New entropy conditions for scalar conservation laws with discontinuous flux

We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under assumptions on the flux which provide the maximum principle. In particular, we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions.

In the current contribution, we consider the following problem where u is the scalar unknown function; u 0 is a function such that a ≤ u 0 ≤ b, a, b ∈ IR; H is the Heaviside function; and f, g ∈ C 1 (R) are such that f (a) = f (b) = g(a) = g(b) = 0. Problems such as (1) are non-trivial generalization of scalar conservation law with smooth flux, and they describe different physical phenomena (flow in porous media, sedimentation processes, traffic flow, radar shape-from-shading problems, blood flow, gas flow in a variable duct...). Therefore, beginning with eighties (probably from [35]), problems of type (1) are under intensive investigations.
As usual in conservation laws, the Cauchy problem under consideration in general does not possess classical solution, and it can have several weak solutions. Since it is not possible to directly generalize standard theory of entropy admissible solutions [23], in order to choose a proper weak solution to (1) many admissibility conditions were proposed. We mention minimal jump condition [17], minimal variation condition and Γ condition [10,11], entropy conditions [19,1], vanishing capillary pressure limit [18], admissibility conditions via adapted entropies [6,8] or via conditions at the interface [2,3,12].
But, in every of the mentioned approaches, in order to prove existence or uniqueness of a weak solution to the considered problem, some structural hypothesis on the flux (such as convexity or genuine nonlinearity) or on the form of the solution (see [2,3]) were assumed.
Recently, in [26], we have proved existence and uniqueness in the multidimensional situation. Still, due to certain technical obstacles, admissible solutions selected in that paper are rather special.
Here, we propose admissibility conditions which involve much less restrictions than in previous works on the subject (excluding [26] where there are no restrictions), and we still can make many different stable semigroups depending on the physical situation under considerations.
Since one can find excellent overviews on the subject in many papers [5,3,7,8,12,28] which are easily available via internet (e.g. www.math.ntnu.no/conservation), in this introduction, we shall restrict our attention on papers [19], [21], and [28] which are in the closest connection to our contribution. Later, in Section 2, we shall comment how our admissibility conditions can be considered as a generalization of the entropy solution of type (A, B) given in [8] (see Definition 1.1 in the current paper).
In [19], degenerate parabolic equation with discontinuous flux is considered: where A is non-decreasing with A(0) = 0. Assuming that A ≡ 0 we obtain the problem of type (1). In order to obtain uniqueness of a weak solution to the problem, the Kruzhkov type entropy admissibility condition [23] is used: Definition 0.1. [19] Let u be a weak solution to problem (1). We say that u is an entropy admissible weak solution to (1) if the following entropy condition is satisfied for every fixed ξ ∈ R: Still, merely such entropy condition was insufficient to prove stability of the admissible weak solution to the considered problem. Two more things were necessary.
First, one needs the following technical assumption: Crossing condition: For any states u, v the following crossing condition must hold: f (u) − g(u) < 0 < f (v) − g(v) ⇒ u < v. Geometrically, the crossing condition requires that either the graph of f and g do not cross, or the graph g lies above the graph of f to the left of the crossing point (see Figure 1). The functions f and g appearing in (1) do not necessarily satisfy the crossing conditions, but it is possible to transform them so that the crossing condition is satisfied (see Figure 2 and Figure 4).
Next, in [21] existence of strong traces at the interface x = 0 was necessary. We provide appropriate definition.
Definition 0.2. Let W : IR×IR + → IR be a function that belongs to L ∞ (IR×IR + ). By the right and left traces of W (·, t) at the point x = 0 we understand functions t → W (0±, t) ∈ L ∞ loc (IR + ) that satisfy for a.e. t ∈ IR + : Assuming the crossing condition and the existence of traces, we have the following theorem: Theorem 0.3. [19] Assume that weak solutions u and v to (1) with the initial conditions u 0 and v 0 , respectively, satisfy entropy admissibility conditions from Definition 0.1 and admit left and right strong traces at the interface x = 0.
Then for any T, R > 0 there exist constants C,R > 0 such that: Remark 1. It is important to notice that Theorem 0.3 remains to hold if in (1), Indeed, since we did not put a function depending on t ∈ IR + under the derivative ∂ t , and since α and β are increasing bijections (we can extract all the information on u knowing only β(u) or α(u)), we can safely use results from [21] on the equation First, we shall explain how to force the crossing condition and existence of traces. We shall use the idea from [28]. In [28], the following problem was considered where α is a function discontinuous in x ∈ IR and strictly increasing with respect to u. Then, we can write:

Problem (3) becomes
Thus, the discontinuity in x is removed out of the derivative in x, and we can apply standard vanishing viscosity approach: to obtain the sequence (v ε ) strongly converging in L 1 loc (IR × IR + ) to a unique Kruzhkov admissible weak solution v of (4) which immediately gives uniqueness of appropriate weak solution to (3).
It is important to notice that the existence and uniqueness are actually obtained thanks to the appropriate choice of the viscosity term. Such choice enables the author to control the flux corresponding to (3).
Using this observation, we shall propose new admissibility conditions which will enable us to control the flux corresponding to (1) in an extent which will provide uniqueness in a rather general situation. Informally speaking, we shall consider the following vanishing viscosity regularization to (1): and denoting f α = f • α and g β = g • β, we have from (6): So, instead of dealing with the flux H(x)f (u)+H(−x)g(u), we deal with the new flux As we shall see later, by choosing appropriate functions α and β we can always make the new flux to satisfy "the crossing condition" at least in the range of the solution (see Figure 2 and Figure 4 as important special cases). Now, we can introduce the definition of admissibility that we shall use.
From the previous analysis, appealing on [19], we conclude that we need only existence of traces to obtain the uniqueness. The question of existence of traces is rather serious in itself [24,27,36], but it was shown in [27] that they exist practically in all relevant situations . In order to formulate a necessary theorem, we need the notion of the quasi-solution.
Definition 0.5. We say that the function u ∈ L ∞ (IR d ) is a quasi-solution to the scalar conservation law div where γ k is a locally bounded Borel measure.
Next theorem can be found in [27]. We adapt it to our situation.
Suppose that the function u is a quasi-solution to where the vector (h, f ) is such that the mappings λ → h(λ) and λ → f (λ) are not constant on any non-degenerate interval.
-- In the case of a scalar conservation law with a smooth flux, the proof of existence is based on the BV-estimates for a sequence of solutions to the corresponding Cauchy problem regularized with the vanishing viscosity. Such estimates are not available if the flux is discontinuous. Therefore, we need to apply more subtle arguments involving singular mapping [35], local variation bounds [9], compensated compactness [20, 21,22,33], difference schemes [3,19,22] or H-measures [15,16,30,34].
In general, using e.g. the compensated compactness, it is possible to prove that the sequence (u ε ) of solutions to (6) weakly converges to a weak solution u of (1). However, it is not possible to state that the weak solution satisfies wanted admissibility conditions. In order to be sure that u is admissible, in principle, we need to prove that the corresponding sequence (u ε ) strongly converges strongly in L 1 loc (IR + × IR) to u (still, not necessarily; see [29]) which, at least in the framework of the compensated compactness (or the H-measures whose consequences we are going to use), can be proved only by assuming the genuine nonlinearity condition given by the following definition.
The latter condition provides the following theorem to hold.
Then, the following statement holds: So, our last obstacle is the genuine nonlinearity condition. In order to overcome it we shall use an idea from [21] which is further developed in [4]. In [21,4], existence of solution to a Cauchy problem of type (1) is proved. Roughly speaking, the key point of the proof is based on a lemma stating that if in (1) we assume u 0 ∈ BV (IR), then, for the sequence (u ε ) of solutions to (6), it holds ∂ t u ε L 1 (I R) ≤ const for every fixed t, ε ∈ IR + . This actually means that for any function h( Next, it is not difficult to prove that it holds for the sequence (u ε ) of solutions to (6) is the sequence bounded in the space of Radon measures, we also have: ) is genuinely nonlinear, we can apply Theorem 0.8 to conclude about strong L 1 loc precompactness of the family (u ε ). It is clear that a L 1 loc limit along a subsequence of the family (u ε ) will represent wanted admissible weak solution to (1). Furthermore, according to Theorem 0.6, we infer about the existence of traces at the interface x = 0 for the previously constructed weak solution which immediately gives uniqueness. Of course, it is not always possible to choose h R and h L so that we have both, the genuine nonlinearity and the crossing conditions fulfilled. Still, as we shall see, using truncation functions s l,k (u) = max{l, min{k, u}}, l < k, l, k ∈ IR, (first used in [27] for this kind of problems; see also [15]), we are able to localize and thus deal with the segments where the genuine nonlinearity is unobtainable.
The paper is organized as follows.
In Section 1, we solve (1) under additional assumptions on the flux. We find the section important since it sheds (another) light on paper [8] where the crossing condition is bypassed by using so called adapted entropies (see [6]). We show that admissibility conditions that we introduced in Definition 0.4 can be considered as a generalization of the approach from [8], which is actually an explanation how adapted entropies enabled avoiding (or maybe better to say forced) the crossing conditions.
In Section 2, by passing to the measure valued solution concept [13], we show existence and uniqueness in the general situation.

New entropy admissibility conditions
The basic purpose of the section is to explain connection between our (α, β)entropy solutions and the entropy solutions of type (A, B) used in [8]. Furthermore, we find that this section represents a good introduction into the general situation considered in Section 3.
We shall consider here (1) under the additional assumptions that the mappings are nonconstant and strictly positive on any subinterval of the interval (a, b) (notice that this assumption is weaker than the appropriate assumption [8, (1.2)] which demands a genuine nonlinearity of f and g).
To proceed, let us briefly recall the concept from [8]. First, we need the function c AB (see [8, (11)]): In [8], the function c AB is used to form the function u → |u − c AB (x)| which is an example of what is in [6] called an adapted entropy. Still, in [6], the existence of infinitely many adapted entropies was necessary to prove uniqueness (see also [28]) while in [8] only the entropy u → |u − c AB (x)| was sufficient (together with the classical Kruzhkov entropies out of the interface). The function c AB is called a connection if it represents a weak solution to (1) for a more precise explanation). We remark that the notion of the connection originated from [2]. The following admissibility conditions were used in [8]: 2) For any test function 0 ≤ ϕ ∈ D([0, T ) × IR), T > 0, which vanishes for x ≥ 0, and any ξ ∈ IR, the following holds: and for any test 3) The following Kruzhkov-type entropy inequality holds for any test function In the next theorem, we state that the (α, β)-entropy admissible solution from Definition 0.4 is, under certain conditions, at the same time an entropy solution of type (A, B) from Definition 1.1. In Remark 2 after the theorem, we shall explain why such conditions are always fulfilled in the case of the flux given in [8].
Theorem 1.2. Assume that the function u is an (α, β)-entropy admissible solution to (1) in the sense of Definition 0.4 where α and β satisfy: ; • the functions f • α and g • β satisfy the crossing conditions. Then, the (α, β)-entropy admissible solution to (1) is at the same time the entropy solution of type (A, B).
Proof. First, notice that, according to the choice of α and β, the function c AB will represent an (α, β)-entropy admissible solution to (1) in the sense of Definition 0.4. Taking another (α, β)-entropy admissible solution to (1) , and applying the procedure from [19] leading to [19, (2.34)] (keep in mind that f α and g β satisfy the crossing conditions), we reach to the following (well known) relation: Since α and β as well as their inversesα andβ are increasing bijections, it holds From here, we see that (11) (1) were such that they admit unique local maxima points u * f ∈ (a, b) and u * g ∈ (a, b), respectively. Then, the pair (A, B) is called a connection if In this case, we can always find functions α and β such that conditions of Theorem 1.2 are satisfied. Indeed, assume that u * f < u * g (other two situations u * f > u * g and u * f = u * g can be resolved similarly). Denote byα andβ inverse functions to the functions α and β, respectively. Chooseα andβ on the intervals [u * f , b] and [a, u * g ] to be linear and such thatα(u * f ) > c >β(u * g ) (see Figure 2; the situation plotted there is more general but completely analogical with the one we are considering at the moment).
To extend the functionα in the interval [a, u * f ], we will construct its inverse α in the interval [a, c]. Take an arbitrary decreasing functionα connecting the points (a, 0) and (c, f α (c)) such thatα ≤ g β on [a, c]. This is always possible since g β > 0 on (a, c); for instance, we can takeα to be the convex hull of g β on [a, c]. Then, put α = f −1 •α i.e.α = α −1 on [a, u * f ] (this is permitted since f −1 is monotonic on (a, c)). We chooseβ on [u * g , b] in the completely same manner (see Figure 2 for further clarification). It is clear that α =α −1 and β =β −1 chosen in such a way satisfy conditions of Theorem 1.2.
Actually, from the latter discussion, we can conclude that the conditions given in Theorem 1.2 are a generalization of the notion of connection. More precisely, we can say that a pair (A, B) is a connection if there exist functions α and β satisfying conditions of Theorem 1.2. As we shall see in Theorem 1.3, such conditions provide existence and uniqueness of the (α, β)-entropy admissible solution to (1). In particular, the function BH(x) + AH(−x) will be the (α, β)-entropy admissible shock.
Also, remark that conditions (12) can be naturally generalized by assuming that where u * g and u * f are the rear right local maximum of the function g and the rear left local maximum of the function f , respectively. Repeating the procedure from the beginning of the remark, we can find the function α and β such that conditions of Theorem 1.2 are satisfied (see Figure 2).
Finally, notice that if A = u * g and B = u * f , we cannot state that the functions α and β satisfying conditions of Theorem 1.2 exist (for instance, if the functions f and g have several local maxima, and all of them have the same values).
The following theorem is the main theorem of the section: There exists a pair of function (α, β) from Definition 0.4 such that there exists a unique (α, β)-entropy admissible solution to (1).
Before we prove the theorem, we shall need several auxiliary statements and explanations.
In order to construct an (α, β)-entropy admissible solution to (1), we use a nonstandard vanishing viscosity approximation with regularized flux. First, introduce the following change of the unknown function u: Then, take the following regularization of the Heaviside function H, H ε (x) = x/ε −∞ ω(z)dz, where ω is a smooth even compactly supported function with total mass one. Let χ ε be a smooth function equal to one in the interval (−1/ε, 1/ε) and zero out of the interval (−2/ε, 2/ε). Consider the following regularized problem: Obviously, for every fixed ε > 0 quasilinear parabolic Cauchy problem (15) will have a unique smooth solution v ε .
Since α and β are strictly increasing functions which map interval [a, b] into itself, slightly modifying the methodology from [21], we obtain the following three lemmas.

Lemma 1.5. [21, Lemma 4.2] [Lipshitz regularity in time]
Assume that the initial function u 0 from (1) has bounded variation. Then, there exists constant c 1 , independent of ε, such that for all t > 0,

Proof:
Denote η ′ (λ) = H(λ − ξ). Define the entropy flux which corresponds to (15): Denote δ ε (x) = H ′ ε (x), i = 1, 2. After multiplying (15) by η ′ (v ε ), we obtain in the sense of distributions: From here, according to the Schwartz lemma for non-negative distributions, we conclude that there exists a positive Radon measure µ ε ξ (t, x) such that: Rewrite expression (19) in the form: Proof: First, notice that the vector (q(x, λ), q(x, λ)) from (16) is genuinely nonlinear. Indeed, for x > 0 the vector reduces to (f 2 α (λ), f α (λ)) and this is obviously genuinely nonlinear vector according to (10). Similarly, we conclude about the genuine nonlinearity for x < 0. Now, from Theorem 0.8 and Lemma 1.8, we conclude that the family (v ε ) of solutions to (15) is strongly precompact in L 1 loc (IR + ×IR). Denote by v the L 1 loc (IR + ×IR) limit along a subsequence of the family (v ε ). Clearly, u = α(v)H(x) + β(v)H(−x) will represent the (α, β)-entropy admissible solution to (1). Now, we can prove the main theorem of the section. Proof of Theorem 1.3: We need to find the functions α and β so that the functions f α and g β satisfy the crossing conditions. As explained in Remark (2), we choose the points A, B ∈ (a, b) satisfying (13), and construct the functions α and β so that for appropriate c ∈ (a, b) it holds α(c) = B, β(c) = A, and f α ≥ g β on [c, b], and f α ≤ g β on [a, c] which is nothing else but the crossing condition for f α and g β .
Similarly, from the construction again and according to the choice of the function α and β, we see that v is an entropy admissible solution in the sense of Definition 0.1 to the Cauchy problem where f α and g β satisfy the crossing condition. According to Theorem 0.3, we conclude that v is a unique entropy admissible solution to (22) in the sense of Definition 0.1 implying that is a unique (α, β)-entropy admissible solution to (1). Now, assume that u 0 / ∈ BV (IR). Approximate the function u 0 by a sequence (u 0δ ) ∈ BV (IR) so that u 0 − u 0δ → 0 as δ → 0 strongly in L 1 loc (IR). Then, we find a unique (α, β)-entropy admissible solution u δ to (1) where u| t=0 = u 0δ (given α and β for which we have uniqueness i.e. such that f α and g β satisfy the crossing conditions). According to Theorem 1.3, the family (u δ ) satisfy the following stability relation: where R and T are arbitrary positive constants, and C,R are constants depending on R, the functions f , g, α and β. Since the right-hand side of the latter expression is uniformly small with respect to δ 1 and δ 2 , from the Cauchy criterion we conclude that there exists u ∈ L 1 loc such that u δ → u strongly in L 1 loc (IR d ). Clearly, the function u will represent an (α, β)-entropy admissible solution to (1).
Since, according to (10) and Theorem 0.6, the function u admits strong traces at x = 0, we conclude that it must be a unique (α, β)-entropy admissible solution to (1).

General case
At the beginning, notice that there are many examples of fluxes from (1) when we can not apply the procedure from the previous section (see Figure 3). Therefore, in this section, we shall demonstrate how to apply the (α, β)-entropy admissibility concept on (1) in a general case. More precisely, we shall only assume that f, g ∈ C 1 (R) are such that f (a) = f (b) = g(a) = g(b) = 0, and, for simplicity, that there exists a finite number of intervals (a rj , a rj +1 ), j = 1, . . . , k r , and (b li , b li + 1), i = 1, . . . , k l , k l , k r ∈ IN , such that the mappings λ → g(λ) and λ → f (λ) are constant on the intervals (b li , b li+1 ), i = 1, . . . , k l , and (a rj , a rj +1 ), j = 1, . . . , k r , respectively.
For a convenience, assume that [a , where n l , n r ∈ IN , and a 1 = b 1 = a, and a nr = b n l = b.
We shall need the notion of Young measures (we will be highly selective and, for an application of Young measures in conservation laws, address a reader on famous paper [13]).  Theorem 2.1. [31] Assume that the sequence (u ε k ) is uniformly bounded in L p loc (IR + × IR d )), p ≥ 1. Then, there exists a subsequence (not relabeled) (u ε k ) and a family of probability measures exists in the distributional sense for all g ∈ C(IR). The limit is represented by the expectation valueḡ for almost all points (t, x) ∈ IR + × IR d .
We refer to such a family of measures ν = (ν (t,x) ) (t,x)∈I R + ×I R as the Young measure associated to the sequence (u ε k ) k∈N . Furthermore, where δ is the Dirac distribution.
Introduce the truncation operator s l,k (u) = max{l, min{k, u}}, l < k, l, k ∈ IR. The following important lemma holds. (15) where u 0 ∈ BV (IR; [a, b]) and f and g satisfy (23). Assume that the mapping λ → f (λ) is not constant on any subinterval of an interval (l, k). Then, the sequence (H(x)s l,k (v ε )) is strongly precompact in L 1 loc (IR + × IR). Similarly, if the mapping λ → g(λ) is not constant on any subinterval of an interval (l, k). Then, the sequence (H(−x)s l,k (v ε )) is strongly precompact in L 1 loc (IR + × IR).

Lemma 2.2. Denote by (v ε ) family of solutions to
Proof. Notice that from Lemma 1.8, it follows that for the family of functions v ε and any k, l ∈ IR, the families ∂ tq (x, H(x)s l,k (v ε )) + ∂ x q(x, H(x)s l,k (v ε )) and where the functionsq, q given by 16, are strongly precompact in W −1,2 loc (IR + × IR). Indeed, notice that Since [30,Theorem 6]), we conclude from (25) that (24) holds.
Furthermore, notice that if the mapping λ → f (λ) is not constant on any subinterval of an interval (k, l) then the vector (q(x, λ), q(x, λ)) from (16) is genuinely nonlinear on the interval (l, k) and x > 0. Indeed, for x > 0 the vector reduces to (f 2 α (λ), f α (λ)) and this is obviously genuinely nonlinear vector since, due to the assumptions of the lemma, for any ξ 0 , ξ 1 ∈ IR, it holds ξ 0 f 2 (λ) = ξ 1 f (λ) for a.e. λ ∈ (k, l). Now, from Theorem 0.8 and Lemma 1.8, we conclude that the family (H(x)s k,l (v ε )) is strongly precompact in L 1 loc (IR + × IR). In the completely same way, we conclude that the family (H(−x)s k,l (v ε )) is strongly precompact in L 1 loc (IR + × IR) if the mapping λ → g(λ) is different from a constant on every subinterval of the interval (k, l).

Next lemma deals with precompactness properties of the family
strongly in L 1 loc (IR + × IR). Moreover, the function v admits left and right traces at the interface x = 0.

Proof. Denoteṽ
according to assumptions (23). Then, notice that According to Lemma 2.2 and the definition of the functionṽ ε , it is easy to see that (ṽ ε ) is strongly precompact in L 1 loc (IR + × IR) (since this property has each of the summands on the right-hand side of (28)). Denote an accumulation point of the family (ṽ ε ) by v. Clearly, the function v satisfies (26).
Now, we can prove the main theorem of the paper. Proof. At the beginning, assume that u 0 ∈ BV (IR; [a, b]) and, as usual, denote by (v ε ) the family of solutions to (15). By applying the standard procedure (see proof of Lemma 1.8), it is not difficult to see that every v ε satisfies for every ξ ∈ IR: where O D ′ (ε) is a family of distributions tending to zero in the sense of distributions as ε → 0. Letting ε → 0 in (30) and taking Lemma 2.3 and Theorem 2.1 into account, we obtain in D ′ (IR + × IR): where ν t,x is a Young measure corresponding to the sequence (v ε ), and v is the function satisfying (26). The Young measure ν t,x and the function v (admitting strong traces at x = 0), we shall call an (α, β)-entropy admissible measure valued solution to (1). Denote by σ t,x a Young measure and by w a function representing an (α, β)entropy admissible measure valued solution to (1) corresponding to initial data v 0 ∈ BV (IR; [a, b]).
Using the classical arguments by DiPerna [13], we conclude that for any test function ϕ ∈ C 1 0 (IR + × (IR\{0})) it holds (keep in mind that α and β are strictly increasing functions): Now, we follow [19]. Take the function , and for an arbitrary ψ ∈ C 1 0 (IR + × IR), put ϕ = (1 − µ h )ψ in (32). We obtain: is the standard Landau symbol. Since v and w admit strong traces at x = 0, and since f α and g β satisfy the crossing conditions, as in [19, Theorem 2.1 ], we conclude that lim h→0 J(h) ≥ 0. From here, after letting h → 0 in (33), we conclude: and from here, using well known procedure [23], we conclude that for any T, R > 0 and appropriate C,R depending on R, the functions f , g, α and β: Taking u 0 = v 0 , we see from (34) that for almost every (t, x) ∈ [0, T ] × IR the Young measures ν t,x and σ t,x are the same and they are supported at the same point (since α and β are increasing functions). This actually means that σ t,x (λ) = ν t,x (λ) = δ(λ − u(t, x)) for a function u, where δ is the Dirac δ function. From Theorem 2.1, we conclude that v ε → u strongly in L 1 loc (IR + × IR) along a subsequence. The function u will obviously represent the (α, β)-entropy admissible solution to (1).
In order to prove that u is a unique (α, β)-entropy admissible solution to (1), we basically need to repeat the procedure from the first part of the proof.
Accordingly, take two (α, β)-entropy admissible solutions u and v to (1) corresponding to initial data u 0 and v 0 , respectively. By using the same argumentation as before, we reach to the relation analogical to (33): Introduce the functions Using the same arguments as in Lemma 2.3, we conclude that the functionsũ and v have strong traces at the interface Having this in mind, we conclude )H(x) + (g β (u) − g β (v))H(−x)) ∂ x ψdxdt ≥ 0, and from here, as usual, Since α and β are increasing functions on the range of u and v, from the above we immediately obtain the L 1 loc stability of the (α, β)-entropy admissible solutions to (1). Now, as in the last part of the proof of Theorem 1.3, we consider the case u 0 / ∈ BV (IR). We recall briefly the arguments providing the statement of the theorem in this case. First, we take a sequence (u 0ε ) of the functions of bounded variation such that u 0ε → u 0 in L 1 loc (IR). Then, we take the sequence (u ε ) of (α, β)-entropy admissible solutions to (1) with u 0 = u 0ε . The sequence (u ε ) satisfy: where R and T are arbitrary positive constants, and C,R are constants depending on R, the functions f , g, α and β. This readily implies that the sequence (u ε ) is convergent in L 1 loc (IR + × IR). Its limit is clearly an (α, β)-entropy admissible solution to (1). Uniqueness of such (α, β)-entropy admissible solution is proved in the completely same way as when u 0 ∈ BV (IR; [a, b]).
Proof. It is enough to notice that, since |u| + = |u|+u 2 , i.e. sign + (u) = (|u| + ) ′ = sign(u)+1 2 , relation (36) holds if we replace there sign by sign + . From that relation, the standard arguments provide From here, the statement of the corollary immediately follows. Now, we shall prove that we can always find α and β so that there exists a unique (α, β)-entropy admissible solutions to (1). Proof. First, notice that it is always possible to find constants k R and k L such that the translation functions α T (u) = u + k R and β T (u) = u + k L make f c αT and g c βT to satisfy the crossing conditions (see Figure 4). Furthermore, the constants a and b represent (α T , β T )-entropy admissible solutions to Indeed, denoting k(x) = k L , x ≤ 0 k R , x > 0 , according to Definition 0.4, we see that we