M Gubinelli

Samy Tindel
email: samy.tindel@univ-lorraine.fr

Iván Torrecilla
email: itorrecillatarantino@gmail.com

CONTROLLED VISCOSITY SOLUTIONS OF FULLY NONLINEAR ROUGH PDES

Keywords: . 2010 Mathematics Subject Classification. 60G22, 34K50 Rough paths theory, Stochastic viscosity solutions, Stochastic PDEs

 and with the recent results of Friz and coauthors [4] on fully non-linear SPDEs with rough drivers.

Introduction

This note focuses on fully nonlinear rough partial differential equations with general form du = F (t, θ, u, Du, D 2 u) dt

+ d l=1 σ l (t, θ, u, Du) dx l t , (t, θ) ∈ [0, T ] × R n u(0, θ) = α(θ), θ ∈ R n , (1)
where T > 0 is a fixed time horizon, F is a function defined on [0, T ] × R n × R × R n satisfying some suitable smoothness and monotonicity conditions, σ is a smooth and bounded coefficient, and α is a regular enough initial condition. When the noisy driver x is Hölder continuous, recent advances have allowed to solve equation [START_REF] Buckdahn | Pathwise Taylor expansions for Itô random fields[END_REF] in several special cases of interest:

• The case of coefficients F (Du, D 2 u) and σ(Du) is handled in [START_REF] Lions | Fully nonlinear stochastic partial differential equations[END_REF], when the driver x is a d-dimensional Brownian motion. • The article [START_REF] Lions | Fully nonlinear stochastic pde with semilinear stochastic dependence[END_REF] focuses on a dependence of the form F (Du, D 2 u) and σ(u), still in the case of a d-dimensional Brownian motion x. • The situation where x is a general Hölder continuous function generating a rough path, with a function F depending on t, θ, Du, D 2 u and a family of functions σ(x, Du) depending linearly on Du, is treated in [START_REF] Caruana | A (rough) pathwise approach to fully non-linear stochastic partial differential equations[END_REF] and further analysed and applied in [START_REF] Diehl | Backward stochastic differential equations with rough drivers[END_REF][START_REF] Friz | Rough path stability of (semi-)linear SPDEs[END_REF][START_REF] Crisan | Robust Filtering: Correlated Noise and Multidimensional Observation[END_REF][START_REF] Friz | On the splitting-up method for rough (partial) differential equations[END_REF]. Let us also point out that the global strategy in all those papers can be briefly outlined in the following way (though it might be somehow implicit in the series of papers [START_REF] Lions | Fully nonlinear stochastic partial differential equations[END_REF][START_REF] Lions | Fully nonlinear stochastic pde with semilinear stochastic dependence[END_REF]): (i) Start from a regularization x ε of x, for which equation [START_REF] Buckdahn | Pathwise Taylor expansions for Itô random fields[END_REF] is interpreted in the deterministic viscosity sense. Let us call u ε its solution.

(ii) Perform some natural change of variables allowing to transform equation [START_REF] Buckdahn | Pathwise Taylor expansions for Itô random fields[END_REF] driven by x ε into a deterministic type equation whose coefficients depend on x ε , or possibly on a rough path above x ε . Call v ε the solution to the transformed equation, which enjoys two crucial properties: v ε = G 1 (u ε ) on the one hand, and v ε = G 2 (X ε ) on the other hand, where X ε stands for a geometric rough path above x ε . We are obviously very loose about the definition of G 1 , G 2 , but let us mention that G 1 should be invertible, and that G 2 should be continuous with respect to the rough path topology. (iii) With those elements in hand, we now assume that X ε converges to a rough path above x as ε → 0, and the solution to equation ( 1) is defined as u = G -1 1 (G 2 (X)). By its very construction, this way of defining a solution is thus an extension of the viscosity solution for equation [START_REF] Buckdahn | Pathwise Taylor expansions for Itô random fields[END_REF] with a smooth driver.

We propose to step back and wonder what is really meant by a solution to equation (1) when x is a rough signal. More specifically, think of (1) as an integral type equation of the form:

u(t, θ) = α(θ) + t 0 F (r, θ, (u, Du)(r, θ)) dr + d l=1 t 0 σ l (r, θ, (u, Du)(r, θ)) dx l r , (2)
where the integral with respect to x will be understood in the rough path sense. We wish to address the following questions:

• Give a general notion of viscosity solution, based on a suitable class of test functions, which will be called T σ . • Show that this notion of viscosity solution is a generalization of the notion of strong solution.

• Also define some related jets, called P + σ and P - σ , and study their compatibility with the set of test functions.

• In some special cases, observe how equation [START_REF] Buckdahn | Pathwise Taylor Expansions for Random Fields on Multiple Dimensional Paths[END_REF] can be transformed into a PDE with noisy coefficients thanks to rough paths methods, and where our transforms also involve the set T σ and P ± σ . We shall thus observe how noisy fully nonlinear equations can be solved within a framework which mimics the deterministic setting, by looking at the effect of the functions G 1 , G 2 alluded to above on test functions and jets. This gives an alternative way to solve equations like [START_REF] Buckdahn | Pathwise Taylor Expansions for Random Fields on Multiple Dimensional Paths[END_REF], but it should be noticed that our aim here is not to come up with new results in this direction (namely, our examples of application will be those of [START_REF] Caruana | A (rough) pathwise approach to fully non-linear stochastic partial differential equations[END_REF][START_REF] Lions | Fully nonlinear stochastic pde with semilinear stochastic dependence[END_REF]). However, we believe that our article fills a gap in the theory, insofar as it describes a natural notion of solution for viscosity solutions to rough PDEs.

Let us give some hints about the way to define our class of test functions T σ . A natural guess in view of equation ( 2) is to assume that those test functions ψ ∈ T σ satisfy the relation:

ψ t (θ) = α(θ) + t 0 σ l (r, θ, ψ r , Dψ r (θ)) dx l r + t 0 ψ t r (θ) dr,
for a given continuous function ψ t . However, this natural guess raises the problem of a proper definition of the rough integral appearing in the r.h.s.. A well-suited framework to settle this issue is that of (strongly) controlled paths as introduced in [START_REF] Gubinelli | Controlling rough paths[END_REF]. Controlled paths are functions whose increments "looks like" those of x (or more generally that can be expanded along a basis of well behaved even if irregular objects). In our particular case the expansion for ψ is specified in a way allowing to take also the drift ψ t into account. With this structure in mind, our main task will be the following: (i) Obtain some space-time Taylor type expansions for processes like ψ, which allows to relate the set of test functions T σ and the jets P ± σ . (ii) Describe the composition of ψ with a noisy flow driven by x, in order to define transformations of T σ into a space of test functions corresponding to a deterministic viscosity equation. These steps will be detailed in the remainder of the paper, but let us mention at this point that our analysis is restricted here to a noise x which can be lifted to a rough path with Hölder regularity greater than 1/3. This is made for sake of clarity, but rougher situations could be handled at the cost of higher order expansions.

Let us point out also that the approach we follow in defining suitable spaces of test functions or jets is quite natural. In the series of papers [1, 2, 3] Buckdahn, Ma and coauthors develop a similar theory using stochastic integration and stochastic space-time Taylor series. Even comparing with these works the controlled approach has the great advantage of not requiring any condition of adaptedness or suitable stopping time to identify the extremal points needed in the viscosity formulation. This simplifies a lot the statement of the results and renders them quite similar to the deterministic theory while at the same time going well beyond the semi-martingale setting.

Here is a sketch of our paper: Section 2 is devoted to recall some basic notions of algebraic integration and rough flows. Section 3 deals with general definitions for viscosity solutions to rough PDEs. We then focus on two particular cases: transport equations at Section 4 and semilinear stochastic dependence at Section 5. Notations: throughout the paper we use the summation convention over repeated indices. For jets, the notation P ± stands for the fact that either P + or P -can be considered in the statement.

Algebraic integration and rough paths equations

2.1. Increments. In this section we present the basic algebraic structures which will allow us to define pathwise integrals with respect to functions of unbounded variation, the reader being referred to [START_REF] Gubinelli | Controlling rough paths[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF] for further details.

For real numbers 0 ≤ a ≤ b ≤ T < ∞, a vector space V and an integer k ≥ 1 we denote by

C k ([a, b]; V ) the set of functions g : [a, b] k → V such that g t 1 •••t k = 0 whenever t i = t i+1 for some 1 ≤ i ≤ k -1.
Such a function will be called a (k -1)-increment, and we will set

C * ([a, b]; V ) = ∪ k≥1 C k ([a, b]; V
). An important operator for our purposes is given by

δ : C k ([a, b]; V ) → C k+1 ([a, b]; V ), (δg) t 1 •••t k+1 = k+1 i=1 (-1) k-i g t 1 ••• ti •••t k+1 , (3)
where ti means that this argument is omitted. A fundamental property of δ is that δδ = 0, where δδ is considered as an operator from

C k ([a, b]; V ) to C k+2 ([a, b]; V ). We will denote ZC k ([a, b]; V ) = C k ([a, b]; V ) ∩ Kerδ and BC k ([a, b]; V ) = C k ([a, b]; V ) ∩ Imδ.
Some simple examples of actions of δ are as follows:

For g ∈ C 1 ([a, b]; V ), h ∈ C 2 ([a, b]; V ) and f ∈ C 3 ([a, b]; V ) we have (δg) st = g t -g s , (δh) sut = h st -h su -h ut and (δf ) suvt = f uvt -f svt + f sut -f suv for any s, u, v, t ∈ [a, b]. Furthermore, it is easily checked that ZC k+1 ([a, b]; V ) = BC k ([a, b]; V ) for any k ≥ 1.
In particular, the following property holds:

Lemma 2.1. Let k ≥ 1 and h ∈ ZC k+1 ([a, b]; V ). Then there exists a (non unique) f ∈ C k ([a, b]; V ) such that h = δf .
Observe that Lemma 2.1 implies in particular that all elements h ∈ C 2 ([a, b]; V ) with δh = 0 can be written as h = δf for some f ∈ C 1 ([a, b]; V ). Thus we have a heuristic interpretation of δ| C 2 ([a,b];V ) : it measures how much a given 1-increment differs from being an exact increment of a function, i.e., a finite difference.

Our further discussion will mainly rely on k-increments with k ≤ 2. We show now how to measure the size of those increments thanks to Hölder norms, and we shall thus specify a little the kind of space V considered for the resolution of equation ( 1): Notation 2.2. For our future consideration, the space V will be either R n , either a space of functions defined on R n with a certain type of regularity like bounded uniformly continuous (BUC) functions, Sobolev spaces W k,p (R n ) or C 2 functions. In case of function valued increments, the dependence in the space variable will often be omitted, and we will write ϕ t instead of ϕ t (ξ) for the value of the increment ϕ t at a point ξ ∈ R n . The norms on our state spaces are all denoted by | • |.

Let us now start the introduction of our Hölder type norms with 1-increments:

for f ∈ C 2 ([a, b]; V ) we set f µ = sup s,t∈[a,b] |f st | |t -s| µ , and C µ 2 ([a, b]; V ) = {f ∈ C 2 ([a, b]; V ); f µ < ∞} .
Observe now that the usual Hölder spaces C µ 1 ([a, b]; V ) are determined in the following way: for a continuous function g ∈ C 1 ([a, b]; V ) set

g µ = δg µ , and C µ 1 ([a, b]; V ) = {f ∈ C 1 ([a, b]; V ); f µ < ∞} . Note that • µ is only a semi-norm on C 1 ([a, b]; V
), but we will work in general on spaces of the type

C µ 1,α ([a, b]; V ) = {g : [a, b] → V ; g a = α, g µ < ∞} , for a given α ∈ V, on which g µ is a norm. For h ∈ C 3 ([a, b]; V )
we define in the same way

h γ,ρ = sup s,u,t∈[a,b] |h sut | |u -s| γ |t -u| ρ (4)
h µ = inf i h i ρ i ,µ-ρ i ; (ρ i , h i ) i∈N with h i ∈ C 3 ([a, b]; V ), i h i = h, 0 < ρ i < µ .
Then • µ is a norm on C 3 ([a, b]; V ), see [START_REF] Gubinelli | Controlling rough paths[END_REF], and we define

C µ 3 ([a, b]; V ) := {h ∈ C 3 ([a, b]; V ); h µ < ∞} . Eventually, let C 1+ 3 ([a, b]; V ) = ∪ µ>1 C µ 3 ([a, b]; V
) and note that the same kind of norms can be considered on the spaces ZC 3 ([a; b]; V ), leading to the definition of the spaces ZC µ 3 ([a; b]; V ) and ZC 1+ 3 ([a, b]; V ). We shall also use a supremum norm on spaces C k , which will be denoted by • ∞ in all cases.

The crucial point in this algebraic approach to the integration of irregular paths is that the operator δ can be inverted under mild smoothness assumptions. This inverse is called Λ. The proof of the following proposition may be found in [START_REF] Gubinelli | Controlling rough paths[END_REF], and in a simpler form in [START_REF] Gubinelli | Rough evolution equations[END_REF].

Proposition 2.3. There exists a unique linear map Λ :

ZC 1+ 3 ([a, b]; V ) → C 1+ 2 ([a, b]; V ) such that δΛ = Id ZC 1+ 3 ([a,b];V ) and Λδ = Id C 1+ 2 ([a,b];V ) . In other words, for any h ∈ C 1+ 3 ([a, b]; V ) such that δh = 0, there exists a unique g = Λ(h) ∈ C 1+ 2 ([a, b]; V ) such that δg = h. Furthermore, for any µ > 1, the map Λ is continuous from ZC µ 3 ([a, b]; V ) to C µ 2 ([a, b]; V ) and we have Λh µ ≤ 1 2 µ -2 h µ , h ∈ ZC µ 3 ([a, b]; V ). (5)
This mapping Λ allows to construct a generalized Young integral:

Corollary 2.4. For any 1-increment g ∈ C 2 ([a, b]; V ) such that δg ∈ C 1+ 3 ([a, b]; V ) set δf = (Id -Λδ)g. Then δf st = lim |Πst|→0 n i=0 g t i t i+1
for a ≤ s < t ≤ b, where the limit is taken over any partition Π st = {t 0 = s, . . . , t n = t} of [s, t], whose mesh tends to zero. Thus, the 1-increment δf is the indefinite integral of the 1-increment g.

We also need some product rules for the operator δ. For this recall the following convention:

for g ∈ C n ([a, b]; R l,d ) and h ∈ C m ([a, b]; R d,p ) let gh be the element of C n+m-1 ([a, b]; R l,p ) defined by (gh) t 1 ,...,t m+n-1 = g t 1 ,...,tn h tn,...,t m+n-1 , (6)
for t 1 , . . . , t m+n-1 ∈ [a, b].

Proposition 2.5. On the spaces of increments C 1 , C 2 , the following relations hold true:

(i) Let g ∈ C 1 ([a, b]; R l,d ) and h ∈ C 1 ([a, b], R d ). Then gh ∈ C 1 (R l ) and δ(gh) = δg h + g δh. (ii) Let g ∈ C 1 ([a, b]; R l,d ) and h ∈ C 2 ([a, b]; R d ). Then gh ∈ C 2 ([a, b]; R l ) and δ(gh) = -δg h + g δh.
Remark 2.6. Proposition 2.5 will also be applied for products gh where both g and h are function valued increments. In this case the spatial product will be understood as a pointwise product, namely [gh](ξ) = g(ξ)h(ξ).

2.2.

Weakly controlled processes. In this section we define the kind of generalized integral we wish to consider in order to give a meaning to equation (2). This notion is based on the concept of weakly controlled process, which we proceed to recall. Let us start with some additional notation and assumptions: first notice that throughout the paragraph we will use both the notations t s f dg and J st (f dg) for the integral of a function f with respect to a given function g on the interval [s, t]. Then recall that the basic assumption we shall use on our driving noise x is the following: Hypothesis 2.7. Let γ be a constant greater than 1/3. The R d -valued γ-Hölder path x admits a Lévy area, i.e. a process

x 2 = J (dxdx) ∈ C 2γ 2 ([0, T ]; R d,d ), which satisfies δx 2 = δx ⊗ δx, that is δx 2;ij sut = δx i su δx j ut , for all s, u, t ∈ [0, T ], i, j ∈ {1, . . . , d}. The couple x = (x, x 2
) is called rough path above x, and we also assume that x is a geometrical rough path, meaning that if we denote by x 2,s the symmetric part of x 2 (namely

x 2,s = 1 2 (x 2 + (x 2 ) * ))
, then the following rule holds true for 0 ≤ s < t ≤ T :

x 2,s st = 1 2 δx st ⊗ δx st . (7)
As mentioned above, we will be concerned with weakly controlled processes, that is paths whose increments can be expressed in a simple way in terms of the increments of x. More specifically:

Definition 2.8. Let a ≤ b ≤ T and let z be a path in C κ 1 ([a, b]; V m ) with κ ≤ γ and 2κ + γ > 1. We say that z is a weakly controlled path based on x, if δz ∈ C κ 2 ([a, b]; V m ) can be decomposed into δz i = z x;il δx l + ρ i , ( 8
) with z x;il ∈ C κ 1 ([a, b]; V ) and ρ i ∈ C 2κ 2 ([a, b]; V ), for any 1 ≤ i ≤ m, 1 ≤ l ≤ d.
The space of weakly controlled paths on [a, b] with Hölder continuity κ will be denoted by

Q κ x ([a, b]; V m ), and a path z ∈ Q κ x ([a, b]; V m ) should be considered in fact as a couple (z, z x ). The norm on Q κ x ([a, b]; V m ) is given by N [z; Q κ x ([a, b]; V m )] = z κ + z x ∞ + z x κ + ρ 2κ .
Let us first see how smooth functions depending on time act on weakly controlled paths, in a proposition which is a mere variation of [START_REF] Gubinelli | Controlling rough paths[END_REF]:

Proposition 2.9. Let z ∈ Q κ x ([a, b]; V m 1 ) with decomposition (8) and initial condition z 0 = α, and G ∈ C τ,2 b ([a, b] × R m 1 ; R m 2 ) for τ > 0. Set ẑt = G(t, z t ), with initial condition α = G(a, α).
Then the increments of ẑ can be decomposed into

δẑ st = ẑx s δx st + ρ + δG(•, z t ) st with ẑx s = ∇ z G(s, z s )z x s and ρst = ∇ z G(s, z s )ρ st + [(δG(s, z)) st -∇ z G(s, z s )(δz) st ] . Furthermore ẑ ∈ Q κ x ([a, b]; V m 2 ), and
N [ẑ; Q κ x ([a, b]; V m 2 )] ≤ c G 1 + N 2 [z; Q κ x ([a, b]; V m 1 )] . (9)
We now turn to a proposition giving an expression for the integration of weakly controlled paths with respect to x, when an additional term involving Lebesgue integrals is present. This is again a mere variation on the results contained in [START_REF] Gubinelli | Controlling rough paths[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF], for which we introduce first some additional notation:

Notation 2.10. Let η be a path in C 0 1 ([a, b]; V ), where C 0 1 ([a, b]; V )
stands for the space of Vvalued continuous and bounded functions defined on [a, b]. For all a ≤ s ≤ t ≤ b we will write I st (η) for the Lebesgue integral t s η u du. We also write t for the identity function on R, whose increments are simply given by δt st = ts. We thus trivially have

I st (η) = η s δt st + o(|t -s|).
Proposition 2.11. For a given γ > 1/3 and κ < γ, let x be a process satisfying Hypothesis 2.7. Furthermore, let

η ∈ C 0 1 ([a, b]; V ), and µ ∈ Q κ x ([a, b]; V d
) whose increments can be decomposed as

δµ l = µ x;lk δx k + ρ l ,
where

µ x ∈ C κ 1 ([a, b]; V d,d ), ρ ∈ C 2κ 2 ([a, b]; V d ). ( 10
)
Define z by z a = α ∈ V and δz = µ l δx l + µ x;lk x 2;kl + I(η) + Λ(ρ l δx l + δµ x;lk x 2;kl ).

Then:

(i) z is well-defined as an element of Q κ x ([a, b]; V ), and δz st coincides with the Lebesgue-Stieljes integral

t s µ i u dx i u + t s η u du whenever x is a differentiable function. (ii) The semi-norm of z in Q κ x ([a, b]; V ) can be estimated as N [z; Q κ x ([a, b]; V )] (12)
≤ c x 1 + β + (b -a) γ-κ β + N [µ; Q κ,β ([a, b]; V d )] + N [η; C 0 1 ([a, b]; V )]
, where β ≡ µ a , and where the constant c x can be bounded as follows:

c x ≤ c[|x| γ + |x 2 | 2γ ], for a universal constant c. (iii) It holds δz st = lim |Πst|→0 n q=0 µ l tq δx l tq,t q+1 + η tq δt tq,t q+1 + µ x;lk tq x 2;kl tq,t q+1 (13)
for any a ≤ s < t ≤ b, where the limit is taken over all partitions Π st = {s = t 0 , . . . , t n = t} of [s, t], as the mesh of the partition goes to zero.

Remark 2.12. A lot of the considerations below will be based on differentiation arguments under signs. They can be justified easily in natural situations e.g. when µ is a controlled process taking values in a space of differentiable functions of the form V = W k,p with k ≥ 1.

In this context, let z be defined by [START_REF] Friz | Multidimensional dimensional processes seen as rough paths[END_REF]. Then Dz verifies: δDz = Dµ l δx l + Dµ x;lk x 2;kl + I (Dη) + Λ Dρ l δx l + δDµ x;lk x 2;kl .

With these considerations in hand we get in particular a space-time expansion for the process z in the case of a space V of differentiable functions. Corollary 2.13. Assume the hypothesis of Proposition 2.11 hold true, and that V = C 2 . Then

z t (ν) -z s (θ) = η s (θ) δt st + µ l s (θ) δx l st + µ x;lk s (θ) x 2;kl st + Dz s (θ) [ν -θ] + Dµ l s (θ) δx l st [ν -θ] + R ((s, θ), (t, ν)) ,
for all s, t ∈ [a, b] and any θ, ν ∈ R n , where

|R ((s, θ), (t, ν))| ≈ o (|t -s| + |ν -θ|) .
Proof. We shall prove the expansion in the case a ≤ s < t ≤ b, the proof in the case a ≤ t < s ≤ b being very similar. Fix then a ≤ s < t ≤ b and θ, ν ∈ R n . We write

z t (ν) -z s (θ) = z t (ν) -z t (θ) + z t (θ) -z s (θ). (15)
The time increment z t (θ)z s (θ) can be treated thanks to Proposition 2.11 and we get

z t (θ) -z s (θ) = µ l s (θ) δx l st + η s (θ) δt st + µ x;lk s (θ) x 2;kl st + R 1 ((s, θ), (t, ν)) , (16)
where

R 1 ((s, θ), (t, ν)) = Λ st ρ l (θ) δx l + δµ x;lk (θ) x 2;kl + o(|t -s|), which yields |R 1 ((s, θ), (t, ν))| ≈ o(|t -s|).
Going back to [START_REF] Hairer | Regularity of Laws and Ergodicity of Hypoelliptic SDEs Driven by Rough Paths[END_REF], the space increment z t (ν)z t (θ) is now handled with usual Taylor arguments:

z t (ν) -z t (θ) = Dz t (θ)[ν -θ] + 1 2 D 2 z t (θ + λ(ν -θ))[ν -θ] ⊗ [ν -θ], (17)
where λ ∈]0, 1[ is a given value. Invoking now relation [START_REF] Gubinelli | Rough evolution equations[END_REF] we can write [START_REF] Lions | Fully nonlinear stochastic partial differential equations[END_REF] in the following form:

z t (ν) -z t (θ) = Dz s (θ)[ν -θ] + (δDz(θ)) st [ν -θ] + 1 2 D 2 z t (θ + λ(ν -θ))[ν -θ] ⊗ [ν -θ] = Dz s (θ)[ν -θ] + Dµ l s (θ) (δx l ) st [ν -θ] + R 2 ((s, θ), (t, ν)) , (18)
where R 2 ((s, θ), (t, ν)) can be decomposed as

Dη s (θ) δt st [ν -θ] + Dµ x;lk s (θ) x 2;kl st [ν -θ] + 1 2 D 2 z t (θ + λ(ν -θ))[ν -θ] ⊗ [ν -θ] + Λ st Dρ l (θ) δx l + δDµ x;lk (θ) x 2;kl [ν -θ] + o(|t -s|)[ν -θ].
We thus easily get the following estimate:

|R 2 ((s, θ), (t, ν))| ≈ o (|t -s| + |ν -θ|) , (19)
where we notice that the term x 2;kl st [νθ] is bounded thanks to the elementary inequality ab ≤ 1 2 (a 2 + b 2 ), valid for all a, b ∈ R. Let us now plug the decompositions ( 16) and ( 18) into expression [START_REF] Hairer | Regularity of Laws and Ergodicity of Hypoelliptic SDEs Driven by Rough Paths[END_REF]. This finishes the proof of our claim.

Strongly controlled processes.

In order to identify solutions to equations of type [START_REF] Buckdahn | Pathwise Taylor Expansions for Random Fields on Multiple Dimensional Paths[END_REF] we shall include the drift term into the definition of controlled processes, and thus push forward our expansion with respect to the increments of our rough signal x. This yields the following definition:

Definition 2.14. Let a ≤ b ≤ T and let z be a path in C κ 1 ([a, b]; V m ) with κ ≤ γ and 2κ + γ > 1. We say that z is a strongly controlled path based on x, if δz ∈ C κ 2 ([a, b]; V m ) can be decomposed into δz i = z x;il δx l + z xx;ilk x 2;kl + I z t;i + z ♭;i , ( 20
)
with z x;il , z xx;ilk ∈ C κ 1 ([a, b]; V ), z t;i ∈ C 0 1 ([a, b]; V ), z ♭;i ∈ C 3κ 2 ([a, b]; V ), for any 1 ≤ i ≤ m, 1 ≤ l ≤ d
and where we assume the further relation between z x and z xx :

δz x;il = z xx;ilk δx k + z ♯;il , with z ♯;il ∈ C 2κ 2 ([a, b]; V ). (21)
The space of strongly controlled paths on [a, b] with Hölder continuity κ will be denoted by S κ

x ([a, b]; V m ), and a path z ∈ S κ x ([a, b]; V m ) has to be considered as a vector (z, [START_REF] Hairer | Regularity of Laws and Ergodicity of Hypoelliptic SDEs Driven by Rough Paths[END_REF]. The definition of strongly controlled processes in case of a driving noise x ∈ C γ with 1/3 < γ ≤ 1/2 corresponds in fact to the definition of weakly controlled processes corresponding to x ∈ C γ with 1/4 < γ ≤ 1/3. Further information on these structures can be found in [START_REF] Cass | Hörmander's theorem for Gaussian rough differential equations[END_REF][START_REF] Gubinelli | Ramification of rough paths[END_REF][START_REF] Tindel | Some differential systems driven by a fBm with Hurst parameter greater than 1/4[END_REF].

z x , z xx , z t ). The norm on S κ x ([a, b]; V m ) is given by N [z; S κ x ([a, b]; V m )] = z κ + z x ∞ + z x κ + z xx ∞ + z xx κ + z t ∞ + z ♭ 3κ + z ♯ 2κ . Remark 2.
The space of V m -valued strongly controlled processes is obviously a subset of the weakly controlled processes Q κ

x ([a, b]; V m ). It turns out that this subset has a degenerate structure, in the sense that strongly controlled processes are in fact integrals of weakly controlled processes with respect to x. This is the meaning of the following proposition: Proposition 2.16. Let z be a strongly controlled process as in Definition 2.14. Then (i) The increment z ♭ is uniquely determined by the other components of the strongly controlled process z.

(ii) for all s, t ∈ [0, T ] we have

δz i st = t s z x;il u dx l u + t s z t;i u du,
where the integral with respect to x is understood in the sense of Proposition 2.11.

Proof. Compute δz ♭;i . According to equation ( 20) and Proposition 2.5, plus the fact that δI(η) = 0, we have δz ♭;i = δ z iz x;il δx lz xx;ilk x 2;lk = δz x;il δx l + δz xx;ilk x 2;klz xx;ilk δx 2;kl .

Invoking now relation [START_REF] Tindel | Some differential systems driven by a fBm with Hurst parameter greater than 1/4[END_REF] and Hypothesis 2.7 we obtain δz ♭;i = z xx;ilk δx k + z ♯;il δx l + δz xx;ilk x 2;klz xx;ilk δx k δx l = z ♯;il δx l + δz xx;ilk x 2;kl , and it is thus easily seen that δz ♭;i ∈ C 3κ 3 . Owing to Proposition 2.3, we can thus write z ♭;i = Λ δz xx;ilk x 2;kl + z ♯;il δx l , which yields our first assertion.

In order to prove our claim (ii), it is thus sufficient to observe that the decomposition of δz i and formula [START_REF] Friz | Multidimensional dimensional processes seen as rough paths[END_REF] coincide up to increments of order 3κ > 1.

The following proposition proves that the composition of two function-valued strongly controlled processes is still a strongly controlled process. This fact will be used in order to define transformations of test functions and jets for viscosity solutions to rough PDEs. Proposition 2.17. Let x be a path satisfying Hypothesis 2.7, and consider two C 3 (R m ; R m )valued strongly controlled processes y, z admitting a decomposition of type [START_REF] Norris | Simplified Malliavin calculus[END_REF]. Then z • y is another strongly controlled process with decomposition

[z • y] x;il 1 (θ) = z x;il 1 (y(θ)) + ∂ θ j z i (y(θ)) y x;jl 1 (θ), [z • y] xx;il 1 l 2 (θ) = z xx;il 1 l 2 (y(θ)) + ∂ θ j z i (y(θ)) y xx;jl 1 l 2 (θ) + ∂ θ j z x;il 2 (y(θ)) y x;jl 1 (θ) + ∂ θ j z x;il 1 (y(θ)) y x;jl 2 (θ) + ∂ 2 θ j 1 θ j 2 z i (y(θ)) y x;j 1 l 1 (θ) y x;j 2 l 2 (θ), and
[z • y] t;i (θ) = z t;i (y(θ)) + ∂ θ j z i (y(θ)) y t;j (θ).
Proof. Write

z t (y t (θ)) -z s (y s (θ)) = [z t (y t (θ)) -z t (y s (θ))] + [z t (y s (θ)) -z s (y s (θ))] ≡ S st (θ) + T st (θ),
and deal with the two terms S st (θ) and T st (θ) separately. Note that in the remainder of the proof we shall denote by r ♭ any remainder of order C 1+ , independently of its particular value.

In order to handle S st (θ), we invoke the fact that z can be differentiated in the space variable plus Remark 2.12, which yield:

S i st (θ) = ∂ θ j z i t (y s (θ)) δy j st (θ) + 1 2 ∂ 2 θ j 1 θ j 2 z i t (y s (θ)) δy j 1 st (θ) δy j 2 st (θ) + r ♭ st . ( 22
)
We now plug the decomposition

δy j st (θ) = y x;jl 1 s (θ) δx l 1 st + y xx;jl 1 l 2 s (θ) x 2;l 2 l 1 st + I st y t;j (θ) + r ♭ st (θ) into (22), replace ∂ θ j z i t (y s (θ)) by ∂ θ j z i s (y s (θ)) and ∂ 2 θ j θ k z i t (y s (θ)) by ∂ 2 θ j θ k z i s (y s (θ))
and only keep track of terms whose Hölder regularity is < 1, which gives

S i st (θ) = ∂ θ j z i s (y s (θ)) y x;jl 1 (θ) δx l 1 st + 1 2 ∂ 2 θ j 1 θ j 2 z i s (y s (θ)) y x;j 1 l 1 (θ)y x;j 2 l 2 (θ) δx l 1 st δx l 2 st + ∂ θ j δz i st (y s (θ)) y x;jl 1 (θ) δx l 1 st + ∂ θ j z i s (y s (θ)) y xx;jl 1 l 2 s (θ) x 2;l 2 l 1 st + I st ∂ θ j z i (y(θ)) y t;j (θ) + r ♭ st .
Next resort to the decomposition of δ∂ θ j z given by [START_REF] Norris | Simplified Malliavin calculus[END_REF] and also invoke the geometric assumption [START_REF] Diehl | Backward stochastic differential equations with rough drivers[END_REF] for the term δx l 1 st δx l 2 st . As the reader might easily check, this yields S x;il 1 (θ) = ∂ θ j z i (y(θ)) y x;jl 1 (θ), S t;i (θ) = ∂ θ j z i (y(θ)) y t;j (θ) and

S xx;il 1 l 2 (θ) = ∂ θ j z i (y(θ)) y xx;jl 1 l 2 (θ) + ∂ θ j z x;il 2 (y(θ)) y x;jl 1 (θ) + ∂ θ j z x;il 1 (y(θ)) y x;jl 2 (θ) + ∂ 2
θ j 1 θ j 2 z i (y(θ)) y x;j 1 l 1 (θ) y x;j 2 l 2 (θ). The term T st (θ) gives the remaining contributions of our claim, by just composing relation [START_REF] Norris | Simplified Malliavin calculus[END_REF] with y. It is also easy to see that relation [START_REF] Tindel | Some differential systems driven by a fBm with Hurst parameter greater than 1/4[END_REF] between

[z • y] x and [z • y] xx holds.
The previous proposition yields a space-time expansion for the composition z•y of strongly controlled processes, which will be used in the definition of jets.

Corollary 2.18. Assume the same hypotheses as in Proposition 2.17. Then

[z • y] i t (ν) -[z • y] i s (θ) = [z • y] t;i (θ) δt st + [z • y] x;il 1 (θ) δx l st + [z • y] xx;il 1 l 2 (θ) x 2;kl st + D[z • y] i s (θ) [ν -θ] + D[z • y] x;il 1 (θ) δx l st [ν -θ] + o (|t -s| + |ν -θ|) , for all 1 ≤ i ≤ n, s, t ∈ [a, b] and any θ, ν ∈ R n .
Proof. It is a direct application of Proposition 2.17 and Corollary 2.13.

2.4.

Preliminary results on rough flows. The existence of the rough flow φ generated by a family of vector fields A 1 , . . . , A d is a well-known result, which can be stated as follows:

Proposition 2.19. Let x be a path satisfying Hypothesis 2.7. For η ∈ R m and a family of smooth and bounded vector fields A 1 , . . . , A d , consider the unique solution φ t (θ) to the following rough equation:

φ t (η) = η + d l=1 t 0 A l (φ s (η)) dx l s . (23)
Then:

(i) For any t ∈ R + , φ t is a C 3 -diffeomorphism.
(ii) φ(η) is well defined as a strongly controlled process in S γ x (R m ), with decomposition:

φ x;il 1 = A i l 1 • φ, φ xx;il 1 l 2 = [A l 2 A i l 1 ] • φ, φ t;i = 0,
where we have used the interpretation of our vector fields A l as derivative operators. Namely, for a smooth function g defined on R m we set

A l g = A k l ∂ θ k g and in particular A l 2 A i l 1 (θ) = A k l 2 (θ)∂ θ k A i l 1 (θ)
. We now give the equivalent of Proposition 2.19 for the inverse of the flow φ.

Proposition 2.20. Let x be a path satisfying Hypothesis 2.7. For η ∈ R m and the family of vector fields A 1 , . . . , A d of Proposition 2.19, consider the unique solution ζ t (η) to the following rough equation:

ζ t (η) = η - d l 1 =1 t 0 A l 1 ζ s (η) dx l 1 s , (24)
where

A l 1 ζ s stands for A i l 1 ∂ θ i ζ s . Then: (i) ζ(η)
is well defined as a strongly controlled process in S γ

x (R m ), with decomposition:

ζ x;il 1 = -A l 1 ζ i , ζ xx;il 1 l 2 = A l 1 A l 2 ζ i , ζ t;i = 0,
where we have used the interpretation of our vector fields A l as derivative operators and

A l 1 A l 2 g = A j 1 l 1 ∂ θ j 1 A j 2 l 2 ∂ θ j 2 g + A j 1 l 1 A j 2 l 2 ∂ 2 θ j 1 θ j 2 g.
(ii) For all t ∈ [0, T ] we have ζ t = φ -1 t , where φ is the solution to (23). Proof. Item (i) can be thought of as an easy exercise, while item (ii) can be proven exactly as in [START_REF] Kunita | Stochastic differential equations and stochastic flows of diffeomorphisms[END_REF].

Viscosity solutions to rough PDEs

We now go back to equation (1) in quite a general context. However, for sake of clarity, we shall restrict our analysis to the case of a dependence of F and σ on u and Du only. Namely, we consider a general equation of the form:

u t (θ) = α(θ) + t 0 F (u r (θ), Du r (θ)) dr + d l=1 t 0 σ l (u r (θ), Du r (θ)) dx l r , (25)
In this section we shall clarify what we mean by strong and viscosity solution to equation (25).

Strong solutions to rough

PDEs. There are several ways to define solutions to rough PDEs driven by a finite dimensional noise (see [START_REF] Caruana | A (rough) pathwise approach to fully non-linear stochastic partial differential equations[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF] among others). For our purposes and with our previous notations in mind, the most natural way to do so is the following:

Definition 3.1. Let u be a weakly controlled process in Q κ x ([0, T ]; V ) with V = C 2 (R n
) and κ > 1/3, and consider a function σ in C 2 (R × R n ). We say that u is a strong solution to the rough PDE (25) if u is in fact a strongly controlled process in S κ

x ([0, T ]; V ), such that u x;l = σ l (u, Du), u t = F (u, Du)

and u xx;l 1 l 2 = ∂ u σ l 1 (u, Du) σ l 2 (u, Du) + ∂ p j σ l 1 (u, Du) ∂ θ j [σ l 2 (u, Du)], (26)
where in the last expression, the notation

∂ θ j [σ k (u, Du)] is a compact expression for the quantity ∂ θ j [σ l 2 (u, Du)] = ∂ u σ l 2 (u, Du) ∂ θ j u + ∂ p i σ l 2 (u, Du) ∂ θ i θ j u. ( 27
)
Heuristics for Definition 3.1. Let us denote by ρ β any generic remainder with Hölder regularity β. Then the weakly controlled structure of the candidate solution u is given by:

δu st = σ l 1 (u s , Du s ) δx l 1 st + ρ 2γ st ≡ u x;l 1 s δx l 1 st + ρ 2γ st . (28)
We now expand further the 2γ-Hölder term in ρ 2γ st : it is given by

t s K l 1 sr dx l 1 r , with K l 1 sr = σ l 1 (u r , Du r ) -σ l 1 (u s , Du s ).
Furthermore, equation (28) yields

δu st = σ l 2 (u s , Du s ) δx l 2 st + ρ 2γ st , and δD l 3 u st = ∂ θ l 3 σ l 2 (u s , Du s ) δx l 2 st + ρ 2γ st .
Plugging these relations into the definition of K we get

K l 1 sr = ∂ u σ l 1 (u s , Du s ) σ l 2 (u s , Du s ) + ∂ p l 3 σ l 1 (u s , Du s ) ∂ θ l 3 σ l 2 (u s , Du s ) δx l 2 st + ρ 2γ st ,
and thus t s K l 1 sr dx l 1 r = u xx;l 1 l 2 x 2;l 2 l 1 + ρ 3γ st , where

u xx;l 1 l 2 s = ∂ u σ l 1 (u s , Du s ) σ l 2 (u s , Du s ) + ∂ p l 3 σ l 1 (u s , Du s ) ∂ θ l 3 σ l 2 (u s , Du s ),
which is our decomposition (26). Remark 3.2. Note that even if we consider differential equations of the first order in u it is clear from the rough formulation (and specifically equation ( 27)) that in general, the equation is of second order. This has links with the well-known phenomenon of super-parabolicity of SPDEs.

We will use the a priori structure of strong solutions to guess the natural form of test functions for the viscosity formulation of our rough PDE. To this aim we first get a spacetime expansion for strong solutions to equation (25):

Proposition 3.3. Suppose that F ∈ C 3 (R × R n ; R) and σ ∈ C 3 R × R n ; R d .
Let u be a strong solution to (25). Then, for any 0 ≤ s, t ≤ T and any θ, ν ∈ R n , it holds that

u t (ν) = u s (θ) + a s (θ) δt st + b l s (θ) δx l st + c lk s (θ) x 2,kl st + p j s (θ) [ν j -θ j ] + q jl s (θ) δx l st [ν j -θ j ] + o (|t -s| + |ν -θ|) , (29) where a s (θ) = F (u s (θ), Du s (θ)), b l s (θ) = σ l (u s (θ), Du s (θ)), p j s (θ) = ∂ θ j u s (θ), q jl s (θ) = ∂ θ j [σ l (u s (θ), Du s (θ))], c lk s (θ) = b k s ∂ u σ l (u s (θ), Du s (θ)) + q jk s (θ) ∂ p j σ l (u s (θ), Du s (θ)).
Proof. We apply Proposition 2.9 to z = (u, Du) and G = σ and then Corollary 2.13 to the weakly controlled process defined by σ(u, Du) and η = F (u, Du).

Viscosity solutions.

As mentioned above, the structure of test functions for our viscosity solutions is based on the a priori structure of strong solution. Specifically, the set of test functions related to a rough coefficient σ belongs to a subset of strongly controlled processes defined as follows:

Definition 3.4. Consider a coefficient σ ∈ C 3 (R × R n ; R d ).
Let ϕ be a strongly controlled process in S κ x ([0, T ]; V ) with V = C 2 (R n ) and κ > 1/3. We say that ψ is an element of the test functions T σ if the coefficients in the decomposition (20) of ψ satisfy ψ x;l = σ l (ψ, Dψ) and

ψ xx;l 1 l 2 = ∂ u σ l 1 (ψ, Dψ) σ l 2 (ψ, Dψ) + ∂ p j σ l 1 (ψ, Dψ) ∂ θ j [σ l 2 (ψ, Dψ)], (30)
where we have used the convention (27) for the definition of ∂ θ j [σ l 1 (ψ, Dψ)].

Remark 3.5. Owing to Proposition 2.16, a generic element of T σ satisfies the relation

δψ st = t s σ l (ψ u , Dψ u ) dx l u + t s ψ t u du,
for a given continuous element ψ t . It is thus not obvious a priori that T σ is a nonempty set, not to mention the fact that it contains enough functions to fully characterize solutions to equation (25). However, this kind of property will be observed on the particular examples of equations treated in the next sections.

Similarly to what is done in [START_REF] Lions | Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations[END_REF], we now define the concept of viscosity solution to our equation (25) in the following manner:

Definition 3.6. A function u ∈ C([0, T ] × R n ) is called a viscosity subsolution (resp. super- solution) of (25) if (i) We have u(0, θ) ≤ α(θ) (resp. u(0, θ) ≥ α(θ)), for all θ ∈ R n .
(ii) For any ϕ ∈ T σ it holds that if uϕ attains a local maximum (resp. minimum) at (t 0 , θ 0 ) ∈ [0, T ] × R n such that u t 0 (θ 0 ) = ϕ t 0 (θ 0 ), then ϕ t t 0 (θ 0 ) ≤ (resp. ≥) F (ϕ t 0 (θ 0 ), Dϕ t 0 (θ 0 )) , If u is both a viscosity subsolution and supersolution, we say that u is a viscosity solution of the rough PDE (25).

Like in [START_REF] Lions | Fully nonlinear stochastic partial differential equations[END_REF][START_REF] Lions | Fully nonlinear stochastic pde with semilinear stochastic dependence[END_REF], we favor here the definition of viscosity solutions through the introduction of the set of test functions T σ . The point of view of [START_REF] Buckdahn | Pathwise Taylor expansions for Itô random fields[END_REF][START_REF] Buckdahn | Pathwise Taylor Expansions for Random Fields on Multiple Dimensional Paths[END_REF][START_REF] Buckdahn | Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs[END_REF] is related instead on the notion of stochastic sub/superjet. This notion stems directly from the expansion of the strong solution to equation (25) given by Proposition 3.3, and can be stated as follows:

Definition 3.7. Let z be a function from [0, T ] × R n into R and let (t 0 , θ 0 ) ∈ [0, T ] × R n .
Then the σ-subjet (resp. σ-superjet) of z at (t 0 , θ 0 ) is the set, denoted by P + σ z(t 0 , θ 0 ) (resp.

P - σ z(t 0 , θ 0 )), of (a 0 , p 0 , X 0 ) ∈ (R × R n × R n,n ) such that z t (θ) ≤ ( resp. ≥ ) z t 0 (θ 0 ) + a 0 (t -t 0 ) + b l 0 δx l t 0 t + c lk 0 x 2,kl t 0 t + p i 0 (θ i -θ i 0 ) + q jl 0 δx l t 0 t (θ j -θ j 0 ) + o (|t -t 0 | + |θ -θ 0 |) , (31)
where for 1 ≤ l, k ≤ d and 1 ≤ i, j ≤ n, we consider the real valued coefficients:

b l 0 = σ l (z t 0 (θ 0 ), p 0 ), c lk 0 = b k 0 ∂ z σ l (z t 0 (θ 0 ), p 0 ) + q jk 0 ∂ p j σ l (z t 0 (θ 0 ), p 0 ), q jl 0 = p j 0 ∂ z σ l (z t 0 (θ 0 ), p 0 ) + X ij 0 ∂ p i σ l (z t 0 (θ 0 ), p 0 )
. Related to the last definition, one can also define viscosity solutions to rough PDEs by means of the jets:

Definition 3.8. A function u ∈ C([0, T ] × R n ) is called a jet-viscosity subsolution (resp. supersolution) of (25) if (i) We have u(0, θ) ≤ ( resp. ≥ ) α(θ), for all θ ∈ R n ; (ii) For any (t 0 , θ 0 ) ∈ [0, T ] × R n and any (a 0 , p 0 ) ∈ P + σ u(t 0 , θ 0 ) (resp. P - σ u(t 0 , θ 0 )), it holds that a 0 ≤ ( resp. ≥ ) F (u(t 0 , θ 0 ), p 0 ) .
If u is both a jet-viscosity subsolution and supersolution, we say that u is a jet-viscosity solution of (25).

Remark 3.9. There is an easy way to relate Definition 3.8 with Definition 3.6. Indeed, each element (a 0 , p 0 ) ∈ P + σ u(t 0 , θ 0 ) defines a function ϕ in a neighborhood of (t 0 , θ 0 ) by:

ϕ t (θ) = z t 0 (θ 0 ) + a 0 (t -t 0 ) + b l 0 δx l t 0 t + c lk 0 x 2,kl t 0 t + p i 0 (θ i -θ i 0 ) + q jl 0 δx l t 0 t (θ j -θ j 0 ).
This function can be considered as an element of T σ , at least locally around (t 0 , θ 0 ). It satisfies ϕ t 0 (θ 0 ) = u t 0 (θ 0 ), ϕ t t 0 (θ 0 ) = a 0 and Dϕ t 0 (θ 0 ) = p 0 . Now, since we start from an element of P + σ u(t 0 , θ 0 ), the function uϕ admits a local maximum at (t 0 , θ 0 ). If u is assumed to be a subsolution, then according to Definition 3.6 we must have ϕ t t 0 (θ 0 ) ≤ F (ϕ t 0 (θ 0 ), Dϕ t 0 (θ 0 )), which is compatible with the condition a 0 ≤ F (ϕ t 0 (θ 0 ), Dϕ t 0 (θ 0 )) of Definition 3.8. This heuristic argument is not as easy to formalize as in the deterministic setting, where a complete identification between jets P ± σ and functions in T σ is possible. However, we shall observe this relation on the particular cases of equations below.

Transport type equation

We investigate here the first example of stochastic equation which can be solved thanks to viscosity techniques. This occurs when, going back to equation (25), we take F (θ, u, Du) = F (θ, Du) and σ l (θ, u, Du) = -∂ θ i u A i l (θ) for some smooth and bounded vector fields A l . Specifically, the equation we shall consider here is of the form:

u t (θ) = α(θ) + t 0 F (r, θ, Du r (θ)) dr - t 0 ∂ θ i u r (θ) A i l (θ) dx l r , (32)
with the same notational conventions as in Section 3. As in [START_REF] Caruana | A (rough) pathwise approach to fully non-linear stochastic partial differential equations[END_REF], equation ( 32) is handled through a composition with the rough flow related to the vector fields A l (θ) dx l r . This is where we will make use of the (presumably) classical results of Section 2.4.

The definition of viscosity solution for equation ( 32) is a particular case of Definition 3.6. However, it is important enough to label the explicit expression of the space of test functions we get in this case (which will be called T A in the remainder of the section) for further use: Definition 4.1. Consider a family of smooth and bounded vector fields A 1 , . . . , A d defined on R n . Let ψ be a strongly controlled process in S κ

x ([0, T ]; V ) with V = C 2 (R n ) and κ > 1/3. We say that ψ is an element of the test functions T A if the coefficients in the decomposition (20) of ψ satisfy ψ x;l = -A l ψ, and

ψ xx;l 1 l 2 = A l 1 A l 2 ψ, (33)
where we recall that

A l 1 A l 2 ψ = A j 1 l 1 ∂ θ j 1 A j 2 l 2 ∂ θ j 2 ψ + A j 1 l 1 A j 2 l 2 ∂ 2 θ j 1 θ j 2 ψ. Remark 4.2.
Obviously, this definition is just obtained by particularizing Definition 3.4 to the case σ l (θ, ψ, p) = -p i A i l (θ). Indeed, we have ∂ ψ σ l (θ, ψ, p) = 0 and ∂ p j σ l (θ, ψ, p) = -A j l (θ). Plugging this information into (30) we end up with

ψ xx;l 1 l 2 = ∂ p j σ l 1 (ψ, Dψ) ∂ θ j [σ l 2 (ψ, Dψ)] = A j l 1 ∂ θ j [D i ψ A i l 2 ] = A l 1 A l 2 ψ.
In order to compare the solution to our equation with the solution of a deterministic equation with random coefficients, we shall also consider the set of test functions T 0 related to the deterministic problem ∂ t u = F (u, Du), which is nothing else than the set of functions with space-time regularity C 1,2 . We now state a proposition which gives a one-to-one correspondence between the set of test functions T A corresponding to our stochastic problem and the set T 0 : Proposition 4.3. Consider a family of smooth and bounded vector fields A 1 , . . . , A d defined on R n , and the set T A introduced in Definition 4.1. Then:

(i) Consider the solution φ to equation (23), considered as a flow. Then for any ψ ∈ T A , the composition ψ • φ is an element of T 0 . (ii) Consider the solution φ -1 to equation (24). Then for any ψ ∈ T 0 , the composition ψ •φ -1 is an element of T A .

Proof. Consider ψ ∈ T A , which means in particular that ψ is a real valued strongly controlled process, and take the solution φ to equation (23). According to Proposition 2.17, ψ • φ is still a strongly controlled process, with

[ψ • φ] x;l 1 = ψ x;l 1 + A j l 1 ∂ θ j ψ • φ and [ψ • φ] xx;il 1 l 2 = ψ xx;l 1 l 2 + A j l 2 ∂ θ j ψ x;l 1 + A j l 1 ∂ θ j ψ x;l 2 + A j 1 l 1 A j 2 l 2 ∂ 2 θ j 1 θ j 2 ψ + A k l 2 ∂ θ k A j l 1 ∂ θ j ψ • φ.
Plugging now the expression (33) for the components of ψ into this last expression, it is readily checked that both [ψ • φ] x;l 1 and [ψ • φ] xx;l 1 l 2 vanish. This shows item (i).

As regards the proof of item (ii), we apply again Proposition 2.17, plus the fact that ψ x;l = 0 and ψ xx;l 1 l 2 = 0. Remark 4.4. As a simple corollary of Proposition 4.3, we can answer a question raised by Remark 3.5. Indeed, we can now assert that T A is as rich as T 0 , which contains all C 1 functions in time.

We can now relate our rough equation (32) to a deterministic problem in the following way:

Proposition 4.5. A function u ∈ C([0, T ] × R n
) is a viscosity subsolution (resp. supersolution) of (32) if and only if, setting û = u • φ A , the function û is a viscosity subsolution (resp. supersolution) of the following equation:

ût (θ) = α(θ) + t 0 F (r, θ, Dû r (θ)) dr, (34)
where we have used the notation F (r, θ, p) = F r, φ r (θ), p, [Dφ -1 r ] |φr(θ) , with p, [Dφ -1 r ] |φr(θ) ≡ p j ∂ θ j φ -1 r (φ r (θ)). Proof. Assume first that the function u ∈ C([0, T ] × R n ) is a viscosity subsolution of (25). This means that for any ψ ∈ T A condition (D) is met, with:

(D) Whenever uψ reaches a maximum at (t 0 , θ 0 ) with the additional assumption u t 0 (θ 0 ) = ψ t 0 (θ 0 ), then ψ t t 0 (θ 0 ) ≤ F (t 0 , θ 0 , Dψ t 0 (θ 0 )). Consider now φ = φ A defined by ( 23) and û = u • φ. We wish to show that û is a viscosity solution of equation ( 34). This problem can be reduced to the following one: show that for any ϕ ∈ T 0 such that ûϕ reaches a maximum at (t 0 , θ 0 ) with the additional assumption ût 0 (θ 0 ) = ϕ t 0 (θ 0 ), then ϕ t t 0 (θ 0 ) ≤ F (t 0 , θ 0 , Dϕ t 0 (θ 0 )). In order to prove this last claim, consider ψ ∈ T A and set ψ = ψ • φ A . According to Proposition 4.3, we have ψ ∈ T 0 . Moreover, condition (D) can be translated as: if û•φ -1 -ψ•φ -1 reaches a maximum at (t 0 , θ 0 ) with the additional assumption

[û•φ -1 ] t 0 (θ 0 ) = [ ψ • φ -1 ] t 0 (θ 0 ), then [ ψ • φ -1 ] t t 0 (θ 0 ) ≤ F (t 0 , θ 0 , D[ ψ • φ -1 ] t 0 (θ 0 )) = F (t 0 , θ 0 , [D ψt 0 ] • φ -1 (θ 0 )).
Owing to the fact that φ t is a diffeomorphism for any t ∈ R + , we can recast the latter condition as: if û-ψ reaches a maximum at (t 0 , θ 0 ) with the additional assumption ût 0 (θ 0 ) = ψt 0 (θ 0 ), then ψt t 0 (θ 0 ) ≤ F (t 0 , θ 0 , D ψt 0 (θ 0 )), this assertion being true for any element ψ of the form ψ • φ with ψ ∈ T A . Thanks to Proposition 4.3, the collection of these elements coincides with T 0 , which proves that û = u • φ is a viscosity subsolution to equation (34).

The other relations are proven exactly along the same lines, and are left to the reader as an easy exercise.

Remark 4.6. With the same kind of considerations, one can also prove that strong solutions to equation (32) are also viscosity solutions.

We can now turn to the main aim of this section, namely an existence and uniqueness result for the solution to equation (32): Proof. Thanks to Proposition 4.5, the existence and uniqueness problem for equation ( 32) is equivalent to the existence and uniqueness problem for equation (34). In order to solve the latter problem one can invoke the classical viscosity theory, which amounts to show comparison properties for F as shown in [START_REF] Caruana | A (rough) pathwise approach to fully non-linear stochastic partial differential equations[END_REF]. This condition is ensured whenever F is bounded and Lipschitz, which ends the proof. Remark 4.8. A wide range of examples for the function F are investigated in [START_REF] Caruana | A (rough) pathwise approach to fully non-linear stochastic partial differential equations[END_REF] (see also [START_REF] Diehl | Stochastic control with rough paths[END_REF]), and would lead to the same results in our setting. We haven't delved deeper into this direction since our aim is to settle a clear formulation for the notion of viscosity solution to rough PDEs rather than getting new results in terms of existence and uniqueness.

We close this section by an expression of the solution to (32) starting from jets. We will first particularize the definition of jets to our linear context for sake of clarity. Definition 4.9. Let z be a function from [0, T ] × R n into R and let (t 0 , θ 0 ) ∈ [0, T ] × R n . Then the subjet of z at (t 0 , θ 0 ) related to equation (32) is the set, denoted by

P + A z(t 0 , θ 0 ), of triples (a 0 , p 0 , X 0 ) ∈ (R × R n × R n,n )
such that (31) holds, where for 1 ≤ l, k ≤ d and 1 ≤ i, j ≤ n, we consider the real valued coefficients:

b l 0 = -p i 0 A i l (θ 0 ), c lk 0 = X ij 0 A i k (θ 0 ) + p i 0 ∂ θ j A i k (θ 0 ) A j l (θ 0 ), q jl 0 = -X ij 0 A i l (θ 0 ) + p i 0 ∂ θ j A i l (θ 0 )
. The corresponding superjet is defined accordingly.

Consider now the jets P ± 0 z(t 0 , θ 0 ) related to the deterministic equation (34), corresponding to A = 0. We are able to relate stochastic and deterministic jets in the following way: F (Du) and σ l (θ, u, Du) = H l (u) for some smooth, bounded and nonlinear functions H l : R → R. Specifically, the equation we shall consider here is of the form:

u t (θ) = α(θ) + t 0 F (Du r (θ)) dr + d l=1 t 0 H l (u r (θ)) dx l r , (35)
with the same notational conventions as in Sections 3 and 4. We shall also follow the same notational conventions and the strategy considered by Lions and Souganidis in [START_REF] Lions | Fully nonlinear stochastic pde with semilinear stochastic dependence[END_REF]. In particular, we shall rely on the following equation, which is just equation (23) where A l has been replaced by H l :

φ t (v) = v + m l=1 t 0 H l (φ u (v)) dx l u . (36)
This equation will be interpreted as a flow from R to R, and its inverse ζ is obviously defined by equation (24) where A l has been replaced by H l . As we did in Section 4, we give the explicit expression of the space of test functions we get in this case (which will be called T H in the remainder of the section) for further use: Definition 5.1. Consider a family of smooth, bounded and nonlinear vector fields H 1 , . . . , H d defined on R. Let ψ be a strongly controlled process in S κ

x ([0, T ]; V ) with V = C 2 (R n ) and κ > 1/3. We say that ψ is an element of the test functions T H if the coefficients in the decomposition (20) of ψ satisfy

ψ x;l = H l • ψ, and ψ xx;l 1 l 2 = [H l 2 H l 1 ] • ψ. (37)
In order to compare the solution to our equation with the solution of a deterministic equation with random coefficients, we shall also consider the set of test functions T 0 related to the deterministic problem ∂ t v = F (t, v, Dv) (where F is a suitable transformation of F to be specified later on), which is nothing else than the set of functions with space-time regularity C 1,2 . As in Proposition 4.3, we establish now a one-to-one correspondence between the set of test functions T H corresponding to our stochastic problem and T 0 : Proposition 5.2. Consider a family of smooth, bounded and nonlinear vector fields H 1 , . . . , H d defined on R, and the set T H introduced in Definition 5.1. Then: (i) Let φ be the solution to equation (36), considered as a flow. Then for any ψ ∈ T 0 , the composition φ • ψ is an element of T H . (ii) Let ζ be the inverse of φ, solution to equation (24) with A l = H l . Then for any ψ ∈ T H , the composition ζ • ψ is an element of T 0 .

Proof. Take ψ ∈ T 0 , which means in particular that ψ is a real valued strongly controlled process with ψ x = 0 and ψ xx = 0. Consider the flow φ defined by equation (36). According to Proposition 2.17, φ•ψ is still a strongly controlled process, with

[φ•ψ] x;l 1 (θ) = H l 1 ([φ•ψ](θ)), [φ • ψ] xx;il 1 l 2 = [H l 2 H l 1 ]([φ • ψ](θ)) and [φ • ψ] t (θ) = ∂ v φ(ψ(θ)) ψ t (

θ). This shows item (i).

Concerning the proof of item (ii), take ψ ∈ T H . We also recall that for the flow ζ related to to equation (24) with A l = H l , we have:

ζ x;l 1 (v) = -H l 1 (v) ∂ v ζ(v), ζ t (v) = 0, ζ xx;l 1 l 2 (v) = H l 1 (v) ∂ v H l 2 (v) ∂ v ζ(v) + H l 2 (v) ∂ 2 v ζ(v) ,
where here v ∈ R. Then, applying again Proposition 2.17, one easily gets that ζ • ψ is a strongly controlled process with This means that ζ • ψT 0 and finishes the proof of our one-to-one correspondence.

In order to relate our rough equation (35) to a deterministic problem, let us label the following easy monotonicity lemma: Lemma 5.3. Let φ be the flow related to equation (36). Then φ : R → R is a strictly increasing function.

Proof. The process φ, seen as a function from [0, T ] × R to R, is differentiable in v (we refer to [START_REF] Friz | Multidimensional dimensional processes seen as rough paths[END_REF] for a precise account on this fact). If we set J t (v) = ∂ v φ t (v), J is solution to the following equation:

J t (v) = 1 + m l=1 t 0 H ′ l (φ u (v)) J u (v) dx l u .
This equation admits an explicit solution, namely:

J t (v) = exp m l=1 t 0 H ′ l (φ u (v)) dx l u ,
which is obviously strictly positive. Thus φ is strictly increasing as a function of v.

The relation between our rough equation (35) and a deterministic problem now takes the following form:

where we have used the notation

F (r, v, p) := 1 ∂ v φ r (v) F (∂ v φ r (v)p) .
Then a function u ∈ C([0, T ] × R n ) is a viscosity subsolution (resp. supersolution) of (35) if and only if, u is a viscosity subsolution (resp. supersolution) of (38).

Proof. Assume first that the function u ∈ C([0, T ] × R n ) is a viscosity subsolution of (35). We want to show that u is a viscosity solution of equation (38). Thus, we need to show that for any ϕ ∈ T 0 such that uϕ reaches a local maximum at (t 0 , θ 0 ) with the additional assumption u t 0 (θ 0 ) = ϕ t 0 (θ 0 ), then ϕ t t 0 (θ 0 ) ≤ F (t 0 , θ 0 , Dϕ t 0 (θ 0 )). In order to prove this, fix ϕ ∈ T 0 such that uϕ reaches a local maximum at (t 0 , θ 0 ) with the additional assumption u t 0 (θ 0 ) = ϕ t 0 (θ 0 ). Then, notice that we have u t (θ) ≥ ϕ t (θ) in some neighborhood of (t 0 , θ 0 ). Owing to Lemma 5.3, we have u t (θ) ≥ [φ • ϕ] t (θ) in some neighborhood of (t 0 , θ 0 ). By Proposition 5.2 we have that φ • ϕ is an element of T H . Therefore, we deduce that uφ • ϕ reaches a local maximum at (t 0 , θ 0 ) with the additional assumption u t 0 (θ 0 ) = [φ • ϕ] t 0 (θ 0 ). Since u is a viscosity subsolution then

[φ • ϕ] t t 0 (θ 0 ) ≤ F (D[φ • ϕ] t 0 (θ 0 )). (39)
Notice that we can write [φ • ϕ] t t 0 (θ 0 ) = ∂ v φ t 0 (ϕ t 0 (θ 0 )) ϕ t t 0 (θ 0 ), with ∂ v φ t 0 (ϕ t 0 (θ 0 )) > 0, and F (D[φ • ϕ] t 0 (θ 0 )) = F (∂ v φ t 0 (ϕ t 0 (θ 0 ))Dϕ t 0 (θ 0 )). Plugging these expressions into (39) we obtain the equivalent inequality

ϕ t t 0 (θ 0 ) ≤ 1 ∂ v φ t 0 (ϕ t 0 (θ 0 ))
F (∂ v φ t 0 (ϕ t 0 (θ 0 ))Dϕ t 0 (θ 0 )) =: F (t 0 , θ 0 , Dϕ t 0 (θ 0 )),

showing the desired property for u.

The other relations are proven exactly along the same lines, and are left to the reader for sake of conciseness.

We can now turn to the main aim of this section, namely an existence and uniqueness result for the solution to equation (35): Proof. Thanks to Proposition 5.4, the existence and uniqueness problem for equation ( 35) is equivalent to the existence and uniqueness problem for equation (38). In order to solve the latter problem one can invoke the classical viscosity theory using the results sketched in [START_REF] Lions | Fully nonlinear stochastic pde with semilinear stochastic dependence[END_REF]. This is the reason why we need to assume F bounded, Lipschitz and satisfying (M).

Theorem 4 . 7 .

 47 Consider a family of smooth and bounded vector fields A 1 , . . . , A d defined on R n , and a bounded Lipschitz function F : [0, T ] × R n × R n . Then equation (32) admits a unique viscosity solution in the sense of Definition 3.6.

[

  ζ • ψ] x;l 1 (θ) = [ζ • φ] xx;l 1 l 2 (θ) = 0, and [ζ • ψ] t (θ) = ∂ v ζ(ψ(θ)) ψ t (θ).

Proposition 5 . 4 . 0 F

 540 Let φ the C 3 -diffeomorphism defined by (36) with A l = H l . Set u = φ • u where u satisfies the equation:u t (θ) = α(θ) + t (r, u r (θ), D u r (θ)) dr,

Theorem 5 . 5 .

 55 Let α ∈ BUC(R n ). Consider a family of smooth, bounded and nonlinear vector fields H 1 , . . . , H d defined on R n , and a bounded Lipschitz function F defined on R n verifying that: (M) There exists a positive constant C such that, either (a) p • DF (p) -F ≤ C or (b) p • DF (p) -F ≥ -C, for a.e. p ∈ R n . Then equation (35) admits a unique viscosity solution in the sense of Definition 3.6.

Torrecilla wishes to acknowledge the support from both BIGS (Biology, Genetics and Statistics) team at INRIA and the project entitled "Explorations des Chemins RUgeux (ECRU)/ Explorations on rough paths" ANR Projet Blanc 2009/2012 by means of a post-doctoral position for six months in each one.

Proposition 4.10. Let u A be the unique viscosity solution to equation (32) and let u 0 be its deterministic counterpart, solution to (34). Then or all couples (t 0 , θ 0 ) ∈ [0, T ] × R n there exists a bijective correspondence between P ± A u A (t 0 , θ 0 ) and P ± 0 u 0 (t 0 , θ 0 ). Proof. Let us regularize the coefficients and the noise of equation (32), producing a strong solution u ε,A . The same can be done for the deterministic equation, and we get a strong solution u ε,0 . Let us also consider the solution φ to equation (23). Then applying Corollary 2.18 we obtain a Taylor-type expansion for the composition u ε • φ ε , whose coefficients satisfy:

The composition with φ transforms thus the unique element (a 0 , p 0 , X 0 ) of P ± A u ε,A (t 0 , θ 0 ) into the unique element of P ± 0 u ε,0 (φ(t 0 , θ 0 )). Reciprocally, let us strart from the solution u ε,0 of the regularized deterministic equation. Recall that ζ designates the solution to equation (24), and let us call ζ ε its regularization. Then, invoking again Corollary 2.18 we obtain a Taylor-type expansion for the composition u ε,0 • ζ ε with the corresponding coefficients:

Therefore, expanding the terms

(θ 0 ) in terms of the derivatives with respect to θ of ζ ε up to order 2, and in terms of the vector fields A l and their first derivatives, we deduce that if (a 0 , p 0 , X 0 ) ∈ P ± 0 u ε,0 (t 0 , θ 0 ) then (a 0 , p i 0 Dζ ε;i t 0 (θ 0 ), X0 ) ∈

t 0 (θ 0 ). We have thus identified the jets P ± 0 u ε,0 (t 0 , θ 0 ) and P ± A u ε,A (t 0 , θ 0 ) of the regularized equations. Taking limits on P ± 0 u ε,0 (t 0 , θ 0 ), this allows to identify the jets related to u 0 and u A .

Remark 4.11. Obviously, the fact that we resort to solutions of equations in order to establish the jet correspondence is less attracting than the result concerning test functions. We shall thus privilege the definition of viscosity solution through the sets T σ of Definition 3.4 in the sequel.

Equation with semilinear stochastic dependence

We investigate here a second example of stochastic equation which can be solved thanks to viscosity techniques. This occurs when, going back to equation (25), we take F (θ, u, Du) =


Michael J Niebling

Ken Tore Tallakstad

Renaud Toussaint

Knut Jørgen Måløy

Direct velocity measurement of a turbulent shear flow in a planar Couette cell

In a plane Couette cell a thin fluid layer consisting of water is sheared between a transparent band at Reynolds numbers ranging from 300 to 1400. The length of the cells flow channel is large compared to the film separation. To extract the flow velocity in the experiments a correlation image velocimetry (CIV) method is used on pictures recorded with a high speed camera. The flow is recorded at a resolution that allows to analyze flow patterns similar in size to the film separation. The fluid flow is then studied by calculating flow velocity autocorrelation functions. The turbulent pattern that arise on this scale above a critical Reynolds number of Re=360 display characteristic patterns that are proven with the calculated velocity autocorrelation functions. The patterns are metastable and reappear at different positions and times throughout the experiments. Typically these patterns are turbulent rolls which are elongated in the stream direction which is the direction the band is moving. Although the flow states are metastable they possess similarities to the steady Taylor vortices known to appear in circular Taylor Couette cells.

INTRODUCTION

Since the beginning of the 20th century the stability of fluid flows has been studied in unlike systems. Typical examples of such systems are natural convection in the atmosphere or convection in the earth mantle and the Rayleigh-Bénard convection when a thin layer of fluid is heated from below [1,2]. When flow instabilities arise in these systems during the transition to turbulence often turbulent rolls appear that are also referred to as convection cells. The shear flow of a thin fluid layer that is studied here also falls into this group of systems that display turbulent rolls often called Taylor vortices when the turbulent regime is reached. The most prominent setup to study the instability of a sheared thin fluid layer is the circular Taylor Couette cell. In this setup two coaxial mounted cylinders with a small difference in radii rotate independently at different angular frequencies. The small difference in the cylinders radii leaves a thin gap that is filled with a fluid. Different angular frequencies cause a shear of the fluid layer. Centrifugal forces act on the fluid layer between the cylinder in this situation. This centrifugal forces depend on the radial position and on the angular frequencies of the cylinders. For some combinations of the angular frequencies and cylinder radii the resulting centrifugal forces give rise to an instability of the steady rotary flow. The instability leads to toroidal Taylor vortices or rolls that are again stable in time and space. These vortices are arranged in a periodic manner along the cylinders as described in [START_REF] Landau | Fluid Mechanics[END_REF] and studied in [START_REF] Andereck | [END_REF]5]. In more recent experiments it was shown that Coriolis forces can as well lead to periodic flow patterns in shear flows as shown and discussed in [6]. In this work a planar Couette cell (see Fig. 1 and [7,8]) was mounted on a rotating table with an angular frequency controlled in-dependently from the shear velocity of the flow channel. The Coriolis force was then introduced by controlling the rotational frequency and direction of the table. These experiments show that the Coriolis force leads to a similar instability and patterns as the centrifugal force in the circular Taylor Couette cell. In fact in [6] a great variety of flow patterns emerged during the transition to the turbulent regime depending on the shear velocity and the Coriolis force. Weather the turbulent flow pattern are stable or meta stable in time and space at a fixed shear velocity depends on the direction and frequency of the table rotation. In both above setups the most simple configuration of turbulent flow patterns are rolls that span through the whole flow channel in the stream direction. Such rolls are stacked on top of each other in a periodic manner where neighboring rolls rotate counter-wise and move in the opposite stream-wise direction. The streamlines in such a roll cell resemble a helix. The helices are stacked one above another in the stream-wise direction with alternating helix axis direction but same chirality. The turbulent patterns originate from small perturbations of the flow that grow exponentially in time and lead to steady flow patterns under the above discussed conditions. When neither centrifugal nor Coriolis forces are present the turbulent patterns lose their stability and become meta stable. This is the case in the setup that is studied in the present work. It is the main purpose of the present paper to quantify and describe the patterns that result as the stabilizing mechanisms of the fictitious forces are removed. We achieve this through a statistical analysis of the flow in terms of velocity correlation functions and velocity probability distributions. The resulting flow patterns are characterized by stream-wise vortices with alternating stream-wise flow direction between neighboring vortices. In simulations and analytical work of this setup by [9][10][11][12][13][14] meta stable elongated turbulent rolls have been reported and studied. In further investigations a low-order model is used to investigate self-sustaining rolls and effects of the boundary conditions in [15,16] and admissible fluids states and traveling wave solutions of the Navier-Stokes equation are simulated in [17][18][19][START_REF] Holstad | Advances in Turbulence XI[END_REF][START_REF] Holstad | [END_REF]. Coexistence of laminar and turbulent regions where investigated numerically by [22][23][24]. Further the flow in a plane Couette cell was characterized into four flow regimes by [12,13] using direct numerical simulations. In preceding experiments of the planar Taylor Couette cell a rheoscopic (aka kalliroscopic) fluid was sheared and pictures of the patterns taken [25,34,35]. Fluid injection into the flow creates turbulent spots which were studied in detail by [22,23,25,26]. In addition in [27][28][29] the flow patterns in a planar Couette cell close to a wire spanning the cell parallel to the walls were investigated experimentally by the use of different techniques such as laser Doppler velocimetry and rheoscopic particles. When the Reynolds number is used as the control parameter and suddenly decreased in quench experiments the decay time of vortex structures have been measured in experiments by [29,30] and experiments are compared to numerical simulations in [31]. The interaction between thermal convection and shear flows in a plane Couette cell is studied through numerical simulations in [START_REF] Busse | Bifurcation sequences in problems of thermal convection and of plane Couette flow[END_REF][START_REF] Clever | [END_REF]. Using these method periodic and elongated flow structures along the stream direction could be visualized and studied. Compared to Taylor vortices found in circular cells these patterns in a planar Couette cell are much shorter in the stream direction and fluctuate in time and space. To achieve a direct measurement of the flow velocity we decided to use a correlation image velocimetry technique (CIV). For this purpose tracer particles with the same mass density as the sheared fluid are added to the fluid. Adding density matched tracer particles is a standard non invasive technique. This technique allows a direct measurement of the fluid velocity that reveals additional characteristic features of the flow patterns which were not accessible using the previous techniques where only the flow direction was visualized [6,25,34,35]. In more recent studies 3d particle tracking velocimetry (PTV) [36][37][START_REF] Lüthi | Advances in Turbulence XII[END_REF][START_REF] Hoyer | [END_REF] was applied to flow experiments in a plane Couette cell and the flow field measured in three dimensions for the first time. In the following the relative arrangement and shape of the appearing flow patterns is further discussed and quantified through velocity autocorrelation functions.

EXPERIMENTAL SETUP

Mechanical setup

The planar Couette Cell is shown in Fig. 1. It consists of a confining volume (∼ 2.5 liters), in which a thin (thickness ∼ 50 µm) Mylar belt (⋄) is rotating around two cylinders (⋆). The left cylinder can be moved in the x-direction so as to tighten the belt, whereas the right cylinder is driven by a DC motor. Velocities in the interval (0 -60) cm/s of the Mylar film can be obtained. Four small steering cylinders (•) are mounted, two at each end of a narrow flow channel. These cylinders are fixed and define the inner spacing between the Mylar film. In the narrow flow channel, the film has a height of 10 cm in the y-direction, an inner separation distance d = 4.5 mm in the z-direction, over a length of 50 cm along the x-direction. Furthermore, the flow channel is made of Plexiglas to ensure light transparency.

The Mylar film was sealed as a belt, with a joint without overlap. The boundary conditions are open to the rolls on the sides and closed to top and bottom of the cell. Turbulence arises first at the sides of the cell due to the noise induced by the rolls as shown with rheoscopic particles during experiments in the same cell by [35].

The initial preparation of the Couette Cell is done by the following routine: 1) the cell is filled with water, 2) three drops of liquid soap are added only to prevent particle aggregation and clustering by reducing the surface tension (this has no effect on the experiments otherwise), 3) 7 grams of fluorescent polyethylene tracer particles (Cospheric Fluorescent Red) with diameter 125 -150 µm are added to the water. These particles are density matched with water (0.995 g/ml). Furthermore, all experiments are performed in room temperature at roughly 22 • C, and constant humidity.

When the film is set in motion, the fluid/particle mixture between the two sides of the film, moving in opposite x-directions, starts to flow. We define the absolute value of the film speed (x-direction) as U . Depending on the value of U , the flow will be either laminar or turbulent. We emphasize that this flow geometry, gives an overall zero average flow velocity.

Optical setup

The tracer particles added to the water are fluorescent 125-150 µm polyurethane microshperes. Thus in order to visualize the flow, we use a beam from a Lexel 95-3 argon ion laser of wavelength 514.5 nm, at up to 3.0 W. The beam is sent through a cylindrical lens, creating a laser sheet (see Fig. 1) of thickness L d ∼ 1.0 mm. As shown in the upper and lower panel of Fig. 1, the laser Top view -laser sheet at angle A mylar film (⋄) is put around two cylinders (⋆), and set in rotation by a motor. Additional four smaller cylinders (•) set the internal gap spacing between the inner planes of the film, d. The entire setup is placed inside a waterproof container with geometry as indicated by the thick solid boundary. The thinner channel in the middle is made of Plexiglas. A laser sheet (⊲) is used to illuminate a chosen cross section of the flow channel. In this particular illustration, the laser sheet enters the channel at an angle α as indicated (cf. Fig. 2). (Lower panel) Front view of the Couette Cell. In this particular illustration, the laser sheet (⊲) enters the cell from below, and spans the xy-plane. The fluid is seeded with density matched fluorescent particles. Indicated by a dashed line is the 4 cm × 4 cm Region Of Interest (ROI), captured by a fast camera in order to follow the fluid motion (cf. Fig. 2). sheet (⊲) can be positioned differently, in order to span different cross sections of the flow. When the laser light meets the tracer particles, they glow in a red color and allow to visualize the flow. The intensity profile of the laser is Gaussian, and thus, the particles scattered intensity will vary accordingly. The particles are only visible as long as they are positioned inside this light intensity profile, resulting in a two dimensional visualization of the three dimensional flow. To record the flow, a fast camera (Photron Fastcam SA5) is positioned normal to the laser sheet, as shown in the upper panel of Fig. 1. The recording rate of the camera is 2000 frames per second (fps), at a spatial resolution of 40 µm/pixel, and roughly 10 000 images are acquired per experiment. The velocity of the Mylar film U is the only tuning parameter, and we define the Reynolds number as

Re = U ν d 2 , (1)
where ν = 1.0 mm 2 /s is the kinematic viscosity of water, and d/2 is half the separation distance between the two sides of the Mylar film. Previous studies have shown [8,35] that the critical Reynolds number, i.e where the flow goes from laminar to turbulent, is Re c ≈ 360 ± 20.

In this study, we have performed three sets of seven experiments (21 in total), at roughly the same range of Re ≈ {300, 400, 460, 610, 860, 1120, 1400}±20. The Re = 300 experiments are laminar, whereas the rest is in the turbulent regime. For each experimental set, the orientation of the laser sheet has been changed, so as to probe different cross sections of the flow. The coordinate system is shown in the upper and lower panel of Fig. 1. We define x as the stream-wise direction, y as the span-wise direction, and z as the wall-normal direction. A rotation of an angle α, of the original coordinate system, around the y-axis, gives the transformed coordinates (x ′ , y ′ , z ′ ):

x ′ = x cos α -z sin α y ′ = y (2) z ′ = x sin α + z cos α .
The different orientations of the laser sheet are shown in Fig. 2. In set A, the laser sheet spans the xy-plane in the center of the cell, at a distance d/2 ≈ 2.3 mm from the moving front film. In set B, the laser sheet spans the xy-plane further towards the front of the flow channel, at a distance 1.3 mm from the moving front film. Set A and B have identical camera position and experimental alignment apart from the position of the laser sheet. To FIG. 2: Our experiments are grouped into three sets, depending on the orientation of the laser sheet. In set A the laser sheet spans the xy-plane, in the center of the model, at a distance 2.3 mm from the moving front film. In set B the laser sheet spans the xy-plane, but now further towards the front of the flow channel, at a distance 1.3 mm from the moving front film. In set C the laser sheet enters the cell at an angle α = 52 • , spanning an x ′ y ′ -plane as defined in Eq. (2). Also indicated along the z-axis is the linear velocity gradient in the case of a low-Re, laminar flow. distinguish between these two sets of experiments, from now on, variables will be denoted by a subscript c and f for center view (set A) and front view (set B) respectively. In both sets A and B, the images acquired spans a 4 cm × 4 cm square in the laser sheet, as shown in the lower panel of Fig. 1, and is the region of interest (ROI) (see also videos [42]). In set C, the laser sheet is rotated around the y-axis, at an angle α = 52 • from the x-axis, and spans now an x ′ y ′ -plane as defined in Eq. ( 2). Coordinates in this plane will from now on be referred to with the superscript ′ . The ROI in set C, spans a 0.55 cm × 4 cm rectangle of the flow cell (see also videos [42]).

Analysis procedure

To obtain the velocity field of the flow, each sequence of images is analyzed using a standard multi-pass correlation image velocimetry (CIV) technique [40,41]. A presision test of this method was done in i.e. [41]. The pictures are recorded at a resolution of 1024x1024 pixels. Using the CIV technique the velocity fields are calculated at a lower resolution than the initial gray scale pictures which are recorded at a resolution of 256 pixels/cm. The final resolution of the velocity field is 16 measurement points per cm in set A and B and 30 measurement points per cm in set C. This resolution is carefully chosen to fit for a statistical analysis of vortices similar in size to the plate separation d. The finite thickness of the laser sheet illuminates particles within a distance interval normal to the region of interest (ROI) of approximately 1mm. The observed ROI is therefore initially 3 dimensional with the dimension 4cm by 4cm by 1mm. Particles can be tracked easier in a thicker laser sheet since they stay illuminated longer even though they move somewhat normal to the ROI in the wall normal z-direction. When the pictures are taken the normal dimension of the recorded volume is projected on a 2 dimensional plane. With a thick laser sheet particles that appear at the same position in the pictures can have different z-coordinates and therefore different velocities in the recorded 2d pictures. For the correlation tracking the local particle constellation must be related between two subsequent pictures. It is therefore important that the laser sheet is not too thin that particles get lost too fast due to movement normal to the film. Neither should the laser sheet be too thick causing the recorded particles to locally have very different velocity and destroying the local constellation of particles.

RESULTS

Front View

In example Figs. 3 and4 three grays scale snapshots of the resulting flow profiles are shown for 3 different Reynolds numbers. All snapshots are recorded in the middle between the two sides of the film and from experiment set A. The gray scale corresponds to a velocity range from -1cm/s (black) to 1cm/s (white). Velocity values are visualized by a linear gray scale inside this range and values below or above this linear scale are represented black or white respectively. Actual values of the flow velocity are much higher in the turbulent cases but this range qualitatively highlights the vortex structure in the flow and increases the contrast. The spatial resolution in these snapshots is higher than the velocity data that was calculated by the CIV technique: this resolution is achieved by linear interpolation to improve the readability of the snapshots. For Re=300 the particles move randomly left or right depending on slight differences in their wall-normal zposition. In direct observations particle trajectories in the laminar regime Re=300 are straight horizontal lines that span through the whole ROI. In contrast to the laminar case the turbulent flow displays horizontal layers or clusters of alternating coherent v xc -direction (see Figs.

3(b)-3(c) )

. The alternating layers are elongated in the x-direction and the flow velocity is much higher as can be also seen in the velocity probability distribution that are shown further down. Direct observations further shows particle trajectories are now strongly curved. The extension of these layers in the y-direction is around 4 mm before the v xc direction is reversed. This separation in y-direction is much shorter than the stream-wise length of the coherent clusters of v x in x-direction. The separation is however comparable to the film separation d. In stream-wise direction these clusters or layers can be much larger than the observed ROI of 4 cm. Comparing Fig. 3(b) to 3(c) we got the impression that the length of the coherent v x -clusters decreases in x-direction while the Reynolds number is increased. This is an observation that later will be confirmed by velocity correlation functions. The distance in y-direction until the flow v xdirection is reversed however does not change noticeably in the snapshot with the Reynolds number Fig. 3(c). These observations will also be shown by velocity correlation functions in the next paragraphs. The snapshots of the v yc -velocity component show much

Cv α (∆β) = β (v α (β) -v α )(v α (β + ∆β) -v α ) β (v α (β) -v α ) 2 . ( 3
)
Where α defines the x, or y-velocity component. This means that there are only very weakly correlated patterns of the v yc -velocity component for the time and space average in the middle of the cell. Instead a rather random pattern distributions must be the case in the turbulent situation. Furthermore the maximal anticorrelation of the v yvelocity component in Fig. 5(d) is located at the same spatial separation in y-direction as for the v x -velocity component at ∆y ≈ 4 mm as shown in Figs. 5(a) and 5(c). The maximal anticorrelation of the v y -velocity component and the v x -velocity component at the same spatial separation shows that neighboring layers preferably rotate counter-wise around the stream-wise direction. This type of movement resembles a roll with streamlines following a helix with an axis in x-direction. In the turbulent regime the autocorrelation functions show an increased probability that multiple helices coexist at the same time in the cell. The axis direction alternates in xdirection between neighboring helices while the chirality of these coexisting helices is the same. The length of the rolls in horizontal stream-wise direc- Slight asymmetries in these distributions result from the fact that if particles inside the 1mm thick laser sheet are located towards the camera side they will be easier detected and recorded than particles on the other side of the laser sheet. Furthermore it is possible that the laser sheet position is marginally shifted from the middle of the cell. Nevertheless compared to these rather symmetric distributions the v x f -component close to the front film loses this symmetric distribution and Fig. 6(c) shows that here the fluid is more likely to move along the direction the film is moving than against the film. In this case the film was moving in the negative x-direction where the v x fditribution also shows a higher probability compared to the movement in positive x-direction.

The previous results from the autocorrelation functions (see Figs. In the experimental setup previously referred to as set C the laser sheet enters the cell under an angle of 52 • . The angle has been calculated by Snell's law and refraction at the plexiglas wall and the refraction of the beam entering the the water has been taken into account. The setup allows to measure a projection of the velocity vector v on the x ′ y ′ -plane at an angle of α =52 • as shown in Fig. 1 and stated in Eq. (2). In this way the projection of the wall normal flow direction can be studied although it is important to state that the recorded v x ′ -component depends on both the old v x and v z -velocity components. The particles tracked at each position by the CIV method are randomly located within the finite thickness of the laser sheet. Between the two extreme positions on both sides of the laser sheet the maximal possible velocity variation are ∆v x ′ = sin(52 • ) cos(52 • )L d U/d = m = ±0.1U but this will average to zero in time. From the recorded pictures the flow field was again calculated by a correlation image velocimetry CIV method. The resulting flow profile is shown in Fig. 7 for the v x ′ -component, in Fig. 8 for the v y ′ -component. The shown velocity field in In all plots it could also been observed that in the turbulent regime at higher Reynolds numbers the anticorrelation of the flow decreased. This showed that at higher fluid velocity the flow patterns become more random and the probability of ordered patterns gradually decreases.

CONCLUSION:

The analysis of the velocity measurements showed a clear distinction between the laminar and turbulent flow at different Reynolds numbers. Further more the measurement allowed to quantify and qualify characteristic vortices that reappear at different times and positions during the experiments with an increased probability. In the stream-wise x-direction such vortices are elongated and reach lengths of more than the observation area of 4 cm. When the Reynold number is increased the streamwise length decreases. In the y and z-direction the size of the vortices is sharply confined to 4 mm which corresponds to the film separation as shown in the idealized Fig. 11. Typically several such vortices are found in the cell at the same time. The vortices are correlated in their relative position and velocity to each other. In y-direction these vortices are separated by 4 mm. Many such vortices can be stacked on top of each other in the xy-plane. The main direction of fluid movement in each vortex is the x-direction. Neighboring vortices move in the opposite x-direction. In Fig. 12 cross sections in the xz-plane of idealized flow patterns are shown in top view. In Fig. 12(a) a typical laminar shear flow is illustrated. The fluid has a linear velocity profile in z-direction with zero overall velocity in the middle marked with a (red) circle. At higher Reynold numbers stream-wise elongated vortices have developed. From our previous analysis we can outline a number of flow field features of these vortices or turbulent rolls in more details. In a first step we could initially show that at the middle between the two film sides vortices have an either positive or negative main x-flow direction (see Figs. 3, 4 and 6(a), 6(b)). Here the main flow velocity of all vortices is equally distributed in positive and negative x-direction in a time and space average this is also shown in simulations [17,18]. In a second step we studied how the main x-flow velocity inside a vortex is influenced by the film boundaries in z-direction. In z-direction each vortex is confined by the two sides of the film. There is always one side that moves along with the main x-flow direction of the vortex and the other film side that moves against the main xflow direction of the vortex. When moving in z-direction from the middle between the film towards the film side that moves along with the main x-flow direction the v xvelocity inside a vortex increases. When moving to the other side from the middle between the film towards the film side that moves against with the main x-flow direction the v x -velocity inside a vortex decreases. This was shown in Fig. 6(c). At last we know that at the film boundary the fluid veloc-ity must be the same as the film velocity. This condition together with the measurements taken can be combined to outline the flow profile of vortices in the xz-plane. A possible interpretation for the v x flow velocity inside turbulent vortices is given in Figs. 12(b) and 12(c).

On one side of these vortices the film moves in the same stream-wise direction as the vortex while on the other side the film moves against the stream-wise flow of the vortex. As shown before when analyzing the velocity distributions (see Fig. 6(c)) the vortices display an asymmetry towards the two sides of the film and an overall shear of the vortices occurs. How this shear profile looks in detail when crossing the vortex in z-direction can not be stated exactly from the measurements discussed in this article. The precise profile shown in the figures is therefore a guess that fulfills the above discussed conditions. However the point of zero x-velocity is always located closer to the side of the film moving opposite to the vortex' main stream-wise flow direction as measured in Fig. 6 In addition to the horizontal flow in x-direction vortices also rotate around the x-axis which is consitence with among others [29,37] (experimental) and [14,17,[START_REF] Holstad | Advances in Turbulence XI[END_REF] (simulation). Neighboring vortices rotate counter-wise. As shown in the idealized Fig. 13 by assuming circular shaped rolls in the yz-plane cross section. Combining the measured flow properties leads to a helix type of fluid motion around the x-axis in each of those vortices or rolls. In summary a number of distinct features of typical vortices in a plane shear flow at high Reynolds number were revealed and discussed. An exact measurement of the shear and rotation of a typical vortex is considered to be a possible future step. y z x FIG. 13: Several rolls are shown in an cross section along the yz-plane. The (red) dashed line between the roll vortices marks zero x-velocity. In this proposed idealized demonstration the rolls are assumed circular in the cross section along the yz-plane while in reality more complex shapes are likely.

FIG. 1 :

 1 FIG. 1: (Upper panel) Top view of the planar Couette Cell.A mylar film (⋄) is put around two cylinders (⋆), and set in rotation by a motor. Additional four smaller cylinders (•) set the internal gap spacing between the inner planes of the film, d. The entire setup is placed inside a waterproof container with geometry as indicated by the thick solid boundary. The thinner channel in the middle is made of Plexiglas. A laser sheet (⊲) is used to illuminate a chosen cross section of the flow channel. In this particular illustration, the laser sheet enters the channel at an angle α as indicated (cf. Fig.2). (Lower panel) Front view of the Couette Cell. In this particular illustration, the laser sheet (⊲) enters the cell from below, and spans the xy-plane. The fluid is seeded with density matched fluorescent particles. Indicated by a dashed line is the 4 cm × 4 cm Region Of Interest (ROI), captured by a fast camera in order to follow the fluid motion (cf. Fig.2).

  Figs. 3(a)-3(c) show the v xc component and the v yc component is shown in Figs. 4(a)-4(c). Figs. 3(a) and 4(a) show snapshots of the laminar flow at Reynolds number of Re=300. The next two snapshots in Fig. 3(b)-3(c) and 4(b)-4(c) are taken at a higher Reynolds number that leads to a turbulent flow Re=400 and Re=1140.

FIG. 3 :FIG. 4 :

 34 FIG. 3: Snapshots of the horizontal velocity vx c -component in experiment set A display typical flow fields in the xcyc-plane of 4 cm x 4 cm ROI area at 2.3 mm from the moving front film.. Reynolds numbers are Re=300 (laminar) in (a), 400 in (b) and 1120 in (c). To enhance the contrast velocities are represented by a linear gray scale only from -1cm/s (black) up to 1cm/s (white) although the actual range of velocity values is much larger. Values below or above this linear scale are represented black or white respectively. For smoother pictures the spatial resolution was increased by linear interpolation. Stripes of alternating flow direction are visible that are layered on top of each other.

FIG. 5 :

 5 FIG. 5: Velocity autocorrelation as a function of the separation in y-direction: ∆y. (a) shows the autocorrelation of the vx c -component and (b) of the vy c -component in the middle of the film at a position 2.3mm from the front film. (c) shows the autocorrelation of the vx f -component and (d) of the vy f -component close at a position 1.3 mm from the front film. (e) and (f) show velocity autocorrelation functions of the vx-component as a function of the separation in x-direction: ∆x. (a) shows the autocorrelation functions gathered from experiments with the laser sheet in the middle of the film at a position 2.3 mm from the front film. (b) shows autocorrelation functions from experiments with the laser sheet at position 1.3 mm from the front film.

FIG. 6 :

 6 FIG. 6: Velocity distributions of the two experimental sets A (a)-(b) in the middle between the film and B (c)-(d) close to the front film. (a) and (c) show distributions of the vx component. (b) and (d) show distributions of the vy component. All distributions are time and space averaged.

  tion decreases at higher Reynold numbers. The autocorrelation of the v x -velocity component as function of the horizontal stream-wise spatial separation ∆x is shown in Fig. 5(e) for experiment set A and in Fig. 5(f) for experiment set B. In both sets the horizontal v x -velocity component is correlated over a shorter spatial separation with increasing Reynold numbers. To understand the shape of these rolls velocity histogram are shown in Figs. 6(a) and 6(b) for experiment set A in the center of the sheer flow and in Figs. 6(c) and 6(d) for experiment set B close to the front film. Comparing the four plots shows that the v x -component is distributed over a wider range compared to the v y -component. This shows that the fluid flows faster horizontal along the positive or negative stream-wise x-direction than in span-wise y-direction. The main flow direction of a vortex is therefore the stream-wise x-direction in the xy-plane. In the center of the film (Figs. 6(a), and 6(b)) the v xc and v yc -velocity component is distributed almost symmetric with respect to the y-axis. The same holds for the v y f -component close to the front film, which also follows an almost symmetric distribution with respect to the y-axis in Fig. 6(d).

FIG. 7 :

 7 FIG. 7: Snapshots of the horizontal v x ′ -velocity component in the 52 • view. The displayed area shows the whole cross section of the film and has a dimension of 0.55 cm by 4 cm. Reynolds numbers are Re=300 (laminar) in (a), 400 in (b) and 1120 in (c). A clear zero velocity line is visible at x ′ ≈ 3 mm which is fluctuating for the two turbulent Reynolds numbers in plot (b) and (c). To enhance the contrast and visualize the flow direction velocities are represented by a linear gray scale only from -1cm/s (black) up to 1cm/s (white) although the actual range of velocity values is much larger. Values below or above this linear gray scale are represented black or white respectively. For smother pictures the spatial resolution was increased by linear interpolation.

FIG. 8 :

 8 FIG. 8: Corresponding to Fig. 7. Snapshots of the horizontal v y ′ -velocity component in the 52 • view. The displayed area shows the whole cross section of the film and has a dimension of 0.55 cm x 4 cm. Reynolds numbers are Re=300 (laminar) in (a), 400 in (b) and 1120 in (c). To enhance the contrast and visualize the flow direction velocities are represented by a linear gray scale only from -1cm/s (black) up to 1cm/s (white) although the actual range of velocity values is much larger. Values below or above this linear scale are represented black or white respectively. For smother pictures the spatial resolution was increased by linear interpolation.

FIG. 9 :

 9 FIG. 9: Figs. (a) and (c) show autocorrelation functions for the v x ′ -velocity component and Figs. (b) and (d) show autocorrelation functions for the v y ′ -velocity component as a function of ∆y ′ . (a) and (b) are measured at x ′ = 1 mm away from the back film. (c) and (d) are measured at x ′ = 3 mm in the middle between the two films.

FIG. 10 :FIG. 11 :

 1011 FIG.10: At several x ′ -positions velocity autocorrelations as a function of ∆y ′ have been calculated. All of these functions have a first local minimum for increasing ∆y ′ . This value of the first maximum velocity anticorrelation is plotted for the tested x ′ -positions. This is shown in plot (a) for the v x ′ -velocity component and in (b) for the v y ′ -velocity component. The laminar flow experiment at a Reynolds number of Re = 300 does not show a consistent local minimum in the velocity autocorrelation functions and is therefore not part of the plots.

  (c) and shown in Figs. 7(b) and 7(c). In Figs. 12 this point is always marked with a (red) circle. Moving along the y-direction the position of zero streamwise flow velocity travels from one side of the film to the other between two neighboring vortices. In a cross section along the yz-plane zero stream-wise flow velocity is indicated in Fig. 13 by a dashed (red) line. The snapshots of experimental set C show similar curved lines of zero x ′ -velocity in Figs. 7(b) and 7(c) which primarily show the x-velocity.

FIG. 12 :

 12 FIG. 12: (Color online) Idealized preferred flow configurations are shown from the top for a cross section in the horizontal x,z-plane. In (a) the typical laminar flow is shown and marked with the (red)-circle where the fluid has zero x-velocity in the middle between the film sides. In (b) and in (c) possible interpretations of horizontal roll cross sections are shown corresponding to a turbulent flow. The point in the flow of zero x-velocity is marked with a (red) circle. This point moves along the (red) dashed line when the x-flow direction of the roll is reversed.

Acknowledgements: We acknowledge the support of the Norwegian research council through the Frinat grant 205486 of NEEDS-MIPOR, the regional network RE-ALISE, the ANR LANDQUAKE and the ITN FLOW-TRANS. We thank Walter Goldburg, Eirik G. Flekkøy, Luiza Angelutha, Joachim Bergli, Joachim Mathiesen, Mogens Jensen, Mickael Bourgoin and Romain Volk for fruitful discussions.