Measure - valued solutions: Convergence of the empirical process ¯ u Relaxation of the spin flip model: Let H

where u denotes the unique entropy solution to 1.16 with initial value u . 5.2. Microscopic block averages: By means of Lemma 3.5 we can fill in the gap between large microscopic and small macroscopic block averages of the evolved configuration, see [FN06]. Let ¯ u ǫ,l t, x := ¯ ω l,k tǫ if |x − ǫk| ǫ2 , where l ∈ N does not depend on ǫ , then for all τ, r 0 we have lim l →∞ lim sup ǫ→0 E Z τ Z r −r |¯u ǫ t, x − ¯u ǫ,l t, x| d x d t = 0 , whence lim l →∞ lim sup ǫ→0 E Z τ Z r −r |ut, x − ¯u ǫ,l t, x| d x d t = 0 .

5.3. Measure - valued solutions: Convergence of the empirical process ¯ u

ǫ,l to a set of measure- valued solutions holds true under fairly general conditions. Let us consider CRANNI, that is the process generated by L ∗ = L o + β G ∗ with a fixed value of β 0 . Let ψ ∈ C 1 c R 2 , then following the proof of Proposition 2.1 we write lim t →∞ Z ∞ −∞ ψt, x¯ u ǫ,l t, x d x = 0 = Z ∞ −∞ ψ0, x¯ u ǫ,l t, x d x + ¯ M ǫ ∞, ψ + Z ∞ Z ∞ −∞ ψ ′ t t, x¯ u ǫ,l t, x d x d t + Z ∞ Z ∞ −∞ ∇ ǫ ψt, x ¯j ǫ,l t, x d x d t , where ¯ M ǫ is the terminal value of a martingale, and ¯j ǫ,l is a block average, ¯j ǫ,l t, x := ¯j ωo l,k tǫ + βǫ¯j ω∗ l,k tǫ if |x − ǫk| ǫ2 ; remember that l ∈ N does not depend on ǫ 0 . It is easy to show that E ¯ M 2 ǫ → 0 as ǫ → 0 , and the replacement of ¯j ǫ,l with f ¯ u ǫ,l follows by the standard One - Block Lemma of [GPV88]. For τ, r 0 we have lim l →∞ lim sup ǫ→0 Z τ Z r −r E |¯j ǫ,l t, x − f ¯u ǫ,l t, x| d x d t = 0 because the usual entropy argument shows that every translation invariant stationary measure is a superposition of product measures of type λ ∗ u . Therefore all limit distributions of the Young rep- resentation of ¯ u ǫ,l are concentrated on a set of measure - valued solutions, cf. 1.22; these limit distributions are obtained by sending ǫ → 0 first, and l → +∞ next. The weak Lax inequality 2.11 can not be proven in this easy way, we are facing with a non - gradient problem there.

5.4. Relaxation of the spin flip model: Let H

κ ρ, u := 12u − κρ 2 , 0 ≤ ψ ∈ C 1 c R 2 , and consider the evolution of ¯ H κ t, ψ := Z ∞ −∞ ψt, xH κ ¯ v ǫ t, x d x , where the empirical process ¯ v ǫ is the same as in Section 3.4, but it is now generated by L κ = L o + αǫG κ + σǫS . We follow the lines of the derivation of 3.7, the only difference consists in 256 the evaluation of the effect of G κ . Due to G κ η k = 0 , these computations are considerably simpler than those of Section 3.4. Denote ¯ ω j l,k the block average of the sequence ω j , then G κ H ¯ η l,k , ¯ ω l,k = 1 2 X j ∈Z η j − κω j ¯ ω j l,k − κ ¯ η l,k 2 − ¯ ω l,k − κ ¯ η l,k 2 = 1 l k+l −1 X j=k κη j − ω j ¯ ω j l,k + ¯ ω l,k − 2κ ¯ η l,k = −2 ¯ ω l,k − κ ¯ η l,k 2 + 2l ¯ η l,k − κ ¯ ω l,k is an identity, whence by a direct calculation we obtain the final a priori bound: ǫ 2 X |k|rǫ Z τǫ Z ¯ ω l,k − κ ¯ η l,k 2 d µ ǫ,t d t ≤ ¯ C 4  r τ l + r τσ αl 2 ‹ . Let us remark that the second LSI in [FN06] yields a considerably better order, O 1l + ǫα , see Lemma 3.5 there. Our present bound is effective if αǫ 0 is not too small, namely lim ǫ→0 σǫ αǫl 2 ǫ = 0 and lim ǫ→0 ǫαǫl 2 ǫ = +∞ . This means ǫαǫσ 2 ǫ → +∞ as ǫ → 0 ; the condition αǫσǫ → +∞ of Theorem 1.1 in [FN06] has not been improved in this way. 5.5. Entropy - flux pairs of the LeRoux system: Regular entropy - flux pairs of 1.13 can easily be constructed because besides the trivial S a ρ, u := 1 − ρ + au − a 2 and Φ a ρ, u := −a + uS a , also |S a ρ, u| + and −a + u|S a ρ, u| + are convex entropy - flux pairs for all a ∈ R , where |S| + := max {S, 0} . Observe that S a ρ, u 0 means w a z , where z and w are the Riemann invariants of the LeRoux system: −1 ≤ w := u 2 − 1 2 p u 2 + 4 − 4ρ ≤ 0 ≤ u 2 + 1 2 p u 2 + 4 − 4ρ =: z ≤ 1 ; notice that u = z + w and ρ = 1 + zw , moreover S z ρ, u = S w ρ, u = 0 in this case. Therefore S ν ρ, u = Z |S a ρ, u| + νd a = Z z w S a ρ, u νd a is again an entropy with flux Φ ν ρ, u := − Z z w a + uS a ρ, u νd a whenever ν is a finite signed measure on [−1, 1] . Suppose that ν is absolutely continuous on the set [ −1, 0 ∪ 0, 1] , then ∂ ρ S ν ρ, u = − Z z w νd a = − € ν {0} + G + z + G − w Š , 257 where G + a = ν0, a if a ≥ 0 , and G − a = νa, 0 if a 0 . To demonstrate relaxation to 1.16, we need ∂ ρ S ν ρ, uCρ, u ≤ −b ρ − Fu 2 5.1 with some universal constant b 0 , which implies ∂ ρ S ν ρ, u = 0 if ρ = F u . In terms of the Riemann invariants this means 0 = C ρ, u = 14z 2 + w 2 + 3z 2 w 2 − 1 , whence z 2 = 1 − w 2 1 + 3w 2 . Therefore setting G + z = z 2 , ν {0} = −1 and G − w = 1 − 1 − w 2 1 + 3w 2 , we obtain ∂ ρ S ν ρ, u = 1 − z 2 − w 2 − 3z 2 w 2 1 + 3w 2 = − 4C ρ, u 1 + 3w 2 5.2 by a direct computation. However, w 2 ≤ 1 and |Cρ, u| ≥ 12|ρ − Fu| , consequently the above choice of ν yields 5.1 with b = 14 .

5.6. The second relaxation inequality: Using S