Yves Heri
email: yvesheri@gmail.com

Bonaventure Banza

Wa Banza

Banza Wa

Price dynamics for Net electric energy metering on a distribution network : Ising's model approach

Keywords: Price dynamics, Energy, Ising model; Distribution network Highlight:  Net electric energy metering  Reactive power control  Voltage regulation

I. INTRODUCTION

The increase in the integration of renewable energies into electricity grids is bringing a new dynamic to their operation. Subscribers who were until now passive in regulating the operation of the network have now become active and can supply or consume energy on the network [START_REF] Fortenbacher | Optimal sizing and placement of distributed storage in low voltage networks[END_REF][START_REF] Sims | Integration of Renewable Energy into Present and Future Energy Systems[END_REF].

A noteworthy effect of the integration of renewable energies is the significant increase in the exchange of reactive energy on the networks, thus pushing operators (DSO: Distribution System Operator) to review their billing models on the exchange of this energy. [START_REF] Fetzner | Cost Cost-optimized Voltage Control in Distribution Networks[END_REF][START_REF] Swissgrid | [END_REF]. Maintaining the voltage on the nodes depends on the reactive energy compensation made by the active subscriber (agent), the DSO can pay or charge the agent depending on whether they supply or consumes reactive energy on the network.

In [5 -7] we find several approaches to model the price of energy, and we notice that these studies are based either on a singular vision of the agent, or on a global vision of the network.

Several studies have studied economic phenomena in the light of statistical physics by building models that reveal the effects of local interactions of agents on the global structure of the market [8 -11]. Cont and Bouchaud (CB) have created one of the simplest financial market models providing interesting predictions on the microscopic origin of macroscopic phenomena such as crashes or bubbles [START_REF] Cont | Herd behavior and aggregate fluctuations in financial markets[END_REF]. The structure of their model is close to a percolation model [START_REF] Chowdhury | A generalized spin model of financial markets[END_REF][START_REF] Stauffer | Introduction to Percolation Theory[END_REF]. A reformulation of the CB model can be found in [START_REF] Chowdhury | A generalized spin model of financial markets[END_REF] including the notion of super-spin in order to study the evolution over time of the general structure of the market. While in [START_REF] Paluch | Holyst Hierarchical Cont-Bouchaud model[END_REF] the CB model is modified to integrate a hierarchy of agents.

Another approach using statistical physics is introduced by Bornholdt [START_REF] Bornholdt | Expectation bubbles in a spin model of markets: Intermittency from frustration across scales[END_REF][START_REF] Bornholdt | Stability of money: Phase transitions in an Ising economy[END_REF]. Here he uses the well-known Ising model [18 -20] in order to model the interaction between the closest agents and the action of the general structure of the market on a single agent. The Bornholdt model (BI: Bornholdt -Ising) validly predicted several stylized facts such as volatility or the distribution of relative price change [START_REF] Kaizoji | Dynamics of price and trading volume in a spin model ofstock markets with heterogeneous agentsPhys[END_REF]. Sieczka and Holyst modified Bornholdt's model in order to refine its predictions by changing the agent's dynamics, taking into account the agent's opinion [START_REF] Sieczka | A threshold model of financial markets[END_REF].

We have found in the literature only a single application of Ising's model on electrical networks [START_REF] Prieto-Castrillo | An Ising Spin-Based Model to Explore Efficient Flexibility in Distributed Power Systems[END_REF] which addresses the notion of flexibility.

In this article we reconcile these two previous visions by establishing a price model considering the interactions between agents that must lead to a global configuration on the network, we base ourselves on a model well known in statistical physics, namely the Ising's model.

In section 2 of this article, we will present the electrical network model, it will be followed in section 3 by the presentation of the active agent' operation and the billing mechanism. Section 4 will give the application of the BI model on the power grid. Then will come the results of the simulations in section 5 before concluding.

II. NETWORK STRUCTURE

We consider, an electrical distribution network having Nnodes in the form of an undirected graph 𝒢 = (𝒱 𝑁 , ℒ) where 𝒱 𝑁 is the set of nodes and ℒ the set of lines between these nodes. The nodes are denominated by the index 𝑖 = 0, 1, … , 𝑁 -1. If there is a line between nodes 𝑖 and 𝑗, we will denote by (𝑖, 𝑗) ∈ ℒ. The lines are considered for our study to have very close admittances so that we can consider that there is a basis such that 𝑦 𝑖𝑗 ≈ 1 𝑝𝑢, ∀(𝑖, 𝑗) ∈ 𝐿. This hypothesis is plausible for most urban distribution networks [START_REF]Benchmark systems for network integration of renewable and distributed energy resources[END_REF]. To each node 𝑖 corresponds a vector 𝒳 𝑖 giving the state of the node,

𝒳 𝑖 = ( 𝒰 𝑖 𝒬 𝑖 ) (1)
Where 𝒰 𝑖 and 𝒬 𝑖 are respectively the voltage and the reactive power on node 𝑖.

From the Loadflow equations, we can notice a connection between voltage and reactive energy flow on a node [START_REF] Duncan | Power system analysis and design[END_REF][START_REF] Bucher | Analysis and simulation of distribution grids with photovoltaic[END_REF]. So it can be said that reactive energy production causes an increase in voltage, while reactive energy consumption tends to lower voltage (Fig. 1). Thus an active agent can regulate the voltage on its node depending on whether they produce or consume a given amount of reactive energy (RPC method: Reactive Power control) [START_REF] Bucher | Analysis and simulation of distribution grids with photovoltaic[END_REF].

III. THE ACTIVE AGENT

An active agent located at node 𝑖 is a subscriber having the possibility of producing or consuming reactive energy on the network in order to maintain the voltage (𝒰 𝑖 𝑘 ) at his node within a certain tolerance margin around a voltage of reference (𝒰 𝑟 ) fixed by the DSO with 𝑘 which represents a time interval [START_REF] Fetzner | Cost Cost-optimized Voltage Control in Distribution Networks[END_REF]. Its operation can be summarized as follows: if 𝒰 𝑖 𝑘 < 𝒰 𝑟 and 𝒬 𝑖 < 0 (producer), the DSO pays the agent for the amount of energy it provides while if 𝒰 𝑖 𝑘 > 𝒰 𝑟 et and 𝒬 𝑖 < 0, the agent is billed because the surplus reactive energy that they produce will destabilize the network, etc. In Fig. 2, the operating zones, namely the cost zone and the profit zone, are repeated. 𝒰 𝑡𝑜𝑙 is the tolerance on the measurement of the voltage 𝒰 𝑖 𝑘 . An in depth explanation can be found in the work of Fetzner [START_REF] Fetzner | Cost Cost-optimized Voltage Control in Distribution Networks[END_REF]. The cost zone (in red on figure 2) is the zone

ℳ = ℳ 1𝑖 ∪ ℳ 2𝑖 such that ℳ 1,𝑖 ≝ {𝒳 𝑖 ∈ ℝ 2 |𝒬 𝑖 > 0, 𝒰 𝑖 𝑘 -𝒰 𝑟 < -𝒰 𝑡𝑜𝑙 } (2)
and

ℳ 2,𝑖 ≝ {𝒳 𝑖 ∈ ℝ 2 |𝒬 𝑖 < 0, 𝒰 𝑖 𝑘 -𝒰 𝑟 > 𝒰 𝑡𝑜𝑙 } (3)
If the agent's operating point is in zone ℳ, the agent must pay a price 𝒞(𝑡) per unit of energy consumed, otherwise they are paid at price ℬ(𝑡) per unit of energy supplied. In this work we consider an identical pricing 𝒫(𝑡) in both cases such 𝒫(𝑡) = 𝒞(𝑡) = ℬ(𝑡).

Thus the price that the agent pays or is paid is given by

𝑝𝑟𝑖𝑐𝑒 = { 𝒫(𝑡), ∀ 𝒳 𝑖 ∈ ℳ -𝒫(𝑡), ∀𝒳 𝑖 ∉ ℳ (4)
We can thus consider the agent as having two modes of operation namely buyer or seller. We will call the operating mode of the agent "spin" and we associate each spin with a sign. We retain the convention taken in [START_REF] Bornholdt | Expectation bubbles in a spin model of markets: Intermittency from frustration across scales[END_REF] such that if the agent is in buyer mode his spin will be equal to +1, if he is in seller mode his spin will be -1. This consideration brings our system back to an Ising model [16 -18].

IV. APPLICATION OF THE ISING MODEL TO THE DISTRIBUTION NETWORK

We will now formalize the process for establishing the spin of the agent according to its operating point. We consider a membership function 𝑓 𝑖 (𝒳 𝑖 ) such that

{ 𝑓 𝑖 (𝒳 𝑖 ) < 0, ∀ 𝒳 𝑖 ∈ ℳ 𝑓 𝑖 (𝒳 𝑖 ) > 0, ∀ 𝒳 𝑖 ∉ ℳ (5)
In [START_REF] Fetzner | Cost Cost-optimized Voltage Control in Distribution Networks[END_REF], the author notices that the following function takes this property

𝑓 𝑖 (𝒳 𝑖 ) = 𝒬 𝑖 |𝒬 𝑖 | (𝒰 𝑖 𝑘 -𝒰 𝑟 ) + 𝒰 𝑡𝑜𝑙 (6)
The spin 𝜎 𝑖 ( 𝒳 𝑖 ) which provides information on the operating mode of the agent is given by

𝜎 𝑖 ( 𝒳 𝑖 ) = { +1, ∀ 𝒳 𝑖 ∈ ℳ (𝑏𝑢𝑦𝑒𝑟 𝑚𝑜𝑑𝑒) -1, ∀ 𝒳 𝑖 ∉ ℳ (𝑠𝑒𝑙𝑙𝑒𝑟 𝑚𝑜𝑑𝑒) (7)
From relations ( 5) and ( 6), we can consider the spin as opposite of the sign of the function 𝑓 𝑖 (𝒳 𝑖 ) or 𝜎 𝑖 ( 𝒳 𝑖 ) = -sign(𝑓 𝑖 (𝒳 𝑖 ))

From this relationship, it is easy to make a bridge between the electrical parameters of the network nodes and the configuration of an Ising model. We base ourselves on the work of Bornholdt [START_REF] Bornholdt | Expectation bubbles in a spin model of markets: Intermittency from frustration across scales[END_REF][START_REF] Bornholdt | Stability of money: Phase transitions in an Ising economy[END_REF] who constructs an Ising model applicable to a financial market. We call configuration of the Ising model an element 𝜔 ∈ Ω 𝑁 ≝ {-1, +1} 𝒱 𝑁 ; Ω 𝑁 is the configuration space, a variable 𝜎 𝑖 : Ω 𝑁 → {-1, +1}, 𝜎 𝑖 (𝜔) ≝ 𝜔 𝑖 is called spin at vertex 𝑖 of the considered graph.

At each vertex 𝑖 of the network graph, the quantity ℋ 𝑖 (𝑡) called local field is defined.

ℋ 𝑖 (𝑡) = ∑ 𝐽 𝑖𝑗 𝜎 𝑗 -𝛼𝜎 𝑖 𝑁 𝑖=1 | 1 𝑁 ∑ 𝜎 𝑗 𝑁 𝑖=1 | (9)
From this expression 𝐽 𝑖𝑗 = 1 ∀ (𝑖, 𝑗) ∈ 𝐿 and 𝐽 𝑖𝑗 = 0 otherwise. 𝛼 is a parameter coupling the local field with the overall state of the network. We understand that the local field on an agent 𝑖 depends on:

 the state of its direct neighbors, we can consider this to be a consequence of Millmann's theorem;

 the general state of the network, given by the parameter 𝒟(𝑡) called magnetization

𝒟(𝑡) = 1 𝑁 ∑ 𝜎 𝑗 𝑁 𝑖=1 (10)
The probability 𝜇 𝑖,𝛽 of local transition (tilting) of a spin as a function of the state of its neighbors defines the very dynamics of the spin, a review of the different possible dynamics can be found in [START_REF] Am | Herrmannt Mixing heat-bath and Glauber dynamics: damage spreading in the Ising model[END_REF]. In the BI model we consider a Heat -Bath dynamic.

𝜇 𝑖,𝛽 (𝜎

𝑖 = 1|𝜎 𝑗 = 𝜔 𝑗 , ∀(𝑖, 𝑗) ∈ ℒ) ≝ 1 1+exp (-2𝛽ℋ 𝑖 (𝜔)) (11) 𝛽 ∈ ℝ + is called inverse temperature.
By the hypothesis of efficient financial markets, we can consider that the price 𝒫(𝑡) is a function of the overall state of the market here quantified by the magnetization 𝒟(𝑡) [START_REF] Chowdhury | A generalized spin model of financial markets[END_REF][START_REF] Bornholdt | Expectation bubbles in a spin model of markets: Intermittency from frustration across scales[END_REF]. Sieczka and Holyst [START_REF] Sieczka | A threshold model of financial markets[END_REF] then Denys, Gubiec and Kutner [START_REF] Denys | Reinterpretation of Sieczka-Hołyst financial market model[END_REF], consider the following expression to be a good approximation

𝒫(𝑡) = 𝑃 0 exp (𝒟(𝑡)) (12)
With 𝑃 0 the cost at equilibrium. We can consider the relative change in price over a period τ by

𝑟(𝑡) = ln 𝒫(𝑡) 𝒫(𝑡-𝜏) = 𝒟(𝑡) -𝒟(𝑡 -𝜏) (13)
These two expressions define the whole dynamics of the price of energy. From the expression (12), we establish that the total cost of the flow of reactive energy on a network can be expressed at time 𝑡 by

𝒞 𝑇𝑂𝑇 (𝑡) = ∑ |𝒬 𝑖 |𝒫(𝑡 𝑁 𝑖 )𝜎 𝑖 (𝑡) (14)
V. RESULTS OF SIMULATION AND DISCUSSION

We are going to test our model on the modified CIGRE test network (Fig. 3), this network has 18 active nodes, the parameters of this network are identical to those used in [START_REF] Bucher | Analysis and simulation of distribution grids with photovoltaic[END_REF]. For simulation we allocate in a uniform random way to each node an active power between 18 and 4 kW under a power factor of 0.9. It is also considered that each agent can supply or consume only 10% of its nominal reactive power. The simulation is carried out over 24 hours in an interval of 15 minutes. The parameters 𝛼 and 𝛽 ∈ ℝ + of the Ising model are respectively 8 and 1 [START_REF] Bornholdt | Expectation bubbles in a spin model of markets: Intermittency from frustration across scales[END_REF]. Fig. 3. The modified CIGRE test network [START_REF] Bucher | Analysis and simulation of distribution grids with photovoltaic[END_REF] The Heat-Bath algorithm is used for the simulation, it is a Monte-Carlo type algorithm. In this one we pass from one configuration 𝜔 to another configuration 𝜔' as follows: we draw a number 𝑢 ∈ [0, 1] according to a uniform probability, and a vertex 𝑖 ∈ 𝒱 𝑁 also uniformly. We then ask

𝜔′ 𝑗 = { 𝜔 𝑗 𝑠𝑖 𝑖 ≠ 𝑗 +1 𝑠𝑖 𝑢 ≤ 𝜇 𝑖,𝛽 (𝜎 𝑖 = 1|𝜎 𝑗 = 𝜔 𝑗 , ∀(𝑖, 𝑗) ∈ ℒ) -1 𝑠𝑖 𝑢 > 𝜇 𝑖,𝛽 (𝜎 𝑖 = 1|𝜎 𝑗 = 𝜔 𝑗 , ∀(𝑖, 𝑗) ∈ ℒ) (15)
The simulation of the Ising model on the network randomly assigns different spins to the nodes of the network, our simulation being carried out over 24 hours at 15-minute intervals. We thus obtain 96 simulated states. We show in Fig. 4 three of these states of the network obtained during the simulation of the Ising model.

In Fig. 5, we show the variation of the magnetization 𝒟(𝑡) over time, there is mainly an imbalance between the number of buyers and the number of sellers on the network. The effect of the variation of the magnetization on the price 𝒫(𝑡) is perceptible, implying that if there are more buyers on the network the energy price increases and conversely if there are more sellers the price decreases.

We compare this price variation with a fixed reference price 𝑃 0 by simplification this price is maintained 1 monetary unit. The total cost of energy considering a dynamic price is lower, around -0.1769 currency unit than in the case of a fixed price where we have -0.6160 currency unit. In Fig. 6 we present the distribution of the relative price variation for different values of 𝜏. We notice a flattening of the distribution following the increase in 𝜏. This observation shows the validity of the model [START_REF] Takaishi | Multiple Time Series Ising Model for Financial Market Simulations[END_REF][START_REF] Kaizoji | Dynamics of price and trading volume in a spin model ofstock markets with heterogeneous agentsPhys[END_REF][START_REF] Chowdhury | A generalized spin model of financial markets[END_REF][START_REF] Sieczka | A threshold model of financial markets[END_REF][START_REF] Denys | Reinterpretation of Sieczka-Hołyst financial market model[END_REF]. By considering 𝜏 ranging from 1 to 4 we analyze the price variation respectively over 15, 30, 45 and 60 minutes. Fig. 7 shows a general overview of the simulation, showing the variation in the state of each agent, the variation in the power consumed as well as the variation in the cost / benefit of each agent. And some important observations can be drawn from these results presented in Fig. 5:  We note from the proposed pricing method that the price remains higher than the fixed price if there are more buyers than sellers and it is lower than the fixed price if there are more sellers than buyers.

 This pricing increases the profits of the DSO and decreases losses.

VI. CONCLUSION

The simulation of an Ising model on the CIGRE test distribution network demonstrates that dynamic billing of the exchange of reactive energy between the active agents of the network and the DSO allows a reduction in the total cost of energy compared to fixed-price invoicing. The state of each agent as well as the general state of the network can be known through smart-meters, the DSO can thus adapt the price of energy over a given interval.

Fig. 1 .

 1 Fig.1. Voltage variation depending on the direction of reactive energy[START_REF] Fetzner | Cost Cost-optimized Voltage Control in Distribution Networks[END_REF]

Fig. 2 .

 2 Fig.2. Active agent operating zones[START_REF] Fetzner | Cost Cost-optimized Voltage Control in Distribution Networks[END_REF]

Fig. 4 .Fig. 5 .

 45 Fig. 4. Variation of the state of the nodes (agents) during the simulation of the Ising model

Fig. 6 .

 6 Fig. 6. Histogram of r (t) for τ 1 to 4

Fig. 7 .

 7 Fig. 7. Temporal variation for each agent -of the state -of the reactive power -cost / benefit


Antoine Leblond

Well-posedness of the Stokes-transport system in bounded domains and in the innite strip

Keywords: Incompressible viscous uid, active scalar equation, global well-posedness, steady Stokes equation, transport equation

We consider the Stokes-transport system, a model for the evolution of an incompressible viscous uid with inhomogeneous density. This equation was already known to be globally well-posed for any L 1 ∩ L ∞ initial density with nite rst moment in R 3 . We show that similar results hold on dierent domain types. We prove that the system is globally wellposed for L ∞ initial data in bounded domains of R 2 and R 3 as well as in the innite strip R × (0, 1). These results contrast with the ill-posedness of a similar problem, the incompressible porous medium equation, for which uniqueness is known to fail for such a density regularity.

Introduction

This contribution is dedicated to the study of the following coupling between the transport equation and the Stokes equation,

       ∂ t ρ + u • ∇ρ = 0 -∆u + ∇p = -ρe z div u = 0 ρ| t=0 = ρ 0 , (1.0)
where ρ, u and p respectively stand for the density, velocity and pressure of a uid, ρ 0 is the initial density prole and e z is the vertical upward unitary vector. This system is a model of evolution of an incompressible and viscous uid having an inhomogeneous density or buoyancy subject to the gravity directed by -e z . It diers from the classical Boussinesq equation by neglecting the velocity self-advection term and the diusion of the density. It is especially derived as the mesoscopic model of a cloud of inertialess particles sedimenting in a Stokes uid, see for instance [START_REF] Richard | Sedimentation of inertialess particles in Stokes ows[END_REF][START_REF] Mecherbet | Sedimentation of particles in Stokes ow[END_REF]. This system belongs to a broad family of transport equations with non-local velocity eld, the active scalar equations, meaning that u depends on ρ in a non-local way. The vorticity equation, the surface quasi-geostrophic equation and the incompressible porous medium equation (IPM) are exemples of extensively studied systems having this structure, see [START_REF] Bae | Global existence for some transport equations with nonlocal velocity[END_REF] for an overview. In particular, the IPM is a well-known model for the evolution of an incompressible * Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France, mail: leblond@ljll.math.upmc.fr 1 inhomogeneous uid inside a porous material, which writes as the system (1.0) where one replaces the Stokes equation by Darcy's law, namely u + ∇p = -ρe z .

To our knowledge, (1.0) has been shown to be well-posed in the whole space R 3 by Höfer [START_REF] Richard | Sedimentation of inertialess particles in Stokes ows[END_REF] and Mecherbet [START_REF] Mecherbet | On the sedimentation of a droplet in Stokes ow[END_REF]. In particular, Mecherbet proved that for any ρ 0 ∈ L 1 ∩ L ∞ having a nite rst moment, the system (1.0) admits a unique solution, global in time,

(ρ, u) ∈ L ∞ (R + ; L 1 (R 3 ) ∩ L ∞ (R 3 )) × L ∞ (R + ; W 1,∞ (R 3 )).
We thereafter state the well-posedness of (1.0) for initial data ρ 0 in L ∞ (Ω), for any regular enough bounded subdomain Ω of R 2 and R 3 , as well as in the innite strip R × (0, 1). These results stand out with the ill-posedness of IPM, for which uniqueness fails on various domains for weak solutions associated to L ∞ initial data, see [START_REF] Cordoba | Lack of uniqueness for weak solutions of the incompressible porous media equation[END_REF][START_REF] Székelyhidi | Relaxation of the incompressible porous media equation[END_REF]. Even for Sobolev data, the question of global in time well-posedness of IPM is still open, see [START_REF] Kiselev | Small scale formations in the incompressible porous media equation[END_REF], although the particular case of a small and smooth pertubation of a stationary density prole does lead to a global solution, as proved in [START_REF] Castro | Global existence of quasi-stratied solutions for the conned IPM equation[END_REF].

In both Höfer and Mecherbet works, the proof of existence of a solution relies on a xed point argument, respectively a contracting map and a Picard iteration. The latter consists in solving successively the Stokes and the transport equations, providing function sequences that appear to be convergent. This convergence is proven thanks to a stability estimate on the solutions of the transport equation, combined with an energy estimate for the Stokes equation. In the full space R 3 , one can express the solution of the Stokes equation as the convolution with the adequate Green kernel. Thanks to this explicit formula, Mecherbet established the stability estimate on solutions of the Stokes equation, controled by the Wasserstein distance between their source terms. Combined with a stability estimate for the transport equation, expressed with the Wasserstein distance, this provides the contraction inequality allowing to apply the Picard argument. In particular, the Wasserstein distance allows to state the stability estimate without any derivability assumption on the density. In the following, we will emphasize the fact that the shape of the spatial domain on which one solves (1.0) strongly conditions the way one have to address the Stokes part. Apart from that, the classical transport theory is not as much sensitive to the geometry of the domain, as long as the velocity eld is bounded and Lipschitz, with extra condition on the possible boundaries. Moreover it appears that ρ is the push-forward of ρ 0 by the ow of u. In particular, if ρ 0 is a patch, the density ρ will remain a patch at all times.

In this paper, we follow a strategy similar to the one adopted by Mecherbet [START_REF] Mecherbet | On the sedimentation of a droplet in Stokes ow[END_REF] in order to prove the well-posedness of (1.0) on bounded domains of R 2 and R 3 . Since the Green kernel associated to the Stokes equation is no longer explicit for such domains in general, the remaining exploitable tools are the energy estimate and an elliptic gain of regularity due to Stokes equation.

We also replaced the Wasserstein distance, the use of which requires an explicit kernel, by some negative Sobolev norms. Notice that both metrics are relatable, see for instance [14, subsection 5.5.2]. We also impose the Dirichlet boundary condition on the velocity, considering more precisely the system

           ∂ t ρ + u • ∇ρ = 0 in R + × Ω, -∆u + ∇p = -ρe z in R + × Ω, div u = 0 in R + × Ω, u = 0 in R + × ∂Ω, ρ| t=0 = ρ 0 in Ω. (1.1)
Our rst result is the following well-posedness theorem for this system, which comes together with a stability estimate left in Proposition 2.7.

Theorem 1.1. Let Ω be a bounded domain of class C 2 of R d for d = 2 or 3. For any ρ 0 ∈ L ∞ (Ω) there exists a unique solution (ρ, u) to (1.1) in

L ∞ (R + ; L ∞ (Ω)) × L ∞ (R + ; W 1,∞ (Ω)).
We further deal with the innite strip domain R × (0, 1). The main dierence lies in the analysis of the Stokes equation. To work with L ∞ densities requires to deal with velocity proles that are not square-integrable. To overcome this observation we work with Kato spaces, also called uniformly local Sobolev spaces. Using some classical tools we provide a proof of the wellposedness of the Stokes equation in this framework, since we could not nd this result in the literature. In particular we prove that a L ∞ datum induces a W 1,∞ solution, as in the bounded domain case, which will provide handy Lipchitz velocity elds to deal further with the transport part. The method consists in considering an increasing covering of bounded open subsets of the strip, and to solve the Stokes problem on each of these subdomains. This provides a sequence of functions on R × (0, 1). By managing carefully the interior estimates of the elements of this sequence, we prove its boundedness in the uniformly locally H 1 function space, denoted H 1 uloc and dened in paragraph 3.1.1. Therefore we obtain a H 1 uloc solution by a compact argument.

Uniqueness comes from the well-posedness in the classical H 1 framework. Using the elliptic regularity of the Stokes equation in bounded domains, we prove the H 2 uloc regularity of the solution on the strip. Hence we obtain a leverage to establish existence of a unique solution in W 2,q uloc for L q uloc data for exponents 2 < q < ∞, and to conclude to the well-posedness of the problem in W 1,∞ for L ∞ data by Sobolev embeddings. The proof of the well-posedness of the coupling then lies in the extension of results used in the bounded domain case to the innite strip through the uniformly local topology. Precisely, we consider in Ω = R × (0, 1) the system

               ∂ t ρ + u • ∇ρ = 0 in R + × Ω, -∆u + ∇p = -ρe z in R + × Ω, div u = 0 in R + × Ω, u = 0 in R + × ∂Ω, u 1 dz = 0 in R + × R, ρ| t=0 = ρ 0 in Ω, (1.2)
where the extra condition on the ux u 1 dz is introduced and discussed in paragraph 3.1.2.

In the end, the well-posedness of (1.2) almost writes as in Theorem 1.1, with a similar stability result stated in Proposition 3.10.

Theorem 1.2. Let Ω = R × (0, 1). For any ρ 0 ∈ L ∞ (Ω) there exists a unique solution (ρ, u) to (1.2) in

L ∞ (R + ; L ∞ (Ω)) × L ∞ (R + ; W 1,∞ (Ω)).
It is rather natural to wonder if this result extends to the case of the layer domain R 2 × (0, 1). It appears that the Stokes problem in this unbounded domain raises some additional diculties, among those nding a functional space in which the problem admits a unique solution, in accordance with the handling of the interior estimates mentionned in the previous paragraph, see Remark 3.6 for further details. We abstain from considering this case in the present work.

The paper is organized in Section 2, dedicated to the bounded domain case, and Section 3, dedicated to the innite strip case. Both sections are ordered in the same way. Subsections 1 recall and prove the necessary prerequisites about the Stokes equation, including the wellposedness complete proof in the innite strip in Section 3. Subsections 2 contain preliminary results concerning the transport theory, and in particular the stability estimates proofs. Subsections 3 are dedicated to the proofs of Theorem 1.1 and Theorem 1.2, respectively. Ultimately, we state and prove the stability estimates of the coupling in Subsections 4.

Denitions and notations

The dimension d is always 2 or 3. A domain Ω is a non-empty open and simply connected subset of R d . The Bochner spaces are denoted by L q ([0, T ); B(Ω)) with 1 ≤ q ≤ ∞, T ∈ [0, ∞] and B(Ω) a Banach or a Fréchet space of functions dened in Ω. It is endowed with its classical norm denoted here

• L q (0,T ;B) , the space domain being specied only when diering from the whole domain Ω. A vector valued map is denoted by a bold symbol, implicitly of size d. We note u| ∂Ω the trace of u on the boundary of a domain Ω, when the boundary is regular enough to dene it. We sometimes write f ≡ c to signify that a function f is constant to c with respect to time. We denote by C any non-negative constant that is adjusted from one line to another, independent of the data and we specify its eventual space or exponent dependencies when necessary. We sometimes write f g, meaning there exists such a constant C such that f ≤ Cg, as well as f g, meaning f g and g f . 2 Well-posedness of the coupling in a bounded domain In this section, and unless stated otherwise, Ω denotes a bounded domain of R d with Lipschitz boundary.

Preliminaries on the Stokes problem in a bounded domain

Let us recall the Stokes problem on Ω, dened in the weak sense,

   -∆u + ∇p = f in Ω, div u = g in Ω, u = 0 on ∂Ω, (2.1)
where f and g belong to functional spaces specied further. Notice that it is necessary for g to satisfy the following compatibility condition due to the homogeneous assumption,

Ω g = Ω div u = ∂Ω u • n = 0. (2.2)
The well-posedness of this problem is well known for Sobolev data, see in particular [7, Theorem IV.6.1 and Exercise I.6.3], reported below. We especially use that L ∞ data induce W 1,∞ solutions, see the following corollary, which will provide velocity elds easy to deal with in the transport part.

Theorem 2.1 (Galdi). Let Ω be a bounded domain of R d of class C 2 and let 1 < q < ∞.

1. For any f ∈ L q (Ω) and g ∈ W 1,q (Ω) satisfying (2.2), there exists a unique 1 pair (u, p) in W 2,q (Ω) × W 1,q (Ω)/R satisfying (2.1), which moreover obeys the inequality

u W 2,q + p W 1,q ≤ C ( f L q + g W 1,q ) ; (2.3)
1 Unless explicitly stated otherwise, the canonical representative of p considered is the one having zero average over Ω.

2. For any f ∈ W -1,q (Ω) there exists a unique pair (u, p) in W 1,q (Ω) × (L q (Ω)/R) satisfying (2.1) with g = 0, which moreover obeys the inequality

u W 1,q + p L q ≤ C f W -1,q . (2.4) The L ∞ to W 1,∞ regularity of the problem is deduced from the Sobolev embeddings of W 2,4 (Ω) in W 1,∞ (Ω) and W 1,4 (Ω) in L ∞ (Ω) for d = 2 and 3. Corollary 2.2. Let Ω be a bounded domain of R d of class C 2 . For any f ∈ L ∞ (Ω), there exists a unique pair (u, p) in W 1,∞ (Ω) × (L ∞ (Ω)/R) satisfying (2.

1), which moreover obeys the inequality

u W 1,∞ + p L ∞ ≤ C f L ∞ . (2.5)

Preliminaries on the transport equation and stability estimates

Let us consider the transport equation, in the weak sense, for a given vector eld u ∈ L ∞ (R + ; W 1,∞ (Ω)) satisfying the Dirichlet condition u| ∂Ω ≡ 0,

∂ t ρ + u • ∇ρ = 0 in R + × Ω, ρ(0, •) = ρ 0 in Ω. (2.6)
We recall the denition of the characteristic (map) or ow X associated to the vector eld u,

as the solution of ∀s, t ∈ R + , ∀x ∈ Ω, ∂ t X(t, s, x) = u(t, X(t, s, x)) X(s, s, x) = x.
The Cauchy-Lipschitz theory ensures that X is well dened, and that for any

s, t ∈ R + , X(t, s, •) is a homeomorphism from Ω onto itself, satisfying the composition principle ∀r, s, t ∈ R + , X(t, s, •) • X(s, r, •) = X(t, r, •).
In particular we have the relation X(t, 0, •) -1 = X(0, t, •). From now on, we use indierently the

following notations ∀t ∈ R + , X(t) = X(t, •) = X(t, 0, •), X(-t) := X(0, t, •).
Let us enumerate a few classical properties of the ow. These are elementary conseqences of Duhamel formula and Gronwall inequality.

Lemma 2.3. Let u ∈ L ∞ (R + ; W 1,∞ (Ω)) with u| ∂Ω ≡ 0. The associated characteristic map X satises, ∀t ∈ R, ∀x, y ∈ Ω, |X(t, x) -X(t, y)| ≤ e C|t| ∇u L ∞ (0,t;L ∞ ) |x -y|.
(2.7)

In particular X(t) is bi-lipschitzian for any t. Recall that Liouville theorem ensures that if div u ≡ 0 the jacobian determinant of X(t) is identically equal to 1 with respect to t. Besides, let us introduce a classical stability estimate on the characteristics. Lemma

2.4. Let u i ∈ L ∞ (R + ; W 1,∞ (Ω)) with u i | ∂Ω ≡ 0 and div u i ≡ 0 for i = 1, 2. If Ω is bounded, the associated characteristic maps X i satisfy, for any 1 ≤ q ≤ ∞, ∀t ∈ R + , X 1 (t) -X 2 (t) L q ≤ te Ct ∇u1 L ∞ (0,t;L ∞ ) u 1 -u 2 L ∞ (0,t;L q ) .
(2.8)

If q = ∞, the inequality holds true for unbounded Ω.

Proof. Let us consider Ω bounded. From Duhamel formula we write for any t ∈ R + and x ∈ Ω,

X 1 (t, x) -X 2 (t, x) = t 0 u 1 (τ, X 1 (τ, x)) -u 2 (τ, X 2 (τ, x)) dτ = t 0 u 1 (τ, X 1 (τ, x)) -u 1 (τ, X 2 (τ, x)) dτ + t 0 u 1 (τ, X 2 (τ, x)) -u 2 (τ, X 2 (τ, x)) dτ.
The Lipschitz regularity of u i and the Minkowski inequality provide, for any 1 ≤ q ≤ ∞,

X 1 (t) -X 2 (t) L q ≤ C ∇u L ∞ (0,t;L ∞ ) t 0 X 1 (τ ) -X 2 (τ ) L q dτ + t 0 u 1 (τ, X 2 (τ )) -u 2 (τ, X 2 (τ )) L q dτ.
As a consequence of Liouville theorem, one has for 1 ≤ q < ∞ and τ ∈ [0, t],

u 1 (τ, X 2 (τ )) -u 2 (τ, X 2 (τ )) L q = u 1 (τ, •) -u 2 (τ, •)) L q .
The case q = ∞ holds naturally true. Then we have

X 1 (t) -X 2 (t) L q ≤ C ∇u 1 L ∞ (0,t;L ∞ ) t 0 X 1 (τ ) -X 2 (τ ) L q dτ + t u 1 -u 2 L ∞ (0,t;L q ) ,
which yields (2.8) by Gronwall inequality.

The classical characteristics method provides the well-posedness of the transport equation.

Proposition 2.5. Let Ω be a Lipschitz domain of

R d , not necessarily bounded. Let u ∈ L ∞ (R + ; W 1,∞ (Ω))
with u| ∂Ω ≡ 0, and let ρ 0 ∈ L ∞ (Ω). There exists a unique ρ in L ∞ (R + ; L ∞ (Ω)) satisfying (2.6), which is moreover the push-forward of ρ 0 by the characteristic X of u, namely

∀t ∈ R + , ρ(t) = ρ 0 • X(-t).
In particular the L q norm of the solution ρ is constant in time, for any 1 ≤ q ≤ ∞. Besides, we state the following estimate of the evolution of the dierence of two solutions of (2.6) associated to distinct velocity elds.

Proposition 2.6. Let

u i ∈ L ∞ (R + ; W 1,∞ (Ω)), with u i | ∂Ω ≡ 0 and div u i ≡ 0, for i = 1, 2.
Let ρ 0 ∈ L ∞ (Ω) and ρ i be the solutions of (2.6) associated to u i with initial datum ρ 0 . For any 1 < q < ∞ there exists T ( ∇u i L ∞ ) > 0 and C(Ω, q) > 0 such that for any T ∈ [0, T ],

ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) ≤ C ρ 0 L ∞ T e CT ∇u1 L ∞ (0,T ;L ∞ ) u 1 -u 2 L ∞ (0,T ;L q ) .
Proof. Let t ∈ R + and ϕ ∈ C ∞ c (Ω). Since the vector elds u i are divergence-free, the Liouville theorem ensures the following change of variable,

I ϕ := Ω ρ 1 (t, x) -ρ 2 (t, x) ϕ(x) dx = Ω ρ 0 (X 1 (-t, x)) -ρ 0 (X 2 (-t, x)) ϕ(x) dx = Ω ρ 0 (x) ϕ(X 1 (t, x)) -ϕ(X 2 (t, x)) dx.
Since ϕ is smooth, we can write

I ϕ = Ω ρ 0 (x) X 1 (t, x) -X 2 (t, x) • 1 0 ∇ϕ(X θ (t, x)) dθ dx,
where we set X θ (t, x) := θX 1 (t, x) + (1 -θ)X 2 (t, x). Then, Hölder's inequality provides

|I ϕ | ≤ ρ 0 L ∞ X 1 (t) -X 2 (t) L q 1 0 ∇ϕ(X θ (t)) L q dθ.
(2.9)

Let us show that X θ (t) is bi-lipchitzian from Ω onto its range. Consider the derivative of the Duhamel formula satised by X i ,

∇X i (t, •) -I d = t 0 ∇u i (τ, X i (τ, •)) • ∇X i (τ, •) dτ
and deduce thanks to inequality (2.7) the uniform estimate

∇X i (t) -I d L ∞ ≤ C ∇u i L ∞ (0,t;L ∞ ) e Ct ∇ui L ∞ (0,t;L ∞ ) -1 .
Therefore for some arbitrary constant C > 1 there exists T (

∇u i L ∞ ) > 0 such that ∀t ∈ [0, T ], C -1 ≤ det ∇X i (t) ≤ C,
and such that the Lipschitz constants of X i (t) are uniformly bounded with respect to t, by a constant smaller than 1, as follows,

L T := max i sup t∈[0, T ] Lip(X i (t) -Id) < 1.
The latter inequality ensures the injectivity of X θ (t), since for any x, y ∈ Ω, the equality

X θ (t, x) = X θ (t, y) is equivalent to x -y = θ X 1 (t, y) -y -X 1 (t, x) -x + (1 -θ) X 2 (t, y) -y -X 2 (t, x) -x ,
and implies |x -y| ≤ L T |x -y|, so that x = y since L T < 1. In the end we have proved that X θ (t) is bi-lipschitzian for any θ ∈ [0, 1] and t ∈ [0, T ], with a uniform bound on its jacobian determinant, independent of θ and t. Therefore we have for any 1 < q < ∞, ∇ϕ(X θ (t)) L q ≤ C q ∇ϕ L q .

(2.10)

Combining (2.9) and (2.10) leads to ∀t ∈ [0, T ],

Ω (ρ 1 (t, x) -ρ 2 (t, x))ϕ(x) dx ≤ C ρ 0 L ∞ X 1 (t) -X 2 (t) L q ϕ W 1,q .
Plugging the stability estimate (2.8) and taking the supremum over the test functions provides the following bound on the negative Sobolev norm,

∀t ∈ [0, T ], ρ 1 (t) -ρ 2 (t) W -1,q ≤ C ρ 0 L ∞ te Ct ∇u1 L ∞ (0,t;L ∞ ) u 1 -u 2 L ∞ (0,t;L q ) .
It remains to consider the supremum over t ∈ [0, T ] for any T ≤ T to get the result.

Proof of Theorem 1.1

The strategy of the proof is inspired from the one adopted by Mecherbet in [START_REF] Mecherbet | On the sedimentation of a droplet in Stokes ow[END_REF] for the space domain R 3 . In [START_REF] Mecherbet | On the sedimentation of a droplet in Stokes ow[END_REF] Local existence : Set ρ 0 ≡ ρ 0 in L ∞ (R + ; L ∞ (Ω)). Theorem 2.1 and Proposition 2.5 allow to dene the following sequences by induction on N ∈ N,

ρ N ∈ L ∞ (R + ; L ∞ (Ω)), u N ∈ L ∞ (R + ; W 1,∞ (Ω)), satisfying for any N ∈ N the Stokes problem    -∆u N + ∇p N = -ρ N e z in R + × Ω, div u N = 0 in R + × Ω, u N = 0 in R + × ∂Ω, (2.11)
and the transport equation

∂ t ρ N +1 + u N • ∇ρ N +1 = 0 in R + × Ω, ρ N +1 | t=0 = ρ 0 in Ω. (2.12)
Let us denote B := C ρ 0 L ∞ with adjustable constant. Since ρ N is the push-forward of ρ 0 by the ow of u N -1 , we have the uniform bound

∀N, ρ N L ∞ (R+;L ∞ ) = ρ 0 L ∞ ≤ B.
Using althemore Stokes estimate (2.5), we obtain

∀N, u N L ∞ (R+;W 1,∞ ) ≤ C ρ N L ∞ (R+,L ∞ ) ≤ B.
Hence ρ N , u N and ∇u N converge in w * -L ∞ (R + × Ω) up to the extraction of subsequences. Besides, estimates from Proposition 2.6 and Theorem 2.1 ensure that there exists T (

ρ 0 L ∞ ) > 0 such that ∀T ∈ [0, T ], ρ N +1 -ρ N L ∞ (0,T ;H -1 ) ≤ BT e BT u N -u N -1 L ∞ (0,T ;H 1 ) ≤ BT e BT ρ N -ρ N -1 L ∞ (0,T ;H -1 )
.

(2.13)

We see that for T > 0 small enough we have BT e BT < 1 so that (ρ N ) N is a Cauchy sequence in L ∞ (0, T ; H -1 (Ω)). As a consequence of the Stokes estimate (2.3), we have that (u N ) N is also a Cauchy sequence, in L ∞ (0, T ; H 1 (Ω)). Its limit belongs to L ∞ (0, T ; W 1,∞ (Ω)) since it converges for the weak * topology. In particular, u N converges in L 1 (0, T ; W 1,1 (Ω)), which together with the weak * convergence of (ρ N ) N allows to pass to the limit in the weak formulations of both (2.11) and (2.12). Therefore the limit (ρ, u) satises (1.1) on a short time, with regularity

L ∞ (0, T ; L ∞ (Ω)) × L ∞ (0, T ; W 1,∞ (Ω)).
Local uniqueness : Let (ρ i , u i ) be two such solutions of (1.1). The estimate (2.13) adapts in

ρ 1 -ρ 2 L ∞ (0,T ;H -1 ) ≤ BT e BT ρ 1 -ρ 2 L ∞ (0,T ;H -1 ) ,
up to the choice of a smaller T > 0. We deduce that ρ 1 = ρ 2 on [0, T ], and that u 1 = u 2 thanks to the Stokes estimate.

Globality : By existence and uniqueness of a solution to (1.1) locally in time, we know that there exists a unique maximal solution (ρ, u) on some interval [0, T * ) with T * ∈ [0, ∞]. Remark that T * depends only on ρ 0 L ∞ . Since ρ L ∞ ≡ ρ 0 L ∞ , a classical continuation argument implies that T * = ∞, which proves that the solution is global.

Stability estimate for the system in a bounded domain

We prove a stability estimate for the Stokes-transport system, inherited from Proposition 2.6.

Proposition 2.7. Let ρ 0,i ∈ L ∞ (Ω) and set ρ i the solution of (1.1) with initial datum ρ 0,i for i = 1, 2. For any 1 < q < ∞ there exists C(Ω, q, ρ 0,i L ∞ ) > 0 such that

∀T ∈ R + , ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) ≤ Ce CT ρ 0,1 -ρ 0,2 W -1,q .
(2.14)

Proof. Let us set u i the velocity elds associated to ρ i for i = 1, 2. Set ρ 1,2 the solution of (2.6)

with initial datum ρ 0,1 and vector eld u 2 . Hence, consider the triangular inequality

∀T ∈ R + , ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) ≤ ρ 1 -ρ 1,2 L ∞ (0,T ;W -1,q ) I1 + ρ 1,2 -ρ 2 L ∞ (0,T ;W -1,q ) I2
From Proposition 2.6 and estimate (2.5) we know that there exists some T ( ρ 0,1 L ∞ ) > 0 and C(Ω, q) > 0 such that ∀T ∈ [0, T ],

I 1 ≤ C ρ 0,1 L ∞ e CT ∇u1 L ∞ (0,T ;L ∞ ) u 1 -u 2 L ∞ (0,T ;L q ) .
Let us denote B := C max i ρ 0,i L ∞ . Stokes estimates (2.4) and (2.5) respectively provide here

u 1 -u 2 L ∞ (0,T ;L q ) ≤ C ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) , ∇u 1 L ∞ (0,T ;L ∞ ) ≤ B, which yields ∀T ∈ [0, T ], I 1 ≤ BT e BT ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) .
To bound I 2 let us apply again Liouville theorem for any t ∈ R + and ϕ ∈ C ∞ c (Ω), to get

Ω ρ 1,2 (t, x) -ρ 2 (t, x) ϕ(x) dx = Ω (ρ 0,1 (x) -ρ 0,2 (x))ϕ(X 2 (t, x)) dx.
Now we have, by denition of the Sobolev norm,

Ω ρ 1,2 (t, x) -ρ 2 (t, x) ϕ(x) dx ≤ ρ 0,1 -ρ 0,2 W -1,q ϕ(X 2 (t)) W 1,q .
From estimates (2.7) and (2.5) together with the bound ∇X i L ∞ (0,t;L ∞ ) ≤ Ce Bt provided by Lemma 2.4, it follows ϕ(X 2 (t)) W 1,q ≤ Ce Bt ϕ W 1,q .

Passing to the supremum over the test functions gives

I 2 ≤ Ce BT ρ 0,1 -ρ 0,2 W -1,q .
In the end we have shown that

∀T ∈ [0, T ], ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) ≤ BT e BT ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) + Ce BT ρ 0,1 -ρ 0,2 L ∞ (0,T ;W -1,q ) .
Up to the choice of a small enough T > 0, we have

∀T ∈ [0, T ], ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) ≤ Ce B T 1-B T e B T C ρ 0,1 -ρ 0,2 W -1,q ,
Notice that the choice of T depends only on B, and recall that ρ i (t) L ∞ = ρ 0,i L ∞ for any t ∈ R + . Therefore, one obtains by induction

∀T ∈ R + , ρ 1 -ρ 2 L ∞ (0,T ;W -1,q ) ≤ C T / T ρ 0,1 -ρ 0,2 W -1,q ,
which also writes (3.18).

3 Well-posedness of the system in the innite strip

In this section, Ω stands for the innite strip R × (0, 1). We denote by (e x , e z ) the canonical of R 2 in which u has coordinates (u 1 , u 2 ).

Regarding our problem, the transport theory does not depend on the nature of the domain, and the related results presented in the previous section are still valid in the strip. The main dierence lies in the tools and methods required to solve the Stokes equation in Ω. In particular, we state that this equation is still well-posed for L ∞ data, with W 1,∞ solution. To do so, we consider Kato spaces, also known as uniformly local Sobolev spaces. This framework allows to consider uniformly bounded densities ρ and non globally integrable velocity elds u, having innite energy u H 1 , but admitting locally a nite energy, uniformly bounded with respect to any compact stallion subdomain of the strip. We rst solve Stokes equation in a L 2 framework, then recover elliptic regularity and use Sobolev injection to prove its well-posedness for bounded data. Even if it requires known methods, we did not nd the precise proof of this latter result in the uniformly local framework.

The Subsection 3.1 is dedicated to the statements and proofs related to the Stokes equation.

In particular we introduce Kato spaces in paragraph 3.1.1, then discuss the ux condition in

The Stokes problem in the strip

The functional spaces

Let us set the following subdomains of Ω, for any k ∈ Z,

U k = {(x, z) ∈ Ω : k < x < k + 1} U * k = {(x, z) ∈ Ω : k -1 < x < k + 2}.
Dene a smooth map χ : Ω → [0, 1], depending on x only, equal to 1 on U 0 and to 0 outside U * 0 .

Set its tranlations χ k := χ(• -ke x ) so that χ k is equal to 1 on U k and supported in U * k . For convenience we choose χ so that

+∞ k=-∞ χ k = 2 in Ω. Let us set for m ∈ Z and 1 ≤ q ≤ ∞ the uniformly local norm ∀u ∈ W m,q loc (Ω), u W m,q uloc := sup k∈Z χ k u W m,q ,
and dene the Kato space as the set of locally Sobolev maps having a nite uniformly local norm, W m,q uloc (Ω) := {u ∈ W m,q loc (Ω) : u W m,q uloc < ∞}. This is a Banach space, that does not depend on the choice of χ, see for instance [1, 2.2]. The following result provides in particular some handy norms equivalences. Lemma 3.1. For m ∈ N and 1 < q ≤ ∞, the following quantities are equivalent,

u ∈ W m,q loc (Ω), sup k∈Z u W m,q (U k ) sup k∈Z u W m,q (U * k ) u W m,q uloc .
For any m ∈ N * and 1 < q < ∞, the following quantities are comparable,

u ∈ W -m,q loc (Ω), sup k∈Z u W -m,q (U k ) sup k∈Z u W -m,q (U * k ) u W -m,q uloc .
Although this result presents no diculty, we provide a short proof and a comment, about the missing inequality for the last comparison, in appendix A.1 for sake of completeness.

Flux condition

In general, the homogeneous Stokes system as formulated in (2.1) admits non-trivial solutions in domains with unbounded boundaries, called Poiseuille solutions, see for instance [7, Section IV]. In our case, these are described as follows

u φ (x, z) = 6φz(1 -z) 0 , p φ (x, z) = 12φx, φ ∈ R.
Let us introduce the denition of the ux of u throw the section of abscissa x ∈ R of Ω,

u 1 dz := 1 0 u 1 (x, z) dz = 1 0 u(x, z) • e x dz.
Notice that the divergence-free and the homogeneous Dirichlet condition ensure that any solution of (2.1) on Ω has ux independent of x, which we will denote further by u 1 dz. Indeed,

d dx 1 0 u 1 (x, z) dz = 1 0 ∂ x u 1 (x, z) dz = - 1 0 ∂ z u 2 (x, z) dz = u 2 (x, 0) -u 2 (x, 1) = 0. (3.1)
For instance, the ux of the Poiseuille solution u φ is φ. We will see that a choice of ux value prescribes a unique Poiseuille solution, and provides uniqueness of a solution in Kato spaces.

Since the Stokes equation is linear, we can choose one value without loss of generality. From now on we consider the Stokes problem with the zero ux condition,

       -∆u + ∇p = f in Ω, div u = 0 in Ω, u = 0 on ∂Ω, u 1 dz = 0 in R. (3.2)

Well-posedness of the Stokes problem in the strip

We show that the system (3.2) is well-posed for H -1 uloc (Ω) and L 2 uloc (Ω) data f , with some elliptic regularity gain. We then deduce that this system is also well-posed for L ∞ (Ω) data with W 1,∞ (Ω) solutions, by adapting the steps of the proof of the bounded domain case. The general technique presented here is originally due to Ladyzhenskaya and Solonnikov [START_REF] Ladyzhenskaya | Determination of the solutions of boundary value problems for stationary stokes and Navier-Stokes equations having an unbounded Dirichlet integral[END_REF]. A proof in a framework closer to ours can be found in [START_REF] Gérard | Relevance of the slip condition for uid ows near an irregular boundary[END_REF]Theorem 3]. Although these are classical tools, we have not found the proof of this result in the literature. Theorem 3.2. Let f ∈ H -1 uloc (Ω). There exists a unique u in H 1 uloc (Ω) satisfying (3.2), which moreover obeys the inequality

u H 1 uloc ≤ C f H -1 uloc . (3.3)
Let us a few more notations. Set for any k ∈ Z,

Ω k := {(x, z) ∈ Ω : -k < x < k}.
Let us dene for any k ∈ N * a smooth map η k : Ω → [0, 1], depending on x only, equal to 1 in Ω k and supported in Ω k+1 . Remark that its derivatives are supported in Ω k+1 \Ω k . We can choose η k such that there exists a constant C > 0 independent of k and satisfying

η k L ∞ + η k L ∞ ≤ C.
Finally, let us observe the following estimate linking uniformly local and classical Sobolev norms over Ω n . We report its proof in appendix A.2.

Lemma 3.3. Let n ∈ N * . There exists a constant C > 0 such that for any f in L 2 uloc (Ω), resp. in H -1 uloc (Ω), one has

f L 2 (Ωn) ≤ Cn 1/2 f L 2 uloc , resp. f H -1 (Ωn) ≤ Cn 1/2 f H -1 uloc .
Proof of Theorem 3.2. Let us set for any n ∈ N * the unique couple (u

n , p n ) in H 1 (Ω n ) × L 2 (Ω n )/R satisfying the system    -∆u n + ∇p n = f in Ω n , div u n = 0 in Ω n , u n = 0 on ∂Ω n ,
existence and uniqueness of which is ensured by [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations andRelated Models[END_REF]Theorem IV.5.1]. Then dene for any integers

1 ≤ k ≤ n the energy of u n on the subdomain Ω k , E n,k := u n 2 H 1 (Ω k ) = k -k 1 0 |∇u n | 2 + |u n | 2 .
By evaluating (3.1.3) in the test function u n we nd

n -n 1 0 |∇u n | 2 = f , u n Ωn ≤ f H -1 (Ωn) u n H 1 (Ωn) .
Using Lemma 3.3 and Poincaré's inequality, for which the constant involved can be chosen

independent of n, one nds E n,n ≤ Cn 1/2 f H -1 uloc .
Our goal is to show that there exists C > 0 independent of n and f , such that

E n,1 = u n 2 H 1 (Ω1) ≤ C f 2 H -1 uloc . (3.4)
This allows to conclude to the existence of a solution u ∈ H 1 uloc (Ω) obeying the estimate (3.3), by translation invariance of the domain and compactness considerations. To prove (3.4) we will

x n ∈ N * and show by descending induction over k that there exists C > 0 independent of u, f , n and k such that

∀ 1 ≤ k ≤ n, E n,k ≤ Ck f 2 H -1 uloc .
Let us evaluate the variational formulation of (3.1.3) in the test function η k u n , which yields

Ωn η k |∇u n | 2 = f , η k u n Ω k+1 - Ωn η k u n • ∂ x u n + Ωn η k p n u n,1 . (3.5)
By Poincaré's inequality we bound from below the left hand side by E n,k , up to a multiplicative constant. Let us bound from above all the right hand side terms. Lemma 3.3 provides

f , η k u n Ω k+1 ≤ C f H -1 (Ω k+1 ) u n H 1 (Ω k+1 ) ≤ C(k + 1) 1/2 f H -1 uloc E 1/2 n,k+1 . (3.6)
Since η k is supported in Ω k+1 \Ω k and uniformly bounded independently of k, we have

Ωn η k u n • ∂ x u n ≤ C Ω k+1 \Ω k |∇u n | 2 + |u n | 2 = C(E n,k+1 -E n,k ). (3.7)
Let us split the remaining integral of 3.5 as follows,

Ωn η k p n u n,1 = U k η k p n u n,1 + -U k η k p n u n,1 .
Remark that

U k η k u n,1 = k+1 k η k u n,1 dz = 0,
since the ux u n,1 dz is independent of x for the same reason as in (3.1), and equal to 0 because of the homogenous Dirichlet condition in {x = ±n}. Hence, let us denote by p n U k the average of p n over U k , and nd

U k η k p n u n,1 = U k η k (p n -p n U k )u n,1 ≤ C p n -p n U k L 2 (U k ) u n L 2 (U k ) .
Let us apply Ne£as inequality, see [3, Lemma IV.1.9], and get

p n -p n U k L 2 (U k ) ≤ C ∇p n H -1 (U k ) ≤ C ∆u n + f H -1 (U k ) ≤ C ∇u n L 2 (U k ) + f H -1 (U k ) .
Hence we deduce, using Lemma 3.3, that

U k η k p n u n,1 ≤ C (E n,k+1 -E n,k ) 1/2 + f H -1 uloc E 1/2 n,k . (3.8)
The very same considerations hold true for the integral over -U k . Bounding (3.5) thanks to (3.6), (3.7) and (3.8), plus applying Young's inequality, we obtain that for any integers k, n such

that 1 ≤ k ≤ n, E n,k ≤ C E n,k+1 -E n,k + (k + 1) f 2 H -1 uloc . (3.9)
This relation implies (3.4), as stated in the following lemma, proven apart in appendix A.3.

Lemma 3.4. Let (E n,k ) k,n be a non-negative family indexed by all couples (k, n) ∈ N 2 satisfying 1 ≤ k ≤ n, non-decreasing with respect to k, obeying (3.9) and such that

∀n ∈ N * , E n,n ≤ Cn f 2 H -1 uloc .
There exists C 0 > 0 and k 0 ∈ N * independent of f such that for any k, n ∈ N satisfying k 0 ≤ k ≤ n, we have

E n,k ≤ C 0 k f 2 H -1 uloc .
This result implies the expected inequality,

∀n ≥ k 0 , E n,1 ≤ E n,k0 ≤ C 0 k 0 f 2 H -1 uloc .
By extending f | Ωn and u n to Ω by 0 outside Ω n , we can perform a similar analysis and nd the same energy estimates over each subdomain U , namely,

∀n ≥ k 0 , ∀ ∈ Z, u n 2 H 1 (U ) ≤ C f 2 H -1 uloc ,
where C = C 0 k 0 with C 0 and k 0 independent of n and . Therefore, for any n ≥ k 0 , u n belongs to H 1 uloc (Ω) and satises

u n H 1 uloc ≤ C f H -1 uloc .
Since bounded subsets of H 1 (U ) are weakly relatively compact, there exists a subsequence of (u n ) n converging weakly in H 1 loc (Ω) toward some u ∈ H 1 uloc (Ω), with u satisfying estimate (3.3).

The limit also veries u 1 dz = 0 since every u n has zero ux. Hence, it is a solution of (3.2). To prove uniqueness of such an element, let us consider some u ∈ H 1 uloc (Ω) satisfying (3.2) with f = 0. Dene the energy E k := u 2 H 1 (Ω k ) and proceed to the same computations as previously to nd

E k ≤ C(E k+1 -E k + 1).
Notice that the zero ux condition is necessary to bound the pressure term and obtain such an estimate. Since E k+1 -E k is bounded by u 2

H 1 uloc , we have ∀k ∈ Z, E k ≤ C( u 2 H 1 uloc + 1) < ∞,
which means that u belongs to H 1 (Ω). Then we conclude by well-posedness of the Stokes system over Ω in H 1 (Ω), see for instance [START_REF] Temam | Navier-Stokes equations: theory and numerical analysis[END_REF].

Remark 3.5. The pressure does not belong to L 2 uloc (Ω) in general; observe for instance the following triplet, satisfying (3.2),

f = e x , u = 0, p(x, z) = x.
Nevertheless, we have thanks to Ne£as inequality some similar estimate as (3.3) on the pressure,

sup k∈Z p -p U k L 2 (U k ) ≤ C f L 2 uloc .
Remark 3.6. This proof does not adapt straightforwardly to the case of the layer domain R 2 × (0, 1). The rst issue one needs to deal with is to determine conditions on u ensuring uniqueness of a solution. Also, the descending induction on the energy estimates no longer holds in this form. Indeed, one needs to replace the slices

[k, k + 1] × (0, 1) in R × (0, 1) by chunks [k, k + 1] × [ , + 1] × (0, 1) in R 2 × (0, 1).
To bound the energy on [-k, k] 2 × (0, 1) by the energy on [-(k + 1), k + 1] 2 × (0, 1) makes appear some quadratic terms in k in (3.9 instead of the linear ones present for the strip case, making the induction fail. Under dierent boundary assumptions, it is however possible to adapt it in a non-trivial way and to conclude; see for instance [START_REF] Dalibard | Well-posedness of the stokescoriolis system in the half-space over a rough surface[END_REF]Section 3].

Since L ∞ (Ω) ⊂ L 2 uloc (Ω) ⊂ H -1 uloc (Ω), we already have existence of a solution to (3.2) for L ∞ data. Recall that we need to establish the W 1,∞ regularity of this solution. We rst show that the system satises some elliptic regularity property in the hilbertian framework.

:= χ k (p -p U * k ), which satisfy the system      -∆u k + ∇q k = F k in U k , div u k = χ k u 1 in U k , u k = 0 on ∂ U k , (3.11)
where we set

F k := χ k f -2χ k ∂ x u -χ k u + χ k p -p U * k e x
for any smooth bounded subdomain U k of Ω containing U * k . Let us set ( U k ) k a family of such domains given by the choice of a smooth U 0 containing U * 0 and its translations U k = U 0 +ke x . The regularity of u and p implies that F k belongs to L 2 ( U k ) and that χ k u 1 satises the compatibility condition (2.2). Therefore, Theorem 2.1 ensures that u k is the only solution of (3.11) on U k , with estimate

u k H 2 ( U k ) ≤ C F k L 2 ( U k ) , (3.12)
where the constant C > 0 can be chosen independent of k since the subdomains U k are translations of each other. A few computations lead to

F k L 2 ( U k ) ≤ C( u H 1 uloc + f L 2 uloc + p -p U * k L 2 (U * k ) ). (3.13)
Ne£as inequality and Lemma 3.3 provide

p -p U * k L 2 (U * k ) ≤ C ∇p H -1 (U * k ) ≤ C ∆u + f H -1 (U * k ) ≤ C( u H 1 uloc + f L 2 uloc ).
The latter estimate combined with (3.3) and (3.13) in (3.12) leads to

u H 2 (U k ) ≤ u k H 2 ( U k ) ≤ C f L 2
uloc , which proves that u belongs to H 2 uloc (Ω) and satises inequality (3.10).

From the latter result and Sobolev embeddings we obtain existence of W 1,q uloc solutions for L q data. Then elliptic regularity is once again recovered and we show that these solutions are in W 2,q uloc . Finally Sobolev embeddings once again yield a unique solution in W 1,∞ for L ∞ data.

Theorem 3.8. Let f ∈ L ∞ (Ω). There exists a unique u ∈ W 1,∞ (Ω) satisfying (3.2), which obeys

u W 1,∞ ≤ C f L ∞ . (3.14) Proof. Let f ∈ L ∞ (Ω). We always have f L 2 uloc ≤ C f L ∞ .
The Sobolev embeddings in bounded domains adapts into the continuous inclusion

H 2 uloc (Ω) → W 1,q uloc (Ω), 2 ≤ q < ∞.
Since Theorem 3.7 ensures the existence of a solution to (3.2) in H 2 uloc (Ω), we also have existence of a solution in W 1,q uloc (Ω). Besides, the inclusions of Lebesgue spaces imply

W 1,q uloc (Ω) → H 1 uloc (Ω), 2 ≤ q < ∞.
Hence, the uniqueness of a solution in H 1 uloc , ensured by Theorem 3.2, implies that there exists at most one solution of (3.2) in W 1,q uloc (Ω). In the end, (3.2) admits a unique solution u ∈ W 1,q uloc (Ω)

for 2 ≤ q < ∞, with estimate u W 1,q uloc ≤ C q f L ∞ .
Now the method is exactly the same as in Theorem 3.7 to prove that u belongs to W 2,q uloc (Ω). To do so, the only extra result we require is Ne£as inequality in the general L q framework, see [7, Ex. III.3.4, p. 175], which provides the very same pressure estimates as for q = 2. In the end we obtain the well-posedness of the problem in W 2,q uloc (Ω), with estimate

u W 2,q uloc ≤ C f L ∞ , 2 ≤ q < ∞. Now use that W 2,4 uloc (Ω) → W 1,∞ (Ω) → W 1,4 uloc (

Ω).

As previously, the rst embedding provides existence of a solution u ∈ W 1,∞ (Ω), together with estimate (3.14), and the second one ensures uniqueness.

Stability estimate for the transport in the strip

The transport equation (2.6) is still well-posed on Ω = R × (0, 1) and Proposition 2.5 still applies. The lemmas related to the properties of the characteristics are also valid still, up to minor adaptations mentioned when required in the following. The only adaptation demanding particular attention is the stability estimate from Proposition 2.6, stated as follows. Proposition 3.9. Let u Ω) and set ρ i the solution of (2.6) associated to u i with initial datum ρ 0 . There exists T ( ∇u i L ∞ ) > 0 such that for any T ∈ [0, T ],

i ∈ L ∞ (R + ; W 1,∞ (Ω)) with u i | ∂Ω ≡ 0 and div u i ≡ 0, for i = 1, 2. Let ρ 0 ∈ L ∞ (
ρ 1 -ρ 2 L ∞ (0,T ;H -1 uloc ) ≤ BT (1 + M T ) 1/2 e CT ∇u1 L ∞ (0,T ;L ∞ ) u 1 -u 2 L ∞ (0,T ;L 2 uloc ) ,
where

B := C ρ 0 L ∞ and M := max i u i L ∞ .
Proof. The goal is to bound the following quantity for any test function ϕ ∈ C ∞ c (Ω) uniformly in k ∈ Z and with respect to t ∈ [0, T ] where T is determined further. To apply Liouville theorem gives

I ϕ,k := Ω ρ 1 (t, x) -ρ 2 (t, x) (χ k ϕ)(x) dx = Ω ρ 0 (x) (χ k ϕ)(X 1 (t, x)) -(χ k ϕ)(X 2 (t, x)) dx = Ω ρ 0 (x)(X 1 (t, x) -X 2 (t, x)) • 1 0 ∇(χ k ϕ)(X θ (t, x)) dθ dx,
where X θ (t, x) := θX 1 (t, x) + (1 -θ)X 2 (t, x). Since ρ i are the push-forwards of ρ 0 by X i , the respective transports occur at nite speed, bounded by

M . Since χ k is supported in U * k , the support of (χ k ϕ) • X i (t) is included in U M,t k := {k -1 -M t < x < k + 2 + M t}.
Hence Hölder's inequality applies as follows

|I ϕ,k | ≤ ρ 0 L ∞ X 1 (t) -X 2 (t) L 2 (U M,t k ) 1 0 ∇(χ k ϕ)(X θ (t)) L 2 dθ.
We saw in Proposition 2.6 that there exists T ( ∇u i L ∞ ) > 0 such that X θ (t) performs a change of variable with jacobian determinant uniformly bounded with respect to t ∈ [0, T ] and θ ∈ [0, 1], meaning there exists a constant C > 0 such that ∀t ∈ [0, T ], θ ∈ [0, 1],

∇(χ k ϕ)(X θ (t)) L 2 ≤ C ∇(χ k ϕ) L 2 ≤ C χ ϕ H 1 .
Besides, Lemma 2.4 applies on the bounded domain U M,t k , providing X 1 (t) -X 2 (t) L 2 (U M,t k ) ≤ te Ct ∇u1 L ∞ (0,t;L ∞ ) u 1 -u 2 L ∞ (0,t;L 2 (U M,t k )) .

From considerations similar to those of Lemma 3.3 we have u 1 -u 2 L ∞ (0,t;L 2 (U M,t k

)) ≤ C(1 + M t) 1/2 u 1 -u 2 L ∞ (0,t;L 2 uloc ) ,

with nite right hand side, since u i ∈ L ∞ (R + ; W 1,∞ (Ω)). Combining these last equalities lead to ∀t ∈ [0, T ], |I ϕ,k | ≤ C ρ 0 L ∞ t(1 + M t) 1/2 e Ct ∇u1 L ∞ (R + ;L ∞ ) u 1 -u 2 L ∞ (0,t;L 2 uloc ) ϕ H 1 .

Taking the supremum over the test functions, k ∈ Z and t ∈ [0, T ], we see that for any T ∈ [0, T ]

we have ρ 1 -ρ 2 L ∞ (0,T ;H -1 uloc ) ≤ BT (1 + M T ) 1/2 e CT ∇u1 L ∞ (0,T ;L ∞ ) u 1 -u 2 L ∞ (0,T ;L 2 uloc ) .

Proof of Theorem 1.2

The proof essentially follows the same path as in Theorem 1.1. For this reason we recall briey the similar steps and focus on the parts that dier from this former case.

Finally, the reciprocal inequality is proved by noticing that χ k ϕ belongs to W m,q 0 (U * k ) for any ϕ ∈ W m,q 0 (Ω);

χ k u Ω = sup ϕ 0,Ω =1 u, χ k ϕ Ω ≤ sup ϕ 0,Ω =1 u U * k χ k ϕ 0,U * k ≤ C(χ, m) u U * k sup ϕ 0,Ω =1 ϕ U * k ≤ C u U * k .
Remark A.1. For m ∈ N * , we do not have in general

sup k∈Z u W -m,q (U k ) sup k∈Z u W -m,q (U * k ) .
Indeed, consider the Dirac mass δ (0,1/2) belonging to H -2 (Ω) and therefore to H -2 uloc (Ω), with δ H -2 uloc > 0. Nevertheless, for any k ∈ Z we have

∀ϕ ∈ H 1 0 (U k ), δ, ϕ U k = 0.
The reason is that the support of an element of the negative Sobolev spaces can be included in the complementary of ∪ k U k . This does not happen when the considered subdomains family covers the whole domain, as does (U * k ) k .

A.2 Proof of Lemma 3.3

The case f ∈ L 2 uloc (Ω n ) is straightforward,

f 2 L 2 (Ωn) = n-1 k=-n f 2 L 2 (U k ) ≤ 2n f 2 L 2 uloc .
The case f ∈ H -1 uloc (Ω n ) requires a little more care. We use notations from the proof of Lemma

f H -1 (U * ) χ ϕ H 1 f H -1 uloc n =-n-1 χ ϕ H 1 f H -1 uloc n =-n-1 ϕ H 1 (U * ) f H -1 uloc (2n + 2) 1/2 n =-n-1 ϕ 2 H 1 (U * ) 1/2
, where we used Lemma 3.1. Now, bound 2n + 2 by 4n and notice that the last sum is equivalent to ϕ H 1 to complete the proof.

k 0 k 0 + 1 ≤ C C + 1 1 + 1 α =: C α .
Up to the choice of a greater α, we can assume that C α < 1. This implies that k 0 ≤ Cα 1-Cα , independently of n and F . Therefore, we conclude that for any k, n such that k 0 ≤ k ≤ n we have E k ≤ αF k.

paragraph 3 . 1 .

 31 2 and nally prove the related well-posedness theorems in paragraph 3.1.3. The Subsection 3.2 concerns the stability estimate for the transport in the strip. Subsection 3.3 contains the proof of the well-posedness of the coupling, and we state its stability estimate in Subsection 3.4.

3. 1 .

 1 Notice that n =-n-1 χ = 2 on Ω n , so for any ϕ ∈ H 1 0 (Ω n ) we have f, ϕ Ωn n =-n-1 f, χ ϕ U * n =-n-1

20A. 3 2 H - 1 uloc.

 321 Proof ofLemma 3.4 Set (E n,k ) n,k a family of positive real numbers, indexed by the couples (n, k) ∈ N 2 such that 1 ≤ k ≤ n, non-decreasing according to k for a xed n, obeying∀1 ≤ k ≤ n, E n,k ≤ C E n,k+1 -E n,k + F (k + 1) (A.1)and satisfying ∀n, E n,n ≤ CF n,(A.2)where F is a constant playing the role of f Let us show that there exists α > 0 andk 0 ∈ N * such that ∀k, n, k 0 ≤ k ≤ n =⇒ E k ≤ αF k. (A.3)By (A.2) we already know that (A.3) is satised for any n ∈ N * and k = n, with α = C. For a xed n, let k 0 be the greatest index such that (A.3) is not satised, meaning E n,k0 > αF k 0 .(A.4) Therefore, plugging (A.4) in (A.1) and using the denition of k 0 provides α(1 + C)F k 0 ≤ CF (α + 1)(k 0 + 1),which is equivalent to

  the author solves successively Stokes and transport problems, providing a contracting sequence of velocity elds and density proles. Here the contracting property is obtained by combination of the Stokes estimate from Theorem 2.1 and the stability estimate for the transport from Proposition 2.6. Both interplay in the case of the whole space thanks to the Green kernel, also called Oseen tensor, representation of the Stokes solution and the stability estimates formulated with the Wasserstein distance. Since we work in general open bounded domains, we do not use Green kernels but only rely on the variational estimates. We also replace the Wasserstein distance, handled by Mecherbet, by negative Sobolev norms. Both are known to be related, see[14, subsection 5.5.2], and in our case present the same asset to allow the statement of stability estimates without any derivability assumption concerning the density.

  Proof. The demonstration consists in truncating the global solution u within some bounded subdomains, and to use the elliptic regularity in these bounded domains provided by Theorem 2.1. Let f ∈ L 2 uloc (Ω) and set (u, p) ∈ H 1 uloc (Ω) × L 2 loc (Ω)/R , the associated solution to (3.2). For any k ∈ Z, set u k := χ k u and q k

	Theorem 3.7. Let f ∈ L 2 uloc (Ω). The associated solution u ∈ H 1 uloc (Ω) of (3.2) belongs to H 2 uloc (Ω) and obeys the inequality
	u H 2 uloc ≤ C f L 2 uloc .	(3.10)

Acknowledgement

The author thanks Anne-Laure Dalibard and Julien Guillod for introducing him to this problem, for their guidance and the quality of their supervision. The author also thanks Amina Mecherbet for giving him early access to one of her production, and for her useful comments about the present work.

Fundings This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC Mathematical Study of Boundary Layers in Oceanic Motion. This work was also supported by the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR).

Local existence :

Set ρ 0 ≡ ρ 0 in L ∞ (R + ; L ∞ (Ω)). Dene, thanks to Proposition 2.5 and Theorem 3.8, the following sequences

satisfying the partial problems

and

The uniforms bounds, with

(3.17)

Therefore we still have weak * convergence of ρ N , u N and ∇u N , up to the extraction of subsequences. Beside, Proposition 3.9 ensures the existence of a T ( ρ 0 L ∞ ) > 0 such that for any T ∈ [0, T ],

where we have plugged (3.17). Therefore, up to the choice of a small enough T > 0, (ρ

, with limit denoted u. The weak * convergence of (u N ) N and (∇u N ) N ensures that u also belongs to L ∞ (0, T ; W 1,∞ (Ω)). In particular, u N and its derivatives converge in L 1 loc ([0, T ] × Ω), which, together with the weak * convergence of (ρ N ) N , is enough to pass to the limit in the weak formulation of partial problems (3.15) and (3.16). We obtain a local in time solution (ρ, u) of (3.2), with regularity

Local uniqueness : Let (ρ i , u i ) be two such solutions of (3.2). The contraction adapts thanks to Proposition 3.9 in

, which implies uniqueness for T > 0 small enough.

Globality : The extension proves just as in the bounded case, see the proof of Theorem 1.1.

Stability estimate for the system in the strip

The result and the proof are identical to the ones of Proposition 2.7, replacing the stability estimate (1.2) of the bounded case by the one in the strip from Proposition 3.9. Proposition 3.10. Let ρ 0,i ∈ L ∞ (Ω) and ρ i be the solution of (1.1) with initial datum ρ 0,i , for i = 1, 2. There exists

The support of χ k being U * k , one has

One can split this last norm as follows

These three inequalities prove the rst assertion,

Let m ∈ N * and 1 < q < ∞. For readability we adopt the following notations in the rest of this proof. For any u ∈ W -m,q (U ) and ϕ ∈ W m,q 0 (U ) where U is a subdomain of Ω, denote

and the duality brackets u, ϕ U = u, ϕ W -m,q (U ),W m,q 0 (U ) .

The inclusion {ϕ : ϕ 0,U k = 1} ⊂ {ϕ : ϕ 0,U * k = 1} provides the rst inequality u W -m,q (U k ) ≤ u W -m,q (U * k ) .

Let us show the remaining direct inequality. We use the denition of the dual norm, and recall that (χ k ) k is a partition of the unity, up to a factor 2. Also notice that a product χ ϕ with ϕ ∈ W m,q 0 (U * k ) has possibly non-empty support only if | -k| ≤ 3. These remarks justify each step of the following computations, for any k ∈ Z,

| -k|≤3 χ u, ϕ Ω ; u uloc .

Conict of interest

The author declares that there is no conict of interest regarding the publication of this article.