Conservation laws and Hamilton–Jacobi equations with space inhomogeneity

Conservation laws with an x-dependent flux and Hamilton–Jacobi equations with an x-dependent Hamiltonian are considered within the same set of assumptions. Uniqueness and stability estimates are obtained only requiring sufficient smoothness of the flux/Hamiltonian. Existence is proved without any convexity assumptions under a mild coercivity hypothesis. The correspondence between the semigroups generated by these equations is fully detailed. With respect to the classical Kružkov approach to conservation laws, we relax the definition of solution and avoid any restriction on the growth of the flux. A key role is played by the construction of sufficiently many entropy stationary solutions in L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf{L}}^\infty }$$\end{document} that provide global bounds in time and space.


Introduction
This paper provides a framework where Cauchy problems for x-dependent scalar conservation laws, such as and Cauchy problems for x-dependent scalar Hamilton-Jacobi equations, such as are globally well posed and a complete identification between the two problems is possible.
The well-posedness of both (CL) and (HJ) is here proved under the same assumptions on the function H , which is the flux of (CL) and the Hamiltonian of (HJ). These assumptions define a framework included neither in the one outlined by Kružkov in his classical work [27] devoted to (CL), nor in the usual assumptions on (HJ) found in the literature, e.g., [3,4,14,25]. The identification of (CL) with (HJ) is then formalized, extending to the non-homogeneous case [26,Theorem 1.1], see also [10,Proposition 2.3]. This deep analogy also stems out from the direct identification of the constants appearing in the various stability estimates for the 2 equations, compare, for instance, (2.13) with (2.18).
For completeness, we add that a standard truncation argument could be used to extend Kružkov result to Example 1.1, as soon as the initial datum attains values between the stationary solutions u(t, x) = 0 and u(t, x) = 1. Note, however, that the a priori estimates and qualitative properties in Sect. 2.1 as well as the construction of stationary solutions in Sect. 2.2 are in general preliminary to any truncation argument. Technically, it is essentially due to our adopting (UC) that we can avoid truncation arguments. Moreover, such an argument applies to (CL) but hinders our simultaneous treatment of (CL) and (HJ). Thus, we provide an existence proof alternative to that by Kružkov and explicitly state the correspondence between (CL) and (HJ) in Sects. 2.3, 2.4 and 2.5.
To our knowledge, only few results in the literature focus on the (CL) ↔ (HJ) connection. The homogeneous, x independent, stationary case is considered in the BV case in [26] (by means of wave front tracking), see also [8, § 6] for the case of fractional equations. An extension to L ∞ is in the more recent [10] (where Dafermos' [15] theory of generalized characteristics play a key role). The stationary x dependent case is considered in [6] (using semigroups generated by accretive operators). Here, we deal with the non-stationary x dependent case, relying on vanishing viscosity approximations and on the compensated compactness machinery. In this connection, note that the techniques developed in [32,33] cannot be directly applied here, due to our need of passing to the limit also in the Hamiltonian.
Remark that in Kružkov's paper [27], the latter condition in (1.1) is essential to obtain uniform L ∞ and BV bounds on the sequence of viscous approximations in the case u o ∈ L ∞ (R; R). In our approach, which does not rely on (1.1), the L ∞ bound on viscous solutions depends on the fact that u o ∈ W 1,∞ (R; R). We thus need to devise new additional bounds, provided by the stationary solutions to (CL), see Sect. 2.2, which are specific to the non-viscous case, and allow to pass from data in W 1,∞ to data in L ∞ at the non-viscous level.
In the literature, a recurrent tool in existence proofs for (CL) is the (parabolic) Maximum Principle, see for instance [23,Theorem B.1,Formula (B.3)] or [24, § 3.2], which provides an a priori uniform bound on vanishing viscosity approximate solutions, which is an essential step in passing to the vanishing viscosity limit. More precisely, only in the homogeneous case where ∂ x H ≡ 0, the Maximum Principle ensures that 1. vanishing viscosity approximate solutions have a common L ∞ bound, and 2. this bound only depends on the L ∞ norm of the initial datum.
In the present-non-homogeneous-case, we replace (1) obtaining L ∞ bounds on vanishing approximate solutions by means of a, here suitably adapted, Bernstein method, see [39, § 6] for a general introduction. This requires a higher regularity of the initial datum and (2) above is irremediably lost.
However, in the homogeneous case, one also takes advantage of the fact that constants are stationary solutions, ensuring 2. easily. This fact fails in the nonhomogeneous case. Below, we exhibit (sort of) foliations of R × [U, +∞[ and R×] − ∞, −U] (for a sufficiently large U) consisting of graphs of stationary solutions to (CL), each contained in a level curve of H . Then, solutions to (CL) are well known to preserve the ordering [16,Formula (6.2.8)] and 2. follows. Note that these stationary solutions need to be merely L ∞ . Therefore, in their construction, the choice of jumps deserves particular care to ensure that they turn out to be entropy admissible. In general, the solutions to (HJ) corresponding to stationary solutions to (CL) may well be non-stationary.
The differences between the construction below and the classical one by Kružkov [27] arise from the different choices of the assumptions but are not limited to that. Indeed, the two procedures differ in several key points. In [27], uniform L ∞ "parabolic" bounds on vanishing viscosity approximate solutions to (CL) are obtained and L 1 compactness follows from Kolmogorov criterion. Here, the stationary solutions constructed as described above allow to obtain L ∞ "hyperbolic" bounds directly on the solutions to (CL), while it is an application of the compensated compactness machinery that ensures the existence of a limit, thanks to our modified (weakened) definition of solution. Under (WGNL), also the kinetic approach in [30,34] is likely to allow for analogous results. Moreover, in [27] the term −∂ x H is essentially treated as a contribution to the source term. Here, we exploit the conservative form of (CL), thus respecting the analogy between (CL) and (HJ). Our weakening of Kružkov definition, motivated also by our use of compensated compactness, avoids any requirement on the trace at time 0+. It is of interest that this construction actually relies also on a sort of stability with respect to the flux H , where condition (WGNL) appears essential.
However, continuity in time, not proved in [27], is recovered in weak- * L ∞ loc (R; R) in Proposition 2.5 and in L 1 loc in Theorem 2.6, always relying exclusively on condition (C3). Differently from [7,42], condition (WGNL) plays here no role. Thus, in the present setting, the trace at 0+ condition [27,Formula (2.2)] can be omitted from the definition of solution to (CL) without any consequence.
Throughout this paper, we alternate considering (CL) and (HJ), simultaneously gathering step by step results on the two problems. When H does not depend on the space variable x, [26,Theorem 1.1] and [10, Proposition 2.3] ensure the equivalence between (CL) and (HJ). In the space homogeneous case, the correspondence between (CL) and (HJ) is exploited in [5,31] and it is particularly effective in the characterization of the initial data evolving into a given profile at a given time, see [10,28]. Below, we extend this equivalence to the x dependent case, while [11] is devoted to the inverse design problem in the x-dependent case. This correspondence may also suggest new properties of (CL) or (HJ), proving them in the present framework, posing the question of an intrinsic proof in more general settings, see Remark 2. 22. As a matter of fact, our original goal was the detailed description of the relation between (CL) and (HJ), but such a correspondence requires the two Cauchy problems to be settled in the same framework.
In this paper, results are presented in the paragraphs in Sect. 2, while all proofs are collected in the corresponding paragraphs in Sect. 3. The main goal of this paper are the results in Paragraph 2.5.

Main results
Throughout this work, T denotes a strictly positive time or +∞.

Definitions of solution, local contraction and uniqueness
In this paragraph, we let u o ∈ L ∞ (R; R) while we require exclusively (C3) on H . No genuine nonlinearity condition is assumed, not even (WGNL), differently from [7,42] (that have different goals and motivations).
Concerning the notion of solution to (CL), we modify that in the sense of Kružkov [27,Definition 1]. Indeed, in view of the compensated compactness technique used below, we do not require continuity in time in the sense of [27,Formula (2.2)]. On the contrary, full L 1 loc continuity in time is here proved, merely on the basis of (C3). With reference to (CL), the following quantity often recurs below, where x, u, k ∈ R: In (2.2), the integral term on the last line allows to avoid requiring the existence of the strong trace at 0+, as required in [ We recall what we mean by entropy-entropy flux pair for (CL).
and F ∈ Lip(R 2 ; R) is an entropy-entropy flux pair with respect to H if for all x ∈ R and for a.e. u ∈ R The classical Kružkov choice in (2.4) amounts to set, for k ∈ R, where k ∈ R, which applies also when E is merely in C 0 (R; R for any test function in the sense of distributions. As a first step, we prove that Definition 2.1 ensures the weak- * L ∞ loc (R; R) time continuity.
Proposition 2.5. Let H satisfy (C3). Fix the initial datum u o ∈ L ∞ (R; R). Assume that the Cauchy Problem (CL) admits the distributional solution u in the sense of Remark 2.2. Then, for all a, b ∈ R with a < b, setting Even without the nonlinearity condition (WGNL), we can single out a particular representative of any solution, so that we obtain the continuity in time in the (strong) L 1 loc topology, the uniqueness of solutions and their stability with respect to initial data for all times. Indeed, the next theorem shows that (2.9) and (2.10) hold at every time and with the same K C L , provided at all times suitable representative u * (t, ·) is carefully chosen.   13) and assume L < +∞. Then, all representatives u * and v * satisfying Item 1 above are such that for all t ∈ [0, T ] and for all R > 0 In particular, Remark that Definition 2.1 implies that C < +∞ in (2.14). Then, condition (CNH), if assumed, ensures that L is finite.
Turning to the Hamilton-Jacobi equation (HJ), recall the apparently entirely different framework of the standard Crandall-Lions definition of viscosity solutions.
and for all ( We have for all t 1 , t 2 ∈ [0, T ] Remark that the Lipschitz continuity assumptions in Item 2 of Theorem 2.8 precisely mean that C < +∞. Requiring also condition (CNH), then ensures that L is finite. We underline the evident deep analogy between Theorem 2.6 referring to the conservation law (CL) and Theorem 2.8 referring to the Hamilton-Jacobi equation (HJ). The definitions (2.13) and (2.18) are essentially identical. Note moreover that the factor 2 appearing in (2.8) and not in (2.17) is a mandatory consequence of the correspondence between the two equations formalized in Sect. 2.5.

A bounding family of stationary solutions
Essential to get the necessary global in time L ∞ bounds on the solutions to (CL) is Theorem 2.9. In the homogeneous case, a sufficient supply of stationary solutions is immediately provided by constant functions, which are clearly also entropic. Here, we need to find L ∞ solutions that, first, are entropic and, second, are sufficiently many to ensure the necessary L ∞ bounds, together with the order preserving property (2.15) in Theorem 2.6.
The proof begins with a careful construction of piecewise C 1 stationary entropic solutions by means of the Implicit Function Theorem and Sard's Lemma for a particular class of fluxes whose level sets enjoy suitable geometric properties. Then, compensated compactness allows to pass to the limit on the fluxes, essentially showing a stability of solutions with respect to the flux, thus getting back to the general case. In this connection, we recall that already in [1,2] stationary solutions are assigned a key role in selecting solutions.
In the correspondence between (CL) and (HJ), the stationary solutions to (CL) constructed in Theorem 2.9 have as counterpart viscosity solutions to (HJ) that may well be non-stationary, see (2.28), and are Lipschitz continuous but, in general, not differentiable.

Vanishing viscosity approximations
We now proceed toward existence results both for (CL) and for (HJ), obtained through vanishing viscosity approximations, under the assumptions (C3)-(CNH)-(UC). Thus, we consider the Cauchy problems (2.20) and A classical solution to (2.21) on ]0, T [×R is a function Note that (2.23) in Definition 2.10 requires 3 space derivatives in U , although the third derivative does not appear in (2.21).
We now prove that the Cauchy problems (2.20) and (2.21) are equivalent.
is the solution to ( Since T is arbitrary both in Theorem 2.11 and in Theorem 2.12 and moreover M in (2.25) is independent of T (and ε), both results apply also to the case T = +∞.  Thanks to Theorem 2.11, applied with I = R, the proof of Corollary 2.15 is a direct consequence of Theorem 2.14 and is hence omitted.

Existence of vanishing viscosity limits
We now deal with the vanishing viscosity limit of the solutions constructed in the previous Paragraph. Differently from [27], we complete this step in the case of more regular initial data, i.e., in the case where Theorem 2.12 and Corollary 2.13 apply.
. Let ε n be a sequence converging to 0. Then, the sequence U ε n of the corresponding classical solutions to (2.21) on R converges uniformly on all compact subsets of R + × R to a function U * ∈ Lip(R + × R; R) which is a viscosity solution to (HJ).
Striving to treat (CL) and (HJ) in parallel, the next statement mirrors the previous one. The proof, entirely different from that of Theorem 2.16, by means of (WGNL), relies on an ad hoc adaptation of classical compensated compactness arguments, see [16,Chapter 17] or [38, Chapter 9].

The limit semigroups and their equivalence
Here, we complete all previous steps obtaining the main results, stated in terms of the existence of the semigroups generated by (CL) and (HJ), their properties and their connection. There exists a unique semigroup S C L : in the sense of Definition 2.1 and enjoys the properties: 3.a For all u o ∈ L ∞ (R; R), the map t → S C L t u o is Lipschitz continuous with respect to the weak- * L ∞ loc (R; R) topology in the sense that there exists a K > 0 such that for all a, b ∈ R with a < b and for all t 1 ,

3.b
For all u o ∈ L ∞ (R; R), the map t → S C L t u o is continuous with respect to the L 1 loc (R; R) topology, in the sense that for allt ∈ R + and for all R > 0 , define L as in (2.13). Then, for all t ∈ R + and for all R > 0,   (2.18). Then, for all t ∈ R + and for all R > 0,

Theorem 2.20. Let H satisfy assumptions (C3)-(CNH)-(UC)-(WGNL). Let the
Then, problems (CL) and (HJ) are equivalent in the sense that for all t ∈ R + and for a.e. x ∈ R, Remark 2.21. In the same setting of Theorem 2.20, formally, as a consequence of (2.27), for a fixed x o ∈ R, we can write (2.28) The latter integral on the right hand side in (2.28) is meaningful only under further regularity conditions, such as in the case H is convex in u, which ensures that S C L t u o ∈ BV(R; R).
We can rephrase the above relations with the following commutative diagrams.
We do not know of a proof of this bound for (HJ) independent from (CL).

Analytical proofs
Throughout, 1 we have The expression on the right in (3.2) is relevant when u = p k . Indeed, it allows to prove that the bound on the derivatives in (3.3) holds at every u and not only at a.e. u. Proof of Lemma 3.1. Let δ be a modulus of uniform continuity of E on the interval [−r, r ] corresponding to min{ε, ε/(2r )}, so that Choose n in N such that n ≥ 2r/δ. Define the points p k and the map α : R → R by dv so that the condition on the left in (3.3) is satisfied byη, as well as the one on the right for u = p k . Requiring the weights w 0 , . . . , w n to solve the (n+1)×(n+1) The matrix of the above system is i.e., a i j = 2r n |i − j| for i, j = 1, . . . , n + 1 and straightforward calculations show that its determinant is (−1) n r 2 n . Hence, this matrix is invertible, so that the weights w 0 , . . . , w n are uniquely defined. Moreover, differentiatingη we getη ( We are left to prove that the expression for η in (3.2) satisfy (3.3) also at u = p k . Since w k ≥ 0, by the choice (3.1) and by the construction above, we Possibly erasing the terms vanishing because w k = 0, the proof is completed.
By the linearity in the entropy/entropy flux and by the positivity of the weights, To estimate the term (3.5), recall that from (2.6) so that also using (3.2) and (3.7) Adding the resulting estimates, we obtain where O(1) depends only on ϕ and on H . The proof of Claim 1 is completed.
Clearly, E ε is C ∞ , F ε is C 1 and are an entropy-entropy flux pair in the sense of Note that (3.8) and (3.9) ensure the uniform convergence on compact sets of E ε to E and of F ε to F as ε → 0+. Therefore, it is immediate to pass to the limit ε → 0+ in (3.10) and (3.12). Indeed, with the notation (2.1), Consider now (3.11). Definition (3.9), (2.6) and (C3) ensure that ∂ x F ε converges uniformly on compact sets to ∂ x F. To deal with the term E ε , write Since ρ ε is even, we have that E ε converges pointwise everywhere to E as ε → 0+, with E ≤ 1. Thus, the Dominated Convergence Theorem [22,Theorem (12.24)] allows to pass to the limit also in (3.11): Combining the obtained estimates of the limit ε → 0+ of the terms (3.10)-(3.11)-(3.12) we get (2.2), completing the proof of Claim 2 and of Proposition 2.4.
Proof of Proposition 2.5. We adapt the arguments in [15,Lemma 3.2]. Therein, a similar result is obtained in a different setting: a source term is present, the flux is also time dependent but convex in u. Furthermore, the definition of solution in [15] requires the existence of both traces is required at any point for all time.
Proof of (2.10). Fix a, b ∈ R with a < b and t 1 , Recall the Definition (2.8) of K C L , so that the first line above is estimated as follows: (3.14) To compute the limit as ε → 0 of the left hand side in (3.14), observe first that An entirely similar procedure yields . Thus, if t 1 and t 2 are Lebesgue points [19, The latter relations, together with the limits (3.15) and (3.16), inserted in (3.14) complete the proof of (2.10).
and ψ ε is as in (3.13). Repeat a procedure analogous to the one above choosing fort a Lebesgue point of the map t → b a u(t, x)dx. The use of equality (2.3) in Remark 2.2 allows to let u o appear explicitly.
The proof of Proposition 2.5 is completed.
Proof of Theorem 2.6. Fix a representative u of a solution to (CL) in the sense of Definition 2.1.

Claim 1:
There exists a u * such that u * = u a.e. and u * satisfies (a) and (b) in Item 1. By (2.9)-(2.10), for all a, b ∈ R with a < b, there exists a negligible set N a,b ⊆ [0, T ] such that (2.10) holds for all t 1 , t 2 ∈ R + \N a,b and (2.9) holds for allt ∈ R + \ N a,b . Define which is also negligible by the definition of the L ∞ norm and by Fubini Theorem [22,Theorem 21.13] (set on the left) and by the choice of N a,b (union on the right). Note that for allt, t 1 , t 2 ∈ [0, T ]\N and for all a, b ∈ Q, u satisfies (2.9) and (2.10).
Fix now a, b ∈ R with a < b. Choose an increasing sequence a n and a decreasing sequence b n , both of rational numbers, such that lim n→+∞ a n = a, lim n→+∞ b n = b and a n < b n . Then, b n a n u(t, x) − u o (x) dx and b n a n (u(t 2 , x) − u(t 1 , x)) dx are uniformly bounded by the right hand sides in (2.9) and in (2.10). The Dominated Convergence Theorem [22,Theorem (12.24)] thus applies proving that u satisfies (2.9) and (2.10) for allt, t 1 , t 2 ∈ [0, T ] \ N and also for all a, b ∈ R.
Hence, for any real bounded interval I , R for a constant C I depending on I . This bound then holds also for all piecewise constant functions and, by further approximations, we know that for all is uniformly continuous with respect to the weak- * L ∞ loc (R; R) topology. Apply now Proposition A.1, which is possible since L ∞ (R; R) is weakly- * complete (as it follows, for instance, from Banach-Alaoglu Theorem [37, Theorem 3.15 and Theorem 3.18]), and obtain an extensionū of u which is defined on all [0, T ], attains values in L ∞ (R; R) and is continuous with respect to the weak- * L ∞ loc (R; R) topology. The bound (2.9) also ensures that lim t→0+ū (t) = u o in the weak- * topology of Fix a strictly convex entropy E ∈ C 1 (R; R). Choose a corresponding entropy flux F by means of (2.4). With reference to (2.7), introduce the function G ∈ L ∞ (R 2 ; R) For n ∈ N\{0} and τ > 0, choose the test function ϕ n, Proceed now as in the Proof of Proposition 2.5. If τ ∈ P ψ , then By (3.19), for all τ ∈ P as defined in (3.18) Fix a positive R. Choose a sequence τ n ∈ P with τ n −→ n→+∞ 0. By [19, Chapter 1, § 9, Theorem 1.46], the sequence u * (τ n , ·) admits a subsequence u * (τ n k , ·) and, for a.e. x ∈ R, a Young measure [19, Chapter 1, § 9, Since G τ ψ ≥ 0 and thanks to the Dominated Convergence Theorem [22,Theorem (12.24) On the other hand, by Claim 1, . The choice of the τ n is arbitrary, up to the set P, as is the choice of R. Hence, Claim 2 is proved.
Claim 3: For all R > 0 and for all t 1 ∈ P, For ε > 0 and t 2 > t 1 > 0, choose the test function χ ε as in (3.13) and define Proceed now as in the Proof of Proposition 2.5 and as in Claim 2. If t 1 , t 2 ∈ P as defined in (3.18), then Proceed now exactly as in the previous Claim 2 to complete the proof of Claim 3.

Claim 4: For allt
Use ϕ ε as a test function in (2.2) in Definition 2.1. Then, where in the last line above we used Claim 3. Claim 4 is proved.

Claim 5: (c) in Item 1 holds.
For any R > 0 define also in the sense of [ where we set The above inequality shows that the map is uniformly continuous. Hence, it can be uniquely extended to a continuous map defined on all of [t 1 , t 1 + 1/ R ]. Since Claim 1 ensures that u * is continuous in the weak- * L ∞ loc (R; R) topology, this extension coincides with u * . Claim 5 follows becausē t ∈]t 1 , t 1 + 1/ R [.

Claim 6: Item 2 holds.
Let u * , v * be solutions to (CL) with data u o and v o , satisfying (c) in Item 1, proved in Claim 5. Then, u * and v * are also solutions to (CL) in the sense of [27,Definition 1]. By [27, Theorem 1 and Theorem 3], which we can apply thanks to (C3), we have that if L in (2.13) is finite, for all R > 0 and for almost all t ∈ [0, T ] the following estimates hold: Use the L 1 loc (R; R) continuity to obtain the above inequalities for all t ∈ [0, T ], proving Claim 6 and thus completing the proof of Theorem 2.6.
Claim 3: Let C, L be as in (2.18). Then, Formula (3.23) can be rewritten as which, passing to the infimum over q, also proves the third line in (3.24). The first line is analogous and the middle one follows from Claim 1, completing the proof of Claim 3.
Passing to the limits x → x o ±, we get |∂ x ϕ(t o , x o )| ≤ C hence, by Claim 1 and using the fact that U is a subsolution of (HJ), To complete the proof of Claim 4, repeat the same procedure with the supersolution V . Choose and define, for A > 0, Claim 5: U A is a strict subsolution of ∂ t w +H (x, ∂ x w) = 0 on as defined in (3.22).
Let ϕ ∈ C 1 ( ; R), (t o , x o ) ∈˚ such that U A −ϕ has a point of maximum at (t o , x o ). Then, γ ∈ C 1 ( ; R), since by the Definition (3.25) of χ , γ locally vanishes near x = 0 for t < τ. The regularity of ϕ combined with that of (t, together with Claim 4, ensures that where Claim 2 was used. Recall that by (3.26) completing the proof of Claim 5. For all (t, x) ∈ , by the compactness of and the continuity of U , V . Introduce a maximizing sequence If |x| = R + L (τ −t), then, by (3.22), we have the bound that would once again imply U A (t n , x n ) − V (t n , x n ) −→ n→+∞ − ∞, which is not acceptable, since (t n , x n ) is a maximizing sequence, completing the proof of Claim 6.
Let ω V be a modulus of continuity of V in (t, x) on and compute: proving the first limit in Claim 8. To prove the second one, refine the computations (3.28)-(3.29) above as completing the proof of Claim 8.
Using Claim 9, we can introduce a sequence ε n converging to 0, such that 1 ε n 2 (x ε n − y ε n ) →p for a suitablep ∈ [−C, C] and so that t ε n −→ n→+∞t and x ε n −→ n→+∞x for a suitable (t,x) ∈ . By Claim 8, we also have that s ε n −→ n→+∞t and y ε n −→ n→+∞x . Then, we are proceeding by contradiction,t > 0 and for all n sufficiently large, also t ε n > 0, so that (t ε n , x ε n ) ∈˚ and also (s ε n , y ε n ) ∈˚ .
Let now n be sufficiently large and consider the maps The former one admits a maximum at (t ε n , x ε n ), while the latter admits a minimum at (s ε n , y ε n ). Since U A is a subsolution and V is a supersolution, by (3.27) in the proof of Claim 5 and Claim 4 we have Take the difference between the last lines above, let n → +∞ and we get the contradiction: A/(τ −t) 2 ≤ 0, proving Claim 10.

Conclusion.
For Hence, using Claim 10, for fixed (t, x) ∈ , and in the limit By the continuity of U − V , the latter inequality holds for all (t, x) ∈ , completing the proof of Item 2 in Theorem 2.8. Let ϕ ∈ C 1 (R 2 ; R) and fix (t, x) ∈ R 2 such that V − ϕ has a point of minimum at (t, x). For all ε ∈ R, if |ε| is sufficiently small, then

Proof of Item 1 in Theorem
Again for |ε| is sufficiently small, x)) ≥ 0, proving Claim 1. The proof of this claim is entirely analogous to that of the previous one.

Conclusion.
We apply Item 2 in Theorem 2.8, which was proved above, on [s, +∞[×R to the couples of subsolution-supersolution (U, V ) and (W, U ) to get for all (t, x) ∈ [s, +∞[×R and by the arbitrariness of (s, y) we complete the proof of Item 1 in Theorem 2.8

Existence of helpful stationary solution
Here, we prove Theorem 2.9, which yields, for all U ∈ R, 2 stationary entropic solutions u − and u + to (CL) such that |u ± | > U . We detail the case of u + , that of u − is similar. Further information and visualizations of the solutions constructed below, together with hints to their role as asymptotic states, can be found in [12].
Claim 1: There exist increasing sequences a n and b n converging to 0 such that for all n ∈ N, (a n , b n ) is a regular value for G and a o > −1, b o > −1.
This claim follows from Sard's Lemma A.3 applied with f = G, k = 1, n 1 = n 2 = 2. Remark that here condition (C3) is fully exploited. The assumption (3.31) allows to introduce

Claim 2: Y is negligible and P is countable.
The former statement directly follows from Sard's Lemma A.3 applied first with f = H then with f = H n and k = 3, n 1 = 2, n 2 = 1. Fix n ∈ N and define Recall that (a n , b n ) is a regular value for G, so we have that Q n is discrete, hence countable. As a consequence, also H n (Q n ) is countable.
This holds for all n ∈ N, hence P = n∈N H n (Q n ) is countable, proving Claim 2. Define, using (CNH), |H (x, u)| and note that the set ]H 1 +U + 1 2 U 2 , +∞[\(Y ∪P) is not empty by Claim 2 and (3.32). ChooseH in this set and with this choice, items 1, 2 and 5 hold by construction.
We have H (x, u) = |H (x, u)| >H and since for all n ∈ N, a n < 0, b n < 0, we also have H n (x, u) ≥ H (x, u) > 0. Claim 4 is proved, as is Lemma 3.2.
Call π x : R × R → R the canonical projection π x (x, u) = x. Introduce the set (corresponding to the diamonds in Fig. 1, right) Define y * := inf Y where, denoting co(A) the convex hull of A and using the notation (2.1), (3.41) Above, u piecewise C 1 on [y, X ] means that that there exist finitely many pairwise disjoint open intervals I such that [y, X ] = I , u |I ∈ C 0 (I ; R) and u |I ∈ C 1 (I ; R).

Claim 3: y * ∈ Y.
The Implicit Function Theorem and Claim 1 ensure that Y contains a left neighborhood of X , so that Y = ∅. Moreover, Y ⊆ [−X, X ], so that y * = inf Y is finite.
If X = ∅, defineȳ := X . Otherwise, note that there existsȳ ∈ Y such thatȳ < min(X ∩]y * , X ]), since X is finite by Claim 2 and by the properties of the infimum. In both cases, there exists a map u satisfying (i), (ii), (iii) and (iv) in (3.41) defined on [ȳ, X ]. An application of the Implicit Function Theorem, since ]y * ,ȳ] ∩ X = ∅, allows to extend u down to y * so that u |[y * ,ȳ] is C 1 . Hence, y * ∈ Y, proving Claim 3.
Call u + the map corresponding to y * ∈ Y as defined in (3.41) and set u * := u + (y * ).
Claim 4: y * = −X Assume y * > −X . Then, consider first the case ∂ u H (y * , u * ) = 0. The Implicit Function Theorem ensures that u + can be extended toward left in a C 1 way (so that the properties defining Y remain trivially satisfied), contradicting the above construction.
Case 1: Suppose that H (y * , u) <H for all u ∈]u * , u * + ε * [. Introduce , V is bounded above by V and we can introduce v * := sup V, which is finite. Note that for u near to v * showing that v * is neither an isolated point of maximum nor an isolated point of minimum of u → H (y * , u). By (3.39), it then follows that ∂ u H (y * , v * ) = 0 and, hence, ∂ u H (y * , v * ) > 0. Apply now the Implicit Function Theorem on the level set Clearly, u is piecewise C 1 . Moreover, it satisfies (i), (ii) and (iii) because u * and ψ (thanks to the definition of v * as the supremum of V) satisfy them. Concerning (iv): if y < y * , simply note that ψ is C 1 ; for y > y * , u + satisfies (iv) and at y = y * we have u (y * +) = u * , u (y * −) = v * and by the definition of v * , v * > u * and for all k ∈]u * , v * [ by (3.42), H (y * , k) ≤H = H (y * , v * ). This implies y * − η ∈ Y, which contradicts the choice y * := inf Y. Case 2: Otherwise, since u → H (y * , u) is continuous, a connectedness argument ensures that H (y * , u) >H for all u ∈]u * , u * + ε * [. We have ∂ u H (y * , u * ) = 0, so ∂ 2 uu H (y * , u * ) ≥ 0 and by (3.39), ∂ 2 uu H (y * , u * ) > 0. Thus, for all u ∈]u * − ε * , u * [, H (y * , u) >H . We now proceed as in the case above. Introduce (3.36), V is bounded below by U and we can introduce v * := inf V, which is finite. Note that for u near to v * showing that v * is neither an isolated point of maximum nor an isolated point of minimum of u → H (y * , u). By (3.39), it then follows that ∂ u H (y * , v * ) = 0 and, hence, ∂ u H (y * , v * ) > 0. Apply now the Implicit Function Theorem on the level set H (x, u) =H in a neighborhood of (y * , v * ), obtaining a map x → ψ(x) defined on ]y * − η, y * + η[. Define Clearly, u is piecewise C 1 . Moreover, it clearly satisfies (i), (ii) and (iii) because u * and ψ satisfy them. Concerning (iv): for y < y * , ψ is C 1 ; for y > y * , u + satisfies (iv) and at y = y * we have u (y * +) = u * , u (y * −) = v * and by the definition of v * , v * < u * and for all k ∈]u * , v * [, H (y * , k) >H = H (y * , v * ). This implies y * −η ∈ Y, which contradicts the choice y * := inf Y. Claim 4 is proved

Conclusion.
First, extend u + on ]−∞, −X ] setting it to be constant and, separately, on [X, +∞[ also setting it to be constant. Note that u + is of class C 1 both on a neighborhood of −X and on a neighborhood of X , since by (CNH), ∂ x H (±X, u) = 0 for all u and thanks to (ii) in (3.41).
Call p 1 , p 2 , . . . , p n (with p i < p i+1 ) the points of jump in x → u + (x), they are finitely many by the Definition (3.41) of Y and that of u + . For later use, let p 0 := − X and p n+1 := X . We know that u + ∈ C 1 . . . , n. When x is different from all p 1 , . . . , p n and, using [27,Lemma 3], compute since, by the definition of u + , H (x, u + (x)) ≡H . Fix t ∈ R + and compute: We thus obtain and we compute the generic i-th term of the latter sum as where we used H (x, u + (x)) =H for all x. Clearly, if k ∈ co {u + ( p i −), u + ( p i +)}, the latter term vanishes. Assume k ∈ co {u + ( p i −), u + ( p i +)}. Then, property (iv) in (3.41) ensures that sgn (u On the other hand, being k between u + ( p i −) and u + ( , so that the difference (3.44) is nonnegative and so is the test function ϕ. The proof of Lemma 3.3 is completed.

Lemma 3.4. Let H satisfy (C3)-(CNH)-(UC) and moreover
Let U and V be positive real numbers andH be negative such that Then, there exist a stationary solution u When (3.35) is replaced by (3.45), the above procedure can be repeated with essentially only technical modifications. We list below the various steps, omitting the details. We stress that it is critical that the case below be treated "from left to right", i.e., from −X to X , corresponding, with the terminology of the previous proof, to y * = sup Y.
Proof of Lemma 3.4. Referring to the proof of Lemma 3.3, we only describe below the necessary modifications when (3.45) substitutes (3.35).
Introduce the set Claim 2 is modified to: X is finite. Define y * = sup Y, where, using the notation (2.1), is a positive measure and satisfies for all Given an entropy E ∈ C 2 (R; R), we can introduce by means of (2.6) the corresponding flux We apply [38, Lemma 9.2.1], which we adapt here to the present (stationary) situation. By (3.51), using Proposition 2.4, straightforward computations yield: u n (x)) .
for any g ∈ C 0 (R; R) and for any ϕ ∈ L 1 (R; R). Clearly, we also obtain that for any Recall that u n ∈ L ∞ (R; [U, V ]). In view of our later use of Fubini Theorem, we use Stone-Weierstrass Theorem [22,Corollary 7.31] so that for every δ > 0 there exist a ν ∈ N and functions f 1 , . . (3.55) Since G satisfies (CNH), for = 1, . . . , ν, introducing the functions we can extend the latter statement (3.55) to Recall that the support of ν x is included in [U, V ] for a.e. x. Then, and each term in the latter sum above converges to 0 by (3.54), since eachf ϕ is in L 1 ([0, T ] × R; R). Passing to the lim sup and using the arbitrariness of δ, Claim 2 is proved.

Claim 3: For any
The above assumptions ensure that G satisfies the hypotheses of Claim 2. Therefore, where we used (3.51) and Claim 2, completing the proof of Claim 3.

Claim 4: For any entropy
where F is an entropy flux corresponding to E with respect to H , according to Definition 2.3.

Consider the vector fields
and assume preliminarily that E is convex. Call F n the flux corresponding to E with respect to H n as defined by (3.52).
where F is an entropy flux corresponding to E with respect to H . Since R is arbitrary, equality (3.58) ensures that (3.57) is proved in the case of a convex entropy for all (t, x) ∈ˆ E , for a setˆ E such that ([0, T ] × R)\ˆ E is negligible.
Note that equality (3.57) is independent of time and ([0, T ] × R) \ˆ E is negligible, hence we may assume that (3.57) holds for all x ∈ E , where R \ E is negligible. Claim 4 is proved in the case of a convex entropy.
Assume now that E is not necessarily convex. Then, we can introduce two convex functions E + , E − of class C 2 (R; R) such that These functions are not uniquely defined, since adding/subtracting affine functions of w does not alter the validity of the latter requirements. Repeating the argument above, for all x ∈ E + ∩ E − , equality (3.57) holds also for the not necessarily convex entropy E, the set R\( E + ∩ E − ) being negligible. Claim 4 is proved.
Call E the countable set of all polynomials with rational coefficients and define (3.59)

Claim 5:
The set is such that R\ is negligible and for all E ∈ C 0 (R; R) and for all x ∈ , equality (3.57) holds, where F k is given by (2.6), for any k ∈ R.. For any E ∈ E and for all x ∈ , by Claim 4 equality (3.57) holds, R \ being negligible.
Let now E ∈ C 0 (R; R) be fixed. By the classical Stone-Weierstrass Theorem [22,Corollary 7.31], there exists a sequence E n in E converging to E uniformly on [U, V ]. Clearly, the sequence of fluxes F k n corresponding to E n defined by (2.6) converges to the flux F k , also defined by (2.6). Since (3.57) holds in for each pair (E n , F k n ), it also holds for (E, F k ). By the arbitrariness of E, Claim 5 is proved.
Define for all  H (x, w) − H (x, u(ξ ))), see also (2.6). By Claim 5, using (2.1), we get that for all x ∈ Rearranging the terms, one gets Either the first factor vanishes, or ν ξ is Dirac delta at u(ξ ). In both cases, using (3.60) and the arbitrariness of ξ , Claim 6 is proved.
Claim 7: The sequence u n converges to u, as defined in (3.60), a.e. in R.
(The content of this step is heavily inspired by [24,Section 5.4]). From Claim 5 and from (3.61) in Claim 6, we obtain that for all x ∈ , as defined in (3.59), and for all E ∈ C 0 (R; R) where F is as in (2.6), for any k. For a.e. x ∈ R, ν x is a probability measure and the In the above equality, E can be any C 1 function, E can be any continuous function, hence Furthermore, we have that Call The latter equality contradicts (WGNL) unless a = b, ensuring that, for a.e. x ∈ R, ν x is a Dirac measure, which in turn implies pointwise convergence up to a subsequence by (3.54), see [38, Proposition 9.1.7]. Claim 7 is proved.

Conclusion.
By Claim 7, up to a subsequence, we have the pointwise a.e. convergence u n → u as n → +∞. The L ∞ bound u n (x) ∈ [U, V ] for a.e. x ∈ R allows to use the Dominated Convergence Theorem [22,Theorem (12.24)] in (2.7). By Proposition 2.4, we get that u is a weak entropy stationary solution (Definition 2.1) attaining values between U and V . This accomplishes the construction of u + , that of u − is entirely similar. The proof of Theorem 2.9 is completed.

Vanishing viscosity approximations
Proof of Theorem 2.11. Let u be a classical solution to (2.20) on I . Clearly, U as defined by (2.24) satisfies (2.23), simple computations yield U (0, x) = U o (x) and thus U is a classical solution to (2.21) on I , proving Item (1). Verifying Item (2) is immediate, completing the proof of Theorem 2.11. (3.67) By the boundedness assumption on u + , it follows that v η attains its global maximum at a point (t η , x η ) ∈ [0, T ] × [X, +∞[. Three possible cases are in order. (3.68) Case 3: t η ∈]0, T ] and x η > X . Then, by the choice of (t η , To obtain a strictly positive lower bound for the right hand side (3.70), recall that v η (t η , x η ) ≥ v η (X, 0) which, together with (3.67), implies that Passing to the limit η → 0, we complete the proof of Lemma 3.6.  Proof. By Theorem 2.11, with I = R, it is sufficient to apply Corollary 3.7 to ∂ x U .
Below, we exploit the fact that (3.84) actually depends on u o only through its L ∞ norm. By Claim 1, we know that T m is well defined and that T m ≥ T as defined in (3.84).
We prove that T m = +∞ assuming that T m < +∞. Let C be the constant given by Corollary 2.13, which can be applied since u o is actually required Lipschitz continuous. Fix τ > 0 so that and note that , we can construct a solution u τ in the sense of Definition 2.10 to The concatenation  [18, § C.7] can be applied on every compact subset of R + × R. Use a diagonal argument to obtain U * as the limit of a convergent subsequence. Clearly, U * is Lipschitz continuous with the Lipschitz constant provided by (2.25). Proving that U * satisfies Definition 2.7 is classical, we refer, for instance, to [4,Chapter 2] or [18,Chapter 10].
By Theorem 2.8, U * is independent of the particular subsequence, hence the whole sequence U ε n converges to U * .

Proof of Theorem 2.17. Claim 1: The map
We now prove that for every positive T and R there is a constant C T,R such that For all ϕ ∈ C 1 c (R; R) and for all t ∈]0, T [, by (2.20) we have By direct computations, using also (2.20), from (3.85) we get: so that, integrating also over t on [0, T ] and using the definition of ψ R , we have Multiply by ψ R 2 (x), integrate over [0, T ] × R and take the absolute value: where M is as in Corollary 2.13. The two summands on the lines (3.88)-(3.89) are both independent of ε. Concerning (3.90) above, integrate by parts and use (3.87) to obtain which, again, is a quantity independent of both ε and u ε . The latter bound inserted together with (3.88) in (3.86) provides the desired L 2 loc (R + × R; R) bound. Claim 1 is proved. Claim 2: For any T, R > 0 and for any entropy E ∈ C 2 (R; R), let F be a flux satisfying (2.4). Then, the set This Claim essentially amounts to an application of Murat Lemma [38, Lemma 9.2.1], which we adapt here to the present situation.
Using (2.20), straightforward computations yield: We now verify the following 3 assumptions to apply Murat Lemma [38, Lemma 9.2.1]:  corresponding to a subsequence ε n k , meaning that for each (t, x) ∈ [0, T ] × R, ν t,x is a Borel probability measure on R such that for any g ∈ C 0 (R; R) and for any ϕ ∈ L 1 ([0, T ] × R; R), we have Remark 3.9. Following a standard habit, to simplify the notation, in the sequel we write ε for ε n k , ε → 0 for k → +∞ and, correspondingly, refer to u ε as to a sequence.
By Corollary (3.93) By (CNH), introducing for = 1, . . . , m the functions Hence, for any where, to get to the last inequality, we used the inclusion spt ν t,x ⊆ [−M, M]. Moreover, each term in the latter sum above converges to 0 by (3.92), since each F ϕ is in . Then, Claim 3 is proved.

Claim 4: For any entropy
where F is any entropy flux corresponding to E with respect to H , according to Definition 2.3.
(The content of this step closely follows Claim 5 in the proof of Theorem 2.9).
Consider the vector fields and where E is in C 2 (R; R) and F is a corresponding flux defined by (2.4 Hence, we may now intend ( (2.26). Clearly, the sequence of fluxes F k n corresponding to E n defined by (2.6) converges uniformly to the flux F k , also defined by (2.6). Since (3.94) holds in for each pair (E n , F k n ), repeating the same argument as in the proof of Claim 3, one proves that it also holds for (E, F k ). By the arbitrariness of E, Claim 5 is proved.
Hence, either the first factor vanishes, or ν τ,ξ is Dirac delta at u(τ, ξ ). In both cases, Claim 6 is proved. Claim 7: Up to a subsequence, the sequence u ε converges to u, defined in (3.97), a.e. in defined in (3.96). (This step, similarly to Claim 7 in the proof of Theorem 2.9, is inspired by [24,Section 5.4]).
Recall (3.98) from Claim 6. From (3.94) using Claim 5, we get that for (t, x) ∈ ,   u(t, x))) A t,x (w) = (w − u(t, x)) B t,x (w) since the two sides have the same distributional derivative in w by (3.101) and the definitions (3.100) of A t,x , B t,x . Inserting (3.101) in the last equality, we have (H (x, w) − H (x, u(t, x))) A t,x (w) = (w − u(t, x)) ∂ w H (x, w) A t,x (w) for a.e. w ∈ R.
Call [a, b] the minimal (with respect to set inclusion) interval containing the support of ν t,x . Note that A t,x (w) = 0 for w ∈]a, b[. Indeed, by the definition of A t,x (w) and since ν t,x is nonnegative, the map w → A t,x (w) vanishes for w < a, weakly decreases for w ∈]a, u(t, x)[, weakly increases for w ∈]u(t, x), b[ and vanishes for w > b. At the same time, the minimality of [a, b] ensures that A t,x is nonzero in both a right neighborhood of a and a left neighborhood of b. Simplifying, we thus obtain H (x, w) − H (x, u(t, x)) = (w − u(t, x)) ∂ w H (x, w) for all w ∈]a, b[, and differentiating this equality with respect to w we contradict (WGNL), unless a = b, which in turn ensures that, for a.e. (t, x) ∈ [0, T ] × R, ν t,x is a Dirac measure. We thus have the pointwise a.e. convergence, up to a subsequence, of the vanishing viscosity solutions, see [38,Proposition 9.1.7]. Claim 7 is proved. Let (E, F) be an entropy-entropy flux pair in the sense of Definition 2.3, with E of class C 2 and convex. Using (2.20), thanks to the regularity of u ε , simple computations give x x E(u ε ) − ε n E (u ε ) (∂ x u ε ) 2 so that by the convexity of E Fix a test function ϕ ∈ C 2 c (R 2 ; R + ), multiply both sides in (3.102) by ϕ and integrate to get By (2.26), we have the L ∞ boundedness of u ε uniformly in ε. Using Claim 7 and the Dominated Convergence Theorem [22,Theorem (12.24)] we obtain (2.7) for any test function ϕ ∈ C 2 c (R 2 ; R + ). A standard approximation argument allows to relax (2.7) to any test function ϕ ∈ C 1 c (R 2 ; R + ). The proof of Claim 8 follows by Item 2 in Proposition 2.4.

Conclusion
By Claim 8, u solves (CL) in the sense of Definition 2.1 and thus its uniqueness follows from Theorem 2.6. Recall that the sequence ε n , prior to the simplification in the notation in Remark 3.9, is an arbitrary sequence converging to 0. Above, we proved that there exists a subsequence ε m k such that the corresponding subsequence u ε n k converges to a limit u, independent of the choice of the initial sequence ε n . The arbitrariness of the choice of ε n ensures that u ε , now understood as a continuous family, converges to u.
The proof of Theorem 2.17 is completed. An alternative approach allowing to pass from weak to strong convergence might be adapted from [20, Items 2 and 3 in the proof of Theorem 4.1].

Properties of the limit semigroups
Proof of Theorem 2.18. Theorem 2.17 ensures the existence of a solution in the sense of Definition 2.1 globally in time, for all initial data in W 1,∞ (R; R), proving 1. and 2. for such data. The uniqueness of this solution follows from estimate (2.14) in Theorem 2.6.
Define pointwise (S C L t u o )(x) := u * (t, x), where u * is as in Theorem 2.6. We thus have the existence of a map S C L defined on R + × W 1,∞ (R; R) attaining values in (a precise representative in) L ∞ (R; R), satisfying 3.a and 3.b for all u o ∈ W 1,∞ (R; R) and 4. for all u o , v o ∈ W 1,∞ (R; R), thanks to Theorem 2.6.
Fix an initial datum u o in L ∞ (R; R). Use Theorem 2.9 to find two stationary solutionsǔ andû such that for all x ∈ R, Take a sequence u n o ∈ W 1,∞ (R; R) converging to u o in L 1 loc (R; R) and such that u n o (x) ∈ [ǔ(x),û(x)] for all x ∈ R.
By the contraction property (2.14), for all t ∈ R + , S C L t u n o is a Cauchy sequence in L 1 loc (R; R). Define S C L t u o as this limit and note that (2.14) also shows that S C L does not depend on the choice of the sequence (u n o ). Nevertheless, by (2.15), for all t ∈ R + , (S C L t u n o )(x) ∈ [ǔ(x),û(x)] for a.e. x ∈ R, so that (t, x) → (S C L t u o )(x) is in L ∞ (R + × R; R). Moreover, (t, x) → (S C L t u n o )(x) is a converging sequence in L 1 loc (R + × R; R). Up to the extraction of a subsequence, we have that the sequence (t, x) → (S C L t u n o )(x) converges pointwise a.e. to (t, x) → (S C L t u o )(x). Since we have the L ∞ bound (S C L t u n o )(x) ∈ [ǔ(x),û(x)] we can pass to the limit in (2.2), apply the Dominated Convergence Theorem [22,Theorem (12.24)] and obtain that (t, x) → (S C L t u o )(x) solves (CL) in the sense of Definition 2.1.
By this construction, we immediately have that the map u(t, x) := (S C L t u o )(x) satisfies 1. and 2., while S C L satisfies 3.a and 4..
Fix u o ∈ L ∞ (R; R). Applying again Theorem 2.6, we see that the map (t, x) → (S C L t u o )(x) admits a representative that satisfies 3.b. Since S C L satisfies 3.a, we can conclude that for all t ∈ R + and for a.e. x ∈ R that (S C L t u o )(x) equals this representative. Hence, S C L satisfies 3.b.
To complete the proof, note that S C L is a semigroup, thanks to the uniqueness and L 1 loc (R; R) continuity proved in Theorem 2.6 and since S C L t (L ∞ (R; R)) ⊆ L ∞ (R; R).  Note that Theorem 2.16 also proves 3., while 4. is a consequence of Item 2 in Theorem 2.8. sequential closedness of V and the sequential continuity of f proved above ensures that for all t 1 , t 2 ∈ A with |t 2 − t 1 | < δ, we have f (t 2 ) − f (t 1 ) ∈ V ⊆ U , completing the proof.
Proposition A.2. (The set C 1 c is separable.) There exists a countable set S ⊂ C 1 c (R; R) with the following property: for any ψ ∈ C 1 c (R; R) there exists a compact set K ⊂ R such that for all ε > 0 there exists a map σ ∈ S satisfying spt σ ⊆ K and σ − ψ C 1 (R;R) ≤ ε.