The initial-boundary value problem for general non-local scalar conservation laws in one space dimension

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The initial-boundary value problem for general non-local

scalar conservation laws in one space dimension

Cristiana de Filippis, Paola Goatin

To cite this version:

Cristiana de Filippis, Paola Goatin. The initial-boundary value problem for general non-local scalar conservation laws in one space dimension. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2017, 161, pp.131-156. �hal-01362504�

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The initial-boundary value problem for general non-local scalar

conservation laws in one space dimension

Cristiana De Filippis∗ Paola Goatin† September 8, 2016

Abstract

We prove global well-posedness results for weak entropy solutions of bounded variation (BV) of scalar conservation laws with non-local flux on bounded domains, under suitable regularity assumptions on the flux function. In particular, existence is obtained by proving the conver-gence of an adapted Lax-Friedrichs algorithm. Lipschitz continuos dependence from initial and boundary data is derived applying Kruˇzhkov’s doubling of variable technique.

Key words: Scalar conservation laws, Non-local flux, Initial-boundary value problem, Lax-Friedrichs scheme.

1 Introduction

Given a bounded open interval I = ]a, b[ ⊂ R, we consider the following initial-boundary value problem ∂tρ + ∂xf (t, x, ρ, ρ ∗ η) = 0 , (t, x) ∈ R+× I , (1.1a) ρ(0, x) = ρ0(x) , x ∈ I , (1.1b) ρ(t, a) = ρa(t) , t ∈ R+, (1.1c) ρ(t, b) = ρb(t) , t ∈ R+, (1.1d) where f ∈ C2(R+× ¯_{I × R × R; R) satisfies} f (t, x, 0, R) = 0 ∀t, x, R, (1.2a) sup t,x,ρ,R ∂ρf (t, x, ρ, R) < L, (1.2b) sup t,x,R ∂xf (t, x, ρ, R) < C|ρ|, sup t,x,R ∂Rf (t, x, ρ, R) < C|ρ|, (1.2c) sup t,x,R ∂ 2 xxf (t, x, ρ, R) < C|ρ|, sup t,x,R ∂ 2 xRf (t, x, ρ, R) < C|ρ|, sup t,x,R ∂ 2 RRf (t, x, ρ, R) < C|ρ|, (1.2d)

for some constants L > 0 and C > 0, and η ∈ (C1 ∩ W1,∞_{)(R; R) is a convolution kernel (not}

necessarily with compact support) such that Z

R

η(x)dx = 1.

Equations of type (1.1a) arise in several applications, and have made the object of a large literature in recent years. Space-integral terms appear for example in models for granular flows

∗

University of Milano-Bicocca, Italy, & Inria Sophia Antipolis - M´editerran´ee, France, E-mail: c.defilippis@campus.unimib.it

†

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[3], sedimentation [7], supply chains [19], conveyor belts [18], weakly coupled oscillators [2], struc-tured populations dynamics [24], or more general problems like gradient constrained equations [4]. Equations with non-local flux have been recently introduced also in traffic flow modeling to ac-count for the reaction of drivers or pedestrians to the surrounding density of other individuals, see [8, 10, 11, 26].

General analytical results on non-local conservation laws, proving existence and eventually unique-ness of solutions of the Cauchy problem for (1.1a), can be found in [5] for scalar equations in one space dimension, in [12] for scalar equations in several space dimensions and in [1, 13, 14] for multi-dimensional systems of conservation laws. Besides, specific finite volume numerical methods have been developed recently in [1, 17, 21]. To our knowledge, initial-boundary value problems of the form (1.1) have not been rigorously studied yet, the difficulties lying in the presence of the non-local term, which may exceed the boundaries of the space domain. Nonetheless, real applications (confined environments, networks, etc.) and numerical computations require a precise account for boundary conditions.

The scope of the present article is to propose an approach for a rigorous treatment of boundary conditions, in the case of one space-dimensional problems. The strategies we employ are inspired by classical results on scalar conservation laws with boundary conditions. In particular, we refer to [6, 9, 27]. Our results are based on the extension of the solution outside the domain, set to be constantly equal to the corresponding boundary condition values. It is far from obvious to generalize this technique to problem in several space-dimensions.

As in the classical case, we assume that boundary conditions can not generally be satisfied in strong sense. Therefore, we introduce the following notion of weak entropy solution for (1.1), which extends to problems with boundaries the definition of solution given in [5] for the corresponding Cauchy problem. This formulation, based on semi Kruˇzhkov entropies [23, 27], has the advantage of not using explicitly the traces of the solution at the boundaries of the domain, which turns particularly useful in the existence proof, provided in Section 2.

Definition 1 Let ρ0 ∈ L∞(I; R) and ρa, ρb ∈ L∞(R+; R). A map ρ ∈ L∞ R+× I; R is a weak entropy solution to (1.1) if for every test function ϕ ∈ C1_c(R2; R+) and for every κ ∈ R

Z +∞ 0 Z b a (ρ − κ)± ∂tϕ + sgn(ρ − κ)± f (t, x, ρ, R(t, x)) − f (t, x, κ, R(t, x)) ∂xϕ (1.3) − sgn(ρ − κ)± d dxf (t, x, κ, R(t, x)) ϕ dx dt + Z b a (ρ0− κ)± ϕ(0, x) dx +Lip(f ) Z +∞ 0 ρa(t) − κ ± ϕ(t, a) dt + Lip(f ) Z +∞ 0 ρb(t) − κ ± ϕ(t, b) dt ≥ 0 , (1.4) where R(t, x) := (ρ(t, ·) ∗ η)(x) = Z R ρ(t, y)η(x − y) dy = Z b a ρ(t, y)η(x − y) dy + ρa(t) Z a −∞ η(x − y) dy + ρb(t) Z +∞ b η(x − y) dy = Z b a ρ(t, y)η(x − y) dy + ρa(t) Z +∞ x−a η(y) dy + ρb(t) Z x−b −∞ η(y) dy . (1.5) Above, we have noted sgn+(s) := max{s/|s|, 0}, sgn−(s) := − sgn+(−s), s+ := s sgn+(s) and s− := (−s)+ for s ∈ R. In the paper, we will also denote I(r, s) :=min{r, s}, max{r, s} for any r, s ∈ R.

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The Definition 1 is equivalent to the one provided in [6] (for a proof of equivalence we refer the reader to [22, Theorem 7.31]). This second definition will be used in Section 3 to prove Lipschitz continuous dependence of solution with respect to initial and boundary data.

Definition 2 Let ρ0 ∈ L∞(I; R) and ρa, ρb ∈ L∞(R+; R). A map ρ ∈ BV R+× I; R is a weak

entropy solution to (1.1) if for every test function ϕ ∈ C1_c(R2; R+) and for every κ ∈ R Z +∞ 0 Z b a |ρ − κ|∂_tϕ + sgn (ρ − κ)f (t, x, ρ, R(t, x)) − f (t, x, κ, R(t, x)) ∂xϕ − sgn (ρ − κ) d dxf (t, x, κ, R(t, x)) ϕ dx dt + Z b a |ρ0− κ|ϕ(0, x) dx + Z +∞ 0

sgn (ρa− κ)f (t, a, ρ(t, a+), R(t, a)) − f (t, a, κ, R(t, a)) ϕ(t, a) dt

+ Z ∞

0

sgn (ρb− κ)f (t, b, κ, R(t, b)) − f (t, b, ρ(t, b−), R(t, b)) ϕ(t, b) dt ≥ 0 . (1.6)

We remark that to ensure that the traces of ρ at x = a, b, are well defined, we need to assume that the solutions have bounded variation, see [6, Lemma 1]. Moreover, following [6, 15], we recall that the entropy condition (1.6) implies that the traces of the solution at the boundary satisfy

• On the left boundary x = a: for all κ ∈ R

sgn ρ(t, a+) − κ − sgn ρa(t) − κ

f (t, a, ρ(t, a+), R(t, a)) − f (t, a, κ, R(t, a)) ≤ 0, (1.7) • On the right boundary x = b: for all κ ∈ R

sgn ρ(t, b−) − κ − sgn ρb(t) − κ

f (t, b, ρ(t, b−), R(t, b)) − f (t, b, κ, R(t, b)) ≥ 0. (1.8) Our main result states the global well-posedness of (1.1).

Theorem 1 Let hypotheses (1.2) hold. If ρ0 ∈ (L∞∩ BV) I; R+ and ρa, ρb ∈ (L∞∩ BV) R+; R+,

then for all T > 0 problem (1.1) has a unique weak entropy solution ρ ∈ BV [0, T ] × I; R+ in the sense of Definitions 1, 2. Moreover, the following estimates hold:

kρ(T, ·)k_L1_(I) ≤ kρ₀k_L1_(I)+ α kρ_ak_L1_{([0,T ])}+ kρ_bk_L1_{([0,T ])} , (1.9) kρ(T, ·)k_L∞_(I)≤ eLTkρ₀k_L∞_(I), (1.10) TV ρ(T, ·); I ≤ eK1T_{TV (ρ} 0; I) + K₂ K1 eK1T _{− 1}_{+ TV ρ} a; [0, T ] + TV ρb; [0, T ] , (1.11) kρ(T, ·) − ρ(T − τ, ·)k_L1_(I) ≤ C_t(T )τ , τ > 0 , (1.12)

with L as in (2.12), K1,2 as in (2.18) and (2.24), and Ct as in (2.28).

Finally, let ρ, σ ∈ C0 R+; L1(I; R+) ∩ BV∞ [0, T ] × I; R+ be two weak entropy solutions to (1.1), with initial data ρ0, σ0 ∈ L∞ I, R+

and boundary data ρa, ρb, σa, σb ∈ L∞ R+; R+ respectively. Then the following estimate holds:

kρ(T, ·) − σ(T, ·)k_L1_(I) ≤ eST kρ₀− σ₀k_L1_(I)+ (L + S0) kρ_a− σ_ak_L1_{([0,T ])}+ kρ_b− σ_bk_L1_{([0,T ])} , where the constants S, S0 are defined by (3.10).

The above result can be easily generalized to unbounded domains I = ]a, b[, with a = −∞ or b = +∞, under the assumption that the initial datum also belongs to L1(I).

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2 Existence of weak entropy solutions

The proof of existence is based on the following strategy: we construct a sequence of approximate solutions using a finite volume algorithm, we prove the convergence of a subsequence and, finally, we show that the limit is indeed a weak entropy solution in the sense of Definition 1. The procedure follows closely [1, 5].

Let us fix a space grid in [a, b] of size ∆x = (b − a)/N , N ∈ N, and choose a time step ∆t (satisfying some stability conditions which will be detailed later). We introduce the usual notation

tn= n ∆t, n ∈ N ; xj = a + j − 1 2 ∆x, x_j+1/2= a + j∆x, j = 1, . . . , N ; λ = ∆t ∆x. Throughout, an initial datum ρ0∈ (L∞∩ BV)(R; R) is fixed and we denote

ρ0_j = 1 ∆x

Z x_j+1/2

xj−1/2

ρ0(x) dx dy for j = 1, . . . , N.

We define a piecewise constant approximate solution ρ∆ to (1.1) as

ρ∆(t, x) = ρnj for ( t ∈ [tn, tn+1[ , x ∈ [xj−1/2, xj+1/2[ , where n ∈ N, j = 1, . . . , N, through the following adapted Lax-Friedrichs scheme

ρn+1_j = ρn_j − λhF_j+1/2n (ρn_j, ρn_j+1) − F_j−1/2n (ρn_j−1, ρn_j)i, (2.1) where F_j+1/2n (ρn_j, ρn_j+1) := 1 2 h f (tn, xj, ρnj, Rnj) + f (tn, xj+1, ρnj+1, Rnj+1) + α(ρnj − ρnj+1) i (2.2) is the numerical flux (for some α ∈ R, α > 0) and

Rn_j := ∆xX

k∈Z

η(xj−k)ρnk, j = 1, . . . , N,

are the quadrature formulae approximating the convolution terms. Remark that, due to the bound-edness of the domain ]a, b[, we can set

Rn_j := ∆x N X k=1 η(xj−k)ρnk+ ρna∆x X k≤0 η(xj−k) + ρnb ∆x X k>N η(xj−k) = ∆x N X k=1 η(xj−k)ρnk+ ρna∆x X k≥j η(xk) + ρnb ∆x X k<j−N η(xk) .

The proof of the convergence of approximate solutions is divided in several steps, which are intended to show that the sequence verifies the hypotheses of Helly’s compactness theorem.

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2.1 Positivity

The following lemma ensures the positivity of approximate solutions corresponding to positive initial and boundary data.

Lemma 1 Let ρ0 ∈ L∞ I; R+ and ρa, ρb ∈ L∞ R+; R+. Moreover, assume that

α ≥ L, λ ≤ 1 3min 1 α, 1 L (1 + ∆x) . (2.3)

Then ρ∆(t, x) ≥ 0 for all x ∈ I, t > 0.

Proof. We rearrange (2.1) as ρn+1_j = ρn_j − λhF_j+1/2n (ρn_j, ρn_j+1) ± F_j+1/2n (ρn_j, ρn_j) ± F_j−1/2n (ρn_j, ρn_j) − F_j−1/2n (ρn_j−1, ρn_j)i = 1 − αn_j − β_jnρn_j + αn_jρn_j−1+ β_jnρ_j+1n − λF_j+1/2n (ρn_j, ρ_jn) − F_j−1/2n (ρn_j, ρn_j) , where, for j ∈ {1, · · · , N }, αn_j :=    λF n j−1/2(ρnj,ρnj)−Fj−1/2n (ρnj−1,ρnj) ρn j−ρnj−1 if ρ n j 6= ρnj−1, 0 if ρn_j = ρn_j−1, (2.4) and β_jn:=    −λF n j+1/2(ρnj,ρnj+1)−Fj+1/2n (ρnj,ρnj) ρn j+1−ρnj if ρ n j+16= ρnj , 0 if ρn_j+1= ρn_j . (2.5)

We consider the following estimates: F n j+1/2(ρnj, ρnj) − Fj−1/2n (ρnj, ρnj) = = 1 2 f (t n_{, x} j+1, ρnj, Rj+1n ) − f (tn, xj−1, ρnj, Rj−1n ) = 1 2 f (t n_{, x} j+1, ρnj, Rj+1n ) ± f (tn, xj−1, ρnj, Rj+1n ) − f (tn, xj−1, ρnj, Rj−1n ) ≤ 1 2 f (t n_{, x} j+1, ρnj, Rj+1n ) − f (tn, xj−1, ρnj, Rj+1n ) +1 2 f (t n_{, x} j−1, ρnj, Rnj+1) + 1 2 f (t n_{, x} j−1, ρnj, Rnj−1) ≤ L ρ n j ∆x + 1 2 f (t n_{, x} j−1, ρnj, Rj+1n ) − f (tn, xj−1, 0, Rnj+1) +1 2 f (t n_{, x} j−1, ρnj, Rnj−1) − f (tn, xj−1, 0, Rnj−1) ≤ L ρ n j (1 + ∆x) .

Moreover, we observe that, whenever ρn_j 6= ρn

j−1 and ρnj+1 6= ρnj, α_jn= λ 2ρn j − ρnj−1 f (tn, xj−1, ρnj, Rnj−1) − f (tn, xj−1, ρnj−1, Rnj−1) + α ρn_j − ρn_j−1

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= λ 2 ∂ρf (tn, xj−1, ξj−1/2n , Rnj−1) + α (2.6) and β_jn= − λ 2ρn j+1− ρnj f (tn, xj+1, ρnj+1, Rnj+1) + α ρn_j − ρn_j+1− f (tn, xj+1, ρnj, Rnj+1) = λ 2 α − ∂ρf (tn, xj+1, ξnj+1/2, Rnj+1) , (2.7)

for some ξ_j−1/2n ∈ I(ρn

j−1, ρnj) and ξj+1/2n ∈ I(ρ n j, ρnj+1). Assuming that α ≥ L, λα ≤ 1 3, λL (1 + ∆x) ≤ 1 3, we get αn_j, β_jn∈ 0,1 3 , 1 − αn_j − β_jn∈ 1 3, 1 , λF_j+1/2n (ρn_j, ρn_j) − F_j−1/2n (ρn_j, ρn_j)≤ 1 3 ρ n j , which allow us to recover the sought estimate

ρn+1_j ≥1 − αn_j − β_jnρ_jn+ α_jnρn_j−1+ βn_jρn_j+1−1 3 ρ n j , ≥ 2 3 − α n j − βjn ρn_j + αn_jρn_j−1+ β_jnρn_j+1 ≥ 0 . 2.2 L1 bound

Lemma 2 Let hypotheses (1.2) and conditions (2.3) hold. If ρ0 ∈ L∞ I; R+

and ρa, ρb ∈

L∞ R+; R+, then for all T > 0

kρ∆(T, ·)kL1_(I)≤ kρ₀k_L1_(I)+ α

kρakL1_{([0,T ])}+ kρ_bk_L1_{([0,T ])}

=: C1(T ) . (2.8)

Proof. Thanks to the positivity of the discrete solution, using the definition of the scheme, we compute kρn+1k_L1_(a,b)= ∆x N X j=1 ρn+1_j = ∆x N X j=1 ρn_j − λF_j+1/2n (ρn_j, ρn_j+1) − F_j−1/2n (ρn_j−1, ρn_j) = ∆x N X j=1 ρn_j − λ∆xF_{N +1/2}n (ρn_N, ρn_b) − F_1/2n (ρn_a, ρn₁) = ∆x N X j=1 ρn_j −∆t 2 f (tn, xN, ρnN, RnN) + f (tn, xN +1, ρnb, RnN +1) + α ρnN− ρnb

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+∆t 2 f (tn, x0, ρna, Rn0) + f (tn, x1, ρn1, Rn1) + α ρna− ρn1 = ∆x N X j=1 ρn_j −∆t 2 ∂ρf (tn, xN, ξN,0n , RnN)ρnN + ∂ρf (tn, xN +1, ξN +1,0n , RnN +1)ρnb + α ρnN − ρnb +∆t 2 ∂ρf (tn, x0, ξa,0n , R0n)ρna+ ∂ρf (tn, x1, ξ1n, Rn1)ρn1 + α ρna− ρn1 = ∆x N X j=1 ρn_j +∆t 2 ∂ρf (tn, x0, ξa,0n , Rn0) + α ρn_a +∆t 2 −∂ρf (tn, xN +1, ξN +1,0n , RnN +1) + α ρn_b +∆t 2 −∂ρf (tn, xN, ξN,0n , RnN) − α ρn_N +∆t 2 ∂ρf (t n_{, x} 1, ξ1n, Rn1) − α ρn1.

Being the last two coefficients in the previous estimate non positive, we can conclude that

∆x N X j=1 ρn+1_j ≤ ∆x N X j=1 ρn_j + α∆t ρn_a+ ρn_b ,

thus ending the proof.

2.3 L∞ bound

Lemma 3 Let hypotheses (1.2) and conditions (2.3) hold. If ρ0 ∈ L∞ I; R+

and ρa, ρb ∈

L∞ R+; R+, then for all T > 0

kρ_∆(T, ·)k_L∞_(I) ≤ eLTkρ₀k_L∞_(I), (2.9)

where L is given by (2.12). Proof. We observe that

R n j+1− Rnj−1 ≤ ≤ ∆x ρn a X k≤0 η(xj+1−k) − η(xj−1−k) + ∆x N X k=1 ρn_kη(xj+1−k) − η(xj−1−k) + ∆x ρn_b X k≥N +1 η(xj+1−k) − η(xj−1−k) = ∆x ρn_aX k≤0 Z j+1−k xj−1−k η0(s) ds + ∆x N X k=1 ρn_k Z j+1−k xj−1−k η0(s) ds + ∆xρn_b X k≤0 Z j+1−k xj−1−k η0(s) ds ≤ 2∆x2_kη0_k L∞_(R)  ρn_a+ N X j=1 ρn_j + ρn_b  

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≤ 2∆x kη0k_L∞_(R) kρ0kL1_(I)+ α kρakL1_{([0,T ])}+ kρ_bk_L1_{([0,T ])} ≤ 2T ∆x , (2.10)

where we have set

T := kη0k_L∞_(R) kρ₀k_L1_(I)+ α kρ_ak_L1_{([0,T ])}+ kρ_bk_L1_{([0,T ])} (2.11) and for the latest bound we have applied Lemma 2.

Proceding as in Lemma 1, we can rearrange (2.1) as ρn+1_j =1 − αn_j − βn

j

ρn_j + αn_jρn_j−1+ β_jnρn_j+1− λF_j+1/2n (ρn_j, ρn_j) − F_j−1/2n (ρn_j, ρn_j),

where αn_j and β_jn are as in (2.6) and (2.7) respectively. Using (2.10), we get F n j+1/2(ρ n j, ρnj) − Fj−1/2n (ρ n j, ρnj) = = 1 2 f (t n_{, x} j+1, ρnj, Rnj+1) − f (tn, xj−1, ρnj, Rnj−1) = 1 2 ∂xf (t n_{, ˜}_x j, ρnj, ˜Rnj) xj+1− xj−1 + ∂xf (t n_{, ˜}_x j, ρnj, ˜Rnj) R n j+1− Rj−1n ≤ 2C ρ n j ∆x + C ρ n j R n j+1− Rnj−1 ≤ 2C∆x ρ n j (1 + T ) . Thus, ρn+1_j ≤ 1 − αn_j − β_jnρ_jn+ αn_jρn_j−1+ β_jnρn_j+1+ λ F n j+1/2(ρ n j, ρnj) − Fj−1/2n (ρ n j, ρnj) ≤ 1 − αn_j − βn j kρn_k

L∞_(I)+ αn_jkρnk_L∞_(I)+ β_jnkρnk_L∞_(I)+ 2Cλ∆xkρnk_L∞_(I)(1 + T )

= kρnk_L∞_(I)(1 + ∆tL) ≤ eL∆tkρn_k L∞_(I), for j = 1, · · · , N , being L := 2C 1 + kη0k_L∞_(R) kρ₀k_L1_(I)+ α kρ_ak_L1_{([0,T ])}+ kρ_bk_L1_{([0,T ])} ! . (2.12)

A standard iterative argument completes the proof.

2.4 BV estimates

Proposition 1 (BV estimate in space) Let hypotheses (1.2) and conditions (2.3) hold. If ρ0 ∈ L∞ I; R+ and ρa, ρb ∈ L∞ R+; R+, then ρ∆satisfies the following Total Variation estimate

N X j=0 ρ n j+1− ρnj ≤ Cx(t n_{) ,} _(2.13)

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for all n ∈ N, where C_x(tn) := eK1tn N X j=0 ρ 0 j+1− ρ0j + K₂ K₁ eK1tn_{− 1}₊ n X m=1 ρ m a − ρm−1a + n X m=1 ρ m b − ρm−1b ,

with K1 and K2 positive constants defined in (2.18), (2.19), (2.24).

Proof. We consider separately the central and boundary terms. For j = 1, · · · , N − 1, n ∈ N, we consider the difference

ρn+1_j+1 − ρn+1_j = ρn_j+1− ρn_j − λhF_j+3/2n (ρn_j+1, ρn_j+2) − F_j−1/2n (ρn_j−1, ρn_j) ± F_j+1/2n (ρn_j, ρn_j+1) i ∓ λF_j+3/2n (ρn_j, ρn_j+1) + F_j+1/2n (ρn_j−1, ρn_j) = ρn_j+1− ρn_j − λF_j+3/2n (ρn_j+1, ρ_j+2n ) − F_j+1/2n (ρn_j, ρn_j+1) − F_j+3/2n (ρn_j, ρn_j+1) + F_j+1/2n (ρn_j−1, ρn_j) − λF_j+3/2n (ρn_j, ρn_j+1) − F_j+1/2n (ρn_j, ρ_j+1n ) + F_j−1/2n (ρn_j−1, ρn_j) − F_j+1/2n (ρn_j−1, ρn_j) = An_j − λBn_j, (2.14)

where we have set An

j := ρnj+1− ρnj − λ

F_j+3/2n (ρn_j+1, ρn_j+2) − F_j+1/2n (ρn_j, ρn_j+1) − F_j+3/2n (ρn_j, ρn_j+1) + F_j+1/2n (ρn_j−1, ρn_j), Bn_j := F_j+3/2n (ρn_j, ρn_j+1) − F_j+1/2n (ρn_j, ρ_j+1n ) + F_j−1/2n (ρn_j−1, ρn_j) − F_j+1/2n (ρn_j−1, ρn_j) .

Concerning the first term An_j and recalling (2.4) and (2.5), after suitable rearrangements we get An_j = ρn_j+1− ρn_j × " 1 + λF n j+1/2(ρ n j, ρnj+1) − Fj+1/2n (ρ n j, ρnj) ρn j+1− ρnj − λF n j+3/2(ρ n j+1, ρnj+1) − Fj+3/2n (ρ n j, ρnj+1) ρn j+1− ρnj # +ρn_j+2− ρn_j+1 −λF n j+3/2(ρ n j+1, ρnj+2) − Fj+3/2n (ρ n j+1, ρnj+1) ρn j+2− ρnj+1 ! +ρn_j − ρn_j−1 λF n j+1/2(ρ n j, ρnj) − Fj+1/2n (ρ n j−1, ρnj) ρn j − ρnj−1 ! = 1 − β_jn− γn j+1 ρn_j+1− ρn j + β_j+1n ρn_j+2− ρn j+1 + γ_jnρn_j − ρn j−1 , (2.15) where γ_jn:= λF n j+1/2(ρnj, ρnj) − Fj+1/2n (ρnj−1, ρnj) ρn_j − ρn j−1 , and the bounds γn

j ∈ [0, 1/3] can be proved exactly as it has been done for αnj, thanks to (2.3).

From (2.15) we recover N −1 X j=1 |An_j| ≤ N −1 X j=1 1 − βn_j − γ_j+1n ρ n j+1− ρnj + β n j+1 ρ n j+2− ρnj+1 + γ n j ρ n j − ρnj−1

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= N −1 X j=1 |ρn_j+1− ρn_j| − β₁nρ n₂ − ρn₁+ γ₁n ρn₁ − ρn_a+ β_Nn ρn_b − ρn_N− γ_Nn ρn_N − ρn_{N −1}. (2.16)

We now focus on the term Bn_j in (2.14):

Bn j = 1 2 f (tn, xj+2, ρnj+1, Rnj+2) − f (tn, xj+1, ρnj+1, Rnj+1) −1 2 f (tn, xj, ρnj−1, Rnj) − f (tn, xj−1, ρnj−1, Rnj−1) =∂xf (tn, ˜xj+3/2, ρnj+1, ˜Rj+3/2n )∆x + ∂Rf (tn, ˜xj+3/2, ρnj+1, ˜Rnj+3/2)(R n j+2− Rnj+1) −1 2 ∂xf (tn, ˜xj−1/2, ρnj−1, ˜Rnj−1/2)∆x + ∂Rf (tn, ˜xj−1/2, ρnj−1, ˜Rnj−1/2)(Rnj − Rnj−1) = ∆x 2 ∂_xx2 f (tn, ˆxj, ˆρjn, ˆRnj)(˜xj+3/2− ˜xj−1/2) + ∂ρx2 f (tn, ˆxj, ˆρnj, ˆRnj)(ρnj+1− ρnj−1) + ∂_Rx2 f (tn, ˆxj, ˆρjn, ˆRnj)( ˜Rnj+3/2− ˜R n j−1/2) +1 2 ∂_xR2 f (tn, ¯xj, ¯ρjn, ¯Rnj) ˇRnj+1/2(˜xj+3/2− ˜xj−1/2) + ∂2ρRf (tn, ¯xj, ¯ρnj, ¯Rnj) ˇRnj+1/2(ρ n j+1− ρnj−1) + ∂_RR2 f (tn, ¯xj, ¯ρnj, ¯Rnj) ˇRnj+1/2( ˜Rj+3/2n − ˜Rnj−1/2) + ∂Rf (tn, ¯xj, ¯ρnj, ¯Rjn)(Rnj+2− Rnj+1− Rnj + Rnj−1) , with ˜x_j−1/2∈ (x_j−1, xj), ˆρnj, ¯ρnj ∈ I(ρnj−1, ρnj+1), ˜Rnj−1/2∈ I(R

n j−1, Rnj), ˆRnj, ¯Rnj ∈ I( ˜Rnj−1/2, ˜R n j+3/2), ˆ xj, ¯xj ∈ I(˜xj−1/2x˜j+3/2), ˇR_j+1/2n ∈ I(Rnj − Rnj−1, Rnj+2− Rnj+1).

Notice that as in (2.10) we can estimate R n j+2− Rnj+1 ≤ T ∆x , R n j − Rnj−1 ≤ T ∆x , R n j+2− Rnj ≤ 2T ∆x . Moreover, by their very definition,

R˜ n j+3/2− ˜R n j−1/2 = λ n j+3/2R n j+2+ (1 − λnj+3/2)R n j+1− µnj−1/2R n j−1− (1 − µnj−1/2)R n j ≤ R n j+2− Rj+1n + R n j − Rnj−1 + R n j+1− Rnj ≤ 3T ∆x , and | ˇR_j+1/2n | = δ_j+1/2n R_jn− R_j−1n + (1 − δn_j+1/2)Rn_j+2− Rn_j+1 ≤ δn_j+1/2 R n j − Rnj−1 + (1 − δ n j+1/2) R n j+2− Rnj+1 ≤ T ∆x , for some δn j+1/2, λnj+3/2, µnj−1/2 ∈ [0, 1]. Finally, |ˆρn_j| = ˆ n jρnj−1+ (1 − ˆnj)ρnj+1 ≤ ˆ n j ρ n j−1 + (1 − ˆ n j) ρ n j+1 ,

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|¯ρn_j| = ¯ n jρnj−1+ (1 − ¯nj)ρnj+1 ≤ ¯ n j ρ n j−1 + (1 − ¯ n j) ρ n j+1 , with ˆn j, ¯nj ∈ [0, 1]. Therefore |Bn_j| ≤ 3 2C(∆x) 2_|ˆ_ρn j| + 1 2∆xk∂ 2 ρxf kL∞ ρ n j+1− ρnj−1 +3 2CT (∆x) 2_|ˆ_ρn j| + 3 2CT (∆x) 2_|¯_ρn j| + 1 2T k∂ 2 ρRf kL∞∆x ρ n j+1− ρnj−1 +3 2CT 2_(∆x)2 ρ¯ n j + CT (∆x) 2 ρ¯ n j ≤ 1 2∆x ρ n j+1− ρnj−1 k∂_ρx2 f k_L∞_(R)+ T k∂_ρR2 f k_L∞ +3 2C (1 + T ) (∆x) 2_|ˆ_ρn j| + 1 2CT (5 + 3T ) (∆x) 2_|¯_ρn j|. Thus N −1 X j=1 λ|Bn_j| ≤ 1 2∆t k∂2 ρxf kL∞+ T k∂_ρR2 f k_L∞ N −1X j=1 ρ n j+1− ρnj−1 +1 2C∆t 3 (1 + T ) + T (5 + 3T ) ∆x N −1 X j=1 (ˆn_j + ¯n_j) ρ n j−1 + (2 − ˆ n j − ¯nj) ρ n j+1 ≤ K₁∆t N X j=0 ρ n j+1− ρnj + eK2∆t∆x N X j=1 ρ n j+1 , (2.17)

where we have set

K₁ = k∂_ρx2 f kL∞+ T k∂_ρR2 f k_L∞, (2.18)

e K2 = C

3 + 8T + 3T2. (2.19)

Collecting (2.16) and (2.17), we conclude that

N X j=1 ρ n+1 j+1 − ρ n+1 j ≤ N X j=1 |An j| + λ N X j=1 |Bn j| ≤ (1 + K₁∆t) N X j=1 ρ n j+1− ρnj + eK2∆t∆x N X j=1 ρ n j+1 + 1 2K1∆t ρn₁ − ρn_a ≤ (1 + K1∆t) N X j=1 ρ n j+1− ρnj + eK2C1(t n_{)∆t +} 1 2K1∆t ρn₁ − ρn_a . (2.20)

We now take into account the boundary terms. From the definition of the scheme we know that ρn+1₁ − ρn+1_a = 1 − αn₁ − β₁n ρn 1 + αn1ρna+ βn1ρn2 − λ F_3/2n (ρn₁, ρn₁) − F_1/2n (ρn₁, ρn₁) − ρn+1_a ± ρn_a = βn₁ ρn₂ − ρn₁ + 1 − αn 1 ρn₁ − ρn_a +ρn_a − ρn+1_a

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− λF_3/2n (ρn₁, ρn₁) − F_1/2n (ρn₁ − ρn₁) = βn₁ ρn₂ − ρn 1 + 1 − γ1n ρn₁ − ρn a + ρn_a− ρn+1 a − λF_3/2n (ρn_a, ρ₁n) − F_1/2n (ρn_a, ρn₁) . Indeed, by definition of αn₁ and of γ₁n we get

αn₁ ρn₁ − ρn_a + λF_3/2n (ρ₁n, ρn₁) − F_1/2n (ρn₁, ρn₁)= = λ F_3/2n (ρn₁, ρn₁) − F_1/2n (ρ_an, ρn₁) ± F_3/2n (ρn_a, ρn₁) = λF n 3/2(ρn1, ρn1) − F3/2n (ρna, ρn1) ρn₁ − ρn a ρn₁ − ρn_a + λF_3/2n (ρ_an, ρn₁) − F_1/2n (ρn_a, ρn₁) = γ₁n ρn₁ − ρn_a + λF_3/2n (ρn_a, ρn₁) − F_1/2n (ρn_a, ρn₁) . Since λ F_3/2n (ρn_a, ρn₁) − F_1/2n (ρn_a, ρn₁) = = λ 2 f (t n_{, x} 1, ρna, Rn1) − f (tn, x0, ρna, Rn0) + λ 2 f (t n_{, x} 2, ρn1, Rn2) − f (tn, x1, ρn1, Rn1) = λ 2 ∂xf (tn, ˜x1/2, ρna, ˜Rn1/2)∆x + ∂Rf (tn, ˜x1/2, ρna, ˜R1/2n ) Rn1 − Rn0 + λ 2 ∂xf (tn, ˜x3/2, ρn1, ˜Rn3/2)∆x + ∂Rf (tn, ˜x3/2, ρn1, ˜Rn3/2) R n 2 − Rn1

(where we used obvious notations for ˜x_1/2, ˜x_3/2, ˜Rn

1/2 and ˜Rn3/2), we can conclude that

λ F n 3/2(ρna, ρn1) − F1/2n (ρna, ρn1) ≤ ∆t C 2 (1 + T ) |ρ n a| + |ρn1| .

Thus, because of the positivity of the coefficients involved, we get ρ n+1 1 − ρn+1a ≤ β n 1 ρn₂ − ρn₁ + 1 − γ₁n ρn₁ − ρn_a + ρ n a− ρn+1a + ∆tC 2 (1 + T ) |ρ n a| + |ρn1| . (2.21)

Concerning the right boundary data, we have ρn+1_N − ρn+1 b = 1 − α n N− βNn ρnN+ αnNρnN −1+ βNnρnb − λ F_{N +1/2}n (ρn_N, ρn_N) − F_{N −1/2}n (ρn_N, ρn_N) − ρn+1_b ± ρn_b = − αn_N ρn_N − ρn_{N −1} + 1 − βn N ρn_N− ρn_b +ρn_b − ρn+1_b + αn_N ρn_N− ρn N −1 − γNn ρnN − ρnN −1 − λF_{N +1/2}n (ρn_{N −1}, ρn_N) − F_{N −1/2}n (ρn_{N −1}, ρn_N) = 1 − β_Nn ρn_N− ρn_b +ρ_bn− ρn+1_b − γ_Nn ρn_N − ρn_{N −1} − λF_{N +1/2}n (ρn_{N −1}, ρn_N) − F_{N −1/2}n (ρn_{N −1}, ρn_N). (2.22)

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We can justify the above equalities as follows. Taking into account of the expressions of α_Nn and of γ_Nn, we can rearrange −λF_{N +1/2}n (ρn_N, ρ_Nn) − F_{N −1/2}n (ρn_N, ρn_N) = = − λ F_{N +1/2}n (ρn_N, ρn_N) − F_{N −1/2}n (ρn_N, ρn_N) ± F_{N −1/2}n (ρn_{N −1}, ρn_N) ± F_{N +1/2}n (ρn_{N −1}, ρn_N) = − λF_{N −1/2}n (ρn_{N −1}, ρn_N) − F_{N −1/2}n (ρn_N, ρn_N)− λF_{N +1/2}n (ρn_N, ρn_N) − F_{N +1/2}n (ρn_{N −1}, ρn_N) − λF_{N +1/2}n (ρn_{N −1}, ρ_Nn) − F_{N −1/2}n (ρn_{N −1}, ρn_N) = αn_N ρn_N − ρn N −1 − γNn ρnN− ρnN −1 − λ F_{N +1/2}n (ρn_{N −1}, ρn_N) − F_{N −1/2}n (ρn_{N −1}, ρn_N).

Let us estimate the last term in (2.22): λ F n N +1/2(ρ n N −1, ρnN) − FN −1/2n (ρ n N −1, ρnN) = = λ 2 f (tn, xN, ρnN −1, RnN) − f (tn, xN −1, ρnN −1, RN −1n ) + f (tn, xN +1, ρnN, RnN +1) − f (tn, xN, ρnN, RnN) = λ 2 ∂xf (t n_{, ˜}_x N −1/2, ρnN −1, ˜RnN −1/2)∆x + ∂Rf (t n_{, ˜}_x N −1/2, ρnN −1, ˜RnN −1/2) R n N − RnN −1 +∂xf (tn, ˜xN +1/2, ρnN, ˜RnN +1/2)∆x + ∂Rf (t n_{, ˜}_x N +1/2, ρnN, ˜RnN +1/2) R n N +1− RnN ≤ ∆tC 2(1 + T ) |ρ n N −1| + |ρnN| . Therefore we get ρ n+1 N − ρ n+1 b ≤ 1 − β n N ρn_N − ρn_b + γ_Nn ρn_N − ρn_{N −1} + ρ n b − ρn+1b + ∆tC 2 (1 + T ) |ρ n N −1| + |ρnN| . (2.23)

Collecting estimates (2.20), (2.21) and (2.23), we obtain

N X j=0 ρ n+1 j+1 − ρ n+1 j = ρ n+1 1 − ρ n+1 a + ρ n+1 b − ρ n+1 N + N −1 X j=1 A n j − λBnj ≤ β₁n ρn₂ − ρn₁ + 1 − γ₁n ρn₁ − ρn_a + ρ n a − ρn+1a + ∆t C 2 (1 + T ) |ρ n a| + |ρn1| + 1 − β_Nn ρn_b − ρn N + γ_Nn ρn_N − ρn N −1 + ρ n b − ρ n+1 b + ∆t C 2 (1 + T ) |ρ n N −1| + |ρnN| + N −1 X j=1 ρ n j+1− ρnj − β n 1 ρn₂ − ρn₁ + γ₁n ρn₁ − ρn_a + β_Nn ρn_b − ρn_N − γ_Nn ρn_N − ρn_{N −1} + λ N −1 X j=1 |B_jn|

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≤ ρ n a− ρn+1a + ρ n b − ρn+1b + N X j=0 ρ n j+1− ρnj + ∆tC 2 (1 + T ) |ρ n a| + |ρn1| + |ρnN −1| + |ρnN| + K1∆t N X j=0 ρ n j+1− ρnj + eK2∆t∆x N X j=1 ρ n j+1 ≤ ρ n a− ρn+1a + ρ n b − ρn+1b + (1 + K1∆t) N X j=0 ρ n j+1− ρnj + eK2C1(t n_)∆t + C (1 + T ) 3 2e Ltn kρ0kL∞_(I)+1 2kρakL∞([0,T ]) ∆t . Setting K2:= eK2C1(tn) + C (1 + T ) 3 2e Ltn kρ0kL∞_(I)+ 1 2kρakL∞[0,T ] , (2.24)

we deduce from the previous estimate

N X j=0 ρ n j+1− ρnj ≤ (1 + K1∆t) tn/∆t N X j=0 ρ 0 j+1− ρ0j + K2 (1 + K1∆t)t n_/∆t − 1 K1 + n X m=1 ρ m a − ρm−1a + n X m=1 ρ m b − ρm−1b ≤ eK1tn N X j=0 ρ 0 j+1− ρ0j + K₂ K1 eK1tn_{− 1}₊ n X m=1 ρ m a − ρm−1a + n X m=1 ρ m b − ρm−1b ,

thus concluding the proof.

Corollary 2 (BV estimate in space and time) Let hypotheses (1.2) and conditions (2.3) hold. If ρ0 ∈ L∞ I; R+

and ρa, ρb ∈ L∞ R+; R+, then ρ∆ satisfies the following Total Variation

estimate in space and time

n−1 X m=0 N X j=0 ∆t ρ m j+1− ρmj + n−1 X m=0 N X j=0 ∆x ρ m+1 j − ρ m j ≤ Cxt(n∆t) , (2.25) with Cxt(n∆t) given by (2.30).

Proof. The spatial BV estimate (2.13) yields

n−1 X m=0 N X j=0 ∆t ρ m j+1− ρmj ≤ n∆t Cx(n∆t) . (2.26)

In order to bound the second term in (2.25), we make use of the definition of the numerical scheme (2.1), (2.2). In fact, by (1.2) and (2.10) we have the following estimate

ρ m+1 j − ρ m j ≤ λα 2 ρ m j − ρmj−1 + ρ m j+1− ρmj

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+λ 2 " 2 ∂xf (t m_{, ˜}_x j, ξjm, ˜Rmj ) ∆x + ∂ρf (tm, ˜xj, ξmj , ˜Rmj ) ρm_j+1− ρm_j−1 + ∂Rf (tm, ˜xj, ξjm, ˜Rmj ) Rm_j+1− R_j−1m # ≤ λ 2(α + L) ρ m j − ρmj−1 + ρ m j+1− ρmj + C∆t|ρm_j+1+ (1 − )ρm_j−1| + T ∆t ρ m j+1+ (1 − )ρmj−1 ≤ λ 2(α + L) ρ m j − ρmj−1 + ρ m j+1− ρmj + ∆t (C + T ) |ρm_j+1| + (1 − )|ρm_j−1| for j = 1, · · · , N − 1, where ξm j = ρmj+1+ (1 − )ρmj−1 ∈ I(ρmj−1, ρmj−1), ∈ [0, 1], ˜xj ∈ [xj−1, xj+1] and ˜Rm_j ∈ I(Rm j−1, Rmj+1). Therefore, N X j=0 ∆x ρ m+1 j − ρ m j = ∆x ρ m+1 a − ρma + ρ m+1 b − ρ m b + N −1 X j=1 ∆x ρ m+1 j − ρ m j ≤ ∆x ρ m+1 a − ρma + ρ m+1 b − ρ m b + ∆t (α + L) N −1 X j=0 ρ m j+1− ρmj + 2∆x∆t (C + T ) N X j=0 |ρm_j | ≤ ∆x ρ m+1 a − ρma + ρ m+1 b − ρ m b + ∆t (α + L) Cx(m∆t) + 2∆x∆t (C + T ) C1(m∆t) , (2.27)

due to estimates (2.13) and (2.8). In particular,

N X j=0 ∆x ρ m+1 j − ρmj ≤ ∆t Ct(m∆t) , where Ct(m∆t) := λ ρ m+1 a − ρma + ρ m+1 b − ρ m b + (α + L) Cx(m∆t) + 2∆x (C + T ) C1(m∆t) , (2.28)

which allows to derive the L1 Lipschitz continuity in time (1.12). Summing over = 0, · · · , n − 1, we get

n−1 X m=0 N X j=0 ∆x ρ m+1 j − ρ m j ≤ ∆x n−1 X m=0 ρ m+1 a − ρma + ρ m+1 b − ρ m b + n∆t (α + L) Cx(n∆t) + 2n∆t∆x (C + T ) C1(n∆t) . (2.29)

Summing (2.26) and (2.29) we get (2.25) with Cxt(n∆t) := n∆t (α + L + 1) Cx(n∆t)

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+ ∆x  2∆t (C + T ) C1(n∆t) + n−1 X m=0 ρ m+1 a − ρma + ρ m+1 b − ρ m b  , (2.30)

thus completing the proof.

2.5 Discrete entropy inequalities

We adopt the following notation F_j+1/2n (u, v) := 1 2 f (tn, xj, u, Rnj) + f (tn, xj+1, v, Rnj+1) + α (u − v) , Hj(u, v, z) := v − λ F_j+1/2n (v, z) − F_j−1/2n (u, v), Gκ_j+1/2:= F_j+1/2n (u ∧ κ, v ∧ κ) − F_j+1/2n (κ, κ) . The approximate solution ρ∆satisfies the following inequalities.

Lemma 4 Under the hypotheses (1.2) and the conditions (2.3), the following discrete entropy inequalities hold for j = 1, · · · , N , n ∈ N, κ ∈ R:

ρn+1_j − κ+−ρn_j − κ++ λ Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j−1/2(ρn_j−1, ρn_j) +λ 2 sgn +_ρn+1 j − κ f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) ≤ 0. (2.31) Proof. Let us consider the map (u, v, z) 7→ Hj(u, v, z). By (2.3), it holds

∂H ∂u(u, v, z) = − λ 2 ∂ρf (tn, xj−1, u, Rnj−1) − α ≥ 0 , ∂H ∂v(u, v, z) = 1 − αλ 2 ≥ 0 , ∂H ∂z (u, v, z) = − λ 2 ∂ρf (tn, xj+1, z, Rnj+1) − α ≥ 0 . Notice that Hj(κ, κ, κ) = κ − λ 2 f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) .

For κ ∈ R, j = 1, · · · , N , noticing that ρn+1j = Hj(ρnj−1, ρnj, ρnj+1) and using the monotonicity

above, we get Hj(ρnj−1∧ κ, ρnj ∧ κ, ρnj+1∧ κ) − Hj(κ, κ, κ) ≥ ≥ Hj(ρnj−1, ρnj, ρnj+1) ∧ Hj(κ, κ, κ) − Hj(κ, κ, κ) = Hj(ρnj−1, ρnj, ρnj+1) − Hj(κ, κ, κ) + = ρn+1_j − κ +λ 2 f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) + . On the other hand we obtain

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=ρn_j − κ+ − λF_j+1/2n (ρn_j ∧ κ, ρn_j+1∧ κ) − F_j−1/2n (ρn_j−1∧ κ, ρn_j ∧ κ) − F_j+1/2n (κ, κ) + F_j−1/2n (κ, κ) = ρn_j − κ+− λGκ_j+1/2(ρ_jn, ρn_j+1) − Gκ_j−1/2(ρn_j−1, ρn_j) . Thus, ρn_j − κ+− λGκ_j+1/2(ρ_jn, ρn_j+1) − Gκ_j−1/2(ρn_j−1, ρn_j) ≥ ≥ ρn+1_j − κ + λ 2 f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) + = sgn+ ρn+1_j − κ + λ 2 f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) × ρn+1_j − κ + λ 2 f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) ≥ sgn+_ρn+1 j − κ ρn+1_j − κ + λ 2 f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rj−1n ) = ρn+1_j − κ++λ 2sgn +_ρn+1 j − κ f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) , which proves (2.31).

2.6 Convergence towards a weak entropy solution

The estimates given by Lemmas 3 and Corollary 2 allow to apply Helly’s compactness theorem, ensuring the existence of a subsequence, still denoted {ρ∆} converging to a function ρ ∈ L∞([0, T ]×

I) in the L1-norm, for all T > 0 (see for example [16, Section 5.3.5]). We need now to prove that the limit of approximate solutions is indeed a weak entropy solution, in the sense of Definition 1. Lemma 5 Let hypotheses (1.2) and conditions (2.3) hold. If ρ0∈ (L∞∩ BV) I; R+ and ρa, ρb∈

(L∞∩ BV) R+_{; R}+_{, then the piecewise constant approximate solutions ρ}

∆ resulting from the

adapted Lax-Friedrichs scheme (2.1) converge, as ∆x & 0, towards a weak entropy solution of the initial boundary value problem (1.1).

Proof. We follow closely [27]. Adding and subtracting Gκ_j+1/2(ρn_j, ρn_j), we rearrange (2.31) as

0 ≥ ρn+1_j − κ+−ρn_j − κ++ λ Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+1/2(ρn_j, ρn_j) + λGκ_j+1/2(ρn_j, ρn_j) − Gκ_j−1/2(ρn_j−1, ρn_j) +λ 2sgn +_ρn+1 j − κ f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) . (2.32)

Let ϕ ∈ C_c1([0, T )×[a, b], R+) for some T > 0. Multiplying (2.32) by ∆x ϕ(tn, xj) ≥ 0, and summing

over j = 1, · · · , N , n ∈ N, we get the inequality 0 ≥ ∆x +∞ X n=0 N X j=1 η+_κ(ρn+1_j ) − η+_κ(ρn_j)ϕ(tn, xj)

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+ ∆t +∞ X n=0 N X j=1 Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+1/2(ρn_j, ρn_j)−Gκ_j−1/2(ρn_j−1, ρn_j) − Gκ_j+1/2(ρn_j, ρn_j) ϕ(tn, xj) +∆t 2 +∞ X n=0 N X j=1 sgn+ρn+1_j − κ f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) ϕ(tn, xj) . (2.33)

Summing by parts in (2.33) we obtain

T1 := ∆x +∞ X n=0 N X j=1 η_κ+(ρn+1) − η+_κ(ρn_j)ϕ(tn, xj) = −∆x N X j=1 ϕ(0, xj)ηκ+(ρ0j) − ∆x∆t +∞ X n=1 N X j=1 η_κ+(ρn_j)ϕ(t n_{, x} j) − ϕ(tn−1, xj) ∆t −→_∆x&0+ − Z b a ϕ(0, x)ρ0(x) dx − Z +∞ 0 Z b a ∂tϕ(t, x)ηκ+(ρ(t, x)) dx dt , and T3:= ∆t 2 +∞ X n=0 N X j=1 sgn± ρn+1_j − κ f (tn, xj+1, κ, Rnj+1) − f (tn, xj−1, κ, Rnj−1) ϕ(tn, xj) = ∆t∆x +∞ X n=0 N X j=1 sgn+ ρn+1_j − κf (t n_{, x} j+1, κ, Rnj+1) − f (tn, xj−1, κ, Rj−1n ) 2∆x −→_∆x&0+ Z +∞ 0 Z b a sgn+ρn+1_j − κ∂xf (t, x, κ, R(t, x))ϕ dx dt ,

by the Dominated Convergence Theorem. Furthermore

∆t +∞ X n=0 N X j=1 Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+1/2(ρn_j, ρn_j) −Gκ_j−1/2(ρn_j−1, ρ_jn) − Gκ_j+1/2(ρn_j, ρn_j) ϕ(tn, xj) = ∆t +∞ X n=0 N X j=1 Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+1/2(ρn_j, ρn_j)ϕ(tn, xj) − ∆t +∞ X n=0 N −1 X j=0 Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+3/2(ρn_j+1, ρn_j+1)ϕ(tn, xj+1) = ∆t +∞ X n=0 N −1 X j=1 Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+1/2(ρn_j, ρn_j)ϕ(tn, xj) −Gκ_j+1/2(ρn_j, ρn_j+1) − G_j+3/2κ (ρn_j+1, ρn_j+1)ϕ(tn, xj+1) + ∆t +∞ X n=0 Gκ_{N +1/2}(ρn_N, ρn_b) − Gκ_{N +1/2}(ρn_N, ρn_N)ϕ(tn, xN) − Gκ_1/2(ρn_a, ρn₁) − Gκ_3/2(ρn₁, ρn₁)ϕ(tn, x1) =: T₂int+ T₂b =: T2. (2.34)

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Let us define T20= − ∆t∆x X n∈N N X j=1 Gκ_j+1/2(ρn_j, ρn_j)ϕ(t n_{, x} j+1) − ϕ(tn, xj) ∆x − α∆tX n∈N ϕ(tn, b)η_κ+(ρn_b) + ϕ(tn, a)η_κ+(ρn_a). Being Gκ_j+1/2(ρn_j, ρn_j) = F_j+1/2n (ρn_j ∧ κ, ρn_j ∧ κ) − F_j+1/2n (κ, κ) = 1 2 f (tn, xj, ρnj ∧ κ, Rjn) − f (tn, xj, κ, Rnj) +1 2 f (tn, xj+1, ρnj ∧ κ, Rj+1n ) − f (tn, xj+1, κ, Rnj+1) = 1 2sgn +_ρn j − κ f (tn, xj, ρnj, Rjn) − f (tn, xj, κ, Rnj) +1 2sgn +_ρn j − κ f (tn, xj+1, ρnj, Rj+1n ) − f (tn, xj+1, κ, Rj+1n ) , it is straightforward to see that

T20−→∆x&0+ − Z +∞ 0 Z b a sgn+(ρ − κ) f (t, x, ρ, R(t, x)) − f (t, x, κ, R(t, x)) ∂xϕ(t, x) dx dt − Lip(f ) Z +∞ 0 ρa(t) − κ + ϕ(t, a) dt − Lip(f ) Z +∞ 0 ρb(t) − κ + ϕ(t, b) dt . We decompose T20 as T20= − ∆t +∞ X n=0 N X j=1 Gκ_j+1/2(ρn_j, ρn_j) ϕ(tn, xj+1) − ϕ(tn, xj) − α∆t +∞ X n=0 ϕ(tn, b)η_κ+(ρn_b) + ϕ(tn, a)η+_κ(ρn_a) = − ∆t +∞ X n=0   N X j=1 Gκ_j+1/2(ρn_j, ρn_j)ϕ(tn, xj+1) − N −1 X j=0 Gκ_j+3/2(ρn_j+1, ρn_j+1)ϕ(tn, xj+1)   − α∆t +∞ X n=0 ϕ(tn, b)η_κ+(ρn_b) + ϕ(tn, a)η+_κ(ρn_a) = − ∆t +∞ X n=0 N −1 X j=1 Gκ_j+1/2(ρn_j, ρn_j) − Gκ_j+3/2(ρn_j+1, ρn_j+1)ϕ(tn, xj+1) − ∆t +∞ X n=0 Gκ_{N +1/2}(ρn_N, ρ_Nn)ϕ(tn, xN +1) − Gκ3/2(ρ n 1, ρn1)ϕ(tn, x1) − α∆t +∞ X n=0 ϕ(tn, b)η_κ+(ρn_b) + ϕ(tn, a)η+_κ(ρn_a) =: T₂₀int+ T₂₀b .

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We rewrite T₂₀int= − ∆t +∞ X n=0 N −1 X j=1 Gκ_j+1/2(ρn_j, ρn_j) ∓ Gκ_j+1/2(ρ_jn, ρn_j+1) − Gκ_j+3/2(ρn_j+1, ρn_j+1) ϕ(tn, xj+1) = ∆t +∞ X n=0 N −1 X j=1 ϕ(tn, xj+1) Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+1/2(ρn_j, ρn_j) − ∆t +∞ X n=0 N −1 X j=1 ϕ(tn, xj+1) Gκ_j+1/2(ρn_j, ρn_j+1) − Gκ_j+3/2(ρn_j+1, ρn_j+1). Hence T int 2 − T20int ≤ ∆t +∞ X n=0 N −1 X j=1 ϕ(tn, x_j) − ϕ(tn, x_j+1) G κ j+1/2(ρ n j, ρnj+1) − Gκj+1/2(ρ n j, ρnj) . Notice that G κ j+1/2(ρ n j, ρnj+1) − Gκj+1/2(ρ n j, ρnj) = = 1 2 f (tn, xj+1, ρnj+1∧ κ, Rnj+1) − f (tn, xj+1, ρnj ∧ κ, Rj+1n ) + α ρn_j ∧ κ − ρn j+1∧ κ ≤ L + α 2 ρ n j+1− ρnj ≤ α ρ n j+1− ρnj ,

according to conditions (2.3). Therefore T int 2 − T20int ≤ α∆x∆tkϕxkL∞ +∞ X n=0 N −1 X j=1 ρ n j+1− ρnj ≤ α∆x T kϕxkL∞ max 0≤n≤T /∆tTV(ρ∆(t n_{, ·)) = O(∆x) ,}

thanks to the uniform BV bound (2.13). We now compare the terms T₂b and T₂₀b :

T₂₀b − Tb 2 = − ∆t +∞ X n=0 Gκ_{N +1/2}(ρn_N, ρn_N)ϕ(tn, xN +1) − Gκ3/2(ρn1, ρn1)ϕ(tn, x1) − α∆t +∞ X n=0 ϕ(tn, b)η+_κ(ρn_b) + ϕ(tn, a)η_κ+(ρn_a) − ∆t +∞ X n=0 Gκ_{N +1/2}(ρn_N, ρn_b) − Gκ_{N +1/2}(ρn_N, ρn_N) ϕ(tn, xN) + ∆t +∞ X n=0 Gκ_1/2(ρn_a, ρn₁) − Gκ_3/2(ρn₁, ρn₁)ϕ(tn, x1) = ∆t +∞ X n=0 Gκ_1/2(ρn_a, ρn₁)ϕ(tn, x1) − αηκ+(ρna)ϕ(tn, a)

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− ∆t +∞ X n=0 αη+_κ(ρn_b)ϕ(tn, b) + Gκ_{N +1/2}(ρn_N, ρn_b)ϕ(tn, xN) − ∆t +∞ X n=0 Gκ_{N +1/2}(ρn_N, ρn_N) ϕ(tn, xN +1) − ϕ(tn, xN) = ∆t +∞ X n=0 Gκ_1/2(ρn_a, ρn₁) − αη+_κ(ρn_a)ϕ(tn, a) −αη_κ+(ρn_b) + Gκ_{N +1/2}(ρn_N, ρn_b)ϕ(tn, b) + O(∆x) ,

owing to the regularity of ϕ. Indeed, we observe that

∆t X n∈N Gκ_{N +1/2}(ρn_N, ρn_N) ϕ(tn, xN +1) − ϕ(tn, xN) ≤ ∆t∆xkϕ_xk_L∞ X n∈N G κ N +1/2(ρ n N, ρnN) = ∆t∆xkϕxkL∞ X n∈N F n N +1/2(ρ n N ∧ κ, ρnN ∧ κ) − FN +1/2n (κ, κ) ≤ ∆t∆xkϕxkL∞ 2 X n∈N f (tn, x_N, ρn_N∧ κ, Rn_N) − f (tn, x_N, κ, Rn_N) +∆t∆xkϕxkL∞ 2 X n∈N f (tn, x_{N +1}, ρn_N∧ κ, Rn_{N +1}) − f (tn, x_{N +1}, κ, R_{N +1}n ) ≤ L∆t∆xkϕ_xk_L∞ X n∈N ρn_N − κ+ ≤ LT kϕ_xk_L∞eLTkρ₀k_L∞_(I)∆x = O(∆x) ,

thanks to the L∞-bound (2.9). Moreover, since

Gκ_j+1/2(u, v) = F_j+1/2n (u ∧ κ, v ∧ κ) − F_j+1/2n (κ, κ) ≥ F_j+1/2n (κ, v ∧ κ) − F_j+1/2n (κ, κ) = 1 2 f (tn, xj+1, v ∧ κ, Rnj+1) − f (tn, xj+1, κ, Rj+1n ) − α (v ∧ κ − κ) ≥ −1 2 L|v ∧ κ − κ| + α (v − κ)+ = −1 2 L (v − κ)++ α (v − κ)+ ≥ −α (v − κ)+ and Gκ_j+1/2(u, v) = F_j+1/2n (u ∧ κ, v ∧ κ) − F_j+1/2n (κ, κ) ≤ Fn j+1/2(u ∧ κ, κ) − Fj+1/2n (κ, κ) = 1 2 f (tn, xj, u ∧ κ, Rnj) − f (tn, xj, κ, Rnj) + α (u ∧ κ − κ) ≤ 1 2 L|u ∧ κ − κ| + α (u − κ)+

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= 1 2 L (u − κ)++ α (u − κ)+ ≤ α (u − κ)+, we conclude that T₂₀b − Tb 2 ≤ O(∆x) . Therefore we get 0 ≥ T1+ T3+ T2 = T1+ T3+ T2int+ T2b ≥ T₁+ T3+ T2int+ T20b − O(∆x) = T1+ T3+ T20− O(∆x) ,

thus concluding the proof.

3 Stability

Proposition 2 Under hypotheses (1.2), let ρ, σ ∈ C0 R+; L1(I; R+) ∩ BV [0, T ] × I; R, T > 0, be two weak entropy solutions to (1.1), with initial data ρ0, σ0 ∈ L∞ I, R+ and boundary data

ρa, ρb, σa, σb∈ L∞ R+; R+ respectively. Then the following estimate holds:

kρ(T, ·) − σ(T, ·)k_L1_(I) ≤ eST kρ₀− σ₀k_L1_(I)+ (L + S0) kρ_a− σ_ak_L1_{([0,T ])}+ kρ_b− σ_bk_L1_{([0,T ])} , (3.1) where the constants S, S0 are defined by (3.6), (3.7) and (3.10), and L is as in (2.28).

Proof. Let ρ, σ be two weak entropy solutions to (1.1), with fluxes f (t, x, ρ, R(t, x)) and f (t, x, σ, S(t, x)) respectively, where R(t, x) := Z R ρ(t, y)η(x − y) dy and S(t, x) := Z R σ(t, y)η(x − y) dy . In particular from Definition 2, they satisfy

∂t|ρ − κ| + d dx h sgn (ρ − κ) f (t, x, ρ, R(t, x)) − f (t, x, κ, R(t, x)) i + sgn (ρ − κ) d dxf (t, x, κ, R(t, x)) ≤ 0, (3.2) ∂t|σ − κ| + d dx h sgn (σ − κ) f (t, x, σ, S(t, x)) − f (t, x, κ, S(t, x))i + sgn (σ − κ) d dxf (t, x, κ, S(t, x)) ≤ 0, (3.3)

in distributional sense on R+× I. Rearranging (3.2) we get

0 ≥ ∂t|ρ − κ| + sgn (ρ − κ)

d

dx f (t, x, κ, R(t, x)) ± f (t, x, κ, S(t, x))

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+ d dx h sgn (ρ − κ) f (t, x, ρ, R(t, x)) − f (t, x, κ, R(t, x)) ± f (t, x, ρ, S(t, x)) ± f (t, x, κ, S(t, x)) i = ∂t|ρ − κ| + d dx h sgn (ρ − κ) f (t, x, ρ, S(t, x)) − f (t, x, κ, S(t, x))i + sgn (ρ − κ) d dxf (t, x, κ, S(t, x)) + d dx sgn (ρ − κ) f (t, x, ρ, R(t, x)) − f (t, x, ρ, S(t, x)) − f (t, x, κ, R(t, x)) − f (t, x, κ, S(t, x)) + sgn (ρ − κ) d dxf (t, x, κ, R(t, x)) − f (t, x, κ, S(t, x)) , thus ∂t|ρ − κ| + d dx h sgn (ρ − κ) f (t, x, ρ, S(t, x)) − f (t, x, κ, S(t, x)) i + sgn (ρ − κ) d dxf (t, x, κ, S(t, x)) ≤ d dx sgn (ρ − κ) h f (t, x, ρ, S(t, x)) − f (t, x, ρ, R(t, x)) − f (t, x, κ, S(t, x)) − f (t, x, κ, R(t, x))i + sgn (ρ − κ) d dxf (t, x, κ, S(t, x)) − f (t, x, κ, R(t, x)) . (3.4) Notice that from (1.5) we can bound

R(t, x) − S(t, x) ≤ kηk_L∞ Z b a ρ(t, x) − σ(t, x) dx + ρ_a(t) − σ_a(t) + ρ_b(t) − σ_b(t) , ∂xR(t, x) − ∂xS(t, x) ≤ k∂_xηk_L∞ Z b a ρ(t, x) − σ(t, x)dx + kηk_L∞ ρa(t) − σa(t) + ρb(t) − σb(t) . Therefore we recover the following estimate:

d dx f (t, x, κ, S(t, x)) − f (t, x, κ, R(t, x)) ≤ ≤∂_xf (t, x, κ, S(t, x)) − ∂_xf (t, x, κ, R(t, x)) +∂Rf (t, x, κ, S(t, x)) ∂xS(t, x) − ∂Rf (t, x, κ, R(t, x)) ∂xR(t, x) ± ∂Rf (t, x, κ, S(t, x)) ∂xR(t, x) ≤ ∂ 2 xRf (t, x, κ, ˜R1(t, x)) S(t, x) − R(t, x) +∂Rf (t, x, κ, S(t, x)) ∂xS(t, x) − ∂xR(t, x) +∂Rf (t, x, κ, S(t, x)) − ∂Rf (t, x, κ, R(t, x)) ∂xR(t, x) ≤ ∂ 2 xRf (t, x, κ, ˜R1(t, x)) + ∂ 2 RRf (t, x, κ, ˜R2(t, x)) ∂_xR(t, x) S(t, x) − R(t, x) +∂_Rf (t, x, κ, S(t, x)) ∂_xS(t, x) − ∂_xR(t, x) ≤ C|κ|1 +∂xR(t, x) S(t, x) − R(t, x)+ C|κ| ∂xS(t, x) − ∂xR(t, x) = C|κ| 1 + Z R ∂xη(x − y)ρ(t, y)dy ! S(t, x) − R(t, x) + C|κ| ∂_xS(t, x) − ∂_xR(t, x) ≤ C|κ|1 + ρ(t, ·) L∞k∂xηkL1 S(t, x) − R(t, x)+ C|κ| ∂xS(t, x) − ∂xR(t, x) ≤ C|κ|1 + ρ(t, ·) L∞k∂xηkL1 " kηk_L∞ Z b a ρ(t, x) − σ(t, x)dx +ρa(t) − σa(t) + ρb(t) − σb(t) # + C|κ| " k∂_xηk_L∞ Z b a ρ(t, x) − σ(t, x)dx + kηk_L∞ ρa(t) − σa(t) + ρb(t) − σb(t) #

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≤ C|κ| kηk_L∞ 1 + ρ(t, ·) L∞k∂xηkL1 + k∂xηkL∞ Z b a ρ(t, x) − σ(t, x)dx + C|κ|1 + ρ(t, ·) L∞k∂xηkL1+ kηk_L∞ ρa(t) − σa(t) + ρb(t) − σb(t) ≤ S1 Z b a ρ(t, y) − σ(t, y) dy + S₁0 ρ_a(t) − σ_a(t) + ρ_b(t) − σ_b(t) , (3.5) where S₁= C sup t∈[0,T ] ( maxn ρ(t, ·) L∞, σ(t, ·) L∞ o kηk_L∞ 1 + ρ(t, ·) L∞k∂xηkL1 + k∂xηkL∞ ) , (3.6) S₁0 = C sup t∈[0,T ] maxn ρ(t, ·) L∞, σ(t, ·) L∞ o 1 + ρ(t, ·) L∞k∂xηkL1 + kηk_L∞ , (3.7)

which are bounded by assumption. Moreover d dx sgn (ρ − κ) h f (t, x, ρ, S(t, x)) − f (t, x, ρ, R(t, x)) − f (t, x, κ, S(t, x)) − f (t, x, κ, R(t, x))i = ∂xf (t, x, ρ, S(t, x)) − ∂xf (t, x, ρ, R(t, x)) + ∂xf (t, x, κ, R(t, x)) − ∂xf (t, x, κ, S(t, x)) + ∂Rf (t, x, ρ, S(t, x)) ∂xS(t, x) − ∂xR(t, x) + ∂xR(t, x) ∂xf (t, x, ρ, S(t, x)) − ∂xf (t, x, ρ, R(t, x)) + ∂Rf (t, x, κ, S(t, x)) ∂xR(t, x) − ∂xS(t, x) + ∂xR(t, x) ∂xf (t, x, κ, R(t, x)) − ∂xf (t, x, κ, S(t, x)) o = ∂ 2 xRf (t, x, ρ, ˜R1) S(t, x) − R(t, x) + ∂xR2 f (t, x, κ, ˜R2) R(t, x) − S(t, x) + ∂Rf (t, x, ρ, S(t, x)) ∂xS(t, x) − ∂xR(t, x) + ∂xR(t, x) ∂xR2 f (t, x, ρ, ˜R3) S(t, x) − R(t, x) + ∂Rf (t, x, κ, S(t, x)) ∂xR(t, x) − ∂xS(t, x) +∂xR(t, x) ∂xR2 f (t, x, κ, ˜R4) R(t, x) − S(t, x) ≤ C kρ(t, ·)k_L∞+ |κ| 1 + kρ(t, ·)k_L∞k∂_xηk_L1 R(t, x) − S(t, x) + C kρ(t, ·)kL∞+ |κ| ∂_xR(t, x) − ∂_xS(t, x) = S2 Z b a ρ(t, y) − σ(t, y) dy + S₂0 ρa(t) − σa(t) + ρb(t) − σb(t) , (3.8) being S2 = 2S1 and S20 = 2S10.

Inserting estimates (3.5) and (3.8) in (3.4) we get ∂t|ρ − κ| + d dx h sgn (ρ − κ) f (t, x, ρ, S(t, x)) − f (t, x, κ, S(t, x))i+ sgn (ρ − κ) d dxf (t, x, κ, S(t, x)) ≤ S Z b a ρ(t, y) − σ(t, y) dy + S0 ρa(t) − σa(t) + ρb(t) − σb(t) , (3.9) with S = S₁+ S2 = 3S1 and S0 = S10 + S 0 2 = 3S 0 1. (3.10)

Following [6, Theorem 2] and [25, Theorem 15.1.5], we apply the standard Kruˇzhkov doubling of variable technique [20, Section 3] to (3.9) and (3.3), with a test function ϕ ∈ C_c1(R × I; R+). We obtain the following Kato inequality

Z b a ρ0(x) − σ0(x) ϕ(0, x) dx

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+ Z +∞ 0 Z b a ρ(t, x) − σ(t, x)∂tϕ + sgn (ρ − σ)f (t, x, ρ, S(t, x)) − f (t, x, σ, S(t, x)) ∂xϕ dx dt + Skϕk_L∞ Z +∞ 0 Z b a ρ(t, x) − σ(t, x)dx ! dt + S0kϕk_L∞ Z +∞ 0 ρa(t) − σa(t) + ρb(t) − σb(t) dt ≥ 0 . (3.11)

We now consider in (3.11) a test function of the form ϕ(t, x) = ψ(t)θδ(x), where θδ ∈ C1([a, b]) be

such that

θδ(a) = 0, θδ(b) = 0,

kθ_δ0k∞≤ K/δ,

θδ≡ 1 on [a + δ, b − δ],

0 ≤ θδ(x) ≤ 1 for all x ∈ [a, b],

where C does not depend on δ, and ψ ∈ C_c1([0, T [). In this case, (3.11) becomes Z b a ρ0(x) − σ0(x) ψ(0)θδ(x) dx + Z +∞ 0 Z b a ρ(t, x) − σ(t, x) θ_δ(x)ψ0(t) +ψ(t)θ0_δ(x) sgn (ρ − σ)f (t, x, ρ, S(t, x)) − f (t, x, σ, S(t, x))dx dt + Skϕk_L∞ Z T 0 Z b a ρ(t, x) − σ(t, x) dx ! dt + S0kϕk_L∞ Z T 0 ρ_a(t) − σ_a(t) + ρ_b(t) − σ_b(t) dt ≥ 0 . (3.12)

Integrating by parts in (3.12) and letting δ & 0 we obtain Z b a ρ₀(x) − σ₀(x) ψ(0) dx + Z +∞ 0 Z b a ρ(t, x) − σ(t, x)ψ0(t) dx dt + Skϕk_L∞ Z T 0 Z b a ρ(t, x) − σ(t, x)dx ! dt + S0kϕk_L∞ Z T 0 ρa(t) − σa(t) + ρb(t) − σb(t) dt + Z +∞ 0 ψ(t) sgnρ(t, a+) − σ(t, a+) hf (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a))i − sgnρ(t, b−) − σ(t, b−) hf (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b))i dt ≥ 0 . (3.13) From the weak boundary conditions (1.7) and (1.8), we earn

sgn ρ(t, a+) − σ(t, a+) h f (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a)) i = 1 2sgn ρ(t, a+) − σ(t, a+) h f (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a)) i

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+ 1 2sgn ρ(t, a+) − σ(t, a+) h f (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a)) i ≤ 1 2sgn ρa(t) − σ(t, a+) h f (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a)) i + 1 2sgn ρ(t, a+) − σa(t) h f (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a))i = 1 2 sgnρa(t) − σ(t, a+) + sgnρ(t, a+) − σa(t) h f (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a))i ≤ sup s,r∈I(ρa(t),σa(t)) f (t, a, s, S(t, a)) − f (t, a, r, S(t, a)) ≤ sup s,r∈I(ρa(t),σa(t)) L|s − r| ≤ Lρ_a(t) − σ_a(t) , (3.14) and sgn ρ(t, b−) − σ(t, b−) h f (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b)) i = 1 2sgn ρ(t, b−) − σ(t, b−) h f (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b)) i + 1 2sgn ρ(t, b−) − σ(t, b−) h f (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b)) i ≥ 1 2sgn ρb(t) − σ(t, b−) h f (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b)) i + 1 2sgn ρ(t, b−) − σb(t) h f (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b))i = 1 2 sgnρb(t) − σ(t, b−) + sgnρ(t, b−) − σb(t) h f (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b))i ≥ − sup s,r∈I(ρb(t),σb(t)) f (t, b, s, S(t, b)) − f (t, b, r, S(t, b)) ≥ − sup s,r∈I(ρb(t),σb(t)) L|s − r| ≥ − Lρb(t) − σb(t) . (3.15)

Collecting (3.14) and (3.15) we conclude that Z +∞ 0 ψ(t) sgn ρ(t, a+) − σ(t, a+) h f (t, a, ρ(t, a+), S(t, a)) − f (t, a, σ(t, a+), S(t, a)) i − sgnρ(t, b−) − σ(t, b−) h f (t, b, ρ(t, b−), S(t, b)) − f (t, b, σ(t, b−), S(t, b)) i dt ≤ L Z +∞ 0 ψ(t)ρa(t) − σa(t) + ρb(t) − σb(t) dt. Thus (3.13) becomes Z b a ρ₀(x) − σ₀(x) ψ(0) dx + Z +∞ 0 Z b a ρ(t, x) − σ(t, x)ψ0(t) dx dt + Skϕk_L∞ Z T 0 Z b a ρ(t, x) − σ(t, x)dx ! dt

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+ S0kϕk_L∞ Z T 0 ρ_a(t) − σ_a(t) + ρ_b(t) − σ_b(t) dt + L Z +∞ 0 ψ(t)ρa(t) − σa(t) + ρb(t) − σb(t) dt ≥ 0 . We choose the test function ψ = ψ as

ψ(t) =        1 if t ∈ [0, T − [ ψ(t) ∈ [0, 1] for all t ∈ [0, T ] |ψ0 (t)| ≤ K/ for all t ∈ [0, T ] . As & 0, we get kρ(T, ·) − σ(T, ·)k_L1_(I) ≤ kρ₀− σ₀k_L1_(I)+ L kρb− σbkL1_{([0,T ])}+ kρ_a− σ_ak_L1_{([0,T ])} +S Z T 0 kρ(t, ·) − σ(t, ·)k_L1_(I)dt + S0 Z T 0 ρa(t) − σa(t) + ρb(t) − σb(t) dt ,

and Gronwall’s lemma allows us to recover (3.1).

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