Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave

We study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the ‘N-wave’ entropy solution of the Burgers equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.

Keywords: nonlocal conservation laws, nonlocal flux, Burgers equation, approximation of local conservation laws, N-waves, source-type solutions, entropy solutions Mathematics Subject Classification numbers: 35L65 (Some figures may appear in colour only in the online journal)

Introduction
Let us consider the following nonlocal regularisation of the Burgers equation: supplemented by the nonlocal term which also satisfies the identity In what follows, we assume that the initial datum satisfies and introduce the notation M := ´R ρ 0 (x) dx for its L 1 -mass.Under these assumptions, the nonlocal conservation law (1.1) has a unique global nonnegative weak solution ρ.In particular, in contrast to the case of the local Burgers equation, no entropy condition is required to select a unique weak solution; moreover, the regularity and integrability of the initial datum is essentially preserved along the evolution (see [29] and section 2 below).The aim of this paper is to study its asymptotic behaviour when t → +∞.The main result asserts that, as t → +∞, the solution ρ(t, •) of (1.1) converges to the (unique) N-wave solution (or source-type solution) w of the local Burgers equation (see [37, equation (2.1)]), i.e. the solution of the Burgers equation with a Dirac delta as initial data, which is given explicitly by see figure 1.We refer to [37] (and to section 2 below) for the proof that (1.5) does indeed have a unique entropy solution (which is given by (1.6)) under suitable assumptions.More precisely, our main theorem on the long-time behaviour of the solution of (1.1) can be stated as follows.Plot of an N-wave solution (1.6) (with M = 1) for t = 0.5 (blue), t = 1 (red), and t = 2 (yellow).
For local conservation laws, the long-time convergence of the entropy solution to the corresponding N-wave profile, as well as existence and uniqueness of entropy solutions in case of measure initial data, was first established rigorously in [37].In [17-20, 32, 33], the analysis was extended to classes of viscous conservation laws with flux f(ξ) = ξ q (for ξ ∈ R).In case 1 < q < 2, as t → +∞, the solutions converge to the N-wave profile of the inviscid conservation law; on the other hand, if q = 2, the limit profile is given by the fundamental solution of the viscous conservation law and, if q > 2, it is of Gaussian type (i.e. the fundamental solution of the heat equation with mass ´R u 0 dx).Similar results have been also obtained for scalar conservation laws with fractional diffusion (see [3,16,28]).For the multi-dimensional setting, we also refer to the recent work [38], where is was proved that a multidimensional Burgers-type equation with a Dirac delta distribution as initial data is not well-posed (despite the L 1 -L ∞ smoothing effect established in [39]).
For nonlocal conservation laws, this problem has not been considered in the literature.However, interestingly, it can be reduced to a type of nonlocal-to-local singular limit problem that has attracted much attention in recent years.Indeed, following [37], given λ > 0, we consider the rescaled function with which satisfies We shall prove that, for a fixed t > 0, ρ λ (t, •) → w(t, •) in L 1 (R) as λ → ∞, which, in turn, will be shown to yield with w defined in (1.6) (and, by interpolation, the claim in theorem 1.1).This type of singular limit problem has been intensively studied in the case of initial data that are uniformly bounded with respect to the scaling parameter.First, in [2], it has been observed that, at least numerically, there is some hope that the solution of a nonlocal conservation law converges to the entropy solution of the corresponding local problem when the nonlocal term approaches a Dirac delta.Positive results in this direction were obtained in [30] for a large class of nonlocal conservation laws under the assumption of having monotone initial data; in [13] under the assumption that the initial datum has bounded total variation, is bounded away from zero, and satisfies a one-sided Lipschitz condition.For the case of an exponential weight, in [7,8], Bressan and Shen proved a convergence result under the assumption that the initial datum is bounded away from zero and has bounded total variation.The core of their argument is the observation that, under suitable changes of variables, the nonlocal problem can be rewritten as a hyperbolic system with relaxation terms.The assumption on the initial data being bounded away from zero played a key role in showing a uniform total variation bound for the solution of the nonlocal problem.Indeed, in [13], a counterexample shows that the total variation of the solution may blow up if the data is not bounded away from zero.
On the other hand, in [11], by arguing on the nonlocal term W rather than on the solution of the conservation law, it was possible to remove the additional assumption on the initial datathe key observation being that the nonlocal term W enjoys further regularity and, in particular, its total variation remains uniformly bounded.From the compactness of the sequence of nonlocal terms, it is then possible to deduce the convergence for the sequence of solutions as well.This approach was later adapted in [14] to classes of weights more general than the exponential one. 7Note that Then . The difference and substantial added difficulty of the present contribution compared to the above-mentioned works is that, under the scaling transformation, we are considering initial data that concentrate to a Dirac delta: i.e.
in the sense of distributions as λ → +∞.That is, with respect to λ, the only uniform bound for the initial data ρ 0 is given in terms the L 1 -mass.
To overcome this difficulty, we take advantage of an Oleinik-type inequality satisfied by the nonlocal term W λ [ρ λ ].Indeed, from [11], it is known that we can rewrite (1.9) as a conservation law with nonlocal source formulated purely in W λ [ρ λ ] (see (2.2) below).This motivates using the notation W λ instead of W λ [ρ λ ] in what follows.From (2.2), arguing as in [10], we can deduce the Oleinik-type estimate (see theorem 3.2): Combining it with the uniform this inequality yields an L ∞ -bound for t > 0 (see lemma 3.3): With these ingredients, the approach of [17] leads to the claimed convergence of {W λ } λ>0 towards the N-wave solution of the (local) Burgers equation and, thanks to (1.3), to the convergence of {ρ λ } λ>0 as well.
The paper is organised as follows.In section 2, we recall the necessary preliminaries on the well-posedness of (1.9) (for fixed λ > 0).In section 3, we prove the key and a priori estimates on W λ sketched above.Then, in section 4, we combine them and establish the convergence of {ρ λ } λ>0 and {W λ } λ>0 to the N-wave solution of the local Burgers equation as λ → +∞; or, equivalently, of {ρ(t, •)} t>0 and {W(t, •)} t>0 as t → +∞.This convergence result is illustrated by several numerical simulations in section 5 (together with some further conjectures).Finally, in section 6, we conclude the paper by presenting some open problems.

Preliminaries
For the nonlocal conservation law in (1.9), we recall the following well-posedness result and some fundamental properties of the solution.We refer to [11, [10] for the proof of a similar statement.

Theorem 2.1 (existence and uniqueness of weak solutions and maximum principle).
Let us suppose that the initial datum ρ 0 satisfies (1.4).Then, for every λ > 0, there exists a unique weak solution of the nonlocal Burgers equation (1.9) and the following maximum principle holds: Moreover, for the nonlocal term W λ , the following properties hold: Furthermore, W λ satisfies the following conservation law with nonlocal source in the strong sense: (2.2) For the limit problem (1.5), we rely on a more general well-posedness result from [37, theorem 1.1 and remark 1.1].

Theorem 2.2 (non-negative entropy solutions with measure initial data). Let us consider the local conservation law
and that µ is a non-negative finite measure on R. Then there exists at most one nonnegative solution u ∈ C (0, +∞); , for all τ ∈ (0, +∞), which satisfies the Kružkov entropy condition, i.e.
and achieves the initial datum in the narrow (or weak) sense of measures 8 , In particular, in our setting, theorem 2.2 yields the uniqueness of the N-wave entropy solution (1.6) of (1.5).

A priori estimates
Before presenting our key a priori estimates, let us recall the following stability result of the nonlocal conservation law (1.1) with respect to the initial datum (see [10]).Lemma 3.1 (stability of the nonlocal term with respect to the initial datum).Given ρ 0,1 , ρ 0,2 ∈ L 1 (R), let us denote by W 1 , W 2 ∈ L ∞ ((0, T); W 1,∞ (R)) the nonlocal terms associated to the corresponding solutions of (1.9).Then, the following stability result holds: for all where C is a suitable constant that depends only on the quantities mentioned above.
Proof.From the results in [12,29], we know that the solution of (1.9) can be written as where ξ W1 and ξ W2 solve the characteristic ODEs (here written as Volterra-type integral equations).In particular, we recall that the nonlocal terms corresponding to the initial data ρ 0,1 and ρ 0,2 satisfy the following fixed-point equations for (t, x) ∈ (0, T) × R: Taking the absolute value of the difference, we have To conclude, we need to study the stability of the characteristics with respect to W 1 and W 2 .For (t, x, τ ) ∈ (0, T) × R × (0, T), we compute Gronwall's inequality yields Plugging this into (3.2),we get Applying again Gronwall's inequality on W 1 − W 2 and recalling that (thanks to the maximum principle in theorem 2.1), we conclude the proof.
As a first step, we prove an Oleinik-type inequality on the nonlocal term W λ .The result is essentially contained in [10] (in a more general form).We present the proof below for the sake of completeness.

Theorem 3.2 (Oleinik-type inequality for
for all λ > 0. Proof of theorem 3.2.We consider a smoothed initial datum ρ ε 0,λ (for ε > 0) and call the corresponding smooth nonlocal term W ε λ .We then compute, differentiating the PDE in (2.2) with respect to x, For t > 0 fixed, considering m(t) = sup y∈R ∂ y W ε λ (t, y) and assuming-without loss of generality-that m(t) ⩾ 0, we estimate the right-hand side of (3.5) as follows: .
We have that, for every t > 0, there exists a maximum point of ∂ y W ε λ (t, y) (by choosing, e.g., a compactly supported ρ ε 0,λ and relying on the regularity results of [29]).Using [15, theorem 2.1], we consider x(t) ∈ R such that m(t) = ∂ x W ε λ (t,x(t)), evaluate the previous expression at x = x(t), and compute Since m(t) = 1/t is a solution of the above Riccati-type differential inequality and m(0) = ∞, we use the comparison principle for ODEs to conclude that m(t) ⩽ 1/t and thus Taking the limit ε → 0 + , thanks to lemma 3.1, we conclude the proof.
As a by-product of (3.4), we prove (arguing as in [17, lemma 1.3]) that a L ∞ -bound holds for all t > 0 (which blows up as t → 0 + ).

Corollary 3.4 (BV-regularisation effect).
The function W λ (t, •) belongs to BV loc (R) for every t > 0 and uniformly with respect to λ > 0: namely, for every compact interval K ⋐ R, Taking lemma 3.2 into account, we note that which implies Integrating over K h and taking the absolute values on both sides yields

Long-time behaviour
As a first step towards finishing the proof of theorem 1.1, we shall show next that {W λ } λ>0 is compact in the canonical C [t 0 , T]; L 1 loc (R) topology.Note that the time-interval does not include t = 0 because the L ∞ -estimate from lemma 3.3 blows up as t → 0 + .

Lemma 4.1 (compactness of {W
Proof.Arguing as in [11, theorem 4.1], we shall apply the compactness result in [40, lemma 1]: given a Banach space B, a set In our case, let us fix a compact interval K ⋐ R and define B = L 1 (K) and F(t Thanks to lemma 3.2, we know that W λ (t, •) has a uniform total variation bound; thus, by [35, theorem 13.35], the set It remains to show the second point, the uniform equi-continuity.To this end, we again replace the initial datum ρ 0,λ by a smooth ρ ε 0,λ , with ε > 0, and call the corresponding smooth nonlocal term W ε λ .Then, we can estimate where we used Fubini-Tonelli's theorem to exchange the order of integration and estimate the last term.Thanks to lemmas 3.2 and 3.3, we have that this is a uniform bound in λ > 0 and ε > 0. This yields the uniform equi-continuity so that we obtain indeed the claimed compactness.
We can now complete the proof of theorem 1.1 arguing as in [17, section 2].
Proof of theorem 1.1.The core of the proof consists in showing that the family {ρ λ } λ>0 converges to the N-wave defined in (1.6).We shall divide the argument of this theorem in several steps.
Step 1. Compactness of the family {W λ } λ>0 in C [t 0 , T]; L 1 loc (R) .For any 0 < t 0 < T, by lemma 4.1, we have that W λ converges (up to extracting a subsequence) to a limit point w * strongly in C [t 0 , T]; L 1 loc (R) ; hence, we also have T] and W λ → w * pointwise (again up to subsequences) for all t ∈ [t 0 , T] and a.e.x ∈ R.
Thanks to (1.11), we can deduce that ρ λ also converges to w * along the same subsequence.Indeed, first we observe that and thus we also obtain Step 2a.Tail control and convergence of the family {ρ λ } λ>0 in C([t 0 , T], L 1 (R)).In order to pass from the convergence ρ λ → w * strongly in C [t 0 , T]; L 1 loc (R) to the convergence in C [t 0 , T]; L 1 (R) , we need a uniform bound on the 'tail' of the functions {ρ λ } λ>1 .We shall prove that there exists a constant Since ρ 0 ∈ L 1 (R), the right-hand side of (4.1) can be made arbitrarily small choosing R large enough.Then, from (4.1), the convergence In order to prove (4.1), let us consider a test function ϕ ∈ C ∞ (R) such that 0 ⩽ ϕ ⩽ 1, ϕ ≡ 1 for |x| > 2, and ϕ ≡ 0 for |x| ⩽ 1; we consider the rescaling ϕ Let us multiply the PDE in (1.9) by ϕ R , integrate in (0, t) × R (for some t > 0), and perform an integration by parts (to rigorously justify this computation, we can use a smoothing argument based on lemma 3.1): We remark that where we used lemma 3.3 in the last line.
Step 2b.Tail control and convergence of the family which yields, thanks to (4.1), As a byproduct of Steps 1 and 2, we note that the limit point w * satisfies Step 3. Identification of the initial condition.We now identify the initial datum taken by the limit point w * , i.e. we verify that the initial condition Mδ 0 is achieved in the weak sense of non-negative measures on R. We need to prove that, for all ϕ ∈ C b (R), To this end, arguing as in [17, pp 52-54], we shall split the argument into two steps.First, we consider a smaller class of test functions ϕ ∈ C ∞ c (R; [0, 1]) and secondly ϕ ∈ C b (R).
We start by estimating, for a test function Then, letting λ → +∞, we obtain which, in turn, goes to zero as t → 0 + .As a second step, let us consider the case of a bounded continuous function ϕ ∈ C b (R).We shall rely on an approximation argument and on the tail control of ρ λ in (4.1).Let us consider a regularised test function ϕ ε obtained as ϕ ε := ϕ * η ε (where η ε denotes a standard mollifier; see [21, The control of the first term follows by the same argument developed above.For the second and third term, we estimate which can both be made arbitrarily small provided that ε > 0 is small enough and R > 0 is large enough. A similar argument can be used for {W λ } λ>0 .Indeed, for ϕ ∈ C ∞ c (R; [0, 1]), we estimate For the term I 1 , we compute where, in the last line, we used lemma 3.3.For I 2 , using Fubini-Tonelli's theorem, we compute integrating by parts on the term x → exp(λ(y − x)) yields integrating by parts on the term y → W λ (s, y)∂ y W λ (s, y) and using the fact that lim x→±∞ W λ (t, •) = 0 (which is a consequence of the fact that W λ (t, •) ∈ L 1 (R) ∩ BV loc (R) for t > 0 and λ > 0), we then get where, in the last line, we used lemma 3.3.Thus, for any ε > 0, we can choose τ > 0 and The rest of the argument for ϕ ∈ C b (R) goes through as above.
Step 4. Entropy admissibility of the limit point.The limit point w * is actually the unique entropy admissible N-wave solution w of the Burgers equation (1.5) defined in (1.6).This follows immediately from passing to the limit pointwise in the Oleinik inequality (3.4).Thanks to Urysohn's subsequence principle, from the uniqueness of the entropy solution of (1.5), we also deduce that the whole families {ρ λ } λ>0 and {W λ } λ>0 converge to w (not just up to extracting a subsequence).
Step 5. Conclusion of the proof.From the steps above, we have that where w denotes the N-wave solution entropy of (1.5).For p = 1, (1.7) is a consequence of the fact that (and that the same would hold true replacing t = 1 by any fixed t > 0), i.e. letting λ → +∞ for a fixed time t > 0 is equivalent to fixing λ = 1 and letting t → +∞.
(still observable for t = 10 at x ≈ 1).This can be understood when recalling that around x ≈ 1 the velocity of the dynamics is smaller than for x < 1 so that the density increases between both points and the jump decreases (which is visible in particular for t = 1 and t = 10).
Secondly, in figure 3, we consider γ(x) := 1 (0,1) (x), x ∈ R, instead of an exponential weight in (1.2), i.e. we study The numerical simulation shows that, even in this case (which is not covered by the results of the present paper or by the ones on the singular limit problem contained in [11,14]), a convergence result can be observed.However, the convergence seems to occur 'less regularly' as the constant kernel generates more and more points where the solution is not differentiable.
Indeed, in contrast to the exponential kernel case, the regularity of the solution for piece-wise constant kernels depends points-wise and locally (on the trace of backward characteristics) on initial data, kernel, and their interplay.Finally, we present some simulations illustrating the case of a more general power-type velocity: namely, for some for q ⩾ 2. In this case, the explicit N-wave solution of the corresponding local conservation law is given by x qt that is, in the rescaled variables wq := t 1/q w q , y := xt −1/q , wq (y) =      y q 1 q−1 if y ∈ 0, q M q−1 q−1 q , 0 otherwise (see [37, equation (2.1)]).In particular, in figure 4 (for q = 3), the convergence result seems to hold.In this case, none of the previously established results hold.However, the numerical experiments point to the fact that we may still observe the L 1 -convergence to the N-wave profile.The behaviour of the rescaled solution, which explodes at x = 0, is particularly noteworthy.It can be explained as follows.For the conservation law ∂ t ρ(t, x) + ∂ x W[ρ] 2 (t, x)ρ(t, x) = 0, (t, x) ∈ (0, T) × R, we can compute, along characteristics (see [12,29] As W 'looks' to the left and the solution vanishes on the left half-space for all time t > 0, we have that W[ρ](t, 0) = 0 for all t > 0; thus, the value of the solution at x = 0 never changes, i.e. lim x↘0 ρ(t, x) = ρ 0 (0) for all t > 0, which yields the long-time behaviour at x = 0 observed in figure 4 upon rescaling.

Conclusions
In this contribution, we have proved the convergence of the solution of the nonlocal conservation law (1.1) with bounded, integrable, and non-negative initial datum to the N-wave solution of the Burgers equation as t → +∞.
Several open problems and possible generalizations of this result could be of interest for future work.We mention a few below.

1033 of 17
December 2022 adopted by the Italian Ministry of University and Research, CUP: D93C22000410001, Centro Nazionale per la Mobilità Sostenibile and the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP-D93C23000100001). L Pflug has been supported by the Deutsche Forschungsgemeinschaft (DFG)-Project-ID 41 622 9255-SFB 1411.E Zuazua has been funded by the Alexander von Humboldt-Professorship program, the ModConFlex Marie Curie Action, HORIZON-MSCA-2021-DN-01, the COST Action MAT-DYN-NET, the Transregio 154 Project 'Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks' of the DFG, Grants PID2020-11 2617GB-C22 and TED2021-13 1390B-I00 of MINECO (Spain), and by the Madrid Goverment-UAM Agreement for the Excellence of the University Research Staff in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).