Consistency and stability

which concludes the proof of the proposition.

Remark. Lemma 2.4.12 yielded a-priori bounds in H¹(T^d), which is stronger in d = 1 dimension than the L^∞(T^d)–bound from the maximum principle. FurthermoreL^∞(T^d) is just the critical case, in which simple interpolation arguments do not yield a bound onD^kf_t of the form of Lemma 2.4.13. Thus, without any stronger bounds, we needed to restrict ourselves to d = 1. This dichotomy in regularity between low and high dimensions is natural for quasilinear parabolic equations like the filtration equation (2.3). The regularity of the filtration equation in d = 1,2 was known even before de Giorgi’s and Nash’s famous results on the H¨older-continuity of the solutions to parabolic equations. Indeed our proposed approach in Section 2.7 depends on the regularity result of de Giorgi and Nash.

for all f , f˜ ∈H^k(T^d) such that R

T^d

f(u)du˜ =R

T^df(u)du. The factor Λ(f) is given by (2.41) Λ(f) = 1 +kfk_L^∞ 1 +kD^kfk_L² +kfk^2k_L∞²k∇fk^2k_L2^2+k

_2k+2^d+4

for all f ∈H^k(T^d).

(ii) Furthermore S_t^∞ :H →H satisfies the continuity estimates kS_t^∞f−fk_H⁻¹ ≤Ckfk_L²√

t and kS_t^∞f−fk_H⁻¹ ≤Ck∇fk_L²t for all f ∈H and t ≥0.

(iii)For any t≥0, the mapT_t^∞Ψ :H →Ris differentiable with respect to theH⁻¹–norm in the sense detailed below and it holds that

(2.42) D(T_t^∞Ψ) (f) = F⁰(hS_t^∞f, ϕi_L²)DS_t^∞(f)^∗ϕ∈H¹(T^d)

for all f ∈H. The function w_t:=DS_t^∞(f)^∗ϕ∈H is given as the weak solution of (2.43) ∂_tw_t=σ⁰(S_t^∞f)∆w_t,

such that w₀ =ϕ. We have the estimates

|T_t^∞Ψ( ˜f)−T_t^∞Ψ(f)| ≤ kF⁰k_L^∞kϕk_H¹kf˜−fk_H⁻¹ and

|T_t^∞Ψ( ˜f)−T_t^∞Ψ(f)−DT_t^∞Ψ(f)( ˜f −f)|

≤ 1

2kF⁰k_C¹kϕk²_H1kf˜−fk²_H−1 +Cmax{Λ( ˜f),Λ(f)}kF⁰k_L^∞kϕk_H¹kf˜−fk^1+θ_H−1

for all f , f˜ ∈ H^k(T^d) such that R

T^d

f˜(u)du = R

T^df(u)du, where Λ(·) denotes the same function as in (i).

(iv) Furthermore, it holds that DS_t^∞(f)^∗ϕ ∈ L^∞(0,∞;H¹(T^d))∩L²(0,∞;H²(T^d)) with uniform bounds

k∇DS_t^∞(f)^∗ϕk_L² ≤ k∇ϕk_L², and Z t

0

k∆DS_s^∞(f)^∗ϕk²_L2 ds ≤Ck∇ϕk²_L2

for all ϕ∈H¹(T^d), f ∈H, and t≥0.

Proof. (i) The filtration equation (2.3) yields d

dtkf˜t−ftk²_H−1 =−2 Z

(σ( ˜ft)−σ(ft))( ˜ft−ft)du≤0

because σ is monotonous. This contraction property in weak measure distance is key to most stability estimates.

Let now additionallyvt denote the solution to (2.40). The respective equations forft, ˜ft, and v_t yield

d

dtkf˜_t−f_t−v_tk²_H−1 =−2 Z

(σ( ˜f_t)−σ(f_t)−σ⁰(f_t)v_t)( ˜f_t−f_t−v_t)du.

The mean value theorem implies that the right hand side is bounded by

−2 Z

σ⁰(f_t)( ˜f_t−f_t−v_t)²+ 1

2σ⁰⁰(ξ)( ˜f_t−f_t)²( ˜f_t−f_t−v_t)

du

for some ξ(u) (measurable in u) in the interval between ft(u) and ˜ft(u). Therefore the bound (2.24) on σ⁰ yields that

d

dtkf˜_t−f_t−v_tk²_H−1 ≤ −ckf˜_t−f_t−v_tk²_L2 − Z

σ⁰⁰(ξ)( ˜f_t−f_t)²( ˜f_t−f_t−v_t)du for a positive constantc >0. Now, Lemma 2.4.1 yields|σ⁰⁰(ρ)| ≤C(1 +ρ). Hence Young’s inequality yields a bound

(2.44) d

dtkf˜_t−f_t−v_tk²_H−1

≤ −c

2kf˜_t−f_t−v_tk²_L2 −C 1 +kf˜_tk²_L∞ +kf_tk²_L∞

Z

T^d

( ˜f_t−f_t)⁴ du.

Now the Cauchy-Schwarz inequality in Fourier space and the standard Gagliardo-Nirenberg- Sobolev inequality yield

kfk_L² ≤Ckfk

1 1+k

H^k kfk

k k+1

H⁻¹

kfk_L⁴ ≤Ckfk_H^4kdkkfk¹⁻_L2^4kd

for all f ∈H^k. Taken together these inequalities yield kfk⁴_L4 ≤Ckfk

d+4 k+1

H^k kfk

4k−d 1+k

H⁻¹ . Since d= 1, Lemma 2.4.13 implies

d

dtkf˜_t−f_t−v_tk²_H−1 ≤Cmax{Λ( ˜f),Λ(f)}kf˜_t−f_tk

4k−d 1+k

H⁻¹ ,

where Λ(·) is given in (2.41). Thus the H⁻¹–contractivity yields the desired result.

(ii) Recall that we set ft =S_t^∞f. Since R

T^dft(u)du=R

T^df(u)du, it holds that d

dtkf_t−fk²_H−1 =−2 Z

T^d

σ(f_t)(f_t−f)du.

Adding and subtractingσ(f) from σ(ft) yields (2.45) d

dtkf_t−fk²_H−1 =−2 Z

T^d

(σ(f_t)−σ(f))(f_t−f)du−2 Z

T^d

σ(f)(f_t−f)du.

We apply ellipticity ofσon the first summand and the Cauchy-Schwarz inequality on the second summand to obtain that

d

dtkf_t−fk²_H−1 ≤ −ckf_t−fk²_L2 +Ckfk_L²kf_t−fk_L². for some finite constantsc, C >0. Young’s inequality yields

d

dtkf_t−fk²_H−1 ≤Ckfk²_L2

which yields

kf_t−fk²_H−1 ≤Ckfk²_L2t.

On the other hand, estimate (2.45) yields d

dtkft−fk²_H−1 ≤ −ckft−fk²_L2 +Ck∇fk_L²kft−fk_H⁻¹.

by interpolation betweenH¹ and H⁻¹, see (2.27). Here we have used again that Z

T^d

f_t(u)du= Z

T^d

f(u)du.

Hence it holds that

d

dtkft−fk_H⁻¹ ≤Ck∇fk_L², which yields the desired estimate by integration over t.

(iii) First we want to show thatDS_t^∞(f)^∗ is the adjoint inL²(T^d) of the operatorDS_t^∞(f).

This is standard except for the time-dependence through f_t of the coefficients of the linearized equation (2.40). Let us denote for all 0 ≤ s < t the propagator from time s to time t of the evolution equation (2.40) with time-dependent coefficients by S(t, s).

Likewise the propagator from s to t of equation (2.43) is denoted by S(t, s)^∗. To be precise, we denote by S(·, s)f the solution to equation (2.40) such that at time s the solution equalsf and similarly for S(t, s)^∗. We show thatS(t,0)^∗ is indeed the adjoint of S(t,0). For all 0≤s < t, it holds that

d ds

S(t, s)( ˜f−f), S(s,0)^∗ϕ

L²

=

S(t, s)( ˜f −f), σ⁰(fs)∆[S(s,0)^∗ϕ]

L² −

∆ σ⁰(fs)[S(t, s)( ˜f −f)]

, S(s,0)^∗ϕ

L²

= 0.

Integrating with respect to s yields

(2.46)

DS_t^∞(f)( ˜f−f), ϕ

L² =f˜−f, DS_t^∞(f)^∗ϕ

L².

The differentiability estimates are a consequence of the chain rule and (i) as follows.

Lipschitz continuity of F and (i) yield the first estimate. Using equation (2.46), the second term to be estimated can be rewritten as

F hS_t^∞f , ϕi˜

−F hS_t^∞f, ϕi

−F⁰ hS_t^∞f, ϕi

hDS_t^∞(f)( ˜f −f), ϕi . We bound this term by the sum of the following two terms:

F⁰ hS_t^∞f, ϕiD

S_t^∞f˜−S_t^∞f−DS_t^∞(f)( ˜f−f), ϕE and

F hS_t^∞f , ϕi˜

−F hS_t^∞f, ϕi

−F⁰ hS_t^∞f, ϕi

hS_t^∞f˜−S_t^∞f, ϕi . The results of part (i) yield a bound on the first term by

CkF⁰k_L^∞kϕk_H¹max{Λ( ˜f),Λ(f)}kf˜−fk^1+θ_H−1, whereas the second term is bounded by

1

2kF⁰k_C¹

hS_t^∞f˜−S_t^∞f, ϕi

2 ≤ 1

2kF⁰k_C¹kϕk²_H1kf˜−fk²_H−1. (iv) Equation (2.43) yields

(2.47) d

dt Z

T^d

|∇w_t(u)|² du=−2 Z

T^d

σ⁰(S_t^∞f(u))|∆w_t(u)|² du≤0,

since σ⁰ >0. Since furthermore∇w0(u) = ∇ϕ(u), integration of this equation yields 2

Z ∞ 0

Z

T^d

σ⁰(S_t^∞f(u))|∆wt(u)|² dudt≤ k∇ϕk²_L2, uniformly and f and ϕ. Thus we obtain that

Z ∞ 0

Z

T^d

|∆w_t(u)|² dudt≤Ck∇ϕk²_L2

using the uniform ellipticity of σ⁰ again, see (2.24).

Next provide some large–time decay estimates which will enable us to provide uniform in time bounds in the hydrodynamic limit.

Lemma 2.4.15 (Spectral gap). Let F ∈ C_b²(R) and ϕ ∈ C³(T^d) and define Ψ as in (2.19). Using the notation of the stability lemma 2.4.14, let f , f˜ ∈ H. Furthermore let f∞ =R

f(u)du denote the spatial average of f and set

(2.48) T1( ˜f , f; Ψ) := Ψ( ˜f)−Ψ(f)−DΨ(f)( ˜f−f).

(This is the difference of the functionΨevaluated atf˜∈H and its first Taylor polynomial around f ∈H.) Then there exist finite, positive constants c, C such that

kS_t^∞f −f∞k^p_Lp ≤Ce^−ctkf −f∞k^p_Lp, kS_t^∞f −f∞k_H⁻¹ ≤Ce^−ctkf −f∞k_H⁻¹, kT₁( ˜f , f;S_t^∞)k²_H−1 ≤Ce^−ct

kf −f∞k⁴_L4 +kf˜−f˜∞k⁴_L4

, and k∇DS_t^∞(f)^∗ϕk_L² ≤Ce^−ctk∇ϕk_L²

for all 2≤p < +∞. Furthermore it holds that

|T₁( ˜f , f;T_t^∞Ψ)| ≤Ce^−ct

kf−f_∞k²_L4 +kf˜−f˜_∞k²_L4

where Ψ is defined in (2.19) with F ∈C_b²(R) and ϕ∈C²(T^d).

Proof. We will use the notation of the stability lemma 2.4.14, letting f_t :=S_t^∞f and f˜_t:=S_t^∞f˜

denote two solutions of the filtration equation and

v_t:=DS_t^∞(f)( ˜f −f) and w_t:=DS_t^∞(f)^∗ϕ the linearization around f_t as well as its L²-dual.

First of all, conservation of mass yieldsR

f_t(u)du =f∞. Set ¯f_t:=f_t−f∞, which solves the equation

∂_tf¯_t=∇ · σ⁰( ¯f_t+f∞)∇f¯_t .

Note that in contrast to f_t, the function ¯f_t is no longer non-negative everywhere. The equation for ¯f_t yields

d dt

Z

T^d

|f¯_t|^p du=p Z

T^d

|f¯_t|^p−2f¯_t∇ · σ⁰( ¯f_t+f∞)∇f¯_t du.

Now integration by parts yields d

dt Z

T^d

|f¯_t|^p du=−p(p−1) Z

T^d

σ⁰( ¯f_t+f∞)|f¯_t|^p−2|∇f¯_t|² du.

Since

|f¯_t|^p−2|∇f¯_t|² =

|f¯_t|^p/2−1∇f¯_t

2 = 4 p²

∇|f¯_t|^p/2

2, it holds that

d dt

Z

T^d

|f¯_t|^p du=−4(p−1) p

Z

T^d

σ⁰( ¯f_t+f_∞)

∇|f¯_t|^p/2

2 du ≤ −c Z

T^d

∇|f¯_t|^p/2

2 du.

Now Poincar´e’s inequality in the L²–norm applied to |f¯_t|^p/2 yields d

dt Z

T^d

|f¯t|^p du≤ −c Z

T^d

|f¯t|^p du,

which yields the decay of the L^p–norms of ¯f_t =f_t−f_∞. Note thatc, C can be taken to be independent of the choice of p≥2.

The decay of the H⁻¹–norm follows from d

dtkf_t−f∞k²_H−1 =−2 Z

T^d

(σ(f_t(u))−σ(f∞))(f_t(u)−f∞)du≤ −ckf_t−f∞k²_H−1

by uniform ellipticity and (2.27).

The third statement follows directly from equation (2.44) and the exponential decay of the L⁴–norm.

To prove the fourth statement, recall equation (2.47). The lower bound on σ⁰ yields d

dt Z

T^d

|∇w_t(u)|² du≤ −c Z

T^d

|∆w_t(u)|² du.

Applying once again Poincar´e’s inequality yields d

dt Z

T^d

|∇w_t(u)|² du≤ −c Z

T^d

|∇w_t(u)|² du,

and hence exponential decay. Consequently, the proof of Lemma 2.4.14 yields

|T1( ˜f , f;T_t^∞Ψ)| ≤Ce^−ct

kf−f∞k²_L4 +kf˜−f˜∞k²_L4 +kf −f∞k²_H−1 +kf˜−f˜∞k²_H−1

, and the result follows since the H⁻¹(T^d)–norm is bounded by the L⁴(T^d)–norm.

Many of the differentiability properties we have just shown fit in the framework of dif- ferentiable functions on C_b(H) whose derivative is bounded uniformly with respect to a weight function. The following definition formalizes this. It is an adaptation of Definition 2.10 in [67].

Definition 2.4.16. Let ˜H₁ and ˜H₂ be any two metric spaces and consider two Banach spaces (H₁,k · k_H1),(H₂,k · k_H2) such that ˜H_i−H˜_i ⊂H_i,i= 1,2. In general the metric of

the subspace ˜Hi is stronger than the norm of Hi. We can understand Hi to be a tangent space of ˜H_i. Let

Λ : ˜H₁ →[0,+∞) be a weight function. Abusing notation, we also set

Λ( ˜f , f) := max{Λ( ˜f),Λ(f)}.

Then we denote byC_Λ^1,θ( ˜H1, H1; ˜H2, H2), the space of continuously differentiable function from ˜H₁ to ˜H₂ whose derivative approximates the original function to the order 1 +θ and is uniformly bounded with respect to the weight function Λ. Specifically, a function

Φ : ˜H₁ →H˜₂ is in C_Λ^1,θ( ˜H₁, H₁; ˜H₂, H₂) if and only if there exists a continuous function

DΦ : ˜H₁ →L(H₁, H₂) and finite constantsC₁, C₂, and C₃, such that it holds

kΦ( ˜f)−Φ(f)k_H2 ≤C₁Λ( ˜f , f)kf˜−fk_H1 (2.49)

kDΦ(f)( ˜f−f)k_H2 ≤C₂Λ( ˜f , f)kf˜−fk_H1 and (2.50)

kΦ( ˜f)−Φ(f)−DΦ(f)( ˜f−f)k_H2 ≤C₃Λ( ˜f , f)kf˜−fk^1+θ_H (2.51) 1

for all ˜f , f ∈ H˜₁. Let C_i^opt, i = 1,2,3, be the optimal constant in each of (2.49)-(2.51), i.e. the infimum over allC_i, i= 1,2,3, such that each of the inequalities holds. Then we set

[Φ]_C^0,1

Λ :=C₁^opt, [Φ]_C^1,0

Λ :=C₂^opt, and [Φ]_C^1,θ

Λ :=C₃^opt, which are seminorms associated toC_Λ^1,θ( ˜H1, H1; ˜H2, H2).

Note that the proof of the hydrodynamic limit can be understood without using the above notation, since in principle all we need to do is keep track of various differences.

We include it to provide context and simplify notation, especially in order to distinguish clearly between stability and consistency estimates. Let us translate the stability result (i) and (ii) of Lemma 2.4.14 to the language of Definition 2.4.16.

Corollary 2.4.17 (Stability in terms of differentiability). Let R ≥0, k ∈N and denote H_R:=

f ∈H^k(T^d)|R

T^df(u)du=R .

Furthermore, let F ∈ C_b²(R) and ϕ∈C²(T^d), and define Ψ as in equation (2.19). Then there exists a constant C=C(F, ϕ) independent oft andR such that the following holds.

(i) For any t ≥0 and k >(d+ 2)/2, it holds that

S_t^∞∈C_Λ^1,θ(H_R, H⁻¹(T^d);H_R, H⁻¹(T^d)) and [S_t^∞]_C^1,θ

Λ ≤C

in the sense of Definition 2.4.16, where θ = 2k−d−2

2k+ 2 andΛ(f) = 1 +kfkL^∞ 1 +kD^kfk_L² +kfk^2k_L^∞2k∇fk^2k_L2^2+k

_2k+2^d+4 .

(ii) For any t ≥0, it holds that

T_t^∞Ψ∈C_Λ^1,θ(H_R, H⁻¹(T^d);R,R) and [T_t^∞Ψ]_C^1,θ

Λ ≤C

where θ and Λ are given as above.

Proof. Again, part (ii) is a direct consequence of the corresponding estimates on part (i) and the regularity of F ∈ C_b²(R) and ϕ ∈ C²(T^d). To show (i), just let ˜H₁ = ˜H₂ = H_R and H₁ =H₂ =H⁻¹(T^d). Note that since all functions in H_R have the same integral over T^d, it holds that indeed

H_R−H_R⊂H⁻¹(T^d).

The function S_t^∞ maps H_R to H_R according to Lemma 2.4.13. Note that the (distribu- tional) solution

v_t =DS_t^∞(f)( ˜f−f)

to (2.40) also exists for initial data ˜f −f ∈ H⁻¹(T^d) and satisfies the bounds given in Lemma 2.4.14. Hence [Ψ]_C^0,1

Λ and [Ψ]_C1,θ Λ

of Definition 2.4.16 are indeed finite. We just need to estimate [Ψ]_C^1,0

Λ . Calculations similar to the proof of Lemma 2.4.14 yield d

dtkv_tk²_H−1 =−2 Z

T^d

v_tσ⁰(f_t)v_tdu≤0,

if the initial datum satisfies ˜f−f ∈L²(T^d). This shows kDS_t^∞(f)( ˜f −f)k_H⁻¹ ≤ kf˜−fk_H⁻¹,

if ˜f −f ∈L²(T^d), and by approximation in general. In particular, it holds that DS_t^∞(f)∈L(H⁻¹(T^d), H⁻¹(T^d)).

It remains to prove the continuity of DS_t^∞(f) ∈ L(H⁻¹(T^d), H⁻¹(T^d)) with respect to f ∈H_R(even though we shall not make use of this property in this thesis). This is easiest to see in the dual setting. Let ˜f , f ∈ H_R. As in the stability result, Lemma 2.4.14, we

setwt:=DS_t^∞(f)^∗ϕand we also set ˜wt:=DS_t^∞( ˜f)^∗ϕ. Then the continuity estimate DS_t^∞(f)−DS_t^∞( ˜f)

L(H⁻¹,H⁻¹)→0 as kf˜−fk_H^k →0 corresponds by duality to showing that

(2.52) k∇( ˜w_t−w_t)k_L² ≤ω(kf˜−fk_H^k)k∇ϕk_L²

for some modulus of continuity ω, i.e. some function ω : [0,∞) → [0,∞) such that ω(τ)→0 as τ → 0. Note that we may allow ω to depend on kf˜k_H^k, kfk_H^k, and t. The equations for ˜w_t and w_t yield

d dt

Z

T^d

|∇( ˜w_t−w_t)|² du=−2 Z

T^d

∆( ˜w_t−w_t) σ⁰( ˜f_t)∆ ˜w_t−σ⁰(f_t)∆w_t du.

Hence it holds that d

dt Z

T^d

|∇( ˜w_t−w_t)|² du=−2 Z

T^d

|∆( ˜w_t−w_t)|²σ⁰( ˜f_t)du

−2 Z

T^d

∆( ˜w_t−w_t) σ⁰( ˜f_t)−σ⁰(f_t)

∆w_tdu.

Young’s inequality yields a bound on the right hand side by

(2.53) C

(σ⁰( ˜f_t)−σ⁰(f_t))∆w_t

2 L². The mean value theorem and Lemma 2.4.1 yield

σ⁰( ˜f_t)−σ⁰(f_t)

≤C (kf˜_tk_L^∞+kf_tk_L^∞)kf˜_t−f_tk_L^∞.

By the maximum principle and since k > (d + 2)/2 yields an embedding H^k(T^d) ,→ L^∞(T^d), we can therefore bound (2.53) by

C kf˜k_Hk,kfk_Hk

kf˜_t−f_tk_L^∞k∆w_tk²_L2,

whereC(kfk˜ _Hk,kfk_Hk) denotes some constant depending on its arguments, but not on t orw_t. Again sincek > (d+ 2)/2, there exists a bounded embedding H^k−1(T^d),→L^∞(T^d) and therefore

kf˜_t−f_tk_L^∞ ≤Ckf˜_t−f_tk_H^k−1. Hence interpolation yields

kf˜_t−f_tk_H^k−1 ≤ kf˜_t−f_tk

k k+1

H^k kf˜_t−f_tk

1 k+1

H⁻¹

Since Lemma 2.4.13 yields uniform-in-time bounds on the H^k–norms of ˜f and f, the contractivity of S_t^∞ with respect to the H⁻¹–norm yields

kf˜_t−f_tk

k k+1

H^k kf˜_t−f_tk

1 k+1

H⁻¹ ≤C kfk˜ _H^k,kfk_H^k

kf˜−fk

1 k+1

H⁻¹,

for some (possibly different) constantC(kfk˜ _H^k,kfk_H^k). In summary, we have proved that d

dtk∇( ˜w_t−w_t)k²_L2 ≤C kfk˜ _H^k,kfk_H^k

kf˜−fk

1 k+1

H⁻¹k∆w_tk²_L2. Since ˜w₀ =ϕ=w₀, integration with respect to time yields

k∇( ˜w_t−w_t)k²_L2 ≤C kfk˜ _H^k,kfk_H^k

kf˜−fk

1 k+1

H⁻¹

Z t 0

k∆w_sk²_L2 ds.

Lemma 2.4.14 (iv) yields

k∇( ˜wt−wt)k²_L2 ≤C kf˜k_H^k,kfk_H^k

kf˜−fk

1 k+1

H⁻¹k∇ϕk²_L2, which shows the desired estimate (2.52).

Now let us identify an expression for the time-derivative dT_t^∞Ψ(f)/dt.

Lemma 2.4.18. Let F ∈ C_b¹(R) and ϕ ∈ C²(T^d), and define Ψ as in equation (2.19).

Furthermore let S_t^∞f solve the filtration equation with initial datum f ∈ H^k(T^d) with k > (d+ 2)/2. Then we obtain the following characterizations of the derivative in t of the limit evolution.

(i) It holds that

d

dtS_t^∞f =DS_t^∞(f)(∆σ(f)).

(ii) We can lift this result to the level of observables. Recalling the definition G^∞T_t^∞Ψ(f) := d

dtT_t^∞Ψ(f), the result (i) translates to T_t^∞ as

G^∞T_t^∞Ψ(f) = F⁰(hS_t^∞f, ϕi_L²)h∆σ(f), DS_t^∞(f)^∗ϕi_L² =:T_t^∞G^∞Ψ(f).

for all f ∈H^k(T^d) and all t≥0, where Ψ is given in (2.19).

Proof. (i) Consider

I := 1 s

S_t+s^∞ f −S_t^∞f

−DS_t^∞(f)(∆σ(f))

The semigroup property of S_t^∞ (a direct consequence of uniqueness of solutions to the filtration equation (2.3)) yields

I = 1 s

S_t^∞S_s^∞f −S_t^∞f

−DS_t^∞(f)(∆σ(f)).

Therefore, the stability result of Lemma 2.4.14 yields (2.54) kIk_H⁻¹ =

1 s

DS_t^∞(f)(S_s^∞f−f)

−DS_t^∞(f)(∆σ(f)) H⁻¹

+O s⁻¹Λ(f, S_s^∞f)kS_s^∞f−fk^1+θ_H−1

, whereθ = (2k−d−2)/(2k+ 2)>0. The maximum principle and the improved regularity of Lemma 2.4.13 yield that Λ(f, S_s^∞f) ≤ C(f) independent of s. Furthermore Lemma 2.4.14 (ii) yields

kS_s^∞f −fk_H⁻¹ ≤Ck∇fk_L²s, and hence

O s⁻¹Λ(f, S_s^∞f)kS_s^∞f−fk^1+θ_H−1

=O(s^θ)

for fixedf ∈H^k(T^d). Since DS_t^∞(f)∈ L(H⁻¹(T^d), H⁻¹(T^d)) is a contraction, the other summand on the right hand side of (2.54) equals

DS_t^∞(f)

S_s^∞f −f s

−DS_t^∞(f)(∆σ(f)) _H−1

≤

S_s^∞f−f

s −∆σ(f)

_H−1

, where we have used that

Z

T^d

S_s^∞f(u)−f(u)

du= 0 = Z

T^d

∆σ(f(u))du.

The right hand side vanishes sincef_t is a solution to equation (2.3).

(ii) This part is a direct consequence of (i) and the chain rule.

For the statement of the consistency result, we will need a replacement lemma, which in the spirit of an ergodic theorem (or a law of large numbers) allows us to replace locally the spatial average of a function of the number of particles over a small box with its expectation value with respect to the local density in this box. Since we are interested in obtaining qualitative results, we will prove a quantitative L²–version with an explicit upper bound on the rate of convergence. The proof is deferred until Section 2.6.

Lemma 2.4.19 (A quantitative replacement lemma). Assuming that the initial data possess bounded relative entropy and are bounded with respect to some Gibbs measure, i.e.

H^N µ^N₀ |ν_ρ^N

≤CN^d, and µ^N₀ ≤ν_ρ^N

for some ρ >0. Then it holds that 1

T Z T

0

Z

T^d

D µ^N_t ,

(g◦η)⁽⁾(uN)−σ(η⁽⁾(uN))

2E dudt

!1/2

≤r_RL(%, l, , N),

where we recall definition (2.34). The rate function r_RL satisfies r_RL(%, l, , N)≤C

(N⁻¹²l¹² +¹²l¹⁴)%l^d4 +%l^−d4 +%⁻¹⁴ + l N

for all N ∈N, 1/N < <1, l < N, and % >0.

Now we are ready to state the consistency result.

Lemma 2.4.20 (Consistency). Let F ∈ C_b²(R) and ϕ ∈ C³(T^d), and define Ψ as in equation (2.19). Furthermore assume that the initial data µ^N₀ of the ZRP have bounded relative entropy and are bounded relative to some Gibbs measure

H^N µ^N₀ |ν_ρ^N

≤CN^d and µ^N₀ ≤ν_ρ^N for all N ∈N. Then it holds that

Z t 0

µ^N_s , G^Nπ^N −π^NG^∞

T_t−s^∞ Ψ ds

≤r_C(T, %, l, , N)

for all 0 ≤ t < +∞, T > 0, N ∈ N, and 1/N < < 1. The consistency bound rC(T, %, l, , N) is given explicitly by the function

(2.55) C

^−dθ2 N^1−θ(d+1)T sup

0≤s≤t

sup

x∼y

µ^N_t−s, T_s^∞Ψ

C^1+θΛ(α^N,_ηx,y, α^N,_η )N^−dP

zη(z) +^−(3+d2) sup

η∈X_N

Z ∞ T

k∇(DS_s^∞(α^N,_η )^∗ϕ)k_L2(T^d) ds +N^2+dsup

t≥T

Z t T

D

µ^N_t−s,sup

x∼y

T₁(α^N,_ηx,y, α^N,_η ;T_s^∞Ψ) E

ds +

∆(DS_t^∞(α^N,_η )^∗ϕ) L^∞_η L²_t,u

√

T r_RL(%, l, , N) +^−(2+d2)N⁻² . for any θ =θ(k) andk > 0. Here θ, Λ, and [T_s^∞Ψ]_C1+θ

Λ are given in Corollary 2.4.17 and r_RL(%, l, , N) stems from Lemma 2.4.19. The notation T₁ is explained in Lemma 2.4.15 and the function DS_s^∞(α^N,_η )^∗ϕ is given in Lemma 2.4.14.

Note that in contrast to Section 2.3, the form (2.19) of Ψ is not conserved under the application T_t^∞, hence we need to keep track of T_t−s^∞. This also explains why we needed to derive the above fairly complex stability results.

Proof of Lemma 2.4.20. Let us first assume that t ≤ T. Denote the quantity to be estimated by

I :=

Z t 0

D

µ^N_t−s, G^Nπ^N −π^NG^∞

T_s^∞ΨE ds

.

Inserting the expressions for the generators G^N and G^∞, cf. Lemma 2.4.18, we obtain that

I =

Z t 0

µ^N_t−s, N²X

x∼y

g(η(x))h

T_s^∞Ψ(α^N,_ηx,y)−T_s^∞Ψ(α^N,_η )i

− hDT_s^∞Ψ(α^N,_η ),∆σ(α^N,_η )i_L²

ds ,

whereDT_s^∞Ψ is defined in equation (2.42). Linearizing T_s^∞Ψ around α^N,_η yields

I ≤R1+

Z t 0

µ^N_t−s, N²X

x∼y

g(η(x))DT_s^∞Ψ(α^N,_η )(α^N,_ηx,y −α^N,_η )

− hDT_s^∞Ψ(α_η^N,),∆σ(α^N,_η )i_L²

ds with an error term

(2.56) R1 = Z t

0

µ^N_t−s, N²X

x∼y

g(η(x))

T_s^∞Ψ(α^N,_ηx,y)−T_s^∞Ψ(α^N,_η )

−DT_s^∞Ψ(α^N,_η )(α^N,_ηx,y −α^N,_η )

ds.

Substituting the definition (2.29) of the empirical measure into the right hand side then yields

I ≤R1+

Z t 0

µ^N_t−s, N^2−dX

x∼y

g(η(x))DT_s^∞Ψ(α^N,_η )(δ⁽⁾y N

−δ⁽⁾x N

)

− hDT_s^∞Ψ(α^N,_η ),∆σ(α^N,_η )i_L²

ds .

The explicit expression for DT_s^∞Ψ, see (2.42), then yields I ≤R1

+

Z t 0

*

µ^N_t−s, F⁰

S_s^∞α^N,_η , ϕ

L²

N^−dX

x

g(η(x)) D

DS_s^∞(α^N,_η )^∗ϕ,∆Nδ^()x

N

E

L²

−

DS_s^∞(α^N,_η )^∗ϕ,∆σ(α^N,_η )

L²

+ ds

.

Next, up to an error R2, we replace the discrete Laplacian ∆_N by its continuous version

∆ and, after an integration by parts, obtain that

I ≤R1+R2+

Z t 0

µ^N_t−s, F⁰ hS_s^∞α_η^N,, ϕi_L² Z

T^d

N^−dX

x

g(η(x))δ⁽⁾x N(u)

−σ(α^N,_η (u))

∆ DS_s^∞(α^N,_η )^∗ϕ

(u)du

ds .

The explicit expression for the error term is

(2.57) R2 =

Z t 0

µ^N_t−s, F⁰ hS_s^∞α^N,_η , ϕi_L²

"

N^−dX

x

g(η(x))hDS_s^∞(α^N,_η )^∗ϕ,∆_Nδ⁽⁾x N

−∆δ⁽⁾x N

i_L²

# ds

Recall that DS_s^∞(α^N,_η )^∗ϕ ∈ L²(0,∞;H²(T^d)). Thus we apply H¨older’s inequality to obtain

I ≤R1+R2+kF⁰k_L^∞× Z t

0

Z

T^d

D

µ^N_t−s, 1 N^d

X

x

g(η(x))δ⁽⁾x N

(u)−σ(α^N,_η (u))2E duds

1/2

×

∆_u(DS_s^∞(α^N,_η )^∗ϕ)

L^∞_ηL²_s,u. Since t≤T, Lemma 2.4.19 yields

I ≤R1+R2+√ T

∆_u(DS_s^∞(α^N,_η )^∗ϕ)

L^∞_η L²_s,ur_RL(%, l, , N).

It remains to find a bound on R1 and R2 to finish the proof of the consistency result.

Since

Z

T^d

α^N,_η (u)du= Z

T^d

α^N,_η^x,y(u)du,

the first error term (2.56) is bounded from above by R1 ≤N^2+dT sup

t∈[0,T]

sup

s∈[0,t]

hµ^N_t−s, T_s^∞Ψ

C_Λ^1,θΛ(α^N,_η^x,y, α^N,_η )kα^N,_η^x,y −α^N,_η k^1+θ_H−1i,

where we have used the notation of Definition 2.4.16. To calculate the differencekα^N,_ηx,y− α^N,_η k^1+θ_H−1, we test with a function ω ∈H¹(T^d). Thus we obtain that

hω, α^N,_ηx,y −α^N,_η i_L² = Z

ω(u) 1 N^d

δ⁽⁾y

N

−δ⁽⁾x N

du

= Z 1

N^d

ω(u+ y

N)−ω(u+ x N)

δ⁽⁾₀ du

≤ C

N^d+1k∇ωk_L²kδ₀⁽⁾k_L², and hence

kα^N,_ηx,y −α^N,_η k_H⁻¹ ≤C^−d/2N^−(d+1). The second error term (2.57) is bounded from above by

R2 ≤g^∗kF⁰k_L^∞ Z t

0

D

µ^N_t−s, X

x∈T^d_N

η(x) N^d

E2

ds ¹₂

× k∆DS_s^∞(α^N,_η )^∗ϕk_L^∞

η L²_s,uk(−∆)⁻¹(∆_N −∆)δ₀⁽⁾k_L²

u. Now it holds that

(2.58) k(−∆)⁻¹(∆N −∆)δ₀⁽⁾k_L² ≤CN⁻²kD²δ⁽⁾₀ k_L² ≤ C N^22+d/2. Hence we obtain that

R2 ≤C√

Tk∆DS_s^∞(α^N,_η )^∗ϕk_L^∞_{η L}²_s,uN^{−2−(2+d/2)},

which completes the proof of Lemma 2.4.20 if t ≤ T. If t > T, all bounds remain valid

for

Z T 0

D

µ^N_t−s, G^Nπ^N −π^NG^∞

T_s^∞ΨE ds

, and it remains to estimate

II :=

Z t T

D

µ^N_t−s, G^Nπ^N −π^NG^∞

T_s^∞ΨE ds

.

Again we will have three contributions to this quantity. The first one equals

Re1 = Z t

T

µ^N_t−s, N²X

x∼y

g(η(x))

T_t^∞Ψ(α_η^N,x,y)−T_t^∞Ψ(α^N,_η )

−DT_t^∞Ψ(α^N,_η )(α^N,_ηx,y −α^N,_η )

ds.

It holds that

Re1 ≤CN^2+dsup

t≥T

Z t T

D

µ^N_t−s,sup

x∼y

T₁(α_η^N,x,y, α^N,_η ;T_s^∞Ψ)

N^−dX

z

η(z)E ds,

where the second supremum is taken over all neighbours x and y in T^dN. The second contribution to II is given by

Re2 ≤

Z t T

µ^N_t−s, F⁰(hS_s^∞α^N,_η , ϕi_L²)

N^−dX

x

g(η(x))hDS_s^∞(α_η^N,)^∗ϕ,∆_Nδ⁽⁾x

N −∆δ⁽⁾x N i_L²

ds .

Since the mass is conserved and bounded, it follows that Re2 ≤CkF⁰k_L^∞k(∆−∆_N)δ⁽⁾₀ k_H⁻¹ sup

η∈XN

Z ∞ T

k∇DS_s^∞(α^N,_η )^∗ϕk_L² ds

Similarly to (2.58), we obtain

k(∆−∆N)δ₀⁽⁾k_H⁻¹ ≤CN⁻²kD³δ₀⁽⁾k_L² ≤C^−(3+d/2)N⁻². Thus we have shown

Re2 ≤C^−(3+d/2)N⁻² sup

η∈X_N

Z ∞ T

k∇DS_s^∞(α^N,_η )^∗ϕk_L² ds Finally, the remaining term is bounded by

Z t T

µ^N_t−s, F⁰(hS_s^∞α^N,_η , ϕi_L²) Z

T^d

∇

N^−dX

x

g(η(x))δ⁽⁾x N(u)

−σ(α_η^N,(u))

∇ DS_s^∞(α^N,_η )^∗ϕ (u)du

ds

.

H¨older’s inequality yields a bound by kF⁰k_L^∞

Z t T

µ^N_t−s,

∇ N^−dX

x

g(η(x))δ⁽⁾x N

(u)−σ(α^N,_η (u)) L²

×

∇ DS_s^∞(α^N,_η )^∗ϕ L²

ds.

Sinceg and σ are uniformly Lipschitz-continuous, the first term is bounded by

∇ N^−dX

x

g(η(x))δ⁽⁾x

N −σ(α^N,_η ) L²

≤CN^−dX

x

η(x)k∇δ⁽⁾x N k_L².

By the conservation of particles, this term is uniformly bounded by C^−(1+d/2), which completes the proof.

Remark 2.4.21. Similarly to the proof of Lemma 2.4.18, where we needed to take the time-derivative att = 0 and which can be understood as proving

π^NG^∞T_t^∞Ψ =π^NT_t^∞G^∞Ψ =DT_t^∞Ψ(∆σ(α^N,)), the term R1 can be seen as capturing the “commutation relation”

G^Nπ^NT_t^∞Ψ(η)≈DT_t^∞Ψ(G^Nα^N,_η ).

Closer inspection of the proof shows that in order to prove the former relationship between G^∞ andT_t^∞, we just needθ >0 in the stability result, Lemma 2.4.14. On the other hand, we needθ >1/(d+ 1) in order to guarantee that the error termR1 coming from the latter relationship betweenG^N and T_t^∞ vanishes in the limit.

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