In this paper we consider the Cauchy problem
where , is a nonnegative finite Radon measure on , and , , is a Lipschitz continuous function (see assumption H1). Therefore, φ grows at most linearly.
Problem (P) with a superlinear φ of the type , , was studied in , proving existence and uniqueness of nonnegative entropy solutions (see also ). By definition, in that paper the solution for positive times takes values in , although the initial data 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 , in which the same definition of Radon measure-valued solutions used below (see equality (3.8)) was proposed.
When (), problem (P) is the Cauchy problem for the linear transport equation
whose solution is trivially the translated of along the lines (). In particular, the singular part of the solution is nonzero for if and only if the same holds for .
It is natural to ask what happens if φ is sublinear. To address this case we must consider solutions of problem (P) which, for , possibly are finite Radon measures on as the initial data . Therefore, throughout the paper we consider solutions of problem (P) as maps from to the cone of nonnegative finite Radon measures on , which satisfy (P) in the following sense: for a suitable class of test functions ζ, we have
(see Definition 3.3). Here the measure is defined for a.e. , is the density of its absolutely continuous part, denotes the duality map, and
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 of an entropy solution of problem (P) does not increase along the lines (see Proposition 3.8). In particular, if , the map is nonincreasing.
Concerning the case when φ is sublinear, the following example is particularly instructive:
with , and
The function in (1.2) is increasing and concave, with , 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 us define the waiting time for solutions u of (P):
(by abuse of language, we call “waiting time” even if ). Then, by Proposition 1.1,
More precisely, if , the singular part persists until the waiting time at which it disappears, whereas for , the singular part vanishes for all , thus – an instantaneous regularizing effect. Instantaneous regularization also occurs if (see  and Remark 3.24), whereas, as already remarked, in the linear case , we have if .
Since () is bounded if and only if , and , statement (*) could be rephrased as follows.
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 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 , of a waiting time was already pointed out in [10, Proposition 2.1] (see also Theorem 3.8 (ii)).
Namely, the regular part diverges when approaching from the right the point , where 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 is bounded in and is a finite superposition of Dirac masses (see Proposition 3.17 and Theorem 3.22). In  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 ).
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, 25, 24]). Mathematical modelling of the process leads to the Hamilton–Jacobi equation in one space dimension
where 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 , which formally solves (P) with . In this way, discontinuous solutions of (HJ) correspond to Radon measure-valued solutions of (P) having a Dirac mass concentrated at any point , where is discontinuous . 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 . We stress that this concept of “statistical solutions” is completely different from that of Radon measure-valued solutions, introduced by Demengel and Serre , 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 [6, 4, 5, 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 ), and an appropriate generalization of our results to the case of solutions with changing signs, when additional nonuniqueness phenomena (such as N-waves, see ) may occur; in this regard, the general case of an initial signed Radon measure 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 but the limit as 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.
2.1 Function spaces and Radon measures
We denote by the Banach space of finite Radon measures on , with norm . By , we denote the cone of nonnegative finite Radon measures; if , then we write if . We denote the convex set of probability measures on by . We have for .
We denote by the space of continuous real functions with compact support in . The space of the functions of bounded variation in is denoted by , where is the distributional derivative of u. It is endowed with the norm . We say that if for every open bounded subset .
The Lebesgue measure, either on or , is denoted by . Integration with respect to the Lebesgue measure on or on S will be denoted by the usual symbols dx, respectively . A Borel set E is null if . The expression “almost everywhere”, or shortly “a.e.”, means “up to null sets”. For every measurable function f defined on and , we write if there is a null set such that for any sequence , . We set for every measurable function f on .
We denote the duality map between and by . By abuse of notation, we extend to any μ-integrable function ρ. A sequence converges strongly to μ in if as . A sequence of (possibly not finite) Radon measures on converges weakly to a (possibly not finite) Radon measure μ, i.e., , if for all . Similar definitions are used for (possibly not finite) Radon measures on , with .
Every has a unique decomposition , with absolutely continuous and singular with respect to the Lebesgue measure. We denote by the density of . Every function can be identified to a finite absolutely continuous Radon measure on ; we shall denote this measure by the same symbol f used for the function.
The restriction of to a Borel set is defined by for any Borel set . Similar notations are used for the spaces of finite Radon measures , with , and , where .
We shall use measures which, roughly speaking, admit a parametrization with respect to the time variable.
We denote by the set of finite nonnegative Radon measures such that for a.e. , there is a measure with the following properties:
if , the map belongs to and
the map belongs to .
Accordingly, we set
The definition implies that for all , the map is measurable, thus the map is weakly measurable (e.g., see [22, Section 6.7]). For simplicity, we prefer the notation to the more correct one , which is used in .
If , then also and, by (2.1),
for . One can easily check that for a.e. ,
where denotes the density of the measure . For , we have
For any and , the translated measure is defined by
for any , where (). Clearly, and
2.2 Young measures
We recall the following result .
Let be Lebesgue measurable, let be closed, and let be a sequence of Lebesgue measurable functions such that
for any open neighborhood U of K in . Then there exist a subsequence and a family of nonnegative measures on , depending measurably on , such that
for a.e. ,
for a.e. ,
for every continuous function satisfying , we have
Suppose further that satisfies the boundedness condition
for every , where . Then
is a probability measure for a.e. ,
given any measurable subset , we have
for all continuous functions such that is sequentially weakly compact in .
Below we shall always refer to the family of probability measures given by the previous theorem as the disintegration of the Young measure τ (or briefly Young measure) associated to the sequence . We denote the set of Young measures on by ; in particular, denotes the set of Young measures on , with .
Condition (2.5) is very weak. It is equivalent to the statement that for any , there is a continuous nondecreasing function such that
Therefore, Theorem 2.3 applies to bounded sequences in (in which case ).
If is bounded and is a bounded but not uniformly integrable sequence in , 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).
Let be a bounded sequence in , where is a bounded open set. Moreover, let and be the subsequence and the Young measure given in Theorem 2.3, respectively. Then there exist a subsequence and a decreasing sequence of measurable sets of Lebesgue measure such that the sequence is uniformly integrable and
where is called the barycenter of the disintegration .
3 Main results
Throughout the paper we assume that . Concerning φ, we always suppose that
, , , and exists.
Hence, there exists such that
3.1 Definition of solution
In the following definitions, we denote by
the derivative of any along the vector .
By a solution of problem (P) in the sense of Young measures, we mean a pair such that
for a.e. , and
where is the disintegration of τ,
for all , with in , we have
where 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, , and for every pair , , such that
E is convex, , in , and , exist.
In (3.6), for a.e. , we set
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 as above, with E nondecreasing (respectively nonincreasing).
A measure is called a solution of problem (P) if for all , in , we have
The following proposition states that for any solution of (P) in the sense of Young measures, the map , possibly redefined in a null set, is continuous up to with respect to the weak topology of . In particular, it explains in which sense the initial condition is satisfied.
3.2 Existence and monotonicity
The existence of solutions is proven by an approximation procedure. If , then there exist such that
(e.g., see [23, Lemma 4.1]). Consider the approximating problem
A function is called an entropy solution of problem (Pn) if for every , with in and , and for any couple , with E convex and , we have
Entropy solutions are weak solutions if , in and
By studying the limiting points of the sequence , we shall prove the following result.
Let (H1) be satisfied. Then problem ( (P) ) has a solution u , which is obtained as a limiting point of the sequence of entropy solutions to problems ( (Pn) ). In addition, u is an entropy solution of problem ( (P) ) in the sense of Young measures.
Let (H1) and the following assumption be satisfied:
, and for every , there exist , , such that is strictly monotone in .
Then u is an entropy solution of problem ( (P) ).
The following proposition shows that the singular part of an entropy subsolution of (P) does not increase along the lines .
Let (H1) be satisfied.
3.3 Waiting time and regularity
It is convenient to distinguish two cases: (sublinear growth at infinity) and (linear growth at infinity), with defined by (H1).
3.3.1 Sublinear growth
Beside (H1), we will use the following assumption:
, , there exist , such that for all .
By (H2) the map is strictly negative and continuous in , hence two cases are possible: either (a) , , or (b) , in . In case (a), we have in , since and . Similarly, in case (b), we have plainly in . In particular, in both cases (H2) implies (C1). Moreover, if also (H1) holds, thus , we have in if and only if .
The following examples show that all values of may occur in (H2):
If φ is bounded in and, moreover, , , then
If φ is unbounded in , then .
Concerning estimates (3.21) and (3.22), it is worth considering the case in which and , . By explicit calculations, in Proposition 1.1, we show that in this case the waiting time defined in (1.5) is . Hence, in this case, estimates (3.21)–(3.22) are sharp, since
In part (ii) of Theorem 3.11, it is enough to require condition (H2) for large values of u. More precisely (see Remark 6.10), Theorem 3.11 (ii) remains valid if instead of (H2), for some , the following holds:
the function , , satisfies (H2).
In this connection, observe that the conditions and exclude the function . The same conditions also exclude the function , where . However, in this case, we can use hypothesis (H3) for , which is satisfied with and .
Let us finally mention the following regularization result.
It suffices to prove Proposition 3.10, Theorem 3.11 and Proposition 3.14 by assuming in (H2) (hence, , by (H2) and the assumption ). Otherwise, it can be easily seen that if is a solution of problem (P), the map defined by setting
for every is a solution of the problem
Here for all , and the function satisfies (H2) with . The same holds for entropy solutions.
3.3.2 Linear growth
Let φ satisfy the following assumption:
and there exist , such that
It is easily seen that if u is a solution (respectively an entropy solution) of problem (P), then , defined by
By Remark 3.16, the above results for the case can be generalized as follows.
Let be bounded in . If and , then
Let be unbounded in . Then .
In connection with equality (3.11), observe that if and the waiting time is equal to 0, then the map is not continuous at in the strong topology of (otherwise we would have , a contradiction). Instead, continuity along the lines may occur if the waiting time is positive.
Let (H1) be satisfied. Let be bounded in , and let satisfy
The following uniqueness result will be proven in Section 7.
Let (H1) be satisfied and let be bounded and monotonic in . Let satisfy (3.26). Then there exists at most one entropy solution u of problem (P) which satisfies either (3.24) or (3.25), and the condition
Conditions (3.24)–(3.25) in Theorem 3.22 cannot be omitted. In fact, there exist entropy solutions of problem (P) which do not satisfy either (3.24) or (3.25), depending on φ. Therefore, by Proposition 3.17, they are different from those given by Theorem 3.7, thus uniqueness fails.
For example, let and . Let be defined by
where is the unique entropy solution of problem (P) with replaced by . Since , one easily checks that (3.8)–(3.9) are satisfied, thus u is an entropy solution of (P). On the other hand, , so for a.e. , and (3.24)–(3.25) fails.
If is unbounded and satisfies assumptions (H1) and (H4), by [19, Theorem 1.1] and Theorem 3.18, for every there exists a unique entropy solution of problem (P) with waiting time equal to 0. In fact, every entropy solution u given by Theorem 3.18 is a solution according to . This follows if we show that
To prove (3.29) we fix . By (1.5) we may assume that and for all . By standard approximation arguments, we may substitute in the entropy inequality (3.9) , with , and . Hence, for a.e. and (3.29) follows.
4 Approximating problems
Let be a standard mollifier, let for , and set
(here for and for ). The regularized problem associated with (Pn) is
(where , ), has a unique strong solution , (e.g., see ). Some properties of the family are collected in the following lemmata. Up to minor changes, the proof is standard (e.g., see ), thus is omitted.
Let be the solution of problem (4.3). Then, for every and ,
Let φ satisfy (3.1). Then there exists , which only depends on , such that for all , and ,
Let , in , and set
Multiplying the first equation in (4.3) by gives
Hence, for all ,
By (3.1) and the definition of the function , for all ,
Choose , , with , and
for all and . Passing to the limit as , we obtain (4.7). ∎
Let φ satisfy (3.1) and let be such that
Then there exists such that for all and ,
If, moreover, , then the family , where
and , is bounded in .
Then it follows from (4.15) that
On the other hand, by (4.5) and since , we have
whence the result follows. ∎
From the above lemmata, we get the following convergence results.
By (4.4), in , where , and a.e. in S. The a.e.-convergence of and part (ii) follow from (4.19), and since converges uniformly to the continuous function φ on compact subsets of , we also obtain the a.e.-convergence of .
Set for and let . By Lemma 4.3, with , the sequence is bounded in and has a subsequence (not relabeled) such that
for some . Since in ,
whence for a.e. , and the convergence in (4.24) is satisfied along the whole sequence . Hence, for all , there exists a null set such that
Since is dense in and is separable, the choice of the set N can be made independent of ρ. Hence, we have proven (4.23).
Let . For all , problem (Pn) has an entropy solution , which is unique if φ is locally Lipschitz continuous. For a.e. , we have
Let ζ and E be as in Definition 3.6, and . Then
Let be such that for , if , and . Setting in (4.29) and letting , we get
since . On the other hand, by the monotone convergence theorem,
with as in (4.16). This completes the proof. ∎
5 Existence and monotonicity: Proofs
We proceed with the proof of Theorem 3.7.
For all , there exists a decreasing sequence of Lebesgue measurable sets, with as , such that
where is the Young measure associated with , and
for a.e. (here is the disintegration of τ).
Then the result follows by Theorem 2.5 and a standard diagonal procedure. ∎
The function in (5.2) is defined for a.e. in , since τ is globally defined in S. In addition, by (4.20) and the arbitrariness of L in Proposition 5.1, a routine proof shows that and a.e. in S. Therefore, the Radon measure (see (5.3)) is defined on S, , and
then, for all ,
where is defined by
If satisfies (5.5), there exists such that
Moreover, if , since and
Proof of Proposition 5.3.
For all , there exist such that
For any , , let be such that , , in , and . Then, by (5.8),
Since , it follows that is uniformly integrable in . Hence, by Theorem 2.3, for all ,
in . Here belongs to , since, by (5.7),
by letting in (5.13), we get plainly
From the above inequalities, the conclusion follows. ∎
as for . Moreover, for all , there exist a null set and a subsequence of (not relabeled), such that for all ,
Choosing in (5.15), we obtain that
If satisfies (5.5), and is bounded in (see (4.20) and (5.7)). Since every can be uniformly approximated in bounded sets by finite sums , with , bounded and continuous functions (; e.g., see [12, Théorème D.1.1]), it follows from (5.14) that, as , for all ,
Proof of Proposition 5.5.
where is defined by (4.21) and . Since, by Proposition 4.5, is bounded in if , there exists a subsequence which converges in . Combined with (5.18), this yields that in and in for all . Since the sequence is bounded in and , the condition may be relaxed to , and we have found (5.14).
(ii) Next we prove (5.14) for all (in this case ). To this end, let for any , where is a sequence of standard mollifiers ). Then , uniformly on compact subsets of and . By part (i) and (4.20), for all and , ,
where we have used Chebychev’s inequality and the inequality
Letting , since uniformly on compact sets in , we obtain
Since is a probability measure, we have as for a.e. , thus, by the dominated convergence theorem,
and, by (5.8), for all and ,
Since , the function belongs to . Then, by part (ii),
Then we obtain that
To complete the proof of (5.14), we show that
for a.e. . Since and as for a.e. , (5.24) follows from the dominated convergence theorem.
and (5.14) follows from the arbitrariness of ε.
Let U be a convex function with and . By (3.16),
for all and a.e. , where
as , where
belong to . In particular, setting , we have that
By (5.15) and a diagonal argument, there exist a null set and a subsequence, denoted again by , such that for all and ,
Since is bounded in and converges a.e. to , it follows from (3.15) that
for all and . Since for all (see (3.1)),
we have that , , and (as ) a.e. in S. Thus, by the dominated convergence theorem and (5.31), for all ,
The following result is based on the concept of compensated compactness (e.g., see ).
Let (H1) hold. Then a.e. in S.
for all and , and up to a subsequence,
for some . By the lower semicontinuity of the norm,
Let . Then (see (4.9))
for some , so for fixed , the family is uniformly bounded in . Similar results hold for V and , and letting in (5.35) along some subsequence (see the proof of Proposition 4.5), it follows from by (5.33) that for all and ,
Let be a bounded open set and let be defined by
Since are bounded in , the sequences , , and are bounded in and uniformly integrable, and, by Theorem 2.3,
in , where denotes the disintegration of the Young measure τ associated with . Since the sequences , , and are bounded in , they also converge weakly in , so
By a similar argument,
For every U as above with in , by a standard approximation argument, we may choose , so and, by (5.40),
Let satisfy (4.12) and
thus is bounded in for every . We claim that, as ,
and Proposition 5.8 follows from the arbitrariness of A.
On the other hand,
for some and sufficiently large, where . Hence,
To prove the second part of Theorem 3.7, we need the following result which characterizes the disintegration of the Young measure τ.
there exist , such that is constant in and, if , then is strictly monotone in and for some and .
Then, for a.e. , the following hold:
If , then .
If is strictly monotone in , with , , then
If is constant in the above interval for some , , then
where is the maximal interval where .
So let . Let , and
for and sufficiently large . Then as , and
Letting and arguing as in the proof of Proposition 5.8, we obtain that
Similarly, let and set
Letting in (5.40), with as above and , we obtain that
This implies that . Since is a probability measure and ,
Hence, and (5.49) follows since is a probability measure.
Similarly, if φ is strictly convex or strictly concave in , it follows from (5.52) that (we omit the details). Thus, , and arguing as above we obtain (5.49). (b) If φ is affine in for some , let be the maximal interval containing , where . If , (5.50) is satisfied. If , by (C3) and the maximality of I, φ is strictly convex (or concave) in for some (and affine in ). By (5.51), with , we obtain that
It follows that , whence . Similarly, if , by (C3) and the maximality of I, φ is strictly convex (or concave) in for some (and affine in ). Arguing as before, we obtain from (5.52), with , that (we omit the details). Summing up, we obtain (5.50): . ∎
If (C1) is satisfied for all , it follows from (5.49) and standard properties of narrow convergence of Young measures (see ) that in measure, where is the subsequence in Proposition 5.1. Therefore, up to a subsequence, a.e. in S. Hence, if φ is bounded, it follows from the dominated convergence theorem that in for all .
Now we can prove Theorem 3.7.
Proof of Theorem 3.7.
Let , with in , and let be such that . By (5.17), with and ,
(in this regard, see also (3.15)). Thus, the function given by Proposition 5.1 is an entropy solution of problem (P) in the sense of Young measures. By Proposition 5.8, it is also a solution in the sense of Definition 3.3. This proves the first part of the theorem. The second part is an immediate consequence of Proposition 5.9; in fact, (3.9) follows from (3.6) and (5.49). ∎
Let us end this section by proving Proposition 3.8.
Proof of Proposition 3.8.
For every , , we set and in the entropy inequalities (3.6) (). Then we get
where, for a.e. ,
As in the proof of Proposition 5.7, we have and as , whence
Let . By Definition 2.1 (for ), the map belongs to . Hence,
and . Letting in (5.53), we obtain that
Similarly, let . Setting in (5.53) and letting , we obtain that
(ii) It follows from (3.8) that for a.e. and ,
where is such that in , , and in , and . Since and , a routine proof shows that
Since for all , we also get that and as . Letting in (5.58), we obtain claim (ii). ∎
6 Regularity: Proofs
The first regularity result which we prove is Proposition 3.5. Hence, we need the following lemma.
Since , there exists a null set such that the spatial disintegration is defined for every . Arguing as in the proof of [23, Lemma 3.1], we can show that there exists a null set , , such that for every and ,
Proof of Proposition 3.5.
Let be the null set given by Lemma 6.1. Let , with as . Since, by (3.7), and , it follows from (6.1) that for all . Since, by Definition 2.1 (ii), , there exist and a subsequence such that in as . By standard density arguments, this implies that . Hence, along the whole sequence , and (3.11) follows from (6.1) and the arbitrariness of .
To prove (3.13), we observe that, given and two sequences and contained in and converging to , we have for all . Hence, if , the continuous extension of from with respect to the weak topology is well-defined. ∎
Let us now prove the results of Section 3.3. As explained there, replacing x by we may assume, without loss of generality, that ; namely, it suffices to prove Proposition 3.10, Theorem 3.11 and Proposition 3.14. Moreover, replacing x by and φ by , it suffices to do so by assuming that (H2) is satisfied with , in (see Remark 3.15). Therefore, we make use of the following assumption:
, , , and there exist , such that
(Recall that in this case and in .)
First we prove some estimates of the constructed entropy solutions. As already said, these estimates are analogous to the Aronson–Bénilan inequality for the convex case , (see ).
there exists such that
then , , and for every bounded open set and .
where 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 ). In particular, there exist a sequence and such that in and for all ,
Arguing as in the proof of Proposition 4.5 and letting , we obtain that
So satisfies (3.16) and, by Kružkov’s uniqueness theorem, . Hence, we have shown the following lemma.
Let (H1) and (H5) be satisfied, and let be the unique entropy solution of problem (Pn) given by Proposition 4.5. Then there exists a subsequence of solutions of (6.6) such that in and satisfies (6.7).
for all , and . Moreover, if (C4) is satisfied, then
For convenience, we set , thus in S. Let
It follows from (H5) and a straightforward calculation that
Proof of Proposition 6.2.
for all and . Hence, by Lemma 6.4,
for a.e. . Since is strictly decreasing in (recall that φ is concave by assumption H5), possibly extracting another subsequence (denoted again by ), a.e. in S (see Remark 5.10). Letting in (6.11)–(6.12) (with ), we obtain (6.4)–(6.5).
Since a.e. in S, there exists such that
for all , and , and, by (3.1),
whence, by (6.14) and the lower semicontinuity of the total variation,
and, by (6.14) and the lower semicontinuity of the total variation,
It remains to prove that . Observe that for all , , and , in , in Ω,
By (5.16) and the lower semicontinuity of the total variation,
So we may define for all such that . Since is arbitrary, the proof is complete. ∎
To prove Proposition 3.10, we need the following lemma.
Let (H1) be satisfied, and let u be the solution of problem (P) given by Theorem 3.7. Let be as in the proof of Theorem 3.7. Then, for a.e. and all , there exist a sequence and a subsequence of such that and as .
Let . We may assume that the convergence in (5.16) is satisfied for this t. Since , there is no neighborhood such that the sequence lies in a bounded subset of . Otherwise, up to a subsequence, in for some , . However, this would imply that in , a contradiction.
Setting , we obtain that for all . Hence, for all , there exists such that . ∎
Proof of Proposition 3.10.
By the proof of Lemma 6.5, for all ,
where . For every , let , . Multiplying (6.16) by , integrating by parts and setting , we find that
Let , and let , be as in Lemma 6.6, for a.e. . Let be fixed. Since , there exists such that for all . Consider any sequence , , in . Without loss of generality, we may assume that both and are Lebesgue points of for all . Setting and in (6.17), and letting , we find that
whence, by the invertibility of Ψ,
Letting in the previous inequality, we obtain (3.20). ∎
To prove Theorem 3.11, we need the following result.
the map belongs to ,
for all , ,
Proof of Proposition 6.7.
Letting , it follows from (6.23) that
Hence, the distributional derivative belongs to .
(ii) We set, for and ,
By standard regularization arguments, we can choose in (6.24) to obtain
By the dominated convergence theorem, as (), whereas, by part (i),
with Φ defined by (6.22).
Proof of Theorem 3.11.
(i) By (6.20), for a.e. ,
whence if . Hence, (3.21) follows.
Letting , by (6.26), we obtain, for a.e. and a.e. ,
Letting , we find in both cases that
(recall that we have assumed if φ is bounded; otherwise, if φ is unbounded, we have , since and in by H5). If , and , the sequence lies in a bounded subset of (thus, by (5.16) and ) for a.e. such that
This proves claim (ii) (a).
As we claimed in Remark 3.13, in Theorem 3.11 (ii), we may relax hypothesis (H2) to (H3), with . To prove this, for every , let be any sequence as in (3.14)–(3.15), and let be the entropy solution of problem (Pn). Set , where for every , and let be the entropy solution of the following problem:
(). A standard calculation shows that is an entropy subsolution of the above problem, whence
Following the proof of Theorem 3.7, the sequence converges to an entropy solution v of problem (P) with initial datum . Moreover, by assumption (H3), satisfies (H2) and we may apply Theorem 3.11 (ii) to v. Therefore, the conclusion follows from (6.28).
Proof of Proposition 3.14.
If , let and set . Since , we have that
We continue this construction recursively as long as , with : there exists such that, setting ,
Hence, this construction stops at some , and . Therefore,
Since as , the claim follows. ∎
7 Uniqueness: Proofs
Again, without loss of generality, we may assume that in the following proofs (see Remark 3.16).
Proof of Proposition 3.20.
(i) The first step of the proof consists in showing that
Let and , . Let be such that if and . Let . Setting and , we apply the -contraction property to the parabolic equation
First we let and then