Well-posedness of general 1D Initial Boundary Value Problems for scalar balance laws

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax-Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kru\v{z}kov's doubling of variables technique, together with a careful treatment of the boundary terms.


Introduction
Consider the following general Initial-Boundary Value Problem (IBVP) for a one dimensional scalar balance law on the bounded interval ]a, b[⊂ R x, u) = g(t, x, u), (t, x) ∈ I×]a, b[, u(0, x) = u o (x), x ∈ ]a, b[, u(t, a) = u a (t), t ∈ I, u(t, b) = u b (t), t ∈ I, (1.1) where I =]0, T [ for a positive T and we introduce the notation Aim of the present work is to prove the well-posedness of (1.1) and the stability of its solutions with respect to variations in the flux and in the source functions.
IBVPs for balance laws in several space dimensions were originally studied by Bardos, le Roux and Nédélec [1]. However, the existence and uniqueness result proved in [1] is limited to rather smooth initial data, namely functions of class C 2 , while the boundary datum is assumed to be zero. An extension of this result to more general, although smooth, boundary data is carried out in [3]. Other contributions in the field are due to [12], see also [10,Chapter 2], and more recently [14] and [11]. In particular, in this latter article, the author proves the well-posedness of an IBVP for a multi-dimensional balance law with L ∞ data. However, a restrictive hypothesis on the flux and the source functions is needed, in order to get a maximum principle on the solution.
In all the above cited references, the vanishing viscosity technique is used to get the existence of solutions. Here, existence is obtained by proving the convergence of a Lax-Friedrichs type numerical scheme, together with operator splitting to account for the source term. The idea of the proof comes from [7,14]. It is remarkable how the L ∞ and total variation estimates on the solution obtained in the present work (see Theorem 2.4) are more accurate with respect to those presented in [3], allowing moreover for less regular data.
As far as it concerns the uniqueness, the Lipschitz continuous dependence on initial and boundary data of solutions to general multiD IBVP, proved in [3,Theorem 4.3], applies to the present setting. Indeed, this result is valid in a generality wider than that assumed to prove existence of solutions in [3]: its proof follows directly form the definition of solution, which requires initial and boundary data of class L ∞ ∩ BV.
The investigation of stability results for IBVPs for balance laws has begun only recently. At the present time, only partial results, namely considering particular classes of equations, are available. For instance, in the multi-dimensional case, [5] presents a stability estimate for a class of multiD linear conservation laws in a bounded domain, with homogeneous boundary conditions and initial data of class L ∞ ∩ BV. In one space dimension, only conservation laws with a flux function not explicitly dependent on the space variable x have been considered, see [4,6]. Our result is more general: a stability estimate for 1D IBVPs for balance laws with general flux and source functions. An adaptation of the doubling of variables technique by Kružkov, together with a careful treatment of the boundary terms, allows to obtain the desired result.
The paper is organised as follows. Section 2 presents the assumptions needed throughout the paper, the definitions of solution to problem (1.1) and the main analytic results, namely the well-posedness of the problem and the stability estimate. Section 3 is devoted to the introduction and analysis of the numerical scheme. The estimates necessary to prove its convergence, as well as the convergence result, constitute the contribution of this section. Finally, Section 4 contains the proofs of the main results. Following [11,14], we set In the rest of the paper, we will denote I(r, s) = [min {r, s} , max {r, s}], for any r, s ∈ R.
We introduce the constants Concerning the definition of solution to problem (1.1), we refer below to the following extension of [14,Definition 1] presented in [11] for the multi dimensional case.
We introduce a second definition of solution to problem (1.1). This is an adaptation of the definition of solution presented in [1] to the one dimensional case, where the domain is an open bounded interval.
We remark that, for functions in (L ∞ ∩ BV)([0, T [×]a, b[; R), Definition 2.1 and Definition 2.2 are equivalent, see [13] for further details. Remark 2.3. In Definition 2.2, to ensure that the traces of u at the boundary u(t, a + ) and u(t, b + ) are well defined, we need the solution to be of bounded total variation. Moreover, we recall the well-known BLN-boundary conditions ([13, Lemma 5.6]), linking the boundary data to the traces of the solution at the boundary: • at x = a: for all k ∈ R, for a.e. t ∈ ]0, T [ or, equivalently, for all k ∈ I(u(t, a + ), u a (t)) and a.e. t ∈ ]0, T [ The following Theorem contains the existence and uniqueness result, as well as some a priori estimates on the solution to the IBVP (1.1).
• The functions C 1 (t) and C 2 (t) are clearly strictly related to the source term and to the space dependent flux. Consider, for instance, problem (1.1) with u o = 0 and u b = 0, f (t, x, u) = −x and g = 0. The solution is u(t, x) = t, and from (3.8) we obtain C 1 (t) = 1, so that (2.4) now reads u(t) L ∞ (]a,b[) ≤ t.
• Compare (2.4) with the L ∞ estimate on the solution presented in [3,Formula (2.5)]. At a glance, one could notice that the present estimate is more accurate. Moreover, here the boundary data should be of class L ∞ ∩ BV, thus they are not required to be as regular as in [3,Theorem 2.7].
Consider, for instance, the following example: and, as boundary datum in x = a, an increasing smooth function (at least of class C 3 , as required by [3,Theorem 2.7]) such that u a (0) = 0, u a (t) = 0.4 for t > 0.1. Now compare the two L ∞ estimates at a time t > 0.1. As already observed in the first point of this remark, C 1 (t) = C 2 (t) = 0, and the estimate (2.4) reads On the other hand, with the notation of [3], where all the norms on the right hand side of the inequality are evaluated on [0, t].
The following Theorem presents a stability estimate with respect to the flux and the source functions. A particular case of the IBVP (1.1) is considered, for instance, in [6], where a flux function independent of the space variable is taken into account. There, a stability estimate with respect to the flux function is provided.
Theorem 2.6. Let f 1 , f 2 satisfy (f ), g 1 , g 2 satisfy (g), (u o , u a , u b ) satisfy (D). Call u 1 and u 2 the corresponding solutions to the IBVP (1.1). Then, for all t ∈ [0, T [, the following estimate holds The proof is postponed to Section 4.

Existence of weak entropy solutions
Consider a space step ∆x such that b − a = N ∆x, N ∈ N, and a time step ∆t subject to a CFL condition which will be specified later. For j = 1, . . . , N , introduce the following notation and let x j+1/2 = a + j∆x = a + y j+1/2 be the cells interfaces, for j = 0, . . . , N , and x j = a + (j − 1/2)∆x = a + y j the cells centres, for j = 1, . . . , N . Moreover, set N T = ⌊T /∆t⌋ and, for n = 0, . . . , N T , let t n = n∆t be the time mesh. Set λ = ∆t/∆x and let α ≥ 1 be the viscosity coefficient. Approximate the initial datum u o and the boundary data as follows: Introduce moreover the notation u n 0 = u n a and u n N +1 = u n b . We define a piecewise constant approximate solution u ∆ to (1.1) as through a Lax-Friedrichs type scheme together with operator splitting, in order to treat the source term.
In particular, the algorithm is defined as follows: for n = 0, . . . N T − 1 for j = 1, . . . , N F n j+1/2 (u n j , u n j+1 ) = We require, moreover, the following CFL condition: The proof of the convergence of approximate solutions consists of several steps, whose aim is to show that the sequence verifies the hypotheses of Helly's compactness theorem.

L ∞ bound
where Proof. Fix j between 1 and N , n between 0 and N T − 1, and rewrite (3.3) as follows: (3.12) Using the explicit expression of the numerical flux (3.2) and the hypotheses on f (f ), we observe that, whenever u n j = u n j−1 , with r n j−1/2 ∈ I u n j−1 , u n j . Similarly, whenever u n j = u n j+1 , with r n j+1/2 ∈ I u n j , u n j+1 . By (3.5) we get β n j , γ n j ∈ 0, +∞ for all t ∈ I. Inserting the above estimate into (3.10) and exploiting the bounds (3.13) on β n j and γ n j , we get .
An iterative argument yields the thesis.

BV estimates
where Proof. For the sake of simplicity, introduce the space with the notation introduced in (3.7). By the definition of the scheme (3.4), observe that, for all j = 1, . . . , N − 1, where ∂ x g L ∞ (Σn) is bounded, thanks to (g). On the other hand, for j = 0 and j = N we have respectively Therefore, .

Focus first on
where we set Rearrange A n j as follows: while γ n j is as in (3.12). It can be proven that δ n j ∈ 0, 1/3 . Thus, Focus now on B n j : Notice that Hence, .

(3.19)
Therefore, collecting together the estimates (3.18) and (3.19) we get: . (3.20) Focus now on the terms involving the boundary data in (3.16). With the notation (3.11), (3.12) and (3.17), we observe that Hence, from the definition of the scheme (3.3),we have [,ũ n a ∈ I(0, u n a ) and u n 1 ∈ I(0, u n 1 ), we conclude By the positivity of the coefficients involved in (3.21), we obtain Similarly as before, compute Therefore, concerning the other boundary term, we have Observing that By the positivity of the coefficients involved in (3.23)-(3.24), we obtain Insert (3.20), (3.22) and (3.25) into (3.16): where .
and C 1 (t) as in (3.8), we deduce by a standard iterative procedure the following estimate where actually the norms appearing in C 2 (t) in (3.9) can now be computed on Σ t instead of Then, for n between 1 and N T , the following estimate holds where C xt (t n ) is given by (3.35). In particular, the first term on the r.h.s. of (3.30) can be estimated as follows

Proof. By Proposition 3.3, we have
and the norm of g appearing above is bounded thanks to (g). Concerning the second term on the r.h.s. of (3.30), by (3.2) and (3.3), we obtain ). Remark that, by (f ), the norm of ∂ x f appearing above is bounded. Therefore, by Proposition 3.3, where we set which, summed over m = 0, . . . , n − 1, yields concluding the proof.
Proof. Consider the map (u, v, z) → H n j (u, v, z). By the CFL condition (3.5), it holds By the monotonicity properties obtained above, we have H n j (u n j−1 ∧ k, u n j ∧ k, u n j+1 ∧ k) − H n j (k, k, k) ≥ H n j (u n j−1 , u n j , u n j+1 ) ∧ H n j (k, k, k) − H n j (k, k, k) = H n j (u n j−1 , u n j , u n j+1 ) − H n j (k, k, k) Moreover, we also have Hence, by (3.38) and (3.39), proving (3.36), while (3.37) is proven in an entirely similar way.

Convergence towards an entropy weak solution
The uniform L ∞ -bound provided by Lemma 3.2 and the total variation estimate in Corollary 3.4 allow to apply Helly's compactness theorem, ensuring the existence of a subsequence of u ∆ , still denoted by u ∆ , converging in We need to prove that this limit function is indeed an MV-solution to the IBVP (1.1), in the sense of Definition 2.1. Proof. We consider the discrete entropy inequality (3.36), for the positive semi-entropy, and we follow [7], see also [14]. The proof for the negative semi-entropy is done analogously. Add and subtract G n,k j+1/2 (u n j , u n j ) in (3.36) and rearrange it as follows 0 ≥ (u n+1 j − k) + − (u n j − k) + + λ G n,k j+1/2 (u n j , u n j+1 ) − G n,k j+1/2 (u n j , u n j ) + λ G n,k j+1/2 (u n j , u n j ) − G n,k j−1/2 (u n j−1 , u n j ) − ∆t sgn + (u n+1 j − k) g t n , x j , u n+1/2 j ; R + ) for some T > 0, multiply the inequality above by ∆x ϕ(t n , x j ) and sum over j = 1, . . . , N and n = 0, . . . , N T − 1, so to get (3.44) Consider each term separately. Summing by parts, we obtain G n,k j+1/2 (u n j , u n j+1 ) − G n,k j+3/2 (u n j+1 , u n j+1 ) ϕ(t n , x j+1 ) = ∆t where we set (u n a − k) + ϕ(t n , a) + (u n b − k) + ϕ(t n , b) .

(3.45)
Observe that It follows then easily that Let us rewrite S in (3.45) as follows: where we set Focus on S int : by adding and subtracting G n,k j+1/2 (u n j , u n j ) in the first brackets, we can rewrite this term as We evaluate now the distance between T int and S int : Observe that G n,k j+1/2 (u n j , u n j+1 ) − G n,k j+1/2 (u n j , u n j ) Therefore, thanks to the uniform BV estimate (3.14). Pass now to the terms T b and S b : meaning that the numerical flux is increasing with respect to the first variable and decreasing with respect to the second one. Thus, Hence,

Uniqueness
The uniqueness of the solution to the IBVP (1.1) follows from the Lipschitz continuous dependence of the solution on initial and boundary data, proved for the multidimensional case in [3,Theorem 4.3]. Then, for all t > 0, the following estimate holds where L f (t) and L g (t) are defined in (2.1).

Proofs of Theorem 2.and Theorem 2.6
Proof of Theorem 2.4. The existence of a unique solution to the IBVP (1.1) is ensured by the results presented in Section 3, see in particular § 3.4 and § 3.5.
The estimates on the solution to the IBVP (1.1) are obtained by passing to the limit in the corresponding discrete estimates, namely (3.6) for the L ∞ -bound, (3.14) for the bound on the total variation, and (3.30)-(3.32) for the Lipschitz continuity in time.