József Fritz
3. Compensated compactness via artificial viscosity
vk =uk+1−uk, u˙k =J(χk+1, vk+1)−J(χk, vk) andχ˙k=J(χk, vk)(uk+1−uk), whence
βkπ˙k = (γk−γk−1) + (βk−1−βk)Jk−1,
˙
γk = (βk+1πk+1−βkπk)Jv0(χk, vk) + (βk−βk+1)Jv0(χk, vk), β˙k = (βkπk−βk+1πk+1)Jχ0(χk, vk) + (βk+1−βk)Jχ0(χk, vk) follow by a direct computation.
As a consequence of these calculations, we see the expected cancelation of the sum of all leading terms on the right hand side of (2.10), while the remainders can be estimated by means of the entropy inequality. The summary of these computations results in (2.12), thus the proof can be terminated as it was outlined in the previous two paragraphs.
3. Compensated compactness via artificial viscosity
As we have already explained, randomness in the above modifications of the anhar-monic chain implies convergence to a classical solution of the macroscopic system (2.5) or (2.8) by the strong ergodic hypothesis, but in a regime of shocks much more information is needed to pass to the hydrodynamic limit. Effective coupling techniques that we have for attractive models are not available in the case of two-component systems, compensated compactness seems to be the only tool we can use. The microscopic dynamics can not admit non-classical conservation laws be-cause it should be ergodic in the strong sense, therefore a nontrivial Lax entropy is not conserved by the microscopic dynamics. In general, the flux of a Lax en-tropy exhibits a non-gradient behavior, but the standard spectral gap estimates of Varadhan (1994) are not sufficient for its control in this case, alogarithmic Sobolev inequality(LSI) is needed. The effective LSI is due to thestrong artificial viscosity of our next model, we will consider a Ginzburg–Landau type stochastic system:
dpk = (V0(rk)−V0(rk−1))dt+σ(ε) (pk+1+pk−1−2pk)dt +p
2σ(ε) (dwk−dwk−1)
and
drk= (pk+1−pk)dt+σ(ε) (V0(rk+1) +V0(rk−1)−2V0(rk))dt +p
2σ(ε) (dw˜k+1−dw˜k),
where {wk : k ∈ Z} and {w˜k : k ∈ Z} are independent families of independent Wiener processes. Of course, the macroscopic viscosity εσ(ε) vanishes as ε→ 0, but we also needεσ2(ε)→+∞to suppress extreme fluctuations of Lax entropies.
To have a standard existence and uniqueness theory for this infinite system of stochastic differential equations, we are assuming thatV00is bounded. The gener-ator of theFeller processdefined in this way reads asL=L0+σGp+σGr, where Gr is also elliptic. Additional conditions on the interaction potential V are listed below.
3.1. Conditions and main result
Just as in the case of (2.7), the same{λπ,γ :π, γ∈R} is the family of stationary product measures, and the converse statement, i.e. the strong ergodic hypothesis can be proven in the same way. Therefore again (2.8) is expected to govern the macroscopic behavior of the system under hyperbolic scaling. The first crucial problem is the evaluation of L0hwhen his a Lax entropy, we have to show that its dominant part is a difference of currents. These probabilistic calculations are based on a logarithmic Sobolev inequality. In view of theBakry–Emery criterion, see Deuschel–Stroock (1989), we have to assume thatV is strictly convex, i.e. V00 is bounded away from zero. On the other hand, the existence of weak solutions to (2.8) requires the condition of genuine nonlinearity: the third derivative S000 can not have more than one root,see DiPerna (1985), Shearer (1994) and Serre–Shearer (1994). In terms ofV this is a consequence of one of the following assumptions.
(i)V0 is strictly convex or concave onR.
(ii)V is symmetric andV0(r)is strictly convex or concave forr >0.
The very same properties of the flux S0 follow immediately by the theory of total positivity. Of course, small perturbations of such potentials also imply the required genuine nonlinearity of the macroscopic flux,V(r) :=r2/2−alog cosh(br) is an explicitly solvable example ifa >0 is small enough.
A technical condition: asymptotic normality requires the existence of positive constantsα,V+00,V−00andRsuch that|V00(r)−V+00| ≤e−αr ifr≥R, while|V00(r)− V−00| ≤eαr ifr≤ −R.
Since we are not able to prove the uniqueness of the hydrodynamic limit, our only hypothesis on the initial distribution is an entropy bound: S[µ0,ε,n|λ0,0] = O(n).
LetPεdenote the distribution of the empirical process(uε, vε), then the simplest version of our main result reads as:
Theorem 3.1. Pε is a tight family with respect to the weak local topology of the L2 space of trajectories, and its limit distributions are all concentrated on a set of weak solutions to (2.8).
The notion of weak convergence changes from step to step of the argument. We start with the Young measure of the block-averaged process, and at the end we get tightness in the strong localLp(R2+)topology forp <2;R2+ :=R+×R. This strong form of our result is proven for a mollified version(ˆuε,vˆε)of the empirical process, it is defined a bit later, after (3.2). Compensated compactness is the most relevant keyword of the proofs.
3.2. On the ideas of the proof
We follow the argumentation of the vanishing viscosity approach. In a concise form (2.8) can be written as∂tz+∂xΦ(z) = 0, where z := (u, v), Φ(z) :=−(S0(v), u), and its viscid approximation reads as ∂tzδ +∂xΦ(zδ) = δ ∂x2zδ. This parabolic system admits classical solutions ifδ >0, and the original hyperbolic equation can be solved by sendingδ→0. The argument is not trivial at all, see e.g. Dafermos (2005). Our task is to extend this technology to microscopic systems.
Energy inequality: Observe first that the space integral ofW(z) :=u2/2 +S(v) is constant along classical solutions to the wave equation (2.8), moreover its viscid approximation satisfiess
∂tW(zδ) =∂x(uδS0(vδ)) +δ ∂x(uδ∂xuδ+S0(vδ)∂xvδ)
−δ (∂xuδ)2+S00(vδ)(∂xvδ)2 .
SinceS is strictly convex, we have got a standard energy inequality: anL2bound forδ1/2∂xzδ. In a regime of shocks, however, this bound does not vanish asδ→0, consequently a strong compactness argument is not available.
Young family: Nevertheless, a very weak form of compactness holds true at the level of the Young measure. The approximate solutionzδ can be represented by a measure Θδ onR2+×R2 such that dΘδ := dt dx θt,xδ (dz), whereθt,xδ is the Dirac masssitting at the actual valuezδ(t, x)ofzδ. Sincezδ is locally bounded inL2(R2+), we can select weakly convergent sequences fromΘδ asδ→0. Of course, theYoung family{θt,x: (t, x)∈R2+}of a limiting measureΘofΘδneeds not be Dirac, thus we only have convergence to measure valued solutions: ∂tθt,x(z) +∂x(θt,x(Φ(z))) = 0 in the sense of distributions, where the abbreviationθt,x(ϕ(z)) :=R
ϕ(z)θt,x(dz)is used; we writeθt,x(z)ifϕ(z)≡z. F The identification of measure valued solutions as weak solutions is the subject of the theory of compensated compactness, in fact the Dirac property of the limiting Young measure should be verified.
Compensated factorization: It is crucial that (2.8) admits a rich family ofLax entropy pairs (h, J), these are characterized by the conservation law: ∂th(z) +
∂xJ(z) = 0along classical solutions. Let us now turn to the viscid approximation.
We see that entropy production
Xδ:=∂th(zδ) +∂xJ(zδ) =δ ∂x(h0u∂xuδ+h0v∂xvδ)
−δ h00uu(∂xuδ)2+ 2h00uv∂xuδ∂xvδ+h00vv(∂xvδ)2
decomposes as Xδ =Yδ+Zδ, where Yδ vanishes inH−1, whileZδ is bounded in the space of measures. As a first consequence we get the Lax entropy inequality:
Xδ ≤0 as a distribution if his convex, but the famousDiv-Curl Lemma is more relevant at this point. Letθt,xdenote the Young family of a weak limit pointΘof the sequence of Young measuresΘδ asδ→0, then for couples(h1, J1)and(h2, J2) of Lax entropy pairs we have a compound factorization property:
θt,x(h1J2)−θt,x(h2J1) =θt,x(h1)θt,x(J2)−θt,x(h2)θt,x(J1) (3.1) almost everywhere on R2. In his pioneering papers Ronald DiPerna managed to show that (3.1) implies the Dirac property of the Young family, at least if the sequence of approximate solutions is uniformly bounded, see DiPerna (1985) with further references.
The microscopic evolution: The Ito lemma yields a parabolic energy inequality
∂tEHk(ω(t)) =E(pk+1V0(rk)−pkV0(rk−1)) +σ(ε)E(pk(pk+1+pk−1−2pk))
+σ(ε)E(V0(rk)(V0(rk+1) +V0(rk−1)−2V0(rk)))
at the microscopic level. If εσ(ε)remains positive asε→0, then the tightness in the local topology ofL2(R)of the distributions of the time averaged process might follow from this bound in much the same way as it is done in PDE theory.5 However, εσ(ε)→0 as ε→0, thus the bound degenerates in the limit, consequently there is no hope to get tightness in L2. That is why we say that a direct compactness argument does not work, the method of compensated compactness is needed.
In our case a difficult step of the usual non-gradient analysis can be avoided by considering the Lax entropy pairs(h, J)as functions of the block averaged empirical process (ˆuε,vˆε). Entropy productionXε:=∂th(ˆuε,vˆε) +∂xJ(ˆuε,vˆε)is defined as a generalized function, without the conditionεσ2(ε)→+∞its fluctuations might explode in the limit even if we define the empirical processes in terms of block averages. The main difficulty is to identify the macroscopic flux in the microscopic expression ofL0h, and to show that the remainders do vanish in the limit. This is achieved by replacing block averages of the microscopic currents of momenta with their equilibrium expectations, a logarithmic Sobolev inequality plays a decisive role in the computations. This substitution transforms the evolution equation ofh into a fairly transparent form: we can recover essentially the same structure which appears when the vanishing viscosity limit for (2.8) is performed. At this point can we launch the stochastic theory of compensated compactness, and the proof is terminated by referring to known results from PDE theory. Unfortunately we can not find bounded, positively invariant regions in stochastic situations as DiPerna (1985) did at the PDE level, but the results of Shearer (1994) and Serre–Shearer (1994) on anLp theory of compensated compactness are applicable.
5In case of the diffusive models of Fritz (1986) and its continuations, an energy inequality implies this kind of tightness of the process in the space of trajectories. Guo–Papanicolau–
Varadhan (1988) had raised the problem to the level of measuresµt, and instead of energy and theH+1norm of configurations, the relative entropy and its rate of production (Dirichlet form) are estimated to get the required a priori bounds including an energy inequality.
3.3. Stochastic theory of compensated compactness
Most computations involve mesoscopic block averagesof sizel=l(ε)such that
εlim→0
l(ε)
σ(ε) = 0 and lim
ε→0
εl3(ε)
σ(ε) = +∞.
For sequencesξk indexed byZwe define two kinds of block averages:
ξ¯l,k:= 1 l
l−1
X
j=0
ξk−j and ξˆl,k:= 1 l2
Xl j=−l
(l− |j|)ξk+j. (3.2) For example, V¯l,k0 denotes the arithmetic mean of the sequence ξj =V0(rj). We start calculations with the “smooth” averagesξˆl,k, the arithmetic means appear in canonical expectations. The corresponding empirical process (ˆuε,vˆε) and (¯uεv¯ε) are defined according to ˆuε(t, x) := ˆpl,k(t/ε)if|εk−x|< ε/2, and so on. Sinceuˆε
andvˆεare bounded in a mean sense inL2(dt, dx), the distributionsPˆεof the Young measure Θ form a tight family; these are now defined as dΘε := dt dx θεt,x(du), where θt,xε is the Dirac mass at the actual value of (ˆuε,ˆvε). The Young family controls the asymptotic behavior of various functions of the empirical processes.
Given a Lax entropy pair(h, J), the associated entropy production is defined as
Xε(ϕ, h) :=− Z∞ 0
Z∞
−∞
(h(ˆuε,ˆvε)ϕ0t(t, x) +J(ˆuε,vˆε)ϕ0x(t, x))dx dt,
where the test function ϕ is compactly supported in the interior of R2+. We call an entropy pair (h, J) well controlled if its entropy production decomposes as Xε(ϕ, h) = Yε(ϕ, h) +Zε(ϕ, h), and we have two random functionalsAε(φ, h) andBε(φ, h)such that
|Yε(ψφ, h)| ≤Aε(φ, h)kψk+ and |Zε(ψ, h)| ≤Bε(φ, h)kψk,
wherek · kis the uniform norm, whilek · k+ denotes the norm of the Sobolev space H+1. Here the test functionφis compactly supported in the interior ofR2+, its role is to localize the problem. The factors Aε andBε do not depend on ψ, moreover limEAε(φ, h) = 0andlim supEBε(φ, h)<+∞asε→0.
Proposition 3.2. If (h1, J1) and (h2, J2) are well controlled entropy pairs, then (3.1)holds true with probability one with respect to any limit distribution ofPˆεthat we obtain as ε→0.
This is the stochastic version of the Div-Curl Lemma above. The proof is not difficult, by means of the Skorohod Embedding Theoremit can be reduced to the original, deterministic version, see Fritz (2001), Fritz (2004) and Fritz–Tóth (2004). The main problem is the verification of its conditions, the logarithmic Sobolev inequality plays an essential role here.
3.4. The a priori bounds
Following Fritz (1990), our a priori bounds are all based on the next inequality that controls relative entropy and its rate of production. The initial condition implies that
for allt, n, εwith the same constantC, whereD is the Dirichlet form, it is due to the elliptic perturbation of the anharmonic chain:
D[µt,ε,n|λ0,0] := This is the consequence of a system of differential inequalities:
∂tSn+ 2σ(ε)Dn ≤K local entropy bound see Fritz (2011) with further references.
LSI: The logarithmic Sobolev inequality we are going to use, can be stated as follows. Givenr¯l,k=v, letµvl,kandλvl,kdenote the conditional distributions of the variablesrk, rk+1, ..., rk+l−1with respect toµandλ0,0, and setfl,kv :=dµvl,k/dλvl,k,
for all µ, v, k, l with a universal constantClsi depending only on V. Of course, a similar inequality holds true for the conditional distributions of momenta. Com-bining this with the standard entropy inequalityR
ϕ dµ≤S[µ|λ] + logR
eϕdλ, the calculation of expectations reduces to large deviation bounds for the canonical dis-tributions of the equilibrium measure λ0,0. The most important consequence of the local entropy bound and this LSI is the evaluation of the microscopic current of momentum as follows:
X on.Entropy flux: Finally, let us outline the crucial step of the evaluation of entropy production at a heuristic level. Consider a Lax entropy h = h(u, v) with flux
J =J(u, v)and expandJ. The second order terms of the Lagrange expansion can be neglected, thus we have
X0,k:=L0h(ˆpl,k,ˆrl,k) +J(ˆpl,k+1,rˆl,k+1)−J(ˆpl,k,rˆl,k)
≈h0u(ˆpl,k,rˆl,k)( ˆVl,k0 −Vˆl,k−10 ) +h0v(ˆpl,k,rˆl,k)(ˆpl,k+1−pˆl,k) +Ju0(ˆpl,k,rˆl,k)(ˆpl,k+1−pˆl,k) +Jv0(ˆpl,k,ˆrl,k)(ˆrl,k+1−ˆrl,k).
Sinceh0u(u, v)S00(v) +Jv0(u, v) =h0v(u, v) +Ju0(u, v) = 0,
X0,k≈h0u(ˆpl,k,rˆl,k)( ˆVl,k0 −Vˆl,k0 −1)−h0u(ˆpl,k,rˆl,k)S00(ˆrl,k)(ˆrl,k+1−rˆl,k).
Observe now that ξˆl,k+1−ξˆl,k = (1/l)( ¯ξl,k+l−ξ¯l,k), thus the substitution V¯l,k0 ≈ S0(¯rl,k)results inl X0,k≈0 as
l X0,k≈h0u(ˆpl,k,rˆl,k) (S0(¯rl,k−1+l)−S0(¯rl,k−1)−S00(ˆrl,k)(¯rl,k+l−r¯l,k)). Of course, the precise computation is much more complicated because in the formula Xε of entropy production the terms X0,k have a factor 1/ε. In fact, (εl(ε)σ(ε))−1 is the order of the replacement error; that is why we needεσ2(ε)→ +∞and the sharp explicit bounds provided by the logarithmic Sobolev inequality.