Strong solution for Korteweg system in bmo$^{-1}(\mathbb{R}^N)$ with initial density in $L^\infty$

In this paper we investigate the question of the local existence of strong solution for the Korteweg system in critical spaces in dimension $N\geq 1$ provided that the initial data are small. More precisely the initial momentum $\rho_0 u_0$ belongs to $\mbox{bmo}_{T}^{-1}(\mathbb{R}^N)$ for $T>0$ and the initial density $\rho_0$ is in $L^\infty(\mathbb{R}^N)$ and far away from the vacuum. This result extends the so called Koch-Tataru Theorem for the incompressible Navier-Stokes equations to the case of the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any $\rho(t,\cdot)$ with $t>0$. We also prove the existence of global strong solution for small initial data $(\rho_0-1,\rho_0u_0)$ in the homogeneous Besov spaces $(\dot{B}^{\frac{N}{2}-1}_{2,\infty} (\mathbb{R}^N) \cap \dot{B}^{\frac{N}{2}}_{2,\infty} (\mathbb{R}^N) \cap L^\infty(\mathbb{R}^N)) \times (\dot{B}^{\frac{N}{2}-1}_{2,\infty} (\mathbb{R}^N))^N$. This result allows in particular to extend in dimension $N=2$ the notion of Oseen solutions defined for incompressible Navier-Stokes equations to the case of the Korteweg system when the vorticity of the momentum $\rho_0 u_0$ is a Dirac mass $\alpha\delta_0$ with $\alpha$ sufficiently small.


Introduction
We are concerned with compressible fluids endowed with internal capillarity.The model we consider originates from the XIXth century work by J. F. Van der Waals and D. J. Korteweg [26,21] and was actually derived in its modern form in the 1980s using the second gradient theory (see [8,18,25]).Korteweg-type models are based on an extended version of nonequilibrium thermodynamics, which assumes that the energy of the fluid not only depends on standard variables but also on the gradient of the density.We are now going to consider the so-called local Korteweg system which is a compressible capillary fluid model, it can be derived from a Cahn-Hilliard like free energy (see the pioneering work by J.-E.Dunn and J. Serrin in [8] and also [1,4,13]).The conservation of mass and of momentum write: (1.1) Here u = u(t, x) ∈ R N stands for the velocity field with N ≥ 1, ρ = ρ(t, x) ∈ R + is the density, D(u) = 1 2 (∇u + t ∇u) is the strain tensor and P (ρ) is the pressure (we will only consider regular pressure law).We denote by µ > 0 the viscosity coefficients of the fluid.
We supplement the problem with initial condition (ρ 0 , u 0 ).The Korteweg tensor reads as: where κ(ρ) = κ 2 ρ is the coefficient of capillarity with κ 2 ∈ R + .The term divK allows to describe the variation of density at the interfaces between two phases, generally a mixture liquid-vapor.In our case the capillary term is also called quantum pressure.
We briefly mention that the existence of global strong solutions for the system (1.1) with small initial data for N ≥ 2 is known since the works by Hattori and Li [17] in the case of constant capillary coefficient κ(ρ).Danchin and Desjardins in [7] improved this result by working with initial data (ρ 0 − 1, ρ 0 u 0 ) belonging to the homogeneous Besov spaces 1 2,1 is embedded in L ∞ which allows to control the L ∞ norm of the density, it is even better since it implies that ρ 0 is necessary a continuous function).In [16], it is proved the existence of global strong solution with small initial data provided that (ρ 0 − 1, . This result extends [7] but does not allow to deal with general shocks on the initial density. We wish now rewrite the system (1.1) using the formulation introduced in [2,19] (where the existence of global energy weak solution is proved), we consider then the following effective velocity with c > 0: which enables us to rewrite the system (1.1) as follows when κ 2 ≤ µ 2 (we will only consider this case in the sequel) using the fact that div(ρ∇∇ ln ρ) = 2ρ∇( (1.4) with κ 2 1 = κ 2 − 2µc + c 2 .We specify now the value of c and we want to deal with a c such that κ 2 1 = 0, we take then: Setting now v 1 = u + c 1 ∇ ln ρ and v 2 = u + c 2 ∇ ln ρ , we have: − µ 2 − κ 2 ∇div(ρv 1 ) + ∇P (ρ) = 0, + µ 2 − κ 2 ∇div(ρv 2 ) + ∇P (ρ) = 0. (1.5) In this paper our main goal consists in proving the existence of global or local strong solutions for the system (1.1) with minimal regularity assumption on the initial data.More precisely since this system models a mixture liquid vapor with different density, we wish to show the existence of strong solution for initial density ρ 0 belonging only to the set L ∞ (R N ).In particular it implies that our functional setting will include the case of initial density admitting shocks what is also fundamental both in the physical theory of non-equilibrium thermodynamics as well as in the mathematical study of inviscid models for compressible flow.In addition we wish also to deal with momentum ρ 0 u 0 exhibiting specific structure, typically we have in mind the case of initial vorticity belonging to the set of finite measure in dimension N = 2 (we will recall later that there is such a theory for Navier-Stokes equations and even explicit such solutions, the so called Oseen tourbillon).In order to obtain such results, it seems necessary to work in space with minimal regularity assumptions.Typically the space with third index r = +∞ is a good candidate for the initial momentum ρ 0 u 0 .To do this, let us now recall the notion of scaling for the Korteweg's system (1.1).Such an approach is now classical for incompressible Navier-Stokes equations and yields local well-posedness (or global well-posedness for small initial data) in spaces with minimal regularity.In our situation we can easily check that, if (ρ, u) solves (1.1), then (ρ λ , u λ ) solves also this system: provided the pressure laws P have been changed to λ 2 P .
Remark 1 It is very important to point out that since there is only a scaling invariance up to the pressure term, we can not hope to show the existence of global strong self similar solution for the Korteweg system.Indeed in comparison with the Navier-Stokes system, we know that it exists global in time self similar solution provided that the initial velocity u 0 is homogeneous of degree −1 (such initial data exists for example in the set In particular in the sequel we will able to deal with initial data (ρ 0 , ρ 0 u 0 ) such that ρ 0 is homogeneous of degree 0 and ρ 0 u 0 of degree −1, however the associated strong solution will be a priori not self similar.
The previous transformation (1.6) suggests us however to choose initial data (ρ 0 , u 0 ) in spaces whose norm is invariant for all λ > 0 by the transformation (ρ 0 , u As it was mentioned, this invariance was also used initially by Kato [20] to prove that the Navier-Stokes system is locally well-posed for arbitrary data in L N (R N ) (when N ≥ 2) and globally well-posed for small initial data.Kato's result was extended to larger scale invariant function spaces (one interest of dealing with larger function spaces is that they may contain initial data which are homogeneous of degree −1 and therefore give rise to self-similar solutions).In particular in [5] Cannone, Meyer and Planchon proved the existence of global strong solution with small initial data in B N p −1 p,∞ with p < +∞.A similar analysis was carried out for the vorticity equation in Morrey spaces by Giga and Miyakawa [12].Finally this approach has been generalized by Koch and Tataru in [22] when the initial data is small in BM O −1 (R N ).These results allow in particular to obtain the existence of global self similar solution for small initial data when N ≥ 2 for the incompressible Navier-Stokes equations.In dimension N = 2, it proves the existence and the uniqueness of solution for initial data u 0 satisfying curlu 0 = αδ 0 (which is equivalent to u 0 = α x ⊥ |x| 2 ) with α small enough.These solutions are the so-called Lamb-Oseen solutions which are self-similar.Let us emphasize in addition that we have even an explicit formula for these solutions even when |α| is large, the Lamb-Oseen vortex are given by: where: with ξ ⊥ = (−ξ 2 , ξ 1 ).In passing, we mention that the existence of global weak solution with initial vorticity in the set M(R 2 ) of all finite real measures on R 2 was first proved by Cottet [6] and independently by Giga, Miyakawa and Osada [12].In [12], the authors proved also the uniqueness when the atomic part of the initial vorticity is sufficiently small.The uniqueness for any curlu 0 ∈ M(R 2 ) is proved in [10], it allows in particular to obtain the existence and the uniqueness of global self similar solution for large initial data when N = 2.
In this paper we are interested in extending the technics of Koch-Tataru to the case of the Korteweg system (1.5).More precisely, we wish to prove the existence of strong solution in finite time for (1.1) provided that ρ 0 v 1 (0, •) and ρ 0 v 2 (0, •) are sufficiently small in norm for T > 0. Furthermore we will assume that the initial density is far away from the vacuum and is bounded in L ∞ norm .We recall that bmo −1 (R N ) is the set of temperated distribution u 0 for which for all T > 0 we have: We define the norm • bmo −1 T (R N ) by: , with e t∆ u 0 be the solution to the heat equation with initial data u 0 : ).
In the sequel we will denote by E T the space of temperated distribution associated to the following norm: In the sequel, we will show that ρv 1 and ρv 2 verifying (1.5) belong to E T for T > 0 the time of existence and that (ρ, . We recall that ρv 1 = ρu + c 1 ∇ρ and ) and we deduce from the definition of E T that it exists C > 0 independent on t such that for all 0 < t ≤ T we have: (1.9) Combining (1.9) and the fact that ρ remains in L ∞ T (L ∞ (R N )) it implies that instantaneously the density ρ is regularized and becomes necessary Lipschitz even if the initial density ρ 0 admits shocks.The second point is that with our choice on the initial data, it implies that ρ 0 u 0 is in bmo −1  T .In other way we can work with Dirac mass αδ 0 for the initial vorticity associated to the momentum ρ 0 u 0 in dimension N = 2 provided that α is sufficiently small, it consists to prescribe the initial momentum as follows ρ 0 u 0 (x) = α x ⊥ |x| 2 with x ∈ R2 .We can then extend the notion of Lam-Oseen tourbillon to the case of the Korteweg system at least when α is sufficiently small.It is obvious that in this compressible framework the divergence of such Lamb-Oseen tourbillon does not remain null all along the time, this is due to the coupling between vorticity and divergence of the velocity.Similarly we can deal with vorticity vortex and divergence vortex if we take the following initial momentum ρ 0 u 0 (x) = α x ⊥ |x| 2 + α 1 x ⊥ |x| 2 with x ∈ R 2 and |α|, |α 1 | sufficiently small.Finally we will extend the previous result in dimension N ≥ 2 by proving the existence of global strong solution provided that the initial data (ρ 0 − 1, ρ 0 u 0 ) are small enough in ( It enables us again to obtain global strong Oseen solutions for the Korteweg system provided that the initial data is sufficiently small.

Mathematical results
In this section we state our main result.
Then there exists T > 0 sufficiently small such that there exists a unique solution (ρ, ρu) of the system (1.1) on [0, T ] provided that for ε 1 > 0 small enough we have : In addition it exists C > 0 such that: It is also interesting to observe that there is no smallness assumption on ρ 0 when κ 2 = µ 2 since in this case We deduce also that in this case ρ 0 u 0 is not necessary small in bmo −1 but the smallness carries on the momentum of the effective velocity ρ 0 v 1 (0, •) which describes the coupling between the velocity u 0 and the density ρ 0 .
Remark 3 It is important to mention that the initial data are defined by the momentum ρ 0 v 1 (0, •) and ρ 0 v 2 (0, •), indeed the initial velocity u 0 is a priori not defined (however the momentum m 0 which is equal roughly speaking to ρ 0 u 0 is well defined).
then there exists T > 0 such that there exists a unique solution (ρ, ρu) of the system (1.1).We have in addition the estimate (1.11).
We prove now a result of global strong solution with small initial data.
Theorem 1.2 Let 0 < κ 2 ≤ µ 2 , ρ > 0, N ≥ 2 and P ′ (ρ) > 0. We assume that There exists ε 0 such that if: then there exists a global strong solution (ρ, u) of the system (1.1).In addition it exists Remark 4 As previously, this theorem allows to prove the existence of global strong Oseen solution provided that curl(ρ 0 u 0 ) = αδ 0 with |α| small enough in dimension N = 2.When κ 2 < µ 2 it would be possible to extend this result by working with Besov spaces constructed on L p Lebesgue spaces in high frequencies as in [14] for compressible Navier-Stokes equations.

Proof of the Theorem 1.1 and the Corollary 1
We are going now to prove the Theorem 1.1 in the case 0 < κ 2 < µ 2 .The case κ 2 = µ 2 is similar except that v = v 1 = v 2 and we apply the estimates in bmo −1 T on ρv.From (1.5), we observe that we have for t > 0: We are going now to work in the following space: for T small enough.We observe in particular that for any ρ ∈ L ∞ T (L ∞ (R N )) we have: We shall use in the sequel a contracting mapping argument with the function ψ defined as follows: (2.14)More precisely we want to prove that ψ is a contractive map from E R,M,T to E R,M,T with R, M > 0: E R,M,T is endowed with the following norm: with β > 0 sufficiently large that we will defined later and E R,M,T is a Banach space.Let c > 0, we know that it exists C > 0 (depending on c) such that for all 0 < t ≤ T we have (see [23] p163): (2.17) Similarly we have (see [22]) for C, C 1 > 0: |P (ρ)(s, y)|dsdy (2.18) We deduce from (2.13), (2.17), (2.18) and the definition of the bmo −1 T that it exists C > 0 such that for any T > 0 we have: (2.19) Let us estimate now the L ∞ norm of (ψ 1 (ρ, ρv 1 , ρv 2 ), 1 ψ 1 (ρ,ρv 1 ,ρv 2 ) ), we have then for any t ∈ [0, T ] and using the maximum principle: (2.20) We have now using integration by parts and for 0 < t ≤ T it exists C, C 1 , C 2 , C 3 > 0 such that: (2.21) We deduce now from (2.20) and (2.21) that for C > 0 large enough we have for t ∈ [0, T ]: Let us prove now that for M , T , R well chosen, we have: (2.23) ) and where min x∈R N ρ 0 (x) ≥ c > 0, we deduce from (2.22) that for any t ∈ [0, T ] and for any (ρ, Now we deduce from (2.19) that we have for any (ρ, ρv 1 , ρv 2 ) ∈ E R,M,T and C 1 > 0 large enough: ) with and M 1 = sup x∈[0,2 ρ 0 L ∞ ] P (x).We set now: 2 and T such that C 1 M 1 T ≤ R 4 we deduce from (2.24) and (2.25) that we have for any (ρ, ρv 1 , ρv 2 ) ∈ E R,M,T : In conclusion we have chosen R and T such that: Let us prove now that ψ is a contractive map.More precisely let us estimate We have then: Now we have: (2.29) From (2.17) and (2.29), we deduce that for C, C 1 > 0 large enough and for any (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ), (ρ, ρv 1 , ρv 2 ) in E R,M,T we have for C, C 1 > 0 large enough: We proceed similarly for the part Similarly from (2.18), we have for c = c 1 , c 2 and C > 0 large enough: From (2.28), (2.29), (2.30) and (2.31) we deduce that it exists C > 0 sufficiently large such that: From (2.21), it yields that for C > 0 large enough: Combining (2.32) and (2.33), we deduce that we have for C > 0 large enough: It suffices now to choose β, T and R such that: It proves in particular that the map ψ is contractive and it concludes the proof of the theorem 1.2.
Let us prove now the Corollary 1, if then we can observe that: lim Indeed assume that w 0 ∈ (D(R N )) N then we have using the maximum principle for the heat equation and C > 0 large enough: By density we can conclude.In particular it implies that for T > 0 small enough we have: (2.37) Using the proof of the Theorem 1.1, we conclude that there exists a strong solution (ρ, ρu) on a finite time interval [0, T ].Indeed it we follow the previous proof, it suffices to fix R and T sufficiently small and verifying the previous estimate of the proof of the Theorem 1.1.Indeed by density it exist w 1 ∈ (D(R N )) N such that: Now we choose T 1 ≤ T sufficiently small such that e t∆ w 1 E T 1 ≤ R 2 , we get: with R, T 1 satisfying the estimates of the proof of the Theorem 1.1.
3 Proof of the Theorem 1.2 In order to prove the Theorem 1.2, we are going to start by studying the linear system associated to (1.5): with c > 0, µ > 0, µ + α > 0 and β > 0. (F, G) are external forces.In the sequel we will note W c,µ,α,β the semigroup associated to the previous system and we have in particular from the Duhamel formula: This system has been studied by Bahouri et al (see [3]) in the framework of the Besov space B s 2,1 when c = 0. We are going now to extend this study to the case of general Besov space of the form B s 2,r .We set now d = Λ −1 divm and Ω = Λ −1 curlm, with Λ s 1 f (ξ) = |ξ| s 1 f (ξ) when s 1 ∈ R and for f a temperated distribution.We now study the following system: (3.40) with ν = (µ + α).We refer to [3] for the definition of the Chemin-Lerner spaces L ρ T (B s p,r ) with (ρ, p, r) ∈ [1, +∞] 3 , T > 0, s ∈ R and to [14] for the definition of the hybrid Besov spaces B s 1 ,s 2 p 1 ,p 2 ,r with (s 1 , s 2 ) ∈ R 2 , (p 1 , p 2 , r) ∈ [1, +∞] 3 .
Proposition 3.1 Let T ∈ (0, +∞].We assume that (q 0 , u 0 ) belongs to N be a solution of the system (3.38), then there exists a universal constant C such that for any T > 0 we have:

.41)
Proof: Let (q, m) be a solution of (3.38), we are going to separate the case of the low and high frequencies, which have a different behavior concerning the control of the derivative index for the Besov spaces.Our goal consists now in studying the system (3.40) and in particular to estimate (q, d) and Ω.We observe that Ω verifies simply an heat equation and classical estimates on the heat equation in Besov spaces give (see [3]): Let us study now the unknowns (q, d).

Case of low frequencies
We assume here that l ∈ Z with l ≤ l 0 (we will determine later l 0 ∈ Z).Applying operator ∆ l (see [3] for the definition of ∆ l ) to the system (3.40) and denoting g l = ∆ l g, we obtain the following system: We set: Taking the L 2 scalar product of the first equation of (3.43) with q l and of the second equation with d l , we get the following two identities: (3.45) We deduce that: From the definition of f 2 l we deduce that it exists α 1 , C > 0 independent on l 0 such that: We have in particular for l ≤ l 0 : 1 2

Case of high frequencies
We consider now the case where l ≥ l 0 + 1 and we define now f l as follows: with A > 0 to be determinated.We apply the operator Λ∆ l to the first equation of (3.40), multiply by Λq l and integrate over R N , so we obtain: Moreover we have in a similar way: (3.50)By linear combination of (3.49)-(3.50)we have: (3.51) We have now in using Young inequalities for all a > 0: (3.52) We have now since l ≥ l 0 + 1 for c 1 > 0: 1 2 (3.53)We choose now a, A and l 0 as follows: with M > 1 sufficiently large to determine later.Now from the definition of f 2 l and using Young inequality, we have: . Using (3.53) and (3.55) we deduce that there exists constants α 2 > 0 and C 1 > 0 such that for l ≥ l 0 + 1 we have: 1 2

Final estimates
Integrating over [0, t] the estimates (3.48) and (3.57) and multiplying by 2 l(s−1) , we have for C 2 > 0 large enough and any l ∈ Z: From the definition of f l we deduce the estimate (3.41) using in particular (3.55).It concludes the proof of the proposition.
Let us study again the system (3.38) and applying div to the momentum equation, we have: We assume now that µ, c, ν, β > 0 and ν = c.The only case where ν = c will be in the sequel the case κ 2 = µ 2 .We will study this case later.When we apply the Fourier transform F, we have then: . The characteristic polynom is: The eigenvalues are:

Low Frequencies
When |ξ| 2 < 4β (ν−c) 2 we have: we have: Finally we obtain: (3.65)We get then: (3.66) In the two next propositions, in order to simplify the notations we will denote by W (t) the semigroup W c,µ,α,β (t) with t > 0.
Proposition 3.2 Let φ be a smooth function supported in the shell There exist two positive constants κ and C depending only on φ and such that for all t ≥ 0 and λ > 0, we have: Proof: From (3.63), we deduce that for |ξ| Fg(ξ) for g a temperated distribution): We have used the fact that Pm satisfies simply an heat equation.Similarly from (3.66), we have for |ξ| 2 < 4β (ν−c) 2 : (3.69) Applying Plancherel Theorem, we observe easily that it exists ε > 0 small enough and C ε , κ ε > 0 such that: Indeed we use the fact that when ξ ∈ R N /C(0, The only difficulty is the behavior of the solution in the region C(0,  3.72), we deduce that it exists ε > 0 small enough and C ε , κ ε > 0 such that: Let us deal now with the case |ξ| 2 ∈ ( 4β |ν−c| 2 , 4β |ν−c| 2 − ε) whoch corresponds to α 1 (ξ) ∈ (0, √ ε), we have then: When tα 1 (ξ) ≤ 1, it exists C 1 , C 2 > 0 large enough such that: When tα 1 (ξ) ≥ 1, we deduce that for C 3 > 0 large enough:  3.75) we obtain finally from the Plancherel Theorem that there exist C, κ > 0 such that for any t > 0 we have: It concludes the proof of the proposition 3.2.
We are now going to prove time decay estimates in Besov spaces for the semi group W (t).
Proposition 3.3 Let s ∈ R, r ∈ [1, +∞] and s 1 > s, (q 0 , m 0 ) ∈ (B s 2,∞ ) N +1 then it exists C s 1 > 0 such that for all t > 0 we have: Proof: From proposition 3.2, we have for κ, C > 0 and any l ∈ Z: We deduce that for any l ∈ Z, we have:

.79)
We use now the fact that x It concludes the proof of the proposition 3.3.

Proof of the Theorem 1.2
We shall use a contracting mapping argument to prove the Theorem 1.23 and we consider the following map ψ 1 defined as follows with q = ρ − 1: (s) ds.
(3.81) with We define finally ψ 3 as follows: Let us prove now that ψ 3 is a map from X N 2 in itself with s 1 ∈ ( 3 4 , 1): The space X N 2 in which we work is more complicated as in [7], indeed we need decay estimate in time in Besov space on the solution in order to control the L ∞ norm of q.In [7], the control of the L ∞ norm of q is a direct consequence of Besov embedding 2,1 ֒→ L ∞ since the third index of the Besov spaces are 1.From the proposition (3.1), we deduce that for C > 0 large enough: . (3.83) Next using classical paraproduct law and composition theorems (see [3]), we get for C 1 > 0 large enough and a continuous function C using the fact that div(ρv (3.84) Combining (3.83), (3.84), interpolation and composition theorems, we obtain for C 1 > 0 large enough and a continuous function C: ) . (3.85) It remains now to estimate the ψ 3 (q, ρv 1 , ρv 2 ) norm, using the definition of ψ 1 and ψ 2 (see (3.81)) we have for t > 0: q 0 ρv 2 (0, •) ds.
and T > 0 to determine later.We are going now to prove that ψ is a map from X N 2 ,R ∩ E R 1 ,M,T in itself with s 1 ∈ ( 3 4 , 1) and 0 < R, R 1 , T < 1 2 sufficiently small that we will define later.We define X N 2 ,R as follows: In the sequel we will note X + We obtain then the stability for the norm X − N 2 using (3.102) and (3.103).In a similar way, we can prove that ψ 4 is contractive.More precisely taking (q, ρv 1 , ρv 2 ), (q 1 , ρ 1 w 1 , ρ 1 w 2 ) in X N 2 ,R ∩ E R 1 ,M,T we have for t > T : ψ 1 (q, ρv 1 , ρv 2 ) − ψ 1 (q 1 , ρ 1 w 1 , ρ (3.104)As previously we show that for C 1 > 0 large enough we have: (3.105) Taking again R sufficiently small we deduce that the map ψ 4 is contractive (it suffices again to repeat the same process on [0, T ] with T > 0 sufficiently small).We proceed similarly for X − N 2 and E R 1 ,M,T .It concludes the proof of the Theorem 1.2 in the case 0 < κ 2 < µ 2 .The proof in the case κ 2 = µ 2 is similar except that v 1 = v 2 , in addition when we study the system (3.59)there is no distinction between high and low frequencies.

N 2 ,N 2 − 1 2 2 ≤ • X − N 2
T and X − N 2 ,T to define the subsets of X N 2 where the norms are respectively only considered on the time interval [T, +∞[ and [0, T ].It is important to mention that we can work in small time on (0, T) in E R 1 ,M,T since (ρv 1 (0, •), ρv 2 (0, •)) are in BM O −1 (R N ), indeed we know that B ,∞ is embedded in BM O −1 (R N ).We have obviously • X N