Local in time results for local and non-local capillary Navier-Stokes systems with large data
aa r X i v : . [ m a t h . A P ] J un Local in time results for local and non-local capillaryNavier-Stokes systems with large data
Fr´ed´eric Charve ∗ Abstract
In this article we study three capillary compressible models (the classical lo-cal Navier-Stokes-Korteweg system and two non-local models) for large initial data,bounded away from zero, and with a reference pressure state ¯ ρ which is not necessarilystable ( P ′ (¯ ρ ) can be non-positive). We prove that these systems have a unique localin time solution and we study the convergence rate of the solutions of the non-localmodels towards the local Korteweg model. The results are given for constant viscouscoefficients and we explain how to extend them for density dependant coefficients. In this article we are interested in the dynamics of a liquid-vapor mixture in the settingof the Diffuse Interface (DI) approach: between the two phases lies a thin region ofcontinuous transition and the phase changes are read through the variations of the density(with for example a Van der Waals pressure). Unfortunately the basic models providean infinite number of solutions and in order to select the physically relevant solutions,following Van der Waals and Korteweg, one penalizes the high variations of the densitythanks to capillary terms related to surface tension.We first consider the classical compressible Navier-Stokes system (NSK) endowedwith an internal local capillarity (this is why we call (NSK) the local Korteweg system).This system, first considered by Korteweg and renewed by Dunn and Serrin, reads: ( ∂ t ρ + div ( ρu ) = 0 ,∂ t ( ρu ) + div ( ρu ⊗ u ) − A u + ∇ ( P ( ρ )) = κρ ∇ D [ ρ ] , where ρ and u denote the density and the velocity ( ρ is a non-negative function and u isa vector-valued function defined on R + × R d ). The general diffusion operator is definedas follows: A u = div (2 µ ( ρ ) Du ) + ∇ ( λ ( ρ )div u ) , where 2 Du = t ∇ u + ∇ u . For simplicity we will present the results in the case of constantviscosity coefficients (we refer to the end of the article for general coefficients) so that A u = µ ∆ u + ( λ + µ ) ∇ div u, with µ > ν = λ + 2 µ > . ∗ Universit´e Paris-Est Cr´eteil, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees (UMR 8050),61 Avenue du G´en´eral de Gaulle, 94 010 Cr´eteil Cedex (France). E-mail: [email protected]
1n the classical Korteweg system (
N SK ), the capillary term is defined by div K , wherethe general Korteweg tensor is given by: K ( ρ ) = κ ( ρ )2 (∆ ρ − |∇ ρ | ) I d − κ ( ρ ) ∇ ρ ⊗ ∇ ρ, The coefficient κ may depend on ρ but in this article it is chosen constant and then thecapillary term turns into κρ ∇ D [ ρ ] with (see [19]): D [ ρ ] = ∆ ρ, The solutions of this system are much more regular than those of the classical compress-ible system (NSC) (that is for D [ ρ ] = 0). We refer to [29, 19, 16] for more informationsabout this model.Another way of selecting the physical solutions consists in defining a non-local capil-lary term involving the density through a convolution and only one derivative (comparedto the numerical difficulties generated by the previous local capillary term with deriva-tives of order 3). In the non-local Korteweg system ( N SRW ), introducing φ , calledinteraction potential, which satisfies the following conditions( | . | + | . | ) φ ( . ) ∈ L ( R d ), Z R d φ ( x ) dx = 1 , φ even, and φ ≥ , (1.1)then D [ ρ ] is the non-local term given by: D [ ρ ] = φ ∗ ρ − ρ. Computing the Fourier transform of these capillary terms, ( b φ ( ξ ) − b ρ ( ξ ) in the non-local model, and −| ξ | b ρ ( ξ ) in the local model, a natural question arises which is to studythe closedness of the solutions of these models when b φ ( ξ ) is formally ”close” to 1 − | ξ | .To answer this question, we chosed in [5] a particular interaction potential and consideredthe following non-local system:( N SRW ε ) ( ∂ t ρ ε + div ( ρ ε u ε ) = 0 ,∂ t ( ρ ε u ε ) + div ( ρ ε u ⊗ u ε ) − A u ε + ∇ ( P ( ρ ε )) = κρ ε ∇ D [ q ε ] , where φ ε = ε d φ ( xε ) with φ ( x ) = π ) d e − | x | ,D [ q ε ] = ε ( φ ε ∗ ρ ε − ρ ε ) . When ξ is fixed, we have b φ ε ( ξ ) = e − ε | ξ | , and when ε is small, b φ ε ( ξ ) − ε is close to −| ξ | .We refer to [34, 31, 11, 30, 23, 24, 5, 8] for more details. The solutions of the non-localmodel have a regularity structure closer to what is obtained for system (NSC) but thenumerical difficulties seem comparable to (NSK) due to the convolution operator.For this reason, C. Rohde introduced in [32] a new model, called the order-parametermodel, and inspired by the work of D. Brandon, T. Lin and R. C. Rogers in [3]. In thisnew system the capillary term α ∇ ( c − ρ ) involves a new variable c called the ”orderparameter”, which is coupled to the density via the following relation coming from the2uler-Lagrange equation from the variational approach ( α controls the coupling between ρ and c ): ∆ c + α ( ρ − c ) = 0 , so that the new system he considered is the following:( N SOP α ) ∂ t ρ α + div ( ρ α u α ) = 0 ,∂ t ( ρ α u α ) + div ( ρ α u α ⊗ u α ) − A u α + ∇ ( P ( ρ α )) = κα ρ α ∇ ( c α − ρ α ) , ∆ c α + α ( ρ α − c α ) = 0 . As emphasized by C. Rohde, from a numerical point of view this system is much moreinteresting (only one derivative in the capillary tensor which is local), and the additionnalequation for the order parameter is a simple linear elliptic equation that can be easilysolved (and numerically fast). Another important feature of this model is that thanks tothe capillary term, the momentum equation can be rewritten with a modified pressure: e P ( ρ ) = P ( ρ ) + κα ρ /
2, so that if α is large enough the derivative will be positive.Moreover as explained in [9] introducing the following interaction potential: ψ α = α d φ ( α · ) with ψ ( x ) = C d | x | d − K d − ( | x | ) , where K ν denotes the modified Bessel function of the second kind and index ν , then thesystem can be rewritten in the shape of the previous non-local capillary model:( N SOP α ) ( ∂ t ρ α + div ( ρ α u α ) = 0 ,∂ t ( ρ α u α ) + div ( ρ α u ⊗ u α ) − A u α + ∇ ( P ( ρ α )) = ρ α κα ∇ ( ψ α ∗ ρ α − ρ α ) , Remark 1
From the previous computations, we immediately get that c α − ρ α = ( − ∆ + α I d ) − ∆ ρ α = ψ α ∗ ρ α − ρ α , that is c α = ψ α ∗ ρ α . This is why the only choice for the initial order parameter is c = ψ α ∗ ρ . We refer to [3, 32, 9] for more details.We focus here on strong solutions with initial data in critical spaces. Let us recallthat a critical space is a space whose corresponding norm has the same scaling invarianceas the (NSC) system: if ( ρ ( t, x ) , u ( t, x )) is a solution corresponding to the initial data( ρ ( x ) , u ( x )), then for each λ >
0, ( ρ ( λ t, λx ) , λu ( λ t, λx )) is also a solution, corre-sponding to the dilated initial data ( ρ ( λx ) , λu ( λx )), provided that the pressure P hasbeen changed into λ P . For example the Sobolev space ˙ H d ( R d ), or the Besov space ˙ B dp p, are critical. We refer to [12, 1, 14, 13, 4] for more details.A natural first step in the study of system (NSC) is to consider initial data in criticalspaces close to an equilibrium state ( ρ,
0) with P ′ ( ρ ) >
0. Assuming that the initialdensity fluctuation q , defined by ρ = ρ (1 + q ), is small, we perform the classicalchange of function ρ = ρ (1 + q ) and expect that q is also small. For simplicity we take3 = 1, then q = ρ − ρ will be bounded away from zero.Simplifying by ρ , the previous systems become (we chosed to drop the subscripts): ∂ t q + u · ∇ q + (1 + q )div u = 0 ,∂ t u + u · ∇ u −
11 + q A u + P ′ (1 + q )1 + q · ∇ q − κ ∇ D [ q ] = 0 , where D [ q ] = 0 for ( N SC ) ,D [ q ] = ∆ q for ( N SK ) ,D [ q ε ] = φ ε ∗ q ε − q ε ε for ( N SRW ε ) ,D [ q α ] = α ( ψ α ∗ q α − q α ) for ( N SOP α ) . (1.2)If we also assume that the equilibrium (1 ,
0) is stable (that is P ′ (1) >
0) then it is usefulto rewrite the system into: ( ∂ t q + u · ∇ q + (1 + q )div u = 0 ,∂ t u + u · ∇ u − A u + P ′ (1) ∇ q − κ ∇ D [ q ] = K ( q ) ∇ q − I ( q ) A u, where K and I are the following real-valued functions defined on R : K ( q ) = (cid:18) P ′ (1) − P ′ (1 + q )1 + q (cid:19) and I ( q ) = qq + 1 . If the density fluctuation q is small, then K ( q ) and I ( q ) are expected to be small so thatthe contribution of the right-hand side term should also be small.Indeed when q is small in ˙ B d − , ∩ ˙ B d , and P ′ (1) >
0, R. Danchin obtained in[12, 14] global existence of a solution for the compressible Navier-Stokes system (
N SC ).The local capillary model has been treated by R. Danchin and B. Desjardins in [16] andwe refer to [23, 5, 8, 9] for the same results in the non-local capillary case. In additionwe proved that the solutions of these non-local models converge towards the solutionof the local model with the same small initial data q , and provided an explicit rate ofconvergence. In [8, 9] we give refined estimates.When the constant state (1 ,
0) is not assumed to be stable anymore, the best we canobtain are local in time existence results. Such results were first obtained by R. Danchinwhen q is small in ˙ B d , (then again 1 + q is bounded away from zero: no vacuum).We refer to [13, 1] for the ( N SC ) system and to [16] for the local capillary model. Weemphasize that no assumption on the pressure law or on the stability are needed, so thatthe case of a Van Der Waals pressure law is covered. From [5, 9] these existence resultscan be very easily adapted to the non-local capillary models as well as the convergence,when ε is small or α is large for a finite lifespan T .A much more difficult question is to study these systems whithout any stability as-sumptions on the state (1 ,
0) and any smallness condition on the initial density fluctuation q only assumed to belong to ˙ B d , , which is naturally the context when two phases co-exist. For example if ρ = (1 − χ ) ρ + χρ with ρ , two constants, and χ is a smoothcut-off function. 4he only assumption is that ρ = 1 + q is bounded away from zero. As explainedmore in details later, in this unfavorable case we cannot rely on the a priori estimatesused in the previous works because of the terms q div u and I ( q ) A u . When q is smallthese terms are harmless because their Besov-norms are easily absorbed by the left-handside. On the contrary, in the present case q has no reason to be small and none of theprevious terms, even in a small intervall of time, can be handled like previously and bothof them obtruct any use of the previous estimates. The idea introduced by R. Danchinto deal with large density fluctuation is basically to decouple ( q, u ) and study a slightlymodified equation on the velocity, where thanks to a frequency truncation, a big part ofthe previous problematic term can be included in the linear system and the rest can bemade small and absorbable by the left-hand side. Roughly speaking, instead of studying: ∂ t u + v · ∇ u − A u = − P ′ (1 + q )1 + q ∇ q − q q A u, R. Danchin, studied in [18]: ∂ t u + u · ∇ u − ˙ S m ( 11 + q ) A u = − P ′ (1 + q )1 + q ∇ q + (cid:18) ( I d − ˙ S m ) 11 + q (cid:19) A u. Using refined estimates (see [18, 1]) on the density fluctuation equation, we can fix m large enough (only depending on the initial data), so that ( I d − ˙ S m ) q is small and thelast term can be absorbed through estimates for the following equation: ∂ t u + v · ∇ u − b A u = F, (1.3)where b is a regular function, bounded away from zero, the first part of the right-handside being small in a small interval of time.To the best of our knowledge there are very few similar results in the capillary caseor for density dependant coefficients (viscosity, capillarity). For special choices on theviscosity and capillary coefficients, the previous method can be simplified as there is noneed for new a priori estimates. For instance we refer to [7] in the case of the shallowwater model, that is when µ ( ρ ) = ρ , λ ( ρ ) = 0, the system turns into: ( ∂ t q + u · ∇ q + (1 + q )div u = 0 ,∂ t u + u · ∇ u − A u − D ( u ) . ∇ (ln(1 + q )) + ∇ ( H (1 + q )) = 0 . And the problematic term can be decomposed into: ∇ ln(1 + q ) · D ( u ) = I + II , with I = ∇ (cid:16) ln(1 + q ) − ln(1 + ˙ S m q ) (cid:17) · D ( u ) , II = ∇ ln(1 + ˙ S m q ) · D ( u ) . As before, the first one is small for large fixed m (we refer to [18, 1, 7]), and the secondone will be proved to be small thanks to the smoothness of ˙ S m q : for 0 < α < k II k ˙ B d − , ≤ k∇ ln(1 + ˙ S m q ) k ˙ B d α − , k D ( u ) k ˙ B d − α , ≤ C ( k q k L ∞ )2 mα k q k ˙ B d , k u k ˙ B d − α , , (1.4)5hich is partly absorbed by the left-hand side when t ∈ [0 , T ] with T small enough, theother part being dealt thanks to the Gronwall lemma. We emphasize that this couldnot have been performed with q instead of ˙ S m q . Let us precise that in [7] there is animprovement on the assumptions on the initial data velocity which is taken in a biggerspace: u ∈ ˙ B d − , ∩ ˙ B − ∞ , , div u ∈ ˙ B d − , .We also refer to [25, 26] where B. Haspot obtains local results for large data for thelocal Korteweg model with special choices of the pressure and the viscosity and capillaritycoefficients: κ ( ρ ) = ρ , µ ( ρ ) = ρ , λ ( ρ ) = ρ or 0 and P ( ρ ) = ρ . We emphasize that anotherimportant feature of [25, 26] is that the initial data are taken in Besov spaces with thirdindex 2 or infinite. In these works it is more convenient to introduce q = log ρ and theeffective velocity v = u + ∇ ln ρ which has important regularity properties. We refer to[21, 6] for other use of the notion of effective velocity. The present work is devoted to the study of the capillary models with large data in thecase of constant viscosity and capillarity coefficients without any stability assumptionon the state (1 , ε or α in ( N SRW ε ) or ( N SOP α ), we cannotafford to treat separatedly q and u : for example in the non-local case, the capillary termis multiplied by a large coefficient ε − that would lead to a lifespan of size O ( ε ) whichis obviously useless for the study of the convergence towards the solution of ( N SK ). Asexplained before, as q is not assumed anymore to be small, using the previous a prioriestimates on the couple ( q, u ) leads to an obstruction.The main ingredient in this paper is a new priori estimate in the spirit of [18] wherewe recouple q and u . Obviously as we need to adapt the idea developped in [18] wherethe key is to study equation (1.3) we cannot hope to use refined estimates obtained fromFourier analysis of the linear system as in [4, 8, 9]. As we recouple q and u , in additionto the term (1 + q ) − A u we will have to deal with q div u in the first equation. When q issmall this term is harmless and easily absorbed by the left-hand side. On the contrary,in our general case this term prevents any convenient estimates that would be useful forthe proof of uniqueness or convergence. It has then to be also included into the linearsystem we will study. This leads us to the following linear system: ( ∂ t q + v · ∇ q + c · div u = 0 ,∂ t u + v · ∇ u − b · A u − κ ∇ D [ q ] = 0 , where b, c are positive real valued functions, bounded away from zero. We refer to the appendix for definitions and properties of the classical and hybrid Besovspaces.
Definition 1
The space E s ( t ) is the set of functions ( q, u ) in (cid:16) C b ([0 , t ] , ˙ B s , ) ∩ L t ˙ B s +22 , (cid:17) × (cid:16) C b ([0 , t ] , ˙ B s − , ) ∩ L t ˙ B s +12 , (cid:17) d ndowed with the norm k ( q, u ) k E s ( t ) def = k u k e L ∞ t ˙ B s − , + k q k e L ∞ t ˙ B s , + k u k L t ˙ B s +12 , + k q k L t ˙ B s +22 , . (2.5) Definition 2
The space E sβ ( t ) (for β > expected to be large) is the set of functions ( q, u ) in (cid:16) C b ([0 , t ] , ˙ B s , ) ∩ L t ˙ B s +2 ,sβ (cid:17) × (cid:16) C b ([0 , t ] , ˙ B s − , ) ∩ L t ˙ B s +12 , (cid:17) d endowed with the norm k ( q, u ) k E sβ ( t ) def = k u k e L ∞ t ˙ B s − , + k q k e L ∞ t ˙ B s , + k u k L t ˙ B s +12 , + k q k L t ˙ B s +2 ,sβ . (2.6) Remark 2
We observe that the parabolic regularization on q occurs for all frequenciesin E s ( t ) and only for low frequencies in E sβ ( t ) . Moreover the threshold between theregularized low frequencies and the damped high frequencies goes to infinity as β goes toinfinity. We refer to [5, 8, 9] for more details about this threshold and the close relationwith the capillary term. Theorem 1
Let ε > , q ∈ ˙ B d , , u ∈ ˙ B d − , and assume that < c ≤ q ≤ c and min( µ, µ + λ ) > . There exist a positive constant C and a time T > , only dependingon the physical parameters d , µ , λ , κ and the initial data ( q , u ) , such that system ( N SK ) has a unique solution ( ρ, u ) with ( q, u ) ∈ E d ( T ) , and system ( N SRW ε ) has aunique solution ( ρ ε , u ε ) with ( q ε , u ε ) ∈ E d /ε ( T ) . Moreover k ( q, u ) k E d ( T ) + k ( q ε , u ε ) k E d /ε ( T ) ≤ C ( k q k ˙ B d , + k u k ˙ B d − , ) . As in [5], we prove that the solution of ( RW ε ) goes to the solution of ( K ) when ε goesto zero. Theorem 2
Assume that min( µ, µ + λ ) > , q ∈ ˙ B d , , u ∈ ˙ B d − , . Then for T givenby the previous result, k ( q ε − q, u ε − u ) k E d /ε ( T ) −→ ε → . Moreover there exists a constant C = C ( η, κ, q , u , T ) > such that for all h ∈ ]0 , (if d = 2 ) or h ∈ ]0 , (if d ≥ ), k ( q ε − q, u ε − u ) k E d − h /ε ( T ) ≤ Cε h , .3 Order parameter model The very same results are true for the order parameter model:
Definition 3
The space F sβ ( t ) (for β > ) is the set of functions ( q, c, u ) in (cid:16) C b ([0 , t ] , ˙ B s , ) ∩ L t ˙ B s +2 ,sβ (cid:17) × (cid:16) C b ([0 , t ] , ˙ B s − , ) ∩ L t ˙ B s +12 , (cid:17) d endowed with the norm k ( q, c, u ) k F sβ ( t ) def = k u k e L ∞ t ˙ B s − , + k q k e L ∞ t ˙ B s , + k c k e L ∞ t ˙ B s , + k u k L t ˙ B s +12 , + k q k L t ˙ B s +2 ,sβ + k c k L t ˙ B s +2 ,sβ . (2.7) Theorem 3
Let α > , q ∈ ˙ B d , , u ∈ ˙ B d − , and assume < c ≤ q ≤ c and min( µ, µ + λ ) > . Let c be defined by − ∆ c + α c = α ρ , that is c = ψ α ∗ ρ . Thereexist a positive constant C and a time T > only depending on the physical parametersand ( q , u ) such that system ( N SK ) has a unique solution ( ρ, u ) with ( q, u ) ∈ E d ( T ) ,and system ( N SOP α ) has a unique solution ( ρ α , c α , u α ) with ( q α , c α , u α ) ∈ F d α ( T ) and c α = ψ α ∗ q α . Moreover: k ( q, u ) k E d ( T ) + k ( q α , c α , u α ) k F d α ( T ) ≤ C ( k q k ˙ B d , + k u k ˙ B d − , ) , and k c α − ρ α k e L ∞ T ˙ B d , −→ α →∞ , and k c α − ρ α k L T ˙ B d , ≤ Cα − . Theorem 4
Assume that min( µ, µ + λ ) > , q ∈ ˙ B d , , u ∈ ˙ B d − , . Then for T givenby the previous result, k ( q α − q, c α − ρ, u α − u ) k F d α ( T ) −→ α →∞ . Moreover there exists a constant C = C ( η, κ, q , T ) > such that for all h ∈ ]0 , (if d = 2 ) or h ∈ ]0 , (if d ≥ ), and for all t ∈ [0 , T ] , k ( q α − q, c α − ρ, u α − u ) k F d − hα ( T ) ≤ Cα − h , We refer to the end of the article for the variable coefficients case, and the case ofBesov spaces ˙ B sp, with p = 2. We will prove theorems 1 and 2. As we only use energy methods, the proofs are strictly thesame for the order parameter model, because in the L -setting, the interaction potentials ψ α and φ ε play exactly the same role. Let us emphasize that this was not the case in[8, 9] as it involved Fourier computations and finite differences representations of Besovnorms. 8 .1 Linear estimates with variable coefficients As announced, these results rely on a priori estimates for solutions of the following system: ( ∂ t q + v · ∇ q + c · div u = F,∂ t u + v · ∇ u − b · A u − κ ∇ D [ q ] = G. (3.8)With A u = µ ∆ u + ( λ + µ ) ∇ div u, and we recall that D [ q ] is given in (1.2). Theorem 5
Let s ∈ ] − d , d + 1] , ν = 2 µ + λ and assume that ν = min( µ, ν ) > , q ∈ ˙ B s , , u ∈ ˙ B s − , . Let T > , and assume that ( q, u ) solves system (3.8) on [0 , T ] ,with v ∈ L ∞ T ˙ B d − , ∩ L T ˙ B d +12 , and b, c : [0 , T ] × R d → R + such that: • b − , c − ∈ C ([0 , T ] , ˙ B d , ) , • ∂ t b, ∂ t c ∈ L T ˙ B d − , , • for all t ≤ T, x ∈ R d , we have < c ∗ ≤ c ( t, x ) ≤ c ∗ and < b ∗ ≤ b ( t, x ) ≤ b ∗ . Then ( q, u ) ∈ E s ( T ) (respectively ( q ε , u ε ) ∈ E s /ε ( T ) , ( q α , u α ) ∈ E sα ( T ) ). Moreover if wedefine g s ( q, u )( t ) = X j ∈ Z j ( s − sup t ′ ∈ [0 ,t ] (cid:0) k u j ( t ′ ) k L + h j ( t ′ ) (cid:1) , with h j ( t ′ ) = ( u j ( t ′ ) | c m u j ( t ′ )) L + κ ( q j ( t ′ ) | D [ q j ( t ′ )]) L + η (cid:18) u j ( t ′ ) |∇ q j ( t ′ )) L + ν ( ∇ q j ( t ′ ) | b m c m ∇ q j ( t ′ )) L (cid:19) , (3.9) where u j = ˙∆ j u (the same for q ) and b m , c m are smooth functions defined by: b m = 1 + ˙ S m ( b − and c m = 1 + ˙ S m ( c − . (3.10) Then there exist m ∈ Z , two constants γ ∗ > and F ∗ ≥ such that if η > is fixedsmall enough (all of them only depending on the bounds b ∗ , c ∗ , b ∗ , c ∗ and the viscous andcapillary coefficients) then for all t ≤ T and m ≥ m , F ∗− g s ( t ) ≤ k u k e L ∞ t ˙ B s − , + k q k e L ∞ t ˙ B s , ≤ F ∗ g s ( t ) , (3.11)9 nd g s ( q, u )( t )+ ν b ∗ k u k L t ˙ B s +12 , + γ ∗ k D [ q ] k L t ˙ B s , ≤ g s ( q , u )(0)+ F ∗ Z t (cid:16) k F k ˙ B s , + k G k ˙ B s − , (cid:17) dτ + F ∗ Z t g s ( q, u )( τ ) " m (cid:16) k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , (cid:17) + (1 + k b − k ˙ B d , + k c − k ˙ B d , ) (cid:16) k∇ v k ˙ B d , + 2 m k v k ˙ B d , + 2 m + k v k B d , (cid:17) dτ + F ∗ Z t k ( I d − ˙ S m )( b − k ˙ B d , + k ( I d − ˙ S m )( c − k ˙ B d , ! k u k ˙ B s +12 , dτ. (3.12) Remark 3
The condition s ∈ ] − d , d + 1] is required by paraproduct and remainderlaws, we refer to (3.45) for details. Remark 4
Let us emphasize that we have: k D [ q ] k L T ˙ B s , ∼ k q k L T ˙ B s +22 , in the local case ( N SK ) , k q k L T ˙ B s +2 ,s /ε in the non-local case ( N SRW ε ) , k q k L T ˙ B s +2 ,sα in the non-local case ( N SOP α ) . We will prove the result in the first non-local case, that is for D [ q ] = φ ε ∗ q − qε . As saidbefore, for the order parameter model everything works the same, and for the local casethe same argument is valid but many steps are much easier thanks to the fact that q is more regular. We will highlight in the following proof what can be simplified for thelocal case ( N SK ).From the definition of b m we have b m − S m ( c − b m − B d , -norm is bounded by the one of b −
1. Moreover b − b m = ( I d − ˙ S m )( b − m large enough so that for all m ≥ m : k b − b m k ˙ B d , ≤ b ∗ k c − c m k ˙ B d , ≤ c ∗ . In this case, thanks to the injection ˙ B d , ֒ → L ∞ , we immediately have for all t ≤ T, x ∈ R d , that0 < c ∗ ≤ c m ( t, x ) ≤ c ∗ + c ∗ < b ∗ ≤ b m ( t, x ) ≤ b ∗ + b ∗ . (3.13)Next, in the spirit of [18], let us first rewrite system (3.8) as follows: ∂ t q + v · ∇ q + c m div u = F + F m ,∂ t u + v · ∇ u − b m A u − κ φ ε ∗ ∇ q − ∇ qε = G + G m , F m = ( c m − c ) · div u = − (cid:16) I d − ˙ S m (cid:17) ( c − · div u,G m = ( b − b m ) · A u = (cid:16) I d − ˙ S m (cid:17) ( b − · A u. Then applying operator ˙∆ j to the system, and using the notation f j = ˙∆ j f , we obtain(following the lines of [18]): ∂ t q j + v · ∇ q j + div ( c m u j ) = f j ,∂ t u j + v · ∇ u j − µ div ( b m . ∇ u j ) − ( λ + µ ) ∇ ( b m div u j ) − κ φ ε ∗ ∇ q j − ∇ q j ε = g j , (3.14)where f j = F j + F m,j + R j + e R j and g j = G j + G m,j + S j + e S j ,R j = v · q j − ˙∆ j ( v · q ) and S j = v · u j − ˙∆ j ( v · u ) , e R j = div ( c m u j ) − ˙∆ j ( c m div u ) = div (cid:16) ( c m − u j (cid:17) − ˙∆ j (cid:16) ( c m − u (cid:17) . e S j = µ (cid:16) ˙∆ j ( b m ∆ u ) − div ( b m ∇ ˙∆ j u ) (cid:17) + ( λ + µ ) (cid:16) ˙∆ j ( b m ∇ div u ) − ∇ ( b m div ˙∆ j u ) (cid:17) . = µ (cid:16) ˙∆ j (( b m − u ) − div (( b m − ∇ ˙∆ j u ) (cid:17) +( λ + µ ) (cid:16) ˙∆ j (( b m − ∇ div u ) − ∇ (( b m − j u ) (cid:17) . (3.15)Let us begin by stating estimates on these external terms (we refer to lemma 3 in theappendix and [18, 1] for details and proofs): Proposition 1
Under the previous assumptions, there exist a positive constant C anda nonnegative summable sequence ( c j ) j ∈ Z = (cid:0) c j ( t ) (cid:1) j ∈ Z whose summation is such thatif we denote by ν = µ + | λ + µ | , then for all j ∈ Z we have: k R j k L ≤ Cc j − js k∇ v k ˙ B d , k q k ˙ B s , , k S j k L ≤ Cc j − j ( s − k∇ v k ˙ B d , k u k ˙ B s − , , k e R j k L ≤ Cc j − js m k c − k ˙ B d , k u k ˙ B s , , k e S j k L ≤ Cνc j − j ( s − m k b − k ˙ B d , k u k ˙ B s , , k F m k ˙ B s , ≤ C k (cid:16) I d − ˙ S m (cid:17) ( c − k ˙ B d , k u k ˙ B s +12 , , k G m k ˙ B s − , ≤ C k (cid:16) I d − ˙ S m (cid:17) ( b − k ˙ B d , k u k ˙ B s +12 , . (3.16)In this proof of theorem 5, the ideas are classical (we refer to [12, 18, 13, 1, 23, 24, 5])and consist of combining innerproducts in L of the equations in order to cancel termsthat we are not able to estimate (too much derivatives or large coefficients). Remark 5
Due to the initial regularity, the most natural way is to study the evolutionof k u j k L and k∇ q j k L , for this we consider the inner product of the gradient of thefirst equation by ∇ q j , and the second by u j . This computation involves the following roblematic term κε − ( φ ε ∗ ∇ q j − ∇ q j | u j ) L that unfortunately, at this level of the study,we are not able to estimate uniformly with respect to ε . We need a way to cancel itand the easiest way to do this is, as in [5], to consider the innerproduct of the densityequation by ε − ( q j − φ ε ∗ q j ) and the velocity equation by c m u j . Then the problematicterm will be neutralized if we sum the results. Keeping in mind the fact that ddt ( u j | c m u j ) L = 2( ∂ t u j | c m u j ) L + ( ∂ t c m .u j | u j ) L , we beginby taking the inner product of the velocity equation with c m u j :( ∂ t u j | c m u j ) L +( v ·∇ u j | c m u j ) L + µ ( b m . ∇ u j |∇ ( c m u j )) L +( λ + µ )( b m div u j | div ( c m u j )) L − κ ( φ ε ∗ ∇ q j − ∇ q j ε | c m u j ) L = ( g j | c m u j ) L . (3.17)We have: ( b m . ∇ u j |∇ ( c m u j )) L = Z R d b m c m . |∇ u j | dx + ( b m . ∇ u j | u j . ∇ c m ) L , ( b m div u j | div ( c m u j )) L = Z R d b m c m . | div u j | dx + ( b m div u j | u j . ∇ c m ) L , with (cid:12)(cid:12)(cid:12) µ ( b m . ∇ u j | u j . ∇ c m ) L + ( λ + µ )( b m div u j | u j . ∇ c m ) L (cid:12)(cid:12)(cid:12) ≤ ν k b m k L ∞ k∇ c m k L ∞ j k u j k L . We estimate the following term like in [12, 5] using integrations by parts: (cid:12)(cid:12)(cid:12) ( v · ∇ u j | c m u j ) L (cid:12)(cid:12)(cid:12) ≤ k div ( c m v ) k L ∞ k u j k L ≤ (cid:16) k c m k L ∞ . k∇ v k L ∞ + k v k L ∞ . k∇ c m k L ∞ (cid:17) k u j k L . (3.18)Thanks to the frequency truncation in the definition of c m and the fact that ∂ t c m = ∂ t ( c m −
1) = ∂ t (cid:16) ˙ S m ( c − (cid:17) = ˙ S m ( ∂ t ( c − S m ( ∂ t c ) , the additional term is estimated by:( ∂ t c m .u j | u j ) L ≤ k ∂ t c m k L ∞ k u j k L ≤ k ∂ t c m k ˙ B d , k u j k L ≤ m k ∂ t c k ˙ B d − , k u j k L . Gathering these estimates we obtain:12 ddt ( u j | c m u j ) L + µ Z R d b m c m . |∇ u j | dx + ( λ + µ ) Z R d b m c m . | div u j | dx − κ ( φ ε ∗ ∇ q j − ∇ q j ε | c m u j ) L ≤ k g j k L k c m k L ∞ k u j k L + 2 m k ∂ t c k ˙ B d − , k u j k L + ν k b m k L ∞ k∇ c m k L ∞ j k u j k L + (cid:16) k c m k L ∞ . k∇ v k L ∞ + k v k L ∞ . k∇ c m k L ∞ (cid:17) k u j k L . (3.19)Thanks to the bounds on b m and c m (see (3.13)), we prove similarly to [12, 5] that (wether λ + µ is negative or not) that: µ Z R d b m c m . |∇ u j | dx + ( λ + µ ) Z R d b m c m . | div u j | dx ≥ ν b ∗ c ∗ j k u j k L , ( ν = min( µ, ν + λ ) ,ν = µ + | µ + λ | . Moreover, ν k b m k L ∞ k∇ c m k L ∞ j k u j k L ≤ ν b ∗ c ∗ j k u j k L + 2 ν ν b ∗ c ∗ k b m k L ∞ k∇ c m k L ∞ k u j k L . Using the bound from (3.13), and the fact that k∇ c m k L ∞ ≤ k∇ ( c m − k ˙ B d , ≤ k∇ ˙ S m ( c − k ˙ B d , ≤ m k c − k ˙ B d , , we end up with:12 ddt ( u j | c m u j ) L + ν b ∗ c ∗ j k u j k L − κ ( φ ε ∗ ∇ q j − ∇ q j ε | c m u j ) L ≤ C ∗ k g j k L k u j k L + C k u j k L " m k ∂ t c k ˙ B d − , + C ∗ (cid:16) k∇ v k L ∞ + 2 m k c − k ˙ B d , k v k L ∞ (cid:17) + C ∗ ν ν m k c − k B d , . (3.20)where C ∗ ≥ b and c Remark 6
In the following we adopt the convention that even if varying from line to linewe will always denote by C ∗ such a constant, and by D ∗ , F ∗ a constant that in additiondepends on the physical parameters ( λ , µ , κ ). Let us now turn to the density fluctuation: as explained, in order to neutralize the capil-lary term, instead of studying k∇ q j k , we study the quantity ( q j | q j − φ ε ∗ q j ε ) L . computingthe inner product of the density equation by q j − φ ε ∗ q j ε , we obtain that:12 ddt ( q j | q j − φ ε ∗ q j ε ) L + ( v · ∇ q j | q j − φ ε ∗ q j ε ) L + (div ( c m u j ) | q j − φ ε ∗ q j ε ) L = ( f j | q j − φ ε ∗ q j ε ) L . (3.21)We refer to [8, 9] and the appendix for the fact that: ( q j | q j − φ ε ∗ q j ε ) L ∼ min( 1 ε , j ) k q j k L , k q j − φ ε ∗ q j ε k L ∼ min( 1 ε , j ) k q j k L . So that, exactly like in [5], we estimate: (cid:12)(cid:12)(cid:12) ( f j | q j − φ ε ∗ q j ε ) L (cid:12)(cid:12)(cid:12) ≤ k f j k L k q j − φ ε ∗ q j ε k L ≤ k f j k L min( 1 ε , j ) k q j k L ≤ j k f j k L r min( 1 ε , j ) k q j k L ≤ C j k f j k L r ( q j | q j − φ ε ∗ q j ε ) L , (3.22)13nd similarly, (cid:12)(cid:12)(cid:12) ( v · ∇ q j | q j − φ ε ∗ q j ε ) L (cid:12)(cid:12)(cid:12) ≤ C k v k L ∞ k∇ q j k L j r ( q j | q j − φ ε ∗ q j ε ) L . Collecting these estimates implies that:12 ddt ( q j | q j − φ ε ∗ q j ε ) L − ( c m u j | ∇ q j − φ ε ∗ ∇ q j ε ) L ≤ C j k f j k L r ( q j | q j − φ ε ∗ q j ε ) L + k v k L ∞ k∇ q j k L j r ( q j | q j − φ ε ∗ q j ε ) L . (3.23)so that, when we compute (3.20)+ κ (3.23), there is a cancellation of the problematicterms ε − (div ( c m u j ) | q j − φ ε ∗ q j ) L and we finally obtain:12 ddt (cid:18) ( u j | c m u j ) L + κ ( q j | q j − φ ε ∗ q j ε ) L (cid:19) + ν b ∗ c ∗ j k u j k L ≤ C ∗ k g j k L k u j k L + Cκ j k f j k L r ( q j | q j − φ ε ∗ q j ε ) L + Cκ k v k L ∞ k∇ q j k L j r ( q j | q j − φ ε ∗ q j ε ) L + C k u j k L " m k ∂ t c k ˙ B d − , + C ∗ (cid:16) k∇ v k L ∞ + 2 m k c − k ˙ B d , k v k L ∞ (cid:17) + C ∗ ν ν m k c − k B d , . (3.24)As in [5], the only way to obtain a regularization for the density fluctuation is to considerthe inner product of the velocity equation by ∇ q j which provides the nonnegative term κε − ( ∇ q j |∇ q j − φ ε ∗ ∇ q j ) L , that is we will study the variation in time of ( u j |∇ q j ) L . As ddt ( u j |∇ q j ) L = ( ∂ t u j |∇ q j ) L + ( u j | ∂ t ∇ q j ) L , like in [5] we are lead to sum the following estimates:( ∂ t u j |∇ q j ) L + ( v · ∇ u j |∇ q j ) L − µ (div ( b m . ∇ u j ) |∇ q j ) L − ( λ + µ )( ∇ ( b m div u j ) |∇ q j ) L − κ ( ∇ q j | φ ε ∗ ∇ q j − ∇ q j ε ) L = ( g j |∇ q j ) L ≤ k g j k L k∇ q j k L , (3.25)and (taking the innerproduct of the equation on ∇ q j with u j )( ∂ t ∇ q j | u j ) L + ( ∇ ( v · ∇ q j ) | u j ) L − (div ( c m u j ) | div u j ) L = ( ∇ f j | u j ) L = − ( f j | div u j ) L ≤ k f j k L j k u j k L . (3.26)A simple computation shows that (we refer to [12, 1, 5] for details): (cid:12)(cid:12)(cid:12) ( v · ∇ u j |∇ q j ) L + ( ∇ ( v · ∇ q j ) | u j ) L (cid:12)(cid:12)(cid:12) ≤ C k∇ v k L ∞ k∇ q j k L k u j k L . Moreover, a rough estimate on the term(div ( c m u j ) | div u j ) L = ( c m div u j + u j · ∇ c m | div u j ) L , (cid:12)(cid:12)(cid:12) (div ( c m u j ) | div u j ) L (cid:12)(cid:12)(cid:12) ≤ k c m k L ∞ k∇ u j k L + k∇ c m k L ∞ j k u j k L ≤ k c m k L ∞ j k u j k L + 2 m k c − k ˙ B d , j k u j k L ≤ C ∗ j k u j k L + 2 m k c − k B d , k u j k L , (3.27)Where, as usual, C ∗ is a constant only depending on the bounds of b and c . Finally: ddt ( u j |∇ q j ) L − µ (div ( b m . ∇ u j ) |∇ q j ) L − ( λ + µ )( ∇ ( b m div u j ) |∇ q j ) L + κ ( ∇ q j | ∇ q j − φ ε ∗ ∇ q j ε ) L ≤ k g j k L k∇ q j k L + 2 j k f j k L k u j k L + C k∇ v k L ∞ k∇ q j k L k u j k L + C ∗ j k u j k L + 2 m k c − k B d , k u j k L . (3.28)This estimate provides regularization for ( q j | q j − φ ε ∗ q j ε ) L (this is the best we can hopefor), but involves terms that cannot be absorbed or neutralized through the Gronwalllemma because they introduce too many derivatives, namely: − µ (div ( b m . ∇ u j ) |∇ q j ) L − ( λ + µ )( ∇ ( b m div u j ) |∇ q j ) L . (3.29)As in [5] we need to compensate them thanks to the density equation: more preciselyin [5] (constant coefficients, no b m ) we could entirely neutralize − µ (∆ u j ) |∇ q j ) L − ( λ + µ )( ∇ div u j |∇ q j ) L with ( ∇ div u j |∇ q j ) L from the estimate on k∇ q j k L thanks to inte-grations by parts. In our case, due to the variable coefficients b m and c m , it will notbe possible to entirely cancel those terms, but we will be able to substract to (3.29) itsmost dangerous parts, that is where all the derivatives pound on u j or q j , the rest be-ing absorbable because at least one derivative pounds on b m or c m (and then producinga harmless 2 m thanks to the Bernstein lemma). For this, we study the variations of( ∇ q j | b m c m ∇ q j ) L . As we have ddt ( ∇ q j | b m c m ∇ q j ) L = 2( ∂ t ∇ q j | b m c m ∇ q j ) L + ( ∇ q j | ∂ t ( b m c m ) ∇ q j ) L , (3.30)we begin by estimating the derivative of ∂ t ( b m c m ): there exists a constant only dependingon the bounds of b and c once again denoted by C ∗ so that: k ∂ t ( b m c m ) k L ∞ = k ∂ t b m c m − b m c m ∂ t c m k L ∞ ≤ k ∂ t b m k ˙ B d , c ∗ + k b m k L ∞ ( c ∗ ) k ∂ t c m k ˙ B d , ≤ C ∗ ( k ∂ t b m k ˙ B d , + k ∂ t c m k ˙ B d , ) ≤ m C ∗ ( k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , ) . (3.31)Next, taking the inner product of the equation on ∇ q j by b m c m ∇ q j ,( ∂ t ∇ q j | b m c m ∇ q j ) L + ( ∇ ( v · ∇ q j ) | b m c m ∇ q j ) L + ( ∇ div ( c m u j ) | b m c m ∇ q j ) L = ( ∇ f j | b m c m ∇ q j ) L . ∇ ( v · ∇ q j ) | b m c m ∇ q j ) L = Z R d d X k,l =1 (cid:18) ∂ k v l .∂ l q j . b m c m ∂ k q j + v l b m c m ∂ l ( ∂ k q j ) (cid:19) dx = Z R d d X k,l =1 ∂ k v l .∂ l q j . b m c m ∂ k q j −
12 div ( b m c m v ) |∇ q j | dx, (3.32)so that (cid:12)(cid:12)(cid:12) ( ∇ ( v · ∇ q j ) | b m c m ∇ q j ) L (cid:12)(cid:12)(cid:12) ≤ C (cid:18) k∇ v k L ∞ k b m c m k L ∞ + k div ( b m c m v ) k L ∞ (cid:19) k∇ q j k L ≤ C (cid:18) k∇ v k L ∞ k b m c m k L ∞ + k v k L ∞ k∇ ( b m c m ) k L ∞ (cid:19) k∇ q j k L (3.33)Similarly to (3.31), there exists a constant C ∗ only depending on the bounds of b and c so that: k∇ ( b m c m ) k L ∞ = k ∇ b m c m − b m c m ∇ c m k L ∞ ≤ k∇ b m k ˙ B d , c ∗ + k b m k L ∞ ( c ∗ ) k∇ c m k ˙ B d , ≤ C ∗ ( k∇ b m k ˙ B d , + k∇ c m k ˙ B d , ) = C ∗ ( k∇ ( b m − k ˙ B d , + k∇ ( c m − k ˙ B d , ) ≤ m C ∗ ( k b − k ˙ B d , + k c − k ˙ B d , ) . (3.34)Therefore, there exists a constant only depending on the bounds of b and c once againdenoted by C ∗ so that: (cid:12)(cid:12)(cid:12) ( ∇ ( v ·∇ q j ) | b m c m ∇ q j ) L (cid:12)(cid:12)(cid:12) ≤ C ∗ k∇ v k L ∞ + 2 m (cid:16) k b − k ˙ B d , + k c − k ˙ B d , (cid:17) k v k L ∞ ! k∇ q j k L . (3.35)In all the cited works, the right-hand side ( ∇ f j |∇ q j ) L is completely harmless if thecoefficients are constant and is estimated the following way (thanks to the fact that q j islocalized in frequency): | ( ∇ f j |∇ q j ) L | = | ( f j | ∆ q j ) L | ≤ C j k f j k L k∇ q j k L . In our case the study of ( ∇ f j | b m c m ∇ q j ) L will be more delicate. Indeed f j = F j + F m,j + R j + e R j and as the last two terms are not localized in frequency, as well as b m c m ∇ q j , wewill have to be much more careful. We remark that in the variable coefficients cases from[1, 18, 25, 26], as the studies of the density and velocity are decoupled, such a term doesnot occur. 16s F j and F m,j are localized in frequency we immediately have: (cid:12)(cid:12)(cid:12) ( ∇ F j + ∇ F m,j | b m c m ∇ q j ) L (cid:12)(cid:12)(cid:12) ≤ j ( k F j k L + k F m,j k L ) k b m c m k L ∞ k∇ q j k L ≤ C ∗ j ( k F j k L + k F m,j k L ) k∇ q j k L . (3.36)The last term e R j is not localized in frequency but is rather easy to estimate: thanks toits definition (we refer to (3.15)) we have: e R lj = ∂ l e R j = d X k =1 ∂ k (cid:16) ( c m − .∂ l u kj (cid:17) − ˙∆ j (cid:16) ( c m − .∂ k ∂ l u k (cid:17) + d X k =1 ∂ k (cid:16) ∂ l ( c m − .u kj (cid:17) − ˙∆ j (cid:16) ∂ l ( c m − .∂ k u k (cid:17) . (3.37)Using lemma 2 from the appendix of [18] (see lemma 3 in the appendix of the presentpaper), there exists a constant C > c ′ j ( t )and c ′′ j ( t ) whose summation is 1 such that: k∇ e R j k L ≤ Cc ′ j − js k c m − k ˙ B d h , k∇ u k ˙ B s − h , + Cc ′′ j − js k∇ ( c m − k ˙ B d h , k u k ˙ B s − h , , (3.38)and thanks to the frequency localization of c m , k∇ e R j k L ≤ Cc ′ j − js mh k c − k ˙ B d , k∇ u k ˙ B s − h , + Cc ′′ j − js m (1+ h ) k c − k ˙ B d , k u k ˙ B s − h , . (3.39)With s = s = s − h = h = 1, there exists a nonnegative summable sequence c j ( t ) whose summation is 1 such that: k∇ e R j k L ≤ Cc j − j ( s − k c − k ˙ B d , (cid:16) m k∇ u k ˙ B s − , + 2 m k u k ˙ B s − , (cid:17) . and then: (cid:12)(cid:12)(cid:12) ( ∇ e R j | b m c m ∇ q j ) L (cid:12)(cid:12)(cid:12) ≤ k∇ e R j k L k b m c m k L ∞ k∇ q j k L ≤ C ∗ c j − j ( s − k c − k ˙ B d , (cid:16) m k u k ˙ B s , + 2 m k u k ˙ B s − , (cid:17) k∇ q j k L . (3.40)Finally, thanks to interpolation estimates we obtain (cid:12)(cid:12)(cid:12) ( ∇ e R j | b m c m ∇ q j ) L (cid:12)(cid:12)(cid:12) ≤ c j − j ( s − k∇ q j k L k u k ˙ B s +12 , + C ∗ m (1 + k c − k ˙ B d , ) k u k ˙ B s − , ! . (3.41)17he real problem comes from ∇ R j = (cid:16) v · ∇ q j − ˙∆ j ( v · ∇ q ) (cid:17) + (cid:16) ∇ v · ∇ q j − ˙∆ j ( ∇ v · ∇ q ) (cid:17) . Indeed, we could use the same estimate as in [12, 5] (see lemma 3) for the first two terms,but not for the last two as there are too many derivatives pounding on v . We are thenforced to make the derivative pound on the second term of the inner product:( ∇ R j | b m c m ∇ q j ) L = − ( R j | div ( b m c m ∇ q j )) L = − ( R j | b m c m ∆ q j )) L − ( R j |∇ q j · ∇ ( b m c m )) L , and then, thanks to (3.34) (cid:12)(cid:12)(cid:12) ( ∇ R j | b m c m ∇ q j ) L (cid:12)(cid:12)(cid:12) ≤ C ∗ j + 2 m ( k b − k ˙ B d , + k c − k ˙ B d , ) ! k R j k L k∇ q j k L . (3.42)When m ≤ j ( m will be more precisely fixed later), (cid:12)(cid:12)(cid:12) ( ∇ R j | b m c m ∇ q j ) L (cid:12)(cid:12)(cid:12) ≤ C ∗ k b − k ˙ B d , + k c − k ˙ B d , ) ! j k R j k L k∇ q j k L . (3.43)which gives a good estimate thanks to (3.16). Unfortunately, due to the fact that weonly have q ∈ ˙ B s , and do not assume q ∈ ˙ B s − , , this will not work in the case j ≤ m .In this case we rewrite the equation on ∇ q j the following way: ∂ t ∇ q j + ∇ div ( c m u j ) = ∇ F j + ∇ F m,j + ∇ e R j − ∇ ˙∆ j ( v · ∇ q ) , and take its innerproduct with b m c m ∇ q j .( ∂ t ∇ q j | b m c m ∇ q j ) L +( ∇ div ( c m u j ) | b m c m ∇ q j ) L = (cid:18) ∇ F j + ∇ F m,j + ∇ e R j − ∇ ˙∆ j ( v · ∇ q ) (cid:12)(cid:12)(cid:12) b m c m ∇ q j (cid:19) L ≤ C ∗ j ( k F j k L + k F m,j k L ) k∇ q j k L + (cid:18) ∇ ˙∆ j ( v · ∇ q ) (cid:12)(cid:12)(cid:12) b m c m ∇ q j (cid:19) L + c j − j ( s − k∇ q j k L k u k ˙ B s +12 , + C ∗ m (1 + k c − k ˙ B d , ) k u k ˙ B s − , ! . (3.44)Thanks to the Bony decomposition, v · ∇ q = T v ∇ q + T ∇ q v + R ( v, ∇ q ) and we can easilyshow, thanks to the paraproduct and remainder laws for Besov spaces that: k ˙∆ j (cid:16) T ∇ q v + R ( v, ∇ q ) (cid:17) k L ≤ Cc j − js k v k ˙ B d , k q k ˙ B s , . (3.45) Remark 7
This is here that, due to the paraproduct and remainder conditions on in-dices, we get the condition s ∈ ] − d , d + 1] in theorem 5. But, T v ∇ q cannot be estimated this way, indeed, the best be can have is: k T v ∇ q k ˙ B s , ≤ k v k L ∞ k q k ˙ B s +12 , . j ≤ m , we will be able toestimate: T v ∇ q = X l ∈ Z ˙ S l − v. ˙∆ l ∇ q. As ˙ S l − v. ˙∆ l ∇ q is localized in frequency in an annulus 2 l C ′ , using that q ∈ ˙ B s , thereexists a summable nonnegative sequence whose summation is 1, ( c l ( t )) l ∈ Z , such that:2 j k ˙∆ j ( T v ∇ q ) k L ≤ j X | l − j |≤ N k ˙ S l − v k L ∞ l k ˙∆ l q k L ≤ C j X | l − j |≤ N k v k L ∞ l − ls c l ( t ) k q k ˙ B s , ≤ C − j ( s − X | l − j |≤ N k v k L ∞ l ( j − l ) s c l ( t ) k q k ˙ B s , ≤ C m + N − j ( s − X | l − j |≤ N ( j − l ) s c l ( t ) k v k L ∞ k q k ˙ B s , ≤ C ′ m − j ( s − d j ( t ) k v k L ∞ k q k ˙ B s , , (3.46)because l ≤ j + N ≤ m + N and, thanks to convolution estimates, ( d j ( t )) j ∈ Z is non-negative, summable with k d k l ≤ C s k c k l ≤ C s . Finally we are able to write the desiredestimates: collecting (3.30), (3.31) and (3.35), (3.36), (3.41), (3.43) (in the case j ≥ m )and (3.36), (3.44), (3.45), (3.46) (in the case j ≤ m ) we end up with, for all j ∈ Z :12 ddt ( ∇ q j | b m c m ∇ q j ) L +( ∇ div ( c m u j ) | b m c m ∇ q j ) L ≤ C ∗ k∇ q j k L " m ( k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , )+ (1 + k b − k ˙ B d , + k c − k ˙ B d , )( k∇ v k ˙ B d , + 2 m k v k ˙ B d , ) + k∇ q j k L " c j − j ( s − k u k ˙ B s +12 , + C ∗ j ( k F j k L + k F m,j k L )+ c j − j ( s − C ∗ m (1 + k c − k ˙ B d , ) k u k ˙ B s − , + c j − j ( s − C ∗ (1 + k b − k ˙ B d , + k c − k ˙ B d , )( k∇ v k ˙ B d , + 2 m k v k ˙ B d , ) k q k ˙ B s , . (3.47)Let us recall that the interest of computing (3.28)+ ν (3.47) is that, as in [5], it allowsto neutralize (3.29), that is the terms consuming too many derivatives (estimated by C ∗ j k u j k L k∇ q j k L ), and in the remaining terms at least one derivative will pound on b m − c m − m j or 2 m instead of19 j . Let us explain this in details: as ν = λ + 2 µ , we have (from (3.28)+ ν (3.47)) B def = − µ (div ( b m . ∇ u j ) |∇ q j ) L − ( λ + µ )( ∇ ( b m div u j ) |∇ q j ) L + ν ( ∇ div ( c m u j ) | b m c m ∇ q j ) L = µ (cid:20) ( ∇ div ( c m u j ) | b m c m ∇ q j ) L − (div ( b m . ∇ u j ) |∇ q j ) L (cid:21) + ( λ + µ ) (cid:20) ( ∇ div ( c m u j ) | b m c m ∇ q j ) L − ( ∇ ( b m div u j ) |∇ q j ) L (cid:21) = µB + ( λ + µ ) B . (3.48)Let us now estimate separatedly B and B : B = (cid:18) b m c m ( ∇ c m div u j + c m ∇ div u j + ∇ u j . ∇ c m + u j . ∇ c m ) (cid:12)(cid:12)(cid:12) ∇ q j (cid:19) L − ( b m ∇ div u j + ∇ b m . div u j |∇ q j ) L = (cid:18) b m c m ∇ c m div u j + b m c m ∇ c m . ∇ u j + b m c m . ∇ c m .u j − ∇ b m . div u j (cid:12)(cid:12)(cid:12) ∇ q j (cid:19) L . (3.49)We emphasize that ( b m ∇ div u j ) |∇ q j ) L , which was mentionned before as an obstructionterm, has disappeared, and all that remain are harmless terms where at most two deriva-tives are applied to u j or q j . Thanks to (3.34), and the fact that ∇ c m = ∇ ( c m −
1) = ∇ ˙ S m ( c −
1) we obtain | B | ≤ C ∗ ( k b − k ˙ B d , + k c − k ˙ B d , ) (cid:0) m j k u j k L + 2 m k u j k L (cid:1) k∇ q j k L . The other term requires a little more attention: B = − (cid:18) div ( c m u j ) (cid:12)(cid:12)(cid:12) div ( b m c m ∇ q j ) (cid:19) L − ( b m ∆ u j + ∇ u j . ∇ b m ) |∇ q j ) L = − (cid:18) c m div u j + u j . ∇ c m (cid:12)(cid:12)(cid:12) b m c m ∆ q j + ∇ q j . ∇ ( b m c m ) (cid:19) L − ( b m ∆ u j + ∇ u j . ∇ b m ) |∇ q j ) L = B + B , (3.50)with B = − ( b m div u j | ∆ q j ) L − ( b m ∆ u j |∇ q j ) L , and B = − (cid:18) u j . ∇ c m (cid:12)(cid:12)(cid:12) b m c m ∆ q j + ∇ q j . ∇ ( b m c m ) (cid:19) L − (cid:18) c m div u j (cid:12)(cid:12)(cid:12) ∇ q j . ∇ ( b m c m ) (cid:19) L − ( ∇ u j . ∇ b m ) |∇ q j ) L . (3.51)In each term of B at most two derivatives act on u j and q j so that we can estimate B with the same arguments used for B , and for the other term we have: B = − (cid:16) ∆( b m div u j ) (cid:12)(cid:12)(cid:12) q j (cid:17) L − ( b m ∆ u j |∇ q j ) L = − (∆ b m . div u j + 2 ∇ b m . ∇ div u j + b m . ∆div u j | q j ) L + ( b m . div ∆ u j + ∆ u j . ∇ b m | q j ) L = − (∆ b m . div u j + 2 ∇ b m . ∇ div u j + ∆ u j . ∇ b m | q j ) L . (3.52)20s expected, the dangerous terms are neutralized and then, as before, we can easilyestimate B and we can finally write: | B | ≤ C ∗ ν ( k b − k ˙ B d , + k c − k ˙ B d , ) × m j k u j k L k∇ q j k L + 2 m (cid:16) k c − k ˙ B d , (cid:17) k u j k L k∇ q j k L ! , (3.53)and then, using the fact that 2 ab ≤ a + b , we finally end up with: | B | ≤ C ∗ j k u j k L + C ∗ ν m (1+ k b − k ˙ B d , + k c − k ˙ B d , ) (cid:0) k u j k L k∇ q j k L + k∇ q j k L (cid:1) . (3.54)Thanks to (3.54), computing (3.28)+ ν (3.47) allows us to obtain (using once more (3.16),and estimating 2 j k e R j k L and k e S j k L like in (3.41)), denoting by D ∗ a constant onlydepending on the bounds of b, c and on the physical parameters µ , λ , κ : ddt (cid:18) ( u j |∇ q j ) L + ν ∇ q j | b m c m ∇ q j ) L (cid:19) + κ ( ∇ q j | ∇ q j − φ ε ∗ ∇ q j ε ) L ≤ D ∗ j k u j k L + D ∗ ( k u j k L + k∇ q j k L ) " m ( k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , )+ (1 + k b − k ˙ B d , + k c − k ˙ B d , ) ( k∇ v k ˙ B d , + 2 m k v k ˙ B d , + 2 m ) + D ∗ ( k u j k L + k∇ q j k L ) " c j − j ( s − k u k ˙ B s +12 , +( k G j k L + k G m,j k L +2 j k F j k L +2 j k F m,j k L )+ c j − j ( s − (1 + k b − k ˙ B d , + k c − k ˙ B d , ) ( k∇ v k ˙ B d , + 2 m ) k u k ˙ B s − , + c j − j ( s − (1 + k b − k ˙ B d , + k c − k ˙ B d , )( k∇ v k ˙ B d , + 2 m k v k ˙ B d , ) k q k ˙ B s , . (3.55)Let us introduce, as stated in theorem 5, the following quantity h j ( t ) = ( u j | c m u j ) L + κ ( q j | q j − φ ε ∗ q j ε ) L + η (cid:18) u j |∇ q j ) L + ν ( ∇ q j | b m c m ∇ q j ) L (cid:19) , (3.56)for some η < η so that h j ∼ k u j k L + k∇ q j k L . Thanks to (3.13), ν b ∗ c ∗ + c ∗ k∇ q j k L ≤ ν ( ∇ q j | b m c m ∇ q j ) L ≤ ν b ∗ + b ∗ c ∗ k∇ q j k L . Using the fact that 2 ab ≤ a + b , we get2 | ( u j |∇ q j ) L | ≤ ν b ∗ c ∗ + c ∗ ) k∇ q j k L + 2(2 c ∗ + c ∗ ) νb ∗ k u j k L
21o that if η satisfies η ≤ min(1 , νb ∗ c ∗ c ∗ + c ∗ ) ) , (3.57)there exists a constant D ∗ such that we have: c ∗ k u j k L + κ ( q j | q j − φ ε ∗ q j ε ) L + νηb ∗ c ∗ + c ∗ ) k∇ q j k L ≤ h j ( t ) ≤ D ∗ ( k u j k L + k∇ q j k L ) + κ ( q j | q j − φ ε ∗ q j ε ) L . (3.58)then, writing (3.24)+2 η (3.55), and using the fact that (in (3.24)): Cκ k v k L ∞ k∇ q j k L j r ( q j | q j − φ ε ∗ q j ε ) L ≤ κη ∇ q j | ∇ q j − φ ε ∗ ∇ q j ε ) L + C κ η k v k L ∞ k∇ q j k L , (3.59)if η satisfies η ≤ ν b ∗ c ∗ D ∗ , (3.60)then for all t ∈ [0 , T ]:12 ddt h j ( t ) + ν b ∗ c ∗
16 2 j k u j k L + κη ∇ q j | ∇ q j − φ ε ∗ ∇ q j ε ) L ≤ D ∗ h j ( t ) " m ( k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , )+ (1 + k b − k ˙ B d , + k c − k ˙ B d , ) ( k∇ v k ˙ B d , + 2 m k v k ˙ B d , + 2 m + k v k B d , ) + D ∗ h j ( t ) " c j − j ( s − η k u k ˙ B s +12 , + ( k G j k L + k G m,j k L + 2 j k F j k L + 2 j k F m,j k L )+ c j − j ( s − (1 + k b − k ˙ B d , + k c − k ˙ B d , ) ( k∇ v k ˙ B d , + 2 m ) k u k ˙ B s − , + c j − j ( s − (1 + k b − k ˙ B d , + k c − k ˙ B d , )( k∇ v k ˙ B d , + 2 m k v k ˙ B d , ) k q k ˙ B s , . (3.61)If we introduce γ ∗ = min( κη D ∗ , ν b ∗ c ∗ D ∗ , η , then for all j ∈ Z , ν b ∗ c ∗
16 2 j k u j k L + κη ∇ q j | ∇ q j − φ ε ∗ ∇ q j ε ) L ≥ γ ∗ min( 1 ε , j ) h j ( t ) , t : h j ( t ) + γ ∗ min( 1 ε , j ) Z t h j ( τ ) dτ ≤ h j (0)+ D ∗ Z t h j ( τ ) " m ( k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , )+ (1 + k b − k ˙ B d , + k c − k ˙ B d , ) ( k∇ v k ˙ B d , + 2 m k v k ˙ B d , + 2 m + k v k B d , ) dτ + D ∗ Z t " c j − j ( s − η k u k ˙ B s +12 , + ( k G j k L + k G m,j k L + 2 j k F j k L + 2 j k F m,j k L )+ c j − j ( s − (1 + k b − k ˙ B d , + k c − k ˙ B d , ) ( k∇ v k ˙ B d , + 2 m ) k u k ˙ B s − , + c j − j ( s − (1 + k b − k ˙ B d , + k c − k ˙ B d , )( k∇ v k ˙ B d , + 2 m k v k ˙ B d , ) k q k ˙ B s , dτ. (3.62)In particular, we obtained a bound for the integral (see remark 5) :min( 1 ε , j ) Z t k∇ q j ( τ ) k L dτ, that is, we can now obtain the parabolic regularization on the velocity exactly as in[12, 5] by taking the innerproduct of the velocity equation from system (3.14) by u j ,finally if we set, as stated in theorem 5, g j ( t ) = k u j ( t ) k L + h j ( t ), then we have (3.11),and multiplying by 2 j ( s − and summing over j ∈ Z , we obtain: g s ( q, u )( t ) + ν b ∗ j Z t k u k ˙ B s +12 , dτ + γ ∗ Z t k φ ε ∗ q − qε k ˙ B s , dτ ≤ g s ( q , u )(0) + D ∗ Z t g s ( q, u )( τ ) " m ( k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , )+ (1 + k b − k ˙ B d , + k c − k ˙ B d , ) ( k∇ v k ˙ B d , + 2 m k v k ˙ B d , + 2 m + k v k B d , ) dτ + D ∗ Z t " η k u k ˙ B s +12 , + ( k G k ˙ B s − , + k G m k ˙ B s − , + k F k ˙ B s , + k F m k ˙ B s , ) dτ. (3.63)Using (3.16) and finally fixing η such that η ≤ ν b ∗ D ∗ , (3.64)we end up with (3.12). Remark 8
Recollecting the various conditions on η , (3.57) , (3.60) and (3.64) , we needthat: < η ≤ min(1 , νb ∗ c ∗ c ∗ + c ∗ ) , ν b ∗ c ∗ D ∗ , ν b ∗ D ∗ ) , nd then we defined γ ∗ = min( κη D ∗ , ν b ∗ c ∗ D ∗ , η > , where D ∗ only depends on the bounds of b and c , the physical parameters (viscosities,capillarity, dimension) and s . Remark 9
As explained, many points of the previous proof are much easier for theKorteweg model.
This part is classical so we will leave some details to the reader (the proofs are verysimilar to those in [13, 14, 18, 1, 5, 25, 26]). As in [18], we will first prove the existencefor more regular initial data (we refer to [13, 14]), then use it for regularized initial data,and finally prove the convergence to a solution of our problem.The special feature of the present models is that, due to the capillary term, we cannotafford to study separatedly the density fluctuation and the velocity, we need to recouplethem via the previous a priori estimate. Again we present the proof in the non-local case(it is easier for the Korteweg system).Let us first assume that the initial data satisfy q ∈ ˙ B d , ∩ ˙ B d +12 , and u ∈ ˙ B d − , ∩ ˙ B d , .We emphasize that this will not be necessary for system (NSK). We introduce ( q L , u L )the unique global solution of system: ∂ t q L + div u L = 0 ,∂ t u L − A u L − κ φ ε ∗ ∇ q L − ∇ q L ε = 0 , ( q L , u L ) | t =0 = ( q , u ) . (3.65)We can easily adapt the linear estimates from [8] in the case where we do not assume P ′ (1) > p = 0 in system ( LR ε ) from [5] or [8]), we obtain the followingglobal in time estimates (let us recall that we use the estimates from [16] for ( N SK ), oradapt those from [9] for the order parameter model): there exists a constant
C > t , and for s ∈ { d , d + 1 } , ν k q L k e L ∞ t ˙ B s , + ν k q L k L t ˙ B s +2 ,s /ε + k u L k e L ∞ t ˙ B s − , + ν k u L k L t ˙ B s +12 , ≤ C ( ν k q k ˙ B s , + k u k ˙ B s − , ) . (3.66)As this system is linear, for all m ∈ Z , thanks to this estimates we also have for all t , ν k ( I d − ˙ S m ) q L k e L ∞ t ˙ B s , + ν k ( I d − ˙ S m ) q L k L t ˙ B s +2 ,s /ε + k ( I d − ˙ S m ) u L k e L ∞ t ˙ B s − , + ν k ( I d − ˙ S m ) u L k L t ˙ B s +12 , ≤ C ( ν k ( I d − ˙ S m ) q k ˙ B s , + k ( I d − ˙ S m ) u k ˙ B s − , ) . (3.67) Remark 10
Of course, we could also use our a priori estimates in the case v = 0 , b = c = 1 on the time interval [0 , T ] provided by theorem 1, the interest of the previousestimates being the precision of the coefficients.
24s in [18] we define the following iterative scheme: for n ≥
0, ( q n +1 , u n +1 ) is the uniqueglobal solution of the following linear system: ∂ t q n +1 + u n · ∇ q n +1 + (1 + q n )div u n +1 = 0 ,∂ t u n +1 + u n · ∇ u n +1 −
11 + q n A u n +1 − κ φ ε ∗ ∇ q n +1 − ∇ q n +1 ε = −∇ ( H (1 + q n ) − H (1)) , ( q n +1 , u n +1 ) | t =0 = ( q , u ) , (3.68)where H is the real-valued function defined on R by H ′ ( x ) = P ′ ( x ) /x . We also definethe difference ( q n +1 , u n +1 ) = ( q n +1 − q L , u n +1 − u L ) satisfies the following system: ∂ t q n +1 + u n · ∇ q n +1 + (1 + q n )div u n +1 = − u n · ∇ q L − q n div u L = F n ,∂ t u n +1 + u n · ∇ u n +1 −
11 + q n A u n +1 − κ φ ε ∗ ∇ q n +1 − ∇ q n +1 ε = −∇ ( H (1 + q n ) − H (1)) − u n · ∇ u L − q n q n A u L = G n , ( q n +1 , u n +1 ) | t =0 = (0 , , (3.69)Let us denote by J the real function J ( x ) = 1 / (1 + x ) and assume that c ≤ q ≤ c (then c ≤ J ( q ) ≤ c ). Let us define F ∗ , γ ∗ and η as provided by theorem 5 applied forfunctions b, c having the following bounds: c ∗ = c ≤ c ≤ c ∗ = c + c , and b ∗ = 1 c ∗ ≤ b ≤ b ∗ = 1 c ∗ . For n ∈ N ∗ , t > m ∈ Z , 0 < β ≤ min( b ∗ , c ∗ ) and two positive constants C, C , wedefine the following proposition: P ( n ) = (1) g s ( q n , u n )( T ) + ν b ∗ k u n k L T ˙ B s +12 , + γ ∗ k q n k L T ˙ B s +2 ,s /ε ≤ C , for s ∈ { d , d + 1 } , (2) g d ( q n , u n )( T ) + ν b ∗ k u n k L T ˙ B d , + γ ∗ k q n k L T ˙ B d , d /ε ≤ β, (3) ν b ∗ k u n k L T ˙ B d , + γ ∗ k q n k L T ˙ B d , d /ε ≤ β, (4) k ∂ t q n k L T ˙ B d − , ≤ C F ∗ (1 + 2 C ) , (5) k ∂ t J ( q n ) k L T ˙ B d − , ≤ C F ∗ (1 + 2 C ) , (6) k ( I d − ˙ S m ) q n k e L ∞ T ˙ B d , ≤ β, (7) k q n − q k e L ∞ T ˙ B d , ≤ β and c ∗ ≤ q n ≤ c ∗ , b ∗ ≤ J ( q n ) ≤ b ∗ , (8) k J ( q n ) − k e L ∞ T ˙ B d , ≤ C F ∗ C , (9) k J ( q n ) − J ( q ) k e L ∞ T ˙ B d , ≤ β, (10) k ( I d − ˙ S m ) ( J ( q n ) − k e L ∞ T ˙ B d , ≤ β (3.70)Let us state the main ingredient for the proof of the existence theorem:25 emma 1 Let C = max( g d ( q , u )(0) , g d +1 ( q , u )(0) , k J ( q ) − k ˙ B d , ) . Let F ∗ , γ ∗ , b ∗ , b ∗ , c ∗ , c ∗ as before, < β < min( b ∗ , c ∗ , ν b ∗ F ∗ ) small, and assume that m ∈ Z is largeenough so that: k ( I d − ˙ S m ) q k e L ∞ t ˙ B d , + k ( I d − ˙ S m ) ( J ( q ) − k e L ∞ t ˙ B d , ≤ β. (3.71) If β then T are small enough so that: e C F ∗ (1+2( C +1) C ) (1+2 C ) β ≤ , (1 + F ∗ T ) e C F ∗ (1+2( C +1) C ) (cid:16) m +1 T +2 m T (cid:17) ≤ , (3.72) and C F ∗ k q L k L T ˙ B d , + k u L k L T ˙ B d , + k u L k L T ˙ B d , ! ≤ β, (3.73) then P ( n ) is true for all n ≥ . Remark 11
We emphasize that as for all n , q n and J ( q n ) have the same bounds,the constants F ∗ and γ ∗ provided by theorem 5 are the same for all n . Remark 12
Note that more precisely, in (1), in the case s = d , the constant C onlydepends on norms with regularity index d . Proof:
We will prove this result by induction. Assuming that the proposition is truefor n ∈ N ∗ , we first apply theorem 5 to system (3.68). We have F = 0 and thanks tocomposition estimates in Besov spaces (see the appendix), for s ∈ { d , d + 1 }k G k ˙ B s − , ≤ C ( k q n k L ∞ ) k q n k ˙ B s , ≤ C ( C. k q k L ∞ ) F ∗ C , so that for all t ∈ [0 , T ] (thanks to (1 , , , , g s ( q n +1 , u n +1 )( t ) + ν b ∗ k u n +1 k L t ˙ B s +12 , + γ ∗ k q n +1 k L t ˙ B s +2 ,s ε ≤ g s ( q , u )(0)+ F ∗ C t + 3 β F ∗ k u n +1 k L t ˙ B s +12 , + F ∗ Z t g s ( q n +1 , u n +1 )( τ ) C (1 + 2 C ) m + (1 + 2( C + 1) C ) h m k u n k ˙ B d , + k u n k B d , + k u n k ˙ B d , + 2 m i! dτ, (3.74)then using that 3 β F ∗ ≤ ν b ∗ and the Gronwall lemma, we obtain: g s ( q n +1 , u n +1 )( t ) + ν b ∗ k u n +1 k L t ˙ B s +12 , + γ ∗ k q n +1 k L t ˙ B s +2 ,s ε ≤ ( g s ( q , u )(0) + F ∗ C t ) × e F ∗ C (1+2( C +1) C ) R t h m +1 +2 m +2 m k u n k ˙ B d , + k u n k B d , + k u n k ˙ B d , i dτ . (3.75)26sing (1) for s = d and (3) from P ( n ), we obtain (denoting again by F ∗ the constants): g s ( q n +1 , u n +1 )( t ) + ν b ∗ k u n +1 k L t ˙ B s +12 , + γ ∗ k q n +1 k L t ˙ B s +2 ,s ε ≤ C (1 + F ∗ t ) e F ∗ C (1+2( C +1) C ) h m +1 t +2 m t β + β (1+2 C ) i dτ . (3.76)When β > T > m being fixed moreprecisely later) we obtain point (1) of P ( n + 1). Applying theorem 5 to system (3.69)and doing the same leads to (thanks to (3.72)): g d ( q n +1 , u n +1 )( t ) + ν b ∗ k u n +1 k L t ˙ B d , + γ ∗ k q n +1 k L t ˙ B d , d ε ≤ e F ∗ C (1+2( C +1) C ) h m +1 t +2 m t β + β (1+2 C ) i dτ F ∗ Z t k F n k ˙ B d , + k G n k ˙ B d − , ! dτ ≤ F ∗ Z t k F n k ˙ B d , + k G n k ˙ B d − , ! dτ. (3.77)Estimating the forcing terms: Z t k F n k ˙ B d , + k G n k ˙ B d − , ! dτ ≤ k u n k L t ˙ B d , k q L k L t ˙ B d , + k q n k L ∞ t ˙ B d , k u L k L t ˙ B d , + tC ( k q k L ∞ ) k q n k L ∞ t ˙ B d , + C k u n k L t ˙ B d , k u L k L t ˙ B d , + C ( k q k L ∞ ) k q n k L ∞ t ˙ B d , k u L k L t ˙ B d , ≤ C F ∗ k q L k L t ˙ B d , + k u L k L t ˙ B d , + k u L k L t ˙ B d , ! . (3.78)Thanks to condition (3.73), we obtain the second point. Remark 13
This is here (precisely for u n ·∇ q L ) that we needed the additional regularityon the initial data. The proof of (3) immediately follows for T small enough so that: ν b ∗ k u L k L T ˙ B s +12 , + γ ∗ k q L k L T ˙ B s +2 ,s ε ≤ β. For the estimates on the time derivative of q n +1 (point (4)) we simply use the firstequation of the iterative scheme and the estimates from P ( n ). This implies the result for J ( q n ) thanks to composition estimates (see for example [1] or [12] lemma 1.6) as: ∂ t J ( q n ) = − ∂ t q n (1 + q n ) . k ( I d − ˙ S m ) q n +1 k e L ∞ T ˙ B d , ≤ k q n +1 k e L ∞ T ˙ B d , + k ( I d − ˙ S m ) q L k e L ∞ T ˙ B d , . Using the estimates on q L and on q n +1 (given by (2) n +1 , we obtain (6). The followingpoint is simply given by estimating the equation: ∂ t ( q L − q ) = − div u L , which implies for T small enough k q L − q k e L ∞ t ˙ B d , ≤ k u L k L t ˙ B d , ≤ β, so that, writing q n +1 − q = q n +1 + ( q L − q ) we obtain (7). For the last estimates on J ( q n +1 ): the first one is obtained thanks to composition estimates in Besov spaces on: J ( q n +1 ) − − q n +1 q n +1 . Next we estimate the equation on J ( q n +1 ) − J ( q ) (we refer to [18] proposition 8 fortransport estimates): ∂ t ( J ( q n +1 ) − J ( q )) + u n · ∇ ( J ( q n +1 ) − J ( q ))= (cid:18) q n (1 + q n +1 ) − (cid:19) div u n +1 − u n · ∇ ( J ( q ) − , (3.79)Thanks to the classical transport estimates, P ( n ) and the Gronwall lemma, we obtainthat: k J ( q n +1 ) − J ( q ) k e L ∞ t ˙ B d , ≤ C t e CC t " C (1 + 2 CC ) + k J ( q ) − k ˙ B d , . Then for T small enough as stated in the lemma we get the result. The last point isobtained writing that:( I d − ˙ S m ) ( J ( q n +1 ) −
1) = ( I d − ˙ S m ) ( J ( q n +1 ) − J ( q )) + ( I d − ˙ S m ) ( J ( q ) − , and using (3.71) and (8) n . (cid:4) The next step in the proof of the existence theorem is then to prove that ( q n , u n ) isa Cauchy sequence in E d /ε ( T ) for T small enough as below. For that we define:( δq n , δu n ) = ( q n +1 − q n , u n +1 − u n ) = ( q n +1 − q n , u n +1 − u n ) . It solves the following system: ∂ t δq n + u n · ∇ δq n + (1 + q n )div δu n = δF,∂ t δu n + u n · ∇ δu n −
11 + q n A δu n − κ φ ε ∗ ∇ δq n − ∇ δq n ε = δG, ( δq n , δu n ) | t =0 = (0 , , ( δF = − δu n − . ∇ q n − δq n − div u n ,δG = −∇ ( H (1 + q n ) − H (1 + q n − )) − δu n − · u n + (cid:16) q n − q n − (cid:17) A u n . Then using theorem 5 we prove that for sufficiently small β >
T > k ( δq n , δu n ) k E d /ε ≤ β < . This part is classical (see for example [18, 25, 26]) so we shall not give details (we alsorefer to the next part, the present computations are close to the ones to estimate theconvergence rate). This implies that ( q n , u n ) is a Cauchy sequence in E d /ε ( T ) for T smallenough, and then this gives a solution for system ( N SR ε )in the case of smooth initialdata: q ∈ ˙ B d , ∩ ˙ B d +12 , and u ∈ ˙ B d − , ∩ ˙ B d , .In the case q ∈ ˙ B d , and u ∈ ˙ B d − , we do as in [18]: we begin by regularizing theinitial data: ( q p , u p ) = ( ˙ S p q , ˙ S p u ) , and thanks to what precedes, we get a sequence of smooth solutions ( q p , u p ) p ∈ N of ( N SR ε )on a bounded intervall [0 , T ] (uniformly in p ). We then use classical arguments (estimateson the time derivatives, compactness, Fatou lemma) to prove that the sequence converges,up to an extraction, towards a solution of ( N SR ε ) with initial data ( q , u ) and satisfyingthe energy estimates. We refer to [18] for details: in our case the method is very closeexcept that we mainly use our a priori estimates from theorem 5. Remark 14
We could simplify the proof of (1) n +1 by using also (1) n for s = d + 1 , thenit would not have been necessary to ask for the condition on β in (3.72) , but we choosedto present it this way because for ( N SK ) this is the only way to do as we do not ask formore regularity. Remark 15
The proof for (NSK) follows the same lines but is easier: thanks to thegreater regularity of q , we can immediately prove that ( q n , u n ) is a Cauchy sequence.There is no need to regularize, and therefore P ( n ) is simpler: (1) is only proved for s = d . Remark 16
A very quick method consists in using what is proved for ( N SC ) to oursystem ( N SR ε ) . The problem is that the method from [18] only provides a lower boundfor the maximal lifespan T ∗ ε in O ( ε ) , but we can use our a priori estimates to bound T ∗ ε from below by the same small T > exhibed in what precedes. This method is valid forthe non-local systems but fails for ( N SK ) . The method is similar to what is done in [18], except that we use our a priori estimates.As ε is fixed we can also simply use the estimates from [18] to get the uniqueness.29 Proof of the convergence
Let us now turn to the convergence rate. Once we obtained (3.12) the end of the proofis very close to the one from [5]. Let ( q , u ) ∈ ˙ B d , × ˙ B d − , and ( q ε , u ε ) and ( q, u ) thecorresponding solutions of systems ( N SR ε ) and ( N SK ) both of them existing on [0 , T ]where T is provided by theorem 1: ∂ t q ε + u ε · ∇ q ε + (1 + q ε )div u ε = 0 ,∂ t u ε + u ε · ∇ u ε −
11 + q ε A u ε + ∇ ( H (1 + q ε ) − H (1)) − κ φ ε ∗ ∇ q ε − ∇ q ε ε = 0 , and let us write ( q, u ) as a solution of system ( N SR ε ) with an additionnal external force: ∂ t q + u · ∇ q + (1 + q )div u = 0 ,∂ t u + u · ∇ u −
11 + q A u + ∇ ( H (1 + q ) − H (1)) − κ φ ε ∗ ∇ q − ∇ qε = R ε , where the remainder is given by R ε def = − κε ∇ ( φ ε ∗ q − q − ε ∆ q ). Let us define ( δq, δu ) =( q ε − q, u ε − u ), who solves: ∂ t δq + u ε · ∇ δq + (1 + q ε )div δu = δF + δF ,∂ t δu + u ε · ∇ δu −
11 + q ε A δu − κ φ ε ∗ ∇ δq − ∇ δqε = − R ε + δG + δG + δG , ( δq, δu ) | t =0 = (0 , , (4.80)where ( δF = − δu · ∇ q,δF = − δq. div u, and δG = − δu · ∇ u,δG = (cid:16) q ε − q (cid:17) A u,δG = −∇ ( H (1 + q ε ) − H (1 + q )) . Remark 17
We emphasize that in the external force terms from the first equation, thereis no such term as q · div δu , which was one of the aims of our methods. Applying (3.12) to this system for s = d − h for h ∈ ]0 , ∩ ]0 , d −
1[ (we refer to [5] forexplainations of such a choice, mainly due to paraproduct and remainder endpoints) and: b = 11 + q ε = J ( q ε ) , c = 1 + q ε , v = u ε ,
30e obtain g d − h ( δq, δu )( t ) + ν b ∗ k δu k L t ˙ B d − , + γ ∗ k δq k L t ˙ B d − h +2 , d − h ε ≤ g d − h (0 , F ∗ Z t (cid:16) k δF + δF k ˙ B s , + k R ε + δG + δG + δG k ˙ B s − , (cid:17) dτ + F ∗ Z t g d − h ( δq, δu )( τ ) " m (cid:16) k ∂ t b k ˙ B d − , + k ∂ t c k ˙ B d − , (cid:17) + (1 + k b − k ˙ B d , + k c − k ˙ B d , ) (cid:16) k∇ u ε k ˙ B d , + 2 m k u ε k ˙ B d , + 2 m + k u ε k B d , (cid:17) dτ + F ∗ Z t k ( I d − ˙ S m )( b − k ˙ B d , + k ( I d − ˙ S m )( c − k ˙ B d , ! k u ε k ˙ B s +12 , dτ. (4.81)Thanks to Theorem 1 we have: k c − k e L ∞ t ˙ B d , = k q ε k e L ∞ t ˙ B d , ≤ C ′ , (4.82)if T is small enough. Thanks to composition estimates (see for example [1], [12] lemma1.6 or [5] appendix), we get: k b − k e L ∞ t ˙ B d , = k q ε q ε k e L ∞ t ˙ B d , ≤ C ( k q ε k L ∞ ) k q ε k e L ∞ t ˙ B d , ≤ C ′ . (4.83)Moreover, thanks to the equation on q ε , k ∂ t c k ˙ B d − , = k ∂ t q ε k ˙ B d − , ≤ (1 + k q ε k ˙ B d , ) k u ε k ˙ B d , , and by interpolation we obtain for all t ≤ T : k ∂ t c k L t ˙ B d − , ≤ (1 + k q ε k L ∞ t ˙ B d , ) k u ε k L ∞ t ˙ B d − , k u ε k L t ˙ B d , ≤ C ′ V ε ( t ) , (4.84)with V ε ( t ) = Z t k u ε k ˙ B d , dτ. Similarly, we obtain k ∂ t b k ˙ B d − , = k (cid:20) (cid:16) q ε ) − (cid:17)(cid:21) ∂ t q ε k ˙ B d − , ≤ (1 + k q ε ) − k ˙ B d , ) k ∂ t q ε k ˙ B d − , ≤ (1 + C ( k q ε k L ∞ ) k q ε k e L ∞ t ˙ B d , ) k ∂ t q ε k ˙ B d − , . (4.85)31ollecting these estimates and thanks to the estimates on ( q, u ) and ( q ε , u ε ) given bytheorem 1, we conclude that there exists a constant C ′ > k b − k e L ∞ t ˙ B d , + k c − k e L ∞ t ˙ B d , ≤ C ′ , k ∂ t b k L t ˙ B d − , + k ∂ t c k L t ˙ B d − , ≤ C ′ V ε ( t ) . (4.86)As q ε satisfies a transport equation, thanks to Proposition 9 from [18] we can write: k ( I d − ˙ S m )( c − k e L ∞ t ˙ B d , = k ( I d − ˙ S m ) q ε k e L ∞ t ˙ B d , ≤ k ( I d − ˙ S m ) q k ˙ B d , + (1 + k q k ˙ B d , )( e CV ε ( t ) − . (4.87)Similarly, as b = J ( q ε ) satisfies: ∂ t J ( q ε ) + u ε · ∇ J ( q ) = J ( q )div u ε , we obtain: k ( I d − ˙ S m )( b − k e L ∞ t ˙ B d , = k ( I d − ˙ S m )( J ( q ε ) − k e L ∞ t ˙ B d , ≤ k ( I d − ˙ S m )( J ( q ) − k ˙ B d , + (1 + k J ( q ) − k ˙ B d , )( e CV ε ( t ) − . (4.88)If m is large enough, and T is small enough (both of them only depending on the initialdata and the physical parameters and not on ε ), we have: k ( I d − ˙ S m )( b − k e L ∞ t ˙ B d , + k ( I d − ˙ S m )( b − k e L ∞ t ˙ B d , ≤ γ ∗ F ∗ , so that (4.81) reduces to ( m is now fixed) g d − h ( δq, δu )( t ) + 3 ν b ∗ k δu k L t ˙ B d − , + γ ∗ k δq k L t ˙ B d − h +2 , d − h ε ≤ F ∗ Z t (cid:16) k δF + δF k ˙ B s , + k R ε + δG + δG + δG k ˙ B s − , (cid:17) dτ + F ∗ Z t g d − h ( δq, δu )( τ ) (cid:16) k∇ u ε k ˙ B d , + 2 m k u ε k ˙ B d , + 2 m + k u ε k B d , (cid:17) dτ. (4.89)The last step is now to estimate the external forcing terms. This is similar to [5] so wewill skip details: using Besov product laws and interpolation we obtain: k δF k ˙ B d − h , ≤ C k δu k ˙ B d − h , k q k ˙ B d , ≤ ν b ∗ F ∗ k δu k ˙ B d − h +12 , + F ∗ k δu k ˙ B d − h − , k q k ˙ B d , k q k ˙ B d , , (4.90)32nd k δF k ˙ B d − h , ≤ C k δq k ˙ B d − h , k∇ u k ˙ B d , , k δG k ˙ B d − h − , ≤ C k δu k ˙ B d − h − , k∇ u k ˙ B d , , k δG k ˙ B d − h − , ≤ C k I ( q ε ) − I ( q ) k ˙ B d − h , kA u k ˙ B d , ≤ C ′ k δq k ˙ B d − h , kA u k ˙ B d , , k δG k ˙ B d − h − , ≤ C ( k q ε k L ∞ , k q k L ∞ ) k δq k ˙ B d − h , . Plugging this into (4.89) and using the Gronwall lemma allows us to write for all t ≤ T : g d − h ( δq, δu )( t ) + 3 ν b ∗ k δu k L t ˙ B d − h +12 , + γ ∗ k δq k L t ˙ B d − h +2 , d − h /ε ≤ F ∗ k R ε k L t ˙ B s − , e F ∗ R t (cid:16) k∇ u ε k ˙ B d , +2 m k u ε k ˙ B d , +1+2 m + k u ε k B d , + k∇ u k ˙ B d , + k q k ˙ B d , k q k ˙ B d , (cid:17) dτ (4.91)We can estimate R ε thanks to Corollary 1 from [5] (section 4): there exists a constant C h > k R ε k L T ˙ B d − h − , ≤ C h κε h k q k L T ˙ B d , . We conclude using the estimates on q given by theorem 1. Remark 18
As in [5, 9], in the case h = 0 , using the same estimates we have: k R ε k L t ˙ B d − , ≤ κ X j ∈ Z j d e − ε C j − ε C j ε k ˙∆ j q k L t L , and thanks to the Lebesgue theorem it goes to zero as ε goes to zero. In this section we explain how our results can be extended for density dependant viscositycoefficients. Let us recal that in this case the diffusion operator writes: A u = div (2 µ ( ρ ) Du ) + ∇ ( λ ( ρ )div u ) , where 2 Du = t ∇ u + ∇ u . Then we obtain:1 ρ A u = µ ( ρ ) ρ ∆ u + µ ( ρ ) + λ ( ρ ) ρ ∇ div u + 2 Du · µ ′ ( ρ ) ρ ∇ ρ + div u · λ ′ ( ρ ) ρ ∇ ρ. The last two terms are of the form ∇ u · ∇ L ( ρ ) and can be dealt the same way as we didin [7] writing: ∇ u · ∇ L (1 + q ) = ∇ u · ∇ (cid:16) L (1 + q ) − L (1 + ˙ S m q ) (cid:17) + ∇ u · ∇ (cid:16) L (1 + ˙ S m q ) − L (1) (cid:17) . µ (1 + q )∆ u + (cid:16) µ (1 + q ) + λ (1 + q ) (cid:17) ∇ div u, with µ ( x ) = µ ( x ) /x , λ ( x ) = λ ( x ) /x . Instead of dealing with the operator b (1+ q )( µ ∆ u +( λ + µ ) ∇ div u ) in the proof of theorem 5, we have to deal with b (1+ q )∆ u + b (1+ q ) ∇ div u where we will require estimates on ∂ t b , ∂ t b , ∂ t c , c − b − b (1) and b − b (1).The last point we need to carefully check is the transport estimates involving µ (1+ q )or λ (1 + q ) like in (5) (8) (9) (10) from P ( n ) or (4.87). This relies on the compositionestimates for Besov norms (see lemma 1.6 from [12] or [1]) which require that µ ( x ) /x, µ ′ ( x ) , λ ( x ) /x, λ ′ ( x ) ∈ W [ d ]+2 , ∞ local . We emphasize that for special cases µ ( ρ ) = µρ and λ ( ρ ) = λρ we only need the estimatesfrom [5]. Our method relies on innerproducts based on L and cannot be adapted in the settingof ˙ B sp,r spaces with p = 2. In the case of particular coefficients (viscosity, capillarity) wecan use the decoupled methods (we refer to [18, 7, 25, 26]) and obtain results in this case.For a summation index r = 1, the methods we present here can be adapted in thesame way as in [7, 25, 26]. The Fourier transform of u with respect to the space variable will be denoted by F ( u ) or b u . In this subsection we will briefly state (as in [4]) classical definitions and propertiesconcerning the homogeneous dyadic decomposition with respect to the Fourier variable.We will recall some classical results and we refer to [1] (Chapter 2) for proofs and acomplete presentation of the theory.To build the Littlewood-Paley decomposition, we need to fix a smooth radial function χ supported in (for example) the ball B (0 , ), equal to 1 in a neighborhood of B (0 , ) andsuch that r χ ( r.e r ) is nonincreasing over R + . So that if we define ϕ ( ξ ) = χ ( ξ/ − χ ( ξ ),then ϕ is compactly supported in the annulus { ξ ∈ R d , c = ≤ | ξ | ≤ C = } and wehave that, ∀ ξ ∈ R d \ { } , X l ∈ Z ϕ (2 − l ξ ) = 1 . (7.92)Then we can define the dyadic blocks ( ˙∆ l ) l ∈ Z by ˙∆ l := ϕ (2 − l D ) (that is d ˙∆ l u = ϕ (2 − l ξ ) b u ( ξ ))so that, formally, we have u = X l ˙∆ l u (7.93)We can now define the homogeneous Besov spaces used in this article:34 efinition 4 For s ∈ R and ≤ p, r ≤ ∞ , we set k u k ˙ B sp,r := (cid:18) X l rls k ˙∆ l u k rL p (cid:19) r if r < ∞ and k u k ˙ B sp, ∞ := sup l ls k ˙∆ l u k L p . We then define the space ˙ B sp,r as the subset of distributions u ∈ S ′ h such that k u k ˙ B sp,r isfinite. Once more, we refer to [1] (Chapter 2) for properties of the inhomogeneous and homo-geneous Besov spaces.In this paper, we mainly work with functions or distributions depending on both thetime variable t and the space variable x , and we introduce the spaces L p ([0 , T ]; X ) (resp. C ([0 , T ]; X )).The Littlewood-Paley decomposition enables us to work with spectrally localized(hence smooth) functions rather than with rough objects. We naturally obtain bounds foreach dyadic block in spaces of type L ρT L p . Going from those type of bounds to estimatesin L ρT ˙ B sp,r requires to perform a summation in ℓ r ( Z ) . When doing so however, we donot bound the L ρT ˙ B sp,r norm for the time integration has been performed before the ℓ r summation. This leads to the following notation (after J.-Y. Chemin and N. Lerner in[10]): Definition 5
For
T > , s ∈ R and ≤ r, ρ ≤ ∞ , we set k u k e L ρT ˙ B sp,r := (cid:13)(cid:13) js k ˙∆ q u k L ρT L p (cid:13)(cid:13) ℓ r ( Z ) . Let us now recall a few nonlinear estimates in Besov spaces. Formally, any productof two distributions u and v may be decomposed into uv = T u v + T v u + R ( u, v ) , where (7.94) T u v := X l ˙ S l − u ˙∆ l v, T v u := X l ˙ S l − v ˙∆ l u and R ( u, v ) := X l X | l ′ − l |≤ ˙∆ l u ˙∆ l ′ v, where for j ∈ Z , the operator ˙ S j is defined by:˙ S j u = X l ≤ j − ˙∆ l u = χ (2 − j D ) u. The above operator T is called a “paraproduct” whereas R is called a ”remainder”.The decomposition (7.94) has been introduced by J.-M. Bony in [2]. We refer to [1] forproperties, and also to [5] or [8] for paraproduct and remainder estimates for externalforce terms.In this article we will frequently use the following estimates (we refer to [1] Section 2.6,[12], [22] for general statements, more properties of continuity for the paraproduct andremainder operators, sometimes adapted to e L ρT ˙ B sp,r spaces): under the same assumptionsthere exists a constant C > k ˙ T u v k ˙ B s , ≤ C k u k L ∞ k v k ˙ B s , ≤ C k u k ˙ B d , k v k ˙ B s , , (7.95)35 ˙ T u v k ˙ B s + t , ≤ C k u k ˙ B t ∞ , ∞ k v k ˙ B s , ≤ C k u k ˙ B t + d , k v k ˙ B s , ( t < , k ˙ R ( u, v ) k ˙ B s s , ≤ C k u k ˙ B s ∞ , ∞ k v k ˙ B s , ≤ C k u k ˙ B s d , k v k ˙ B s , ( s + s > , k ˙ R ( u, v ) k ˙ B s s − d , ≤ C k ˙ R ( u, v ) k ˙ B s s , ≤ C k u k ˙ B s , k v k ˙ B s , ( s + s > . Let us now turn to the composition estimates. We refer for example to [1] (Theorem2 .
59, corollary 2 . Proposition 2
1. Let s > , u ∈ ˙ B s , ∩ L ∞ and F ∈ W [ s ]+2 , ∞ loc ( R d ) such that F (0) =0 . Then F ( u ) ∈ ˙ B s , and there exists a function of one variable C only dependingon s , d and F such that k F ( u ) k ˙ B s , ≤ C ( k u k L ∞ ) k u k ˙ B s , .
2. If u and v ∈ ˙ B d , and if v − u ∈ ˙ B s , for s ∈ ] − d , d ] and F ∈ W [ s ]+3 , ∞ loc ( R d ) , then F ( v ) − F ( u ) belongs to ˙ B s , and there exists a function of two variables C onlydepending on s , d and G such that k F ( v ) − F ( u ) k ˙ B s , ≤ C ( k u k L ∞ , k v k L ∞ ) | G ′ (0) | + k u k ˙ B d , + k v k ˙ B d , ! k v − u k ˙ B s , . We end this subsection with two commutator estimates proven in [18] (we also referto [1]).
Lemma 2 let σ ∈ ] − d , d + 1] . There exists a sequence ( c j ) j ∈ Z ∈ l ( Z ) with summation1, and a constant C only depending on d and σ such that for all j ∈ Z , k [ v · ∇ , ˙∆ j ] a k L ≤ Cc j − jσ k∇ v k ˙ B d , k a k ˙ B σ , . Lemma 3
Let σ ∈ ] − d , d + 1] and h ∈ ]1 − d , , k ∈ { , ..., d } and R j = ˙∆ j ( a∂ k w ) − ∂ k ( a ˙∆ j w ) . There exists a constant C only depending on σ, h, d such that: X j ∈ Z jσ kR j k L ≤ C k a k ˙ B d h , k w k ˙ B σ +1 − h , . As explained, in the study of the compressible Navier-Stokes system with data in criticalspaces, the density fluctuation has two distinct behaviours in low and high frequencies,separated by a frequency threshold. This leads to the notion of hybrid Besov spaces and36e refer to R. Danchin in [12] or [1] for general hybrid spaces . In this paper we only willuse the following hybrid norms: k f k ˙ B s +2 ,sβ def = X j ∈ Z min( β , j )2 js k ˙∆ j f k L = X j ≤ log β j ( s +2) k ˙∆ j f k L + X j> log β β js k ˙∆ j f k L . (7.96)In this formulation, we obviously remark the threshold frequency log β which separateslow (parabolically regularized) and high (damped with coefficient β ) frequencies. Butas we prove in [8, 9], the frequency transition is in fact continuous and the followingequivalent formulations show that these norms are completely tailored to our capillaryterm. We refer to [8] for the case of the first non-local model: k f k ˙ B s +2 ,s /ε ∼ X j ∈ Z − e − cε j ε js k ˙∆ j f k L ∼ k φ ε ∗ f − fε k ˙ B s , and we refer to [9] for the order parameter model: k f k ˙ B s +2 ,sα ∼ X j ∈ Z j j α js k ˙∆ j f k L ∼ k α ( ψ α ∗ f − f ) k ˙ B s , . References [1] H. Bahouri, J.-Y. Chemin, R. Danchin. Fourier analysis and nonlinear partial dif-ferential equations,
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