Lagragian methods for a general inhomogeneous incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients

We study the inhomogeneous incompressible Navier-Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index p is different from 2) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin and Mucha for the incompressible Navier-Stokes system with constant coefficients.

Remark 1 Due to the incompressibility condition the usual second viscous term, namely ∇(λ(ρ)div u), vanishes in the pressure term and for the same reason we directly write the capillary term as a general divergence form (the case of a constant function k ≡ 1 corresponds to the classical capillary coefficients ρ∇∆ρ and −∇ρ∆ρ which are equivalent as their difference is a gradient, absorbed by the pressure term).
Let us first give a few words about the classical compressible Navier-Stokes-Korteweg system in the case of a constant coefficient k(ρ): ∂ t ρ + div (ρu) = 0, ∂ t (ρu) + div (ρu ⊗ u) − Au + ∇(P (ρ)) = kρ∇∆ρ. (NSK) As pointed out by Danchin and Desjardins in [12] there is a strong coupling between the compressible part of the velocity and the density fluctuation ρ 0 −1 that helps regularizing ρ 0 − 1 (the incompressible part of the velocity being totally decoupled with it). This is why ρ 0 − 1 features a parabolic regularization for all frequencies, unlike for the classical compressible Navier-Stokes system where this occurs only for low frequencies. More precisely they prove that if ρ 0 − 1 ∈Ḃ 2,1 ), near a stable constant state (P ′ (1) > 0) they obtain a global solution. Let us also mention the works of Haspot (see [18,19,20,21]) where the Korteweg system is studied in general settings, or with minimal assumptions on the initial data (or Besov indices) in the case of particular viscosity or capillarity coefficients.
Another way of modelling the capillarity (still in the diffuse interface setting) consists in considering a non-local capillary term, featuring only one derivative. This was first suggested by Van der Waals and re-discovered by Rohde (see [25,26]). As studied in [19,7,8,9] for small perturbations of a stable constant case (in critical spaces for integrability index p = 2) the regularity structure of the density fluctuation is closer to the classical compressible Navier-Stokes case: parabolic regularization in low frequencies, damping in the high frequency regime (the threshold depending on µ 2 /κ). Moreover there is also a strong coupling between the density and the gradient part of the velocity, and in the cited works is also studied the transition from the non-local to the local capillary models. We also mention [10] for the same study without smallness and stability assumptions.
In the case of the inhomogeneous incompressible Korteweg, the velocity is divergencefree so that no part of it can combine with the density fluctuation and improve its regularization. Any energy method with use of symmetrizers is completely useless. This system is much less studied than the compressible version. We can for example refer to [5,27] both of them in the case of constant viscosities and constant capillary coefficient ∇ρ∆ρ. The first article provides local solutions for bounded domains and regular initial data in Sobolev spaces W s,p (p ≥ 1) and requires L 2 -assumptions. The second article requires regular initial data in energy spaces (H 3 ) and obtains local solutions as well as convergence towards the Euler system as κ and µ go to zero. It is crucial in their article to take advantage of the L 2 -structure to deal with the capillary term as (∇ρ · ∆ρ|u) = (u · ∇ρ|∆ρ) = −(∂ t ρ|∆ρ).
In the present article, we wish to obtain well-posedness results for initial data in critical spaces (minimal regularity assumptions), for general variable capillary and viscosity coefficients and for general integrability index (p not necessarily equal to 2) and we will follow Lagrangian methods developped by [15,16] (and now frequently used, we refer for example to [11,23,24]). Let us precise that this method was first introduced by Hmidi in [22] and extended by Danchin in [14]. The incompressibility condition in our system leads us to adapt the arguments of [15] but the additional capillary term will force us to mix them with arguments from [16] that extends the methods from [15] in the compressible setting and therefore deals with additional external force terms that are only bounded in time (which imposes bounds for the lifespan).

Remark 2
We present here results for general capillary and viscosity terms in the incompressible setting but various non-local capillary terms can also be considered in compressible or quasi-incompressible cases (see for example [1,2]).

Statement of the results
The main goal of this article is to state results for general smooth viscosity and capillary coefficients, without any smallness assumptions on the initial density fluctuation ρ 0 − 1, and in truly critical spaces, that is without any extra low frequencies assumptions, all of this in a general L p -setting (without energy methods). Thanks to the Lagrangian methods we will be able to give a very short proof of these results.
For this we will use the new class of estimates obtained by the first author in [6] instead of the classical maximal regularity estimates obtained by Danchin and Mucha in [15,16]. As in [6] our result will be given for n = 2 with p ∈ (1, 4) and n = 3 with p ∈ (6/5, 4).
If we wish to recover the full usual range for n, p (p ∈ [1, 2n[) we need to assume as in [15,14] that ρ 0 − 1 is small. Like in [15] we obtain a local solution. Finally if we also assume that the initial velocity is small, although we are unable to obtain global existence as in [15], we are able to prove that the lifespan can be bounded from below by C κ , which goes to infinity asκ goes to zero. Moreover the solution converges on any time interval [0, T ] to the global solution of the inhomogeneous incompressible Navier-Stokes system given by [15].
Let us now state in detail these results: Theorem 1 Assume that n = 2 and p ∈ (1, 4) or n = 3 and p ∈ ( 6 5 , 4). Let the initial data (ρ 0 , u 0 ) satisfy: Then there exists a positive time T > 0 and a unique solution Remark 3 At the end of the article we give more details about the dependency of T in terms of the parameters.
Theorem 2 Assume that n ≥ 2 and p ∈ [1, 2n). Let the initial data (ρ 0 , u 0 ) satisfy: There exists a constant c > 0 such that if ρ 0 − 1 Ḃ Moreover, if in addition u 0 ≤ cμ, then T is bounded from below by some Tκ = Cμ κ . Finally, if we denote by (ρκ, uκ, ∇Pκ) the previous solution and by (ρ, u, ∇P ) the global solution of the inhomogeneous incompressible Navier-Stokes system given by [15] there where the space E s T is defined in (2.9).
The article will be organized as follows. Sections 2 and 3 are devoted to the proof of the theorems. As the results adapt the methods from [15,16] we will skip details and mainly focus on what is new, namely the capillary term. The end of this section is devoted to some precisions and extensions of the results. Section 4 is a an appendix gathering definitions and main properties for homogeneous Besov spaces, Lagrangian change of variables, and estimates for the Lagrangian flow and its derivatives.
2 Proof of Theorem 1

Rescaling of the system
If we introduce ( ρ, u, P ) as follows: Then this allows us to study the case (1,κ µ 2 ) instead of (μ,κ) in the sense that (ρ, u, ∇P ) Remark 4 We emphasize that the ratioκ µ 2 also plays an important role in the compressible system, as observed for example in [8].
From now on we will focus on (N SIK2) and a solution for (N SIK2) on some [0, T 1 ] will produce a solution for (N SIK) on [0, T 1 µ ] thanks to the reverse change of variable:

Lagrangian formulation
The method, first used in the incompressible case by T. Hmidi in [22] then developped by Danchin (see for example [14,16]), Danchin and Mucha in [15,17], is based on the following observation: if (ρ, u, ∇P ) solves (N SIK2) and if u is smooth enough (say Lipschitz), we introduce the flow X associated to u, that is the solution to: or equivalently X(t, x) = x + t 0 u(τ, X(τ, x))dτ . As the jacobean determinant satisfies det(DX(t, x)) = e t 0 div u(τ,X(τ,x))dτ , the incompressibility condition implies in our case det(DX(t, .)) ≡ 1. Then, introducing for any function f ,f (t, x) def = f (t, X(t, x)), with the computations from [15] we obtain that (ρ,ū, ∇P ) solves the following system (we refer to the appendix for the transformation of the capillary term, the rest is dealt as in [15,16] We emphasize that the new system presents two important features: • there are no advection terms anymore, • the density becomes constant (this is due to incompressible setting, in the compressible setting the density is multiple of the inverse of the Jacobean determinant, we refer to [16] for more details).
The price to pay is reasonnable as it consists in dealing with Matrix A which is close to I d when the time or the initial data are small as outlined in Proposition 8. The previous system can then be rewritten into the form we will finally study (now denoting (Xū, Aū) instead of (X, A)): Remark 5 Observe that with this notation Xū(t, x) = x + t 0ū (τ, x)dτ . Next our stategy will be the same as in [15,16,6], letv satisfying on [0, T ] for some where the small bound ε 0 comes from (4.42) (see Proposition 7). Let us define the flow: then all the work consists in proving that the following system (we recall that adj(A) is has a unique solution (ū, ∇P ) on some [0, T 0 ] for anyv, which we reformulate in terms of fixed point: for any (v, ∇Q) the function Φ : (v, ∇Q) → (ū, ∇P ) has a unique fixed point that solves (2.5). Next, thanks to the properties ofū we will be able to prove that Xū is a C 1 -diffeomorphism on R n and we can perform the inverse change of variable (ρ, u, P ) = (ρ,ū,P ) • X −1 u . As a fixed pointū satisfies div (adj(DXū)ū) which garantees that div u = 0 and as a consequence Xū (which is the flow of u) is measure preserving and adj(DXū) = Aū, which implies thatū solves (2.5). Finally (ρ, u, P ) solves (N SIK2) on some [0, T 0 ] which provides a unique solution of (N SIK) on the time interval [0, T 0 µ ].

Remark 6
We emphasize that the given functionv genuinely depends on the Lagrangian variables, it is not the change of variable of a given function v. In other words, the system is solved in Lagrangian variables then the inverse change of variables is performed.

A priori estimates
As in [15] the main ingredient in the proof of Theorem 2 is the following maximal regularity estimates proved in [15,17]: admits a unique solution (u, ∇P ) in the following space and there exists a positive constant C (independant of µ, T ) such that Similarly, the main ingredient in the proof of Theorem 1 (no smallness assumptions or additional low frequency regularity) is the new estimates recently obtained by the first author in [6]: Proposition 2 Let n = 2 and p ∈ (1, 4) or n = 3 and p ∈ ( 6 5 , 4). Let a, b two functions such that there exists positive constants (a * , a * , b

First step: well-posedness for (2.7)
With a aim of conciseness, we will only give details for terms and computations related to the capillary term as the other terms are the same as in [6,15]. Letv and T given as in (2.6), for (w, ∇Q) ∈ E T let us consider the following system: Proposition 2 implies there exists a unique solution (ū, ∇P ) ∈ E T and a constant C ρ 0 such that for all t ≤ T , The first two external terms are the same as in [15,6], and using Condition (2.6) and (4.43) from Proposition 8, we easily obtain that: The third term is dealt with the classical product laws in Besov spaces (see section 4.1), using the fact that k(ρ 0 ) = 1 + k(ρ 0 ) − k(1) (recall that k(1) = 1) and (4.43): ).
So that according to condition (2.6) we end up for all t ≤ T with, and the application Ψ : (w, ∇Q) → (ū, ∇P ) is well defined E T → E T . Next let us prove that this is a contraction: δw), div δū = div Mv(δw), δū |t=0 = 0.
(2.14) Similar computations as before imply that for all t ≤ T , (δū, ∇δP ) Et ≤ C ρ 0 εe Cρ 0 (t+1) (δw, ∇δQ) Et , and Ψ is 1 2 -Lipschitz when for example T ≤ 1 and ε < 1 2Cρ 0 e −2Cρ 0 and has a unique fixed point (ū, ∇P ) ∈ E T which solves (2.7). Moreover the fixed point belongs to E T and for all t ≤ T , Remark 7 We could also ask T ≤ B and ε < 1 2Cρ 0 e −(1+B)Cρ 0 for some B > 0, but as we will see later, taking a large B will not help for the final lifespan.

Let us introduce the space
We just proved that given any (v, ∇Q) ∈ F ε T and under Condition (2.6), System (2.7) has a unique solution (ū, ∇P ) = Φ(v, ∇Q). Let us recall that our aim is to prove that (2.5) has a unique solution by proving that Φ has a unique fixed point. Proving that Φ is a contraction will not be difficult but the problem is that as u 0 is not assumed to be small, there is no reason forū to satisfy the following condition (which is crucial to go back to the original variables): even if there exists some Tū > 0 such that the integral is bounded by ε, we do not know if Tū ≥ T . In other words, we are not able to prove that Φ maps F ε T into itself due to this integral condition.

Second step: well-posedness for (2.5)
As in [15,6], to overcome this problem, the idea is to introduce the free solution, let us define (u L , ∇P L ) the unique global solution of the following system: The a priori estimates also provide the fact that for all t, We are now able to precise the parameters: assume that ε and T satisfy: Thanks to Condition (2.16), for any ( v, ∇ Q) ∈ G ε T , then (v, ∇Q) def = (u L + v, ∇P L + ∇ P ) belongs to F ε T and we easily check that Condition (2.6) is satisfied so there exists a unique solution (ū, ∇P ) = Φ(v, ∇Q). If we set ( u, ∇ P ) = (ū − u L , ∇P − ∇P L ) then if T, ε are small enough, we can prove that ( u, ∇ P ) ∈ G ε T : indeed they satisfy the following system and Mv(ū) are the same as in (2.11), and the last term is: As before, we obtain thanks to Propositions 2 and 8: ).
As we wish to take advantage of the smallness of the L 1 T norm, we cannot do as before for the last term (as the u L L ∞ TḂ n p −1 p,1 norm may not be small and would prevent any absorption by the left-hand side) and the idea is (as in [15,6]) to use the L 2 tḂ n p p,1 -norm of u L instead: ). (2.22) Thanks to the fact that ( v, ∇ Q) ∈ G ε T and condition (2.16) we end up for all t ≤ T with (we do not give details as the computations are the same as before), and then, as εC ρ 0 e Cρ 0 (T +1) ≤ 1 2 (thanks to (2.16)), Finally, we proved that for any ( v, ∇ Q) ∈ G ε T there exists a unique solution ( u, ∇ P ) ∈ G ε T of System (2.17). We will abusively denote once more this solution by Φ( v, ∇ Q). To prove that Φ is a contraction is very close to what we did for the application Ψ so we will not give details. All that remains is to get back to the original variables: as the unique fixed point of Φ,ū satisfies div (adj(DXū)ū) = 0 and as we also have ∇ū Let us end this section with precisions about the dependancy of T in terms of the physical parameters: a more precise use of Proposition 2 gives that (u L , ∇P L ), solution of (2.15), satisfies: and due to the exponential term we can see that if we asked T ≤ B for some large B as in Remark 7, the above estimates implies that t ≤μ 2 κ εe −(1+B)Cρ 0 4Cρ 0 , which explains why B = 1 is sufficient as outlined in Remark 7.

Proof of Theorem 2
Let us now turn to the cases when ρ 0 − 1 is small (allowing the full ranges for n, p). As announced in the previous section, instead of System (2.7), we will take advantage of Proposition 1 (As div D(u) = ∆u + ∇div u we are in the scope of this result) and study: where Mv(ū) is the same as before and the external force terms are very close to those from (2.11) (recall that µ(1) = 1): As before, if (v, T ) satisfies Condition (2.6), if (w, ∇Q) ∈ E T , let us introduce the following system: (3.24) Thanks to Proposition 1, with similar computations as before, we obtain that for all t ≤ T , and the application Ψ : Similarly as before we easily show this is a contraction if C( 1 − ρ 0 Ḃ n p p,1 + ε) ≤ 1 2 for example and then we obtain a unique fixed point satisfying for all t ≤ T : Remark 8 Contrary to the result from [15], a smallness condition only on ρ 0 −1 will not be sufficient because the capillary term features (k(ρ 0 ) − k(1) to be estimated in Besov spaces.
We then have to split in two cases wether u 0 is small or not. First case : if u 0 and T are so small that that is for example when then (ū, ∇P ) E T ≤ ε and the application Φ, associating to (v, ∇Q) the unique solution (ū, ∇P ) is well defined F ε T → F ε T . As no free system is required (and no time exponential appears in the a priori estimates) we can simply take T = ε 4CCρ 0μ 2 κ . We prove similarly that Φ is a contraction and then we obtain a unique fixed point for Φ which provides the unique solution for System (2.5) and concludes, thanks to the inverse Lagrangian change of variable and inverse scale change, the proof of the last part of Theorem 2 (giving a lifespan bounded from below by a multiple ofμ κ ).
Second case : if u 0 is not assumed to be small, due to the L ∞ T -norm nothing garantees that (ū, ∇P ) ∈ F ε T anymore and the idea is, as before, to introduce the unique solution (u L , ∇P L ) of the free system (simpler than the one in the previous section): With the same arguments as in the previous section we prove the existence of a constant C ρ 0 > 0 such that if: (3.27) then (similarly to what we did in the previous section, we leave the details to the reader), the application Φ mapping ( v, ∇ Q) to the unique solution ( u, ∇ P ) of the following system: (3.28) is well defined from G ε T to itself, and contractive. Then there exists a unique fixed point which ends the proof of the rest of Theorem 2.

Convergence whenκ goes to zero
If (ρ, u, P ) and (ρκ, uκ, Pκ) are defined as in Theorem 2, performing both of the Lagrangian changes of variables, we obtain that the difference (δu, ∇δP ) def = (ūκ −ū,Pκ −P ) satisfies the following system: where all the right-hand side terms are the same as in (3.23) except: Using the same arguments as before we obtain that under Condition (3.25), for all t ≤ And for the density as for all t, x : which ends the proof of the theorem.

Precisions about the lifespan
In both proofs, we had to introduce, for some fixed small ε (smaller than ε 0 from (4.42) from Proposition 7), the time: Let us give more details for example in the second case (small ρ 0 − 1): as (u L , ∇P L ) solves System (3.26), projecting thanks to the Leray orthogonal decomposition (P is the orthogonal projector on divergence-free vectorfields, and Q = I d − P is the orthogonal projector on gradients) and denoting As the external force term is independant of t, we immediately obtain that: And concerning the velocity, following classical localization methods, for all j ∈ Z (see [3]):∆ j u L (t) = e t∆ 1 µ∆ taking the L p -norm, thanks to the frequency localization and Lemma 2.4 from [3] and using that F ρ 0 does not depend on t: Then taking the L 1 t -norm and multiplying by 2 j( n p −1) leads us to the classical refined estimate: , and taking the L 2 t -norm in (3.2): Thanks to the assumptions on ρ 0 , we obtain that: , and we end up with p,1 , using that for all α ≥ 0, 1 − e −α ≤ α, the condition in (3.30) is satisfied when: 3 ) (equal to 1 in a sub-annulus), and satisfy that for all ξ ∈ R 3 \ {0}, q∈Z ϕ(2 −q ξ) = 1.
Then for all tempered distribution u we define for all q ∈ Z: The homogeneous Besov spaces are defined as follows: , with lim q→−∞Ṡ q u = 0 and u Ḃs p,r def = 2 qs ∆ q u L p q∈Z ℓ r < ∞}.
Remark 9 Due to the support of ϕ, we easily obtain thaṫ Let us now turn to the Bony decomposition, coming from the fact that for all distributions u, v, we can write (at least formally) the product as follows: A more efficient way to write this product is the following Bony decomposition, where we basically set three parts according to the fact that the frequency l of u is of smaller, comparable or bigger size than the frequency j of v: where • T is the paraproduct : T u v := lṠ l−1 u∆ l v (for each l, the term has its frequencies in an annulus of size 2 l ), • R is the remainder : R(u, v) = l |α|≤1∆ l u∆ l+α v (the term has its frequencies in a ball of size 2 l ).
In this article we will often use the following estimates for the paraproduct and remainders in order to deal with nonlinear terms (we refer to [3] Section 2.6 for general statements, more properties of continuity for the paraproduct and remainder operators: Proposition 3 For any (s, p, r) ∈ R × [1, ∞] 2 and t < 0, there exists a constant C such that T u v Ḃs p,r ≤ C u L ∞ v Ḃs p,r and T u v Ḃ s+t p,r ≤ C u Ḃt ∞,∞ v Ḃs p,r .
2. under the same assumptions, for any u, v ∈Ḃ s p,r ∩ L ∞ , then there exists a constant C such that

Lagrangian change of variables
We gather in this section the main properties that we use in the process of the Lagrangian change of variable. For more details and proofs we refer to [15] (the compressible version and more general results can be found in [16]).
Proposition 5 Let X be a globally defined bi-lipschitz diffeomorphism of R n and (s, p, q) with 1 ≤ p < ∞ and − n p ′ < s ≤ n p . Then a → a • X is a self-map overḂ s p,1 (R n ) in the following cases:  which serves as a crucial ingredient in [15,16,6].