A Whitney extension for modelled distributions

In this subsection we present the solution to a more delicate extension problem.

Suppose we are given a regularity structure T “ pA,T, Gq with a scaling vector s P r1,8q^d equipped with a model pΠ,Γq and satisfying Assumption 2.3.12. Take a sector V Ď T and some γ P R. Suppose further we are given a non-empty set ΩĎR^d and some F P D^γpΩ;V,Γq. Can we find an extension E^ΩF : R^d ÑT such that }EΩF}D^γpR^d;Tq À }F}D^γpΩ;Vq with the involved constant being independent of Ω?

The answer is yes, provided that pV,Tq is γ-regular under some product ‹ on T. We are grateful to M. Hairer for his suggestion to generalize the ideas from [Whi34, Ste70], which we will do from now on.

Since for α P AV the map R^2d Q px, yq ÞÑ Γ^α_yxτ, τ P T and F^α : Ω Ñ Vα are Hölder continuous with some (possibly small) exponent, we can extendF first to Ω such that F P D^γpΩ;Vq with }F}_D^γ_pΩ;Vq “ }F}D^γpΩ;Vq. We then decompose similar as in [Ste70]Ω^c in a countable family of closed sets pQⁿqnPNĎΩ^c, which are in the form ofx`2<sup>´ms</sup>p1`εqr0,1s^d for some varyingxPR^d, mPNand some fixed εą0, such that the following properties are satisfied for nPN ²

diam_sQⁿ «dist_spQⁿ,Ωq, |tn¹ PN|Q_n¹ XQ_n ‰ Hu| ďC , Ω^c“ ď

n¹PN

Qⁿ¹,

where C “Cpdq ą0 is some fixed number, independent of Ω, and where diam_s and dist_s are defined as in Section 2.1. We will denote by pⁿ some arbitrary chosen point pⁿ P Ω such that distspQⁿ,Ωq “ distspQⁿ,Ωq “ distspQⁿ, pⁿq and we will use the setNpΩq:“ tnPN|dist_spQⁿ,Ωq ď 1u.

2In [Ste70] the author uses the isotropic scaling s “ p1, . . . ,1q. However the construction generalizes readily to the anisotropic case.

138 5.3 Singular spaces and extensions There is an adapted partition of unity φⁿ P C_c⁸pR^d;r0,1sq, supp φⁿ Ď Qⁿ such that for every xPΩ^c, kP N^d

ÿ

nPN

φⁿpxq “1, |B^kφⁿpxq| À pdiam_sQⁿq^´|k|s « pdist_spQⁿ,Ωqq^´|k|s. We denote by Φⁿ :“ř

kPN^d 1

k!B^kφⁿX^k the lift of φⁿ to T, so that

}Φⁿ}_D^γ1_pR^d_;Tq À pdist_spQⁿ,Ωqq^´γ1 (5.68) forγ¹ ą0. We then finally define

EΩFpxq:“EΩ^ΓFpxq:“

#F_x^ăγ , xPΩ

ř

nPNpΩqpΓ_xpⁿF_pⁿ ‹Φⁿ_xq^ăγ , xPΩ^c . (5.69) The upper index “Γ” will be omitted if there is no risk of confusion. Note that by the choice of φⁿ{Φⁿ the sum on the right hand side for xP Ω^c is finite in every subspace T^α as the sum actually only runs over the finite set

N_xpΩq:“ tnPNpΩq |xP Qⁿu

since supp Φⁿ Ď Qⁿ. In fact, by our choice of pQⁿqnPN we have |N_xpΩq| ď C with C“Cpdq ą0 as above.

We will sometimes write E^ΩF even if F is defined on some bigger set Ω¹ Ě Ω, by what we then mean EΩpF|Ωq. As usual, we have an abbreviated notation for the components in T^α for αP A, namely

EΩ^αF :“ pEΩFq^α and define the notationEΩ^ăβ “ř

αăβEΩ^α forβ PR. Our main result in this subsection is the following.

Theorem 5.3.16. E^Ω is a continuous linear operator from D^γpΩ;Vq to D^γpR^d;Tq, .i.e.

}EΩF}D^γpR^d;Tq ÀK₁¨ }F}D^γpΩ;Vq, (5.70) where the involved constant is independent of Ω.and where K₁ is some polynomial in }Γ}_γ. Given a second model pΠ,ˆ Γqˆ we have in addition

}EΩ^ΓF;EΩ^ΓˆFˆ}_DγpR^d;T,Γ,Γqˆ ÀK₂¨ p}F; ˆF}_DγpΩ;V,Γ,Γqˆ ` }Γ´Γ}ˆ _γq, (5.71) where K₂ is a polynomial in the norms of F, F ,ˆ Γ and Γ.ˆ

Proof. We first show (5.70), since this step contains all the necessary ideas. We will writeEF instead of E^ΩF and assume without loss of generality that}F}D^γpΩ;Vq ď1.

ForxPR^d we use the notation E^xF :“ pEFqx.

Let us first prove }E^αxF}Tα À1 for α P A, α ăγ and x P R^d. For x P Ω this is obvious. ForxP Ω^c with dist_spx,Ωq ą 1this follows from a rather short estimate:

}Ex^αF}Tα “›

› ÿ

nPNxpΩq

ÿ

µ1`µ2“α

Γ^µ_xp¹nF_pⁿ ˚ pΦⁿ_xq^µ2›

›T_α

(5.68)

À ÿ

nPNxpΩq

ÿ

µ1`µ2“α

}Γ^µ_xp¹nF_pⁿ}T_µ1 ¨dist_spQⁿ,Ωq^´µ2 À1,

where we used in the last step that }Γ^µ_xp¹nF_pⁿ}T_µ1 À1by definition of pⁿ and N_xpΩq and that dist_spQⁿ,Ωq Ádist_spx,Ωq ě 1by definition of Qⁿ.

Let therefore x P Ω^c be such that dist_spx,Ωq ď 1. Pick a x¹ P Ω such that dist_spx,Ωq “ dist_spx,Ωq “ }x´x¹}s. We can then estimate

}Ex^αF}Tα ď ÿ

nPNxpΩq

ÿ

µ1`µ2“α

}Γ^µ_xp¹ⁿpFpⁿ ´Γpⁿx¹Fx¹q ‹ pΦⁿ_xq^µ2}Tα ` }Γ^α_xx1Fx¹}Tα

À ÿ

nPNxpΩq

ÿ

µ1`µ2“α

ÿ

µ1ďβăγ

}pⁿ´x}^β´µ_s ¹}x¹´pⁿ}^γ´β_s dist_spx,Ωq^´µ2 `1

Àdist_spx,Ωq^γ´α`1À1, (5.72)

where we used in the first step ř

nPNxpΩqΦⁿpxq “ ř

nPNΦⁿpxq “ 1 due to 0 ă dist_spx,Ωq ď 1 and in the last step }x¹´pⁿ}s ď }x¹´x}_s` }x´pⁿ}s À dist_spx,Ωq since by our choice ofpQⁿqnPN we have

}x´pⁿ}s Àdistspx,Ωq forn PN_xpΩq.

We now prove for x, y P R^d the estimate }Ey^αF ´Γ^α_yxE^xF}T^α À }y´x}^γ´α_s for α P A, α ă γ. Since for x, y P Ω this is clear, we are left with the following four cases:

1. y PΩ^c, xPΩ, 2. y PΩ, xP Ω^c,

3. x, y P Ω^c with dist_sprx, ys,Ωq ă 2}x´y}_s, 4. x, y P Ω^c with }x´y}_s ď ¹₂dist_sprx, ys,Ωq,

140 5.3 Singular spaces and extensions whererx, ys “x` r0,1s py´xq. Note that we allow in case 3 the distance

dist_sprx, ys,Ωq to be 0, so that rx, ys is allowed to intersect withΩ and there is no implicit assumption on convexity ofΩ^c used here.

For case 1, note that is enough to considerywithdist_spy,Ωq ď1, since otherwise we can apply the boundedness of components of F withinΩ and the definition of a model. Assume first that xP Ωis such that dist_spy,Ωq “ }y´x}_s, we then rewrite forα ăγ

Ey^αF ´Γ^α_yxE^xF “ ÿ

nPNypΩq

ÿ

µ1`µ2“α

Γ^µ_yp¹ⁿFpⁿ‹ pΦⁿ_yq^µ2 ´Γ^α_yxFx

“ ÿ

nPNypΩq

ÿ

µ1`µ2“α

pΓ^µ_yp¹nF_pⁿ ´Γ^µ_yx¹F_xq ‹ pΦⁿ_yq^µ2, (5.73)

where we usedř

nPNypΩqΦⁿpyq “ 1in the second step. Since we have pΓ^µ_yp¹ⁿFpⁿ´ Γ^µ_yx¹F_xq “ ř

µ1ďνăγΓ^µ_yp¹npF_p^νn ´Γ^ν_pnxF_xqwe can bound this expression by ÿ

nPNypΩq

ÿ

µ1`µ2“α

ÿ

µ1ďνăγ

}y´pⁿ}^ν´µ_s ¹}pⁿ´x}^γ´ν_s pdiam_sQⁿq^´µ2 À }y´x}^γ´α_s ,

where we used that }y´pⁿ}s À dist_spy,Ωq “ }y´x}_s, }x´pⁿ}s ď }x´y}_s ` }y´ p<sup>n</sup>}s À }y ´x}<sub>s</sub> and }y´ x}<sub>s</sub> “ dist<sub>s</sub>py,Ωq « diam<sub>s</sub>Q<sup>n</sup> for n P N<sub>y</sub>pΩq. In the general case choose first x<sup>1</sup> P Ω such that dist<sub>s</sub>py,Ωq “ }y ´x<sup>1</sup>}s and split Ey<sup>α</sup>F ´Γ<sup>α</sup><sub>yx</sub>F<sub>x</sub> “ Ey<sup>α</sup>F ´Γ<sup>α</sup><sub>yx</sub>1F<sub>x</sub><sup>1</sup> `Γ^α_yxpΓ_xx¹F_x¹ ´F_xq and the statement follows due to }y´x¹}s “ dist_spy,Ωq ď }y´x}_s,}x´x¹}s ď }x´y}_s` }y´x¹}s À }y´x}_s. For the case 2 observe, similar as above, that it is sufficient to considerxP Ω^c with dist_spx,Ωq ď 1, since otherwise we can use the boundedness of the componentsE^αF, already shown above. Choosey¹ PΩ such thatdist_spx,Ωq “ }x´y¹}s and split

Ey^αF ´Γ^α_yxE^xF “F_y^α´Γ^α_yy1Fy¹ `Γ^α_yxΓxy¹Fy¹´Γ^α_yxE^xF

“F_y^α´Γ^α_yy1F_y¹ `Γ^α_yx ÿ

n:xPQⁿ

ÿ

ν1`ν2ăγ

Γ^ν_xp¹npΓ_pⁿ_y¹F_y¹´F_pⁿq ‹ pΦⁿq^ν_x², (5.74) where we used ř

n:xPQⁿΦⁿ_x “1 in the second step. The desired estimate readily follows. The case 3 is then a consequence of 1 and 2 if we choose ζ P Ω such that dist_spζ,rx, ysq “dist_spΩ,rx, ysq and reshape Ey^αF ´Γ_yxE^xF_x “Ey^αF ´Γ_yζF_ζ ` Γ_yζpF_ζ´Γ_ζxExFq. Let’s now turn to case 4. Note that we now have

distsprx, ys,Ωq « distspx,Ωq «distspy,Ωq «distspQⁿ,Ωq

forn PN_xpΩqYN_ypΩq. First consider pairsx, ywithdist_spy,Ωq ď1anddist_spx,Ωq ď 1. Choose ζ PΩ such thatdist_spζ,rx, ysq “dist_spΩ,rx, ysqand reshape

Ey^αF ´Γ^α_yxE^xF “ ÿ

nPNxpΩqYNypΩq

ÿ

µ1`µ2“α

ÿ

µ1ďνăγ

Γ^µ_yx¹Γ^ν_xpnFpⁿ‹ ppΦⁿ_yq^µ2 ´Γ^µ_yx²pΦⁿ_xq^ăγ´νq (5.75)

p˚q“ ÿ

µ1`µ2“α

ÿ

µ1ďνăγ

ÿ

nPNxpΩqYNypΩq

Γ^µ_yx¹pΓ^ν_xpnFpⁿ´Γ^ν_xζFζq ‹ ppΦⁿ_yq^µ2 ´Γ^µ_yx²pΦⁿ_xq^ăγ´νq (5.76) where we usedγ-regularity of pV,Tqin the first equality and in p˚q the identity

ÿ

nPNxpΩqYNypΩq

Φⁿpxq ´ ÿ

nPNxpΩqYNypΩq

Γ^µ_yx²Φⁿpyq “1´1“0

to sneak in a term Γ^µ_yx¹Γ^ν_xζF_ζ, independent of n. Since Γ^µ_yx¹pΓ^ν_xpnF_pⁿ ´ Γ^ν_xζF_ζq “ Γ^µ_yx¹pΓ^ν_xpnF_pⁿ ´F_x^ν `F_x^ν ´Γ^ν_xζF_ζq can be estimated by }y´x}^ν´µ1}x ´pⁿ}^γ´ν À }y´x}^ν´µ1dist_spx,Ωq^γ´ν « }y´x}^ν´µ1dist_spΩ,rx, ysq^γ´ν we obtain together with }Φⁿ}D^γ´νpR^dq À dist_spQⁿ,Ωq^ν´γ « dist_spΩ,rx, ysq^ν´γ (for n P N_xpΩq YN_ypΩq) the total bound

ÿ

µ1`µ2“α

ÿ

µ1ďνăγ

}y´x}^ν´µ_s ¹dist_spΩ,rx, ysq^γ´νdist_spΩ,rx, ysq^ν´γ}y´x}^γ´ν´µ_s ² À }y´x}^γ´α_s .

It remains to analyze case 4 when either x or y has distance more than 1 from Ω, so that we now have dist_sprx, ys,Ωq ą ¹₂. We can assume that }x´ y}_s ď 1 since otherwise we can simply use the boundedness of the components of EF, but then we have }x´pⁿ}s À1 for all terms in (5.75) that do not vanish. Estimating then}Γ^ν_xpnF_pⁿ}T_ν À1yields for (5.75) the boundř

µ1`µ2“α

ř

µ1ďνăγ }y´x}^ν´µ_s ¹}y´ x}^γ´ν´µ_s ² À }y´x}^γ´α_s , due to }Φⁿ}D^γ´νpR^dqÀ1 forn PN_xpΩq YN_ypΩq.

To prove (5.71) for EF “ EΩ^ΓF, EˆF :“ EΩ^ΓˆF we have to estimate for α P A the objects

}Ex^αF ´Eˆx^αF}Tα, }Ey^αF ´Γ^α_yxExF ´ pEˆy^αF ´Γˆ^α_yxEˆxFq}Tα

The first norm can once more be bounded similar as in (5.72), for the second quantity we use once more (5.73), (5.74) and (5.75)/(5.76) depending on the position of x, y.

142 5.3 Singular spaces and extensions

Schauder theory for singular SPDEs based on paraproducts

In this chapter we give an application of the theory presented in Chapter 5. We want to prove Schauder estimates for singular SPDEs of the form

apDqu:“ pB_t´ppD¹qqu“Fpu, ξq, up0q “ u₀ (6.1) on r0, Ts ˆR^d´1. Here, F is some nonlinearity that might also act non-local on u by depending on derivatives Bxu for instance. The object ppD¹q denotes some homogeneous polynomial of spatial derivatives of even degree θ. Let us denote the Green’s function of the operator apDq byA.

We want to describe the solutions of (6.1) via a regularity structure as in Chapter 2. In [Hai14] similar equations such as (6.1) were considered. We here give a distinct proof of the Schauder estimates for modelled distributions using the machinery we introduced in Chapter 5.

However, we will skip one important step in the solution theory for (6.1) via regularity structures: We will not give a renormalization theory for the considered models as in [Hai14, Section 9] or [BHZ16, CH16]. This is merely out of convenience, as such a step could probably be done in some similar way as in the references but would be quite elaborate without shedding any light on the usefulness of the theory from Chapter 5.

We will therefore instead assume we are already given a model pΠ,Γq on some regularity structure T “ pA,T, Gq and show how to build a map

K:D^rη,γspΩ^T;T,Γq ÑD^{rη`θ,γ`θs}pΩ^T;T,Γq (6.2) that corresponds to integration against the Green’s functionA ofapDqon the level of modelled distributions. The map (6.2) is constructed by the main result of this chapter, Theorem 6.2.3 below. Actually, in order to obtain estimates that can be

143

Im Dokument Refinements of the Solution Theory for Singular SPDEs (Seite 149-156)