Regularity structures and inhomogeneous models

The purpose of regularity structures, introduced in [Hai14] and motivated by [Lyo98, Gub04], is to generalise Taylor expansions using essentially arbitrary functions or distributions instead of polynomials. The precise definition is as follows.

Definition 3.2.1. Aregularity structureT = (T,G)consists of two objects:

• Amodel spaceT, which is a graded vector spaceT =L

α∈ATα, where each Tα is a (finite dimensional in our case) Banach space andA ⊂ Ris a finite set of “homogeneities”.

• Astructure groupGof linear transformations ofT, such that for everyΓ ∈ G,

everyα∈ Aand everyτ ∈ Tα one hasΓτ−τ ∈ T<α, withT<α

def

=L

β<αTβ. In [Hai14, Def. 2.1], the setA was only assumed to be locally finite and bounded from below. Our assumption is more strict, but does not influence anything in the analysis of the equations we consider. In addition, our definition rules out the ambiguity of topologies onT.

Remark 3.2.2. One of the simplest non-trivial examples of a regularity structure is given by the “abstract polynomials” ind+ 1indeterminatesXi, withi= 0, . . . , d. The setAin this case consists of the valuesα∈_Nsuch thatα ≤r, for somer <∞ and, for eachα ∈ A, the spaceTαcontains all monomials in theXi of scaled degree α. The structure groupGpoly is then simply the group of translations inRd+1 acting

We now fixr > 0to be sufficiently large and denote byTpoly the space of

such polynomials of scaled degreerand byFpolythe set{Xk : |k|s ≤r}. We will

only ever consider regularity structures containingTpolyas a subspace. In particular,

we always assume that there’s a natural morphismG → Gpoly compatible with the

action ofGpolyonTpoly,→ T.

Remark 3.2.3. Forτ ∈ T we will writeQατ for its canonical projection ontoTα, and definekτkα

def

=kQατk. We also writeQ<αfor the projection ontoT<α, etc. Another object in the theory of regularity structures is a model. Given an abstract expansion, the model converts it into a concrete distribution describing its local behaviour around every point. We modify the original definition of model in [Hai14], in order to be able to describe time-dependent distributions.

Definition 3.2.4. Given a regularity structure T = (T,G), an inhomogeneous

model(Π,Γ,Σ)consists of the following three elements:

• A collection of mapsΓt:_Rd×_Rd_{→ G, parametrised byt∈}

R, such that

Γt_xx = 1 , Γt_xyΓt_yz= Γt_xz , (3.7) for anyx, y, z ∈_Rd_{andt ∈}

R, and the action ofΓtxy on polynomials is given as in Remark 3.2.2 withh= (0, y−x).

• A collection of mapsΣx :R×R → G, parametrized byx ∈ Rd, such that, for anyx∈_Rd_{ands, r, t∈}

R, one has

Σtt_x = 1, Σsr_x Σrt_x = Σst_x , Σst_xΓt_xy = Γs_xyΣst_y , (3.8) and the action of Σst

x on polynomials is given as in Remark 3.2.2 with h = (t−s,0).

• A collection of linear mapsΠt_x :T → S0₍

Rd), such that

Πt_y = Πt_xΓt_xy , Πt_xX(0,¯k)(y) = (y−x)¯k , Πt_xX(k0,¯k)_{(y) = 0 , (3.9)}

for allx, y ∈_Rd_{,t ∈}

Moreover, for anyγ > 0 and every T > 0, there is a constant C for which the analytic bounds hΠt_xτ, ϕλ_xi ≤Ckτkλl, kΓt_xyτkm ≤Ckτk|x−y|l−m , (3.10a) kΣst_xτkm ≤Ckτk|t−s|(l−m)/s0 , (3.10b) hold uniformly over allτ ∈ Tl, withl ∈ Aandl < γ, allm ∈ Asuch thatm < l, all λ ∈ (0,1], all ϕ ∈ Br

0(Rd) with r > −bminAc, and all t, s ∈ [−T, T] and

x, y ∈_Rd_{such that|t−s| ≤1and|x−y| ≤1.}

In addition, we say that the mapΠhas time regularityδ >0, if the bound

h(Πt_x−Πs_x)τ, ϕλ_xi

≤Ckτk|t−s|δ/s0λl−δ, (3.11)

holds for allτ ∈ Tland the other parameters as before.

Remark 3.2.5. For a model Z = (Π,Γ,Σ), we denote by kΠkγ;T, kΓkγ;T and kΣkγ;T the smallest constants C such that the bounds on Π, Γ and Σ in (3.10a) and (3.10b) hold. Furthermore, we define

|||Z|||γ;T

def

=kΠkγ;T +kΓkγ;T +kΣkγ;T .

IfZ¯= ( ¯Π,Γ¯,Σ)¯ is another model, then we also define the “distance” between two models

|||Z; ¯Z|||γ;T

def

=kΠ−Π¯kγ;T +kΓ−Γ¯kγ;T +kΣ−Σ¯kγ;T . (3.12) We note that the norms on the right-hand side still make sense withΓandΣviewed as linear maps onT. We also setkΠkδ,γ;T

def

=kΠkγ;T +C, whereCis the smallest constant such that the bound (3.11) holds, and we define

|||Z|||δ,γ;T

def

=kΠkδ,γ;T +kΓkγ;T +kΣkγ;T . Finally, we define the “distance”|||Z; ¯Z|||δ,γ;T as in (3.12).

Remark 3.2.6. In [Hai14, Def. 2.17] the analytic bounds on a model were assumed to hold locally uniformly. In the problems which we aim to consider, the models are periodic in space, which allows us to require the bounds to hold globally.

Remark 3.2.7. For a given model(Π,Γ,Σ)we can define the following two objects ˜ Π(t,x)τ (s, y) = Πs_xΣst_xτ(y), Γ˜(t,x),(s,y) = ΓtxyΣ ts y = Σ ts xΓ s xy , (3.13)

forτ ∈ T. Of course, in general we cannot fix the spatial pointyin the definition ofΠ˜, and we should really write Π˜(t,x)τ

(s,·)(ϕ) = Π_xsΣst_xτ(ϕ)instead, for any test functionϕ, but the notation (3.13) is more suggestive. One can then easily verify that the pair( ˜Π,Γ)˜ is a model in the original sense of [Hai14, Def. 2.17].