exists). Mild
solutions were obtained in [18] as the limit of Euler approximations. Let the approximating sequence corresponding to fn be indexed by k, and let µfn ,k be one such Euler
approximation. The question is now whether the limits k → ∞ and n → ∞ commute.
The following scheme shows the four limit processes involved:(A)µfn ,k
(C)

n→∞

−→

↓ k→∞
µfn

µf,k
k→∞ ↓

n→∞

−→

(B)

µf

(D)

Note that each of the limits should be understood as convergence in the metric given by(4.5).
At this moment we are not yet able to prove that this scheme represents reality,
but the results of this paper yield additional insight. Regarding the limit processes (A),(B), (C) and (D) in the scheme above, the following can be said:

199(A) For fixed k ∈ N, there is an obvious candidate for limn→∞ µfn ,k , namely µf,k , theEuler approximation corresponding to f . Note that, for fixed k, µf,k is well-definedby (4.2) and Theorem 3.2 (that is, [16, Propositions 3.1 and 3.3]). However, it is∗nontrivial to actually show that supt∈[0,T ] kµft n ,k − µf,kt kBL → 0 as n → ∞.
The most straight-forward way to prove this, would be to look at the interval (tkj , tkj+1 ],∗k kestimate kµfτn ,k − µf,kτ kBL for arbitrary τ ∈ (tj , tj+1 ], take the supremum over τ andfinally the maximum over j. We will now point out what the problem is with thisstrategy.
Like in (4.2), let the semigroup Q denote the operator that maps initial data to thesolution in the sense of Definition 3.1. From now on, we use Qu,g to denote thesemigroup associated to velocity u ∈ BL([0, 1]) and right-hand side Fg , where g ispiecewise bounded Lipschitz. For any τ ∈ (tkj , tkj+1 ], we havenµfn ,k ,µfτn ,k = Qu,f
τ −tk tk
j

j

ū,f
f,k
µf,k
τ = Qτ −tk µtk ,jj∗]. To estimate kµfτn ,k −µf,kwith u = v[µftkn ,k ] and ū = v[µf,kτ kBL from above, one wouldtkj
juse the triangle inequality and obtain three terms of the formν̄k∗BL ,ν − Qw̄,ḡ
kQw,g
τ −tk
τ −tk
j

(5.1)

j

for the appropriate choices of]},w, w̄ ∈ {v[µftkn ,k ], v[µf,ktkjjg, ḡ ∈ {fn , f },}.
ν, ν̄ ∈ {µftkn ,k , µf,ktk
jjSpecifically, we would use the triangle inequality in such a way that in each term on
the right-hand side two of these three pairs of variables are the same (e.g. w = w̄,g = ḡ and ν 6= ν̄). An estimate for each term of the form (5.1) follows e.g. from
[16, Proposition 3.5], [16, Proposition 4.2], [18, Corollary 2.9] or [18, Lemma 2.10].
There is freedom in how exactly we apply the triangle inequality, and thus in which
specific terms of the form (5.1) appear on the right-hand side. However, each of theseapproaches results in an upper bound that depends on the Lipschitz constant of fn(mostly via the bounded Lipschitz norm of fn ), which is unbounded as n → ∞ if f
is discontinuous.
For the moment, it is therefore still an open question whether µfn ,k converges to µf,k
as n → ∞.
(B) We emphasize that the results of Theorems 3.5 and 4.8 in this paper, do not hinge on
the assumption that f ∈ BL([0, 1]). Theorem 3.5 is stated explicitly to hold for any fthat is piecewise bounded Lipschitz. Theorem 4.8 is only restricted to f ∈ BL([0, 1]),
because it builds on Theorem 4.5, which we managed to prove only for continuous f
in [18, Theorem 3.10]. Let us assume that Theorem 4.5 does provide the convergenceof Euler approximations even for f that is piecewise bounded Lipschitz. Note that
Theorem 3.2 demands that the discontinuities of f and zeroes of v do not coincide;
for the sake of the argument here, ignore this complication and assume that we can

200
generalize Theorem 4.5 to piecewise bounded Lipschitz functions f . In that case thestatement of Theorem 4.8 still holds: the resulting mild solution is a weak solution.
The current proof of Theorem 4.8 requires a slight modification, though, since in(4.10) we used that f ∈ BL([0, 1]) to obtain immediately an estimate against kf kBL .
The convergence in (4.12), can however be obtained for f piecewise bounded Lipschitz, using arguments very much like the ones leading to (4.23). These arguments
involve the portmanteau theorem (in a slightly more general form than Lemma 4.7)
and the dominated convergence theorem. See Appendix B for more details.
(C) Since n is fixed in this step and since fn ∈ BL([0, 1]) for each n, this convergence
result is covered by [18, Theorem 3.10]. In the current work, we show that the limitµfn is a weak solution; see Theorem 4.8.
(D) We assumed above that µf can be obtained as the limit of the Euler approximationsµf,k , and we stress here that this is only an assumption. When trying to relate themild solutions µfn to the mild solution µf by letting n tend to infinity, one encountersthe following problem: a mild solution is defined as the limit of a sequence of Euler
approximations, but this does not provide a useful characterization of the limit itself.
We suggested in [18] to use the weak formulation of the problem as an alternative
characterization. In the current paper we show that µfn is a weak solution. A next
step would be to show that this solution converges in some sense (e.g. weakly) as
n → ∞.
Even if the addressed problems in steps (A)–(D) would be resolved, an additional
argument is needed to conclude that the two limits k → ∞ and n → ∞ commute – andthus the scheme above is fully correct.
Under the aforementioned assumption that µf,k → µf as k → ∞, the mild solution
obtained via the route (A)-(B) is a weak solution, as argued above. On the other hand,
our considerations regarding route (C)-(D) lead to an alternative µf , obtained as the(weak?) limit of the weak solutions µfn . It might be possible to show (easily) that thislimn→∞ µfn is also a weak solution.
Finally, assume that we want to compare limk→∞ µf,k , resulting from (A)-(B), tolimn→∞ µfn , resulting from (C)-(D), based on the fact that they are both weak solutions.
We did not prove in this paper that weak solutions are unique. To be able to identify
limk→∞ µf,k with limn→∞ µfn , therefore an extra uniqueness criterion or selection criterion
might be needed. The issue of uniqueness is nontrivial and it probably plays a role which
specific weak formulation is used and which space of test functions is chosen. This topic
is ‘work in progress’ and will (hopefully) be the subject of a follow-up paper.

Acknowledgements
I thank Sander C. Hille (Leiden University, The Netherlands) for fruitful discussions andhis valuable suggestions, and I thank the anonymous reviewer, whose comments helped
to improve the manuscript.

201
Until 2015, I was a member of the Centre for Analysis, Scientific computing and Applications (CASA), and the Institute for Complex Molecular Systems (ICMS) at Eindhoven
University of Technology, The Netherlands, supported financially by the Netherlands Organisation for Scientific Research (NWO), Graduate Programme 2010. Some of the resultspresented here, were obtained during my time in Eindhoven.

A

Basics of measure theory

We denote by M([0, 1]) the space of finite Borel measures on the interval [0, 1] and by
M+ ([0, 1]) the convex cone of positive measures included in it. For x ∈ [0, 1], δx denotes
the Dirac measure at x. LetZhµ, φi :=
φ dµ
(A.1)
[0,1]

denote the natural pairing between measures µ ∈ M([0, 1]) and bounded measurable
functions φ. The push-forward or image measure of µ under Borel measurable Φ : [0, 1] →[0, 1] is the measure Φ#µ defined on Borel sets E ⊂ [0, 1] by(Φ#µ)(E) := µ Φ−1 (E) .

(A.2)

One easily verifies that hΦ#µ, φi = hµ, φ ◦ Φi.
The total variation norm k · kTV on M([0, 1]) is defined bykµkTV := sup {hµ, φi : φ ∈ Cb ([0, 1]), kφk∞ 6 1} ,where Cb ([0, 1]) is the Banach space of real-valued bounded continuous functions on [0, 1]
equipped with the supremum norm k·k∞ . It follows immediately that for Φ : [0, 1] → [0, 1]
continuous, kΦ#µkTV 6 kµkTV .
In [16, 18] and in this paper, we mainly use a different norm on M([0, 1]) that wewill introduce now. Let BL([0, 1]) be the vector space of real-valued bounded Lipschitz
functions on [0, 1], equipped with the norm
kφkBL := kφk∞ + |φ|L

(A.3)

for which this space is a Banach space [19, 14]. Here,|φ|L := sup|φ(x) − φ(y)|: x, y ∈ [0, 1], x 6= y|x − y|

is the Lipschitz constant of an arbitrary φ ∈ BL([0, 1]). With this norm BL([0, 1]) is a
Banach algebra for pointwise product of functions:kφ · ψkBL ≤ kφkBL kψkBL .

(A.4)

202
Let k · k∗BL be the dual norm of k · kBL on the dual space BL([0, 1])∗ , i.e. for any x∗ ∈BL([0, 1])∗ its norm is given bykx∗ k∗BL := sup {| hx∗ , φi | : φ ∈ BL([0, 1]), kφkBL 6 1} .
A linear embedding of M([0, 1]) into BL([0, 1])∗ is provided by the map µ 7→ Iµ withIµ (φ) := hµ, φi; see [14, Lemma 6]. Thus k · k∗BL induces a norm on M([0, 1]), which isdenoted by the same symbols. It is called the dual bounded Lipschitz norm or Dudleynorm. Generally, kµk∗BL 6 kµkTV for all µ ∈ M([0, 1]). For positive measures the two
norms coincide:kµk∗BL = µ([0, 1]) = kµkTV

for all µ ∈ M+ ([0, 1]).

(A.5)

In general, the space M([0, 1]) is not complete for k · k∗BL . We denote by M([0, 1])BL its
completion, viewed as closure of M([0, 1]) within BL([0, 1])∗ . The space M+ ([0, 1]) iscomplete for k · k∗BL , hence closed in M([0, 1]) and M([0, 1])BL .
The k · k∗BL -norm is convenient also for integration. In [16, Appendix C] sometechnical results about integration of measure-valued maps were collected.

B

Proof of convergence statement (4.12) for discontinuous f

The proof is based on the dominated convergence theorem and makes use of theportmanteau theorem.
For each ν ∈ M([0, 1]),
kFf (ν)kTV 6 kf k∞ · kνkTV(B.1)

holds, even for f piecewise bounded Lipschitz, hence not necessarily in Cb ([0, 1]). Thisinequality was previously used in the proof of [16, Proposition 3.1]; a proof can be foundin [15, Lemma 4.3.1]. The proof makes use of an approximation of f by Cb -functions.
Note that each mild solution (νt )t∈[0,T ] corresponding to a fixed v ∈ BL([0, 1]) and with fpiecewise bounded Lipschitz satisfieskνt kTV 6 kν0 kTV exp(kf k∞ t),(B.2)

according to [16, Proposition 3.3]; the proof of that claim was left to the reader there.
The argument is based on Gronwall’s Lemma applied to the estimateZtkνt kTV 6 kν0 kTV +kf k∞ · kνs kTV ds,

(B.3)

0

while the latter inequality follows from the variation of constants formula and theestimate kPτ µkTV 6 kµkTV that holds for any µ ∈ M([0, 1]), due to (2.4). Moreover,203(B.1) is used to obtain (B.3).
The arguments in [18, Lemmas 3.4 and 2.8(i)] build on (B.2), and thus the resultof Lemma 4.2 in this paper is still valid if f is piecewise bounded Lipschitz. That is, theuniform estimate(B.4)
kµkt kTV 6 kν0 kTV exp(kf k∞ T )holds for all k and t.
For each t ∈ [0, T ] fixed, note that ψ(·, t) is in Cb ([0, 1]) and thus,Ff (µkt ), ψ(·, t)

6 sup kψ(·, τ )k∞ · kFf (µkt )kTV 6 sup kψ(·, τ )k∞ · kf k∞ · kµkt kTV ,τ ∈[0,T ]τ ∈[0,T ](B.5)
for each k ∈ N, where the last step is due to (B.2). Combining (B.5) with (B.4), we
obtain that the function t 7→ Ff (µkt ), ψ(·, t) is bounded uniformly in k and for eacht ∈ [0, T ] by the (constant) function t 7→ supτ ∈[0,T ] kψ(·, τ )k∞ · kf k∞ · kν0 kTV exp(kf k∞ T ).
Next we prove that t 7→ Ff (µkt ), ψ(·, t) converges pointwise. We emphasize thatin the rest of this proof we work here under the (unproved) hypothesis that the Euler
approximations µk converge to some unique mild solution µ even if f is piecewisebounded Lipschitz.
For each t ∈ [0, T ], the function x 7→ f (x) · ψ(x, t) is bounded and measurable, because f is piecewise bounded Lipschitz, and ψ(·, t) ∈ Cb1 ([0, 1]). Trivially,
Ff (µkt ), ψ(·, t) = µkt , f · ψ(·, t) holds.
By hypothesis µk → µ in the metric (4.5), hence we know in particular that
µkt , φ → hµt , φi as k → ∞ for all φ ∈ BL([0, 1]). Instead of the simple version of theportmanteau theorem presented here in Lemma 4.7, we use [23, Theorem 13.16], whichstates that convergence against BL([0, 1]) is equivalent to convergence against boundedmeasurable functions that are discontinuous only on a nullset. One such function is
x 7→ f (x) · ψ(x, t), since f is assumed to have finitely many discontinuities. Hence,µkt , f · ψ(·, t) → hµt , f · ψ(·, t)i as k → ∞ for all t ∈ [0, T ], or equivalently
Ff (µkt ), ψ(·, t) → hFf (µt ), ψ(·, t)i , as k → ∞.

(B.6)

Due to the uniform bound on t 7→ Ff (µkt ), ψ(·, t) and the pointwise convergence (B.6),
the dominated convergence theorem yields in particular that (4.12) holds even for f thatis piecewise bounded Lipschitz:
ZT
0

k→∞

Ff (µkt ), ψ(·, t) dt −→ZT
hFf (µt ), ψ(·, t)i dt.

(B.7)

0

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