Saturated Fronts in Crowds Dynamics

: We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided

The second-order term in equation (1.1), which accounts for diffusion, is formally and so the product g(u)Φ  (u x ) represents the diffusion coefficient or diffusivity; because of assumptions (H2) and (H3) we have g(u)Φ  (u x ) ≥ 0. As a consequence, equation (1.1) is a nonlinear forward parabolic equation which degenerates both when u = 0 or u = 1 (because g vanishes) and when u x = ±∞ (because the diffusion term Φ is saturated), i.e., when the tangent to the graph of u( ⋅ , t) is vertical.An example of function Φ satisfying assumption (H3) is In this case, the right-hand side of (1.1) becomes a degenerate version of the mean-curvature operator.
In this paper we are interested in the existence and uniqueness of a suitable class of globally defined, monotone, non-constant traveling-wave solutions to equation (1.1).With this name we mean solutions u of (1.1) of the form u(x, t) = ψ(x − σt), with σ ∈ ℝ and ψ : ℝ → [0, 1]; moreover, ψ is not identically constant and it is monotone in the sense that ξ 1 < ξ 2 implies either ψ(ξ 1 ) ≤ ψ(ξ 2 ) or ψ(ξ 1 ) ≥ ψ(ξ 2 ).Strictly monotone functions are characterized by strict inequalities.The function ψ is the profile of u and σ is the speed of the traveling wave.Such a function u is called a wavefront solution, WF for short, and we denote in the following with ℓ ± the limits of its profile at ±∞, i.e., For brevity we extend the name of WF also to ψ.
In the case f = 0, the existence and uniqueness of solutions to the initial-value problem for equation (1.1) with initial data u 0 is proved in [4,23] in the non-degenerate case g > 0, for u 0 ∈ L ∞ (ℝ), u 0 strictly increasing; the case g(0) = 0 (indeed, in several space dimensions) is instead considered in [2,11,13], where analogous results are proved for u 0 ∈ L ∞ ∩ L 1 (ℝ) or u 0 ∈ BV(ℝ), the space of functions with bounded variation.In both cases the solutions are obtained as vanishing viscosity limits.Recall that if u ∈ BV, then Du is a Radon measure that can be decomposed as Du = D a u + D j u + D c u, where on the right-hand side we have the absolutely continuous part (with respect to the Lebesgue measure), the jump part, and the Cantor part of Du, respectively [1].The solutions provided by the above authors can be discontinuous; since the term Φ(u x ) has no meaning in D  for such functions, it is understood [12] that equation (1.1) holds in D  with Φ(u x ) replaced by Φ(D a u).Under this terminology, a WF ψ in our class turns out to be a distributional solution to the equation (g(ψ)Φ(D a ψ))  + σψ  − (f(ψ))  = 0, (1.4) where we omitted for simplicity the independent variable ξ = x − σt in the arguments of the functions.With ψ  := Dψ we denoted here and in the following the (full) distributional derivative of ψ.The important issue concerning uniqueness is that it fails, in general, in presence of discontinuous solutions.However, uniqueness is recovered by requiring that solutions are entropic, a notion that is strictly related to that with the same name in the hyperbolic theory of conservation laws [6,22,28], see Section 4 for further details.We refer to [12,13] for the general but rather technical definition of entropic solution for equation (1.1), while several additional informations and a bibliography on the subject can be found in the comprehensive survey [8].In one space dimension, this definition reduces to the more explicit condition in [4].
As far as wavefronts are concerned, they can be discontinuous, too.Traveling waves for equation (1.1) have been studied in [15] (in the case g(u) ∼ u n , g  (u) ≥ 0) and [27,29] (for constant g), see also [32], with a particular emphasis to examples, physical motivations and numerics.An intuitive comparison between the case with convection f and the case without convection (f = 0) is given in [14].The references quoted in the papers mentioned just above also give some information for traveling waves in the latter case.We point out that equation (1.1) enters in the framework of none of those papers because of the assumption (H2).
Traveling waves have also been studied in [9] in the case f = 0 (see also [5,24]), while [10] deals with the same case but adds a monostable source term to the equation; see also [30] for source terms with more than two zeros.In both cases, the results were provided for particular functions Φ. Smooth fronts in presence of both convection and reaction, essentially in the case g = 1, were studied in [25].To the best of our knowledge, the case when f ̸ = 0 has never been rigorously considered in this framework.This paper, as the recent articles [17][18][19][20], aims at a better understanding of the wavefront solutions to degenerate parabolic equations modeling collective movements.The main motivation is to provide rigorous results to an issue motivated by [7], namely, roughly speaking, that profiles are smooth if |ℓ + − ℓ − | is small and possibly discontinuous otherwise.Such a result was justified in a special case in [29] and proved in [32] in the case g = 1.
Content of the Paper.Sections 2 to 4 provide a rigorous background to the class of solutions we are dealing with, without relying on the general setting of [2,11,13].The main results are given in Section 5. First, we prove the existence and uniqueness of wavefronts in a simple setting; this allows us to completely characterize their singularities.The result is then extended to a general framework.Then, we show with an example that the entropy condition is necessary for the uniqueness.Indeed, this is known for general solutions since [23].From a geometric point of view, the loss of uniqueness is due to multiple intersections of the functions f − g and s ± , where s ± is the line joining the points (ℓ − , f(ℓ − )) and (ℓ + , f(ℓ + )).This example motivates a further result: we prove that an entropic wavefront ψ is the limit of smooth wavefronts ψ ε , for ε → 0. The latter profiles correspond to the equation where the artificial viscosity εu xx has been added to the original equation (1.1).This result also holds for general solutions [4], but we propose a different and simpler proof which is focussed on wavefronts and includes the presence of the term f .We recall that an analogous vanishing-viscosity criterion is used to uniquely select shock waves in a hyperbolic framework [6,22,28].Since the profiles ψ ε are strictly monotone, this result justifies a posteriori our choice of focussing on monotone profiles since the beginning of the paper.Section 6 contains the proofs of the statements in Sections 3 and 5.At last, Section 7 shows an application to crowds dynamics suggested by a model in [7].

Classical Versus Singular Wavefront Solutions
In the following we always assume conditions (H1)-(H3) without any further mention.We begin with the definition of classical solution to equation (1.4), see [26], and briefly discuss its main properties; the parameter σ in (1.4) is for the moment an arbitrary real value.Then we show how discontinuous solutions can arise and give some simple examples.For an open interval I ⊂ ℝ we denote by AC loc (I) the set of locally absolutely-continuous functions in I, see [31].
In this case A σ,c ⊃ (ℓ − , u 1 ) ∪ (u 2 , ℓ + ), and then no classical solution exists because ℓ − and ℓ + belong to two different intervals contained in A.
If h(u 1 ) = h(u 2 ) = −1, with u 1 ̸ = u 2 , then the corresponding profile ψ is decreasing; in this case we have lim ξ → ξ ψ  (ξ) = −∞ and the terms involving g in the expression of σ change sign; we now have The case u 1 = u 2 , i.e., when the function h has a strict local maximum at u 1 with h(u 1 ) = 1 or strict local minimum with h(u 1 ) = −1, gives rise to a continuous profile whose graph has a vertical tangent with either ψ  ( ξ ) = ∞ or ψ  ( ξ ) = −∞.We refer to Proposition 3.1 for a rigorous statement.The previous discussion is naive because it does not consider the case of several solutions to the equation h(u) = 1 and avoids the points where g vanishes.Moreover, it bypasses the fact that equation (2.5) is not balanced where ψ has a jump discontinuity: the right-hand side is a bounded function (at least if u i ∉ {0, 1} for i = 1, 2) while the left-hand side ψ  is a delta-like distribution.The same problem arises for the equation (g(ψ)Φ(ψ  ))  + σψ  − f(ψ)  = 0, because in this case the term Φ(ψ  ) has no meaning in the distribution sense.(i) Consider the case of an absolutely continuous function ψ with a singularity in the derivative at ξ and ψ( ξ ) ∈ (0, 1); assume moreover that ψ is a classical solution in ( ξ − δ, ξ ) and in ( ξ , ξ + δ) for some δ > 0, see Figure 3 (i).In this case D a ψ = ψ  and both ψ  ( ξ ± ) exist by the equation.
(ii) Let ψ be a classical solution to (1.4) in its maximal existence interval ( ξ , β); assume ξ ∈ ℝ and ψ( ξ + ) = 0. We cannot have ψ  ( ξ + ) = 0, because then ψ could be extended to the left of ξ as a classical solution.Then either ψ  ( ξ + ) does not exist or it is positive (possibly ∞) and different from zero.In these cases, consider the extension ψ of ψ defined by see Figure 3 (ii) for the cases ψ  ( ξ + ) = ∞ and ψ  ( ξ + ) ∈ ℝ + .The function ψ is not a classical solution to (1.4) in (−∞, β); however, we claim that ψ is a solution in the sense of distributions.In fact, let η ∈ C ∞ 0 ( ξ − δ, ξ + δ) with δ ∈ (0, β); we have The term Φ(ψ  (ε)) is bounded with respect to ε and then Also the integral vanishes because ψ is a classical solution.This proves the claim.We emphasize that these solutions are missing in the case g does not vanish at 0.
(iii) Consider a continuous function ψ defined on (−∞, β) and satisfying for some How can ψ be extended to the whole of ℝ as a solution to (1.4)?By items (i) and (ii) above we deduce that ψ can be extended either as a constant function or as a classical non-constant solution, but then ψ( ξ ) ∈ {0, 1} and ψ  ( ξ ) = 0 (bifurcation of a classical solution); we refer to the following Example 5.1 for the latter case.Moreover, if ψ( ξ ) ∈ {0, 1}, then ψ can also be extended as a non-classical solution, see Figure 3 (ii) (bifurcation of a non-classical solution).

Admissible Wavefront Solutions
In this section we characterize, according to their singularities, the WFs we are going to deal with; we call them admissible WFs.The previous section provides us the motivations.The underlying idea is to deal with classical solutions as long as it is possible; otherwise, motivations have been provided in Section 2.More precisely, a typical profile ψ under consideration is smooth except possibly at the finitely many points of its singular set S ψ ; at these points, either ψ is continuous but ψ  is infinite, or ψ has a jump discontinuity (the side limits of ψ are finite but differ).For ξ ∈ ℝ, we write for short Omitting for simplicity the subscript ψ, for n ∈ ℕ we denote where the subsets C and J are defined below.
• We have ξ i ∈ C if ψ is continuous at ξ i and the following holds: has a jump discontinuity at ξ i and the following holds: The two alternatives above (either . . .or) depend on whether the WF is increasing or decreasing, respectively.The requirement ψ  (ξ 0 ) ̸ = 0 in (C 2 ) leaves open the possibilities that either ψ  + (ξ 0 ) does not exist or ψ  + (ξ 0 ) ∈ (0, ∞]; analogous possibilities hold for ψ  − (ξ n ).These requirements exclude smooth WFs; in fact, in that situation, smooth profiles satisfy lim ξ →ξ + 0 ψ  (ξ) = 0 or lim ξ →ξ − n ψ  (ξ) = 0. Notice the asymmetry of conditions (C 2 ) and (J 2 ): in the former case, only the value 0 for the derivative is excluded (otherwise ψ is smooth); in the latter, the WF ψ must have a vertical tangent at the discontinuity point.This depends on the fact that discontinuities in the profile can arise only if the diffusion is saturated to ±1, and in turn this only happens if ψ  = ±∞.Condition (J 1 ) implies, for WFs, the requirement [4, (2.1)], which is used there to give a meaning to discontinuous solutions to (1.1).Condition (J 2 ) (which is missing in the non-degenerate case considered in [4]) is introduced to cope with the vanishing of g at 0 and 1.Both conditions together coincide with the definition given in [8, p. 187].
Example 3.1.We show that the case when ψ  + (ξ 0 ) does not exist can indeed occur in (C 2 ).Define Assume ℓ − = 0 and ℓ + = 1, so that from (2.5) and (2.6) we have σ = c = 0, and is easily shown to exist.Moreover, we claim that the solution ψ reaches the value 0 for some finite ξ 0 ; this claim can be proved as follows.Let ψ be the solution of ψ  = Φ −1 (h(ψ)) satisfying ψ(0) = ψ for some ψ ∈ (0, 1).We notice that ψ  (ξ) > 0 for every ξ such that ψ(ξ) ∈ (0, 1), and then we consider the inverse function ξ = ξ(ψ).Then we define ξ 0 = inf{ξ : This implies that the integral in (3.2) is convergent; then ξ 0 is a real number and the claim is proved.In this case the limit lim u→0 + Φ −1 (h(u)) does not exist.Since 0 < h(u) ≤ 1 2 , we have S = C = {ξ 0 } for some ξ 0 ∈ ℝ with ψ(ξ + 0 ) = 0, but the limit lim 3), (ii) there are points as in (3.1) such that ψ is a classical solution to (1.4) Remark 3.1.The motivation for considering monotone profiles in Definition 3.1 relies on Remark 2.1: any profile ψ in every interval as above and contained in J has to be strictly monotone.Classical wavefront solutions are then monotone.On the other hand, at any point ξ i with ψ(ξ i ) ∈ (0, 1) the sign of the derivative does not change, and we shall see in Theorem 5.3 that solutions of the augmented problem (1.5) single out, in the vanishing-viscosity limit, only entropic (and then monotone) solutions.So, non-monotone profiles do not seem to provide interesting solutions.
In the following we always deal with admissible WFs.Remark that ψ is strictly monotone in the interval (ξ 0 , ξ n ).Moreover, according to Definition 3.1 we have D c ψ = 0, because ψ is assumed to be smooth outside S. The smoothness of the profiles in the above class is straightforward: if S = 0, then ψ ∈ C(ℝ) ∩ C 2 (J); otherwise ψ ∈ C 2 (J \ S), being continuous at points in C. The jumps of the WF at point in J are called subshocks [15].Remark 3.2.Let ψ be an admissible WF.We claim that if ℓ − ∈ (0, 1), then we have ψ(ξ) ̸ = ℓ − for every ξ ∈ ℝ; an analogous statement holds for ℓ + .In fact, every admissible WF ψ is classical in the half-line and cannot be constant there by Examples 2.1 (iii) and 3.2.If ℓ ± ∈ (0, 1), then ψ(ξ) ∈ (ℓ − , ℓ + ) for every ξ ∈ ℝ and ψ is strictly monotone in ℝ.
In particular, see Example 2.1 (iii), if ψ satisfies (2.9) and has a jump discontinuity at ξ , then necessarily ℓ − ∈ {0, 1}, see Figure 4.This is also excluded by condition (J 1 ).The possibility that ψ only assumes the values 0 and 1 is not excluded.We anticipate that these solutions are missing in the case g does not vanish at 0, see Remark 3.4.
Also notice that cuspon "solutions" with a cusp either at 0 or 1, see Example 2.1 (i), are ruled out by Definition 3.1 because they are not monotone.
In the following proposition we characterize condition (iii) in Definition 3.1.Let ψ be a WF of (1.1) with wave speed σ.In every interval where ψ is classical, the equation can be integrated, see (2.1); hence, there exist finitely many c i ∈ ℝ such that where the sign + occurs for decreasing profiles and the sign − for increasing profiles.(c) There exists a unique γ ∈ ℝ such that ψ satisfies Example 3.2.Assume that ψ satisfies (2.9) and has a jump discontinuity at ξ , see Remark 3.2 and Figure 4.By passing to the limit in (2.1) for ξ → ξ ± we deduce σψ( ξ by difference.This value coincides with (3.3) if and only if ℓ − ∈ {0, 1}; if ℓ − ∈ (0, 1), then Proposition 3.1 confirms that ψ is not a solution in the sense of distributions.

Entropic Wavefront Solutions
As we shall see below, profiles are not unique in the class of admissible WFs.As for hyperbolic conservation laws, this depends on the presence of discontinuities and, more precisely, on the occurrence of more than two points where the function h assumes the values 1 or −1.This was first noticed in [4,23], where an entropy condition was introduced, in the case f = 0, to recover the uniqueness.For the case f ̸ = 0, we now provide an analogous condition.= u 2 , be two points such that h(u i ) = 1 (or h(u i ) = −1) for i = 1, 2 and ψ has a jump discontinuity from u 1 to u 2 .Then ψ is entropic if (See Figure 5.) We now comment on this definition by focusing on the case h(u) > 1 for u ∈ (u 1 , u 2 ); the case h(u) < −1 is analogous.In this case condition (4.1) is equivalent to for s ± as in (3.7).This means that the graph of the function f − g must lie above the line s ± .Recall that if we have a discontinuity between u 1 and u 2 , then necessarily the line s ± passes through the points (u 1 , (f − g)(u 1 )), (u 2 , (f − g)(u 2 )) (see Section 2 and Figure 2 (iii)).Hence, condition (4.2) becomes see (2.7).We refer to Figure 6 for a geometrical interpretation of conditions h > 0, h ≷ 1 in terms of the functions f and g; recall If f = 0, then (4.3) reduces to (a) which coincides with the definition of entropy solution in [4] (for wavefront solutions).On the other hand, if g = 0, then we find the usual entropy condition exploited in hyperbolic conservation laws [6,Theorem 4.4].This shows that (4.1) fits to both parabolic and hyperbolic equations, and then the term "entropic" seems particularly suited to design this condition.Notice that condition (4.1) does not appear explicitly in some aforementioned papers, for instance in [8, Assumptions 13.2, p. 186], because g is required to be convex there, and then (4.1) trivially holds.

Main Results
In this section we state and comment our main results.They concern the existence, uniqueness and the smooth approximation of entropic wavefront solutions to (1.1).We deal with the case of increasing profiles; analogous results for decreasing profiles can be obtained as well, see Remark 5.1.Since we focus on increasing profiles, we fix ℓ ± ∈ [0, 1] with ℓ − < ℓ + ; as a consequence, this choice defines a function h as in (3.8).
In the first result, Theorem 5.1, we essentially assume that the function h either is valued in [0, 1] or it is larger than 1 only in an interval.In this case the corresponding WFs are clearly entropic and their singular set S either is empty or contains only one point.This simple framework gives us the possibility of analyzing in detail all possible subcases.Theorem 5.1.We make assumptions (H1)-(H3); fix ℓ ± ∈ [0, 1] with ℓ − < ℓ + and assume h > 0 in (ℓ − , ℓ + ). (5.1) Under the following conditions, equation (1.1) has a unique (up to shifts), increasing and entropic wavefront ψ, which satisfies (1.3), with σ given by (3.6) 1 and S specified below. (5.3) In this case we have S = 0.
We collect here several comments on Theorem 5.1.
(i) Condition (5.1)only depends on f and is the well-known necessary and sufficient condition for the existence of wavefronts in the case the diffusion term in (1.1) has the form (g(u)u x ) x , with g as in (H2) but with g(0) and g( 1) not necessarily 0, see [26,Theorem 9.1].It is always satisfied if f is strictly concave in the interval (ℓ − , ℓ + ).Indeed, condition (5.1) also has a hyperbolic counterpart (i.e., when g = 0), which establishes that the piecewise constant discontinuous solution assuming the values ℓ − for x < 0 and ℓ + for x > 0 is entropic (in the Oleinik sense, see [6,Theorem 4.4]).
(v) The shape of the profiles can be easily deduced from the conditions above.
(viii) The case when g is strictly positive in [0, 1] can be easily treated by dropping condition (J 2 ).We did not include this case in the paper to avoid long statements with enumeration of several cases.Then, our results extend and make precise those in [32], where g is constant and f strictly convex.The two latter conditions make (4.3) trivially satisfied.If moreover f ≡ 0, then the discontinuous profile of the previous item cannot occur, and so no entropic WFs exist.This result matches with what was pointed out in [4, below formula (1.8)].
Our last result shows that entropic WFs are not only singled out uniquely but, moreover, are the limits of classical WFs that correspond to the equation where the non-degenerate diffusive term εu xx has been added to the right-hand side of equation (1.1).The non-entropic profile ψ 1 in (5.11) has not this property.Notice that now the second-order term, which accounts for diffusion, is (g(u)Φ  (u x ) + εu x ) x , which is no more degenerate.This result was first proved in [4] in the case f = 0; we provide here a different proof in the case of wavefront solutions.We first state a lemma about the existence of profiles to equation (5.12).
Here follows our result.Notice that if ψ is an increasing entropic WF as in Theorem 5.2 and J = 0, then ψ only assumes values 0 and 1, with a single jump at some ξ ∈ ℝ.
In conclusion, since the profiles ψ ε are always strictly monotone, it follows that also ψ is monotone.Then, a posteriori, this result rigorously justifies the choice of considering monotone profiles of (1.1), see Remark 3.1.

Proofs
In this section we provide the proofs of the above statements, in particular of Theorems 5.1 and 5.3.

Proofs of Results in Section 3
Proof of Proposition 3.1.We split the proof into three parts.For simplicity we assume ψ increasing.

and we deduce
3) clearly still holds.In the same way we compute By (6.1), (6.3) and (6.4) we obtain Hence, (b) is satisfied.(b) ⇒ (c) If S = 0, then ψ is a classical solution and (c) follows by integration.Otherwise, consider ξ i ∈ S for some i = 1, . . ., n.Then there exists δ > 0 such that ψ is a classical solution to (1.4) in both [ξ i − δ, ξ i ) and (ξ i , ξ i + δ], and hence there are c ± ∈ ℝ such that and If ξ i ∈ C, then we deduce c − = c + by passing to the limit in (6.5) for ξ → ξ − i and in (6.6) for ξ → ξ + i .If ξ i ∈ J, by passing again to the limit in (6.5) and (6.6) as above, we obtain ), and again c − = c + by (3.3).

Proofs of Results in Section 5
Proof of Theorem 5.1.First, we deal with the sufficient conditions for the existence of profiles.We consider separately each case in the statement.
Case (b).We further split the proof into three cases.
At last, we prove Theorem 5.3.
From the proof of Lemma 5.1 we deduce R 0 (ψ) ≥ 0 for ψ ∈ (ℓ − , ℓ + ).By the definition of R ε we have for ε > 0 and A(ρ) = ρ; more generally, A : [0, ρ] → ℝ can be a suitable weight function.In general, one may assume a(ρ) := A  (ρ) > 0, ρ ∈ (0, ρ), so that the deviation term is still directed against ∇ρ.We have V(ρ; ⋅ ) = 0.If ε < 1, this means that in (7.1) the diffusion cannot counterbalance the movement toward the preferred direction: pedestrians stop rather than reversing their direction.This a consequence of the saturation of the diffusion and of the smallness of ε.
In the case ν is a constant vector and Ω = ℝ 2 we can look for plane-wave solutions, which are solutions of the form ρ(x 1 , x 2 , t) = u(μ ⋅ ⃗ x, t), where u = u(x, t) is some function in ℝ 2 and μ ∈ ℝ 2 is a unit vector.In this case u must satisfy the equation which is of the form (1.1) for f(u) = μ ⋅ νuv(u), g(u) = εuv(u), Φ(w) = Φ(A(w)) and ρ = 1.Notice that the convection term disappears if μ is orthogonal to ν.It is immediate to see that ρ(x 1 , x 2 , t) = ψ(μ ⋅ ⃗ x − σt) is a WF solution of (7.1) if and only if u(x, t) = ψ(x − σt) is a WF solution of (7.2).
We now focus on an issue that was motivated by [7], namely, that a WFs u to (1.1) with profile ψ is (a) continuous when |ℓ + − ℓ − | is sufficiently small, (b) possibly discontinuous when |ℓ + − ℓ − | is large.Partial answers to this issue have been reported in the Introduction.From an hyperbolic point of view this means, roughly speaking, that small shock waves have smooth viscous profiles, while large shock waves have possibly discontinuous profiles.To this aim, motivated by [7], in addition to (H1)-(H3) we further assume (H4) f is strictly concave in [0, 1].Assume ℓ ± ∈ [0, 1]; we claim that, under (H4), every entropic WF ψ connecting ℓ − with ℓ + is increasing.In fact, if ψ is an entropic WF then f(u) − s ± (u) > 0, u ∈ (ℓ − , ℓ + ), by (3.6) and (H4).Since ψ satisfies (3.8) in every interval where it is classical, by Definition 3.1, (H2) and (H3) we deduce that it is increasing.The further condition in the case ℓ − = 0 requires, roughly speaking, that the diffusive flux is larger than the convective flux; in other words, is "parabolicity" prevails on "hyperbolicity".
(ii) Assume ℓ − = 0.In this case we have , u ∈ (0, 1), and then by the further assumption in (ii) we deduce lim u→0 + H(0, 0, u) = lim g  (u) = 0, so that the proof of item (i) works again.The proof of the theorem is complete.

1 Figure 1 :
Figure 1: Two typical plots of the functions g and Φ.

Figure 2 :
Figure 2: Formation of singularities in a profile.On the rightmost figure, a geometrical interpretation of (2.6) and (2.7).

Figure 3 :
Figure 3: Candidates for continuous solutions with a singularity in the derivative at ξ .Only cases (c) to (e) may occur.

Example 2 . 1 .
The weaker formulation(1.4) is already sufficient to include or to rule out some patterns of solutions, as we now show by some examples.

Proposition 3 . 1 .
For every monotone function ψ satisfying conditions (i) and (ii) in Definition 3.1 the following three conditions are equivalent: (a) ψ satisfies condition (iii) in Definition 3.1.(b) For every ξ i ∈ J we have σ

. 7 )
By(3.4), Proposition 3.2 and (3.7), we can write the function h in (2.4) in a slightly different way and rewrite the equation for future reference: if ψ is a classical solution to (1.4) satisfying(1.

Figure 5 :
Figure 5: Jumps from a to b, or from c to d, are entropic; a jump from a to d is not.

Figure 6 :
Figure 6: Geometrical interpretation of the conditions h < 1 (left), h = 1 (center) and h > 1 (right), in the case h > 0. The oblique line is the line s ± (u).

Figure 7 : 4 Figure 8 :
Figure 7: Left: the functions f , s ± (thick lines) and g + s ± (dashed).Right: the plot of the function h in (5.8).Unit measures are different in the two figures.

Figure 9 :
Figure 9: For the proof of case (i) of Theorem 7.1.