Radon measure-valued solutions of first order scalar conservation laws

: We study nonnegative solutions of the Cauchy problem where u 0 is a Radon measure and φ : [ 0, ∞) 󳨃→ ℝ is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ , we prove their uniqueness if the singular part of u 0 is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case φ ( u ) = u this happens for all times). In the latter case, we describe the evolution of the singular parts.


Introduction
In this paper we consider the Cauchy problem where T > 0, u 0 is a nonnegative finite Radon measure on ℝ, and φ : [0, ∞) → ℝ, φ(0) = 0, is a Lipschitz continuous function (see assumption (H1)). Therefore, φ grows at most linearly. Problem (P) with a superlinear φ of the type φ(u) = u p , p > 1, was studied in [19], proving existence and uniqueness of nonnegative entropy solutions (see also [8]). By definition, in that paper the solution for positive times takes values in L 1 (ℝ), although the initial data u 0 is a finite Radon measure. Interesting, albeit sparse results concerning (P) with φ at most linear at infinity can be found in the pioneering paper [10], in which the same definition of Radon measure-valued solutions used below (see equality (3.8)) was proposed.
When φ(u) = Cu (C ∈ ℝ), problem (P) is the Cauchy problem for the linear transport equation whose solution is trivially the translated of u 0 along the lines x = Ct + x 0 (x 0 ∈ ℝ). In particular, the singular part u s ( ⋅ , t) of the solution is nonzero for t > 0 if and only if the same holds for t = 0. It is natural to ask what happens if φ is sublinear. To address this case we must consider solutions of problem (P) which, for t > 0, possibly are finite Radon measures on ℝ as the initial data u 0 . Therefore, throughout the paper we consider solutions of problem (P) as maps from [0, T] to the cone of nonnegative finite Radon measures on ℝ, which satisfy (P) in the following sense: for a suitable class of test functions ζ , we have Definition 3.3). Here the measure u(t) is defined for a.e. t ∈ (0, T), u r ∈ L 1 (S) is the density of its absolutely continuous part, ⟨ ⋅ , ⋅ ⟩ ℝ denotes the duality map, and ∂ ν ζ := ∂ t ζ + C φ ∂ x ζ, C φ := lim u→∞ φ (u) u .
Measure-valued entropy solutions are defined similarly (see Definition 3.3).
We use an approximation procedure to construct measure-valued entropy solutions of problem (P) (see Theorem 3.7). In addition, we prove that the singular part u s of an entropy solution of problem (P) does not increase along the lines x = x 0 + C φ t (see Proposition 3.8). In particular, if C φ = 0, the map t → u s ( ⋅ , t) is nonincreasing.
Concerning the case when φ is sublinear, the following example is particularly instructive: The function in (1.2) is increasing and concave, with C φ = 0, and belongs to a class for which the constructed entropy solution of problem (1.1)-(1.2) is unique (see Theorem 3.22). Hence, the following holds. Let More precisely, if p < 0, the singular part u s ( ⋅ , t) persists until the waiting time t 0 = 1 at which it disappears, whereas for 0 < p < 1, the singular part vanishes for all t > 0, thus t 0 = 0 -an instantaneous regularizing effect. Instantaneous regularization also occurs if p > 1 (see [19] and Remark 3.24), whereas, as already remarked, in the linear case p = 1, we have t 0 = T if u 0s ̸ = 0. Since φ(u) = sgn p[(1 + u) p − 1] (p < 1, p ̸ = 0) is bounded if and only if p < 0, and C φ = 0, statement ( * ) could be rephrased as follows. (1.1) if and only if the map u → φ(u) − C φ u, with φ as in (1.2), is bounded in [0, ∞).

Proposition 1.2. Positive waiting times occur in problem
The above result is generalized to problem (P), by Theorem 3.18, for functions φ which satisfy for u large a condition implying either concavity or convexity (see assumption (H4) and Remark 3.13). The proof of Theorem 3.18 makes use of estimates of the density u r of the solution of (P), which are strongly reminiscent of the Aronson-Bénilan inequality for the porous medium equation (see Proposition 6.2). The main results on the waiting time and the regularity of solutions of (P) are collected in Section 3.3. The existence and an upper bound, in terms of φ and u 0 , of a waiting time was already pointed out in [10, Proposition 2.1] (see also Theorem 3.8 (ii)).
Another interesting feature of the solution of (1.1)-(1.2), with p < 0, is that for t ∈ (0, 1), i.e., as long as u s ( ⋅ , t) > 0, we have lim x→0 + u r (x, t) = ∞. Namely, the regular part u r ( ⋅ , t) diverges when approaching from the right the point x 0 = 0, where u s ( ⋅ , t) is concentrated. As we shall see below (see (3.24)- (3.25)), this property can be generalized to entropy solutions of a larger class of problems, characterized by the concavity/convexity property on φ mentioned before. In this class a generalized form of this property will also be used as a uniqueness criterion, provided that φ(u) − C φ u is bounded in [0, ∞) and u 0s is a finite superposition of Dirac masses (see Proposition 3.17 and Theorem 3.22). In [10] it was already observed that Kruzkov's entropy inequalities do not guarantee the uniqueness of solutions (see also Remark 3.23 below), and the formulation of an additional uniqueness criterion was left as an open problem. This problem is addressed in a forthcoming paper, where more general compatibility conditions are given, which ensure uniqueness also for non-convex or non-concave functions φ (see [3]).
Apart from the intrinsic mathematical interest of problem (P), it is worth pointing out its connection with a class of relevant models. Ion etching is a common technique for the fabrication of semiconductor devices, also relevant in other fields of metallurgy, in which the material to be etched is bombarded with an ion beam (see [16,24,25]). Mathematical modelling of the process leads to the Hamilton-Jacobi equation in one space dimension where U = U(x, t) denotes the thickness of the material and φ is bounded, non-convex and vanishing at infinity. Formal differentiation with respect to x suggests to describe the problem in terms of the unknown u := ∂ x U, which formally solves (P) with u 0 = U 0 . In this way, discontinuous solutions of (HJ) correspond to Radon measure-valued solutions of (P) having a Dirac mass δ x 0 concentrated at any point x 0 , where U( ⋅ , t) is discontinuous (t ∈ (0, T)). A rigorous justification of the above argument, relating discontinuous viscosity solutions of (HJ) to Radon measure-valued entropy solutions of (P), is to our knowledge an open problem (in this connection, see [7,14]). In the context of conservation laws, the term "measure-valued solution" usually refers to solutions in the sense of Young measures, after DiPerna's seminal paper [11]. We stress that this concept of "statistical Brought to you by | Lancaster University Authenticated Download Date | 10/3/18 12:25 PM solutions" is completely different from that of Radon measure-valued solutions, introduced by Demengel and Serre [10], and discussed in the present paper. On the other hand, we do use Young measures in this paper, since they are an important ingredient in the construction of Radon measure valued solutions (see Section 3 and, in particular, Section 5).
A number of ideas used in the present paper go back to papers dealing with Radon measure-valued solutions of quasilinear parabolic problems, also of forward-backward type (in particular, see [4-6, 21, 23, 27]).
The results presented in this paper naturally lead to some open problems. Among them we mention a general statement about an instantaneous regularizing effect for fluxes with superlinear growth (singular parts should disappear instantaneously for t > 0), and an appropriate generalization of our results to the case of solutions with changing signs, when additional nonuniqueness phenomena (such as N-waves, see [19]) may occur; in this regard, the general case of an initial signed Radon measure u 0 in problem (P) will be considered in a forthcoming paper. Another open problem is whether new phenomena occur if φ is uniformly Lipschitz continuous on [0, ∞) but the limit φ(s)/s as s → ∞ does not exist.
The paper is organized as follows. In Section 2 we recall several known results used in the sequel and introduce some notation. In Section 3 we present the main results of the paper. In Section 4 we introduce the approximation procedure needed for the construction of solutions. Sections 5-7 are devoted to the proofs of existence, qualitative properties and uniqueness of solutions.
We denote by C c (ℝ) the space of continuous real functions with compact support in ℝ. The space of the functions of bounded variation in ℝ is denoted by BV(ℝ) := {u ∈ L 1 (ℝ) | u ∈ M(ℝ)}, where u is the distributional derivative of u. It is endowed with the norm ‖u‖ BV(ℝ) := ‖u‖ L 1 (ℝ) + ‖u ‖ M(ℝ) . We say that u ∈ BV loc (ℝ) if u ∈ BV(Ω) for every open bounded subset Ω ⊂ ℝ.
The Lebesgue measure, either on ℝ or S := ℝ × (0, T), is denoted by | ⋅ |. Integration with respect to the Lebesgue measure on ℝ or on S will be denoted by the usual symbols dx, respectively dx dt. A Borel set E is null if |E| = 0. The expression "almost everywhere", or shortly "a.e.", means "up to null sets". For every measurable function f defined on ℝ and x 0 ∈ ℝ, we write ess We set f ± := max{±f, 0} for every measurable function f on ℝ.

Young measures
We recall the following result [2].
Suppose further that {u j } satisfies the boundedness condition for every R > 0, where B R := {x ∈ ℝ N | |x| < R}. Then (iv) τ x is a probability measure for a.e. x ∈ Ω, (v) given any measurable subset A ⊆ Ω, we have for all continuous functions f : ℝ → ℝ such that {f(u j )} is sequentially weakly compact in L 1 (A).
If Ω ⊂ ℝ N is bounded and {u j } is a bounded but not uniformly integrable sequence in L 1 (Ω), it is possible to extract a uniformly integrable subsequence "by removing sets of small measure". This is the content of the following "Biting lemma" (e.g., see [17,28] and references therein).
where Z ∈ L 1 (Ω) is called the barycenter of the disintegration {τ x }.

Main results
Throughout the paper we assume that u 0 ∈ M + (ℝ). Concerning φ, we always suppose that

Definition of solution
In the following definitions, we denote by the derivative of any ζ ∈ C 1 (S) along the vector τ ≡ (C φ , 1).
Definition 3.1. By a solution of problem (P) in the sense of Young measures, we mean a pair (u, τ) such that where ∂ ν ζ is defined by (3.2) and By an entropy solution of problem (P) in the sense of Young measures, we mean a solution such that for all ζ as above, ζ ≥ 0, and for every pair Entropy subsolutions (respectively supersolutions) of problem (P) in the sense of Young measures are defined by requiring that inequality (3.6) be satisfied for all ζ and (E, F) as above, with E nondecreasing (respectively nonincreasing).
A solution of problem (P) is also a solution in the sense of Young measures. Moreover, it follows from (3.1) that φ(u r ) ∈ L ∞ (0, T; L 1 (ℝ)). Similar remarks hold for entropy solutions, subsolutions and supersolutions.
The following proposition states that for any solution of (P) in the sense of Young measures, the map t → u(t), possibly redefined in a null set, is continuous up to t = 0 with respect to the weak * topology of M + (ℝ). In particular, it explains in which sense the initial condition is satisfied.
Proposition 3.5. Let (H1) be satisfied, let (u, τ) be a solution of problem (P) in the sense of Young measures, and let ρ ∈ C c (ℝ). Then The map t → u(t) has a representative, defined for all t ∈ [0, T], such that

Existence and monotonicity
The existence of solutions is proven by an approximation procedure. If u 0 ∈ M + (ℝ), then there exist u 0n ∈ L 1 (ℝ) ∩ L ∞ (ℝ) such that Let us recall the definition of entropy solution of problem (Pn) (e.g., see [9]). Definition 3.6. A function u n ∈ L ∞ (0, T; L 1 (ℝ)) ∩ L ∞ (S) is called an entropy solution of problem (Pn) if for every ζ ∈ C 1 ([0, T]; C 1 c (ℝ)), with ζ( ⋅ , T) = 0 in ℝ and ζ ≥ 0, and for any couple (E, F), with E convex and By studying the limiting points of the sequence {u n }, we shall prove the following result.
. Then u is an entropy solution of problem (P).
Hypothesis (C2) fails if for example φ is affine in an interval (a, b) ⊂ (0, ∞). In that case, Proposition 5.9 (iii), which characterizes the limiting Young measure, gives some additional information.
The following proposition shows that the singular part of an entropy subsolution of (P) does not increase along the lines x = C φ t + x 0 .
In particular, The linear case φ(u) = u shows that equality may hold in (3.18). Moreover, if C φ = 0, it follows from (3.18) that the map t → u s ( ⋅ , t) is nonincreasing.

Waiting time and regularity
It is convenient to distinguish two cases: C φ = 0 (sublinear growth at infinity) and C φ ̸ = 0 (linear growth at infinity), with C φ defined by (H1).

Sublinear growth
Beside (H1), we will use the following assumption: is strictly negative and continuous in [0, ∞), hence two cases are possible: either (a) Hφ In particular, in both cases (H2) implies (C2). Moreover, if also (H1) holds, thus φ(0) = 0, we have Hφ The following property of constructed entropy solutions plays an important role as a uniqueness criterion (see its generalized form given by Proposition 3.17 and Theorem 3.22 below).
(ii) Let (H1)-(H2) be satisfied, and let u be the entropy solution of problem (P) given by Theorem 3.7.

Linear growth
Let φ satisfy the following assumption:

Remark 3.16.
It is easily seen that if u is a solution (respectively an entropy solution) of problem (P), for any h ∈ ℝ, is a solution (respectively an entropy solution) of (P) with u 0 replaced by v 0 : By Remark 3.16, the above results for the case C φ = 0 can be generalized as follows.
(ii) Let (H1) and (H4) be satisfied, and let u be the entropy solution of problem (P) given by Theorem 3.7.

Uniqueness
In connection with equality (3.11), observe that if u 0s ̸ = 0 and the waiting time t 0 is equal to 0, then , a contradiction). Instead, continuity along the lines x = x 0 + C φ t may occur if the waiting time t 0 is positive.
(3.26) (i) If condition (C2) holds, then every entropy solution u of problem (P) given by Theorem 3.7 (ii) satisfies Let us mention that the above statement (ii) holds for any u 0 ∈ M + (ℝ) if φ satisfies (H1) and (H4) (see Proposition 6.2). The following uniqueness result will be proven in Section 7.
is the unique entropy solution of problem (P) with u 0 replaced by u 0r .
Let (H5) hold. To prove Proposition 6.2, we use a different regularization of (Pn), that is, where {u ε 0n } satisfies (4.1)-(4.2). The existence, uniqueness and regularity results recalled in Section 4 for problem (4.3), as well as the a priori estimates in Lemma 4.1 and the convergence results in Lemma 4.4 (i), continue to hold for solutions of (6.6) (see [18]). In particular, there exist a sequence {y From (6.6), for every E convex, F = E φ , and ζ as in Definition 3.6, we get Arguing as in the proof of Proposition 4.5 and letting ε m → 0, we obtain that So y n satisfies (3.16) and, by Kružkov's uniqueness theorem, y n = u n . Hence, we have shown the following lemma. whence, by (6.14) and the lower semicontinuity of the total variation, Similarly, by (5.6), (5.25) and Proposition 5.8, and, by (6.14) and the lower semicontinuity of the total variation, It remains to prove that u ∈ C((0, T]; M(Ω)). Observe that for all t 1 , t 2 ∈ (0, T], 0 < τ < t 1 < t 2 , and where we have used (6.13). We let ε = ε m → 0 and use (3.1) and (4.20) to obtain By (5.16) and the lower semicontinuity of the total variation, So we may define u( ⋅ , t) for all t ∈ [τ, T] such that u ∈ C([τ, T]; M(Ω)). Since τ > 0 is arbitrary, the proof is complete.
To prove Proposition 3.10, we need the following lemma. Lemma 6.6. Let (H1) be satisfied, and let u be the solution of problem (P) given by Theorem 3.7. Let {u n j } be as in the proof of Theorem 3.7. Then, for a.e. t ∈ (0, T) and all x 0 ∈ supp u s ( ⋅ , t), there exist a sequence {x 0k } ⊂ ℝ and a subsequence {u n k } of {u n j } such that x 0k → x 0 and u n k (x 0k , t) → ∞ as k → ∞.
Proof. Let x 0 ∈ supp u s ( ⋅ , t). We may assume that the convergence in (5.16) is satisfied for this t. Since x 0 ∈ supp u s ( ⋅ , t), there is no neighborhood I δ (x 0 ) such that the sequence {u n j ( ⋅ , t)} lies in a bounded subset of L ∞ (I δ (x 0 )). Otherwise, up to a subsequence, u n j ( ⋅ , t) * ⇀ f t in L ∞ (I δ (x 0 )) for some f t ∈ L ∞ (I δ (x 0 )), f t ≥ 0. However, this would imply that u s ( ⋅ , t) = 0 in I δ (x 0 ), a contradiction.
Thus, by a variant of the dominated convergence theorem (e.g., see [15, Theorem 4, Section 1.3]), we have F n → |u r − v r | in L 1 (K l ), and we obtain (7.16). This completes the proof of (7.9), thus the result follows.