7.7 Two-phase Navier-Stokes equations with Boussinesq-Scriven surface and gravity
In this section we want to concentrate on the two-phase Navier-Stokes problem with surface viscosity (Boussinesq-Scriven), gravity and free boundary. It describes the motion of two viscous incompressible fluids separated by a closed free interface. The two-phase Navier-Stokes equations were treated by many other authors (cf. [PS09], [PS10], [PS11], [SS11a], and the given references therein) for the caseλs, µs= 0 (i.e. in the absence of surface viscosity). A rigorous derivation of the mathematical model of the two-phase Navier-Stokes equations with Boussinesq-Scriven surface and the associated linearization can be found in a work of D. Bothe and J. Prüss, cf. [BP10]. In [PS11] J. Prüss and G. Simonett proved the well-posedness of the linearized two-phase Navier-Stokes problem with surface tension and gravity but without surface viscosity (i.e. λs, µs = 0). Up to now the well-posedness for λs, µs > 0 has not been considered before in anLp-setting.
Here we present a unified treatment, which can handle both casesλs, µs = 0 andλs, µs >0 at one stroke. In addition our approach simplifies the proofs given in [PS09], [PS10], and [PS11]. We consider the linearized problem for the pressureπ: R+×˙
Rn+1→R, the velocity fieldu= (v, w) :R+×˙
Rn+1→R, and the functionh:R+×Rn→R, which models the boundary dynamics. This problem reads as follows
the gravitational force on the right-hand side to the pressureπ. HereJϕKalways denotes the jump of the functionϕacross the boundary interfaceRn, i.e.JϕK:=γ0,n+1ϕ
Rn+1+ −γ0,n+1ϕ
Rn+1− .
Letb0∈Rnwith|b0| ≤βfor fixedβ >0. Due to the localization argument in [PS11, Theorem 3.1 (v)]
it is crucial to solve (7.33) uniformly in b0, i.e. the norm of the solution operator should be estimated from above uniformly for allb0∈Rn with|b0| ≤β.
Remark 7.25. We consider the two casesλs, µs= 0andλs≥0, µs>0 due to the fact that the highest order on the boundary is not elliptic ifλs>0, µs= 0.
As in the proof of [PS10, Theorem 3.1] the system (7.33) can be reduced to
After formal application of the Laplace and Fourier transform to (7.34) we deduce the system of ordinary
differential equations for(ˆu,ˆπ) = ( ˆu1,πˆ1)χRn+1 use another representation formula for the solution of the ordinary differential equations. Similar to Section 7.6 we use the ansatz
ˆ into the boundary conditions (7.36)-(7.40) yields the following linear system of equations for the unknown ( ˆΦ(2)v ,Φˆ(2)w ,Φˆ(1)w ,ˆh,JˆπK,|ξ|ˆp2)T
7.7: Two-phase Navier-Stokes equations with Boussinesq-Scriven surface and gravity 155 mapping properties. In contrast to [PS09], [PS10], and [PS11] we can avoid many auxiliary problems by our approach. Therefore our proof is shorter and more direct.
Here we can take benefit from the parameter-dependance of the theory developed in the last chapters.
To treat this mixed order system we have to define a complex parameter-dependent matrix. We use the same abbreviations as before in a complex version (i.e. substituteiξ z,|ξ| |z|−,b0 ℘0, and
The constantsθ,δ, andεwill be determined by Lemma 7.27. In the sequel we show that for arbitrary β >0there exists θ, δ, andεsuch that the matrixL is an N-parabolic mixed order system in the sense of Definition 4.11. The symboliξ/|ξ|, which has been replaced by the compact parameter℘1, is related to the Riesz transform, cf. Definition 7.30. At the end of this section we use the Dunford calculus to plug in the Riesz transform for℘1to return to the original system. First, we have to determinedetL[℘]with the next lemma, which shows that we can determine the determinant of a special(n+ 4)×(n+ 4)-matrix by the determinant of a4×4-matrix.
Lemma 7.26. Let n∈N,A:= (aij)i,j=1,...,4∈C4×4,(αj)j=1,...,4∈C,z∈Cn,C >0, andΩ0∈C\ {0}.
Proof. This can be verified easily.
N(P) In the sequel we consider the casesB= 0and B 6= 0separately. If B 6= 0, we have boundary conditions of order 2. Therefore it is obvious that the order structure of P massively depends on the existence of surface viscosity, i.e. on the constantB.
We can show
7.7: Two-phase Navier-Stokes equations with Boussinesq-Scriven surface and gravity 157 (ii) If θ,δ, andεare determined by (i), then detL is N-parabolic with compact parameter. Especially
we have
[O(detL[℘])](γ) =
(max{n+ 3, γ+n+ 2,[n+ 4]/2γ}, λs, µs= 0,
max{3 + 2n, γ+ 2 + 2n,2γ+ 2n,(n+ 4)/2·γ}, λs≥0, µs>0, γ >0 for all℘∈K(β, ε).
Proof. (i) Here we can apply the characterization of Corollary 3.59. This is why we will consider the principal part ofP in the next lines.
(1) Let λs, µs= 0. Then we get
πγP[℘](λ, z) =
2¯µ[σ+ 2¯µh℘i]|z|4−, γ∈(0,1), 2¯µ(2¯µλ+ [σ+ 2¯µh℘i]|z|−)|z|3−, γ= 1, 4¯µ2λ|z|3−, γ∈(1,2),
¯ ρ(√
µ1ρ1+√
µ2ρ2)λ5/2, γ >2 and
πγ=2P[℘](λ, z) = ρ(¯√
µ1ω1+√
µ2ω2)λ+ 4√
µ1µ2ω1ω2|z|−+ ¯ρ¯µλ|z|− +4(µ3/21 ω1+µ3/22 ω2)|z|2−+ 4µ1µ2|z|3−.
It is easily seen that for allγ ∈(1,∞)\ {2} we have πγP[℘](λ, z)6= 0for all non-vanishing tuples (λ, z)∈ (Sθ\ {0})×(Σnδ \ {0}) and ℘∈ K(β, ε), θ > π/2, and δ > 0. But we have to take a closer look atπγP forγ ∈(0,1]∪ {2}. In the following we choose ε >0 such that εβ < σ/[2¯µ]. Hence we get
Re(σ+ 2¯µh℘i) =σ+ 2¯µh℘0,Re℘1i>0, ℘∈K(β, ε) (7.51) due to| h℘0,Re℘1i | ≤εβ < σ/[2¯µ]. Therefore we can chooseδ >0small andθ∈(π/2,2/3π) such that
σ+ 2¯µh℘i ∈Sπ−θ−δ for all℘∈K(β, ε). (7.52) (I) Let γ <1. Then we directly get πγP[℘](λ, z)6= 0for all(λ, z)∈(Sθ\ {0})×(Σnδ \ {0})
and℘∈K(β, ε)by (7.51).
(II) Let γ = 1. Then we have [σ+ 2¯µh℘i]|z|− ∈Sπ−θ according to (7.52). This especially yields
2¯µλ+ [σ+ 2¯µh℘i]|z|−∈Sθ\ {0} (7.53) for all(λ, z, ℘) ∈(Sθ\ {0})×(Σnδ \ {0})×K(β, ε). We obtain π1P[℘](λ, z) 6= 0for all (λ, z, ℘)∈(Sθ\ {0})×(Σnδ \ {0})×K(β, ε).
(III) Letγ= 2and θ∈(π/2,2/3π)be fixed. For simplicity we define the continuous symbol S:X →C, (λ, r, ϕ) 7→ ρ(¯√
µ1ω10 +√
µ2ω02)λ+ 4√
µ1µ2ω01ω02reiϕ+ ¯ρ¯µλreiϕ +4(µ3/21 ω10 +µ3/22 ω20)(reiϕ)2+ 4µ1µ2(reiϕ)3
where X := Sθ×[0,∞)×[−1,1]and ω0j =ω0j(λ, r, ϕ) = µ−1/2j (ρjλ+µjreiϕ)−1/2. Let λ ∈ Sθ with Imλ > 0 and ϕ = 0. Then we also have Imωj > 0 and Im(ωjλ) > 0 and therefore ImS(λ, r,0) > 0. Analogously for λ ∈ Sθ with Imλ < 0 we also get ImS(λ, r,0)<0. For(λ, r)∈[0,∞)2\ {0}we trivially haveS(λ, r,0)>0.
So we have provedS(λ, r,0)6= 0for all(λ, r)∈[Sθ×[0,∞)]\ {(0,0)}. Thus, there exists C0>0 such that
|S(λ, r,0)| ≥C0 (7.54)
for all (λ, r) ∈ V := {(τ, s) ∈ Sθ×[0,∞) : |τ|+s2 = 1}. Due to the compactness of
So we have proved that the symbolP is N-parabolic in both cases.
(ii) According to (i) we haveP ∈SN[K(β, ε)](Sθ×Σnδ). It is easy to see thatΩ0∈SN(Sθ×Σnδ)with
In order to apply Corollary 4.22 we define the order functions t1(γ) :=. . .:=tn(γ) :=
7.7: Two-phase Navier-Stokes equations with Boussinesq-Scriven surface and gravity 159 s1:=s2:= 0, s3(γ) :=−max{1,1/2·γ}, s4:=. . .:=sn+4:= 0.
We easily obtainPn+4
k=1(sk(γ) +tk(γ)) = [O(detL)](γ)forγ >0in both cases. It is easy to verify that sj+tiis an upper increasing, respectively decreasing, order function of Ljifor alli, j= 1, . . . , n+ 4.
So we have proved the following proposition.
Proposition 7.28. The complex matrixL defined in (7.50) is an N-parabolic mixed order system in the sense of Definition 4.11.
For the application of Corollary 4.22 we define forp∈(1,∞) ((r00, s00) := (r01, s01) := (1−1/p,0), (r02, s02) := (0,1/2−1/(2p)),
(F0,K0) := (F1,K1) := (Hp, Bp), (F2,K2) := (Bp, Hp), ifλs, µs= 0 ((r00, s00) := (r01, s01) := (1−1/p,0), (r02, s02) := (r30, s03) := (0,1/2−1/(2p)),
(F0,K0) := (F1,K1) := (Hp, Bp), (F2,K2) := (F3,K3) := (Bp, Hp) ifλs≥0, µs>0.
It is easy to see this scale fulfills all admissibility conditions. Let λs, µs = 0 or λs ≥0, µs > 0. Then Corollary 4.22 and Remark 4.13 (iv) yield the following result.
Theorem 7.29. Let 1< p <∞andβ >0. There existµ0>0 andε >0 such that L[℘] :=h
L[℘](∇(µ)+ )i
|H
∈LIsom(H,F), ℘∈K(β, ε), µ≥µ0,
and k(L[℘])−1kL(F,H) ≤ C for all ℘ ∈K(β, ε). The associated spaces are given by H :=Qn+4 i=1 Hi and F:=Qn+4
j=1 Fj where Hi=
(
0Bp,µ1−1/(2p)(R+, Lp(Rn))∩Lp,µ(R+, Bp2−1/p(Rn)), λs, µs= 0,
0Bp,µ1−1/(2p)(R+, Lp(Rn))∩0B1/2−1/(2p)p,µ (R+, Hp2(Rn))∩Lp,µ(R+, Bp3−1/p(Rn)), λs≥0, µs>0 fori= 1, . . . , nand
Hn+1=Hn+2 = 0B1−1/(2p)p,µ (R+, Lp(Rn))∩Lp,µ(R+, B2−1/pp (Rn)),
Hn+3 = 0B2−1/(2p)p,µ (R+, Lp(Rn))∩0Hp,µ1 (R+, B2−1/pp (Rn))∩Lp,µ(R+, Bp3−1/p(Rn)), Hn+4 = 0B1/2−1/(2p)p,µ (R+, Lp(Rn))∩Lp,µ(R+, B1−1/pp (Rn)),
F1=F2 = 0Bp,µ1/2−1/(2p)(R+, Lp(Rn))∩Lp,µ(R+, Bp1−1/p(Rn)), F3 = 0Bp,µ1−1/(2p)(R+, Lp(Rn))∩Lp,µ(R+, Bp2−1/p(Rn)), F4=. . .=Fn+4 = 0Bp,µ1/2−1/(2p)(R+, Lp(Rn))∩Lp,µ(R+, Bp1−1/p(Rn)).
In the derivation of the matrixL in (7.50) we replaced the symboliξ/|ξ|by the compact parameter
℘1. The symboliξ/|ξ|is related to the Riesz transform, which is introduced below. For returning to the original problem we use the Dunford calculus to plug in the Riesz transform for℘1.
Definition 7.30(Riesz transform). We define the Riesz transform onHp−∞(Rn)by R:Hp−∞(Rn) → [Hp−∞(Rn)]n,
f 7→ (Rjf)j=1,...,n
whereRj :=Tmj(i·),mj(z) :=−i·zj/|z|−forj ∈ {1, . . . , n}andz∈Σnδ (cf. Definition 2.29). For further details we refer to [Ste70, Ch. III].
Remark 7.31. (i) Due to Proposition 2.30 we have
R|Kr(Rn)∈L(Kr(Rn),[Kr(Rn)]n) forK ∈ {Hp, Bp}.
(ii) It is easy to verify thatσ(Rj) = [−1,1]for all j∈ {1, . . . , n}.
We haveQn
j=1σ(iRj)⊆K2(ε)and therefore we find paths of integration inK2(ε)which envelop the spectrum. So we can use the Dunford calculus to define the operators
L(b0) := L[b0, iR+]∈L(H,F), S(b0) := (L[b0,·])−1(iR+)∈L(F,H) for allb0∈K1(β).
Corollary 7.32. We have
L(b0)∈LIsom(H,F), S(b0)∈LIsom(F,H) andL(b0)−1=S(b0). For allβ >0there exists C=C(β)>0 such that
kL(b0)−1kL(F,H)≤C(β) for allb0∈Rn with|b0| ≤β.
At this point we want to mention Remarks 7.23 and 7.24 again. The arguments discussed there can also be applied to the two-phase Navier-Stokes equations in (7.34).