CONSERVATION LAWS WITH NONLOCAL VELOCITY THE SINGULAR LIMIT PROBLEM JAN FRIEDRICH SIMONE G OTTLICHy ALEXANDER KEIMERzAND LUKAS PFLUGx

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CONSERVATION LAWS WITH NONLOCAL VELOCITY – THE

SINGULAR LIMIT PROBLEM

JAN FRIEDRICH∗, SIMONE G ¨

OTTLICH†, ALEXANDER KEIMER‡,AND LUKAS PFLUG§

Key words. Nonlocal conservation law, nonlocal in velocity, convergence, weak entropy solution,

monotonicity preserving, singular limit, singular limit for nonlocal in velocity conservation laws

MSC codes. 35L65,35L99,34A36

Abstract. We consider conservation laws with nonlocal velocity and show for nonlocal weights

of exponential type that the unique solutions converge in a weak or strong sense (dependent on the

regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal

weight approaches a Dirac distribution. To this end, we establish ﬁrst a uniform total variation

estimate on the nonlocal velocity which enables it to prove that the nonlocal solution is entropy

admissible in the limit. For the entropy solution, we use a tailored entropy ﬂux pair which allows

the usage of only one entropy to obtain uniqueness (given some additional constraints). For general

weights, we show that monotonicity of the initial datum is preserved over time which enables it to

prove the convergence to the local entropy solution for rather general kernels and monotone initial

datum as well. This covers the archetypes of local conservation laws: Shock waves and rarefactions. It

also underlines that a “nonlocal in the velocity” approximation might be better suited to approximate

local conservation laws than a nonlocal in the solution approximation where such monotonicity does

only hold for speciﬁc velocities.

1. Introduction. In recent years, the mathematical analysis on nonlocal conser-

vation laws [1,3,24,13,23,36,30] but also its applicability in traﬃc ﬂow modelling

[7,31,28,5,43,45,27,14,12,15,29,11,47], supply chains [33,22,4,44,35],

sedimentation processes [6], pedestrian dynamics [20], particle growth [52,54], crowd

dynamics and population modelling [21,50,49] and opinion formation [53,41] has

drawn increased attention. The theory and in particular the convergence theory when

the nonlocal weight approaches a Dirac and one formally obtains a local conservation

law has been partially understood and several results on this convergence exist to date

[37,18,19,9,10,16,39].

However, what has not been studied for its convergence properties is the quite

related equation where the averaging is not done over the solution but the velocity,

reading as (for the precise deﬁnition of the convolution, see (2.3))

nonlocal in solution local nonlocal in velocity

∂tq+∂xV(γ∗q)q= 0 ∂tq+∂xV(q)q= 0 ∂tq+∂xγ∗V(q)q= 0.(1.1)

This is why we will tackle the problem in this contribution and prove under speciﬁc

conditions the convergence to the local entropy solution. We refer the reader in par-

ticular to the main Theorem 4.7 of this contribution. The convergence is numerically

illustrated in Figure 1 for the exponential kernel and an archetypal initial datum. As

∗RWTH Aachen University, Institute of Applied Mathematics, 52064 Aachen, Germany

(friedrich@igpm.rwth-aachen.de).

†University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany

(goettlich@uni-mannheim.de).

‡UC Berkeley, Institute of Transportation Studies (ITS), Sutardja Dai Hall, Berkeley, US;

Friedrich-Alexander Universit¨at Erlangen-N¨urnberg, Department Mathematik, Cauerstr. 11, 91058

Erlangen, Germany (alexander.keimer@fau.de)

§Friedrich-Alexander Universit¨at Erlangen-N¨urnberg, Competence Unit for Scientiﬁc Computing,

Martensstr. 5a, 91058 Erlangen, Germany (lukas.pﬂug@fau.de)

arXiv:2210.12141v1 [math.AP] 21 Oct 2022

2J. FRIEDRICH, S. G ¨

OTTLICH, A. KEIMER, L. PFLUG

Fig. 1.Exponential kernel γ(·)≡η−1exp(− · η−1),q0≡1

4+1

2χ[−0.5,0.5], Nonlocal in the

velocity, V(·)=1−(·)2, from left to right η∈ {10−1,10−2,10−3},Colorbar: 0 1

can be observed, the right most illustration is not to be distinguished from the local

entropy solution.

1.1. Outline. In section 1 we have motivated the problem setup and have pre-

sented it in the relation to already existing literature. Section 2 presents results on

well-posedness, stability and a maximum principle for the nonlocal (in the velocity)

conservation laws, while section 3 presents the related local conservation laws and

some of their properties particularly the entropy formulation, and more. In section 4

we choose an exponential kernel for the nonlocal velocity and obtain under additional

conditions on the velocity and initial datum a uniform TV bound on the nonlocal

velocity is obtained which is used to pass to the limit. However, this limit is only a

weak solution but the entropy condition (for the limit, the local case) is open. This is

what is established in Theorem 4.5 under slightly more restrictive conditions on the

initial datum and the velocity. The chosen approach is reminiscent to [40]. Although

this result might be generalizable to a variety of other kernels in the spirit of [17] we

do not take this path, but instead look – for general kernels – into the monotonicity

of the proposed dynamics. And indeed, in section 5 we ﬁnd similar as in [37] without

additional restrictions on the velocity that the “nonlocal in the velocity conservation

laws” are monotonicity preserving which makes it possible to pass to the limit for

monotonically increasing and decreasing datum. We thus cover the archetypes of

local conservation laws, rare factions and shock waves. In section 6, the results are

illustrated numerically and the convergence nonlocal in the solution vs. nonlocal in

the velocity is compared when the nonlocal kernel approaches a Dirac distribution.

Section 7 concludes the contribution with a list of open problems and future

research.

2. Basic results on nonlocal (in velocity) conservation laws. In this sec-

tion, we present the general assumptions on the involved data, introduce the consid-

ered problem class rigorously and deﬁne what we mean by weak solutions. Addition-

ally, we provide existence and uniqueness results for the corresponding weak solutions

as well as a stability in the initial datum and a maximum principle.

For the rest of the paper, we make the following assumptions which become

meaningful when taking a look at Deﬁnition 2.2 and Deﬁnition 2.3.

Assumption 2.1 (Nonlocal conservation laws). We assume the following

•initial datum q0∈L

∞(R;R≥0)∩TV(R)

•velocity function V∈C2(R) : V050

•nonlocal weight γ∈L

∞(R;R≥0)∩L

1(R) and γmonotonically decreasing,

•and nonlocal “reach” η∈R>0

CONSERVATION LAWS WITH NONLOCAL VELOCITY – THE SINGULAR LIMIT 3

and set ΩT:= (0, T )×Rfor T∈R>0the considered time horizon.

Having stated the assumptions on the input datum of the nonlocal dynamics we now

specify these dynamics:

Definition 2.2 (Nonlocal dynamics). Given Assumption 2.1, the nonlocal dy-

namics, the conservation law with nonlocal velocity, reads as

∂tq(t, x) = −∂xq(t, x)Wγ, V (q)(t, x),(t, x)∈ΩT,(2.1)

q(0, x) = q0(x), x ∈R,(2.2)

WV(q),γ(t, x):=γ∗V(q)(t, x):=1

ηZ∞

γy−x

ηV(q(t, y)) dy, (t, x)∈ΩT.(2.3)

We call q0:R→Rinitial datum,V:R→Rthe velocity function, γ:R≥0→R≥0

the nonlocal kernel or weight and W: ΩT→Rthe nonlocal velocity or nonlocal

term for the nonlocal reach η∈R>0.

Given the problem setup we will consider in this work we deﬁne what we mean with

weak solutions and then address the questions of existence and uniqueness of these.

Definition 2.3 (Weak solution). Let (T, η)∈R2

>0be given as well as Assump-

tion 2.1, we call qη∈C[0, T ]; L

loc(R)∩L

∞((0, T ); L

∞(R)∩TV(R)) a weak solution of

the nonlocal dynamics in Deﬁnition 2.2 iﬀ ∀φ∈C1

c((−42, T )×R)and for the nonlocal

velocity WV(q), γ∈C[0, T ]; L

loc(R)as in (2.3) it holds that

ZZΩT

q(t, x)∂tφ(t, x) + W[V(q), γ](t, x)∂xφ(t, x)dxdt+ZR

q0(x)φ(0, x) dx= 0.

We then have the following existence and uniqueness result on small time horizon.

Theorem 2.4 (Existence & Uniqueness on small time horizon). Let Assump-

tion 2.1 hold. Then, there is a time T∗∈R>0on which there is a unique weak

solution

q∈C[0, T ∗]; L

loc(R)∩L

∞(0, T ∗); L

∞(R)∩TV(R).

Additionally, the solution is nonnegative.

Proof. The proof is very similar to [5, Theorem 2.15]. The diﬀerence in the

considered setup is that the integral operator of the nonlocal term acts not on qitself

but on V(q) which necessitates to study the related ﬁxed-point problem in the set

W ∈ L

∞((0, T ); L

∞(R)) : ∂2W ∈ L

∞((0, T ); TV(R)). The key idea is to assume

that the velocity W0∈L

∞((0, T ); W1,∞(R)) is given and that we can construct the

solution of the linear conservation law

∂tq(t, x) + ∂xW0(t, x)q(t, x)= 0

q(0, x) = q0(x)

by means of the characteristics as

(2.4) q(t, x) = q0ξW0(t, x; 0)∂2ξW0(t, x; 0)

with the characteristics being the unique solution of

(2.5) ξ(t, x;τ) = x+Zτ

W0(s, ξ(t, x;s)) ds.

4J. FRIEDRICH, S. G ¨

OTTLICH, A. KEIMER, L. PFLUG

Computing the nonlocal term via (2.3) we end up with

W1(t, x) = 1

ηZ∞

γy−x

ηV(q(t, y)) dy

ηZ∞

γy−x

ηVq0ξW0(t, y; 0)∂2ξW0(t, y; 0)dy

(2.6)

and inductively for n∈N≥1

(2.7) Wn(t, x) = 1

ηZ∞

γy−x

ηVq0ξWn−1(t, y; 0)∂2ξWn−1(t, y; 0)dy

with ξWfor W ∈ L

∞((0, T ); W1,∞

loc (R)) as in (2.5).

However, this can be interpreted as a ﬁxed-point problem in Win the topology

∞((0, T ); L

∞(R)) and by applying Lipschitz-estimates of the characteristics with re-

gard to the nonlocal term W(see in particular [38, Theorem 2.4]) and corresponding

TV estimates, we can indeed establish by means of Banach’s ﬁxed-point theorem the

existence and uniqueness of a solution of (2.7). The uniqueness then carries over to

the solution as well (compare again [5, Theorem 2.15]) and the nonnegativity of the

solution follows from the identity (2.4) and the fact that ∂2ξ > 0. We do not detail it

further.

As we will later obtain results on the convergence (compare sections 4 and 5) by means

of approximating the nonlocal solution by smooth solutions as well as the maximum

principle in Theorem 2.6 we present the following stability result with respect to the

initial datum:

Lemma 2.5 (Stability and approximation by strong solutions). Let the datum as

in Assumption 2.1 be given and take a standard molliﬁer {φε}ε∈R>0⊂C∞(R;R≥0)

as in [48, Remark C.18]. Assume that qεis for ε∈R>0the solution of the dynamics

in Deﬁnition 2.2 for the initial datum φε∗q0and qcorrespondingly the solution for

the initial datum q0. Then, it holds on a small enough time horizon T∈R>0

lim

ε→0kqε−qkC([0,T ];L

1(R)) = 0

and {qε}ε∈R>0⊂C1(ΩT).In addition, for V∈C3(R)it holds {qε}ε∈R>0⊂C2(ΩT).

Proof. The stability estimate is very reminiscent to the existence and uniqueness

Theorem 2.4 proof by means of characteristics. However, the uniform TV bound is

crucial to obtain the stability of the solution in C[0, T ]; L

1(R). We do not go into

details. The small time horizon T∈R>0on which the stability result holds can

be extend to any ﬁnite time horizon as long as the corresponding solutions exist on

these.

To be able to extend the solveability to any ﬁnite time horizon but also to demonstrate

the physical reasonability of the model (density is bounded between 0 and a maximal

density) we state the following maximum principle:

Theorem 2.6 (Existence & uniqueness of solutions, maximum principle). Given

Assumption 2.1, the nonlocal conservation law in Deﬁnition 2.2 admits on every ﬁnite

time horizon T∈R>0a unique weak solution

q∈C[0, T ]; L

loc(R)∩L

∞(0, T ); L

∞(R)∩TV(R)

CONSERVATION LAWS WITH NONLOCAL VELOCITY – THE SINGULAR LIMIT 5

in the sense of Deﬁnition 2.3 and the following maximum principle holds

ess- inf

x∈Rq0(x)≤q(t, x)≤ess- sup

x∈R

q0(x),for a.e. x∈R, t ∈[0, T ].

Proof. We only show the maximum principle and only the upper bound. Ap-

proximating the initial datum as in Lemma 2.5 we obtain the corresponding solution

qε∈C1(ΩT) so that it indeed satisﬁes the nonlocal conservation law in the classical

sense. Then, we have for (t, x)∈ΩT

∂tqε(t, x) = −W[V(qε), γ](t, x)∂xqε(t, x)−∂xW[V(qε), γ](t, x)qε(t, x)(2.8)

assuming that for given t∈[0, T ] we are at a maximal point x∈R, i.e. ∂xqε(t, x)=0

=−∂xW[V(qε), γ](t, x)qε(t, x)(2.9)

As qε=0 thanks to Theorem 2.4, we only need to show that ∂xWis positive at the

maximal value x. To this end, write

∂xW[V(qε), γ](t, x) = −1

ηγ(0)V(qε(t, x)) −1

η2Z∞

γ0y−x

ηV(qε(t, y)) dy

and as γ050 and qε(t, x) maximal so that V(qε(t, x)) minimal

≥ − 1

ηγ(0)V(qε(t, x)) −1

η2V(qε(t, x)) Z∞

γ0y−x

ηdy

≥1

ηV(qε(t, x)) limy→∞ γy−x

η= 0

as γ∈TV(R>0)∩L

1(R>0) =⇒limy→∞ γy−x

η= 0 ∀x∈R.

However, this means that in the maximal point x∈Rthe time derivative in (2.8)

is nonpositive, meaning that the maximum cannot increase further. From this, the

upper bound on the smoothed solution follows and as these bounds are uniform, they

also hold in the limit for the non-smoothed version.

The lower bound can be derived analogously, concluding the proof.

Remark 2.7 (Other ways to obtain the previous results). The existence of weak

solutions and the maximum principle can be proven by following the lines of [28].

As in the latter work the problem in Deﬁnition 2.2 is considered with a compactly

supported kernel, we need to restrict the support of the kernel onto a compact interval

(in an appropriate manner) such that the convergence of the numerical scheme in [28]

is ensured. In particular, the estimates obtained in [28] remain valid. We do not go

into details here.

3. Fundamental results on (local) conservation laws. As we will study

the convergence of solutions qηfor the Cauchy problem in Deﬁnition 2.2 we need

to cover the well-established theory of local conservation laws which is for instance

fundamentally described in [8,32,26].

Definition 3.1 (The local conservation law). Given Deﬁnition 2.2 we call the

Cauchy problem, the conservation law with initial datum q0,

∂tq(t, x) + ∂xV(q(t, x))q(t, x)= 0,(t, x)∈ΩT,

q(0, x) = q0(x), x ∈R,

the local conservation law related to the nonlocal (in the velocity) conservation law in

Deﬁnition 2.2.

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CONSERVATIONLAWSWITHNONLOCALVELOCITY{THESINGULARLIMITPROBLEMJANFRIEDRICH,SIMONEGOTTLICHy,ALEXANDERKEIMERz,ANDLUKASPFLUGxKeywords.Nonlocalconservationlaw,nonlocalinvelocity,convergence,weakentropysolution,monotonicitypreserving,singularlimit,singularlimitfornonlocalinvelocityconservationlawsMSCcode...

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CONSERVATION LAWS WITH NONLOCAL VELOCITY THE SINGULAR LIMIT PROBLEM JAN FRIEDRICH SIMONE G OTTLICHy ALEXANDER KEIMERzAND LUKAS PFLUGx

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