Signed Radon measure-valued solutions of flux saturated scalar conservation laws

We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.


Introduction
We study the Cauchy problem for the scalar conservation law: where (A 0 ) H ∈ W 1,∞ (R) , H(0) = 0 (obviously the condition H(0) = 0 is not restrictive). The initial condition u 0 is a signed Radon measure on R. In most of the paper we shall assume that its singular part, u 0s , is a finite superposition of Dirac masses: In that case we denote the support of the singular measure u 0s by F : In [3] we considered the case of nonnegative initial measures u 0 . In the present paper we consider the case of signed measures (see [2,5,7,9] for motivations and related remarks). A specific motivation is the link between measure-valued solutions of (CL) and discontinuous solutions of the Cauchy problem for the Hamilton-Jacobi equation 0 is a Radon measure without singular continuous part, ac , and problems (CL), (HJ) are formally related by the equality u = U x . In a forthcoming paper [4], problem (HJ) will be studied in the context of viscosity solutions.
It is known ( [2]) that (i) the singular part u s of a suitably defined entropy solution may persist for some positive time (see [2,Theorem 3.5]) and (ii) entropy solutions are not always uniquely determined by the initial condition u 0 (see also Remark 3.2). To overcome the latter problem, we introduced in [3] a so-called compatibility condition at those points where u s (⋅, t) is a Dirac mass, and used it as a uniqueness criterion for nonnegative measure-valued solutions.
The starting point of the present paper is the statement that, for general signed initial measures u 0 , the singular part u s of any local entropy solution u of (CL) (in the sense of Definition 3.2) satisfies a monotonicity result: both the positive and negative part of u s , [u(⋅, t)] ± s , are nonincreasing with respect to t (see Theorem 3.1). For the class of initial measures satisfying (A 1 ) this implies that the support of the singular part u s of an entropy solution of problem (CL) is a subset of F × [0, T ] and, in addition, that the sign of [u(⋅, t)] s is determined by that of u 0s . Having this in mind it is rather straightforward to adapt the concept of compatibility condition in [3] to signed measure-valued solutions (see Definition 3.3).
The main result of the paper is that if (A 0 ) and (A 1 ) are satisfied, then (CL) is well-posed in the class of entropy solutions which satisfy the compatibility condition at the p points x j ∈ F .
Existence of a solution is proven by a constructive approach which can be outlined as follows. By (A 0 )-(A 1 ) there exists a positive time τ until which all singularities persist (see [2,Theorem 3.5]), thus the real line is the disjoint union of p+1 intervals. In each interval we solve the initial-boundary value problem for the conservation law in (CL), the initial data being the restriction of u 0r to that interval, with "boundary conditions equal to infinity". Namely, we consider the singular Dirichlet initial-boundary value problems with j = 2, . . . , p, and The choice between u = ∞ and u = −∞ at x j is determined by the sign of c j : we choose ∞ if c j > 0 and −∞ if c j < 0. Existence and uniqueness of an entropy solution to each problem (1.1)-(1.3) is proven in Sections 5-6. In particular, existence follows from an approximation procedure which makes use of BV initial and boundary data, avoiding the L ∞ -theory of initial-boundary value problems developed in [11] (see Section 6).
The function determined by solutions of (1.1)-(1.3) in R × (0, τ ) is, by definition, the regular part of a Radon measure, whose singular part is defined by observing that the variation of mass at each point x j depends on the sweeping effect of the flux across x j (see (7.2), (7.4), (7.6b) and Proposition 5.3). Then it is proven that this measure is the unique entropy solution of (CL) (in the sense of Definition 3.2) which satisfies the compatibility conditions at all x j ∈ F until the time t = τ . Here we use that the required compatibility condition for the solution of the Cauchy problem (CL) at x j is exactly the entropic formulation of the boundary conditions "u = ±∞" for the singular Dirichlet problems (see also Remark 5.3). If τ < T we iterate the procedure in R × (τ, T ) with a smaller number a singularities, thus well-posedness of (CL) follows in a finite number of steps (see Section 7). We observe that the proof of uniqueness of entropy solutions to problem (CL) relies on a general comparison principle between entropy sub and super-solutions of (1.1)-(1.3) (see Definitions 5.2-5.5 and Theorem 5.2 below) which is independent of the above construction procedure. In this sense the comparison results are stronger than those in [3,Theorem 3.2].
The results in the paper can be esaily extended to the case that u 0s is a locally finite superposition of Dirac masses (namely, if the number of Dirac masses in every bounded interval is finite).
For every real function f on R and x 0 ∈ R we say that ess lim For every open subset Ω ⊆ R we denote by C c (Ω) the space of continuous real functions with compact support in Ω and by M + (Ω) the cone of the nonnegative Radon measures on Ω. According to [6, Section 1.3], we say that ν is a (signed) Radon measure on Ω if there exist a (nonnegative) Radon measure µ ∈ M + (Ω) and a locally µ-summable function f ∶ Ω → [−∞, ∞] such that for all compact sets K ⊂ Ω. The space of (signed) Radon measures on Ω will be denoted by M(Ω).
Every µ ∈ M(R) has a unique decomposition µ = µ ac + µ s , with µ ac ∈ M(R) absolutely continuous and µ s ∈ M(R) singular with respect to the Lebesgue measure. We denote by µ r ∈ L 1 loc (R) the density of µ ac . Every function f ∈ L 1 loc (R) can be identified to an absolutely continuous Radon measure on R; we shall denote this measure by the same symbol f used for the function.
For every open subset Ω ⊆ R we denote by BV (Ω) the Banach space of functions of bounded variation in Ω: where z ′ is the first order distributional derivative. The total variation in Ω of z is T V (z; Ω) ∶= z ′ M(Ω) . We say that z ∈ BV loc (R) if z ∈ BV (Ω) for every open subset Ω ⊂⊂ R.
In the remainder of this section Ω denotes an open subset of R, and Q T = Ω × (0, T ). By C([0, T ]; M(Ω)) we denote the subset of strongly continuous mappings Definition 2.1. We denote by L ∞ (0, T ; M + (Ω)) the set of nonnegative Radon measures u ∈ M + (Q T ) such that for a.e. t ∈ (0, T ) there is a measure u(⋅, t) ∈ M + (Ω) with the following properties: (ii) the map t ↦ u(⋅, t) M(K) belongs to L ∞ (0, T ) for every compact K ⊂ Ω.
Moreover, since u + and u − are mutually singular, it follows that for a.e. t the nonnegative measures u + (⋅, t) and u − (⋅, t) are mutually singular, whence
Definition 3.1. Let u 0 be a signed Radon measure on Ω and let (A 0 ) be satisfied.
Solutions of (CL) in S are simply referred to as "solutions of (CL)".
Definition 3.2. Let u 0 be a signed Radon measure on Ω and let (A 0 ) be satisfied. A solution of (CL) in Q τ is called an entropy solution in Q τ if it satisfies the entropy inequality c (Ω)), ζ ≥ 0, ζ(⋅, τ ) = 0 in Ω, and for all k ∈ R; If Q τ ≠ Q T , an (entropy) solution in Q τ can be considered as a local (entropy) solution of (CL). For general initial measures, local entropy solutions satisfy the following monotonicity result.
Theorem 3.1. Let (A 0 ) be satisfied, let u 0 be a signed Radon measure on Ω and let u be an entropy solution u of problem (CL) in Q T . Then, for a.e. 0 < t 1 < t 2 < T , there holds Now we consider the case that u 0s is the sum of a finite number of Dirac masses with support F .  3) holds for any 0 ≤ t 1 ≤ t 2 ≤ T and for every x j ∈ F ∩ Ω there exists t j ∈ (0, T ] such that We observe that the proof of Corollary 3.2 provides an explicit lower bound for t j . If x j ∈ F and t j ∈ (0, T ] as in Corollary 3.2, Theorem 3.1 implies that the support of the singular part of any entropy solution is a subset of F × [0, T ] and that the Delta mass at x j ∈ F does not change sign in the interval [0, t j ). Therefore we may formulate a compatibility condition at x j which depends on the sign of c j , i.e. on the sign of the initial Delta mass at x j : where t j ∈ (0, T ] is defined by Corollary 3.2. We shall prove below (see Remark 5.4) that, if (A 0 )-(A 1 ) hold, for every entropy solution u of (CL) the limits  Remark 3.2. It was already observed in [2] that in general measure-valued entropy solutions are not unique. This is essentially a consequence of the elementary observation that there exists a unique entropy solution for which [u s (t)] = u 0s for a.e. t ∈ (0, T ) (it is enough to set u = u 0 +ũ, whereũ is the entropy solution with initial data u 0r ). But if u 0 satisfies (A 1 ) and F ≠ ∅, one easily checks that if the function H, satisfying (A 0 ), is not constant in intervals of the type (a, ∞) and (−∞, b), then such solution does not satisfy the compatibility condition at x j ∈ F .
In particular, it does not coincide with the solution defined by Theorem 3.3.

Monotonicity of u s .
In this section we prove Theorem 3.1 and Corollary 3.2.
Proof of Theorem 3.1. By (3.1), for every k ∈ R we get . By summing and subtracting the above equality from the entropy inequality (3.2), for every nonnegative ρ and β as above we obtain Letting k → ∞ with "+" and k → −∞ with "-", we obtain that Let 0 < t 1 < t 2 ≤ T . By standard approximation arguments we can choose Arguing as in the proof of Proposition 3.8(i) in [2], there exists a null set N ∈ (0, T ) which does not depend on the function ρ such that, letting n → ∞,

Hence the first inequality in (3.3) follows from the arbitrariness of ρ.
The second inequality in (3.3) can be proved in a similar way, replacing β n by ◻ Proof of Corollary 3.2. Arguing as in the proof of Theorem 3.1, for every ρ ∈ C 1 c (Ω) and t ∈ (0, T ] from (3.1) we get Fix any x j ∈ F ∩ Ω. By standard approximation arguments we can choose in (4.8) Then letting n → ∞, and observing that we obtain Since, by Theorem 3.1, u s (⋅, t) = u s (⋅, t) ⌞ F and F contains p points, we obtain that Then by the monotonicity of the mappings t ↦ u ± s (⋅, t) (see (3.3)) the conclusion follows. ◻

Problem (D): comparison and uniqueness
As already said, to address (CL) we need results concerning singular Dirichlet initial-boundary value problems for the scalar conservation law: where Ω = (a, b) is a bounded interval, m 1 = ±∞, m 2 = ±∞, and u 0 ∶ Ω ↦ R. Similar problems will be considered also for half-lines, either Ω ≡ (a, ∞), or Ω ≡ (−∞, b); obviously, the above condition at {b} × (0, T ) is omitted when Ω ≡ (a, ∞), and that In the case of half-lines problem (D S ) consists only of two cases, namely if Ω = (a, ∞), and if Ω = (−∞, b). We shall write that a statement holds for problem (D S ), if it collectively holds for all problems (D ± ± ). The following definition concerns problem (D R ) (see [13]).
is both an entropy subsolution and an entropy supersolution.
The following definitions for problem (D S ) are formulated for a wider class of initial data.
and for any interval I ⊆ Ω (5.12) lim is called an entropy solution of (D S ) if it is both an entropy subsolution and an entropy supersolution of (D S ).
Remark 5.2. Let us prove that every entropy solution of (D S ) satisfies the weak formulation Let us take the limit as j → ∞ in (5.15). Since u ∈ L 1 (Q) and H is bounded, we have In view of (5.16)-(5.17), letting j → ∞ in (5.15) gives gives, for every ζ as above, Therefore the conclusion follows combining (5.18) and (5.20).
Remark 5.3. The conditions (5.10a-5.10b) and (5.13a-5.13b) are entropy boundary conditions for singular Dirichlet problems and give a meaning, in a hyperbolic sense, to the boundary conditions "u = −∞" and "u = ∞". As already mentioned in the Introduction, they coincide with the compatibility conditions (3.5a) and (3.5b) for entropy solutions of (CL) at points x j where a signed Dirac mass is concentrated.
for any k ∈ R . Since 0 ≤ [u r − k] ± ≤ [u] ± + k , from the above inequality we get for every c ∈ Ω. Hence the distributional derivative of the function is nonpositive. Therefore, the limits The same statement can be applied to entropy solutions of (CL), since they satisfy inequalities (5.21) T ) (recall that by Theorem 3.1 and assumption (A 1 ) the singular part of an entropy solution of (CL) is not supported in these domains).
As for problem (D S ), the following holds.
Theorem 5.2. Let (A 0 ) hold. Let u, u be an entropy sub-and supersolution of (D S ) with the same boundary conditions. Then u ≤ u a.e. in Q. In particular, there exists at most one entropy solution of (D S ).
For future reference we prove the following generalization of [3,Lemma 4.4].
Let us prove (5.32). Clearly, there holds f + a (t) ≤ sup u∈R H(u) for a.e. t ∈ (0, T ). To prove the first inequality, let us choose in (5.8) By standard arguments we can also choose ρ = ασ ǫ with α ∈ C 1 c ([a, b)), α ≥ 0, and Then for every k ∈ R we obtain that Letting ǫ → 0 + and using (5.13a) and (5.31), we get that for every k ∈ R Letting k → ∞ in the above inequality gives whence by the arbitrariness of β inequality (5.32) follows.
To prove (5.34) we argue as for (5.32), using inequality (5.10a), (5.11) and (5.33) instead of (5.8), (5.13a) and (5.31). Then we get for every k ∈ R As k → −∞ in the above inequality, by the arbitrariness of β we obtain thus (5.34) follows. The proof of (5.36) and (5.38) is similar to that of (5.32) and (5.34), using instead of (5.39); we leave the details to the reader.
Finally we prove the following result.
Lemma 5.4. Let u be an entropy solution of (D R ). Then for every t ∈ (0, T ] Proof. By (5.1) and (5.4) there holds for every k ∈ R and ζ as above. By standard arguments we can choose ζ(x, s) = α p (x)β q (s) with for any fixed t ∈ (0, T ] and p, q ∈ N sufficiently large. Then for k = 0 as q → ∞ we get whence as p → ∞ (5.41) follows.
Proof of Theorem 6. Arguing as in [1] shows that u is an entropy solution of problem (D R ), In fact, By standard regularization arguments we can choose in (6.15) E(u ǫ ) = [u ǫ − k] ± , thus obtaining for all k ∈ R and ζ as above, ζ ≥ 0, On the other hand, choosing in (6.16) ζ(x, t) = χ [a, ξ+1 n) (x)β(t) (ξ ∈ Ω, n ∈ N) with β ∈ C 1 c (0, T ), β ≥ 0, and letting n → ∞ plainly gives for every k ∈ R Multiplying the first equation of (D ǫ ) by ζ(x, t) = χ [a, ξ+1 n) (x)β(t) and letting n → ∞, one easily sees that By (6.7), (6.9) and (6.14) we can take the limit as ǫ n → 0 + in (6.17) and (6.20) (written with ǫ = ǫ n ). It follows that the function u in (6.14) satisfies the following inequalities: -for every k ∈ R and for all ζ -for every k ∈ R and β ∈ C 1 c (0, T ), β ≥ 0 and for a.e. ξ ∈ Ω, Letting ξ → a + in the latter inequality and using Remark 5.1 we conclude that u is an entropy solution of (D R ). Hence the result follows. ◻ Remark 6.1. In the proof of Theorem 6.1 when Ω = (a, b) one uses the family of solutions of the problem with m 1 , m 2 ∈ R, H ǫ as above and u 0ǫ defined by a suitable partition of unity; we leave the details to the reader.
Concerning (D S ) the following holds.

Well-posedness of problem (CL)
In this section we prove Theorem 3.3.
It remains to prove (7.15), which is equivalent to showing that u r = v r a.e. in Q j,τ for all j = 1, . . . , p + 1. However, this follows from the uniqueness results provided by Theorem 5.2.. Then the result follows. ◻