Paracontrolled quasilinear SPDEs

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https://hal.archives-ouvertes.fr/hal-01615526

Preprint submitted on 12 Oct 2017

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Marco Furlan, Massimiliano Gubinelli

To cite this version:

Marco Furlan, Massimiliano Gubinelli. Paracontrolled quasilinear SPDEs. 2017. �hal-01615526�

M. Furlan CEREMADE

PSL - Université Paris Dauphine, France Email: [email protected]

M. Gubinelli IAM & HCM Universität Bonn, Germany Email: [email protected]

Abstract. We introduce a non-linear paracontrolled calculus and use it to renor- malise a class of singular SPDEs including certain quasilinear variants of the periodic two dimensional parabolic Anderson model.

1. Introduction

We show how to renormalise a class of general quasilinear equations of which one of the simplest examples is the following parabolic SPDE:

@

u(t; x) ¡ a(u(t; x))u(t; x) = (x); u(0; x) = u

(x); x 2 T

; t > 0; (1) with a: R ! [; 1] for > 0 a uniformly bounded C

diusion coecient, and ka

k

6 1 for k = 0; :::; 3. We assume that 2 C

(T

) with 2/3 < < 1 where C

(T

) is the Besov space B

(T

). This would apply to the space white noise on T

, for example. In this case we only expect that u(t; ) 2 C

(T

) and the term a(u(t; ))u(t; ) is not well dened when 2 ¡ 2 < 0. Eq. (1) is a quasilinear generalisation of the twodimensional periodic parabolic Anderson model (PAM).

Let us remark from the start that the framework we will consider below allows to deal with a class of equations of the form

a

(u(t; x))@

u(t; x) ¡ a

(u(t; x))u(t; x) = (a

(u(t; x)); t; x); x 2 T

; t > 0; (2) where a

; a

are suciently smooth non-degenerate coecients and (z; t; x) is a Gaussian process with covariance

E[(z; t; x) (z

; t

; x

)] = F(z; z

)Q(t ¡ t

; x ¡ x

); x; x

2 T

; t; t

; z; z

2 R;

with F a smooth function and Q a distribution of parabolic regularity > ¡ 4 / 3. This includes the space white noise discussed before or a time white noise with a regular depen- dence on the space variable or some noise mildly irregular in space and time.

Also the scalar character of the equation or of the non-linear diusion coecient will not play any specic role and we could consider vectorvalued equations with general diusion coecients provided the template problem (7) below remains uniformly parabolic.

For the sake of clarity and simplicity we will discuss mainly the basic example (1) since this contains already most of the technical diculties. The fact that one can handle models as general as (2) can be considered a direct byproduct of the techniques we will introduce below.

Recently Otto and Weber [33] and Bailleul, Debussche and Hofmanova [7] investigated quasilinear SPDEs in the context of pathwise methods and in a range of regularities com- patible with the ones we will consider in this paper.

In [33] the authors obtained a priori estimates for equations of the form

@

u(t; x) ¡ a(t; x)@

u(t; x) = f (u(t; x))(t; x); (t; x) 2 T

1

where both space and time variables take values in a one dimensional periodic domain and their noise can be white in time but colored in space, essentially behaving like a distribution of parabolic regularity in ( ¡ 4 / 3; 1). In order to do so they introduce a specic notion of modelled function and related estimates.

Bailleul, Debussche and Hofmanová in [7] obtain local well-posedness for the gen- eralised parabolic Anderson model equation

@

u(t; x) ¡ a(u(t; x))u(t; x) = g(u(t; x))(x) t > 0; x 2 T

: (3) The authors obtain the same result as the one presented in Section 6 of our work, without the machinery of nonlinear paraproducts introduced here, but using only the basic tools of paracontrolled analysis and some clever transformations.

On the other hand, we remark that the apparently innocuous vectorial formu- lation of (3)

@

u(t; x) ¡ a

(u(t; x)) @

@x

@x

u(t; x) = g(u(t; x)) t > 0; x 2 T

is out of reach of the techniques used in [7], while can be treated awlessly in our framework.

Let us state one simple result that can be obtained via the theory developed in this paper:

Theorem 1. Fix 2/3 < < 1. Let 2 C

(T

) be a space white noise with zero average on the torus, u

2 C

an initial condition and a: R ! [; 1] for some > 0, a 2 C

(R) and ka

k

6 1 8k 2 0; :::; 3. Let (

; u

)

be a family of smooth approximations to ; u

obtained by convolution with a rescaled smoothing kernel and u

the classical solution to the Cauchy problem

@

u

¡ a(u

)u

=

+

a

(u

)

a(u

)

; u(0) = u

: (4)

Then we can choose the constants (

)

and a random time T > 0 in such a way that the family of r.v. (u

)

L

(T

) almost surely converge as " ! 0 to a random element u 2 L

(T

), where L

is the parabolic space C([0; T ]; C

(T

)) \ C

([0; T ]; C

(T

)).

This element can be characterised as the solution to a paracontrolled singular SPDE (see below for more details).

In order to devise a suitable formulation of eq. (1) and obtain a theory with u 2 C

we decompose the non-linear diusion term in the l.h.s. with the help of Bony's paraproduct [31] and write

@

u ¡ a(u) u = + (u) (5)

with

(u) := a(u) u + a(u) u (6)

where ; are standard paraproducts and denotes the resonant product (see below for

precise denitions). Now the l.h.s. is always well dened irrespective of the regularity of

the function u and the problem becomes that of controlling the resonant product a(u) u

appearing in the r.h.s. . A key point of the analysis put forward below is that this term

can be expected to be of regularity 2 ¡ 2 > ¡ 2 so better than the leading term .

Our approach can be described as follows. For an equation of the form

@

u ¡ a

(u)u = a

(u) ;

we consider at rst a parametric template problem with constant coecients

@

#(; t; x) ¡

#( ; t; x) =

(t; x); (7) where now = (

;

) are xed numbers. A nonlinear paraproduct

will allow us to modulate the parametric solution # with the coecient a(u) = (a

(u); a

(u)) as to capture the most irregular part of the solution u itself. As a consequence, the paracontrolled Ansatz

u =

(a(u); #) + u

will dene a regular remainder term u

which solves a standard PDE. With this decompo- sition the resonant products appearing in the equation can be estimated along the lines of the standard paracontrolled arguments introduced in [15] and all the arguments introduced there can be extended in a straightforward manner to the quasilinear setting.

This approach has been inspired by the parametric controlled Ansatz of Otto and Weber [33]. At variance with their approach we use the parametric Ansatz in the context of the paradierential calculus and consider more general noise terms.

Usefulness of paraproducts in the analysis of non-linear PDEs is by now well estab- lished: see for example the seminal paper of Meyer [31], the early review of Bony [10], the recent books of Alinhac and Gérard [1] and Bahouri, Chemin, and Danchin [3]. Let us mention also the interesting paper of Hörmander [25] where paradierential operators allows to bypass the NashMoser xpoint theorem in some applications where the loss of regularity prevents straightforward use of standard Banach xpoint theorem. The main observation in that paper is that, with the aid of paradierential operators, it is possible to identify a corrected problem for which standard Banach xpoint applies.

Paracontrolled calculus for singular SPDEs has beed introduced by Gubinelli, Imkeller and Perkowski [15] (see also the lecture notes [17]) and used to study various equations like the KPZ equation [16], the dynamic

model [11] in d = 3 and its global wellposed- ness [32], the spectrum of the continuous Anderson Hamiltonian in d = 2 [2]. By using heatsemigroup techniques paracontrolled calculus has been extended to the manifold context by Bailleul and Bernicot [4].

Nonlinear generalisation of the classic bilinear paraproducts already appeared in the notion of paracomposition introduced by Alinhac [10]. Nonlinear versions of rough paths have been considered by one of the authors in order to study the Kortewegde Vries equation [14]. Nonlinear Young integrals were used by one of the authors in joint work with Catellier [12] to study the the regularising properties of sample paths of stochastic processes processes. See also the related work of Hu and Le [24] on dierential systems in Hölder media. Relevant to this discussion of non-linear variants of rough paths is the work of Bailleul on rough ows [8] and their application to homogeneisation [6]. By looking at the composition f( g(x)) as the action of the distribution

on the function f, nonlinear constructions can be linearised at the price of working in innitedimensional spaces:

this is the approach chosen by Kelly and Melbourne to avoid nonlinear generalisations

of rough path theory in their study of homogeneisation of fastslow system with random

initial conditions [26]. It is worth mentioning also Kunita's theory of semimartingale vector

elds [27] which occupy a place in stochastic analysis quite similar to that which these

recent developments occupy in the landscape of rough paths/paracontrolled distributions

theories.

Paracontrolled calculus is currently limited to rst order computations. This limita- tion is also ubiquitous in the present work. Even if, in practice, this is not a big issue, and the calculus is still able to deal with a large class of problems, it makes the paracontrolled approach less appealing for a general theory of singular SPDEs. Let us remark that recently Bailleul and Bernicot [5] developed an higher order version of the paracontrolled calculus.

However, apart from these recent development, whose impact is still to be assessed, the most general theory for singular SPDEs has been developed by Hairer [19, 20, 13] under the name of regularity structures theory. Regularity structures are a vast generalisation of Lyons' rough paths [28, 30, 29] which give eective tools to describe non-linear oper- ations acting on certain spaces of distributions, their renormalization by subtraction of local singularities and their use to solve singular SPDE. Regularity structures have been successfully applied to all the models mentioned so far [19, 18], to other models like the SineGordon model [22] (which however can also be handled via paracontrolled techniques) and to study weak universality conjectures [21, 23]. In their current instantiation it does not seem possible to solve quasilinear SPDEs via regularity structures. The results of the present paper hint to the fact that a non-linear version of regularity structures is conceivable, at least in principle. Indeed one can imagine models depending on additional parameters and modelled distributions acting as evaluations of the parametric models at certain spacetime dependent values of the parameters. It would be interesting to pursue further this intuition.

The structure of the paper is the following. In Section 2 we introduce our basic tools: the non-linear paraproduct decomposition and some related commutation lemmas. In Section 3 we introduce the paracontrolled Ansatz which allows to transform the singular problem (1) into a wellbehaved PDE. In Section 4 we discuss the apriori estimates, the uniqueness of the solution of the transformed PDE and its continuity w.r.t. the random data and the initial condition, we introduce also the algebraic structure which allows to renormalise the model. Section 5 deals with the renormalization of the stochastic data and the construction of the enhanced noise associated to white noise. Section 6 deals with the extension of the results to more general equations, in particular with equation (3) or with noise whose law depends on the solution itself. Finally Appendix A reviews some reference material on Besov spaces and proves some technical lemmas.

Notations. We will denote C

:= B

(T

) the Zygmund space of regularity 2 R on the torus T

. See Appendix A for the denition of the general Besov spaces B

, the LittlewoordPaley operators (

)

and the basic properties thereof needed in this paper. If V is a Banach space and T > 0, we denote C

V the space of Hölder functions in C

V := C([0; T ]; V ). We introduce parabolic spaces L

:= C

C

_\ C

C

with norm

kf k

= kf k

T

/2C⁰

+ kf k

: (8) Moreover for convenience we denote C

:= C

C

. We will avoid to note explicitly the time span T whenever this does not cause ambiguities. We will need also spaces for functions of (; t; x) where is an additional parameter in [; 1] for 2 (0; 1) which we denote C

V with norm

k F k

= sup

2[;1]

sup

n=0;:::;k

k @

F (; ) k

; (9)

where V is a Banach space of functions on [0; T ] T

, in our case V = C

or V = L

.

We will denote by K

(y) = 2

K(2

(x ¡ y)) the kernel of the LittlewoodPaley operator

and Q

(s) = Q

(t ¡ s) = 2

Q(2

(t ¡ s)) a smoothing kernel at scale 2

in the time direction where Q is a smooth, positive function with compact support in R

and mass 1. We introduce also the shortcut P

= K

= P

j <i¡1

K

. Another notation shortcut widely used in this article is to write R

x;y

for integrals on T

or R with respect to the measures dx and dy without specifying the integration bounds, whenever this does not create ambiguity. Finally, we will note f

= f (t; x) ¡ f (s; y) and

f

= f (s;

y) + (f (t; x) ¡ f (s; y)) for 2 [0; 1].

2. Nonlinear paraproducts

In this section we introduce the nonlinear paraproduct and related results that will be used in Section 3 to analyse equation (1).

Let g: [0; T ] T

! R, and h: R [0; T ] T

! R be smooth functions. We can decompose the composition h(g( ); ) via nonlinear paraproducts as follows. Dene

(g; h)(t; x) := X

q

Z

y;z

P

(y)K

(z)h(g(t; y); t; z) (10)

(g; h)(t; x) := X

kq

Z

y;z

K

(y)K

(z)h(g(t; y); t; z) (11)

(g; h)(t; x) := X

k

Z

y;z

K

(z)P

(y)h(g(t; z); t; y) : (12) This gives a map

(g; h) 7!

(g; h) :=

( g; h) +

(g; h) +

(g; h) = h(g( ); ) (13) that can be uniquely extended to

: C

C

C

_! C

2 (0; 1); 2 R; + > 0;

thanks to the following bounds:

Lemma 2. (Nonlinear paraproduct estimates) Let g 2 C

for some 2 (0; 1), g 2 [; 1], and h 2 C

C

for any 2 R. Then

k

(g; h) k

. k h k

; k

(g; h) k

T^(+)

. k g k

k h k

; and

k

(g

; h) ¡

(g

; h)k

. kg

¡ g

k

khk

; k

( g

; h) ¡

(g

; h)k

T

^(+)

. kg

¡ g

k

(kg

k

+ kg

k

)khk

+ k g

¡ g

k

k h k

:

Moreover if + > 0 we have also

k

(g; h) k

. k h k

k g k

;

k

( g

; h) ¡

(g

; h) k

. k g

¡ g

k

( k g

k

+ k g

k

) k h k

+ k g

¡ g

k

k h k

:

In particular if + > 0 the composition

(g; h) = h(g(); ) is linear in h and locally Lipshitz in g:

k

( g; h)k

. khk

kg k

;

k

( g

; h) ¡

(g

; h)k

. kg

¡ g

k

(1 + kg

k

+ kg

k

)khk

: Proof. Using the fact that

k

h(g(t; y); t; ) k

. 2

k h k

;

k

h( g(t; y); t; ) ¡

h(g(t; y

); t; ) k

. 2

k h k

k g k

j y ¡ y

j

; and

k

h(g

(t; y); t; ) ¡

h(g

(t; y); t; ) k

. 2

k h k

k g

¡ g

k

;

we obtain the bounds on

( g; h),

(g; h),

(g; h) and

(g

; h) ¡

(g

; h). We proceed to estimate the term

(g

; h) ¡

( g

; h). We will use the following notation for brevity:

g

: =g

(t; y) ¡ g

(t; z) and

g

: =g

(t; z) + ( g

(t; y) ¡ g

(t; z)):

Then Z

y;z

K

(z)P

( y)[h(g

(t; z); t; y) ¡ h(g

(t; z); t; y)]

= Z

y;z;2[0;1]

K

(z)P

( y)[@

h(

g

; t; y)g

¡ @

h(

g

; t; y)g

] .

Z

y;z;t2[0;1];2[0;1]

K

(z)P

( y)@

h(

(

g

)

; t; y)(g

¡ g

)g

+

Z

y;z;2[0;1]

K

(z)P

(y)@

h(

g

; t; y)(g

¡ g

) . kg

¡ g

k

(kg

k

+ kg

k

)khk

2

X

q<k¡1

2

+ k g

¡ g

k

k h k

2

X

q<k¡1

2

:

With the same reasoning we can bound the norm of

(g

; h) ¡

( g

; h).

We will need the following time-smoothed nonlinear paraproduct

(g; h)(t; x) := X

i

Z

y;s

Q

(s)P

(y)(

h( g(s; y); t; ))(x); (14) with Q 2 C

(R) with total mass 1, and Q

(s) := 2

Q(2

(t ¡ s)) as specied in the introduction. In (14) we use the convention that a continuous function t 7! g(t) on R

is extended to R by dening g(t) = g(0) for t 6 0. This preserves the Hölder norms of index in [0; 1]. The modied nonlinear paraproduct enjoys similar bounds to the regular one.

Lemma 3. Let g 2 C

L

, g 2 [; 1] and h 2 C

L

for 2 (0; 2). Then k

(g; h)k

. khk

and k

( g; h)k

. khk

: Moreover,

(g; h) is linear in h and:

k

(g

; h) ¡

(g

; h) k

. k g

¡ g

k

k h k

;

k

( g

; h) ¡

(g

; h) k

. k g

¡ g

k

k h k

:

Proof. The norm k

(g; h)k

can be treated in the same way as in Lemma 2. We estimate k

(g; h) k

T/2

C⁰

as follows:

k

(g; h)(t

) ¡

(g; h)(t

)k

. sup

x

Z

z

K

(z) X

ij

Z

y;s

Q

(s)P

(y)[

h( g(s; y); t

; z) ¡

h(g(s; y); t

; z)]

+sup

x

Z

z

K

(z) X

ij

Z

y;s

[Q

(s) ¡ Q

(s)]P

(y)

h( g(s; y); t

; z) . kh(; t

; ) ¡ h(; t

; )k

+ jt

¡ t

j

khk

:

The second inequality can be obtained easily with the same techniques used so far.

Remark 4. Using the Fourier support properties of the kernel P

it is easy to see that it has mass 1. Therefore, the paraproducts (10) and (14) when g(t; x) = g is a constant smooth function become 8 (t; x) 2 R T

:

( g ; h)(t; x) = h(g ; t; x) =

(g ; h);

( g ; h)(t; x) = h(g ; t; x):

2.1. Nonlinear commutator.

The next technical ingredient is a commutator lemma between the non-linear parapro- duct of (14) and the standard resonant product. It will be needed below to analyse a term of the form

( g; h)

(g; h), then we will specialise our discussion to this specic structure. Notice that in the following the various spacetime operators act pointwise in the parameter , in the sense that, for example:

(h h)(; t; x) = (h(; t; ) h(; t; ))(x):

Lemma 5. We dene the map : C

([0; T ]; T

) C

C

([0; T ]; T

) ! C

([0; T ]; T

) by (g; h) := [

(g; h)

(g; h)] ¡

( g; h h):

Then for all 2 (0; 1), < R, " > 0 such that 2 ¡ 2 + ¡ " > 0 and g 2 [; 1], we have k(g; h)k

. kgk

khk

and

k (g

; h) ¡ (g

; h) k

. k g

¡ g

k

( k g

k

+ k g

k

) k h k

+ k g

¡ g

k

k h k

:

As a consequence can be uniquely extended to a locally Lipshitz function : L

C

C

_! C

:

Proof. Let A(t; z) :=

(g; h)

( g; h)(t; z) . We can approximate A(t; z) with its value for a xed g = g(t; z), to obtain

A(t; x) = Z

z

K

(z)(

( g; h)

( g; h))(t; z)

= Z

z

K

(z)(

( g(t; z); h)

(g(t; z); h))(t; z) +

Z

z

K

(z)((

(g; h) ¡

(g(t; z); h))

( g(t; z); h))(t; z) (15) +

Z

z

K

(z)(

(g; h) (

(g; h) ¡

( g(t; z); h)))(t; z): (16)

with the remainders (15) and (16). To estimate the rst term notice that

[

(g(t; z); h)](t; z) =

(g(t; z); h)(t; z) and that, by Remark 4

(g(t; z); h)(t; z) = Z

z⁰

K

(z

)

(g(t; z); h)(t; z

)

= Z

z⁰

K

(z

)h(g(t; z); t; z

)

=

h( g(t; z); t; )(z):

This yields

(

(g(t; z); h)

(g(t; z); h))(t; z) = X

ij

( g(t; z); h)

= X

ij

h(g(t; z); t; )(z)

[h(g(t; z); t; )](z)

=

(g; h h)(t; z):

We proceed then to estimate (15) and (16). We obtain Z

z

K

(z)[(

(g; h) ¡

(g(t; z); h))

(g(t; z); h)](t; z)

= Z

z

K

(z) X

ij&q

(

( g; h)(t; z) ¡

(g(t; z); h)(t; z))

(g(t; z); h)(t; z):

Using Lemma 3 we have

j

(g(t; z); h)(t; z)j . 2

khk

: Lemma 6 gives

j

(

(g; h) ¡

(g(t; z); h))(t; z)j . 2

kg k

khk

; and thus (15) is bounded by 2

k g k

k h k

T

2

.

We can easily bound (16) in the same way, and this proves the rst inequality. For the second inequality, Lemma 3 yields

j

(g

; h)(t; z) ¡

(g

; h)(t; z)j . 2

kg

¡ g

k

khk

; and using the second inequality of Lemma 6 we obtain the desired bound.

The extension of to L

C

L

is standard (see e.g. the proof of the commutator

lemma in [15], Lemma 2.4).

Lemma 6. With the same assumptions of Lemma 5 we have

j

(g; h)(t; z) ¡

(g(t; z); h)(t; z) j . 2

k g k

k h k

and

j

(g

; h)(t; z) ¡

(g

(t; z); h)(t; z) ¡

(g

; h)(t; z) +

(g

(t; z); h)(t; z) j . 2

k g

¡ g

k

k h k

+ k g

¡ g

k

( k g

k

+ k g

k

) k h k

Proof.

[

(g; h) ¡

(g(t; z); h)](t; z)

= X

ki

Z

x; y;s;

K

(x) Q

(s)P

(y)@

h(

g

; t; x) ¡

g

+ g

. X

ki

Z

x; y;s;

j K

(x)Q

(s)P

( y) jk @

h k

j t ¡ s j

k g k

T /2¡"/2

L¹

+ X

ki

Z

x; y;s;

j K

(x)Q

(s)P

( y) jk @

h k

j y ¡ z jk g k

. 2

2

k @

h k

k g k

+ k g k

where we used the notation g

= g(s; y) ¡ g(t; z),

g

= g(t; z) + ( g(s; y) ¡ g(t; z)) and Lemma 22. This proves the rst bound.

The second inequality can be obtained in the same way with the techniques already

used here and in Lemma 2.

2.2. Approximate paradierential problem.

In this section we construct an approximate solution to the equation

(@

¡ g )u = f ; u(0; ) = 0; (17) with data f 2 C

and g 2 L

, for some xed ; 2 (0; 1). The idea is to obtain it via a certain class of paradierential operators. We introduce the operator L acting on functions of (; t; x) by

( L U )(; t; x) := @

U (; t; x) ¡ U (; t; x): (18) We will also use the notation L

:= @

¡

with

2 R.

Observe that if u does not depend on we can dene

(g; L )u :=

( g; L u) (19)

and from denition (10) with h = L u we obtain

(g; L )u = @

u ¡ g u.

We can describe the commutation between the dierential operator L and the para- product

(g; ) via the following estimate:

Lemma 7. Let 2 (0; 1), 2 R. Let U 2 C

C

and g 2 L

such that g 2 [; 1]. Dene (g; U) := R

+ R

with R

and R

as in ( 22), ( 23). Then for every " > 0

k (g; U)k

. (1 + kg k

)kgk

kU k

: (20) Moreover, (g; U ) is linear in U and

k (g

; U) ¡ (g

; U ) k

. k g

¡ g

k

(1 + k g

k

+ k g

k

) k U k

: In particular, we have

(g; U ) =

(g; L U ) ¡

(g; L )

(g; U ) 2 C

C

(21)

whenever this expression makes sense.

Proof. We start considering g 2 C

([0; T ]; T

) and U 2 C

C

([0; T ]; T

), and prove (21) in this setting. Notice that

(g(t; y); L

U ) = L U (g(t; y)). As a consequence, we can estimate

(g; L U )(t; x) ¡

( g; L

(g; U ))(t; x)

=

(g; L U )(t; x) ¡ X

k

Z

y

P

(y)( L

(g; U ))(t; x)

=

(g; L U )(t; x) ¡ X

k

Z

y

P

(y)(@

(g; U ))(t; x) + X

k

Z

y

P

( y) g(t; y)(

(g; U ))(t; x)

=

(g; L U ¡ @

U )(t; x) + X

k

Z

y

P

(y) g(t; y)(

(g; U ))(t; x) + X

k

Z

P

(y)g(t; y)(

[;

(g; )]U )(t; x) ¡ X

k

Z

P

(y)(

[@

;

(g; )]U )(t; x) with the commutators

[;

(g; )]U :=

( g; U) ¡

(g; U );

[@

;

(g; )]U := @

(g; U ) ¡

(g; @

U ):

We have

( g; L U ¡ @

U )(t; x) + X

k

Z

y

P

(y)g(t; y)(

(g; U ))(t; x) = R

(t; x) with the denition

R

(t; x) :=

X

k;i

Z

y;zy0;s

P

(y)K

(z)P

( y

) Q

(s)[g(t; y) ¡ g(s; y

)]

U (g(s; y

); t; z) (22) and

X

k

Z

y

P

(y)[g(t; y)(

[;

(g; )]U )(t; x) ¡ (

[@

;

( g; )]U )(t; x)] = R

(t; x) with the denition

R

(t; x) := X

k;i

Z

y;y⁰;s

P

(y)K

(z) Q

(s) g(t; y)P

(y

)

U (g(s; y

); t; z) +2 X

k;i

Z

y;y⁰;s

P

(y)K

(z) Q

(s) g(t; y) r P

(y

) r

U (g(s; y

); t; z)

¡ X

k;i

Z

y;y⁰;s

P

(y)K

(z)@

Q

(s)P

(y

)

U (g(s; y

); t; z): (23) Indeed:

([@

;

( g; )]U )(t; x) = X

i

Z

y;s

(@

Q

)(s)P

( y)(

U ( g(s; y); t; x));

([;

( g; )]U )(t; x) = X

i

Z

y;s

Q

(s)P

(y)(

U (g(s; y); t; x)) (24) +2 X

i

Z

y;s

Q

(s) r P

(y)( r

U (g(s; y); t; x)):

This shows that (21) holds for smooth functions.

With the techniques used in Lemma 6 we can estimate j

R

(t; x)j . X

kq

2

kgk

T

/2C⁰

+ 2

kg k

2

kU k

: By the spectral support properties of the commutators we have that

k [;

(g; )]U k

. k g k

k U k

; k

[@

;

(g; )]U k

.

2

k g k

T/2

C⁰

+ 2

k g k

k U k

: This yields

k R

k

. (1 + k g k

) k g k

k U k

:

We have so far proved (20) and then (21) follows by continuity. The local Lipshitz depen-

dence on g can be obtained via similar computations.

Remark 8. If f does not depend on we consider the parametric problem

(@

¡ )U

(; t) = f ; U

(; 0) = 0; 2 [; 1]; (25) which is solved by

U

(; t) = Z

0 t

e

f ds:

Remark that

@

U

(; t) = Z

0 t

e

(t ¡ s)f ds and @

U

( ; t) = Z

0 t

e

(t ¡ s)

f ds:

We have, thanks to the well-known Schauder estimates of Lemma 21 (since > ):

kU

k

:= sup

n=0;1;2

sup

2[;1]

k@

U

( )k

. kf k

T

¡2

(26)

We dene then

u(t; x) :=

(g; U

)(t; x) (27)

and observe that u(t; x) is an approximate solution of equation (17), indeed

(@

¡ g )u =

(g; L

(g; U

)) =

(g; L U

) ¡ (g; U

) = f ¡ (g; U

) and the estimation in Lemma 7 together with the bound (26) yield immediately the fol- lowing inequality:

k (g; U

) k

. k g k

(1 + k g k

) k f k

: (28) 3. Paracontrolled Ansatz

In order to give a meaning to the PDE in (5) with initial condition u

2 C

, our initial goal will be to get informations on solutions = (g) of the equation

@

¡ g = ;

for a xed g 2 C

, 2/3 < < 1, g 2 [; 1]. Using the results of Section 2.2, we consider to this eect the parametric problem

(@

¡ )#( ; t) = ;

for 2 [; 1]. We will consider the stationary solution of this problem which has the form

#(; x) = Z

0 1

e

ds = (¡)

(29)

and in order for (29) to be well dened we impose that the noise has zero mean on T

(this is a simplifying assumption which can be easily removed, e.g. at the price of adding a linear term to the equation). We can control (29) by bounding its Littlewood-Paley blocks with a Bernstein lemma for distributions with compactly supported Fourier transform ([3], Lemma 2.1) to obtain:

k # k

= k # k

. k k

: (30) We dene now for every t 2 [0; T ]

(t; x) :=

(a(u); #):

Thanks to Lemma 3 we have the bound kk

. k#k

. k k

. We observe that this denition together with Lemma 7 gives

@

¡ a(u) = ¡ (a(u); #)

with k (a(u); #)k

. ka(u)k

k k

. We expect then (a(u); #) to be bounded in C

for any " > 0. At this point let us introduce the Ansatz

u = + u

: (31)

Remark 9. Notice that we are not making any assumption on the existence of such u, which is the subject of Section 4. Our aim here is to nd the equation that a couple (u; u

) 2 C

C

verifying (31) must solve, in order for u to solve (5).

Observe that

@

u ¡ a(u) u = (@

¡ a(u) ) + (@

¡ a(u) )u

= + (@

¡ a(u) )u

¡ (a(u); #):

It follows that u

must solve

( (@

¡ a(u) )u

= (u) + (a(u); #)

u

(t = 0) = u

:= u

¡

(a(u

); #)(t = 0) 2 C

⁽³²⁾ with (u) = a(u) u +a(u) u, and if we can make sense of the resonant term a(u) u, it is reasonable to expect u

(t; ) 2 C

₈ t 2 (0; T ]. Indeed, take U

:= U

to be the solution of

L U

() := (@

¡ )U

() = Q U

(; t = 0) = 0 (33) for some Q = Q(u

) to be determined and 2 [; 1]. Using again Lemma 7 as shown in Remark 8 we have

(@

¡ a(u) )

(a(u); U

) = Q(u

) ¡ (a(u); U

):

For 2 [; 1] we dene P

u

( ) := e

u

so that L ( P

u

) = 0, with L as in (18).

We set

u

:=

(a(u); U

) +

(a(u); P u

) : (34) Taking

Q(u

): =(u) + (a(u); #) + (a(u); U

) + (a(u); P u

) ;