Fast reaction limit and forward-backward diffusion: a Radon-Nikodym approach

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 where 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) 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 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 (1.5) We also note that for all A ⊂ R + (1.6) In particular, from (1.5) and (1.6) we deduce that for F # µ t,x -a.e.λ we have (1.7) The main result of this paper reads: Theorem 1. (A) Let {µ t,x } t,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 τ 0 = f − , f + we have 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 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: The next result shows that the systems (1.1) and (1.2) are not exactly the same in view of the strong convergence.Indeed, for the first one, we can establish a simple condition on F implying strong convergence of v ε → v that does not exclude piecewise affine functions as in the case of nondegeneracy condition (1.3).Theorem 4. Let {µ t,x } t,x be a Young measure generated by sequence {u ε } ε∈(0,1) solving (1.1).

Suppose that:
• there exists τ Then, v ε → v strongly in L 2 ((0, T ) × Ω).Moreover, there are nonnegative numbers λ 1 (t, x), λ 2 (t, x), 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 g 1 (λ 0 ) + g 2 (λ 0 ) + g 3 (λ 0 ) = 1 to show that for λ 0 ∈ supp F we have F # µ t,x {λ 0 } = 1.Note however that (1.8) is not valid for λ 0 = f − , f + so that some additional care is needed if the support of measure F # µ t,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 Theorem 1. Section 4 is devoted to the simple proofs of Theorems 2, 3 and 4 while in Section 5 we show how to easily adapt Theorems 1-3 to the case of system (1.2).Finally, Appendix A provides necessary background on Young measures, supports of measures and compensated compactness results.
1.3.Technical assumptions and notation.For the sake of completeness, we list here assumptions of technical nature.
Notation 1.1.Let S 1 (λ) ≤ S 2 (λ) ≤ S 3 (λ) be the solutions of equation F (S i (λ)) = λ as already introduced in 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.
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.
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 map 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.
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 : R → R, 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 where 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.
Then, for almost all λ 0 (with respect to F # µ t,x ) and τ 0 = f − , f + we have Proof.We consider φ(τ . Therefore, from Lemmas 3.1 and 3.2 we deduce Using identities from (1.6) and (1.7) we conclude the proof.
For any τ 0 ∈ (λ 1 , λ 2 ) we use Theorem 1 with λ 0 = λ 1 , λ 2 to obtain two equations: Hence, 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.
It follows that F # µ t,x is a Dirac mass and now, we can conclude as in Theorem 2.

5.
Proof of Theorems 1-3 to the forward-backward diffusion system (1.2) We first formulate basic well-posedness result for (1.2).This comes mostly from [31,35] but the compactness estimates are simplified.