Abstract
In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution \(\gamma *q\), we weaken the standard assumption on the kernel \(\gamma \in L^\infty \big ((0,T); W^{1,\infty }({\mathbb {R}})\big )\) to the substantially more general condition \(\gamma \in L^\infty ((0,T); BV({\mathbb {R}}))\), which allows for discontinuities in the kernel.
1 Introduction
Nonlocal balance laws have been extensively used to describe physical phenomena, including traffic flow [4, 15, 23, 25, 28, 41], supply chains [24, 29, 43], crowd dynamics [16], opinion formation [1, 40], chemical engineering [39, 44], sedimentation [5], or conveyor belt dynamics [42].
Existence and uniqueness of solutions of nonlocal conservation laws has been proved in several papers. In [5, 6], existence was established by numerical methods and, in [13], using the vanishing viscosity technique; to ensure uniqueness an entropy condition was prescribed. More recently, existence and uniqueness of weak solutions were established via fixedpoint methods and without requiring an entropy condition (see [30, 33, 34]). For measurevalued solutions, a similar approach had been adopted in [20] requiring specific nonlocal kernels. We also refer to [36, 38] for \(L^p\)valued and measurevalued solutions.
Results on the convergence of nonlocal conservation laws to the corresponding local models have been obtained in [7, 8, 12, 14, 17, 31]. For the study of controllability properties of nonlocal conservation laws, we refer the reader, e.g., to [3, 10, 11, 18, 19] and references therein. The aim of this note is to extend the result on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux established in [30]. In the more simplistic case where the nonlocal term is given by a convolution \(\gamma *q\), we can weaken the standard assumption \(\gamma \in L^\infty \big ((0,T); W^{1,\infty }({\mathbb {R}})\big )\) to the substantially more general condition \(\gamma \in L^\infty ((0,T); BV({\mathbb {R}}))\), which allows for discontinuities in the kernel.
More precisely, we consider the nonlocal conservation law
with
Thereby, for \(T\in {\mathbb {R}}_{>0},\) \(q:{\Omega _{T}}\rightarrow {\mathbb {R}}\) with \({\Omega _{T}}= (0,T) \times {\mathbb {R}}\) denotes the spacetime dependent density of the conservation law, and \(q_0:{\mathbb {R}}\rightarrow {\mathbb {R}}\) the initial datum.
We assume that the following conditions are satisfied:
Assumption 1
(Input datum) For \(T\in {\mathbb {R}}_{>0}\), let the following hypotheses be satisfied:
 (A1):

\(\gamma \in L^\infty ((0,T); BV({\mathbb {R}}))\) and \(\gamma \ge 0\);
 (A2):

\(V \in W_\textrm{loc}^{1,\infty }({\mathbb {R}})\);
 (A3):

\(q_0 \in L^\infty ({\mathbb {R}})\).
Following [2], we recall that the total variation of \(u \in L^1({\mathbb {R}})\) is given by
The norm \(\Vert u \Vert _{BV({\mathbb {R}})}:=\Vert u \Vert _{L^1({\mathbb {R}})}+u_{TV({\mathbb {R}})}\) makes the space of functions of bounded variation BV(\(\mathbb {R}\)) a Banach space.
1.1 Outline
In Sect. 2, we state our main wellposedness result and outline the steps of its proof. In particular, we point out what changes are required to generalize the argument of [30]. In Sect. 3, we obtain the two key lemmata that are needed to extend the proofs of [30] to the more general class of kernels under consideration. Finally, Sect. 4 concludes the paper with some examples and numerical simulations.
2 Main result and outline of the proof
Before stating our main theorem, we recall the notion of weak solution for the nonlocal conservation law in (1.1) stated in [30, Definition 2.13].
Definition 2.1
(Weak solution of the nonlocal balance law) We say that \(q \in C\big ([0,T]; L^1_\text {loc}({\mathbb {R}})\big )\) is a weak solution of the nonlocal conservation law in (1.1) iff for all \(\varphi \in C^1_\text {c}((42,T) \times {\mathbb {R}})\), the following integral equation holds:
with W[q] as in (1.2).
Our main theorem establishes the existence and uniqueness of weak solutions to the nonlocal conservation law in (1.1) given Assumption 1.
Theorem 2.1
(Local wellposedness of nonlocal conservation laws with rough kernels) Let \(T \in {\mathbb {R}}_{>0}\), and let Assumption 1 hold. Then, there exists \(T^* \in (0,T]\) such that the nonlocal initial value problem in (1.1) admits a unique weak solution \(q \in C\big ([0, T^* ]; L^1_{\text {loc}} ({\mathbb {R}})\big )\cap L^{\infty }((0,T^*);L^{\infty }({\mathbb {R}}))\) in the sense of Definition 2.1. Moreover, the weak solution can be written as
where \(w^*\) is the unique solution on \((0,T^*) \times {\mathbb {R}}\) of the fixed point problem in (3.1) and \(\xi _{w^*}\) the characteristics defined in (2.3).
Under physically reasonable additional monotonicity assumptions on the velocity and the kernel (i.e. the further away the density is from the current space location, the less it contributes in the nonlocal term), we obtain the following existence result for larger time (which is established by means of a comparison principle).
Corollary 2.1
(Global existence and comparison principle) Under the assumptions of Theorem 2.1, if, in addition, it holds that
 (A4):

\(V' \leqq 0\),
 (A5):

\({{\,\textrm{supp}\,}}(\gamma (t,\cdot ))\subseteq {\mathbb {R}}_{\ge 0}\) and \(\gamma (t,\cdot )\) monotonically nonincreasing on \({\mathbb {R}}_{>0}\) \( \forall t\in [0,T]\),
 (A6):

\(q_0 \in L^\infty (\mathbb R ; \mathbb R_{\ge 0}),\)
then the initialvalue problem in (1.1) admits, for every \(T\in {\mathbb {R}}_{>0}\), a unique solution
satisfying the comparison principle
2.1 Outline of the proof to Theorem 2.1
As we will mimic the argument in [30] for rough kernels, we first shortly introduce the required steps in this proof.

Step 1. Formulation of the fixedpoint equation in the nonlocal term w. Recalling that for \((t,x)\in {\Omega _{T}}\) we have
$$\begin{aligned} W[q](t,x)=\mathop \int \limits _{{\mathbb {R}}}\gamma (t,xy)q(t,y)\,\textrm{d}y =: w(t,x), \end{aligned}$$(2.1)we assume for now that this nonlocal term is given. Then, the corresponding conservation law is linear with Lipschitz continuous velocity V(w) and we use the method of characteristics to write the solution as
$$\begin{aligned} q_{w}(t,x)=q_{0}(\xi _{w}(t,x;0))\partial _{x}\xi _{w}(t,x;0),\qquad (t,x)\in {\Omega _{T}}, \end{aligned}$$(2.2)where \(\xi _{w}\) solves the characteristic ODE
$$\begin{aligned} \xi _w(t,x;\tau ) = x + \mathop \int \limits _t^\tau V\big (w(s,\xi _w(t,x;s)\big ) \,\textrm{d}s, \quad \tau \in [0,T]. \end{aligned}$$(2.3)This can now be plugged into the nonlocal term in (2.1) once more to obtain, for \((t,x)\in {\Omega _{T}},\)
$$\begin{aligned} w(t,x)&=\mathop \int \limits _{{\mathbb {R}}}\gamma (t,xy)q(t,y)\,\textrm{d}y=\mathop \int \limits _{{\mathbb {R}}}\gamma (t,xy)q_{0}(\xi _{w}(t,y;0))\partial _{y}\xi _{w}(t,y;0)\,\textrm{d}y\\&=\mathop \int \limits _{{\mathbb {R}}}\gamma \big (t,x\xi _{w}(0;y;t)\big )q_{0}(y)\,\textrm{d}y, \end{aligned}$$a fixedpoint problem in w which is then studied for existence and uniqueness of solutions on a sufficiently small time horizon.

Step 2. Local existence for the nonlocal conservation law. Having proven the existence of a \(w^{*}\in L^{\infty }((0,T);W^{1,\infty }({\mathbb {R}}))\) with Banach’s fixedpoint theorem, we can build a solution of (1.1) in terms of characteristics (analogously to (2.2)):
$$\begin{aligned} q(t,x) = q_0(\xi _{w^*}(t,x;0)) \partial _x \xi _{w^*}(t,x;0),\,\,&(t,x)\in (0,T^{*})\times {\mathbb {R}}, \end{aligned}$$which is presented in [30, Theorem 2.20] and [32, Theorem 3.1] in detail.

Step 3. Uniqueness for the nonlocal conservation law. The uniqueness of \(w^*\) is shown to imply the uniqueness of the solution q. The main idea is to prove that any weak solution can be written in the same way as instantiated in (2.2) (see [30, Lemma 3.1 and Theorem 3.2]).

Step 4. Extension of the solution for larger times. Gluing a sequence of initial value problems with initial data equal to the terminaltime solution of the previous one, we can extend the existence result to a longer (but not necessarily arbitrary) timehorizon (as in [30, Theorem 4.1]).

Step 5. Extension to arbitrary timehorizons and comparison principle. Under the stronger assumptions (A4)(A5), we can extend the solution to an arbitrary timehorizon and show that a comparison principle holds. For the detailed argument, we refer to [34, Lemma 5.8]. It mainly consists of studying the time evolution of the maximum/minimum of the solution and deducing that its time derivative is negative implying that the minimum can only increase and the maximum only decrease over time.
Extension of the proof to rough kernels. The only parts of the proof outlined above that need to be adjusted from [30] to extend the wellposedness result to our more general setting are as follows:

1.
proving that for \(t\in [0,T]\) the convolution \((x\mapsto \gamma (t,\cdot ) *q(t,\cdot ))(x)\) is in \(W^{1,\infty }(\mathbb R)\) for \(\gamma \in L^\infty ((0,T); BV({\mathbb {R}}))\);

2.
establishing the analogue of [30, Proposition 2.17], where it was shown that the fixedpoint mapping induced in Step 1 in Sect. 2.1 satisfies the required assumptions of Banach’s fixedpoint theorem by relying on the regularity assumption \(\gamma \in L^\infty \big ((0,T);W^{1,\infty }({\mathbb {R}})\big )\).
To this end, in Sect. 3, we first prove, in Lemma 3.1, that the convolution is Lipschitz in space and then, in Proposition 3.1, we demonstrate the analogue of [30, Proposition 2.17] in our setting.
3 Proof of the properties of the fixedpoint mapping
We start by proving the Lipschitzcontinuity (in space) of the convolution.
Lemma 3.1
(Smoothing via convolution with BV functions) Let \(\gamma \in BV({\mathbb {R}})\) and \(f \in L^\infty ({\mathbb {R}})\). Then \(\gamma *f \in W^{1,\infty }({\mathbb {R}})\).
Proof
For \(h\in {\mathbb {R}}\) by [2, Remark 3.5] or [37, Corollary 2.17], the right htranslation of \(\gamma \), i.e., \(\tau _{h}\gamma (x):=\gamma (x+h)\ \forall x\in {\mathbb {R}}\ \text{ a.e. }\), satisfies
As a consequence, we can estimate, by using Young’s convolution inequality (see [9, Theorem 4.33]),
We thus conclude that \(\gamma *f \in W^{1,\infty }({\mathbb {R}})\). \(\square \)
We now review the proof of the fixedpoint argument contained in [30, Proposition 2.17]. As mentioned in Sect. 2, this is the main step that needs to be taken to adapt the arguments of [30] to the case of a nonlocal term given by the convolution of the density q with a rough kernel \(\gamma \in L^\infty ((0, T); BV({\mathbb {R}}))\).
Proposition 3.1
(Properties of the fixedpoint mapping) Let
be the fixedpoint mapping as introduced in Step 1 of Sect. 2.1 and let \({\tilde{\Omega }}\) be defined by
Then, the fixedpoint mapping defined in (3.1) satisfies the following properties:

1.
\(\exists T^{*}\in (0,T]:\) \(\Vert F[w] \Vert _{L^\infty ((0,T^{*}); L^\infty ({\mathbb {R}}))} \le M\) for all \(w \in \Omega _M^{M'}(T^{*})\);

2.
\(\exists T' \in (0,T]:\) \(\Vert \partial _{2} F[w] \Vert _{L^\infty ((0,T');L^{\infty }({\mathbb {R}}))} \le M'\) for all \(w \in \Omega _M^{M'}(T')\);

3.
F is Lipschitz continuous with respect to the uniform topology, i.e., for \(w, {\tilde{w}} \in \Omega _{M}^{M'}({\bar{T}})\), \({\bar{T}}:=\min \{T^{*},T' \}\),
$$\begin{aligned} \Vert F[w]  F[{\tilde{w}}]\Vert _{L^\infty ((0,{\bar{T}});L^{\infty }({\mathbb {R}}))}&\le \gamma _{L^{\infty }((0,T);TV({\mathbb {R}}))} {\bar{T}}\Vert w  {\tilde{w}} \Vert _{L^\infty ((0,{\bar{T}});L^{\infty }({\mathbb {R}}))}\\&\qquad \cdot \Vert V'\Vert _{L^{\infty }((M,M))} \textrm{e}^{2{\bar{T}}\Vert V'\Vert _{L^{\infty }((M,M))}M'} \end{aligned}$$and thus, for small time \({\hat{T}}\in (0,{\bar{T}}]\), F is a contraction on \(\Omega _{M}^{M'}\big ({\hat{T}}\big )\).
Proof

1.
For \(w \in \Omega _M\) and \(t \in [0,T]\), we estimate—recalling the definition of F in (3.1)—
$$\begin{aligned} \Vert F[w](t,\cdot ) \Vert _{L^\infty ({\mathbb {R}})} = \left\ \mathop \int \limits _{{\mathbb {R}}} \gamma (t,\cdot \xi _w(0,z;t))q_0(z) \,\textrm{d}z \right\ _{L^\infty ({\mathbb {R}})}&\le \Vert \gamma (t,\cdot ) \Vert _{L^1({\mathbb {R}})}\Vert \partial _2 \xi _{w}(t,\cdot ;0) \Vert _{L^\infty }\Vert q_0 \Vert _{L^\infty ({\mathbb {R}})}\\&\le \Vert \gamma (t,\cdot ) \Vert _{L^1({\mathbb {R}})} \textrm{e}^{t\Vert V'\Vert _{L^{\infty }((M,M))}M'} \Vert q_0 \Vert _{L^\infty ({\mathbb {R}})}, \end{aligned}$$where we have used the substitution rule and the properties of the characteristics ([30, Lemma 2.6] and in particular [30, Lemma 2.6(3)], see also [32, Corollary 2.1]): namely,
$$\begin{aligned} \Vert \partial _{2}\xi _{w}(t,\cdot ;0)\Vert \le \textrm{e}^{t\Vert V'\Vert _{L^{\infty }((M,M))}M'}\quad \forall t\in [0,T], \end{aligned}$$(3.3)which is an immediate consequence of differentiating (2.3) with regard to \(x\in {\mathbb {R}}\) to obtain a linear IVP in \(\partial _{2}\xi _{w}\). However, as \(M,M'\) are fixed, we can find a time horizon \(T^{*}\in (0,T]\) such that
$$\begin{aligned} \Vert \gamma \Vert _{L^{\infty }((0,T);L^{1}({\mathbb {R}}))} \textrm{e}^{T^{*}\Vert V'\Vert _{L^{\infty }((M,M))}M'} \Vert q_0 \Vert _{L^\infty ({\mathbb {R}})}\le M \Longleftrightarrow \textrm{e}^{T^{*}\Vert V'\Vert _{L^{\infty }((M,M))}M'}\le 42, \end{aligned}$$which indeed proves the existence of such a \(T^{*}\).

2.
Next, we estimate the spatial derivative of the fixedpoint mapping in (3.1) (which is welldefined according to Lemma 3.1), for \(w\in \Omega _{M}^{M'}(T)\) and \((t,x)\in {\Omega _{T}}\). Some technical details are left out and can be found in [30, Lemma 2.6(2)], however, the argument is as follows (applying once more the substitution rule and the estimate in (3.3)):
$$\begin{aligned} \partial _x F[w](t,x)&\le \mathop \int \limits _{{\mathbb {R}}}\! \big  \partial _x\gamma (t,x\xi _w(0,z;t)) q_0(z)\big  \,\textrm{d}z\\&\le \Vert \partial _2 \xi (t,\cdot ;0)\Vert _{L^\infty ({\Omega _{T}})} \Vert q_0 \Vert _{L^\infty ({\mathbb {R}})}\!\mathop \int \limits _{{\mathbb {R}}}\! \partial _z \gamma (t,xy) \,\textrm{d}y \\&\le  \gamma (t,\cdot ) _{TV({\mathbb {R}})} \textrm{e}^{t\Vert V'\Vert _{L^{\infty }((M,M))}M'} \Vert q_0 \Vert _{L^\infty ({\mathbb {R}})}. \end{aligned}$$Making this uniform in \((t,x)\in {\Omega _{T}}\), and since \(M,M'\) are fixed, we can find a time horizon \(T'\in (0,T]\) so that
$$\begin{aligned} \gamma _{L^{\infty }((0,T);TV({\mathbb {R}}))} \textrm{e}^{T'\Vert V'\Vert _{L^{\infty }((M,M))}M'} \Vert q_0 \Vert _{L^\infty ({\mathbb {R}})}\le M'\Longleftrightarrow \textrm{e}^{T'\Vert V'\Vert _{L^{\infty }((M,M))}M'} \le 42. \end{aligned}$$This proves the existence of such a \(T'\) and we can indeed chose \(T'=T^{*}\). Thus, we can conclude with the two previous results
$$\begin{aligned} F\Big (\Omega _{M}^{M'}\big (T'\big )\Big )\subseteq \Omega _{M}^{M'}\big (T'\big ), \end{aligned}$$i.e., F is a selfmapping on \(\Omega _{M}^{M'}(T')\).

3.
Finally, we approach the contraction property of F in \(L^{\infty }((0,T');L^{\infty }({\mathbb {R}}))\) and estimate for \(w, {{\tilde{w}}}\in \Omega _{M}^{M'}(T')\) (which is due to its uniform bounds on the involved functions and its derivatives closed in \(L^{\infty }((0,T');L^{\infty }({\mathbb {R}}))\)) and \((t,x)\in (0,T')\times {\mathbb {R}}\):
$$\begin{aligned}&F[w](t,x)  F[{{\tilde{w}}}](t,x) \nonumber \\&= \left \mathop \int \limits _{{\mathbb {R}}} \gamma (t,\xi _w(0,z;t))q_0(z) \,\textrm{d}z  \mathop \int \limits _{{\mathbb {R}}} \gamma (t,\xi _{{{\tilde{w}}}}(0,z;t))q_0(z) \,\textrm{d}z \right \nonumber \\ {}&\le \mathop \int \limits _{{\mathbb {R}}} \left \gamma (t,\xi _w(0,z;t))  \gamma (t,\xi _{{{\tilde{w}}}}(0,z;t))\right q_0(z) \,\textrm{d}z\nonumber \\&\le \gamma _{L^{\infty }((0,T);TV({\mathbb {R}}))} \Vert \xi _w  \xi _{{\tilde{w}}}\Vert _{L^\infty ((0,t)\times {\mathbb {R}}\times (0,t))}\Vert q_0\Vert _{L^\infty ({\mathbb {R}})}\textrm{e}^{t\Vert V'\Vert _{L^{\infty }((M,M))}M'} \end{aligned}$$(3.4)$$\begin{aligned}&\le \gamma _{L^{\infty }((0,T);TV({\mathbb {R}}))}t\Vert V'\Vert _{L^{\infty }((M,M))} \Vert w  {\tilde{w}}\Vert _{L^\infty ((0,t);L^{\infty }({\mathbb {R}}))}\Vert q_0\Vert _{L^\infty ({\mathbb {R}})}\textrm{e}^{2t\Vert V'\Vert _{L^{\infty }((M,M))}M'}, \end{aligned}$$(3.5)where we have used, in (3.4), the substitution rule and the uniform bound on \(\partial _{2}\xi _{w},\partial _{2}\xi _{{\tilde{w}}}\) as in (2.2) thanks to the bounds on \(w,{\tilde{w}}\) in \(L^{\infty }\big (W^{1,\infty }\big )\) (compare (3.2)); and, in (3.5), the stability of the characteristics with regard to the nonlocal term (see [30, Lemma 2.6(3)] and [32, Theorem 2.4]). This last stability result can be obtained when comparing the solution with the “perturbed” solution of the IVP in (2.3) in w and a typical Gronwall estimate (see [21, Appendix B k) ii]) can be used to derive it. Making the previous estimate uniform in (t, x) and recalling that \(M,M'\) are fixed there exists \({\hat{T}}\in (0,T']\) so that
$$\begin{aligned} \gamma _{L^{\infty }((0,T);TV({\mathbb {R}}))}{\hat{T}}\Vert V'\Vert _{L^{\infty }((M,M))} \Vert w  {\tilde{w}}\Vert _{L^\infty ((0,{\hat{T}});L^{\infty }({\mathbb {R}}))}\Vert q_0\Vert _{L^\infty ({\mathbb {R}})}\textrm{e}^{2{\hat{T}}\Vert V'\Vert _{L^{\infty }((M,M))}M'}\le \tfrac{1}{2}. \end{aligned}$$From this, it follows that F is also a contraction in \(\Omega _{M}^{M'}\big ({\hat{T}}\big )\) for a sufficiently small \({\hat{T}}\in \big (0,{\bar{T}}\big )\). \(\square \)
4 Conclusions and numerical illustrations
In what follows, we present some numerical simulations (based on a nondissipative discretization scheme using the method of characteristics, see [35]) to illustrate the effect of a discontinuity in the kernel. We consider the Cauchy problem in (1.1) with initial datum
and focus on LWRtype velocity (see [27, Formula (1.26), p. 11]), i.e., \(V(\xi ) := 1 \xi ^2, \xi \in \mathbb R,\) and Burgerstype velocity (see [27, Formula (1.8), p. 3]), i.e., \(V(\xi ) := \xi , \xi \in \mathbb R,\) (see also [22, Section 3.1.2] for the fundamental diagrams and generalized Greenshields [26]). As examples of convolution kernels, we consider
and remark that \(\gamma _1,\gamma _2 \in BV({\mathbb {R}})\).
For the LWRtype velocity, a comparison principle (see Corollary 2.1) is satisfied. Since the initial datum is chosen in a way that it has maximum density and zero velocity in \({\mathbb {R}}_{>1}\), the initial density for \(x<1\) slows down as it gets closer to \(x=1\). We remark that the second illustration in Fig. 1 indicates a disturbance evolving from points where the discontinuities of \(\gamma \) and q “intersect”, i.e., 0.25 left of the spatial discontinuities of q.
For the example involving the Burgerstype velocity (due to the chosen initial datum and the rightlooking nonlocal term), a comparison principle does not hold (see [30, Example 6.1]) and the entire mass concentrates at the point \(x = 0.5\) as time evolves. Thus, the solution ceases to exist for large time. Again, the impact of the discontinuous kernel (the fourth (right) illustration in Fig. 1) is destroying the rather “smooth” structure of the solution which we would obtain when using a smooth kernel.
Possible generalizations of this work may consist of (1) weakening the assumptions on V to be discontinuous in space; (2) determining the precise regularity assumptions on initial datum and weight to have the nonlocal conservation law be wellposed (including measure valued solutions and kernels); and (3) generalizing the results to multidimensional nonlocal balance laws.
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Acknowledgements
G. M. Coclite and N. De Nitti are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). G. M. Coclite has been partially supported by the Research Project of National Relevance “Multiscale Innovative Materials and Structures” granted by the Italian Ministry of Education, University and Research (MIUR Prin 2017, project code 2017J4EAYB and the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUPD94I18000260001). N. De Nitti has been partially supported by the Alexander von Humboldt Foundation and by the TRR154 project of the Deutsche Forschungsgemeinschaft (German Research Foundation, DFG). L. Pflug has been supported by the DFG–ProjectID 416229255– SFB 1411. We thank E. Zuazua for helpful conversations on topics related to this work.
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Coclite, G.M., De Nitti, N., Keimer, A. et al. On existence and uniqueness of weak solutions to nonlocal conservation laws with BV kernels. Z. Angew. Math. Phys. 73, 241 (2022). https://doi.org/10.1007/s00033022017660
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DOI: https://doi.org/10.1007/s00033022017660