Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit

We study the solution u ε to the Navier-Stokes equations in R 3 perforated by small particles centered at ( ε Z ) 3 with no-slip boundary conditions at the particles. We study the behavior of u ε for small ε , depending on the diameter ε α , α > 1, of the particles and the viscosity ε γ , γ > 0, of the fluid. We prove quantitative convergence results for u ε in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain a) the Euler-Brinkman equations in the critical regime, b) the Euler equations in the subcritical regime and c) Darcy’s law in the supercritical regime.

From the application oriented point of view, interest in such homogenization problems arises from the study of flow through porous media and of suspension flows.In the case of such particulate flows, homogenization problems where the particle evolution is frozen or prescribed can be considered as a first step towards the derivation of fully coupled models between the fluid flow and the dispersed phase.
The limiting behavior of solutions to the incompressible (Navier-)Stokes equations with fixed viscosity in perforated domains with no-slip boundary conditions is by now quite well understood.On the microscopic lengthscale of the particles, the fluid inertia becomes negligible.Therefore, in the limit of many small particles, a linear friction relation (Stokes law) prevails, giving rise to an effective massive term, the so-called Brinkman term.Depending on the particle sizes and number density, the Brinkman term becomes negligible, dominant or of order one in the homogenization limit, leading to the (Navier-)Stokes equations, Darcy's law and the (Navier-)Stokes-Brinkman, respectively, see e.g.[All90a; All90b; CH20; DGR08; FNN16; GH19; Giu21; HJ20; HMS19; LY23; Mik91; Tar80].
For the case of the Navier-Stokes equations with vanishing viscosity, only very few results are available though.The problem of considering such fluids in perforated domains with very small viscosity (or more precisely large macroscopic Reynolds numbers) is a very relevant one in applications.Indeed, in the modeling of sprays, it is not unusual to couple kinetic equations for the dispersed phase to the Euler equations (see e.g.[BD06;CDM11]).On the other hand, regarding porous media, understainding flow at large Reynolds number is very important (see e.g.[BMW10]) and nonlinear extensions of Darcy's law, in particular the Darcy-Forchheimer equations, are proposed at very large Reynolds numbers.Although the rigorous derivation of such nonlinear effective models seems currently out of reach, the present work aims at identifying the effective behavior in all scaling limits where a linear friction law prevails.We emphasize that the effective models we obtain are completely different from the ones that result by starting from the Euler equations in perforated domains (see e.g.[HLW22; LLN18; LM16a; MP99] for such models).Instead, correspondingly to the (Navier-)Stokes equations with constant viscosity, we identify and prove homogenization limits in a critical, subcritical and supercritical regime yielding the Euler-Brinkman equations, the Euler equations and Darcy's law, respectively.To the author's knowledge, the Euler-Brinkman equations have not even been formally derived in the literature before.This can be viewed as a first step towards the rigorous justification of spray models like the one analyzed in [CDM11] that couples the incompressible Euler equations to a Vlasov equation through a linear friction force.

Setting and outline of the main results
Let T ⋐ B 1/4 (0), the reference particle, be a fixed closed set with smooth boundary, such that B 1 (0) \ T is connected and 0 ∈ T .For 0 < ε < 1, we consider particles centered at x ε i := εi, i ∈ Z 3 .Moreover, precisely, for α ⩾ 1, we define Then, for some T > 0, γ > 0 and µ 0 > 0, we consider solutions u ε to the Navier-Stokes equations for some given f ε ∈ L 2 (0, T ; L 2 (R 3 )) and u ε 0 ∈ L 2 σ (Ω ε ), where It is well known that then at least one Leray solution u ε exist, i.e. a weak solution which satisfies the energy inequality We focus on the case α > 1 which characterizes the regime where the particle diameters ε α are small compared to the inter-particle distance ε.In a nutshell, the effect of the particles on the fluid can then be described through a superposition of linear friction laws provided that the fluid inertia is negligible on the lengthscale of the particles.More precisely, we consider the particle Reynolds number where U ε , the order of magnitude of the fluid velocity, has yet to be determined.Then, if Re ε part ≪ 1, the influence of each particle on the fluid can be approximated by a friction force determined from the unique solutions (w k , q k ) ∈ Ḣ1 (R 3 ) × L 2 (R 3 ) to the linear Stokes problem which is a positive definite symmetric matrix.Neglecting fluid inertia and particle interaction, classical scaling considerations imply that each particle approximately contributes a friction force Taking into account that the particle number density is ε −3 leads to approximating the fluid velocity u ε by ũε which satisfies the Navier-Stokes equations in the whole space with an additional linear friction term µ 0 ε α+γ−3 Rũ ε , sometimes refered to as Brinkman force.More precisely, provided Re ε part ≪ 1, we expect u ε ≈ ũε where (1.6) From this approximation, we may easily identify the limiting behavior, where we distinguish the critical regime as γ + α = 3, the subcritical regime as γ + α > 3 and the supercritical regime as γ + α < 3. Before writing down the limiting equations, we revisit the constraint Re ε part ≪ 1.In the critical and subcritical regime, the Brinkman force is at most of order one, and therefore the solution ũε , and thus u ε and U ε from (1.3), are expected to be of order 1, provided u ε 0 and f ε are of order 1.Thus, in the critical and subcritical regime, which leads to the condition α > γ.
On the other hand, in the supercritical regime, the Brinkman force dominates thus slows down the fluid velocity to U ε = ε 3−α−γ .Therefore, in the supercritical case, leading to the condition γ < 3/2.
Taking the formal limit in (1.6), assuming f ε → f and u ε 0 → u 0 leads to the following limit systems.The regimes are illustrated in Figure 1.1. (1.7) • In the subcritical regime for α + γ > 3 with α > 1 and α > γ > 0, we obtain the Euler equations (1.8) Since the particles do not create any effective perturbation on the limit system, the asymptotically linear friction law guaranteed by α > γ > 0 is actually not required to obtain this limit case but it instead suffices that Re ε part ⩽ c 0 for some c 0 > 0 independent of ε.This corresponds to the regime α = γ > 3/2 with µ 0 ⩾ M for some M sufficiently large.
• In the supercritical regime, for α + γ < 3 with α > 1 and γ < 3/2, u ε → 0. Thus, we rescale time and velocities to obtain a nontrivial limit.More precisely, if ûε is a solution to (1.1) with µ 0 = 1, we consider the function u ε (t, x) = ε α+γ−3 ûε (ε α+γ−3 t, x).This rescaled velocity 1 One might argue that Euler-Darcy would be a more appropriate name for this system but this is already used for a different system that arises as homogenization limit of the 2-dimensional Euler equations in perforated domains, see e.g.[MP99].

Statement of the main results
The precise results are the following quantitative convergence results for u ε in all three regimes under regularity assumption on the solution u to the respective limit system.Smooth solutions exist at least for short times.Moreover, in the supercritical regime, we obtain in addition a weak convergence result in L 2 (0, T ; L 2 (R 3 )) assuming only a weak solution u ∈ L 2 (0, T ; L 2 (R 3 )) to Darcy's law (1.10).
• The three theorems above imply in particular that for any sequence ε → 0 with Here, f ε , u 0 ε and u ε are to be understood as defined in R 3 through extension by 0. Note that one may choose • The regularity assumptions on u could probably be weakened but we do not pursue to optimize here.
• In the supercritical regime, we do not obtain pointwise estimates in time.Indeed, there are boundary layers in time which prevent pointwise estimates under the stated assumptions.These boundary layers are due to the initial datum u ε but also due to possible jumps in time of the force f ε .

Previous results
The vanishing viscosity limit is a classical problem in the study of incompressible fluids, we refer to [MM18] for a review on the topic.In bounded domains with no-slip boundary conditions, the limiting behavior is not well-understood due to the onset of boundary layers.This is the reason why we consider the whole space in this paper.
In dimensions two and three, the vanishing viscosity limit has been studied in [ILN09] in the presence of a single shrinking body.The convergence to the Euler equations has been established provided that the local Reynolds number is sufficiently small i.e. the same condition a ε ⩽ cµ ε ≪ 1, where a ε and µ ε denote the particle diameter and fluid viscosity, respectively, and c is a sufficiently small constant (depending on the initial data, time, and the reference particle).
There is a vast literature on homogenization in perforated domains.Modeling the fluid velocity u ε by the stationary Stokes equations, Darcy's law has been obtained in [Tar80] in the case of particle of the same size as the inter-particle distance, i.e. α = 1.Later, Allaire [All90a; All90b] proved homogenization results for the Stokes equations for all ranges of α > 1, identifying Darcy's law for α ∈ (1, 3), the Stokes-Brinkman equations for α = 3 and the Stokes equations for α > 3. Allaire's results cover all space dimensions d ⩾ 2 with appropriate adaptations of the ranges of α for d ⩾ 4. In the two-dimensional case, the critical regime corresponds to particle diameters a ε such that ε −2 log a ε ∼ 1.By compactness, Allaire's results also apply to the stationary Navier-Stokes equations (in dimensions d ⩽ 4).
The results of Allaire have been refined in a number of works, for example considering more general distributions of particles, non-homogeneous Dirichlet boundary conditions, the study of higher order approximations and fluctuations.We refer to the recent results [CH20; DGR08; GH19; Giu21; HJ20; HMS19] and the references therein.
The homogenization limits for the full instationary Navier-Stokes for fixed viscosity correspond to the one of the stationary Stokes equations and are displayed in Figure 1.1.Formally they are obtained by setting γ = 0 in (1.6) and taking the limit ε → 0. The critical regime, α = 3, leading to the Navier-Stokes-Brinkman equations, has been considered by Feireisl, Nečasová and Namlyeyeva in [FNN16], whereas the subcritical case α > 3 and the supercritical case α ∈ (1, 3) has been treated recently by Lu and Yang in [LY23].
We emphasize that the Darcy's law in [LY23] and [All90b] is exactly the same as (1.10) whereas the Darcy's law in [Tar80] and [Mik91] differs quantitatively, in terms of a different resistance tensor R per which is obtained analogously as R from (1.5) but by solving the Stokes equations in the torus instead of the whole space.The reason for this difference is that in the case α = 1 the particle diameter is comparable to the interparticle distance.Therefore, the superposition of friction forces through single particle problems in the whole space (cf.(1.4)) must be replaced by studying the collective forces through the problem with periodic boundary conditions.Mathematically, the analysis of the case α = 1 is somewhat easier as it only involves two lengthscales, the microscopic lengthscale ε and the macroscopic lengthscale.Since the study of the case α = 1 requires different corrector problems and is rather well understood, we restrict our attention to α > 1 in the present paper.
Reflecting its importance for applications, there are several works concerning the derivation of non-linear Darcy's laws, especially the Darcy-Forchheimer equations.They seem to focus on the case α = 1, where nonlinear effects are expected to become important for γ ⩾ 3/2.Most of these works do not contain rigorous proofs, we refer to [BMW10] for an overview of the literature.Concerning rigorous results, Mikelić [Mik95] and Marušić-Paloka and Mikelić [MM00] tackled the critical case α = 1, γ = 3/2 in dimensions two and three starting from the stationary Navier-Stokes equations.The obtained limit system is a nonlinear nonlocal Darcy type equation.Moreover, in the subcritical case, α = 1, γ < 3/2, Bourgeat, Marušić-Paloka and Mikelić [BMM95] justified nonlinear versions of Darcy's law as higher order corrections to the linear law.
We also mention that the homogenization of the instationary Stokes equations with vanishing viscosity has been studied by Allaire [All92] for α = 1.In this case, the critical scaling (in any space dimension) is γ = 2 and a Darcy's law with memory effect is obtained as limit system.
The only previous result the author is aware of concerning the homogenization of the Navier-Stokes equations with vanishing viscosities when the particle diameters are much smaller than the interparticle distance (α > 1) is due to Lacave and Mazzucato [LM16b].In dimension two, they recover the unperturbed Euler equations under assumptions on the particle sizes, distances and the viscosity, which guarantuee that the particle Reynolds number is sufficiently small and that the particles do not exert a significant collective force on the fluid (subcritical regime).

Elements of the proof
The proof of the (quantitative) main results is based on an energy argument to estimate u ε − u which is, at its core, classical in the study of vanishing viscosity limits.However, similarly as in [ILN09] and [LM16b], we face the problem, that the limit fluid velocity u does not vanish inside of the particles and thus u is not an admissible testfunction for the PDE of u ε .As in [ILN09] and [LM16b], we therefore consider functions ûε obtained from u by a suitable truncation.In [ILN09], the truncation is performed on the level of the stream function (respectively the vector potential in three dimensions).In [LM16b], the fluid velocity itself is truncated, i.e.
where h ε is a suitable Bogovskii type correction such that ûε is divergence free.
As in [LM16b], we perform the truncation on the level of the fluid velocity itself.However, we need to be more careful, since the truncation needs to contain information of the boundary layers at the particles that produce the Brinkman term in the limit.Thus, instead of the scalar function ϕ ε in [LM16b] that truncate in a ε α neighborhood around the particles, we choose a variant of the matrix-valued oscillating testfunction w ε used by Allaire [All90a; All90b] that are build on the solutions to the resistance problem (1.4).
These functions w ε from [All90a; All90b] (which go back to corresponding functions in [Tar80] and similar functions for the Poisson equations used by Cioranescu and Murat in [CM82]) have been used with some modifications in many related works, see e.g.[GH19; LY23].However, w ε truncates on an ε-neighborhood around the particles, and therefore we could only use them directly in the present context provided the Reynolds number on the ε-lengthscale is small.This is the case if γ < 1 in the (sub-)critical regime and γ < 2 − α/2 in the supercritical regime.To overcome this restriction, we modify the testfunctions of Allaire, to truncate on a lengthscale η ε , ε α ⩽ η ε ⩽ ε.Aside from estimates analogous to their standard versions, we then use a Hardy-type estimate in order to control some error-terms arising from the nonlinear convection term.

Some possible generalizations and open problems
In this paper, we focus on periodic distributions of identical particles for the sake of the clarity of the presentation.The methods of proof do not rely on periodicity, though, and presumably apply to more general settings.
From the viewpoint of applications to suspensions, it would also be interesting to study nonhomogeneous Dirichlet boundary conditions, i.e. u ε = V i on ∂T ε i which have been treated for the corresponding model without vanishing viscosity in [DGR08;FNN16].
As in many related works, we focus here on the three-dimensional case.Extensions to two dimensions are possible with the necessary modifications similar as in [All90a; All90b].As mentioned above, parts of the subcritical regime is treated in [LM16b].There is one important difference between the two-and three-dimensional case, however, that seems to make it more difficult to analyze all the cases in dimensions two where the particle Reynolds number tends to zero.Namely, in three dimensions, the Stokes resistance of a particle of size a ε in the whole space is well approximated by solving Stokes problems in an η ε -neighborhood of the particle, for any lengthscale η ε with η ε ≫ a ε .This allows us to consider the intermediate scale η ε as outlined in the previous subsection.In two dimensions, however, just like for capacities, only relative Stokes resistances are meaningful.As observed in [All90a; All90b], it turns out that the relative resistance in a cell of order of the inter-particle distance ε is the correct object to consider in order to study the collective effect of the particles.2Therefore, the use of an intermediate lengthscale η ε does not seem suitable in 2 dimensions, at least not in the critical and supercritical regimes.As discussed above, this would restrict to assuming that the Reynolds number on the scale ε is of order one, in order that the (accordingly modified) proof given in this paper still works.
It would be of great interest to understand the regimes where the particle Reynolds number Re ε part is not tending to zero, i.e. γ ⩾ max{α, 3/2}, displayed in orange in Figure 1.1.However, as discussed above, the case when the particle Reynolds number is large is not even understood in the case of a single shrinking particle.In the case where the particle Reynolds number is small but fixed, we proved that one still obtains the Euler equations in the subcritical regime.One could still expect convergence to the Euler equations in the subcritical regime.In the critical and supercritical regimes, one could expect the onset of nonlinear behavior similar to the one obtained in [Mik95; MM00] at γ = 3/2.

Outline of the rest of the paper
The rest of the paper is organized as follows.
In Section 2, we define the correctors w ε and prove some useful estimates on them.Mostly, these are standard adaptions of previously established estimates.
Section 3 contains the proofs of the main results.In Section 3.1 we give the proofs of Theorem 1.1 and Theorem 1.2, which are largely analogous.
Section 3.2 contains the proof of Theorem 1.5 and Theorem 1.3.The proof of Theorem 1.3 is very similar to those of Theorems 1.1 and 1.2.For the proof of Theorem 1.5, we first use a well-known Poincaré inequality in the perforated domain (see Proposition 2.4) to get a uniform a-priori estimate of u ε in L 2 (0, T ; L 2 (R 3 )).We use a classical duality argument that allows us to pass to the limit in the weak formulation of the PDE by applying the correctors w ε to smooth testfunctions instead of the solution u of the limit problem as in the proof of the quantitative results.

Corrector estimates
Throughout this section, we write A ≲ B for A, B ∈ R when A ⩽ CB for some constant C that depends only on the reference particle T and possibly the exponent p of some Sobolev space involved in the estimate.
i the open cubes of length ε centered at x ε i that (essentially) cover R 3 .We split each cube Q ε i into four areas, displayed in Figure 2, Then, recalling the definition of (w k , q k ) from (1.4), we define w ε k , q ε k as the ε-periodic functions that satisfy ( Here, e k denotes the k-th unit vector of the standard basis of R 3 .Note that the Stokes equations in D ε i are complemented with inhomogeneous no slip boundary conditions due to the requirement (Ω ε ).We will write w ε for the matrix-valued function with columns w ε k , and q ε for the (row-)vector with entries q ε k .We summarize properties of w ε in the following lemmas.Some of the estimates are very similar to the ones given in [All90a; All90b] and other works.
Lemma 2.1.The functions w ε , q ε satisfy (i) (2.1) (ii) For all compact sets K ⊆ R 3 , we have w ε → Id strongly in L 2 (K).Moreover, for all 3/2 < p < 3 and all φ ∈ W 2,p (R 3 ) (2.6) Step 1: Pointwise estimates and proof of (i). (2.8) The estimates on C ε i follow immediately from standard decay estimates for the Stokes equations in exterior domains (cf. [Gal11, Theorem V.3.2])applied to (w k , q k ) from (1.4) and the definition of w ε , q ε through rescaling on C ε i .Consequently, the estimates on D ε i are deduced from the estimates on ∂D ε i and standard regularity theory for the Stokes equations.Clearly, (i) follows directly from these pointwise estimates.
Step 2: Proof of (ii).Using (2.7) and w ε = Id in K ε i , we compute for one cell, for all p < 3, For any compact K ⊆ R 3 , we can cover K by C(K)ε −3 many cubes Denoting (φ) i = ffl (2.9) Estimates (2.3)-(2.5)are proved analogously.For (2.3) we use in addition that B δε α (x ε i ) ⊆ T ε i for some δ > 0 that depends only on the reference particle T .Therefore . Without loss of generality, we assume x ε i = 0.By the pointwise estimate (2.8) and the fundamental theorem of calculus, we have for all x ∈ C ε i ∪ D ε i with δ > 0 as above This implies ) , as claimed.The proof of the estimate for the term involving q ε is analogous.

Lemma 2.2. We can write
where the matrix R is defined in (1.5).
Proof.We observe that −∆w ε + ∇q ε is supported on i ∂C ε i ∪ ∂D ε i = i ∂D ε i ∪ ∂Ω ε and we define γ ε to be the part supported on ∂Ω ε which consequently satisfies ⟨γ ε , v⟩ = 0 for all v ∈ H 1 0 (Ω ε ).Then (2.10) holds with M ε k , the columns of M ε , being where and where w k , q k are as in (1.4) and n is the unit normal on ∂B ηε/4 (x ε i ).By [All90a, Lemma 2.3.5](which follows from the fact that w k , q k asymptotically behave as the fundamental solution of the Stokes equations), we have To conclude the proof, it suffices to show that for all φ ∈ H 3 (R 3 ) and all (2.15) (2.16) Indeed, η −1 ε ε α ⩽ 1 by assumption and thus (2.12)-(2.16)imply the assertion.To prove (2.14), we begin by observing that for all v ∈ H 1 (Q ε i ) we have due to Sobolev embedding ∥v − (v (2.17) where we recall the notation (v) i = ffl v and where the constant C is universal due to scaling considerations.Similarly, we have the Poincaré-type inequality where the last inequality is shown as in (2.9) We turn to (2.15).We use the pointwise estimates (2.8) to bound It remains to show (2.16).Using again (2.17) and (2.18), we have for Thus, for φ ∈ H 3 (R 3 ) and ψ ∈ H 1 (R 3 ), using (2.13), This finishes the proof.
Lemma 2.3.For all 1 < p < ∞, there exists a linear operator B ε : W 1,p (R 3 ) → W 1,p (R 3 ) such that for all φ ∈ W 1,p (R 3 ) that are divergence free we have and (2.20) Proof.It suffices to construct the linear operator on the subspace of divergence free functions φ ∈ W 1,p (R 3 ).We observe that then as w ε = Id on ∂D ε i .Therefore we may employ a Bogovski operator in A ε i .More precisely, by [DFL17, Lemma 3.1] (which is a consequence of [ADM06] and [DRS10]), there exist operators

19) as well as the first inequality in (2.20). The second inequality in (2.20) follows from the first one and the Poincaré inequality in the domains
For the treatment of the subcritical case, we will rely on the following Poincaré inequality in Ω ε .It is proved in [All90b,Lemma 3.4.1]when Ω ε is a bounded domain.Since the proof is based on a local Poincaré inequality in each of the cubes Q ε i , it still applies here.Proposition 2.4 ([All90b, Lemma 3.4.1]).For all φ ∈ H 1 0 (Ω ε ) (2.21)

Proof of the main results
As outlined in Section 1.4, the strategy for the proof of the main results is based on energy estimates for the difference Here u ε is the solution to (1.1) in the critical and subcritical case and to (1.9) in the supercritical case and u is the solution to (1.7), (1.8) and (1.10), respectively.Moreover, w ε is the matrix valued function defined at the beginning of Section 2 and depends on a parameter ε α ⩽ η ε ⩽ ε that we will choose later.Finally, B ε is the operator from Lemma 2.3.We first observe that the difference ( where the constant C depends only on T and ∥u∥ L ∞ (0,T ;H 3 (R 3 )) .Indeed, this follows immediately from (2.2) and (2.20).

Proof of Theorem 1.1 and Theorem 1.2
Throughout this subsection, we assume that the parameters α and γ are in the range of the critical or subcritical regime specified in Theorem 1.1 and 1.2, respectively, that is α > 3/2 and γ > 0, γ ∈ [3 − α, α) or γ = α and µ 0 ≫ 1.Moreover, v ε is defined by (3.1) where u ε is the solution to (1.1) and u is the solution to (1.8) or (1.7).
The main technical part of the proof of the main results is an energy estimate for v ε stated in the following proposition.Thereafter, we show how Theorem 1.1 and Theorem 1.2 follow from this proposition and Gronwall's inequality.

under the assumptions of Theorem 1.1 we have for all
for some constant C which depends only on (ii) Under the assumptions of Theorem 1.2 we have for all t ⩽ T for some C which depends only on and some C µ 0 which depends additionally on µ 0 .
Proof of Theorem 1.1.We choose η ε = 1 C ε γ such that we may drop the second term on the left-hand side of (3.3).Note that as γ = 3 − α and α ∈ (3/2, 3), the assumption ε α ⩽ η ε ⩽ ε is satisfied for all ε sufficiently small (for ε of order 1, the assertion of the theorem is an immediate consequence of the energy inequality (1.2)).
Then, by Gronwall's inequality, Proposition 3.1 yields and we deduce with (3.2), which only gives a higher order error, that This finishes the proof.Proof of Theorem 1.2.We choose η ε = δε β with β = min{γ, 1} and This choice guarantees ε α ⩽ η ε ⩽ ε is satisfied for all ε sufficiently small.Moreover, choosing M = C, the assumption µ 0 ⩾ M if γ = α allows us to drop the second term on the left-hand side in (3.4) in all cases.Therefore, arguing as in the proof above yields We observe that 2α + γ − 4 ⩽ max{2α − 3, 2α + 2γ − 6} to finish the proof.Proof of Proposition 3.1.We focus on the critical case γ = 3 − α where u solves (1.7).We discuss the necessary adaptions for the subcritical case γ > 3 − α in the last step of the proof.
Throughout the proof we write ≲ for ⩽ C with C as specified in the statement of the proposition.
Step 2: Relative energy inequality: We consider the relative energy 1 2 ∥v ε ∥ 2 L 2 .We estimate using the energy inequality (1.2) for u ε as well as ǔε Using the equation solved by u ε , we have and likewise, using the equation of ǔε where inserting (3.7)-(3.8) in (3.6) and denoting Thus, we deduce where Step 3: Bound of I 1 : We first manipulate the first term in I 1 .Using u ε = ǔε = 0 on ∂Ω ε as well as div u = div u ε = 0 yields by integration by parts ˆt 0 ˆΩε This allows us to rewrite We recall ǔε = w ε u−B ε (u) to estimate by the regularity assumptions of u, (2.6) and (2.20) combined with (2.3) and another integration by parts (3.12) where we used α > 1 in the last estimate.By the regularity assumptions of u and (2.2), we have Similarly, relying additionally on (2.20), Finally, we estimate by another integration by parts We estimate using that u and w ε are uniformly bounded in L ∞ as well as (2.20), (2.2) and Sobolev embedding In summary, we find, Step 4: Bound of I 2 : We split where We estimate We rewrite The first term on the right-hand side is estimated as above.Combining this with (2.11) to estimate the second term on the right-hand side yields for some δ > 0 to be chosen later where we used that η ε ⩾ ε α and α ⩾ 3 − γ to absorb the term η −2 ε ε 2α .Next, we estimate using (2.5) and (2.6) , where we used α + γ ⩾ 3 in the last inequality.
Step 6: Adaptations in the subcritical case: Let now γ > 3 − α and let u solves the the Euler equations (1.8).There are only very little changes in the proof in this case.In Step 1, the only differences are that in the PDE solved by ǔ, (3.5) all inctances of ε γ should be replaced by µ 0 ε γ (in the critical case, we assumed µ 0 = 1) and that (M ε − w ε R)u has to be replaced by µ 0 ε γ+α−3 M ε u .Consequently, estimate (3.10) still holds up to replacing all instances of ε γ by µ 0 ε γ and where in the source F ε (appearing in I 3 ) the term (M ε − w ε R)u is likewise replaced by µ 0 ε γ+α−3 M ε u.
In particular, the estimates for I 1 in Steps 3 still apply, and all the estimates of Step 4 for I 2 are unaffected except for the estimate of I 2 2 which now takes the form Thus, we estimate with Lemma 2.2 and we obtain Combining this estimate as before with the estimates for I 1 , (3.13), yields (3.4).

Proof of Theorem 1.5 and Theorem 1.3
In this subsection, we consider u ε a Leray solution to (1.9) and u the solution to (1.10).
Proof of Theorem 1.3.We follow closely the proof of Proposition 3.1 to obtain an estimate for v ε = ǔε − u ε , where ǔε := w ε u − B ε (u) with w ε as in Section 2 and with B ε as in Lemma 2.3.
Proof of Theorem 1.5.For simplicity of the notation, we write u ε instead of ũε for the extension of u ε by 0 to R 3 .Note that the energy inequality (1.11) does not immediately provide uniform a priori estimates for u ε .The first step of the proof therefore consists in combining the energy inequality with the Poincaré inequality from Proposition 2.4 to deduce a uniform a priori bound for u ε in L 2 (0, T ; L 2 (R 3 ).Then, u ε ⇀ u for some u ∈ L 2 (0, T ; L 2 (R 3 )) along subsequences and it suffices to show that u solves (1.10).
Step 1: Uniform a priori estimate We claim that, Applying Young's inequality, this establishes the estimate for ∇u ε , and the estimate for u ε follows by another application of the Poincaré inequality (2.21).
Step 4: Convergence of I 3 : With M ε as in Lemma 2.2, we rewrite By Lemma 2.2 and (3.18), we have