M ay 2 02 1 FAST REACTION LIMIT AND FORWARD-BACKWARD DIFFUSION : A RADON-NIKODYM

We consider two singular limits: fast reaction limit with nonmonotone nonlinearity and regularization of forward-backward diffusion equation. It was proved by Plotnikov that for cubic-type (nondegenerate) nonlinearities, the limit oscillates between at most three states. In this paper we make his argument more optimal and we sharpen the previous result: we use Radon-Nikodym theorem to obtain a pointwise identity characterizing the Young measure. As a consequence, we establish a simpler condition which implies Plotnikov result for piecewise affine functions. We also prove that the result is true if the Young measure is not supported in the so-called unstable zone, the fact observed in numerical simulations.


Introduction and main results
1.1. Presentation of the problem. In this paper we focus on two interesting limit problems: fast-reaction limit in the reaction-diffusion system and regularization of the forward-backward parabolic equation ∂ t u = ∆F (u) where F is a nonmonotone function, for simplicity assumed to look like as in Fig. 1. Notice that due to nonmonotone character of F , it has three inverses S 1 , S 2 and S 3 . Both problems are posed on some bounded domain Ω ⊂ R d and are equipped with initial conditions and usual Neumann boundary conditions.
System (1.1) is an interesting toy model for studying oscillations in reaction-diffusion systems as they are known to occur in its steady states [28]. For monotonone F the problem is fairly classical and has been studied for a great variety of reaction-diffusion systems, also with more than two components [5,6,14,29] or reaction-diffusion equation coupled with an ODE [21]. In the limit ε → 0 one obtains widely studied cross-diffusion systems [9,10,15,16,22,24] where the gradient of one quantity induces a flux of another one. A slightly different yet connected type of problem deals with the fastreaction limit for irreversible reactions which leads to free boundary problems [11,17,20]. Finally, for nonmonotone F as in this paper, the only available result was established very recently in [33] (see below). We also refer to the recent stability analysis of problems of the type (1.1) [12,13,25].
System (1.2) was extensively studied by Plotnikov [34,35] who identified limits as ε → 0 in terms of Young measures (see below) and by Novick-Cohen and Pego who studied its asymptotics with ε > 0 fixed [31]. Regularization term in (1.2) was also generalized in [3,4,40]. Recently, so called nonstandard analysis was used to study the limit problem in the space of grid functions [7,8].
It is known [33,35] that both systems exhibit the following surprising phenomenon: as ε → 0, F (u ε ) → v and v ε → v converge strongly without any known a priori estimates allowing to conclude so. As a consequence, u ε converges weakly to u(t, x) = λ 1 (t, x) S 1 (v(t, x)) + λ 2 (t, x) S 2 (v(t, x)) + λ 3 (t, x) S 3 (v(t, x)) where 3 i=1 λ i (t, x) = 1. More precisely, if µ t,x is a Young measure generated by {u ε } ε>0 we have which represents oscillations between phases S 1 (v(t, x)), S 2 (v(t, x)) and S 3 (v(t, x)). The proof exploits a family of energies as well as analysis of related Young measures in the spirit of Murat and Tartar work on conservation laws and compensated compactness [30,41]. The numerical simulations suggests that the middle state, referred to as an unstable phase, is not present [19] which motivates research on two-phase solutions to such problems [23,26,39,42] with a result of nonuniqueness when the unstable phase is present [43].
So far, the main assumption on F that allows to deduce strong convergence is the so-called nondegeneracy condition: for (1.1) it reads While it is fairly classical for this type of problems [1,31,35], it is hard to be verified for a given nonlinearity F . Moreover, the nondegeneracy condition excludes piecewise affine functions which allows for more explicit computations as in [26].

1.2.
Main results and outline of the paper. In this paper, we take a slightly different approach to study strong convergence. Although we use family of energy identities to characterize Young measure as Plotnikov [35], we aim at pointwise identities to obtain optimal amount of information from these energy identities, in particular new results. To achieve this, we use Radon-Nikodym Theorem as explained below.
for any bounded function G : R → R we have (up to a subsequence and for a.e. (t, x) ∈ (0, T ) × Ω) see Appendix A.3 if necessary. To analyze amount of µ t,x on the intervals I 1 , I 2 and I 3 , see Figure   1, we introduce restrictions t,x := µ t,x 1 I2 , µ t,x := µ t,x 1 I3 .
The reason we introduce these measures is that in the sequel, we will gain information only about measure F # µ t,x , i.e. push-forward (image) of µ t,x along F defined as Observe that for all i = 1, 2, 3 measures F # µ t,x are absolutely continuous with respect to F # µ t,x . Therefore, Radon-Nikodym theorem implies that there exist densities g (1) (λ), g (2) (λ) and g (3) (λ) such that We also note that for all A ⊂ R + In particular, from (1.5) and (1.6) we deduce that for F # µ t,x -a.e. λ we have The main result of this paper reads: x be Young measure generated by sequence {u ε } ε∈(0,1) solving (1.1).
Then, for almost all λ 0 (with respect to F # µ t,x ) and all where S i are inverses of F as in Notation 1.1 and g i are Radon-Nikodym densities as in (1.5).
x is the Young measure generated by sequence {u ε } ε∈(0,1) solving (1.2) the equalities above holds with functions S ′ i instead of S ′ i + 1.
As F # µ t,x turns out to be the Young measure generated by {v ε } ε>0 cf. Corollary 2.3, strong convergence v ε → v follows from proving that F # µ t,x is a Dirac mass cf. Lemma A.5 (A). Equation (1.8) shows that it is sufficient to find λ 0 in the support such that the sum 3 i=1 (S ′ i (λ 0 ) + 1) g i (λ 0 ) does not vanish (some additional care is needed when λ 0 = f − , f + , cf. Lemma 4.1).
We remark that similar forms of entropy equality as in Theorem 1 are well-known however they are not so easily formulated and they are usually stated without explicitly identified coefficients standing next to (S ′ i (τ 0 ) + 1). First, we show that the form presented in Theorem 1 can be used to recover already known result due to Plotnikov [35] as well as Perthame and Skrzeczkowski [33].
Now, we move to the new results that easily follow from Theorem 1. The first one asserts that if one knows a priori that the Young measure µ t,x is not supported in the interval I 2 where F is decreasing, the strong convergence occurs. The fact concerning support of µ t,x was observed in numerical simulations [19] and so, the next theorem may serve as a tool to prove strong convergence without nondegeneracy condition.
Theorem 3. Suppose that: x is not supported in the interval I 2 (see Figure 1).  Suppose that: As an example of function F satisfying assumptions of Theorem 4 consider Note that F does not satisfy nondegeneracy condition (1.3) that was used in the previous paper on fast reaction limit with nonmonotone reaction function [33].

Proofs of Theorem 3 and 4 are based on equation (1.8), namely one uses
x accumulates only in these points. This is studied in Lemma 4.1 and it requires an additional assumption that does not vanish at least for one value of τ , see also Remark 4.2.
The structure of the paper is as follows. In Section 2 we review (well-known) properties of the fast-reaction system (1.1). Then, in Section 3 we use compensated compactness approach to prove 1.3. Technical assumptions and notation. For the sake of completeness, we list here assumptions of technical nature. Figure 1. These are inverses of F satisfying Their role is too focus analysis on parts of the plot of F where the monotonicity of F does not change. By a small abuse of notation, we extend functions S i by a constant value to the whole of R.
(3) Regularity of inverses: 2. Properties of the fast-reaction system (1.1) We begin from recalling energy equality and well-posedness result from [33] which we prove below for the sake of completeness.
Lemma 2.1 (energy equality). Given a smooth test function φ : R → R, we define Summing up these equations we deduce (2.2).
Lemma 2.2. There exists the unique classical solution u ε , v ε : [0, ∞) × Ω → R of (1.1) which is nonnegative and has regularity Moreover, we have for all smooth φ : Proof. Existence and uniqueness of global solution as well as points (1)-(3) were proven in [33, Theorem 3.1] so we only sketch the argument. First, local well-posedness and nonnegativity follows from classical theory [37]. To extend existence and uniqueness to an arbitrary interval of time, we need to prove a priori estimates as in (1). To this end, we note that thanks to (2.2), the nonnegative is nonincreasing whenever φ ′ ≥ 0. Choosing φ vanishing on (0, M ) and stricly increasing for (M, ∞) we obtain (1) and global well-posedness. Then, (2) and (3) follows from (2.2) with φ(v) = v.
Furthemore, (4) follows from the equality ∂ t u ε + ∂ t v ε = ∆v ε and property (2) while (5) follows from the chain rule for Sobolev functions, boundedness of v ε from (1) and (2). Finally, to see (6) we Thanks to (2.2) we have 3) together with points (2) and (3) to deduce for some possibly larger constant C (independent of ε) 3. Proof of Theorem 1 for fast-reaction system (1.1) We begin with an entropy equality.
Lemma 3.1 (entropy equality). Let Ψ and Φ be defined with (2.1). Let g i be densities given by (1.5). Then, for almost all λ 0 (with respect to F # µ t,x ) we have where S i are inverses of F as in Notation 1.1.
Proof. Thanks to Lemma 2.2 (6), for all smooth φ : As v ε − F (u ε ) → 0 cf. Lemma 2.2 (3), we may replace v ε with F (u ε ) in the identity above to obtain In the language of Young measures, this identity reads We observe that λ = 3 i=1 S i (F (λ)) 1 λ∈Ii . Hence, we may use push-forward measure to write Using (1.5) with densities g 1 (λ), g 2 (λ) and g 3 (λ) we obtain Hence, when λ 0 belongs to the support of the measure F # µ t,x , we obtain To analyze entropy inequality, we need to deal with integrals of the form Si(λ) 0 φ(F (τ )) dτ . This is the content of the next lemma.
Lemma 3.2. We have Proof. For i = 1 we note that F is invertible on (0, S 1 (λ)) so that simple change of variables implies For i = 2 we first split the integral for two intervals (0, α + ), (α + , λ 0 ) cf. Notation 1.1. On each of them F is invertible so we can apply change of variables again: For i = 3 we split the integral for three intervals and apply change of variables again: As S ′ 2 (τ ) = 0 and S ′ 3 (τ ) = 0 for τ ∈ (0, f − ), the proof is concluded.
Proof of Theorem 1. The first part of Theorem 1 is proved in Lemma 3.3. To see the second one, t,x (R + ) − g 1 (λ 0 )) = 0.
For any τ 0 ∈ (λ 1 , λ 2 ) we use Theorem 1 with λ 0 = λ 1 , λ 2 to obtain two equations: It follows that F # µ t,x is a Dirac mass. From Corollary 2.3 we deduce that the Young measure {ν t,x } t,x generated by {v ε } ε is also a Dirac mass so v ε → v strongly and ν t,x = δ v(t,x) , cf. Lemma A.5. The representation formula for µ t,x follows from F # µ t,x = δ v(t,x) .
Before proceeding to the proofs of Theorems 3 and 4, we will state a simple lemma concerning the case when F # µ t,x is supported only at f − and f + . This needs some care as functions S ′ 1 , S ′ 2 and S ′

3
are not continuous at these points.
Proof of Theorem 3. As in the proof of Theorem 2 we may assume that supp F # µ t,x ⊂ [f − , f + ] (this did not use nondegeneracy condition!). By assumption, for any set because g 1 (λ 0 ) + g 3 (λ 0 ) = 1 and S 1 , S 3 are strictly increasing. It follows from Theorem 1 that apply Lemma 4.1.
It follows that F # µ t,x is a Dirac mass and now, we can conclude as in Theorem 2.
Proof of Theorem 4. Mimicking the proof of Theorem 3, we let λ 0 ∈ supp F # µ t,x ∩ (f − , f + ) and we observe that the sum We conclude as in the proof of Theorem 3.
Proof. We observe that equation is equivalent to the following ODE: As long as ε > 0, the (RHS) is Lipschitz continuous, say on L 2 (Ω), so the local well-posedness follows. To obtain global well-posedness, we consider functions Ψ, Φ defined in (2.1). We have so after integration in space, the (RHS) of (5.1) is nonnegative. Hence, ∂ t Ω Ψ(u ε ) ≤ 0. Choosing Lemma 5.2 (entropy equality). Let Ψ be defined with (2.1). Let g i be densities given by (1.5).
Note that Theorem 4 is only true for fast-reaction limit (1.1) because its proof exploits presence of S in L 2 (0, T ; H 1 (Ω)) and {b n } n∈N is uniformly bounded in L 2 (0, T ; L 2 (Ω)). Moreover, assume that the sequence of distributional time derivatives {∂ t b n } n∈N is uniformly bounded in the dual space C(0, T ; H m (Ω)) * for some m ∈ N. Then, if a n ⇀ a and b n ⇀ b we have a n b n → a b in the sense of distributions.
In our case, the considered sequences are also in L ∞ ((0, T ) × Ω) so the resulting convergence is true in the weak * sense.
A.2. Support of the measure. We recall definition of the support of measure on R n [38, Definition 1.14]. For this, let B(x, r) denote a ball of radius r > 0 centered at x ∈ R n .
Definition A.2. Let µ be a nonnegative measure on R n . We say that x ∈ supp µ if and only if µ(B(x, r)) > 0 for all r > 0.
Remark A.3. When given property (like equation) is satisfied for almost every x (with respect to µ) one may worry that it is not true for the particularly chosen value of x. This is not the problem if one takes x ∈ supp µ because in each neighbourhood of x there is y ∈ supp µ such that the property has to be satisfied because the measure of each neighbourhood is nonzero.
A.3. Young measures. Finally, we recall the theory of Young measures introduced by Young [44,45] and recalled in the seminal paper of Ball [2]. Reader interested in modern presentation may consult [18], [32,Chapter 6] or [36,Chapter 4]. For simplicity, we formulate it for sequences of functions {u n } n∈N uniformly bounded in L p (Ω) with some 1 ≤ p ≤ ∞ and Ω ⊂ R n being a bounded domain. We start with the most important result that we cite from [32, Theorem 6.2]: Sketch of the proof. For (A) we consider G(u) = u and G(u) = u 2 to deduce that u n → u in L 2 (Ω) so that up to a subsequence also a.e. The opposite direction is clear because G(u n (x)) → G(u(x)) a.e. For (B) we note that for all bounded and smooth G, weak limits of G(u n (x)) and G(w n (x)) coincide. For (C) we write G(F (u n )) = R G(F (λ)) dµ t,x (λ) =