Ludovic Goudenège
email: goudenege@math.cnrs.fr

Liviu Iulian Palade
email: liviu-iulian.palade@insa-lyon.fr

A New Non-Linear Density Fluctuations Stochastic Partial Differential Equation With a Singular Coefficient of Relevance to Polymer Dynamics and Rheology: Discussions, Proofs of Solution Existence, Uniqueness, and a Conjecture

Keywords: stochastic partial differential equation with a singular coefficient, solution existence results. AMS subject classification: Primary 60H17, Secondary 35B30

In this paper we consider an entirely new -previously unstudied to the best of our knowledge -type of density fluctuations stochastic partial differential equation with a singular coefficient involving the inverse of a probability density. The equation was recently introduced by Schieber [36] while working on a new polymer molecular dynamics approach that pertains to the generally called polymer reptation (aka tube) theory. The corresponding probability density is the solution of an evolution equation (a stochastic transport) defined on a dynamical one-dimensional subspace. A peculiarity of the here studied equation is its very singular pattern, even though it exhibits a well-posed structure. As a first step towards furthering the understanding of this new class of equations we prove the existence and uniqueness of solutions under suitable conditions. Moreover, capitalizing on the assumed gradient structure, we prove that the existing solutions converge to an equilibrium density.

Introduction

Basically there are two dominant, authoritative theories (together with their variants) in the realm of polymer melt molecular dynamics which are of paramount importance to the rheology of those non-Newtonian fluids. Specifically, that of Bird, Curtiss, Armstrong and Hassager [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF] based on early ideas set-up by Kirkwood [START_REF] Kirkwood | [END_REF], and that of Doi and Edwards [START_REF] Doi | The Theory of Polymer Dynamics[END_REF] which incorporates de Gennes' reptation concept [START_REF] De Gennes | Scaling Concepts in Polymer Physics[END_REF]. The former is known as kinetical theory for polymer dynamics while the later is commonly referred to as either the tube model or the reptation theory. In this paper we are concerned uniquely by the second approach, and for sake of clarity we only give a succinct overview of the key ideas and borrow some inspiration from a few of the acclaimed references like e.g. [START_REF] Doi | The Theory of Polymer Dynamics[END_REF], [START_REF] Huilgol | Fluid Mechanics of Viscoelasticity[END_REF], [START_REF] Larson | Constitutive Equations for Polymer Melts and Solutions[END_REF], [START_REF] Macosko | Rheology: Principles, Measurements, and Applications[END_REF], [START_REF] Morrison | Understanding Rheology[END_REF], [START_REF] Lin | Polymer Viscoelasticity: Basics, Molecular Theories and Experiments[END_REF], to which we equally refer the interested reader for detailed presentations.

In a melt polymer material sample, the macromolecular chains are highly entangled with each other, hence the motion of a single chain is strongly influenced by the collective presence and dynamics of the surrounding neighbours. Such an influence is geometrically idealized by considering an individual chain as being confined to a tube with deformable walls -hence the name of the theory. Because the tube largely hinders the transverse motion, the large scale geometric rearrangement of the chain configuration takes place inside the tube walls via a snakelike reptational motion along its own length. As the chain is geometrically approximated as a concatenation of deformable segments (one encounters a many body problem), in the de Gennes -Doi -Edwards theory a chain segment is subjected to a three-stage consecutive motion: a sudden deformation immediately followed by a quick process of retraction to its original length, process continued by a slower reptation motion of relaxation out of the original tube. Actually the reptational diffusion also affects the shape of the both tube extremities in this way: the part of the tube which is emptied as the chain retracts will cease to exist (thus a constraint is released) while a new tube portion is formed at the other extremity and surrounds the diffusing chain. Next, all these dominant interactions are accounted for in a so-called configurational probability density diffusion equation -typically a parabolic partial differential equation (which may be deterministic or stochastic in nature). Its solution constitutes the key ingredient for obtaining the stress tensor which ultimately enters the momentum balance system of equations describing the macroscopic fluid motion. For some recent modeling results of the tube model in applied rheology see [START_REF] Ciuperca | Existence and uniqueness results for the Doi-Edwards polymer melt model: the case of the (full) nonlinear configurational probability density equation[END_REF], [START_REF] Ciuperca | On the IAA version of the Doi-Edwards model versus the K-BKZ rheological model for polymer fluids: A global existence result for shear flows with small initial data[END_REF], [START_REF] Palade | On slow flows of the full nonlinear Doi-Edwards polymer model[END_REF] and references cited therein.

The reptation theory (together with the kinetical approach) is popular in the area of polymer rheology. Since it was published in its original form in 1978-79 (see [START_REF] Doi | The Theory of Polymer Dynamics[END_REF]) it has been subjected to a multitude of modifications, refinements, improvements to make it meet the needs of better predicting an ever wider variety of rheological experimental data; in turn this however made modeling necessarily more complex. Such undertakings include -but not restricted to -tube length diameter fluctuations, constrained release, fluctuating slip links and slip springs, inhomogeneities associated with density fluctuations. Many of these theoretical improvements were tested via advanced Monte Carlo simulations performed on stochastic equations.

For instance, while focusing on the density fluctuations and their relation to quantities like slip spring distribution (see e.g. [START_REF] Ramírez-Hernández | A multichain polymer slip-spring model with fluctuating number of entanglements for linear and nonlinear rheology[END_REF], [START_REF] Ramírez-Hernández | A multi-chain polymer slip-spring model with fluctuating number of entanglements: Density fluctuations, confinement, and phase separation[END_REF]), Schieber [START_REF] Schieber | In a personal communication to L. I[END_REF] has recently authored a density fluctuations stochastic partial differential equation, a simplified version of which is the focus of this work: see the below given problem (1.1)-(1.5). All quantities are dimensionless.

Let ρ(t, s) be a stochastic unknown process, with s ∈ [L -, L + ] being the space variable (related to a tube length being equal to 2L) and t ∈ [0, T ] being the time variable. Let µ(t, s) be the chemical potential, W (t, s) a Wiener process (possibly a space-time white noise, a time white noise or a finite-dimensional brownian motion).

We are interested in the following formal stochastic partial differential equation

∂ t ρ = 1 2 ∂ ∂s 1 ρ ∂µ ∂s dt + ∂ ∂s dW √ ρ , s ∈ [L -, L + ] (1.1)
where the boundary conditions are stochastic in nature:

dL ± = - 1 ρ 1 2ρ ∂µ ∂s T dt + dW √ ρ s=L ± . (1.2)
As is common in thermodynamics, the (• • • )| T notation indicates parameter (temperature) T is kept constant, we will skip this notation later. The free energy (Hamiltonian) E of a generic unknown ρ is given by

E(ρ) = [L -,L + ] ρ(•, s) + 1 ρ(•, s) ds = [L -,L + ] V (ρ(•, s))ds, (1.3)
with V the obvious potential.

The chemical potential µ is usually given by the functional derivative of the free energy

µ = δE δρ = 1 - 1 ρ 2 (+C) (1.4)
but the above chemical potential may be defined up to an additive constant C which is used as a Lagrangian multiplier in various models. However its presence here is impactless for only the gradient of the chemical potential will be used. Equation (1.1) is compatible with total mass conservation law and is also expected to exhibit an equilibrium probability density (invariant measure) P eq given by

dP eq (ρ) = 1 Z exp [-E(ρ)] 1 ρ≥0 dγ(ρ) (1.5)
where γ is a reference measure in some space, and Z is a normalization constant.

The originally proposed free energy [START_REF] Schieber | In a personal communication to L. I[END_REF] contained the extra α ∂ρ(s) ∂s 2 term in the integrand of equation ( 1.3) and the extra -2α ∂ 2 ρ ∂s 2 term in equation (1.4), with α ≥ 0 being a model parameter. In the full model, when a gradient term is added to the free energy, the above introduced singular problem pertains to the Cahn-Hilliard-Cook family of equations (with a singular mobility, singular potential and singular multiplicative noise). Specifically:

E α (ρ) = [L -,L + ] ρ(•, s) + 1 ρ(•, s) + α ∂ρ(•, s) ∂s 2 ds, (1.6) µ = δE α δρ = 1 - 1 ρ 2 -2α ∂ 2 ρ ∂s 2 (+C). (1.7)
In this paper, taking (1.1)-(1.5) as starting point, we undertake to study the n-dimensional generalized problem by considering a regularized expression of the full problem, using an abstract formulation which encodes all the difficulties in the classical approach of SPDEs. We detail the expected concept of solution that could remain unaltered in the singular case. We also explain why a singular version of the 1-dimensional problem is well-posed with the extra-α-term in (1.6)-(1.7) but under the assumption of additive noise.

Our main result is the following existence theorem (with the meaning of the solution concept being detailed later in the text): Theorem 1.1 (Main Result). Let ρ 0 , defined on a given space domain, be a smooth enough initial probability density.

• Given a regular abstract formulation (possibly in the n-dimensional space), then, for all T > 0, there exists a unique solution (ρ t ) t∈[0,T ] to the equation (1.1). Moreover, under strict monotonicity, for T → +∞, the distribution of the solution converges to an equilibrium density (invariant measure).

• Given a singular abstract formulation in the 1-dimensional space but with additive noise and the extra-term with α > 0, then, for all T > 0, there exists a unique solution (ρ t ) t∈[0,T ] to the equation (1.1). Moreover, the distribution of the solution converges to the unique equilibrium density (invariant ergodic measure) given by equation (1.5) with γ a reference Gaussian measure.

This paper is organized as following: Section 2 introduces the concept of solutions based on the general theory of SPDEs. We prove that some a priori estimates are valid for the introduced solutions. It also contains the problem under scrutiny together with the appropriate physical considerations. In Section 3, we introduce a regular abstract formulation to present the essential difficulties which prevent us of using classical techniques about the existence of solutions. In Section 4, where the main result Theorem 4.1 is proved, we introduce a simplified formulation by circumventing selected difficulties which were presented in the previous section. However, we can keep a singular potential which constitutes the core of the physical model. Next, the path-wise uniqueness is given in Section (5) for all the previous here considered variants of the original model. The existence and convergence of invariant measures are presented in Section [START_REF] Cahn | Free Energy of a Nonuniform System. I. Interfacial Free Energy[END_REF]. We eventually draw conclusions for this paper model, introduce a conjecture about the existence and uniqueness of solutions for a general problem in Section [START_REF] Cahn | Spinodal Decomposition: a reprise[END_REF].

Problem Presentation and Main Results

At first sight, this equation could be seen as a Cahn-Hilliard-Cook stochastic partial differential equation, and in simple cases, the existence and uniqueness could be obtained by known results. The Cahn-Hilliard-Cook equation is a model that describes phase separation in a binary alloy (see [START_REF] Cahn | On spinodal decomposition[END_REF], [START_REF] Cahn | Free Energy of a Nonuniform System. I. Interfacial Free Energy[END_REF] and [START_REF] Cahn | Spinodal Decomposition: a reprise[END_REF]) in the presence of thermal fluctuations (see [START_REF] Cook | Brownian motion in spinodal decomposition[END_REF] and [START_REF] Langer | Theory of spinodal decomposition in alloys[END_REF]). It generally reads:

             ∂ t ρ = 1 2 ∇ • (M(ρ)∇µ) + ξ on D ⊂ R n , µ = V ′ (ρ) -2α∆ρ, on D ⊂ R n , ∇ρ • ν = 0 = ∇µ • ν, on ∂D, (2.1)
where t denotes the time variable, α is a small non-negative parameter, ν is the outward normal on the boundary ∂D of the space domain D, ∇• and ∇ are respectively the divergence and gradient operators in the space variable s ∈ D, and ∆ is the Laplace operator. The variable ρ represents the ratio between the two species and the noise term ξ accounts for the thermal fluctuations. For this kind of equation, a key point is that the average D ρ ds is a conserved quantity.

In the Cahn-Hilliard model, the nonlinear term V ′ is based on a double-well shaped potential V (hence non-convex), and the function M represents the mobility (a diffusion velocity). Usually the mobility is a constant or the function M : ρ → (1ρ 2 ). In the deterministic case, the equation is obtained by taking the gradient of the free energy E in H -1 (D) but on a fixed domain D.

Since the gradient operates in H -1 (D), a fourth order equation is obtained. Thus, if α > 0, the presented model is of Cahn-Hilliard-Cook type, but with moving domain, singular mobility and multiplicative noise. Indeed, in this new model, the mobility

M : ρ → 1/ρ, ( 2.2)
is singular leading also to a singular term in front of the noise. Indeed, by fluctuation-dissipation theorem, we expect the square root of the mobility in front of the noise (see Equation (1.1)).

This new model will lead to multiple mathematical difficulties, in particular concerning the multiplicative noise, which could be a hard question. All of these difficulties will not be solved in this paper, but we will give hints for possible direction of improvements. The aim of this article is to give the first mathematical structure as a starting point for future research in tube model and reptation theory.

The deterministic Cahn-Hilliard equation where V is a polynomial function has first been studied in [START_REF] Cahn | Free Energy of a Nonuniform System. I. Interfacial Free Energy[END_REF], [START_REF] Langer | Theory of spinodal decomposition in alloys[END_REF] and [START_REF] Novick-Cohen | Nonlinear aspects of the Cahn-Hilliard equation[END_REF]. The drawback of a polynomial nonlinearity is that the solution is not constrained to remain in the physically relevant interval. Singular nonlinear terms such as the logarithmic or negative power nonlinearity do remedy this problem. Such non smooth functions V have also been considered (see [START_REF] Blowey | The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis[END_REF] and [START_REF] Debussche | On the Cahn-Hilliard equation with a logarithmic free energy[END_REF]). This stochastic Cahn-Hilliard-Cook equation has also been first studied in the case of a polynomial nonlinearity: see [START_REF] Blömker | Spinodal decomposition for the Cahn-Hilliard-Cook equation[END_REF], [START_REF] Blömker | Phase separation in Stochastic Cahn-Hilliard Models[END_REF], [START_REF] Cardon-Weber | Cahn-Hilliard stochastic equation: existence of the solution and of its density[END_REF], [START_REF] Cardon-Weber | Cahn-Hilliard stochastic equation: strict positivity of the density[END_REF], [START_REF] Prato | Stochastic Cahn-Hilliard equation[END_REF], [START_REF] Elezovic | On the stochastic Cahn-Hilliard equation[END_REF]. Again, with a polynomial nonlinear term the solutions do not in general remain in the physical space of interest. It actually gets even worse for in the presence of noise any solution leaves this interval immediately with positive probability. Of notice, however, in the work of Debussche and Goudenège [START_REF] Debussche | Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections[END_REF][START_REF] Goudenège | Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection[END_REF] and that of Debussche and Zambotti [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF], the solution is forced to stay in the physical interval thanks to a singular potential.

In the case considered here, the parameter α may vanish, the mobility is highly singular, the stochastic term is a singular multiplicative noise, and the potential V is strictly convex, thus the equation is not of Cahn-Hilliard-Cook type, and we will need a new abstract formulation to treat this new non-linear density fluctuations equation. On the contrary, in the case of additive noise assumption, with the extra-α-term and regular mobility, we can consider a potential V (ρ) = ρ + 1/ρ in the energy E α since it is related to the model described in [START_REF] Goudenège | Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection[END_REF].

It would be preposterous to assume studying such a model is a classical undertaking. Henceforth conspicuously obtaining a preliminary result with an abstract formulation or simplified term potential is all the more necessary and useful. We now refer the reader to the classical work of Da Prato and Zabczyk [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF] for notations, basic notions and concepts.

We describe the concept of solution for a regular potential V (at least of class C 2 and convex) and a regular mobility M (at least of class C 1 and non-negative). Also observe the concept of solution (valid for smooth potential) may remain unaltered in the singular case, but integrability conditions should be changed to give a clear meaning to all involved terms. In Section 3, we will simplify some assumptions and keep others unchanged. In Section 4, we will also simplify some aspects and keep the others unchanged, and the integrability conditions will be detailed. Definition 2.1. Let ℓ -and ℓ + with ℓ -< ℓ + be fixed. Let ρ 0 ∈ C([ℓ -, ℓ + ]; R * + ) be the initial density in the space of probability measures P. We say that (L

-(t), L + (t), ρ(t), W t ) t∈[0,T ] , defined on a filtered complete probability space Ω, P, F , (F t ) t∈[0,T ] , is a weak solution to (1.1) on [0, T ] for the initial condition ρ 0 if we have (a) a.s. L ± ∈ C ([0, T ]; R) with L -(t) < L + (t) for all t ∈ [0, T ] and L ± (0) = ℓ ± , (b) ρ ∈ L 2 Ω; C [0, T ]; C [L -, L + ]; R * + ∩ L 2 [0, T ]; H 1 [L -, L + ]; R * + and ρ(0) = ρ 0 , (c) µ = V ′ (ρ) ∈ L 2 Ω; C [0, T ]; C 1 ([L -, L + ]; R) and µ(0) = V ′ (ρ 0 ), (d) W is a continuous version of Brownian motion, (e) the process (ρ(t, •), W t ) is (F t )-adapted, (f ) the mobility M is such that M(ρ) ∈ L 2 Ω; L 1 [0, T ]; L 1 ((L -, L + ); R + ) , (g) the operator Q is such that Q(ρ) ∈ L 2 Ω; L 2 [0, T ]; L 2 ((L -, L + ); R + ) , also we assume that Tr(Q * ∆Q) is controlled by C Q,1 + C Q,2 ρ 2 .
(h) for all h ∈ C 4 (R; R) and for all 0 ≤ t ≤ T :

ρ(t, •), h = ρ(0, •), h - 1 2 t 0 M(ρ(r, •))∇µ, ∇h dr - t 0 ∇h, Q(ρ(r, •))dW r -α t 0 ρ(r, •), ∆ 2 h dr.
We point out that the continuity of L ± allows to consider functions on compact sets which simplifies many technical points, and the scalar product are defined on [L -, L + ] assuming the quantity vanishes outside this domain. Since ρ > 0, then its infimum is strictly positive, the chemical potential µ is well defined even if V has singularities at point 0, and M(ρ) is L 1 for the same reason. In the case of a Wiener process in abstract space U at point (d), the condition (g) could be set as: Q(ρ) is Hilbert-Schmidt from U to the Hilbert space H = L 2 (D) with maximal domain D and inner product •, • . Actually in Definition (2.1) we provided details on all the needed properties to make sure all terms in the last point (h) clearly make sense.

We shall now, step by step, proceed to proving the afore announced main result. Before engaging in this undertaking, we pause for several short clarifying comments. On one hand, it is worthy stating the proof may not be particularly challenging if one considers the equation as being a system gradient in infinite dimension, by using the Da Prato and Zabczyk [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF] framework and by additionally assuming the noise to be an additive space-time Wiener process. In this case there will be a unique invariant measure, given by a Gibbs-like formula, which is compatible with the expected equilibrium probability density in equation (1.5). On the other hand now, there are multiple obstacles to be dealt with as one proceeds along those lines. For instance, one is the gradient structure needing more clarifications. Another difficulty concerns the moving boundary, since the space domain is not fixed. This later one seems however not of a hindering nature as the imposed boundary condition has been constructed to cancel supplementary terms coming from integration by parts formula.

In order to explore the concept of solutions, we will detail some a priori estimates and facts based on the existence of smooth solutions. It gives strong hints that the equation is well-posed and satisfied expected physical properties.

Proposition 2.1. Consider a solution in the sense of Definition 2.1 up to time T > 0, then the solution lives in L 2 (Ω; L ∞ ([0, T ]; L 2 ([L -, L + ]))). Moreover, with α = 0, for chemical potential µ given by (1.4) and mobility M given by equation (2.2), then the energy E of the solution is well defined for all t ∈ [0, T ].

Proof. Let α = 0, by Itô's formula, we have

1 2 d ρ 2 = dρ, ρ = - 1 2 M(ρ)∇µ, ∇ρ dt -∇ρ, Q(ρ)dW t + 1 2 Tr(Q * (-∆)Q)dt = - 1 2 M(ρ)∇ρV ′′ (ρ), ∇ρ dt -∇ρ, Q(ρ)dW t + 1 2 Tr(Q * (-∆)Q)dt ≤ -∇ρ(t, •), Q(ρ(t, •))dW t + (C Q,1 + C Q,2 ρ 2 )dt
using that V is a convex potential and M is a non-negative mobility, and C Q,1 and C Q,2 are finite constants invoking that the operator Q has enough boundedness properties (almost linear growing, Lipschitz assumption, etc.). Here, since the noise is a one-dimensional Brownian motion, the assumption in point (g) could be simplified. But remark that the classical assumption about finite trace may be used here to generalize to Hilbert-Schmidt type operators and spacetime Wiener processes. Using that the stochastic term is a martingale with null expectation, by Gronwall's Lemma, we obtain a bound for all t ∈ [0, T ], which leads to a bound for the L ∞ norm such that

E sup t∈[0,T ] ρ(t, •) 2 ≤ E ρ(0, •) 2 + C T = ρ 0 2 + C T < +∞
for some positive constant C T possibly depending on ρ 0 , T, C Q,1 and C Q,2 . At this point it is clear that the initial data ρ 0 can be randomized (with bounded second moment) to obtain exactly the same result. If we have α > 0, the same computation leads to

E sup t∈[0,T ] ρ(t, •) 2 + 2α T 0 ∆ρ(t, •) 2 dt ≤ ρ 0 2 + C T < +∞
which is eventually a stronger result.

Concerning the energy, with α = 0, using that almost surely ρ is a continuous non-negative function on compacts spaces, hence bounded for below by a non-negative constant ε > 0, the Itô's terms may be bounded, such that all the terms are well-defined using explicit truncation, and we obtain by Itô formula

d ρ + 1 ρ = dρ, 1 - 1 ρ 2 = dρ, µ = - 1 2 M(ρ)∇µ, ∇µ dt -∇µ, Q(ρ)dW t + 1 2 T r(Q * ∇ * H(ρ)∇Q)dt
where H is the Hessian operator of the function x → x + 1/x. Using again that the mobility is non-negative, we obtain the following inequality

d ρ + 1 ρ ≤ -∇µ, Q(ρ)dW t + C(Q, ρ, ε)dt (2.3)
with a random function C(Q, ρ, ε) which depends quadratically on Q and ρ, but certainly on the truncated parameters ε in unbounded manner. Using Doob's inequality for the martingale term, the assumptions about integrability in Definition 2.1 will be enough to obtain

sup t∈[0,T ] E(ρ(t, •)) ≤ E(ρ(0, •)) + sup t∈[0,T ] t 0 | ∇µ, Q(ρ)dW t | + t 0 |C(Q, ε, ρ)| dt < +∞.
Assuming that we can take the expectation in Equation (2.3) to suppress the martingale term, we obtain a bound in expectation such that

E sup t∈[0,T ] E(ρ(t, •)) ≤ E [E(ρ 0 )] + E T 0 C(Q, ε, ρ)dt
which is not independent of the truncated parameter ε, thus potentially infinite when passing to the limit or taking expectation. Remark again that the initial data can be randomized (with bounded inverse first moment, i.e. bounded energy) to obtain the same result.

Finally, in the case α > 0, the same computation leads to

E sup t∈[0,T ] E(ρ(t, •)) + 1 2 inf t∈[0,T ] |M(ρ(t, •))| T 0 ∇µ 2 dt ≤ E(ρ 0 ) + E T 0 C(Q, ε, ρ)dt + α T 0 ∆ρ, ∆µ dt
but the last term is not controlled by integrability condition given in Definition 2.1, and it seems unreasonable to expect such controlability.

Regular abstract formulation

In this section, we will assume that the equation can be written in a classical abstract form dX t + A(X)dt = B(X)dW t with operators satisfying assumptions that will be detailed below. Considering this form, the proof of existence and uniqueness of solution is based on classical theorems the detailed presentations of which are given in [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF], [START_REF] Gyongy | Existence of strong solutions for Ito's stochastic equations via approximations[END_REF] or [START_REF] Prévôt | A Concise Course on Stochastic Partial Differential Equations[END_REF]. Precisely, we assume to have a Gelfand triple (V, H, V * ), where V is a reflexive Banach space and H is an Hilbert space, and a space of Hilbert-Schmidt operators L HS (U, H) on some abstract space U to define the Wiener process. We write the operators on these spaces A : V → V * and B : V → L HS (U, H). The notation V * •, • V is the canonical notation for the duality bracket between the spaces V * and V, compatible with the inner product •, • on the space H. We assume that they satisfy the following assumptions:

• (Hemicontinuity) For all u, v, w ∈ V λ → 2 V * A(u + λv), w V is continuous. • (Weak Monotonicity) There exists c ∈ R such that for all u, v ∈ V 2 V * A(u) -A(v), v -u V + B(u) -B(v) 2 HS ≤ c u -v 2 H . • (Coercivity) There exist q ∈ ]1, ∞[, c 1 ∈ R, c 2 ∈]0, ∞[ and an F t -adapted process f ∈ L 1 (0, T ) such that for all u ∈ V 2 V * A(u), u V + B(u) 2 HS ≤ c 1 u 2 H -c 2 u q V + f (t).
• (Boundedness) There exist c 3 ∈ [0, ∞[ and an F t -adapted process g ∈ L q q-1 (0, T ) such that for all u ∈ V A(u) V * ≤ c 3 u q-1 V + g(t). It is important to notice that many problems have been solved assuming that an abstract formulation actually exists. Indeed, the moving domain [L -, L + ] will be certainly encoded in the concept of solution as X t being (L -, L + , ρ t ). Or, under reformulation, we could assume that the domain is fixed and given by the initial domain [ℓ -, ℓ + ], which solves the point (a). Moreover, the fiber bundles in points (b) and (c) of Definition 2.1 are now well-defined as classical spaces. Finally, the mobility M and the operator Q are included in the operator A and B, which will ensure the points (f ) and (g). Even the concept of chemical potential µ has been included in the operator A. Within this framework, the concept of solution introduced in 2.1 will be simplified since we can define a new one.

One has: Definition 3.1. We say that (X(t), W (t)) t∈[0,T ] , defined on a filtered complete probability space Ω, P, F , (F t ) t∈[0,T ] , is a weak solution to (1.1) on [0, T ] for the initial condition X 0 if we have

(b') X ∈ L 2 Ω; C ([0, T ]; H) ∩ L 2 ([0, T ]; V) and X(0) = X 0 , (d')
W is a continuous version of Brownian motion (or Wiener process), (e') the process (X(t), W (t)) is (F t )-adapted, (h') for all h ∈ V * and for all 0 ≤ t ≤ T :

X(t), h = X(0), h - t 0 A(X(r)), h dr + t 0 B(X(r)), hdW r .

Existence of solution

One has the following existence Theorem, standing actually for the first part of Theorem 1.1.

Theorem 3.1. Suppose that there exists a regular abstract formulation satisfying classical assumptions. Given an initial data in the Banach space V, then, for all T > 0, there exists a unique solution (X t ) t∈[0,T ] in the sense of Definition 3.1 up to time T .

Proof. We have supposed that the classical assumptions are satisfied, thus we do not have to verify them. However, we will show how they may be obtained with assumptions on mobility M, potential V and operator Q, or on regularized versions of these objects. Usually, to prove the existence of a solution of a singular equation, we need to prove that a regularized version of the equation is well-posed with estimates being independent of the regularization procedure. The possible regularized version may be some finite dimensional approximation (like Galerkin), a truncation of the non-linearity, a Yosida regularization, or a numerical approximation. Thus, all future approaches will have to use the following calculations and to prove that the estimates are independent of the regularization procedure.

In Theorem 3.1, we proved that some solution could exist in a fixed domain and that the Itô's formula (being used for a priori estimates) suggests that the operator A has the form

A : X → 1 2 M(X)V ′′ (X)∇X,
• plus extra additive terms involving Laplace operator when α = 0. It also suggests that the operator B has the form B : X → (w → •, ∇ * Q(X)w ). Denote a := x → M(x)V ′′ (x) the non-linearity (or, more precisely, a regularized version of it) to simplify the notations in the following computations. It is assumed that it is non-negative, as suggested by the expected physical properties of the convex potential V and the non-negative mobility M. We now illustrate the difficulties in obtaining the expected bounds.

Hemicontinuity.

Suppose that the function a is continuous and bounded from below by non-negative constant (locally on balls of V). Let u, v, w ∈ V. Let (λ k ) k∈N converging to λ as k grows to infinity, then

|2 V * A(u + λ k v) -A(u + λv), w V | = | a(u + λ k v)∇(u + λ k v) -a(u + λv)∇(u + λv), ∇w | ≤ | (a(u + λ k v) -a(u + λv)) ∇(u + λ k v), ∇w | + | a(u + λv)(λ k -λ)∇v, ∇w | .
Since the sequence λ k is bounded, thus the sequence u + λ k v lives in a ball B in V. On this ball, the function a is continuous and bounded. Let ǫ > 0 be fixed, there exists δ ǫ > 0 small enough such that |λ k -λ| < δ ǫ implies |a(u + λ k v)a(u + λv)| < ǫ and thus for large index k ∈ N we have that the first term is bounded by ǫ sup x∈B x V w V and the second term is bounded by

δ ǫ |a(u + λv)| v V w V ,
which ensures convergence to 0 and continuity.

Weak Monotonicity. Let u, v, w ∈ V, then 2 V * A(u) -A(v), v -u V + B(u) -B(v) 2 HS = a(u)∇u -a(v)∇v, ∇(v -u) + ∇ * Q(u) -∇ * Q(v) 2 H = - D a(u)|∇(u -v)| 2 + D (a(u) -a(v))∇v • ∇(v -u) + ∇u Q ′ (u) -∇v Q ′ (v) 2 H .
At this point, in order to obtain the bound c uv 2 H , we see that the first term is not a problem since a is assumed to be non-negative. It can help to control other problematic terms. For the second term, we certainly need some globally Lipschitz regularity for the function a, but which is not precisely clear at this point since mobility and potential are singulars. For the third term, we certainly need the regularity and boundedness of the derivative of Q. Remark that in the additive case (B is a constant), this term does not appear. Remark also that if it exists an extra term with α > 0 involving Laplace operator, then we have an other term with the good sign which can help to control the problematic terms.

Coercivity.

The coercivity corresponds to a control at infinity of the equation. The leading term is c 1 u 2 H , which should absorb the operator V * A(u), u V , B or a supplementary term c 2 u q V . Consider the equation

2 V * A(u), u V + B(u) 2 2 + c 2 u q V = -a(u)∇u, ∇u + ∇u Q ′ (u) 2 H + c 2 ∇u, ∇u q + c 2 u, u q .
We have to bound these terms by c 1 u 2 H with c 1 ∈ R but remember that c 2 > 0. It is clear that the term c 1 u 2 H will not control the terms involving u V since it would imply a Poincaré inequality in reversed order, thus the only term with the good sign is the first one. This term should control the term with the constant c 2 which forces the constant c 2 not to be large, and a should be lower bounded by constant c 2 > 0. The second term involves also the gradient of u, thus it is a supplementary term to control. The last term involves only the norm of u in the Hilbert space H thus it is controlled by c 2 (1 + u 2 ) at least when 1 < q ≤ 2.

Boundedness.

Recall that the norm in V * is defined as the supremum of the quantity

V * A(u), v V when v V ≤ 1. Let v ∈ V,
we are looking for an estimates

| V * A(u), v V | := 1 2 | a(u)∇u, ∇v | ≤ c 3 u q-1 V + g.
It is now clear that the constant c 3 should control the function a which is expected to be bounded from above.

Simplified formulation but with singular potential

In this section, we can study other formulations which are related to the problem introduced earlier. In particular, if we assume to work in dimension 1 in space, the noise is additive and that we have the extra-term containing α > 0, we can prove the existence of unique solution even with a singular potential V of the form ρ + 1/ρ. Furthermore, using formal computations, we can derive an equivalent problem with a modified chemical potential of the form -1/ρ 3 . This equation could also be solved in the same previous simplified framework. Moreover, in both cases, the solution will converge to the unique equilibrium density (invariant measure) given by the formula (1.5) with γ a reference Gaussian measure.

First, assuming that all the following computations are legitimate, we can derive an equivalent formulation of the problem to obtain a similar form.

One has:

Proposition 4.1. Assume the solution is non-negative and continuously differentiable. Then the statement (h) in 2.1 is equivalent, for all h ∈ C 4 (R; R) and for all 0 ≤ t ≤ T , to

ρ(t, •), h = ρ(0, •), h - 1 3 t 0 1 ρ(r, •) 3 , ∆h dr - t 0 ∇h, Q(ρ(r, •))dW r -α t 0 ρ(r, •), ∆ 2 h dr.
Proof. Assuming the solution is non-negative, the quantities M(ρ) = 1 ρ and µ = 1 -1 ρ 2 are well-defined. It will justify the study of the chemical potential µ = Cst -1 ρ 2 when the mobility M(ρ) is fixed (for instance by the previous step in a Picard's fixed-point iteration procedure). Moreover, assuming regularity, we can derive all these quantities. Finally, since we assume the differentials are continuous, all the implied terms are L ∞ and integrable. Let h ∈ C 4 (R; R); we have the following

- 1 2 M(ρ)∇µ, ∇h = - 1 2 1 ρ ∇ - 1 ρ 2 , ∇h = - 1 2 2 ρ 4 ∇ρ, ∇h = - 1 2 2 3 ∇ - 1 ρ 3 , ∇h = - 1 3 1 ρ 3 , ∆h .
Of notice: for the last integration by parts formula (above), it is assumed that boundary term vanishes, fact which is ensured by selecting appropriate Neumann boundary conditions, which means the probability flux vanishes on the domain boundary. Moreover the equation (1.2) is consistent with the fact that ∇h • n vanishes on the boundary.

We can also stop the calculations at the prior to the last line: in this case one observes it corresponds to considering a constant mobility M = 2 3 but a chemical potential µ := Cst -1 ρ 3 , for all constants Cst.

Next, again based on the definition 2.1, we give the definition of a solution based on the form suggested by previous Proposition 4.1 for the chemical potential. Definition 4.1. We say that (ρ(t), W (t)) t∈[0,T ] , defined on a filtered complete probability space Ω, P, F , (F t ) t∈[0,T ] , is a weak solution to (1.1) on [0, T ] for the initial condition ρ 0 , if a.s. we have:

(b') ρ ∈ C ([0, T ] × [0, 1]; R + ) ∩ C ([0, T ]; V -1 ) and X(0) = X 0 , (c') the process µ = Cst -1/ρ 3 (or µ = Cst -1/ρ 2 ) ∈ L 1 [0, T ]; L 1 ((0, 1); R + ) (d') W is a cylindrical Wiener process in L 2 (0, 1), (f ') the mobility M is constant, (g') the covariance is constant (additive noise), (h') for all h ∈ D(∆ 2
) and for all 0 ≤ t ≤ T :

ρ(t), h = ρ 0 , h + M 2 t 0 µ(r), ∆h dr - t 0 ∇h, dW r -α t 0 ρ(r), ∆ 2 h dr
Now we have the following:

Theorem 4.1. Consider the equation (1.1), with constant mobility, and constant additive noise, with initial density ρ 0 ∈ C([0, 1]; R + ) (thus nowhere vanishing) having null Neumann condition on the boundary. Then, for all T > 0, there exists a unique solution (ρ t ) t∈[0,T ] in the sense of Definition 4.1 up to time T . However, in the case µ = Cst -1/ρ 2 , an additional reflection term should be added to force the solution to stay positive. It takes the form Ah, η for η a time-space measure such that supp(η) ⊂ {(t, s) ∈ [0, T ] × (0, 1) : ρ(t, s) = 0}.

Proof. There are two cases to address: µ = Cst -1/ρ p , for p = 2, 3.

The proof is based on a regularization procedure to define approximated solutions of the problem. Precisely, we have to regularize the potential V (ρ) = Cst ρ + 1/ρ (whose derivative is µ = Cst -1/ρ 2 ) rendering it a dissipative potential V n in order to apply the classical results of existence of solution to gradient system. The same applies to potential V (ρ) = Cst ρ + 1/2ρ 2 whose derivative is µ = Cst -1/ρ 3 . For these two approximated problems, the solutions will exist up to any time T > 0.

Denoting ∆ the Laplace operator on (0, 1) with null Neumann boundary condition and

V -1 := D((-∆) 1
2 ) the reference Hilbert space, we are in the context of gradient systems. Applying Theorem 2.2 of [START_REF] Goudenège | Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection[END_REF], we have the existence of solution up to a reflection measure, which should force the solution to stay in the space K of positive solutions.

Moreover, using Theorem 3.1 in [START_REF] Goudenège | Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection[END_REF], in the first case µ := Cst -1/ρ 2 , since the negative power in the associated chemical potential is less than 3, the measure does not vanish.

However, using Theorem 3.1 in [START_REF] Goudenège | Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection[END_REF], in the second case µ := Cst -1/ρ 3 , since the negative power in the associated chemical potential is greater than 3, the measure vanishes.

Path-wise Uniqueness

The path-wise uniqueness will be proven in both cases with the same strategy using Gronwall's Lemma.

Proposition 5.1. Let (ρ i , W ), i = 1, 2 be any pair of weak solutions of (1.1) defined on the same probability space with the same driving noise W , and starting from the same initial data ρ 0 , then ρ 1 (t) = ρ 2 (t) for all t ≥ 0.

Proof. In the abstract formulation, we can follow Proposition 4.2.10 of [START_REF] Prévôt | A Concise Course on Stochastic Partial Differential Equations[END_REF] to obtain that there exists a constant c ∈ R such that we have

E ρ 1 (t) -ρ 2 (t) ≤ exp(ct)E ρ 1 (0) -ρ 2 (0) .
which is valid for a certain c ∈ R under the assumption of weak monotonicity.

Assuming the same initial data ρ 1 (0) = ρ 2 (0) = ρ 0 , the previous result gives the uniqueness, but also a stronger result about sensitivity with respect to the initial data.

In the simplified formulation with singular potential, the uniqueness follows from dissipativity of the non-linearity. However, in Definition 4.1, we only have continuity in the space V -1 , thus the norm should be taken in V -1 (this space is playing the same role than the Hilbert space H of Definition 3.1). Let ρ(t) = ρ 1 (t)ρ 2 (t) the difference between the two solutions starting from initial data ρ 0 . Due to the additive noise assumption, ρ is solution of a random equation (without stochastic integral)

dρ dt = M 2 ∆(µ 1 -µ 2 ) -α∆ 2 (ρ 1 -ρ 2 )
Let now ρ N = [(-∆) -1 P roj N ]ρ with P roj N being a projector operator on first N eigenvectors of (-∆). Taking the scalar product in H with ρ N and integrating in time, we obtain

|ρ N (t)| 2 -1 = |ρ N (0)| 2 -1 -M t 0 µ 1 (r) -µ 2 (r), ρ N (r) dr -2α t 0
(-∆) 1/2 ρ N (r), (-∆) 1/2 ρ N (r) dr.

(5.1) Since µ i , i = 1, 2 are in L 1 , we have for almost all r ∈ [0, t],

µ 1 (r) -µ 2 (r), ρ(r) -ρ N (s) ≤ ρ(r) -ρ N (r) L ∞ ([0,1]) µ 1 (r) -µ 2 (r) L 1 ([0,1]) , (5.2)
where • L ∞ ([0,1]) and • L 1 ([0,1]) are the classical Lebesgue norms on the space [0, 1]. The latter term converges to zero since ρ N (r) converges uniformly to ρ(r) on [0, 1]. Moreover, for both cases µ = Cst -1/ρ p with p = 2, 3, we have

-µ 1 (r) -µ 2 (r), ρ(r) = 1 (ρ 1 (r)) p - 1 (ρ 2 (r)) p , ρ(r) ≤ (ρ 2 (r) -ρ 1 (r))((ρ 2 (r)) p-1 + (p -2)ρ 1 (r)ρ 2 (r) + (ρ 1 (r)) p-1 ) (ρ 1 (r)ρ 2 (r)) p , ρ 1 (r) -ρ 2 (r) ≤ 0 (5.3)
Injecting (5.2) into (5.1), using the sign coming from (5.3), taking the limit in (5.1) as N grows to infinity, we obtain for all t ≥ 0:

|ρ 1 (t) -ρ 2 (t)| 2 -1 ≤ |ρ 1 (0) -ρ 2 (0)| 2 -1 = 0, thus ρ 1 (t) = ρ 2 (t) for all t ≥ 0.

Convergence of Invariant Measures

We say that the measure Π is an invariant measure for the stochastic PDE (or the associated semi-group) if, for a random variable ρ 0 , whose distribution is Π, the solution (ρ(t)) 0≤t≤T is such that its distribution at time t is again Π. Such an invariant measure can be easily calculated (for instance in finite dimensional case) as eigenvector of an adjoint operator P * of the semi-group P of the Markovian evolution.

Usually, we prove that an approximated version (described by some index n ∈ N) of the equation possesses a semi-group which is Strong Feller (see .e.g [START_REF] Peszat | Strong Feller Property and Irreducibility for Diffusions on Hilbert Spaces[END_REF] for details) and irreducible. By Doob theorem (see [START_REF] Doob | Stochastic Processes[END_REF]) we deduce that there exists a unique and ergodic invariant measure Π n defined on the abstract Hilbert space H.

Denoting ρ n (t; x) the solution starting form the initial data x ∈ H, we can define for all Borel bounded test functions φ ∈ B b (H) the quantity

P n t φ(x) := E[φ(ρ n (t; x))
] which is the semi-group of the Markovian evolution.

For instance, using Bismut-Elworthy-Li formula (assuming inversibility of covariance operator and other technical assumptions), it can be seen that (P n t ) t≥0 satisfies, for all φ ∈ B b (H), n ∈ N and t > 0, the following estimate

|P n t φ(x) -P n t φ(y)| ≤ C(n) e λt √ t φ ∞ x -y H , for all x, y ∈ H, (6.1)
where we expect that the constant C(n) could be independent of the parameter n ∈ N. Furthermore, the irreducibility is usually proven using a control-type argument, which is quite clear in the case of (non-degenerate) additive noise.

If the variable λ is non-positive, then there is exponential contraction, and the law of the solution converges exponentially fast to the invariant measure.

Regular abstract formulation

Thanks to the previous Section, we have the existence of a unique solution in the regular abstract formulation. For the second part of the Theorem 1.1 one needs to prove there exists an invariant measure to the equation, and that the solution converges to this invariant measure as T → +∞. For this, we will need supplementary assumptions regarding the abstract formulation, precisely a "strict monotonicity", described by the following Lemma: Lemma 6.1 (Lemma 4.3.8 from [START_REF] Prévôt | A Concise Course on Stochastic Partial Differential Equations[END_REF] -strict monotonicity). Assume that the constant c = -λ in the weak monotonicity assumption for some λ ∈ (0, +∞). Let η ∈ (0, λ). Then there exists δ η ∈ (0, +∞) such that for all v ∈ V 2

V * A(v), v V + B(v) 2 HS ≤ -η v 2 H + δ η .
The above is enough to obtain the following: Theorem 6.1. Assume strict monotonicity. Then there exists an invariant measure P eq for the semi-group P t such that H

x 2 P eq (dx) < +∞. (6.2)

Moreover, for any φ : H → R Lipschitz, for any initial data ρ ∈ H and any invariant measure P eq for the semi-group P t , for all t ∈ [0, +∞)

P t φ(ρ) - H φ(x)P eq (dx) ≤ exp -λ 2 t φ Lip H ρ -x H P eq (dx)
In particular, there exists exactly one invariant measure for the semi-group P t with the property that H

x P eq (dx) < +∞.

Proof. The proof of this theorem comes from Theorem 4.3.9 of [START_REF] Prévôt | A Concise Course on Stochastic Partial Differential Equations[END_REF].

Simplified version with singular potential

To prove the existence of invariant measure, first, we need to have a reference measure in the abstract Hilbert space. It is usually given by the asymptotic law of the solution of the linear part of the equation. For instance, if the operators A and B are linear and constant, we can define explicitly the solution Z(t; x) of the linear equation with initial data x ∈ H which is given by Z(t; x) = e -tA/2 x + t 0 e -(t-s)A/2 BdW s . (6.3)

As easily seen this process is in C([0, +∞[; H) (see [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF]). In particular, the mean of Z is constant and the law of the process Z(t; x) is the Gaussian measure on H: V n (-δ x )1 {x≤-δx} ds ≤ exp (-V n (-δ x )Leb({x ≤ -δ x })) .

γ(t) = N e -tA/
By assumption, this latter term converges to zero as n grows to infinity. Now, for x ∈ K, V n (x(s)) converges to V (x(s)) almost everywhere as n grows to infinity. Moreover 0 ≤ V n (x(s)) ≤ V (sup x ), and by the Dominated Convergence Theorem we deduce (6.6). Finally, (6.5) follows again by the Dominated Convergence Theorem.

We know (see [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF][START_REF] Goudenège | Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection[END_REF]) that the approach described in the previous section will also work in the singular context. Moreover the semi-group P t := lim n→+∞ P n t is Strong Feller, such that for all φ ∈ B b (H), for all x, y ∈ H and t > 0:

|P t φ(x) -P t φ(y)| ≤ φ ∞ √ t
xy H . (6.7)

7 Concluding With A Conjecture

We took on to proving the existence and uniqueness of solutions to a previously unstudied class of non-linear stochastic partial differential equations with a singular coefficient that has recently been introduced [START_REF] Schieber | In a personal communication to L. I[END_REF] within the framework of polymer dynamics theories geared towards polymer physics and rheology.

The equations under scrutiny in this paper are simplified versions of the originally suggested for being studied. We have shown that there exists solutions to these approximated problems under classical assumptions and abstract formulation. It appears at this stage that it is plausible to obtain existence of solution results for the more general singular problem just alluded to by extracting properly defined convergent sequences of approximated solutions, and this upon obtaining enough a priori estimates, and passing afterwards to the limit. For instance, using an ε approximation (by Yosida regularization, or globally Lipschitz potential, or dissipative potential), one may achieve the results by compactness related arguments. Using Picard's fixed-point iteration procedure, it seems also plausible to obtain convergence to a solution.

We can state the following conjecture based on the existence of an abstract formulation which encompasses the space of functions and the non-linearity. Let H be this hypothetical, abstract space of functions and assume there exists a potential derivative V ′ , a mobility M and a covariance operator Q well-defined on H.

Then the following conjecture remains to be proved: Moreover, for T → +∞, the distribution of the solution converges to an equilibrium density.

2 x, Σ 2 t , with Σ 2 t = t 0 e

 20 -sA/2 BB * e -sA/2 ds with N (m, Σ 2 ) the standard notation for the Normal Law of mean m and covariance operator Σ 2 . If we let t → +∞, the law of Z(t; x) converges to the Gaussian measure on H:γ := N 0, ∞ 0 e -sA/2BB * e -sA/2 ds . ≤ exp -1 0

Theorem 7 . 1 ( 2 t 0 ∇ 0 Q

 71200 Conjecture). Let ρ 0 be a smooth enough initial data. There exists a process (ρ(t), W (t)) t∈[0,T ] , defined on a filtered complete probability space Ω, P, F ,(F t ) t∈[0,T ] which is a solution to (1.1) on [0, T ] in the sense that (b") ρ ∈ L 2 (Ω; H) and ρ(0) = ρ 0 , (c") µ := V ′ (ρ) ∈ L 2 (Ω; H), (d") W is a continuous version of Brownian motion, (e") the process (ρ(t), W (t)) is (F t )-adapted, (f ") M(ρ) ∈ L 2 (Ω; H), (g") Q(ρ) ∈ L 2 (Ω; H), (h") for all 0 ≤ t ≤ T : ρ(t) = ρ 0 +1 (M(ρ(r))∇µ(r))dr + t (ρ(r))dW r .

When B commutes with A, we can obtain an explicit representation of the covariance operator Σ 2 := B(-A) -1 B * .

What about the invariant measure of the non-linear equation? It is classic knowledge that if the equation (1.1) may be approximated by a gradient system in H which can be rewritten as:

where ∇ denotes the gradient in the Hilbert space H, then one deduces there exists an unique and ergodic Gibbs-type invariant measure Π n given by:

where Z n is a normalization constant and E n is the energy related to the potential V n (as in (1.3)). We need finish the proof by showing that the sequence (Π n ) n∈N converges to the measure

where for all x ∈ L 2 (0, 1), we have E(x) :=

Of course, we need a strong assumption on the limit measure P eq , precisely we ask that the quantity E(x) is well-defined for γ almost all x ∈ H ∩ K (but possibly infinite with the convention that exp(-∞) = 0).

In the previous section, one can notice that the integrability condition (6.2) of the invariant measure P eq is deduced from the strict monotonicity.

Naturally, we expect that the energy is well-posed on the space of non-negative functions

assuming we have a non-negative potential V . Proposition 6.1. Assume that V n is non-negative, non-increasing and converges from below to V on R + , but converges to +∞ on R -. Then we have the convergence ν n ⇀ n→+∞ ν.

Proof. Let ψ ∈ C b (L 2 (0, 1), R). We want to prove that

We first prove that for (γ-almost surely) x ∈ H we have

Since the reference measure γ(C([0, 1])) = 1, we can restrict our attention to x ∈ C([0, 1]). Then if x / ∈ K there exists δ x > 0 such that Leb({s ∈ [0, 1] : x(s) ≤ -δ x }) > 0, Leb being the Lebesgue measure. Then, since V n (x) is non-negative and non-increasing on R + := (0, ∞), we have: 0 ≤ exp(-E n (x)) ≤ exp -


R Chassagnoux
email: raphaelchassagnoux@protonmail.com

O Lesaint

N Bonifaci

O Gallot-Lavallée

Ch Creusot

A Girodet

IEEE TRANSACTIONS ON DIELECTRICS AND ELECTRICAL INSULATION Breakdown Phenomena in Liquid Nitrogen Under Synchronized Transient Boiling and Impulse Voltage

Keywords: Liquid nitrogen, breakdown, streamer, tape electrode, subcooling, heating, bubbles, impulse voltage

Resistive superconducting fault current limiters constitute a promising solution for high voltage networks. During the resistive current limitation, transient overvoltages and heat dissipation occur on superconducting tapes, inducing fast boiling of surrounding liquid nitrogen (LN2). An experimental system reproducing the fast transient heating of a tape electrode, synchronized with a high voltage impulse is described. Breakdown voltages measurements in both polarities are carried out versus synchronization delay, pressure (0.10 to 0.40 MPa), temperature (77 K and 65 K). Breakdown voltages show complex variations with these parameters. More insight is obtained from the recording of prebreakdown phenomena (streamer). The size of vapor bubbles has a large and counter-intuitive influence on streamer initiation: the smaller bubbles, the lower inception voltage. Subcooling LN2 (by decreasing temperature and/or increasing pressure) sometimes has a beneficial influence (quenching of negative streamer propagation), but deleterious effects can be as well observed in other circumstances (by favouring positive streamer initiation).

I. INTRODUCTION

URING recent years a significant amount of resistive fault current limiter (r-SCFCL) projects emerged [START_REF] Elschner | ENSYSTROB -Design, manufacturing and test of a 3-phase resistive fault current limiter based on coated conductors for medium voltage application[END_REF][START_REF] Tixador | Status of the European Union Project FASTGRID[END_REF][START_REF] Dai | Development and test of a 220 kV/1.5 kA resistive type superconducting fault current limiter[END_REF][START_REF] Moyzykh | First Russian 220 kV superconducting fault current limiter (SFCL) for application in city grid[END_REF][START_REF] Song | Design and tests of a 160-kV/1-kA DC superconducting fault current limiter[END_REF]. This interest is due to the intrinsic advantages of r-SCFCLs: implementing these devices has no influence on HV networks, thanks to the absence of resistance in normal conditions. When a fault involving large over-currents occurs, their resistance suddenly increases and reduces the current in the line down to acceptable level, both for the line itself, the substation equipment and for operation of circuit breakers [START_REF] Tixador | Superconducting Fault Current Limiter: Innovation For The Electric Grids[END_REF]. A new potential application is foreseen for the protection of Multi Terminals DC grids where short circuit currents could be reduced to the order of magnitude of the nominal current and DC circuit breakers design could be highly simplified, and their cost significantly reduced.

The electrical insulation of these high voltage devices is expected to be a mix of solid insulation, gaseous nitrogen (GN2) and liquid nitrogen (LN2). This insulation has to endure quite severe conditions, not found in usual high voltage devices.

First, 2 nd generation high temperature superconductors are often tape-shaped, and this geometry induces large stress to surrounding dielectric materials: a strong electric field reinforcement occurs at sharp edges of the tape. The study of dielectric breakdown in "normal" conditions (i.e. when the tape is superconducting) was studied in a first paper [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF].

Secondly, during resistive current limitation, the electrical insulation faces additional and simultaneous major stresses: LN2 boiling and impulse overvoltage. A strong transient heating occurs due to Joule effect, inducing an immediate and fast temperature rise of the tape, and rapid boiling of surrounding LN2 [START_REF] Maeda | Recovery Characteristics of GdBCO Tape in a Pressurized Liquid Nitrogen for a Resistive SFCL[END_REF][START_REF] Hong | The Structure, Performance and Recovery Time of a 10 kV Resistive Type Superconducting Fault Current Limiter[END_REF]. Since gaseous nitrogen (GN2) shows much weaker breakdown field than LN2, discharges can occur within vapor bubbles, and drastically decrease the overall breakdown voltage [START_REF] Hara | Electrical breakdown mechanism of liquid nitrogen in the presence of thermally induced bubbles[END_REF][START_REF] Fink | Withstand alternating voltage of liquid nitrogen in the presence of gas bubbles[END_REF].

Moreover, in addition to abnormal current conditions that initiated the superconducting transition, and generated a first overvoltage at the beginning of the limitation (due to the transient overcurrent flowing in the resistive r-SCFCL), the interruption of current by external circuit breakers generates another overvoltage at the end of the limitation process. Electrical network simulations show that the overvoltage amplitude and shape depend for a large part on the grid characteristics (including surge arresters, and line inductance). Simulated shapes and amplitudes resemble the standardized Switching Impulse, with amplitude 1.5 to 2 times the DC voltage. This overvoltage induces further additional dielectric stress on insulation, after some delay that depends on the grid structure and protection strategy. Such combination of thermal Olivier Lesaint (e-mail: olivier.lesaint@g2elab.grenoble-inp.fr) and Nelly Bonifaci (e-mail: nelly.bonifaci@g2elab.grenoble-inp.fr), both staff of CNRS, are presently full-time researchers at the G2Elab laboratory (Grenoble, France). Olivier Gallot-Lavallée (e-mail: olivier.gallotlavallee@g2elab.grenoble-inp.fr) is presently lecturer at UGA (Grenoble, France). Christophe Creusot (e-mail: Christophe.Creusot@supergrid-institute.com) and Alain Girodet (e-mail: Alain.Girodet@supergrid-institute.com) are currently research engineers at SuperGrid Institute, Lyon, France.

The experimental results presented here were obtained using the experimental facilities available at the G2Elab laboratory.

D IEEE TRANSACTIONS ON DIELECTRICS AND ELECTRICAL INSULATION

and electrical stresses was seldom investigated in the literature [START_REF] Jin | Keeping Cool Under Pressure: Dielectric Strength of Sheet Insulation in Boiling Liquid Nitrogen[END_REF][START_REF] Hayakawa | Dynamic Breakdown Characteristics of Pancake Coil Model for Resistive-Type Superconducting Fault Current Limiters[END_REF].

In order to reproduce the transient heating of a superconducting tape during resistive limitation, a system using a resistive tape heated by Joule effect (impulse current applied through the tape) was designed [START_REF] Chassagnoux | Study of Turn-to-Turn Electrical Breakdown for Superconducting Fault Current Limiter Applications[END_REF][START_REF] Chassagnoux | Prebreakdown and Breakdown in Liquid Nitrogen under Pulsed Heating for Superconducting Applications[END_REF]. In standard equilibrium conditions (temperature T = 77 K, pressure P = 0.1 MPa), the vaporization of LN2 by impulse heating induces a very large drop of breakdown voltage VBD DC under DC voltage: breakdown occurs through vapor bubbles, and VBD DC is reduced by a factor of nearly four as compared to the case with no heating [START_REF] Chassagnoux | Study of Turn-to-Turn Electrical Breakdown for Superconducting Fault Current Limiter Applications[END_REF]. Increasing the pressure up to P = 0.25 MPa allows recovering values measured in LN2 without heating [START_REF] Chassagnoux | Study of Turn-to-Turn Electrical Breakdown for Superconducting Fault Current Limiter Applications[END_REF].

To represent more closely conditions existing in a real system (transient heating followed by impulse overvoltage) several experiments were then carried out by synchronizing transient heating and high voltage impulse, with P = 0.1 MPa [START_REF] Chassagnoux | Prebreakdown and Breakdown in Liquid Nitrogen under Pulsed Heating for Superconducting Applications[END_REF]. The aim of the present study is to further develop such measurements, by investigating pre-breakdown and breakdown phenomena over a wider range of conditions: LN2 pressure P from 0.10 to 0.40 MPa, temperature T of 77 K and 65 K, variable delay Δt between transient heating and application of the impulse voltage (from 5 to 40 ms), positive and negative impulses.

I. EXPERIMENTAL SETUPS AND METHODS

A. Experimental Setup

The cryostat [START_REF] Frayssines | Prebreakdown phenomena at high voltage in liquid nitrogen and comparison with mineral oil[END_REF][START_REF] Hong | Influence of vapor bubbles initiated by steady heating on the breakdown of liquid nitrogen[END_REF] and the tape-plane electrode setup, used in this experimental study, were presented in previous studies [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF][START_REF] Chassagnoux | Study of Turn-to-Turn Electrical Breakdown for Superconducting Fault Current Limiter Applications[END_REF][START_REF] Chassagnoux | Prebreakdown and Breakdown in Liquid Nitrogen under Pulsed Heating for Superconducting Applications[END_REF]. A schematic view of the electrode system is presented in Fig. 1. The upper electrode is a 40 mm diameter stainless steel plane electrode, connected to high voltage, and the lower grounded electrode is a 100-μm thick Hastelloy ® C276 tape.

This tape electrode, 100 mm in length, can be heated by the application of a strong transient current (about 160 A), inducing an average Joule power dissipation of 2.5 kW for 40 ms, i.e. 100 J total dissipation. All parameters (tape shape, power and duration of transient heating, delay to impulse voltage Δt) were chosen in order to represent closely conditions existing in a real r-SCFCL during operation. The large heating current was obtained by connecting the tape to a capacitor bank for 40 ms, via a series switch made with a power MOSFET transistor. The average temperature Tt(t) of the tape was monitored via precise transient measurements of current and voltage across the tape, providing its resistance Rt(t), and from the knowledge of the Hastelloy ® resistivity versus temperature. Tt raised from 77K up to 400 K after 40 ms (Fig. 1b). Transient tape temperature Tt(t) was also computed by thermal modelling, in good agreement with resistance measurements.

The large and fast temperature rise of the tape induces boiling of surrounding LN2, and growth of a vapor layer on the tape over time. After the onset of heating, a high voltage Step Impulse (STI, 0.4/1400 μs shape) of amplitude Vm was applied to the plane after some pre-selected delay Δt (Fig. 1b). This nonstandard STI impulse shape, already used for the study of prebreakdown phenomena in various liquids and LN2 [START_REF] Frayssines | Prebreakdown phenomena at high voltage in liquid nitrogen and comparison with mineral oil[END_REF] was preferred to the standard Switching Impulse (SI), although it is less representative of the current limiting application. With the long risetime of SI, discharge initiation occurs during the voltage rise at rather unpredictable and scattered instant and instantaneous voltage. Following initiation, discharge propagate while the voltage rapidly varies (during risetime). It is then difficult to properly describe and understand the influence of voltage, that randomly varies from one shot to another. In addition, it is very difficult to synchronize high speed cameras to record phenomena, since initiation time is random. With the short risetime of the STI impulse, initiation instant and voltage are precisely fixed. Thanks to the long fall time, pre-breakdown phenomena develop while the voltage is quasiconstant. This greatly simplifies the interpretation of data, and recording of images. Since the STI impulse provides a better insight into pre-breakdown phenomena, it was preferred to SI at the expense of waveform shape representativity. Notice that a comparative study of STI, LI, and SI waveforms was carried out with non-heated tape in [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF]. In the following text, the quoted polarity of pre-breakdown phenomena is the tape polarity, since discharges always start from this tape electrode.

Additional diagnostic methods such as in [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF] were used: fast video camera to record vapor bubble dynamics, gated image IEEE TRANSACTIONS ON DIELECTRICS AND ELECTRICAL INSULATION intensifier and photomultiplier (PM) to record and visualize the light emitted by discharges within bubbles, or in LN2.

B. Experimental Protocols and Methods

A multiple test protocol was applied in this study: tests were repeated 20 times for each applied impulse of maximum crest voltage Vm, and the number of breakdown (out of 20) was recorded. Then the applied voltage Vm was increased. This procedure was repeated until breakdown occurred at each shot, or until the maximum voltage of the cryostat (160 kV) was reached. From 80 to 160 shots were applied for each set of experimental conditions (constant pressure P, temperature T, delay Δt, and tape polarity).

Breakdown probability versus voltage is supposed to follow the 2-parameter Weibull distribution [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] (1):

𝑝(𝑉) = 1 -𝑒 -( 𝑉 𝑉 0 ) 𝑛 0
(1) with p the breakdown probability, V the applied (crest) voltage, V0 a scale parameter (corresponding to a breakdown probability of 63.2 %), and n0 the shape parameter. Best fits of results according to Weibull distribution were performed with a Bayesian approach, using the RStudio software, and the Rethinking package [START_REF]Statistical Rethinking course and book package[END_REF]. Best fits and raw data are in good agreement, as one can observe with a few examples in [START_REF] Chassagnoux | Prebreakdown and Breakdown in Liquid Nitrogen under Pulsed Heating for Superconducting Applications[END_REF]. In order not to overload the graphs, only best fits will be plotted here. The use of statistical analysis to fit breakdown voltage distributions is supported by the absence of "conditioning" of the system. In contrast with usual hydrocarbon liquids, no degradation of LN2 can occur (i.e. generation of solid, liquid, gaseous by-products), since N2 is the only stable recombination way of species occurring during breakdown. Repeated breakdown measurements in identical conditions showed a good reproducibility, indicating no major evolution of the tape (e.g. creation or destruction of large defects). This was ascertained by SEM imaging after breakdown tests, showing no irregularities of the tape bigger than the typical tape edge radius (5 to 10 µm). This stability of the tape was greatly favored by the 1 kΩ resistor located immediately in series with the plane electrode (Fig. 1a), which considerably limited the energy dissipated during breakdown (< 1J).

III. EXPERIMENTAL RESULTS

A. Vapor Bubble Growth Without Applied Field, Influence of Pressure and Temperature

Without applied high voltage, the recording of bubble growth due to pulse heating (Fig. 2 and3) was obtained with a fast video camera (4000 frames/s) with background illumination. In equilibrium conditions (T = 77 K, P = 0.1 MPa), Fig. 2 shows the development of vapor bubbles at different instants during heating (t = 5, 15, 40 ms) and after heating (t = 100 ms). These shadow-photographs correspond to lateral visualization of the tape (such as the drawing of Fig. 1a), and therefore the vapor layer on the tape surface can not be seen. Only vapor developing in the liquid above the tape edge can be observed. Precise image analysis shows that a "front" of vapor surrounding the tape suddenly appears 3.5 ms after the start of heating. Before vapor appearance, LN2 is superheated. The maximum excess temperature of the liquid above the boiling temperature (77 K) was previously estimated to 33 K [20,[START_REF] Duluc | Boiling incipience in liquid nitrogen induce by a step change in heat flux[END_REF], i.e. a local temperature of 110 K. This value is in reasonable agreement with the tape temperature recorded at t = 3.5 ms from resistance measurement (120 K, see Fig. 1b). Later on, during heating the vapor film grows, and bubbles detach and go up due to buoyancy. Following heating, vapor bubbles eventually fill the gap (t = 100 ms). When either pressure is increased at constant temperature (T = 77 K), or temperature decreased at constant pressure (P = 0.1 MPa), liquid and gaseous phases are not any more in equilibrium. Vapor must eventually condensate, and bubble development under impulse heating considerably changes (Fig. 3). Increasing pressure drastically reduces the volume of vapor, due to two reasons: the increase of gas density while the dissipated thermal energy (100 J) remains constant, and the condensation of vapor generated by the tape at the cold gas/liquid boundary. When LN2 is subcooled at T = 65 K, the same large reduction of vapor volume is observed.

B. Pre-Breakdown and Breakdown Under Impulse Voltage in Equilibrium Conditions: 77 K / 0.1 MPa

In order to investigate the influence of the delay Δt between the start of the heating and the impulse voltage, three different delays Δt of 5, 15 and 40 ms were used. Images of the light emission due to discharges was recorded with a gated image intensifier camera. Opening of the camera gate was synchronized with the beginning of the high voltage impulse, and the shutter was closed a few µs later. The larger size of partial discharges occurring in the vapor at longer delay Δt is linked to the larger bubble size (Fig. 2). As previously observed in [START_REF] Chassagnoux | Prebreakdown and Breakdown in Liquid Nitrogen under Pulsed Heating for Superconducting Applications[END_REF], the sequence leading to breakdown with a heated tape is a 3-step process: partial discharges occurring in vapor, streamer initiation in the liquid from ionized vapor bubbles, and streamer propagation in LN2.

Partial discharges in the vapor layer only.

When impulse voltage Vm is progressively raised, filamentary discharges first occur within vapor around the heated tape electrode (Fig. 4), as soon as the HV impulse is applied. Sensitive light detection with the photomultiplier shows that gas discharges systematically occurred in all measurements, starting from the minimum voltage allowed by the Marx generator (20 kV). At voltages up to about Vm ≈ 50 kV, no light coming from the liquid above the gas/liquid boundary can be seen. Since the vapor volume generated prior to voltage application is small at short Δt (Fig. 2), the extension of the luminous zone also considerably reduces. Although the presence of light emission at short times (Δt = 5, 15 ms) is ascertained from photomultiplier measurements, it becomes hardly observable on photographs (Fig. 4), due to insufficient resolution and sensitivity of images.

Correlated to these gas discharges, the vapor/liquid interface gets destabilized. Photographs obtained after impulse voltage application show that numerous small gaseous "spikes" develop from the vapor/liquid interface (Fig. 5b). Gas discharges impinging the vapor/liquid interface induce surface charges, and presumably local liquid vaporization. Both the vapor generated, and the electrostatic pressure pulling the interface toward the liquid could explain these spikes. To some extent, this phenomenon can be compared to slow "first mode" streamers in liquids, which development also involves internal gas discharges, vaporization, and action of electrostatic forces [START_REF] Watson | The role of electrostatic and hydrodynamic forces in the negative-point breakdown of liquid dielectrics[END_REF][START_REF] Fleszyński | Development of discharges in liquid nitrogen in non-uniform electrical field[END_REF]. This bubble deformation was identical in positive and negative polarities.

Streamers in liquids.

When Vm is raised significantly (typically beyond 50 kV), streamers propagating in the liquid ahead of the gas/liquid boundary occurs. With Δt = 40 ms and large vapor bubbles (Fig. 6d), the light emitted by streamers in liquids sometimes look like the prolongation of a luminous filament in the gas (Fig. 6e,f), but examples without any visible luminous filament in the gas are also frequently observed. At short Δt with small bubbles (Fig. 6a), gas discharges of small size cannot be individually distinguished (although a weakly luminous layer can be observed with strong light intensification), and only streamers developing in the liquid are observed (Fig. 6b,c). Different light intensifier setting were used for each picture to obtain good contrast on streamer, so the amount of discharges cannot be solely be deduced form these pictures. In negative polarity, streamers propagate with a bush-like structure and mean velocity vs ranging from 100 to 200 m/s. This shape and velocity are identical to those of streamers observed with non-heated tape [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF], and point-plane gap [START_REF] Frayssines | Prebreakdown phenomena at high voltage in liquid nitrogen and comparison with mineral oil[END_REF][START_REF] Frayssines | Streamers in liquid nitrogen: characterization and spectroscopic determination of gaseous filament temperature and electron density[END_REF]26]. Minimum voltages to observe streamer development (down to about 60 kV with Δt = 5 ms and d = 10 mm) are considerably reduced as compared to non-heated tape (about 110 kV [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF]).

In positive polarity, two streamer modes were recorded instead of only one with non-heated tape, as shown by timeresolved measurements of emitted light with the photomultiplier, such as in [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF]. Slow streamers with a bush-like structure and typical velocity about 100 m/s were observed at low voltage, typically from Vm = -60 to -80 kV. The observation of positive streamers down to -60 kV shows a large reduction of minimum initiation voltage, compared to non-heated tape. They are very similar to negatives, with emitted light constituted by numerous fast pulses [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF]. Fast filamentary streamers (velocity >> 1 km/s), were also observed, similar to those recorded with non-heated tape [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF] and point-plane gaps in LN2 [START_REF] Frayssines | Streamers in liquid nitrogen: characterization and spectroscopic determination of gaseous filament temperature and electron density[END_REF]26]. In point-plane geometry with very sharp points (tip radius rc = 1 µm, d = 3 mm), slow positive streamers in LN2 were observed only in a narrow voltage range (10 to 20 kV), and switched to fast filamentary above 30 kV [26].

In both polarities, it is obvious that heating may considerably favor streamer initiation, as compared to non-heated tape. When streamers propagate up to the upper plane electrode, then breakdown occurs.

Breakdown measurements.

Breakdown distributions with Δt = 5, 15 and 40 ms and both polarities are presented in Fig. 7. The range of breakdown voltages measured previously without heating (50 % Up and Down protocol [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF]), is added to this figure for comparison (horizontal bars). In any conditions, VBD without heating is higher than with heating. Since breakdown was "initiationcontrolled" without heating (light inception recorded in [START_REF] Chassagnoux | Pre-Breakdown and Breakdown Phenomena in Liquid Nitrogen with a Tape -Plane Electrode Geometry[END_REF] with a photomultiplier was systematically followed by breakdown), this means that streamer initiation is greatly favored by heating. As the delay Δt is increased, the vapor layer existing prior to voltage application gets thicker, but the breakdown voltage surprisingly increases. This rather counter-intuitive effect was already reported in [START_REF] Chassagnoux | Prebreakdown and Breakdown in Liquid Nitrogen under Pulsed Heating for Superconducting Applications[END_REF], and is confirmed here with additional measurements. The shortest delay investigated (Δt = 5 ms) provides the largest drop of breakdown voltage, nearly half that measured without heating.

Field calculations on charged vapor bubbles.

Simple electrostatic calculations can provide a qualitative explanation for this effect. Once a charge layer at the liquid/gas interface is established, with the hypothesis that the voltage drop within bubbles is negligible in the presence of discharges, the charged interface then act as a new "grounded electrode". Fig. 8 shows calculations of the maximum electric field Elmax in the liquid ahead of the interface with these simple assumptions. Two different bubble shapes at the tape edge are considered: either a thin layer with constant vapor thickness h (more relevant for short Δt), and a thick rounded bubble (more relevant of large bubbles observed at large Δt, Fig. 2c). The radius of curvature rc of the tape corner was rc = 5 µm.

Starting from a very thin gaseous layer (h = 5 μm) the maximum electric field Elmax at tape corners decreases as h becomes larger, and further on when large, rounded bubbles appear. Elmax later increases when h exceeds about 3 mm, due to a reducing bubble-plane distance. These calculations confirm that streamer initiation in the liquid should be favored when the vapor thickness h is minimum (i.e. maximum field Elmax), which corresponds to observations. The presence of "spikes" on the interface may induce further local field enhancement, but their influence is not considered in these simple calculations.

Without vapor layer (i.e. metallic tape with rc = 5 µm rounded corner), the maximum field should be even higher IEEE TRANSACTIONS ON DIELECTRICS AND ELECTRICAL INSULATION compared to the case with h = 5 μm (that in fact corresponds to rc + h = 10 µm radius of curvature). However, instead of further decreasing, the measured initiation voltage is conversely about twice that measured with heating and Δt = 5 ms. This shows that streamer initiation in the high field region is considerably favored when the liquid is superheated and contains vapor bubbles of presumably µm size. This agrees with the hypothesis that streamer initiation in liquids starts with liquid heating and bubble nucleation, even in classical point-plane experiments at room temperature [START_REF] Fleszyński | Development of discharges in liquid nitrogen in non-uniform electrical field[END_REF].

C. Influence of Pressure with T = 77 K.

The hydrostatic pressure P was increased to 0.4 MPa at constant temperature T = 77 K, in both polarities for Δt = 15 ms and Δt = 40 ms. Resulting breakdown probabilities are plotted in Fig. 9 and Fig. 10. In negative polarity for pressures higher than P = 0.10 MPa, probability plots could not be obtained when VBD became larger than the voltage limit of the cryostat. In these cases, empirical breakdown frequency defined as the number of breakdown divided by the number of applied impulses (20), were recorded instead on graphs.

Negative polarity

The increase of P leads to a large increase of VBD whatever the delay Δt (15 ms or 40 ms). With Δt = 40 ms and P > 0.10 MPa streamers were sometimes recorded, but very few of them lead to breakdown. With Δt = 15 ms negative streamers were systematically recorded at low breakdown probability (e.g. at 118 kV for P = 0.20 MPa and at 161 kV for P = 0.30 MPa). In such case, breakdown became controlled by streamer propagation: many non-disruptive streamers are recorded at voltages below breakdown voltage. Photographs of non-disruptive negative streamers at 0.20 MPa are shown on Fig. 11a). Although numerous streamers appear, none of them leads to breakdown: negative streamer propagation is very efficiently quenched by the increase of pressure. This observation is in line with previous studies of slow "1 st mode" negative streamers in LN2, strongly influenced by pressure [START_REF] Frayssines | Prebreakdown phenomena at high voltage in liquid nitrogen and comparison with mineral oil[END_REF][START_REF] Fleszyński | Development of discharges in liquid nitrogen in non-uniform electrical field[END_REF].

Positive polarity

The influence of pressure on breakdown in positive polarity is more complex than in negative polarity, mostly because two streamer modes with very different properties can occur with IEEE TRANSACTIONS ON DIELECTRICS AND ELECTRICAL INSULATION heated tape. With Δt = 15 ms, increasing the pressure to 0.4 MPa has little impact on VBD, in marked contrast with negative polarity. Breakdown is due to fast positive filamentary streamers, which propagation is not affected by the moderate pressure used here [START_REF] Frayssines | Prebreakdown phenomena at high voltage in liquid nitrogen and comparison with mineral oil[END_REF]. The initiation of fast streamers is favored by short Δt, which in turn induces thinner vapor layer h and higher fields available for initiation (Fig. 8).

With Δt = 40 ms, the maximum thickness h of vapor bubbles is much larger: it decreases from about h ≈ 4 mm @ 0.1 MPa, down to h ≈ 0.5 mm @ 0.4 MPa. Logically, as in Fig. 7, VBD is much higher than with Δt = 15 ms (larger bubbles induce lower fields for streamer initiation). VBD first increases from P = 0.10 MPa to 0.20 MPa, then decreases from 0.20 MPa to 0.40 MPa. Breakdown probabilities are very similar at 0.10 MPa and 0.40 MPa. However, measurements of breakdown delays tBD from impulse voltage rise to breakdown (Fig. 12), as well as photomultiplier recordings show that prebreakdown phenomena change when pressure is raised. Between 0.10 MPa and 0.40 MPa, median delays tBD drop by around a decade, typically from tBD = 100 µs to 8 µs, indicating marked changes in the pre-breakdown process. At P = 0.1 and 0.2 MPa, long tBD correspond to the propagation of slow "1 st mode" positive streamers. Increasing the pressure efficiently quenches the propagation of such streamers [START_REF] Frayssines | Prebreakdown phenomena at high voltage in liquid nitrogen and comparison with mineral oil[END_REF][START_REF] Fleszyński | Development of discharges in liquid nitrogen in non-uniform electrical field[END_REF], inducing an increase of VBD at 0.2 MPa. It is remarkable that slow streamers are initiated at voltages above Vm = 100 kV, whereas fast streamers already occurred at lower voltage (80 kV) with Δt = 5 ms (Fig. 9). This confirms that fast streamer initiation is favored with low vapor thickness h, and higher fields (Fig. 8).

Increasing the pressure above 0.2 MPa further quenches slow streamer propagation and favors the inception of fast streamers by reducing the vapor thickness h. This explains why breakdown voltage decreases, and becomes predominantly due to fast streamers at P = 0.4 MPa.

Negative polarity

The decrease of T to 65 K at constant pressure leads to an increase of VBD (Fig. 13). Many non-disruptive streamers were recorded at voltages corresponding to low breakdown probability (e.g. Vm = 126 kV for P = 0.10 MPa). Therefore, breakdown remains controlled by streamer propagation at T = IEEE TRANSACTIONS ON DIELECTRICS AND ELECTRICAL INSULATION 65 K. Non-disruptive negative streamers at 65 K are shown on Fig. 11b. To some extent, decreasing the temperature has a similar impact on negative streamers than increasing the pressure: the propagation is impeded due to condensation of vapor while slow streamers develop. Adding both effects (T = 65 K and P = 0.2 MPa) logically further increases the breakdown voltage.

Positive polarity

In positive polarity (Fig. 14), the decrease from T = 77 K to 65 K leads to a large decrease of VBD. Once again, this deleterious effect was unexpected, and can be attributed to the reduction of bubble size due to subcooling (Fig. 3), leading in turn to the increase of maximum electric field Elmax around bubbles.

IV. DISCUSSION: INFLUENCE OF SUBCOOLING ON BREAKDOWN VOLTAGE

A general trend found in the literature about breakdown in LN2 is the increase of breakdown voltage observed when LN2 is "subcooled", i.e. when temperature and/or pressure conditions are adjusted in such a way that the liquid is not any more in boiling conditions. This can be achieved either by increasing the pressure above 0.1 MPa with T = 77 K, or decreasing temperature below 77 K with P = 0.1 MPa, or both. The beneficial effect is that vapor bubbles do not any more exist for long times. A positive effect on breakdown under high voltage was reported, both without heating [START_REF] Blaz | Studies of breakdowns in liquid nitrogen at different pressures between Rogowski electrodes[END_REF][START_REF] Hayakawa | Size effect on breakdown strength in sub-cooled liquid nitrogen for superconducting power apparatus[END_REF] and in the presence of heating [START_REF] Hara | Electrical breakdown mechanism of liquid nitrogen in the presence of thermally induced bubbles[END_REF][START_REF] Sauers | Effect of Bubbles on Liquid Nitrogen Breakdown in Plane-Plane Electrode Geometry From 100-250 kPa[END_REF].

This experimental study was carried out with the objective to simulate as close as possible the conditions found in a real resistive fault current limiter (fast heating of superconductors during limitation), including its interaction with a real energy grid (voltage surge due to current interruption). This situation constitutes a combination of the worst possible conditions for a dielectric liquid: the superconducting tape shape induces a very high local field, the resistive transition induces vapor in this high field region, and once bubbles are developed a high voltage impulse is applied. Several results obtained here with such severe conditions do not agree with the general interest of subcooling. Instead of a positive (or even negligible) influence, a large deleterious effect was sometimes reported when positive high voltage is applied to the tape: decreasing temperature to 65 K can induce a large drop of breakdown voltage (Fig. 14).

The large heating power (10 6 W/m 2 ) used here, and the shape of the tape generating a low field utilization factor ( = Emean/Emax = 0.046 with d = 10 mm, computed with a 3D simulation with a micrometric mesh size around the tape edges) constitute major differences with several previous studies, carried out with smaller heating power and field enhancement. The present study also highlights the key influence of the delay Δt between heating and high voltage impulse. The primary objective was to study the influence of the HV impulse induced by the external circuit breaker. The investigation of the delay Δt over a large range (0 to 40 ms) provides some insight on this key parameter, that can be used to better understand the behavior of a SCFCL depending on its actual working conditions.

With a similar setup in DC [START_REF] Chassagnoux | Study of Turn-to-Turn Electrical Breakdown for Superconducting Fault Current Limiter Applications[END_REF], VBD increased systematically versus pressure. Therefore, heating power and electrode shape are not the only relevant parameters. The voltage shape and synchronization with heating has a large influence with a tape electrode. Altogether, the problem of dielectric withstand during limitation in a real superconducting current limiter is quite complex, requires a careful analysis of several parameters, and probably additional measurements.

V. CONCLUSION

In this study, breakdown voltages measured with heated tape and synchronized high voltage impulse show complex variation with experimental parameters. Although several investigations remains necessary to fully understand and predict the electrical behavior of LN2 in such severe conditions, a few statements can be however derived from the results obtained:  negative streamer propagation is efficiently quenched by subcooling, i.e. by an increase of pressure or a decrease of temperature;  the time delay between start of heating and current interruption appears as a parameter of primary importance. Small delay leading to small vapor bubbles close to the tape leads to lower breakdown voltage. As a result, breakdown voltage in several cases paradoxically decreases when LN2 is subcooled (by increase of pressure or decrease of temperature);  many parameters such as gap distance, voltage polarity, temperature, pressure, synchronization, contribute to a complex and hardly predictable evolution of breakdown voltage.

Fig. 1 .

 1 Fig. 1. a) Schematic drawing of the electrode system used for the experiments. b) High voltage STI impulse applied to the upper plane electrode, heating current through the tape, and transient temperature rise Tt (t) of the tape electrode. Voltage amplitude Vm applied during tests was in the range 20-160 kV.

Fig. 2 .

 2 Fig. 2. Vapor bubbles observed at different instants during (a,b,c) and after (d) heating, without application of high voltage impulse. T = 77 K, P = 0.1 MPa, d = 10 mm. a: t = 5 ms, b: 15 ms, c: 40 ms, d: 100 ms.

Fig. 3 .

 3 Fig. 3. Vapor bubbles observed at the end of heating (t = 40 ms) without applied field, when pressure is increased (left), or temperature decreased (right). d = 10 mm.

Fig. 4 .

 4 Fig. 4. Light emission due to partial occurring in the gaseous phase with high voltage impulse applied after different delays Δt. a) Δt = 5 ms (Ttape = 150 K), Vm = 67 kV. b) Δt = 15 ms (TTape = 240 K). c) Δt = 40 ms (Ttape = 400 K). TLN2 = 77 K, P = 0.1 MPa, d = 10 mm.

Fig. 5 .

 5 Fig. 5. Vapor bubbles observed immediately before (a), and 250 µs after voltage application (b). Δt = 40 ms, T = 77 K, P = 0.1 MPa, d = 10 mm, Vm = 91 kV.

Fig. 6 .

 6 Fig. 6. Streamer propagation in the liquid in both tape polarities for Δt = 5 ms ('a', 'b' and 'c') and Δt = 40 ms ('d', 'e' and 'f'). a,d: no applied voltage, b: 76 kV, b: -71 kV, e: 126 kV, f: -135 kV. T = 77 K, d = 10 mm, P = 0.10 MPa.

Fig. 7 .

 7 Fig. 7. Breakdown probability distributions according to Weibull distribution, for 3 delays Δt and both tape polarities. Breakdown voltages measured without heating [7] are also added on this graph (horizontal bars). T = 77 K, P = 0.10 MPa, d = 10 mm. Points represents experimental data and lines represents best fits.

Fig. 8 .

 8 Fig. 8. 2D electrostatic calculations of the maximum electric field ahead of the gas/liquid interface (cross section view of the tape). a: thin vapor layer of constant thickness. b: thick rounded bubble around the tape edge. c: comparison of maximum calculated electric fields Elmax with Vm = 100 kV applied on the plane electrode.

Fig. 9 .

 9 Fig. 9. Breakdown probability distributions according to Weibull for different hydrostatic pressures P and both tape polarities, with a delay Δt = 15 ms. T = 77 K, d = 10 mm.

Fig. 10 .

 10 Fig. 10. Breakdown probability distributions according to Weibull for different hydrostatic pressures P and both tape polarities, with a delay Δt = 40 ms. T = 77 K, d = 10 mm.

Fig. 11 .

 11 Fig. 11. Photographs of non-disruptive streamers in negative tape polarity. (a) T = 77 K, P = 0.20 MPa, Δt = 15 ms, Vm = 118 kV. (b) T = 65 K, P = 0.10 MPa, Δt = 40 ms, Vm = 135 kV.

Fig. 12 .

 12 Fig. 12. Box and whiskers plot of breakdown delays recorded between Vm = 135 kV and 153 kV for different pressure P. Δt = 40 ms, T = 77 K, d = 10 mm.

D

  . Influence of Temperature: Measurements at 65 K. Temperature was decreased from T = 77 K to 65 K with Δt = 40 ms, at P = 0.10 and 0.20 MPa. Once again, breakdown strongly depends on polarity.

Fig. 13 .

 13 Fig. 13. Breakdown probability distributions according to Weibull for different hydrostatic pressures P, different temperatures T, with a negative tape polarity. Δt = 40 ms, d = 10 mm.

Fig. 14 .

 14 Fig. 14. Breakdown probability distributions according to Weibull for different hydrostatic pressures P, different temperatures T, with a positive tape polarity. Δt = 40 ms, d = 10 mm.

ACKNOWLEDGMENT

The help of S. Flury and J.L. Palenzuela from G2Elab for building experimental systems is greatly acknowledged.

This work was supported by a grant of the French National Research Agency (ANR), a funding from the European Union's Horizon 2020 program n° 721019., and by the Grenoble Electrical Engineering Laboratory (G2Elab, https://g2elab.grenoble-inp.fr/), belonging to the Grenoble-Alpes University (UGA), Grenoble INP Institute, and French National Center for Scientific Research (CNRS).