Analysis of a class of degenerate parabolic equations with saturation mechanisms
aa r X i v : . [ m a t h . A P ] M a y Analysis of a class of degenerate parabolicequations with saturation mechanisms
Juan Calvo ∗ May 11, 2018
Abstract
We analyze a family of degenerate parabolic equations with lineargrowth Lagrangian having the form u t = div ( ϕ ( u ) ψ ( ∇ u/u )). Here | ψ | ≤ ψ, ϕ , under which: 1) these equations fall inthe framework provided by [6, 7] and hence they are well posed, 2) we canensure finite propagation speed for these models, 3) a Rankine–Hugoniotanalysis on traveling fronts is also performed. On the particular case of ϕ ( u ) = u we get more detailed information on the spreading rate of com-pactly supported solutions and some interesting connections with optimalmass transportation theory. The purpose of this article is to analyze a certain class of degenerate parabolicequations having the following general form: u t = div ( ϕ ( u ) ψ ( ∇ u/u )) , (1.1)under a certain set of assumptions on ψ, ϕ . Such equations arise in a number ofinteresting situations in several branches of mathematical physics, as we detailbelow. This includes in particular the “relativistic heat equation” [37] ∂u∂t = ν div u ∇ u q u + ν c |∇ u | (1.2)and some of its porous media variants [11, 18] ∂u∂t = ν div | u | m ∇ u q u + ν c |∇ u | . (1.3) ∗ Centre de Recerca Matem`atica, Edifici C, Campus de Bellatera, 08193 Bellaterra(Barcelona), Spain. ( [email protected] ).
1e will be chiefly interested in the following subclass of (1.1): u t = div ( u ψ ( ∇ u/u )) . (1.4)Note that the prime example of (1.4) is the standard heat equation (also knownas Fokker–Planck or diffusion equation, depending on the context), correspond-ing to the choice ψ ( r ) = r . As it is well known, it lacks of propagating frontsand dissolves immediately any discontinuity initially imposed. On the contrary,equations of the form (1.4) enjoy the property of convecting fronts at a constant(model dependent) speed if ψ provides a suitable saturation mechanism. This isthe kind of behavior we will be interested in: For the sake of various applicationsin heat or mass transfer, plasma diffusion, and hydrodynamics (to name a few)it is reasonable to look for suitable modifications of the standard heat equationfor which heat transfer proceeds by means of convected fronts at large gradientregimes, instead of sheer diffusion. As pointed out in the seminal paper [37], theidea is to have a model that resembles the heat equation at moderate gradientsize, while behaving like a hyperbolic equation at large gradient regimes. Totrack these large gradient regimes we may compute the relative size |∇ u | /u . Inmost applications the spatial variable is measured in units of length. Then theratio |∇ u | /u is measured in units of 1/length.Pushing this idea a bit further leads us to equations of the form (1.4) in anatural way and elucidates how should ψ look like in order to get the desiredbehavior. To see how, let us focus in the case of heat flow and assume that theevolution of heat is described by means of an equation of the following form: u t − div q = 0 . (1.5)What is suggested in [37] is to re-write (1.5) as a transport equation, namely u t − div ( uV ) = 0 . (1.6)The velocity V may be a function depending on x, t, u and its derivatives (evenin a non-local way). The important idea is the following: If the equation isto convect fronts, shocks, etc, then | V | must saturate to a constant value inthe regime in which |∇ u | /u diverges. More specifically, | V | must saturate to thespeed of sound, which is the highest admissible free velocity in a medium. Recallthat V = ν ∇ u/u for the case of the heat equation with diffusion coefficient ν , u t = ν ∆ u. (1.7)We want V to resemble ν ∇ u/u on the regime of moderate gradient size, while | V | must converge to the speed of sound when |∇ u | /u diverges. The easiest wayto achieve this is to impose V to be a function of ∇ u/u alone, on which we fixthe limit behavior at will. A particular instance of this strategy is obtained bysetting V = ν ∇ u q u + ν c |∇ u | , ν, c > V above resembles ν ∇ u/u when |∇ u | ≪ u , while | V | converges to c for |∇ u | /u ր ∞ . Following this rationale set in [37], there are lots of other choicesfor V that would have nearly the same effect (at least at a formal level). Thegeneral strategy that we briefly outlined above was not pursued by Rosenauand coworkers in the subsequent series of papers [20, 21, 30] (in which theydiscuss in particular several variants of (1.2), some of them of the form (1.1)).In fact, the following idea permeates these works: As long as the flux functionis monotone in gradients and saturates above a certain rate, the particulars ofthe flux function are no that important.Our study in the present document can be regarded as a rigorous state-ment of that intuitive idea. Namely, we provide a suitable general frameworkin which this issue can be addressed successfully. Indeed, we show that theclass (1.4) constitutes such an adequate framework: A number of distinctivequalitative properties hold for such class of equations when ψ satisfies somesuitable requirements (to be detailed in Section 2 below), irrespective of theprecise function ψ that is used. First, we show that under such requirementsequations of the form (1.4) fall under the scope of the theory in [6, 7], hencethey are well posed in the class of entropy solutions. Then, combining someresults and techniques in [9, 18] with a number of new ideas, we are able toshow that, under the aforementioned requirements:1. Equations (1.4) can be formally deduced from the point of view of optimaltransport theory, using cost functions with domain contained in a ballhaving the speed of sound as its radius.2. The Rankine–Hugoniot relation holds for propagating discontinuities, whichtransverse the medium at the speed of sound.3. The support of any solution propagates at a finite speed, bounded aboveby the speed of sound. Under some positivity and structure assumptions(including all the relevant examples in the literature so far), we can ensurethat its spreading rate is exactly the speed of sound.To the best of our knowledge, some of these properties have been stated in arigorous way only for (1.2) [7, 9, 15, 18] among all the models having the form(1.4). This makes also clear that so far there is no a priori reason to privilegethe usage of (1.2) over other flux-saturated models that we may come up with.Although our main interest lies in (1.4), some of the previous statementshave suitable generalizations to the more general case of (1.1). Even in somecases this is conceptually simpler, as the underlying ideas appear in a clearerway when we treat the general situation. Thus, the analysis of both types ofequations will be intertwined in the sequel.We felt that the ideas in [37] provide a convenient way to introduce the class(1.4), but this is by no means the only place in which equations of this sort showup. In fact, some particular instances appear already in an unpublished workof J.R. Wilson concerning radiation hydrodynamics (see [35]) and in the works3y Levermore and Pomraning about radiative transfer [31, 32, 33]. Let us alsomention that the class of equations given by (1.4) was already present in [23]for the one-dimensional case, although their reasons in order to introduce it areof a quite different nature. Similar hyperbolic phenomena in a related class ofdegenerate parabolic equations were also observed in [14]. More recently theseideas have also found some applications in mathematical biology [22, 38].Let us detail what is the plan of the paper. In the following section weintroduce the set of assumptions to be considered in order to ensure that equa-tions (1.4) satisfy the aforementioned properties. After that we state the mainresults of the document, Theorems 2.1 and 2.2, which phrase those propertiesin a rigorous way. The section is completed by a list of examples comprisinga number of equations from the literature that fall under the present theory.Section 3 is a summary of the well-possedness theory developed in [6, 7], whichintroduced the fundamental notion of entropy solutions . Although such theoryis the cornerstone in which all the results of the paper are based, this section istechnically involved and may be skipped in a first reading. The purpose of theremaining sections is to supply proofs for the statements in Theorems 2.2 and2.1. Namely, Section 4 deals with the well-posedness of (1.4), Section 5 treatsthe optimal transportation formulation of these problems, Section 6 analyzesthe behavior of the spatial support of solutions during evolution, and Section 7tackles the formulation of Rankine–Hugoniot conditions in this context. Severalof the results that are proved in those sections hold under more general structureassumptions than (1.4), we will comment on this in each precise case.Finally we state some notations that are common to the whole document.The spatial dimension is always denoted by d . We use B ( x, R ) to denote an openball of center x ∈ R d and radius R . The Minkowsky sum of two set A, B ⊂ R d will be written as A ⊕ B = { x ∈ R d /x = a + b with a ∈ A, b ∈ B } . We use cl ( · )for the closure of a set. The indicator function of a set A ⊂ R d is written as χ A .The Kronecker delta is δ ij = 1 if i = j , zero otherwise. We use | · | to denoteeither the modulus of a vector or the absolute value of a number; this will beclear from the context. The scalar product of two vectors u, v ∈ R d is indicatedas u · v . A superscript like v T means transposition. Given an open set Ω ⊂ R d we denote by D (Ω) the space of infinitely differentiable functions with compactsupport in Ω. The space of continuous functions with compact support in Ω willbe denoted by C c (Ω). In a similar way, L p (Ω) and C k (Ω) denote Lebesgue spacesand spaces of functions of class k . We use k · k p to denote the norm in L p (Ω),the base set will be clear from the context. Given u : Ω → R , supp u denotesthe essential support, while u + = max { u, } , u − = − min { u, } are the positiveand negative parts respectively. For any T >
0, we let Q T := (0 , T ) × R d and wewrite u = u ( t, x ) for functions defined in Q T . Partial derivatives with respectto x i are abridged as ∂ i , i = 1 , . . . , d . Sometimes we use subscripts instead, as u t , u x and the like. Finally, O () and o () are the standard Landau symbols, while ∼ indicates asymptotic equivalence. 4 Structure assumptions and main results
Henceforward we deal only with non-negative solutions, which are the relevantones for the applications. The main idea is that | ψ ( ∇ u/u ) | → ψ ∞ ∈ R + when |∇ u/u | → ∞ . In particular, | ψ | cannot be a power law. Hence, we write ourtemplate in the following way ∂u∂t = div (cid:18) s u ψ (cid:18) L ∇ uu (cid:19)(cid:19) . (2.1)Here L > s a constant havingdimensions of speed. Thus s can be regarded as a characteristic speed. Notethat s := c and L := ν/c for the case of (1.2). Rosenau terms c as the speedof sound in [37]. It is the maximum speed of propagation that is allowed in themedium, further justified by optimal transport interpretations of the equation[15, 34].If we compare (2.1) with (1.6) we find out that V = s ψ (cid:18) L ∇ uu (cid:19) . (2.2)Then | V | would converge to sψ ∞ whenever |∇ u/u | → ∞ . Thus, there is noloss of generality in assuming that ψ ∞ = 1 (otherwise we rescale the effectivespeed). These comments are the main reason for the list of assumptions on thefunction ψ below.Before introducing such list of assumptions, we note that a more generalclass of equations can be considered following the same guidelines. Namely, let ∂u∂t = div (cid:18) ϕ ( u ) ψ (cid:18) L ∇ uu (cid:19)(cid:19) (2.3)where ϕ is customarily an even, non-negative convex function such that di-mensions fit (we will be more precise about this below); think for instance in ϕ ( u ) = | u | m for m > ϕ ( z ) = sz .The “velocity” is now given by V = ϕ ( u ) u ψ (cid:18) L ∇ uu (cid:19) . Let us state now what will be required of ψ in order to build up a reasonabletheory. Just before proceeding, note that we may scale out the lengthscale L by In fact we should write it as ∂u∂t = div (cid:18) s | u | ψ (cid:18) L ∇ u | u | (cid:19)(cid:19) in order to deal with signed solutions. Since we are chiefly interested in non-negative solutions,we will make a slight abuse of notation and refer always to (2.1) as the way of writing downthe equation and specific examples. Similar conventions hold for (2.3) below. x = x/L . Then, without any loss of generality we assume L = 1 for therest of the document, except at some places in which we found useful to keepthe original lengthscale. Assumptions 2.1.
Let ψ = ( ψ (1) , . . . , ψ ( d ) ) : R d → R d enjoy the followingproperties1. ψ ∈ C ( R d , R d ) .2. ψ (0) = 0 .3. | ψ ( r ) − r/ | r || ≤ d ( | r | ) for any r ∈ R d and for some continuous function d : R +0 → R +0 such that lim r →∞ d ( r ) = 0 . Thus lim | r |→∞ | ψ ( r ) | = 1 .4. If d = 1 we require that(a) ψ be odd and monotonically increasing,(b) | ψ ′ ( r ) | = o (1 / | r | ) for | r | ≫ ,while the following properties are required for dimension greater than one:(a) ψ is a conservative vector field,(b) ψ ( − r ) = − ψ ( r ) ∀ r ∈ R d ,(c) The Jacobian matrix of ψ , Dψ , is a non-negative definite (symmet-ric) matrix.(d) k Dψ k ∞ ( r ) = O (1 / | r | ) for | r | ≫ . Assumptions 2.1 enable to describe the behavior of our models in the largegradient regime in a coarse way and are independent of the precise form ofthe function ϕ . Note that the fact that these models are nearly isotropic inthe regime in which |∇ u/u | is large is implied. Apart from this, we wouldfind physically reasonable to have | V | ≤ s in (2.2) –which is the case for allthe models of interest, see below–, but this may not be necessarily implied byAssumptions 2.1. What is true is that the radial component of the velocity V is bounded by s . Lemma 2.1.
Being Assumptions 2.1 verified, the following assertions hold true:1. Fix r ∈ R d . Then the map t ψ ( tr ) · r, t > is non-decreasing and hencenon-negative.2. | ψ ( r ) · r | ≤ | r | for every r ∈ R d .Proof. First item is a consequence of Assumptions 2.1. and 2.1. . Thenlim t →∞ ψ ( tr ) · r = | r | and so | ψ ( tr ) · r | ≤ | r | for any t ≥
0. Choosing t = 1 weconclude the proof.A sufficient condition in order to have Assumptions 2.1. when d > ssumptions 2.2. Let d > and assume that ψ ( r ) = rg ( | r | ) for some function g : R +0 → R + such that g ∈ C ( R +0 ) and | rg ′ ( r ) /g ( r ) | ≤ for any r ∈ R +0 . Remark 2.1.
If Assumptions 2.1 and 2.2 are taken at the same time, then thefunction g satisfies lim | r |→∞ | r | g ( | r | ) = 1 . (2.4)The following result shows our claim. Lemma 2.2.
Let Assumptions 2.2 hold true. Then Assumptions 2.1.4 areverified.Proof.
To prove that ψ is conservative, let G be defined as G ( | r | ) := g ( | r | ). If¯ G is a primitive for G , then ¯ G ( | r | ) / ψ ( r ). Assumption (b)follows immediately. To show (c) we use a variant of Sylvester’s determinanttheorem, stating that for an invertible d × d matrix A ,det( A + uv T ) = det( A ) (1 + v T A − u ) , A ∈ M ( R d , R d ) , u, v ∈ R d . (2.5)We rest on Sylvester’s criterion for quadratic forms as well. Under the presentassumptions ∂ i ψ ( j ) = δ ij g ( | r | ) + g ′ ( | r | ) r i r j | r | , i, j = 1 , . . . , d. (2.6)Thus, the k-th principal minor of this matrix can be computed according to(2.5) as g ( | r | ) k g ′ ( | r | ) g ( r ) P ki =1 r i | r | ! and the result follows. As to (d), we resort to (2.6) again. Thus, when | r | ≫ | r | (cid:12)(cid:12)(cid:12) ∂ i ψ ( j ) ( r ) (cid:12)(cid:12)(cid:12) ≤ | r | g ( | r | ) + | r | g ′ ( | r | ) ≤ | r | g ( | r | ) ≤ . The previous set of assumptions will allow to cover the case of (2.1). Inorder to deal with (2.3), let us specify what do we demand of the function ϕ . Assumptions 2.3.
Let ϕ : R R +0 satisfy the following:1. ϕ is Lipschitz continuous.2. ϕ (0) = 0 and lim z → ϕ ( z ) / | z | = ϕ ′ (0) exists and is finite.3. ϕ ( z ) > if z = 0 . Remark 2.2.
The particular case of (2.1) is covered here by the choice ϕ ( z ) = sz as already noted before. Assumptions 2.3 are automatically satisfied in thiscase. 7ll considerations so far set up our framework. We point out first that underAssumptions 2.1 and 2.3 equations of the form (2.3) are well posed. This willbe a consequence of the results in [6, 7], as soon as we show that under theaforementioned set of assumptions equations of the form (2.3) fall under theirframework. We review the class of entropy solutions and related notions like entropy conditions and associated technicalities in Section 3; details on how toconnect (2.3) with that framework are given in Section 4.Taking this for granted, we are now ready to state the main results of thedocument. These collect several extensions of techniques and results in [9, 18]together with some new ideas in order to describe several properties of the class(2.3) and its subclass (2.1). Theorem 2.1.
Let ψ verify Assumptions 2.1 and let ϕ satisfy Assumption 2.3.Then the following assertions hold true:1. ( evolution of discontinuities ) Let u ∈ C ([0 , T ]; L ( R d )) be a distributionalsolution of (2.3) with initial datum ≤ u ∈ L ∞ ( R d ) ∩ BV ( R d ) . Assumethat u ∈ BV loc ( Q T ) and that the singular part of the spatial derivative hasno Cantor part. Assume further that ϕ is a convex function. Then theentropy conditions hold if and only if [ z · ν J u ( t ) ] + = ϕ ( u + ) and [ z · ν J u ( t ) ] − = ϕ ( u − ) . holds at each jump discontinuity (being u + > u − ≥ the lateral tracesof the solution and [ z · ν J u ( t ) ] + , [ z · ν J u ( t ) ] − the lateral traces of the flux).Moreover the speed of any discontinuity front is given by v = ϕ ( u + ) − ϕ ( u − ) u + − u − .
2. ( evolution of the support –see also [27]) Consider a compactly supportedinitial datum ≤ u ∈ L ∞ ( R d ) and let u ( t ) be the associated entropysolution. Thensupp u ( t ) ⊂ cl ( supp u ⊕ B (0 , θt )) , θ = max ≤ z ≤k u k ∞ ϕ ′ ( z ) for every t > . There are some additional properties which are specific of (2.1), as we statenow.
Theorem 2.2.
Let ψ verify Assumptions 2.1. Then the following assertionshold true:1. ( cost functions with bounded domain ) There exists a convex cost function k : R d → R +0 such that (2.1) can be (formally) recovered from the point ofview of optimal mass transport problems associated with k (as detailed inSection 5). Furthermore, k is finite on { v ∈ R d / | v | < s } and assumes thevalue + ∞ on { v ∈ R d / | v | > s } . . ( evolution of the support and strict positivity ) Consider a compactly sup-ported initial datum ≤ u ∈ L ∞ ( R d ) and let u ( t ) be the associatedentropy solution of (2.1) . If u satisfies the positivity assumption (6.14) and either d = 1 or Assumptions 2.2 holds with lim r →∞ | r | g ′ ( r ) = − ,then supp u ( t ) = cl ( supp u ⊕ B (0 , st )) ∀ t ≥ and u ( t ) is strictly positive inside its support for every t > .3. ( persistence of discontinuous interfaces in dimension one ) Consider aninitial datum ≤ u ∈ L ( R ) ∩ L ∞ ( R ) supported on a bounded interval [ a, b ] . Let u ( t ) be the associated entropy solution of (2.1) . Assume thatthere exist some ǫ, α > such that u ( x ) > α for every x ∈ ( b − ǫ, b ) . Ifthere exist some ˜ ǫ > such that d ( r ) = O (1 /r ) and ψ ′ ( r ) = O ( r − − ˜ ǫ ) as r → ∞ , then the left lateral trace of u ( t ) at x = b + st is strictly positive for any t > . A similar statement holds for the left end of the support. We do not expect these results to generalize easily to the setting of Theorem2.1. First, it does not seem to be possible to recast equations of the form (2.3)as equations derived from an optimal mass transportation problem when ϕ isnot linear, not even allowing for any kind of convex entropy. On the contrary, ageneralization of Theorem 2.2. to some of the models presented in [19] seemsfeasible (see for instance [16]). As regards the evolution of the support, equationsof the form (2.3) are expected to display waiting time phenomena, a fact whichhas been already confirmed in some cases [11, 27]. Hence to track the detailedevolution of the support for (2.3) is outside the scope of the techniques we usehere; sub-solutions suited to this task must be able to capture what the waitingtime for a given initial datum would be, which appears to be a very challengingproblem.It is also interesting to notice that the formal limit L → ∞ turns (2.1) intoa diffusion equation in transparent media (see [10] and references therein), ∂u∂t = div (cid:18) s u ∇ u |∇ u | (cid:19) . (2.7)We also note that performing the limit s → ∞ , L → ν := lim s →∞ , L → sL . Then, when d = 1 wearrive to the following equation u t = ψ ′ (0) νu xx . If we assume a structure like that in Assumptions 2.2, the limit equation inhigher dimensions would be u t = νg (0)∆ u. A rigorous analysis of these limit cases, together with the study of regularityproperties of solutions to (2.1)–(2.3) will be the subject of future investigations.9 .1 Examples
We present here a non-comprehensive list of partial differential equations thatare related to (2.1)–(2.3); most of them are already present in the literature.1. The standard heat equation (1.7) can be recast in the form (2.1) with L = ν/s , s = 1 and ψ ( r ) = r . It does not satisfy Assumptions 2.1,though, as the velocity V = ν ∇ u/u is not bounded.2. The porous media equations u t = ν div (( u/κ ) m − ∇ u ) , m > V = ν u m − κ m − ∇ u given by Darcy’s law –but note that it is not bounded. Then they can bewritten as (2.3) with ϕ ( u ) = u m − /κ m − and ψ ( r ) = r , but this does notverify Assumptions 2.1.3. None of Berstch–Dal Passo models [14] falls in our framework.4. The relativistic heat equation (1.2) [37, 15] has been already discussed inthe introduction. So far it is the most popular model in the mathematicalliterature that fits into (2.1).5. Wilson’s model was also mentioned in the introduction. It has the follow-ing form: ∂u∂t = ν div (cid:18) | u |∇ u | u | + νc |∇ u | (cid:19) , ν, c > . (2.8)This fits into (2.1) with V = ν sign ( u ) ∇ u | u | + νc |∇ u | = c νc ∇ uu νc (cid:12)(cid:12) ∇ uu (cid:12)(cid:12) . Thus s := c , L := ν/c and ψ ( r ) := r/ (1 + | r | ).6. We can regard the relativistic heat equation and Wilson’s model as par-ticular instances of a one-parametric family of models that will be usefulin order to probe a number of things in the sequel. Let us introduce afamily of models depending on a parameter p ∈ [1 , ∞ ) by means of ∂u∂t = ν div | u |∇ u (cid:0) | u | p + ν p c p |∇ u | p (cid:1) /p ! , ν, c > . (2.9)This family seems to have been first introduced in the astrophysical liter-ature by E. Larsen (see [36]). Here we have ψ ( r ) = r (1 + | r | p ) /p . p = 2 and p = 1 we recover the relativistic heat equationand Wilson’s model respectively. For any p ∈ [1 , ∞ ), (2.9) falls under thescope of Lemma 2.2 and Assumptions 2.1 are satisfied. The statement inpoint of Theorem 2.2 applies for every p ∈ [1 , ∞ ); the same happens forpoint , except for the case p = 1 (Wilson’s model).7. The following flux-saturated model was also introduced in the astrophys-ical literature [32]: u t = [ s u (coth( Lu x /u ) − u/ ( Lu x ))] x . Here we have ψ ( r ) = coth( r ) − /r . Assumptions 2.1 are also satisfied inthis case. Point in Theorem 2.2 does not apply for this model, though.8. A general family of one-dimensional models having the form (2.1) wasintroduced in [23] as part of a more general program concerning diffusiveapproximations of kinetic models via moment systems. For them ψ is a C ∞ function which is odd and strictly increasing. The simplest instancegiven in [23] is: p t = (( p/ǫ ) tanh( ǫp x / ( γp ))) x . In our notation, we have V = 1 ǫ tanh( ǫp x γp ) . This particular example satisfies Assumptions 2.1; point in Theorem 2.2applies in this case.9. If we choose ψ so that its Jacobian matrix is compactly supported, themodel (2.1) agrees with (2.7) for large values of the ratio |∇ u/u | .10. Equations like (1.3) [11, 18] or more generally those of the form ∂u∂t = ν div ϕ ( u ) ∇ u q u + ν c |∇ u | (2.10)fall under the framework given by (2.3), provided that ϕ satisfies Assump-tions 2.3.11. Models of the form u t = ν div u ∇ u m q ν c |∇ u m | , u t = α div Λ( u ) ∇ Φ( u ) p β |∇ Φ( u ) | ! , which were introduced in [19], do not fall into our framework, althoughsome of their properties are quite similar. More precisely, (2.3) agreesexactly with the second equation above for the choice Φ( u ) = log u , butthis choice is forbidden by the assumptions on Φ set in [19]. Hence thefamilies of models treated in [19] and those discussed here are disjoint.11 A summary about entropy solutions for de-generate parabolic equations with linear growthLagrangian
The class of equations given by (2.3) is a subclass of the set of second orderdiffusion equations in divergence form u t = div a ( u, ∇ u ) in Q T (3.1)which have both a degeneracy with respect to u (more precisely, lim z → + a ( z, ξ ) =0 for any ξ ∈ R d ) and a Lagrangian having linear growth at infinity, in the sensethat 1 | ξ | lim t → + ∞ a ( z, tξ ) ξ = ϕ ( z ) (3.2)for some function ϕ and for every z ≥
0. This is roughly the class of equationsfor which a well-possedness theory was developed in the series of papers [6, 7].The main tool there is the concept of entropy solution , which was introduced inthe previous papers and shown to provide a suitable class of solutions in whichthe well-posedness of the former problem is granted. This notion of solutionis based on a set of Kruzkov’s type inequalities and it requires to define afunctional calculus for functions whose truncations are of bounded variation.The purpose of this section is to collect a number of definitions and results(which we borrow from [6, 7, 9, 18]) that are needed to work with such entropysolutions, which is the concept of solution that we will use in order to deal withour specific class of flux-saturated equations (2.3) in the following sections. Forthat, we introduce functions of bounded variation, several classes of truncationfunctions, a suitable integration by parts formula, lower semicontinuity resultsfor functionals defined on BV and then the functional calculus itself. Thisallows to introduce the concept of entropy solutions and to state an existenceand uniqueness result for such. We conclude the section with a comparisonprinciple for sub- and super- solutions. Denote by L d and H d − the d -dimensional Lebesgue measure and the ( d − R d , respectively.Recall that if Ω is an open subset of R d , a function u ∈ L (Ω) whose gradient Du in the sense of distributions is a vector valued Radon measure with finitetotal variation in Ω is called a function of bounded variation. The class of suchfunctions will be denoted by BV (Ω). For u ∈ BV (Ω), the vector measure Du decomposes into its absolutely continuous and singular parts Du = D ac u + D s u . Then D ac u = ∇ u L d , where ∇ u is the Radon–Nikodym derivative of themeasure Du with respect to the Lebesgue measure L d . We also split D s u intwo parts: The jump part D j u and the Cantor part D c u .12e say that x ∈ Ω is an approximate jump point of u if there exist u + ( x ) = u − ( x ) ∈ R and η u ( x ) ∈ S d − such thatlim ρ ց L ( B + ρ ( x, η u ( x ))) Z B + ρ ( x,η u ( x )) | u ( y ) − u + ( x ) | dy = 0and lim ρ ց L ( B − ρ ( x, η u ( x ))) Z B − ρ ( x,η u ( x )) | u ( y ) − u − ( x ) | dy = 0 , where B + ρ ( x, η u ( x )) = { y ∈ B ( x, ρ ) / ( y − x ) · η u ( x ) > } and B − ρ ( x, η u ( x )) = { y ∈ B ( x, ρ ) / ( y − x ) · η u ( x ) < } . We denote by J u the set of approximate jump points. It is well known (see forinstance [3]) that D j u = ( u + − u − ) ν u H d − J u , with ν u ( x ) = Du | Du | ( x ), being Du | Du | the Radon–Nikodym derivative of Du withrespect to its total variation | Du | . For further information concerning functionsof bounded variation we refer to [3]. We will use in the sequel a number of different truncation functions. For a < b and l ∈ R , let T a,b ( r ) := max { min { b, r } , a } , T la,b = T a,b − l . We denote [6, 7, 9] T r := { T a,b : 0 < a < b } , T + := { T la,b : 0 < a < b, l ∈ R , T la,b ≥ } , T − := { T la,b : 0 < a < b, l ∈ R , T la,b ≤ } . Given any function w and a, b ∈ R we shall use the notation { w ≥ a } = { x ∈ R d : w ( x ) ≥ a } , { a ≤ w ≤ b } = { x ∈ R d : a ≤ w ( x ) ≤ b } , and similarly for thesets { w > a } , { w ≤ a } , { w < a } , etc.We need to consider the following function space T BV +r ( R d ) := (cid:8) w ∈ L ( R d ) + : T a,b ( w ) − a ∈ BV ( R d ) , ∀ T a,b ∈ T r (cid:9) . Using the chain rule for BV-functions (see for instance [3]), one can give a senseto ∇ u for a function u ∈ T BV + ( R d ) as the unique function v which satisfies ∇ T a,b ( u ) = vχ { a
S, T ∈ P such that S ≥ , S ′ ≥ T ≥ , T ′ ≥ p ( r ) = ˜ p ( T a,b ( r )) for some 0 < a < b , being ˜ p differentiable in a neighbor-hood of [ a, b ] and p = S, T . Similarly, we introduce
T SUPER as the class offunctions
S, T ∈ P such that S ≤ , S ′ ≥ T ≥ , T ′ ≤ p ( r ) = ˜ p ( T a,b ( r )) for some 0 < a < b , being ˜ p differentiable in a neighbor-hood of [ a, b ] and p = S, T .Finally, we let J q ( r ) denote the primitive of q for any real function q ; i.e. J q ( r ) := Z r q ( s ) ds. Assume that Ω is an open bounded set of R d with Lipschitz continuous bound-ary. Let p ≥ p ′ its dual exponent. Following [2], let us denote X p (Ω) = { z ∈ L ∞ (Ω , R d ) : div ( z ) ∈ L p (Ω) } . If z ∈ X p (Ω) and w ∈ BV (Ω) ∩ L p ′ (Ω), we define the functional ( z · Dw ) : C ∞ c (Ω) → R by the formula h ( z · Dw ) , ϕ i := − Z Ω w ϕ div ( z ) dx − Z Ω w z · ∇ ϕ dx. Then ( z · Dw ) is a Radon measure in Ω [2], and Z Ω ( z · Dw ) = Z Ω z · ∇ w dx, ∀ w ∈ W , (Ω) ∩ L ∞ (Ω) . We denote by ( z · Dw ) ac , ( z · Dw ) s its absolutely continuous and singular partswith respect to L d . One has that ( z · Dw ) s is absolutely continuous with respectto D s w and ( z · Dw ) ac = z · ∇ w . Moreover, ( z · Dw ) is absolutely continuouswith respect to | Dw | [2].The weak trace on ∂ Ω of the normal component of z ∈ X p (Ω) is defined in[2]. More precisely, it is proved that there exists a linear operator γ : X p (Ω) → L ∞ ( ∂ Ω) such that k γ ( z ) k ∞ ≤ k z k ∞ and γ ( z )( x ) = z ( x ) · ν Ω ( x ) for all x ∈ ∂ Ω–being ν Ω ( x ) the normal vector at x which points outwards–, provided that z ∈ C ( ¯Ω , R d ). We shall denote γ ( z )( x ) by [ z · ν Ω ]( x ). Moreover, the followingGreen’s formula, relating the function [ z · ν Ω ] and the measure ( z · Dw ), for z ∈ X p (Ω) and w ∈ BV (Ω) ∩ L p ′ (Ω), is proved in [2] Z Ω w div ( z ) dx + Z Ω ( z · Dw ) = Z ∂ Ω [ z · ν Ω ] w d H d − . .4 Functionals defined on BV In order to define the notion of entropy solutions of (3.1) we need a functionalcalculus defined on functions whose truncations are in BV . For that we needto introduce some functionals defined on functions of bounded variation [6, 7].Let Ω be an open subset of R d . Let g : Ω × R × R d → [0 , ∞ ) be a Borelfunction such that C ( x ) | ξ | − D ( x ) ≤ g ( x, z, ξ ) ≤ M ′ ( x ) + M | ξ | for any ( x, z, ξ ) ∈ Ω × R × R d , | z | ≤ R , and any R >
0, where M is a positiveconstant and C, D, M ′ ≥ R . Assume that C, D, M ′ ∈ L (Ω). Following Dal Maso [25] we consider thefunctional: R g ( u ) := Z Ω g ( x, u ( x ) , ∇ u ( x )) dx + Z Ω g ( x, ˜ u ( x ) , ν u ( x )) d | D c u | + Z J u Z u + ( x ) u − ( x ) g ( x, s, ν u ( x )) ds ! d H d − ( x ) , for u ∈ BV (Ω) ∩ L ∞ (Ω), being ˜ u the approximated limit of u [3]. The recessionfunction g of g is defined by g ( x, z, ξ ) = lim t → + t g ( x, z, ξ/t ) . (3.3)It is convex and homogeneous of degree 1 in ξ .In case that Ω is a bounded set, and under standard continuity and coercivityassumptions, Dal Maso proved in [25] that R g ( u ) is L -lower semi-continuousfor u ∈ BV (Ω). A very general result about the L -lower semi-continuity of R g in BV ( R d ) can be found on [26].Assume now that g : R × R d → [0 , ∞ ) is a Borel function such that C | ξ | − D ≤ g ( z, ξ ) ≤ M (1 + | ξ | ) ∀ ( z, ξ ) ∈ R d , | z | ≤ R, for any R >
C, D, M ≥ R .Assume also that χ { u ≤ a } ( g ( u ( x ) , − g ( a, , χ { u ≥ b } ( g ( u ( x ) , − g ( b, ∈ L ( R d ) , for any u ∈ L ( R d ) + . Let u ∈ T BV +r ( R d ) ∩ L ∞ ( R d ) and T = T a,b − l ∈ T + .For each φ ∈ C c ( R d ), φ ≥
0, we define the Radon measure g ( u, DT ( u )) by h g ( u, DT ( u )) , φ i := R φg ( T a,b ( u )) + Z { u ≤ a } φ ( x ) ( g ( u ( x ) , − g ( a, dx + Z { u ≥ b } φ ( x ) ( g ( u ( x ) , − g ( b, dx. (3.4)15f φ ∈ C c ( R d ), we write φ = φ + − φ − and we define h g ( u, DT ( u )) , φ i := h g ( u, DT ( u )) , φ + i − h g ( u, DT ( u )) , φ − i . Note that the following is shown in [26]: if g ( z, ξ ) is continuous in ( z, ξ ),convex in ξ for any z ∈ R , and φ ∈ C ( R d ) + has compact support, then h g ( u, DT ( u )) , φ i is lower semi-continuous in T BV + ( R d ) with respect to the L ( R d )-convergence.We can now define the required functional calculus (see [6, 7, 18]). Let S ∈ P + , T ∈ T + . We assume that u ∈ T BV +r ( R d ) ∩ L ∞ ( R d ) and χ { u ≤ a } S ( u ) ( g ( u ( x ) , − g ( a, , χ { u ≥ b } S ( u ) ( g ( u ( x ) , − g ( b, ∈ L ( R d ) . Then we define g S ( u, DT ( u )) as the Radon measure given by (3.4) with g S ( z, ξ ) = S ( z ) g ( z, ξ ).Let us introduce h : R × R d → R defined by h ( z, ξ ) := a ( z, ξ ) ξ, (3.5)being a the flux in (3.1). Under suitable assumptions on a , the measure h S ( u, DT ( u )) will make sense according to the previous functional calculus;see Section 4. Let L w (0 , T, BV ( R d )) be the space of weakly ∗ measurable functions w : [0 , T ] → BV ( R d ) (i.e., t ∈ [0 , T ] → h w ( t ) , φ i is measurable for every φ in the predual of BV ( R d )) such that R T k w ( t ) k BV dt is finite. Observe that, since BV ( R d ) has aseparable predual (see [3]), it follows easily that the map t ∈ [0 , T ] → k w ( t ) k BV is measurable. By L loc,w (0 , T, BV ( R d )) we denote the space of weakly ∗ measur-able functions w : [0 , T ] → BV ( R d ) such that the map t ∈ [0 , T ] → k w ( t ) k BV isin L loc (0 , T ). Definition 3.1.
Assume that ≤ u ∈ L ( R d ) ∩ L ∞ ( R d ) . A measurablefunction u : (0 , T ) × R d → R is an entropy solution of (3.1) in Q T if u ∈C ([0 , T ]; L ( R d )) , T a,b ( u ( · )) − a ∈ L loc,w (0 , T, BV ( R d )) for all < a < b , and(i) u t = div a ( u ( t ) , ∇ u ( t )) in D ′ ( Q T ) ,(ii) u (0) = u , and(iii) the following inequality is satisfied Z T Z R d φh S ( u, DT ( u )) dt + Z T Z R d φh T ( u, DS ( u )) dt ≤ Z T Z R d n J T S ( u ( t )) φ ′ ( t ) − a ( u ( t ) , ∇ u ( t )) · ∇ φ T ( u ( t )) S ( u ( t )) o dxdt, (3.6) for truncation functions S, T ∈ T + , and any smooth function φ of compactsupport, in particular those of the form φ ( t, x ) = φ ( t ) ρ ( x ) , φ ∈ D (0 , T ) , ρ ∈ D ( R d ) . a that are described in [6, 7, 18], which we denotecollectively by (H). We have the following existence and uniqueness result [7]. Theorem 3.3.
Let assumptions (H) hold. Then, for any initial datum ≤ u ∈ L ( R d ) ∩ L ∞ ( R d ) there exists a unique entropy solution u of (3.1) in Q T for every T > such that u (0) = u . Moreover, if u ( t ) , u ( t ) are the entropysolutions corresponding to initial data u , u ∈ L ( R d ) + respectively, then k ( u ( t ) − u ( t )) + k ≤ k ( u − u ) + k for all t ≥ . Existence of entropy solutions is proved by using Crandall-Liggett’s scheme[24] and uniqueness is proved using Kruzhkov’s doubling variables technique[29, 17].
In order to use the comparison principles introduced in [9] a certain technicalcondition is required.
Assumptions 3.4.
Let the function h defined by (3.5) satisfy h ( z, ξ ) ≤ M ( z ) | ξ | for some positive continuous function M ( z ) and for any ( z, ξ ) ∈ R × R d . Definition 3.2. [9] A measurable function u : (0 , T ) × R d → R +0 is an en-tropy sub- (resp. super-) solution of (3.1) if u ∈ C ([0 , T ] , L ( R d )) , a ( u, ∇ u ) ∈ L ∞ ( Q T ) , T a,b ( u ) ∈ L loc,w (0 , T, BV ( R d )) for every < a < b and the followinginequality is satisfied: Z T Z R d φh S ( u, DT ( u )) dt + Z T Z R d φh T ( u, DS ( u )) dt ≥ Z T Z R d n J T S ( u ( t )) φ ′ ( t ) − a ( u ( t ) , ∇ u ( t )) · ∇ φ T ( u ( t )) S ( u ( t )) o dxdt, (3.7) (resp. with ≤ ) for any φ ∈ D ( Q T ) + and any truncations T ∈ T + , S ∈ T − . This implies that u t ≤ div a ( u, ∇ u ) in D ′ ( Q T ) (3.8)(resp. with ≥ ). The following comparison principle was shown in [9]: Theorem 3.4.
Let assumptions (H) and Assumptions 3.4 hold. Given an en-tropy solution u of (3.1) corresponding to an initial datum ≤ u ∈ ( L ∞ ∩ L )( R d ) , the following statements hold true: . if u is a super-solution of (3.1) such that u ( t ) ∈ BV ( R d ) for a.e. t ∈ (0 , T ) , then k ( u ( t ) − u ( t )) + k ≤ k ( u − u (0)) + k ∀ t ∈ [0 , T ] ,
2. if u is a sub-solution of (3.1) such that u ( t ) ∈ BV ( R d ) for a.e. t ∈ (0 , T ) ,then k ( u ( t ) − u ( t )) + k ≤ k ( u (0) − u ) + k ∀ t ∈ [0 , T ] . Some extensions of this result have been shown in [27].
The goal of this paragraph is to show that the class of equations (2.3) (and morespecifically their flux a ( z, ξ )) satisfies the set of assumptions (H) given in [6, 7].This will be the case provided that Assumptions 2.1 and 2.3 below hold true.Hence Theorem 3.3 would apply, ensuring well-posedness for the class (2.3).Comparing (2.3) with (3.1), we let a : R × R d → R +0 be defined as a ( z, ξ ) := ϕ ( z ) ψ ( ξ/ | z | ) if z = 0 , a ( z = 0 , ξ ) := 0 . It follows from the previous that a ( z,
0) = 0 for every z ∈ R . (4.1) Lemma 4.3.
There holds that ∂ a ∂ξ i ∈ C (( R × R d ) \{ , } , R d ) for each i =1 , . . . , d .Proof. This is clear except maybe at z = 0. Note that ∂ a ( j ) ∂ξ i ( z, ξ ) = ϕ ( z ) | z | ∂ i ψ ( j ) ( ξ/ | z | ) if z = 0 . Then lim z → ∂ a ∂ξ i = 0 for any ξ = 0 , thanks to Assumptions 2.3. and Assumptions 2.1. / . Remark 4.3.
Note that assumption (H2) in [6, 7] requires ∂ a ∂ξ i to be contin-uous at every point of R × R d in order to apply their well-posedness results.Nevertheless, it can be shown that the continuity proved in Lemma 4.3 suffices.Now we look for some sort of potential function f ( z, ξ ) such that ∇ ξ f ( z, ξ ) = a ( z, ξ ). We introduce the Lagrangian f ( z, ξ ) := | z | ϕ ( z )Φ( ξ/ | z | ) if z = 0 , f ( z = 0 , ξ ) := 0 , ψ such that Φ(0) = 0. This potential is uniquely givenby Poincare’s Lemma: Φ( r ) := Z γ r ψ dσ with γ r : [0 , → R d , γ r ( t ) := tr for any r ∈ R d . (4.2)Thanks to Assumptions 2.1. , Φ is convex. Hence the vector field ξ ψ ( ξ ) ismonotone. Using Assumptions 2.1. (or equivalently Lemma 2.1) we get that r ψ ( r ) ≥ ∀ r ∈ R d . (4.3)It follows easily that Φ ≥
0. Moreover ξ Φ( ξ/ | z | ) is convex and so ξ f ( z, ξ ) is convex . (4.4) Lemma 4.4.
There holds that Φ( r ) = | r | + o ( | r | ) for | r | ≫ .Proof. Note that given r ∈ R d ,Φ( r ) = Z ψ ( tr ) · r dt = | r | + Z (cid:18) ψ ( tr ) − r | r | (cid:19) · r dt. Clearly (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:18) ψ ( tr ) − r | r | (cid:19) · r dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | r | d ( t | r | ) dt = Z | r | d ( λ ) dλ and the result follows thanks to Assumptions 2.1. .In particular, given ξ ∈ R d ,Φ( ξ/ | z | ) ∼ | ξ | / | z | for | z | ≪ . Thus, thanks to Assumptions 2.3 we deduce that lim z → f ( z, ξ ) = 0. As aconsequence, f ∈ C ( R × R d ) . (4.5)In the same vein, we can compute (recall that f is defined by (3.3)) f ( z, ξ ) = lim t → + tf ( z, ξ/t ) = lim t → + t | z | ϕ ( z )Φ( ξ/ | tz | ) = ϕ ( z ) | ξ | . (4.6)A number of bounds hold for the Lagrangian. The following upper bound iseasily obtained upon using Lemma 2.1. : f ( z, ξ ) = | z | ϕ ( z ) Z ψ ( tξ/ | z | ) · ξ/ | z | dt ≤ | z | ϕ ( z ) Z | ξ | / | z | dt ≤ ϕ ( z )(1 + | ξ | ) . (4.7)19 emma 4.5. There exist suitable constants C , D > such that the Lagrangiansatisfies the following lower bound f ( z, ξ ) ≥ C ϕ ( z ) | ξ | − D | z | ϕ ( z ) for any ( z, ξ ) ∈ R × R d . (4.8) Proof.
Thanks to the convexity of ξ Φ( ξ/ | z | ) we get the following inequalityΦ( ξ/ | z | ) ≥ ξ | z | ψ ( ξ/ | z | ) for any ( z, ξ ) ∈ R × R d . Then we show that ϕ ( z ) ψ ( ξ/ | z | ) ξ ≥ C ϕ ( z ) | ξ | − D | z | ϕ ( z ) for any ( z, ξ ) ∈ R × R d and for some C , D >
0, which would lead us to (4.8). For that we extend aone-dimensional argument presented in [16]. Choose 0 < C <
1. Note that rψ ( r ) = | r | + r (cid:18) ψ ( r ) − r | r | (cid:19) ≥ | r | − | r | d ( | r | ) . Hence, there exists some ˜ r depending on C such that rψ ( r ) ≥ C | r | ∀ r ∈ R d \ B (0 , ˜ r ) . Next, thanks to (4.3) we are able to find D := C ˜ r > rψ ( r ) ≥ C | r | − D ∀ r ∈ R d . Then we choose r = ξ/ | z | above and multiply both sides of the resulting in-equality by | z | ϕ ( z ) to get the desired estimate.Next we introduce h : R × R d → R defined by (3.5) as h ( z, ξ ) := a ( z, ξ ) ξ = ϕ ( z ) ψ ( ξ/ | z | ) ξ if z = 0 , h (0 , ξ ) := 0 . (4.9)It follows from (4.3) and Assumptions 2.1. that h ( z, ξ ) ≥ h ( z, ξ ) = h ( z, − ξ ) (4.11)for every z, ξ ∈ R . Moreover, h exists and coincides with f : h ( z, ξ ) = lim t → + tϕ ( z ) ψ (cid:18) ξ | tz | (cid:19) ξt = lim t → + ϕ ( z ) ξ ξ/ | tz || ξ/ | tz || = ϕ ( z ) | ξ | . (4.12)We have the following inequality relating h and a : a ( z, ξ ) η ≤ h ( z, η ) for every ξ, η ∈ R d and z ∈ R . (4.13)20 emma 4.6. There holds that (cid:12)(cid:12)(cid:12) ( a ( z, ξ ) − a (ˆ z, ξ ))( ξ − ˆ ξ ) (cid:12)(cid:12)(cid:12) ≤ C | z − ˆ z | | ξ − ˆ ξ | (4.14) for any ( z, ξ ) , (ˆ z, ˆ ξ ) ∈ R × R d and for some constant C > depending on | z | , | ˆ z | .Proof. Let us write | a ( z, ξ ) − a (ˆ z, ξ ) | ≤ | ψ ( ξ/ | ˆ z | ) | | ϕ (ˆ z ) − ϕ ( z ) | + ϕ ( z ) | ψ ( ξ/ | ˆ z | ) − ψ ( ξ/ | z | ) | := A + B. Recall that ψ and ϕ are locally Lipschitz. As k ψ k L ∞ ( R d , R d ) < ∞ thanks toAssumptions 2.1. and 2.1. , term A is fine.To deal with B , let us consider first that | z − ˆ z | ≥
1. Then it suffices to showthat B is bounded by a constant, uniformly in ξ and locally in z, ˆ z . This is easilyshown to be the case due to the boundedness of ψ and the local boundednessof ϕ .Let us treat now the case | z − ˆ z | <
1. Without loss of generality, assumethat | ˆ z | ≥ | z | . Using the mean value theorem, ψ ( j ) ( ξ/ | ˆ z | ) = ψ ( j ) ( ξ/ | z | ) + ∇ ψ ( j ) ( θ j )( ξ/ | ˆ z | − ξ/ | z | )for some θ j lying in the segment joining ξ/ | z | and ξ/ | ˆ z | , j = 1 , . . . , d . Thus, B ≤ ϕ ( z ) | z | | ξ || ˆ z | | z − ˆ z | Θ , Θ := sup ≤ λ ≤ ,i, j ∈ { , . . . , d } | ∂ i ψ ( j ) ( λ | ξ | / | z | + (1 − λ ) | ξ | / | ˆ z | ) | . Being ϕ ( z ) / | z | locally bounded, it suffices to bound | ξ | Θ / | ˆ z | independently of ξ and locally in ˆ z . Invoking Assumptions 2.1. / , there are values c, ˜ r > k Dψ k ∞ ( r ) ≤ c/ | r | for any r ∈ R d \ B (0 , ˜ r ). Thus, whenever ˜ r < | ξ/ ˆ z | ≤| ξ/z | , | ξ || ˆ z | Θ ≤ c | ξ || ˆ z | max (cid:26) | z || ξ | , | ˆ z || ξ | (cid:27) ≤ c. If | ξ/ ˆ z | < ˜ r we are also done as the entries of Dψ are bounded.Thanks to (4.1) and (4.4)–(4.14) we can apply the well-posedness theorygiven in [7] (more precisely Theorem 3.3 above). We get the following result. Theorem 4.5.
Consider an initial datum ≤ u ∈ L ( R d ) ∩ L ∞ ( R d ) . LetAssumptions 2.1 be fulfilled. Then the following assertions hold true:1. There exists a unique entropy solution u of (2.1) in Q T for every T > with u as initial datum.2. Let ϕ satisfy Assumption 2.3. Then there exists a unique entropy solution u of (2.3) in Q T for every T > , such that u (0) = u .Moreover, if we are given u, ˆ u two entropy solutions of (2.1) (resp. (2.3) ) corre-sponding to initial data ≤ u , ˆ u ∈ L ( R d ) ∩ L ∞ ( R d ) respectively, then k ( u ( t ) − ˆ u ( t )) + k ≤ k ( u − ˆ u ) + k ∀ t > . Furthermore, using (4.9) we notice at once that Assumptions 3.4 is satisfiedtoo. Hence Theorem 3.4 holds under Assumptions 2.1 and 2.3.21
A connection with optimal transport theory
The use of optimal mass transport problems to solve parabolic equations waspioneered by [28] and further developed by many authors, see [1, 4, 15] forinstance. We give here a brief account on it. Let k : R d → [0 , ∞ ] be a convexcost function and let us define the associated Wasserstein distance between twoprobability distributions ρ and ρ by W hk ( ρ , ρ ) := inf (cid:26)Z R d × R d k (cid:18) x − yh (cid:19) dγ ( x, y ) (cid:30) γ ∈ Γ( ρ , ρ ) (cid:27) , being h >
0. Here Γ( ρ , ρ ) stands for the set of probability measures in R d × R d whose marginals are ρ and ρ .Now let F : [0 , ∞ ) → [0 , ∞ ) be a convex function and let P ( R d ) be the set ofprobability density functions ρ : R d → [0 , ∞ ). Starting from ρ h = ρ ∈ P ( R d ),we can solve iterativelyinf ρ ∈P ( R d ) hW hk ( ρ hn − , ρ ) + Z R d F ( ρ ( x )) dx. Define ρ h ( t ) = ρ hn for t ∈ [ nh, ( n + 1) h ). Then as h → + the solution of thisminimization scheme formally converges to a limit u which solves the followingequation u t = div ( u ∇ k ∗ ( ∇ F ′ ( u ))) . This convergence has been shown to be rigorous in certain cases [28, 1, 34]. Inparticular, the relativistic heat equation (1.2) falls under this general picturefor the choice F ( r ) = ν ( r log r − r ) , with the following cost function: k ( v ) = (cid:16) − p − | v | /c (cid:17) c if | v | ≤ c + ∞ if | v | > c, so that k ∗ ( v ) = c (cid:16)p | v | /c − (cid:17) and ∇ k ∗ ( v ) = v p | v | /c . This was observed in [15] at a formal level and later made rigorous in [34].We notice that this is no particular phenomenon: Equations coming fromsuch minimization schemes may have the form (2.1). Our main concern in thissection is the following: If such a model verifies Assumptions 2.1, what can besaid about the cost function k ?To be able to compare both frameworks we must set F ( r ) = L ( r log r − r ),which would yield an equation of the following form: u t = div ( u ∇ k ∗ ( L ∇ u/u )) . (5.1)22hen, the following result provides an answer to the previous question; we geta new way to describe the role of the constant s . Proposition 5.1.
Let ψ satisfy Assumptions 2.1. Then there exists a convexcost function k : R d → R +0 such that (2.1) can be recast as (5.1) . Furthermore, k is finite on { v ∈ R d / | v | < s } and assumes the value + ∞ on { v ∈ R d / | v | > s } .Provided that the function d in Assumptions 2.1.3 is integrable, k is also finiteon { v ∈ R d / | v | = s } .Proof. Comparing (2.1) with (5.1) we identify sψ ( r ) = ∇ k ∗ ( r ) ∀ r ∈ R d . We can construct k ∗ in a way that it satisfies k ∗ (0) = 0 (as we did with Φ inSection 4). We set k ∗ ( r ) := s Z γ r ψ dσ, γ r as in (4.2) . As previously argued, k ∗ so defined is non-negative, convex and regular enoughso that Fenchel–Moreau’s theorem applies. Then we have the following repre-sentation formula for k : R d → R : k ( v ) = sup p ∈ R d pv − k ∗ ( p ) := sup p ∈ R d Γ v ( p ) . Let us address the properties of k . We will make repeated use of Lemma 2.1 inthe sequel. In order to compute the value of k ( v ), let us note that k ∗ ( r ) ր s | r | for | r | → ∞ irrespective of the direction. Thus, for | p | large enough,Γ v ( p ) ∼ | p | ( | v | cos θ ( p, v ) − s ) , (5.2)being θ ( p, v ) the angle formed by p and v . So, whenever | v | > s , Γ v ( p ) divergesto + ∞ as a function of p along the ray given by the direction of v . Hence k ( v ) = + ∞ for v ∈ R d such that | v | > s .Let us deal now with k ( v ) when | v | < s . In this case we notice that, thanks to(5.2), Γ v ( p ) diverges to −∞ along any ray as a function of p . Then sup p ∈ R d Γ v ( p )is attained at those ¯ p ∈ R d such that ∇ p Γ v (¯ p ) = 0. We are led to solve v = ∇ k ∗ (¯ p ) , that is vs = ψ (¯ p ) . (5.3)Let us write ψ − ( v/s ) for the solution set of (5.3) (if ξ ψ ( ξ ) is strictlymonotone we have a unique solution), so that k ( v ) = sup ¯ p ∈ ψ − ( v/s ) ¯ pv − k ∗ (¯ p ) . This supremum is clearly finite. On the other hand, note that in dimension onewe have k ( v ) = sup ¯ p ∈ ψ − ( v/s ) Z ¯ p v − sψ ( λ ) dλ ≥ sup ¯ p ∈ ψ − ( v/s ) Z ¯ p v − sψ (¯ p ) dλ = 023s ψ is non-decreasing. For higher dimensions, we show that the cost function-to-be is non-negative as follows: k ( v ) = sup ¯ p ∈ ψ − ( v/s ) v ¯ p − s Z ψ ( t ¯ p )¯ p dt ≥ sup ¯ p ∈ ψ − ( v/s ) v ¯ p − s Z vs ¯ p dt = 0 . Hence k ( v ) qualifies as cost function for | v | < s .Finally we study the behavior of k ( v ) when | v | = s . We start doing this indimension one. First, we note thatΓ s ( p ) = s Z p − ψ ( λ ) dλ, then ddp Γ s ( p ) = s (1 − ψ ( p )) ≥ . Then we compute the supremum taking the limit p → + ∞ (this makes senseeven when ψ ′ is compactly supported). Thus k ( s ) = s Z ∞ − ψ ( λ ) dλ. Arguing in a similar way, k ( − s ) is found to have the same value. Let us dis-cuss now the higher dimensional case. Nothing precludes that the solution set ψ − ( v/s ) of (5.3) be non-empty even for | v | = s . This is not troublesome aslong as this set is bounded, as the associated contributions Γ v ( p ) to the value of k ( v ) would be clearly bounded. Then let us discuss what happens for | p | → ∞ .According to (5.2), Γ v ( p ) diverges to −∞ along any ray except maybe alongthe ray determined by v itself. In fact, let p = λ v | v | for λ > | v | = sddλ Γ v (cid:18) λ v | v | (cid:19) = ddλ λs − s Z λ ψ (cid:18) t v | v | (cid:19) v | v | dt ! = s − vψ (cid:18) λvs (cid:19) ≥ . Hence,lim λ → + ∞ Γ v (cid:18) λ v | v | (cid:19) = lim λ → + ∞ s Z λ − v | v | · ψ (cid:18) t v | v | (cid:19) dt = s Z ∞ − v | v | · ψ (cid:18) t v | v | (cid:19) dt. We conclude the proof by noticing that the integrability of d in Assumptions2.1. ensures the convergence of these improper integrals.Let us stress that there is at least a certain subclass of the class of functions ψ satisfying Assumptions 2.1 such that the minimization procedure sketchedat the beginning of the section produces actual solutions of (2.1). See [34] fordetails. The aim of this section is to supply proofs for points and in Theorem 2.2 andpoint in Theorem 2.1. This is done by means of comparison with suitable sub-and super-solutions, in the same vein as [9]. For that we will rest in Theorem3.4, which applies under Assumptions 2.1 and 2.3 as argued in Section 4.24 .1 Upper bounds on support spreading rates We show in this paragraph that dilations of multiples of characteristic functionsof compact sets qualify as super-solutions if their spreading rate behaves in asuitable way. This is an extension of Proposition 1 in [9]. A generalization ofthe results in this paragraph has been independently discovered in [27].
Proposition 6.2.
Let β > and C ⊂ R d a compact set. Let Assumptions 2.1and 2.3 be satisfied. Then u ( t, x ) = βχ B ( t ) , being B ( t ) := C ⊕ B (0 , θt ) with θ = max ≤ z ≤ β ϕ ′ ( z ) , is a super-solution of (2.3) in Q T for every T > .Proof. We start by defining ¯ B ( t ) := C ⊕ B (0 , C ( t )) for some function C ( t ) ≥ C (0) = 0 and C ′ ( t ) ≥
0. Let us introduce now W ( t, x ) := βχ ¯ B ( t ) . Fix
T >
0. We shall determine what extra conditions have to be imposed on C ( t )in order that W be a super-solution of (2.3) in Q T . Note that a ( W, ∇ W ) = 0for such a profile, thanks to (4.1). As W t = βC ′ ( t ) H d − | ∂ ¯ B ( t ) , we get at once that W t ≥ div a ( W, ∇ W ) in D ′ ( Q T ) . (6.1)Next we compute each term in (3.7) of Definition 3.2 separately. Let T ∈ T + and S ∈ T − . Arguing as in [9], Proposition 1, we get that h S ( W ( t ) , DT ( W ( t ))) s + h T ( W ( t ) , DS ( W ( t ))) s = J ( T S ) ′ ϕ ( β ) H d − | ∂ ¯ B ( t ) . (6.2)Note that J ( T S ) ′ ϕ ( β ) = Z β ( T S ) ′ ( r ) ϕ ( r ) dr = − Z β T ( r ) S ( r ) ϕ ′ ( r ) dr + T ( β ) S ( β ) ϕ ( β )= − J T Sϕ ′ ( β ) + ( T Sϕ )( β ) . (6.3)Here we used that ϕ (0) = 0. Apart from this, we notice that J T S ( W ( t )) = J T S ( β ) χ ¯ B ( t ) and so ∂ t J T S ( W ( t )) = C ′ ( t ) J T S ( β ) H d − | ∂ ¯ B ( t ) . Given any 0 ≤ φ ∈ D ′ ( Q T ), we have shown that Z Q T J T S ( W ( t )) φ ′ ( t ) dt = − Z T ( C ′ ( t ) J T S ( β ) Z ∂ ¯ B ( t ) φ d H d − ) dt. (6.4)25ollecting (6.1)–(6.4) and comparing with inequality (3.7), we will be done ifwe can show that the following inequality holds for any T ∈ T + , S ∈ T − and0 ≤ φ ∈ D ′ ( Q T ): Z T ( [( T Sϕ )( β ) − J T Sϕ ′ ( β ) + C ′ ( t ) J T S ( β )] Z ∂ ¯ B ( t ) φ d H d − ) dt ≤ . Here we have that (
T Sϕ )( β ) ≤ S ≤
0. Note also that J T Sϕ ′ ( β ) = Z β T ( r ) S ( r ) ϕ ′ ( r ) dr ≥ θ Z β T ( r ) S ( r ) dr = θJ T S ( β )for θ = max ≤ z ≤ β ϕ ′ ( z ). Thus, in order for W to be a super-solution it is enoughto ask for min t ∈ [0 ,T ] C ′ ( t ) ≥ θ . This implies our result. Remark 6.4.
Tracking the above proof we notice that we do not need a fluxwith structure as in (2.3) in order that the argument works. The main require-ment in order that the above proof goes through while Theorem 3.4 appliesis that the flux must be such ϕ can be defined by means of (3.2), being ϕ aLipschitz-continuous function such that ϕ (0) = 0 , ϕ ( z ) > z = 0 and ϕ ′ (0)exists. This generic point of view is the one that is adopted in [27]. Corollary 6.1.
Let Assumptions 2.1 and 2.3 be verified. Let ≤ u ∈ L ( R d ) ∩ L ∞ ( R d ) be compactly supported and let u ( t ) the entropy solution of (2.3) with u as initial datum. Thensupp u ( t ) ⊂ cl ( supp u ⊕ B (0 , θt )) , θ = max ≤ z ≤k u k ∞ ϕ ′ ( z ) for every t > . To give lower bounds for the spreading rate of solutions to (2.1) we shall lookfor compactly supported sub-solutions. The following result will be helpful inso doing. It is inspired in the proof of Proposition 2 of [9].
Proposition 6.3.
Let W ( t, x ) such that W (0 , · ) is compactly supported and as-sume that B = B ( t ) := supp W ( t, · ) = supp W (0 , · ) ⊕ B (0 , C ( t )) , also satisfying W ( t, · ) | ∂B = γ ( t ) ≥ and the regularity requirements set in Definition 3.2. Let ϕ be defined by (3.2) . Given T > , assume either:1. Relation (3.8) holds inside the support.2. γ ( t ) = 0 for every ≤ t ≤ T .or 1. Relation (3.8) holds inside the support. . γ ( t ) > for every ≤ t ≤ T .3. sup t ∈ (0 ,T ) C ′ ( t ) ≤ inf z ϕ ′ ( z ) .4. [ a ( W, ∇ W ) · ν B ] = − ϕ ( γ ( t )) for every ≤ t ≤ T .Then, W fulfills (3.7) in Definition 3.2. Remark 6.5.
Informally speaking, condition above means that the profile isconcave in a neighborhood of the interface and the contact angle is vertical. SeeRemark 7.8 in that regard. Proof.
Let 0 ≤ φ ∈ D ( Q T ), T ∈ T + and S ∈ T − . We compute each term in(3.7) of Definition 3.2 separately. First, arguing as in the proof of Proposition6.2, h T ( W, DS ( W )) s + h S ( W, DT ( W )) s = [( T Sϕ )( γ ( t )) − J T Sϕ ′ ( γ ( t ))] H d − | ∂B . (6.5)We compute also ∂ t J T S ( W ) = W t T ( W ) S ( W ) χ B + C ′ ( t ) J T S ( γ ( t )) H d − | ∂B . (6.6)Moreover, letting z = a ( W, ∇ W ), Z Q T z ∇ φT ( W ) S ( W ) dxdt = − Z Q T φ div ( z T ( W ) S ( W )) dxdt + Z T Z ∂B [ z T ( W ) S ( W ) · ν B ] φ d H d − dt. (6.7)Collecting (6.5)–(6.7), (3.7) reads now as follows: Z T [( T Sϕ )( γ ( t )) − J T Sϕ ′ ( γ ( t ))] Z ∂B φ ( t ) d H d − dt + Z Q T [ h S ( W, DT ( W )) ac + h T ( W, DS ( W )) ac ] φ dt ≥ Z Q T φ div ( z T ( W ) S ( W )) dxdt − Z T Z B φW t T ( W ) S ( W ) dxdt − Z T Z B [ z T ( W ) S ( W ) · ν B ] φ d H d − dt − Z T J T S ( γ ( t )) C ′ ( t ) Z ∂B φ ( t ) d H d − dt. Our aim is to show that this holds true indeed. It is equivalent to check theabove inequality for the absolutely continuous and singular parts separately. As h S ( W, DT ( W )) ac = S ( W ) h ( W, ∇ T ( W ))= S ( W ) ∇ T ( W ) a ( T ( W ) , ∇ T ( W )) = S ( W ) ∇ T ( W ) a ( W, ∇ T ( W ))27and in the same way for the other term) we have that h S ( W, DT ( W )) ac + h T ( W, DS ( W )) ac = a ( W, ∇ W ) ∇ ( S ( W ) T ( W )) . Thus, the inequality for the absolutely continuous parts reduces to Z Q T φT ( W ) S ( W )div z dxdt − Z Q T φW t T ( W ) S ( W ) dxdt ≤ . Then it suffices to show that W t ≤ div a ( W, ∇ W ) a.e. in B ( t ) , for a.e. t ∈ (0 , T ) . Now we discuss the inequality relating the singular parts (note that when γ ( t ) =0 there is no singular part at all, due to the fact that ϕ (0) = 0). For that wecompute [ z T ( W ) S ( W ) · ν B ] = ϕ ( γ ( t )) T ( γ ( t )) S ( γ ( t ))using condition . Then the inequality for the singular parts is equivalent to − J T Sϕ ′ ( h ( t )) ≥ − J T S ( h ( t )) C ′ ( t ) for a.e. t ∈ (0 , T ) . Thanks to condition , this is automatically fulfilled.In this way we are able to sharpen and extend the program that was in-troduced in [9]. We construct now sub-solutions spreading at any prefixed ratestrictly lower than s . This is crucial as it is less demanding on ψ to constructsuch than to construct sub-solutions attaining the rate given by s , see Remark6.6 below. Proposition 6.4.
Let d = 1 and let ψ satisfy Assumptions 2.1. Let c < s and R > . Then the following statements hold: there exists some A > (depending on ψ , c/s, L, R ) such that W ( t, x ) = e − At p R ( t ) − | x | χ B (0 ,R ( t )) , R ( t ) = R + ct is a sub-solution of (2.1) in Q T for every T > . Let < θ < and assume that lim r →∞ r − θ ψ ′ ( r ) = 0 . (6.8) Fix γ > . Then there exists some A > (depending on ψ , c/s, L, γ , R ) suchthat W ( t, x ) = (cid:8) e − At ( R ( t ) − | x | ) θ + γ e − At (cid:9) χ B (0 ,R ( t )) , R ( t ) = R + ct is a sub-solution of (2.1) in Q T for every T > . roof. Let us define W ( t, x ) = (cid:0) α ( t )( R ( t ) − | x | ) θ + γ ( t ) (cid:1) χ B (0 ,R ( t )) , R ( t ) = R + ct for some functions α, γ > α ′ , γ ′ ≤
0. Thanks toProposition 6.3 we can restrict ourselves to check that W t ≤ ( sW ψ ( W x /W )) x at B (0 , R ( t )) for each t >
0. We are to show that W t ≤ sW x ψ ( W x /W ) + sψ ′ ( W x /W ) (cid:8) W xx − ( W x ) /W (cid:9) (6.9)holds at B (0 , R ( t )) for every t >
0. Neglecting the factor χ B (0 ,R ( t )) in whatfollows, we compute W t = α ′ ( t )( R ( t ) − | x | ) θ + 2 θα ( t ) cR ( t )( R ( t ) − | x | ) − θ + γ ′ ( t ) ,W x = − θα ( t ) x ( R ( t ) − | x | ) − θ ,W xx = 2 θ (2 θ − α ( t ) x − θα ( t ) R ( t )( R ( t ) − | x | ) − θ . Due to parity, it suffices to show (6.9) only for non-negative values of x . Case 1)
To prove the first statement we set γ ( t ) = 0 and θ = 1 /
2. Then weintroduce λ ∈ [0 ,
1) so that x = λR ( t ). In terms of λ , (6.9) reads now as follows: α ′ ( t ) R ( t ) p − λ + cα ( t ) √ − λ ≤ − sλα ( t ) √ − λ ψ (cid:18) − λR ( t )(1 − λ ) (cid:19) + sψ ′ (cid:18) − λR ( t )(1 − λ ) (cid:19) (cid:26) − α ( t )(1 + λ ) R ( t )(1 − λ ) / (cid:27) . This can be rearranged as α ′ ( t ) α ( t ) ≤ − sR ( t ) 1 + λ (1 − λ ) ψ ′ (cid:18) λR ( t )(1 − λ ) (cid:19) + 1 R ( t )(1 − λ ) (cid:18) sλψ (cid:18) λR ( t )(1 − λ ) (cid:19) − c (cid:19) . We introduce a new variable r := r ( λ ) = λR ( t )(1 − λ ) . Note that when λ variesfrom 0 to 1, r varies from 0 to ∞ . Hence, it suffices to show that α ′ ( t ) α ( t ) ≤ s (cid:18) rψ ( r ) − crλs − λ λ r ψ ′ ( r ) (cid:19) for any λ ∈ [0 , − A , then the choice α ( t ) = e − At would suit our pur-poses. The combination of terms at the right hand side is clearly bounded frombelow except maybe when r ≫
1. In order to see what happens in that case,we pick ǫ ∈ (0 ,
1) such that 1 − ǫ = c/s . Let us write rψ ( r ) − crλs = r (cid:18) ψ ( r ) − − ǫ/ λ (cid:19) + ǫr λ . λ > − ǫ/
2, the first term above is non-negative for r largeenough. Thus, if we show thatlim λ → ǫr λ − λ λ r ψ ′ ( r ) ≥ − A for some A > λ → ǫλ λ ≥ lim λ → rψ ′ ( r ) . Recall that ψ ′ ≥ ǫ (thus allowing to get c < s as close to s asdesired). Then we must impose the following condition:lim r →∞ rψ ′ ( r ) = 0 . But this is automatically satisfied thanks to Assumption 2.1. . In this way ourfirst statement follows. Case 2)
We introduce λ ∈ [0 ,
1) so that x = λR ( t ). In terms of λ , (6.9)reads: α ′ ( t ) R θ ( t )(1 − λ ) θ + 2 θcα ( t ) R ( t ) R − θ ( t )(1 − λ ) − θ + γ ′ ( t ) ≤ − θsλα ( t ) R ( t ) R − θ ( t )(1 − λ ) − θ ψ ( I )+ sψ ′ ( I ) × (cid:26) − θ (2 θ − α ( t ) λ R ( t ) − θα ( t ) R ( t ) R − θ ( t )(1 − λ ) − θ − θ α ( t ) λ R ( t ) R − θ ( t )(1 − λ ) − θ [ α ( t ) R θ ( t )(1 − λ ) θ + γ ( t )] (cid:27) , being I = − θα ( t ) λR ( t ) α ( t ) R ( t )(1 − λ ) + γ ( t ) R − θ ( t )(1 − λ ) − θ . Thus, we will be done if we are able to check the following inequality: α ′ α ≤ sR θ ( t )(1 − λ ) θ (cid:26) − γ ′ αs + 2 θR − θ ( t )(1 − λ ) − θ (cid:16) λψ ( I ) − cs (cid:17) + ψ ′ ( I ) (cid:18) θ [(2 θ − λ − R − θ ( t )(1 − λ ) − θ − θ λ R − θ ( t )(1 − λ ) − θ + R − θ ( t )(1 − λ ) − θ γα (cid:19)(cid:27) (6.10)with I = − I = 2 θλR ( t )(1 − λ ) + γα R − θ ( t )(1 − λ ) − θ . As in the previous case, it suffices to ensure that the right hand side of theprevious inequality is bounded from below by some constant − A . Let us choose30 ( t ) = γ α ( t ) with γ >
0. Now we let c/s = 1 − ǫ for ǫ ∈ (0 ,
1) and decompose λψ ( I ) − c/s = λ (cid:18) ψ ( I ) − − ǫ/ λ (cid:19) + ǫ/ . The first term above is non-negative for λ close enough to 1. Taking this intoaccount, it suffices to havelim λ → ψ ′ ( I ) (cid:18) θ [(2 θ − λ − R − θ ( t )(1 − λ ) − θ − θ λ R − θ ( t )(1 − λ ) − θ + R − θ ( t )(1 − λ ) − θ (cid:19) − α ′ γ αs + lim λ → ǫθR − θ ( t )(1 − λ ) − θ ≥ − A . Neglecting the second term above causes no loss ofgenerality. Hence, we ask forlim λ → ǫθR − θ ( t )(1 − λ ) − θ ≥ lim λ → (cid:26) θR − θ ( t )(1 − λ ) − θ ψ ′ ( I ) × (cid:18) θλ R θ ( t )(1 − λ ) θ + 1 − (2 θ − λ − R θ ( t )(1 − λ ) θ (cid:19)(cid:27) . We notice again that the right hand side above is non-negative and that we needto ensure the above inequality independently of the actual value of ǫ . Then thefollowing condition must be imposed:lim λ → ψ ′ (cid:18) − λ ) − θ (cid:19) − λ = 0 . This is the same as (6.8), thus our statement is granted.
Remark 6.6.
Examining carefully the proof of the previous statement we notethe following:1. Provided that d ( r ) = O (1 /r ) and ψ ′ ( r ) = O (1 /r ) as r → ∞ , we can take c = s in the first point of Proposition 6.4.2. Provided that d ( r ) = O (1 /r ) and ψ ′ ( r ) = O ( r θ − − θ ) as r → ∞ , (6.11)we can take c = s in the second point of Proposition 6.4.There is a value of θ such that (6.11) holds for every model of the form (2.9),except for the case of Wilson’s model (2.8).31ome of these results can be extended to higher dimensions under Assump-tions 2.2. Proposition 6.5.
Let d > and let ψ satisfy Assumptions 2.1 and 2.2. Let c < s and R > . Assume in addition that lim r →∞ r g ′ ( r ) = − . (6.12) Then, there exists some
A > (depending on g , c/s, L, R ) such that W ( t, x ) = e − At p R ( t ) − | x | χ B (0 ,R ( t )) , R ( t ) = R + ct is a sub-solution of (2.1) in Q T for every T > .Proof. Let us define W ( t, x ) = α ( t ) p R ( t ) − | x | χ B (0 ,R ( t )) , R ( t ) = R + ct for some function α to be determined, such that α ′ ≤
0. Thanks to Proposition6.3 we can restrict ourselves to check that W t ≤ ( sW ψ ( W x /W )) x at B (0 , R ( t ))for each t >
0. We are to show that W t ≤ s ∆ W g (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∇ WW (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + sg ′ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∇ WW (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:18) ∇ W D W ∇ W T W |∇ W | − |∇ W | W (cid:19) (6.13)holds at B (0 , R ( t )) for every t >
0, being D W the Hessian matrix of W ( t ).Neglecting the factor χ B (0 ,R ( t )) in what follows, we compute W t = α ′ ( t ) p R ( t ) − | x | + α ( t ) cR ( t ) p R ( t ) − | x | ,∂ i W = − α ( t ) x i p R ( t ) − | x | , ∇ WW = − xR ( t ) − | x | ,∂ ij W = − α ( t )[ R ( t ) δ ij − | x | δ ij + x i x j ]( R ( t ) − | x | ) / , ∆ W = − α ( t )[ dR ( t ) + (1 − d ) | x | ]( R ( t ) − | x | ) / . We substitute into (6.13) to obtain, after rearranging a bit, α ′ ( t ) α ( t ) ≤ − cR ( t ) R ( t ) − | x | − sg (cid:18) | x | R ( t ) − | x | (cid:19) dR ( t ) + (1 − d ) | x | ( R ( t ) − | x | ) − sg ′ (cid:18) | x | R ( t ) − | x | (cid:19) | x | + | x | R ( t )( R ( t ) − | x | ) . This depends on x only through | x | . Then we introduce λ ∈ [0 ,
1) so that | x | = λR ( t ). In terms of λ , the inequality to be satisfied reads: α ′ ( t ) α ( t ) ≤ R ( t )(1 − λ ) (cid:26) − c − s d + (1 − d ) λ R ( t )(1 − λ ) g (cid:18) λR ( t )(1 − λ ) (cid:19) − s λ + λR ( t )(1 − λ ) g ′ (cid:18) λR ( t )(1 − λ ) (cid:19)(cid:27) . − A , which would imply our result. Let us write I for the term insidebraces; to get such a bound, it suffices to show that lim λ → I >
0. In fact,lim λ → I = − c − s − s lim λ → λ + λ R ( t )(1 − λ ) g ′ (cid:18) λR ( t )(1 − λ ) (cid:19) = s − c > Remark 6.7.
Note that (6.12) is satisfied by every equation of the form (2.9).The previous results allow us to track the evolution of the support in thesame vein as in [9].
Corollary 6.2. ( Evolution of the support ) Let ψ satisfy Assumptions 2.1. Let C ⊂ R d be an open set and let ≤ u ∈ ( L ∩ L ∞ )( R d ) with support equal to C . Let u ( t ) be the entropy solution of (2.1) with u as initial datum. Assumingthatfor any closed set F ⊂ C, there is α F > such that u ( x ) ≥ α F ∀ x ∈ F, (6.14) and either d = 1 or Assumptions 2.2 holds together with (6.12) , thensupp u ( t ) = cl ( supp u ⊕ B (0 , st )) . Proof.
This is a combination of Proposition 6.4 (resp. Proposition 6.5) andCorollary 6.1. Note that both are invariant under spatial translations. Con-dition (6.14) ensures that for each y ∈ C we can find a suitable radius and asuitable height in order to apply the first point of Proposition 6.4 (resp. Propo-sition 6.5) with a sub-solution centered at y (as argued in the proof of Theorem4 in [9]), whose velocity c can be chosen as close to s as desired. Corollary 6.3. ( Persistence of discontinuous interfaces ) Let d = 1 and let ψ satisfy Assumptions 2.1. Let ≤ u ∈ ( L ∩ L ∞ )( R ) be supported on a boundedinterval [ a, b ] . Let u ( t ) be the entropy solution of (2.1) with u as initial datum.Assuming that there exist some ǫ, α > such that u ( x ) > α > for every x ∈ ( b − ǫ, b ) and that (6.11) holds, the left lateral trace of u ( t ) at x = b + st is strictly positive for every t > . A similar statement holds for the left end ofthe support.Proof. This is similar to the previous one, but we use the second statement inProposition 6.4 this time, taking c = s thanks to Remark 6.6. Under the presentassumptions we may choose R = ǫ/ γ , A > < θ < x = b − ǫ/ u for t = 0. Thanks to Theorem 3.4, u ( t ) ≥ γ e − At for a.e. x ∈ ( b − ǫ, b ) ⊕ B (0 , st ) and any t >
0. This implies inparticular that u ( t ) ∈ BV (( b − ǫ, b ) ⊕ B (0 , st )). Thus, we can compute the leftlateral trace at x = b + st as u ( t, b + st ) − = lim λ → λ Z b + stb + st − λ u ( t, r ) dr ≥ γ e − At for any t >
0. 33
Rankine–Hugoniot relations
The idea of this section is to generalize in a suitable way some results in [18, 19].This will provide a proof for in Theorem 2.1. In so doing we will notice thatsuch results hold for a class of equations which is wider than (2.3). In fact, inorder that the main results in this section hold, what is really essential is thata function ϕ can be defined by means of (3.2) satisfying a number of suitableproperties. No further structure assumptions need to be imposed on the flux.The main results below are Proposition 7.7, which reformulates the entropyinequalities (3.6) as separate requirements on the jump and Cantor parts of thespatial derivative (a fact that was observed in greater generality in [19]), andProposition 7.8, stating that the “jump part” of the entropy inequalities (3.6) isfulfilled if the flux at both sides of the discontinuity satisfies a certain constraint(encoding essentially the fact that contact angles must be vertical) and, giventhat this holds, phrasing the Rankine–Hugoniot relation in terms of ϕ .We start by introducing some notation suited to this purpose (see also [18]).Assume that u ∈ BV loc ( Q T ). Let ν := ν u = ( ν t , ν x ) be the unit normal to thejump set of u and ν J u ( t ) the unit normal to the jump set of u ( t ). We write[ u ]( t, x ) := u + ( t, x ) − u − ( t, x ) for the jump of u at ( t, x ) ∈ J u and [ u ( t )]( x ) := u ( t ) + ( x ) − u ( t ) − ( x ) for the jump of u ( t ) at the point x ∈ J u ( t ) . We assumethat u + > u − in what follows (this determines if ν x points inwards or outwardsaccording to the conventions on Subsection 3.1); we also assume u − ≥
0. Thefollowing result was proved in [18].
Lemma 7.7.
Let u ∈ BV loc ( Q T ) and let z ∈ L ∞ ([0 , T ] × R d , R d ) be such that u t = div z in D ′ ( Q T ) . Then H d ( { ( t, x ) ∈ J u /ν x ( t, x ) = 0 } ) = 0 . Definition 7.3.
Let u ∈ BV loc ( Q T ) and let z ∈ L ∞ ([0 , T ] × R d , R d ) be suchthat u t = div z in D ′ ( Q T ) . We define the speed of the discontinuity set of u as v ( t, x ) = ν t ( t,x ) | ν x ( t,x ) | H d -a.e. on J u . Next we quote a result encoding the Rankine–Hugoniot conditions that canbe found in [18] too.
Proposition 7.6.
Let u ∈ BV loc ( Q T ) and let z ∈ L ∞ ([0 , T ] × R d , R d ) be suchthat u t = div z . For a.e. t ∈ (0 , T ) we have [ u ( t )]( x ) v ( t, x ) = [[ z · ν J u ( t ) ]] + − H d − − a.e. in J u ( t ) , where [[ z · ν J u ( t ) ]] + − denotes the difference of traces from both sides of J u ( t ) . The following statement is a particular case of Proposition 6.8 in [19].
Proposition 7.7.
Let u ∈ C ([0 , T ]; L ( R d )) ∩ BV loc ( Q T ) . Assume that u t = div z in D ′ ( Q T ) , where z = a ( u, ∇ u ) . Assume also that u t ( t ) is a Radon measurefor a.e. t > . Let ϕ defined by (3.2) be a locally Lipschitz continuous function uch that ϕ (0) = 0 . Then u is an entropy solution of (2.3) if and only if forany ( T, S ) ∈ T SUB (for any ( T, S ) ∈ T SUB ∪ T SUPER ) we have h S ( u, DT ( u )) c + h T ( u, DS ( u )) c ≤ ( z ( t, x ) · D ( T ( u ) S ( u ))) c and for almost any t > the inequality [ ST ϕ ( u ( t ))] + − − [ J T Sϕ ′ ( u ( t ))] + − ≤ − v [ J T S ( u ( t ))] + − + [[ z ( t ) · ν J u ( t ) ] T ( u ( t )) S ( u ( t ))] + − (7.1) holds H d − -a.e. on J u ( t ) .Proof. The same proof as in Proposition 7.1 of [18] can be used. The onlynoticeable difference is found when extracting jump parts from the entropyinequalities (3.6). Here the property ϕ (0) = 0 is needed in order to ensure that([ J SϕT ′ ( u ( t ))] + − + [ J T ϕS ′ ( u ( t ))] + − ) H d − | J u ( t ) dt agrees with ([ ST ϕ ( u ( t ))] + − − [ J T Sϕ ′ ( u ( t ))] + − ) H d − | J u ( t ) dt. Then the rest of the proof goes as in [18].Now we state and prove the main result of the Section, which generalizesProposition 8.1 in [18] (see also [19] for a similar statement concerning a relatedclass of flux-limited equations).
Proposition 7.8.
Let u ∈ C ([0 , T ]; L ( R d )) be the entropy solution of (2.3) with ≤ u (0) = u ∈ L ∞ ( R d ) ∩ BV ( R d ) . Assume that u ∈ BV loc ( Q T ) . Assumefurther that ϕ defined by (3.2) is a convex, non-negative function such that ϕ (0) = 0 . Then the entropy conditions (7.1) hold if and only if for almost any t ∈ (0 , T ) [ z · ν J u ( t ) ] + = ϕ ( u + ( t )) and [ z · ν J u ( t ) ] − = ϕ ( u − ( t )) (7.2) hold H d − -a.e. on J u ( t ) . Moreover the speed of any discontinuity front is v = ϕ ( u + ( t )) − ϕ ( u − ( t )) u + ( t ) − u − ( t ) . (7.3) Proof.
The proof is a suitable generalization of that given for Proposition 8.1in [18]. Recall that the Rankine–Hugoniot conditions stated in Proposition 7.6are v [ u ] + − = [[ z · ν J u ( t ) ] + − . Let us show that (7.1) implies (7.2). For that we let ǫ > u −
0, as ϕ is locally Lipschitz.Then (7.1) is written as ǫ ( ϕ ( u + ) − [ z ( t ) · ν J u ( t ) ] + ) ≤ Cǫ − ǫ v, which is a contradiction unless [ z ( t ) · ν J u ( t ) ] + = ϕ ( u + ) (as | [ z ( t ) · ν J u ( t ) ] + | ≤ ϕ ( u + )clearly holds). We show that [ z ( t ) · ν J u ( t ) ] − = ϕ ( u − ) in a similar way. Using theRankine–Hugoniot condition, the speed of the front is given by v = [ z · ν J u ( t ) ] + − [ z · ν J u ( t ) ] − u + − u − = ϕ ( u + ) − ϕ ( u − ) u + − u − . Let us show now the converse implication. Thanks to (7.2) we may write[[ z · ν J u ( t ) ] T ( u ) S ( u )] + − = [ z · ν J u ( t ) ] + T ( u + ) S ( u + ) − [ z · ν J u ( t ) ] − T ( u − ) S ( u − )= ϕ ( u + ) T ( u + ) S ( u + ) − ϕ ( u − ) T ( u − ) S ( u − ) = [ ST ϕ ( u ( t ))] + − . Thus, we recast (7.1) as ϕ ( u + ) − ϕ ( u − ) u + − u − [ J T S ( u ( t )) ] + − ≤ [ J T Sϕ ′ ] + − . (7.4)Let us show that (7.4) holds for any ( T, S ) ∈ T SUB ∪ T SUPER . As argued in[18], to treat the case ( T, S ) ∈ T SUB it suffices to deal with T S ( r ) = p ( r ) = χ ( d, ∞ ) ( r ). There are several sub-cases to consider: • u − ≥ d ≤ u + . Then [ J p ( u ( t ))] + − = [ u ] + − and [ J pϕ ′ ( u ( t )) ] + − =[ ϕ ( u )] + − . Thus (7.4) holds. • u − ≥ u − < d ≤ u + . We compute [ J p ( u ( t ))] + − = u + − d and[ J pϕ ′ ( u ( t )) ] + − = ϕ ( u + ) − ϕ ( d ). Then (7.4) is equivalent to ϕ ( u + ) − ϕ ( u − ) u + − u − ( u + − d ) ≤ ϕ ( u + ) − ϕ ( d )which in turn holds because ϕ is convex. • u − ≥ d > u + . Then [ J p ( u ( t ))] + − = [ J pϕ ′ ( u ( t )) ] + − = 0. Hence (7.4)is trivially satisfied.Similarly, to treat the case ( T, S ) ∈ T SUPER it suffices to deal with T S ( r ) = p ( r ) = c + c ′ χ ( d, ∞ ) ( r ) , c ≤ , ≤ c ′ ≤ | c | . Again, we consider the varioussub-cases: 36 u − ≥ d ≤ u + . Then [ J p ( u ( t ))] + − = ( c + c ′ )[ u ] + − and [ J pϕ ′ ( u ( t )) ] + − =( c + c ′ )[ ϕ ( u )] + − . Thus (7.4) holds. • u − ≥ u − < d ≤ u + . We compute [ J p ( u ( t ))] + − = c [ u ] + − + c ′ ( u + − d )and [ J pϕ ′ ( u ( t )) ] + − = c [ ϕ ( u )] + − + c ′ ( ϕ ( u + ) − ϕ ( d )). Then (7.4) is equivalentto ϕ ( u + ) − ϕ ( u − ) u + − u − ≤ ϕ ( u + ) − ϕ ( d ) u + − d which in turn holds because ϕ is convex. • u − ≥ d > u + . This time [ J p ( u ( t ))] + − = c [ u ] + − and [ J pϕ ′ ( u ( t )) ] + − = c [ ϕ ( u )] + − . Hence (7.4) is satisfied. Remark 7.8.
Under some additional assumptions we may derive from (7.3) avertical contact angle condition, as pointed out in [18]. For that we assume thatfor H d -almost x ∈ J u there is a ball B x centered at x such that either (a) u | B x ≥ α > J u ∩ B x is the graph of a Lipschitz function with B x \ J u = B x ∪ B x ,where B x , B x are open and connected and u ≥ α > B x , while the trace of u on J u ∩ ∂B x computed from B x is zero. In both cases [ ψ ( L ∇ u/u ) · ν J u ( t ) ] + = 1 on J u ∩ B x . If (a) holds, we also have [ ψ ( L ∇ u/u ) · ν J u ( t ) ] − = 1 on J u ∩ B x . Providedthat the Jacobian matrix of ψ is not compactly supported, these relations implyin particular that |∇ u | = ∞ . Remark 7.9.
In case that u − = 0, (7.3) reduces to v = ϕ ( u + ) /u + . As ϕ isconvex, this is compatible with Corollary 6.1.Now we give a sufficient condition to ensure that u t is a Radon measure,which is required in order to use Proposition 7.8. Proposition 7.9.
Let ≤ u ∈ ( L ∩ L ∞ )( R d ) and let u ( t ) be the entropysolution of (2.3) with u as initial datum. If ϕ is homogeneous of degree m > ,then for any t > , u t ( t ) is a Radon measure in R d . Moreover k u t ( t ) k M ( R d ) ≤ m − t k u k .Proof. This is a direct consequence of the results in [12].Next we wonder about the admissible discontinuity gaps for a given speed.
Lemma 7.8.
Assume that ϕ is strictly convex. Then, given values v , u + suchthat ϕ ( u + ) /u + ≤ v < ϕ ′ ( u + ) , there exists an unique value u − ∈ [0 , u + ) suchthat relation (7.3) holds.Proof. Given the value u + , we look for values of u − such that (7.3) holds (pro-vided they exist). We consider the function φ ( x ) = ϕ ( u + ) − ϕ ( x ) u + − x , x ∈ [0 , u + ). It is easily seen that φ (0) = ϕ ( u + ) /u + and φ ( u + ) = ϕ ′ ( u + ) . Next we compute φ ′ ( x ) = ϕ ( u + ) − ( u + − x ) ϕ ′ ( x ) − ϕ ( x )( u + − x ) > x ∈ (0 , u + ) . Thus, φ is a bijection from [0 , u + ] to [ ϕ ( u + ) /u + , ϕ ′ ( u + )], which implies ourresult.Contrary to the situation depicted in the previous result, we have: Corollary 7.4.
Let ϕ ( z ) = sz . Then the only speed of propagation for discon-tinuity fronts that is allowed is precisely s , while any values of u + > u − ≥ areadmissible for such a discontinuity. Acknowledgments
The author acknowledges J.M. Maz´on for his useful comments on a previousversion of this document and the anonymous referees of SIAM J. Math. Anal.,which helped to improve greatly the contents of the paper with their comments.He also thanks F. Andreu, V. Caselles and J.M. Maz´on for their support duringthese years. J. Calvo acknowledges partial support by a Juan de la Cierva grantof the spanish MEC, by the “Collaborative Mathematical Research” programmeby “Obra Social La Caixa”, by MINECO (Spain), project MTM2014-53406-R,FEDER resources, and Junta de Andaluc´ıa Project P12-FQM-954.
References [1]
M. Agueh , Existence of solutions to degenerate parabolic equations via theMonge-Kantorovich theory , PhD Thesis, Georgia Tech, Atlanta, 2001.[2]
G. Anzellotti , Pairings between measures and bounded functions andcompensated compactness , Ann. di Matematica Pura ed Appl. IV, 135(1983), pp. 93–318.[3]
L. Ambrosio, N. Fusco and D. Pallara , Functions of Bounded Varia-tion and Free Discontinuity Problems , Oxford Mathematical Monographs,2000.[4]
L. Ambrosio, N. Gigli and G. Savar´e , Gradient flows in metric spacesand in the spaces of probability measures , Lectures in Mathematics ETHZurich, Birkhauser Verlag, Basel, 2005.[5]
F. Andreu, V. Caselles and J. Maz´on , A Strongly Degenerate Quasi-linear Equation: the Elliptic Case
Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5,Vol III (2004), pp. 555–587. 386]
F. Andreu, V. Caselles and J.M. Maz´on , A Strongly Degener-ate Quasilinear Elliptic Equation , Nonlinear Analysis, TMA, 61 (2005),pp. 637–669.[7]
F. Andreu, V. Caselles and J.M. Maz´on , The Cauchy Problem for aStrongly Degenerate Quasilinear Equation , J. Europ. Math. Soc., 7 (2005),pp. 361–393.[8]
F. Andreu, V. Caselles and J.M. Maz´on , Some regularity results onthe ‘relativistic’ heat equation , Journal of Differential Equations 245 (2008),pp. 3639–3663.[9]
F. Andreu, V. Caselles, J.M. Maz´on and S. Moll,
Finite propaga-tion speed for limited flux diffusion equations , Arch. Rat. Mech. Anal., 182(2006), pp. 269–297.[10]
F. Andreu, V. Caselles, J.M. Maz´on and S. Moll,
A diffusionequation in transparent media , J. Evol. Equ., 7 (2007), pp. 113–143.[11]
F. Andreu, V. Caselles, J.M. Maz´on, J. Soler and M. Verbeni , Radially symmetric solutions of a tempered diffusion equation. A porousmedia flux-limited case , SIAM J. Math. Anal., 44 (2012), pp. 1019–1049.[12]
Ph. Benilan, M.G. Crandall , Regularizing effects of homogeneous evo-lution equations , in Contribution to Analysis and Geometry, 1981, pp. 23–39.[13]
Ph. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierreand J.L. Vazquez , An L1-Theory of Existence and Uniqueness of Solu-tions of Nonlinear Elliptic Equations , Ann. Scuola Normale Superiore diPisa, IV, Vol. XXII (1995), pp. 241–273.[14]
M. Bertsch and R. Dal Passo , Hyperbolic Phenomena in a StronglyDegenerate Parabolic Equation , Arch. Rational Mech. Anal., 117 (1992),pp. 349–387.[15]
Y. Brenier , Extended Monge-Kantorovich Theory , In Optimal Trans-portation and Applications, Lectures given at the C.I.M.E. Summer Schoolhelp in Martina Franca, Lecture Notes in Math. 1813, L.A. Caffarelli andS. Salsa, eds., Springer–Verlag, 2003, pp. 91–122.[16]
J. Calvo, J. Campos, V. Caselles,O. S´anchez and J. Soler , Fluxsaturated porous media diffusion equations and applications , to appear inJ. Eur. Math. Soc. (JEMS)[17]
J. Carrillo and P. Wittbold , Uniqueness of Renormalized Solutions ofDegenerate Elliptic-Parabolic problems , J. Differ. Equ., 156 (1999), pp. 93–121. 3918]
V. Caselles , On the entropy conditions for some flux limited diffusionequations , J. Diff. Eqs., 250 (2011), pp. 3311–3348.[19]
V. Caselles , Flux limited generalized porous media diffusion equations ,Publ. Mat., 57 (2013), pp. 155–217.[20]
A. Chertock, A. Kurganov and P. Rosenau , Formation of discon-tinuities in flux-saturated degenerate parabolic equations , Nonlinearity, 16(2003), pp. 1875–1898.[21]
A. Chertock, A. Kurganov and P. Rosenau , On degeneratesaturated-diffusion equations with convection , Nonlinearity, 18 (2005),pp. 609–630.[22]
A. Chertock, A. Kurganov, X. Wang and Y. Wu , On a chemo-taxis model with saturated chemotactic flux , Kinetic and Related Models, 5(2012), pp. 51–95.[23]
J.-F. Coulombel, F. Golse and T. Goudon , Diffusion Approxima-tion and Entropy-based Moment Closure for Kinetic Equations , AsymptoticAnalysis, 45 (2005), pp. 1–39.[24]
M.G. Crandall and T.M. Liggett , Generation of Semigroups of Non-linear Transformations on General Banach Spaces , Amer. J. Math., 93(1971), pp. 265–298.[25]
G. Dal Maso , Integral representation on BV (Ω) of Γ -limits of variationalintegrals , Manuscripta Math., 30 (1980), pp. 387–416.[26] V. De Cicco, N. Fusco and A. Verde , On L -lower semicontinuity in BV , J. Convex Analysis, 12 (2005), pp. 173–185.[27] L. Giacomelli , Finite speed of propagation and waiting-time phenomenafor degenerate parabolic equations with linear growth Lagrangian . Preprint.[28]
R. Jordan, D. Kinderlehrer and F. Otto , The variational formu-lation of the Fokker–Planck equation , SIAM J. Math. Anal., 29 (1998),pp. 1–17.[29]
S.N. Kruzhkov , First order quasilinear equations in several independentvariables , Math. USSR-Sb., 10 (1970), pp. 217–243.[30]
A. Kurganov and P. Rosenau , On reaction processes with saturatingdiffusion , Nonlinearity, 19 (2006), pp. 171–193.[31]
C.D. Levermore , A Chapman–Enskog approach to flux limited diffusiontheory , Technical Report, Lawrence Livermore Laboratory, UCID-18229(1979).[32]
C.D. Levermore and G.C. Pomraning , A flux-limited diffusion theory ,The astrophysical journal, 248 (1981), pp. 321–334.4033]
C.D. Levermore , Relating Eddington factors to flux limiters , J. Quant.Spectrosc. Radiat. Transfer, 31 (1984), pp. 149–160.[34]
R. Mc Cann and M. Puel , Constructing a relativistic heat flow by trans-port time step , Annales de l’Institut Henri Poincar´e (C) Non Linear Anal-ysis, 26 (2009), pp. 2539–2580.[35]
D. Mihalas, B. Mihalas , Foundations of radiation hydrodynamics , Ox-ford University Press, 1984.[36]
G.L.Olson, L.H.Auer and M.L.Hall , Diffusion, P , and other approx-imate forms of radiation transport , Journal of Quantitative Spectroscopy& Radiative Transfer, 64 (2000), pp. 619–634.[37] P. Rosenau , Tempered Diffusion: A Transport Process with PropagatingFront and Inertial Delay , Phys. Review A, 46 (1992), pp. 7371–7374.[38]