Time-decay estimates for linearized two-phase Navier-Stokes equations with surface tension and gravity

The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane $x_N=0$ in the $N$-dimensional Euclidean space, $N \geq 2$. It is well-known that the Rayleigh-Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of $L_p-L_q$ type for the linearized equations. Our approach is based on solution formulas, given by Shibata and Shimizu (2011), for a resolvent problem associated with the linearized equations.


Introduction
Let us consider the motion of two immiscible, viscous, incompressible capillary fluids, fluid + and fluid − , in the N -dimensional Euclidean space R N for N ≥ 2.
Here fluid + and fluid − occupy Ω + (t) and Ω − (t), respectively, which are given by We denote the density of fluid ± by ρ ± , while the viscosity coefficient of fluid ± by µ ± . Suppose that ρ ± and µ ± are positive constants throughout this paper. That motion of two fluids is governed by the two-phase Navier-Stokes equations where surface tension is included on the interface. In addition, we allow for gravity to act on the fluids. The two-phase Navier-Stokes equations was studied by Prüss and Simonett [4], and they proved that the Rayleigh-Taylor instability occurs in an L p -setting when the upper fluid is heavier than the lower one, i.e., ρ + > ρ − . In the present paper, we assume that the lower fluid is heavier than the upper one, i.e., ρ − > ρ + , and show time-decay estimates of L p − L q type for some linearized system as the first step in proving global existence results for the two-phase Navier-Stokes equations when ρ − > ρ + .
. This paper is concerned with the following linearized system of the two-phase Navier-Stokes equations: where σ is a positive constant called the surface tension coefficient and one has set for the indicator function x, t), . . . , U N (x, t)) T 2 and P = P (x, t) respectively denote the velocity field of the fluid and the pressure field of the fluid at position x ∈Ṙ N and time t > 0, while d = d(x ′ ) and f = f (x) = (f 1 (x), . . . , f N (x)) T are given initial data. Note that e N = (0, . . . , 0, 1) T and I is the N × N identity matrix. Let ∂ j = ∂/∂x j for j = 1, . . . , N . Then while D(U) is an N × N matrix whose (i, j) element is given by ∂ i U j + ∂ j U i . In For the acceleration of gravity γ a > 0, the constant ω is given by ω = −[[ρ]]γ a = (ρ − − ρ + )γ a , which is positive when ρ − > ρ + .
The local well-posedness for the two-phase Navier-Stokes equations with Γ(t) as above was proved in Prüss and Simonett [5,6]. Note that the local well-posedness holds for any positive constants ρ ± , that is, the condition ρ − > ρ + is not required. Those results were extended to a class of non-Newtonian fluids in [1]. In addition, [8] considered the two-phase inhomogeneous incompressible viscous flow without surface tension when gravity is not taken into account, and proved the local wellposedness in general domains including the above-mentioned Ω ± (t). If Ω ± (t) are assumed to be layer-like domains, then it is known that the global well-posedness holds when ρ − > ρ + . In fact, it was shown in [11] in a horizontally periodic setting, and also we refer to [12].

Notation and main results
2.1. Notation. First, we introduce function spaces. Let X be a Banach space. Then X m , m ≥ 2, stands for the m-product space of X, while the norm of X m is usually denoted by · X instead of · X m for the sake of simplicity. For another Banach space Y , we set u X∩Y = u X + u Y . Let N be the set of all natural numbers and N 0 = N ∪ {0}. Let p ≥ 1 or p = ∞. For an open set G ⊂ R M , M ≥ 1, the Lebesgue spaces on G are denoted by L p (G) with norm · Lp(G) , while the Sobolev spaces on G are denoted by H n p (G), n ∈ N, with norm · H n p (G) . Let H 0 p (G) = L p (G). In addition, C ∞ 0 (G) is the set of all functions in C ∞ (G) whose supports are compact and contained in G, and C ∞ (I, X) is the set of all C ∞ functions on an interval I ⊂ R with value X. For any multi-index Let us define a solenoidal space by Furthermore, we set Next, we define the partial Fourier transform with respect to Its inverse transform is also defined by Finally, we introduce some constants and symbols. Let where λ 0 (ε) is given in (1.4). In addition, we set The integral path Γ 0 is defined by Main results. We first introduce the existence of solution operators for (1.1). This immediately follows from the resolvent estimate (1.4) and the standard theory of operator semigroups.
. Let us define projections P 1 , P 2 by One now decomposes the solution (u, p) of (1.2) into a solution of parabolic system and a solution of hyperbolic-parabolic coupled system as follows: It then holds that u = u P + u H and p = p P + p H . In addition, we set which implies For the parabolic part, we have the following theorem by the resolvent estimate (1.5) and the standard theory of operator semigroups.
(3) Let j ∈ N 0 and k = 0, 1, 2. For any t > 0 and f ∈ J r (Ṙ N ), where r ≤ s ≤ ∞ when k = 0, 1 and r ≤ s < ∞ when k = 2. Here C is a positive constant independent of t and f .
To complete time-decay estimates for H(t) and U(t), we further decompose (η, u H , p H ) satisfying (2.6) as follows: for z 1 = d and It then holds that η = η 1 + η 2 , u H = u 1 + u 2 , and p H = p 1 + p 2 . Let ϕ = ϕ(ξ ′ ) be a function in C ∞ (R N −1 ) and satisfy 0 ≤ ϕ ≤ 1 with In addition, we set Together with these cut-off functions, we define for Z ∈ {A 0 , A ∞ , [A 0 , A ∞ ]} and for an integral path Γ Furthermore, we set for S ∈ {H, U} and Γ 0 given by (2.3) Noting (2.7), we see that the formulas in (2.4) satisfy The following two theorems are our main results of this paper. The first one is time-decay estimates for the low frequency part. Theorem 2.3. Let 1 ≤ p < 2 ≤ q ≤ ∞ and suppose that ρ − > ρ + > 0. Then there exists a constant A 0 ∈ (0, 1) such that the following assertions hold.
with some positive constant C independent of t, d, and f .
Then for any t ≥ 1 with some positive constant C independent of t, d, and f . Remark 2.4. Time-decay estimates for higher-order derivatives of the low frequency part will be discussed in a forthcoming paper.
The second one is time-decay estimates for the high frequency part.

Representation formulas for solutions
This section introduces the representation formulas for solutions of (1.1). In this section, we assume that ρ ± are any positive constants except for Lemma 3.4 (2) below. Here we collect several symbols appearing in the representation formulas.
where we have chosen a branch cut along the negative real axis and a branch of the square root so that ℜ √ z > 0 for z ∈ C \ (−∞, 0]. In addition, 2) and also Except for D ± and E, the above symbols are introduced in [ Lemma 3.1. Let ε ∈ (0, π/2) and ρ ± be any positive constants. Then the following assertions hold.
where C is a positive constant depending on ε, s, and α ′ , but independent of ξ ′ and λ.
From Lemma 3.1 and the Bell formula of derivatives of composite functions, we have Lemma 3.2. Let ε ∈ (0, π/2) and ρ ± be any positive constants.
where C is a positive constant depending on ε, s, and α ′ , but independent of ξ ′ and λ.
Our aim is to prove Proposition 3.3. Let ε ∈ (0, π/2) and ρ ± be any positive constants. Suppose that (u P , p P ) is a solution to (2.5) for some λ ∈ Σ ε and f = (f 1 , . . . , f N ) T ∈ C ∞ 0 (Ṙ N ) N . Then there holds Here u P N (ξ ′ , x N , λ) stands for the partial Fourier transform of the N th component of u P and with constants C a,b j,α ′ ,k,l,m and C a,b j,α ′ ,k,l,m independent of ξ ′ and λ. Proof. Let j = 1, . . . , N − 1 and J = 1, . . . , N in this proof. Although we follow calculations of [10, Section 3], we will achieve a set of equations simpler than [10, (3.6)] in what follows, see (3.24) below.
Step 1. We compute w N (ξ ′ , 0 −, λ) for the following resolvent problem: where g = (g 1 , . . . , g N ) T and h = (h 1 , . . . , h N ) T are suitable functions on R N −1 specified in Step 2 below. The restriction of w and r to R N ± are denoted by w ± and r ± , respectively. Let us write the Jth component of w ± by w J± and observe Div(µD(w ± )) = µ ± ∆w ± by div w ± = 0 in R N ± . Then (3.4) can be written as Applying the partial Fourier transform to the last system yields where g J = g J (ξ ′ ) and h J = h J (ξ ′ ). Note that (3.5) and (3.6) are respectively equivalent to From now on, we look for w J± (x N ) and r ± (x N ) of the forms: for ±x N > 0, Inserting these formulas into (3.11), (3.12), and (3.7)-(3.10) furnishes Let us solve the equations (3.14)-(3.18). By (3.15), we have By the first equation of (3.19) and the second equation of (3.14), Multiplying (3.16) by iξ j and summing the resultant formulas yield Combining this equation with the second one of (3.19), furnishes (3.23) We have thus achieved Then the inverse matrix L −1 of L is given by where F (A, λ) is defined in (3.3) and Step 2. We compute the formula of u P be solutions to the following whole space problems without interface: In addition, [[φ]] = 0 as discussed in the appendix below. Thus (u P , p P ) are given by u P = ψ + v and We now see that by (3.27 Combing this property with the above formula of v N (ξ ′ , 0 −, λ) and (A.1) in the appendix below yields the desired formula of u P N (ξ ′ , 0, λ). This completes the proof of Proposition 3.3.

3.2.
Solution formulas for the hyperbolic-parabolic part. In this subsection, we introduce solution formulas of (η k , u k , p k ), k = 1, 2, for (2.8). System (2.8) can be written as In what follows, we apply the argumentation in Step 1 for the proof of Proposition 3.3 in the previous subsection. To this end, we set g = (0, . . . , 0, (ω − σ∆ ′ )η k ) T and h = 0 in (3.4). Then G(g, h) and H(g, h) in (3.23) are given by Combining this relation with (3.26) and (3.30) furnishes and thus (3.13) gives Inserting this formula into (3.29) yields Solving this equation, we obtain where we have used . At this point, we note the following lemma.
(2) The proof is similar to [7,Lemma 3.2], so that the detailed proof may be omitted.
3.3. Representation formulas for (1.1). In this subsection, we give the representation formulas of solutions for (1.1). To this end, we first consider H 1 (2.10). Together with Proposition 3.3, inserting η 1 and η 2 of (3.33) into H 1 Z (t; Γ)d and H 2 Z (t; Γ)f in (2.9), respectively, yields Let us define for ±x N > 0 and m = 1, . . . , N Together with Proposition 3.3, inserting u 1 m± and u 2 m± of (3.39) into U 1 Z,m± (t; Γ)d and U 2 Z,m± (t; Γ)f , respectively, yields the following formulas: for ±x N > 0 and m = 1, . . . , N and furthermore, One defines , and then there holds the following relation between these formulas and U 1 Summing up the above argumentation, we have obtained the representation for- give the representation formulas of solutions for (1.1) by the relation (2.11).
Finally, we introduce another useful formula of L(A, λ).
Lemma 3.5. Let L(A, λ) be given in (3.33) and set where D ± and α are defined in (3.2) and (2.2), respectively. Then Proof. The desired relation follows from an elementary calculation, so that the detailed proof may be omitted.

Analysis of boundary symbol
We assume ρ − > ρ + > 0 throughout this section and analyze mainly the boundary symbol L A (λ) introduced in the last part of the previous section. Note that α in (2.2) is positive by the assumption ρ − > ρ + > 0. 4.1. Analysis of low frequency part. Recalling θ 2 , λ 1 given in (2.1) and and also In addition, we set where ζ ± and L A (A) are given in Theorem 2.3 and Lemma 3.5, respectively. Then and the following lemma holds.
, with a sufficiently small A 1 and a positive constant C independent of A and λ.
It therefore holds that |F A (λ)| > |G A (λ)| for A ∈ (0, A 1 ) when A 1 is sufficiently small. This completes the proof of Lemma 4.1.
By Lemma 4.1 and Rouché's theorem, we immediately have Then L A (λ) has two zeros in K.
Recalling Γ ± res given in Theorem 2.3, we prove Proof. We consider λ ∈ Γ + Res only. Let λ = ζ + + A 6/4 e is (0 ≤ s ≤ 2π). It is clear that , with a sufficiently small A 2 and a positive constant C independent of A and λ.
Next, we estimate |G A (λ)| from above. It holds that Therefore, From this, recalling the definition of β given in (2.2), we have In addition, Combining these two formulas with (4.3) shows that when A 2 is sufficiently small. This completes the proof of Lemma 4.3.
We now obtain (2) Let A ∈ (0, A 3 ) and K ± be the regions enclosed by Γ ± Res , respectively. Then L A (λ) has a simple zero denoted by λ + in K + and another simple zero denoted by Remark 4.5. The zeros λ ± of L A (λ) satisfy When gravity is not taken into account, i.e. α = 0, the asymptotics of the zeros of L A (λ) are obtained in [3].
Proof of Proposition 4.4. (1) The desired properties can be proved by an elementary calculation, so that the detailed proof may be omitted.
We then have Proposition 4.6. Let a, b > 0. Then there exists a sufficiently large positive number A high = A high (a, b) such that the following assertions hold.
(1) For any A ≥ A high and λ ∈ Λ(a, b), where C 1 and C 2 are positive constants independent of A and λ.
(2) For any A ≥ A high and λ ∈ Λ(a, b), where C is a positive constant independent of A, λ, and σ.

Proof. (1) See [7, Lemma 5.3].
(2) First, we estimate |F (A, λ)| from below. Since This implies the desired inequality for |F (A, λ)|. Next, we estimate |L(A, λ)| from below. Recall the formula of L A (λ) in Lemma 3.5. The asymptotics (4.10) gives Thus, from the formula of L(A, λ) in Lemma 3.5, we see that This yields the desired inequality of |L(A, λ)|, which completes the proof of Proposition 4.6.

Next, we consider
where C 1 and C 2 are positive constants independent of A and λ.  Λ(a 1 , b). Thus there exists an a 0 ∈ (0, a 1 ] such that which implies the desired inequality of |F (A, λ)| holds. Analogously, the inequality for |L(A, λ)| follows from Lemma 3.4 (2). This completes the proof of Proposition 4.7.

Time-decay estimates for low frequency part
This section proves Theorem 2.3. Suppose ρ − > ρ + > 0 throughout this section. Let us denote the points of intersection between λ = se ±i(3π/4) (s ≥ 0) and Γ ± 0 given in (2.3) by z ± 3 . Then we define where S ∈ {H, U}. Here we have used Γ ± 1 in (4.2) and the symbols S 1 A0 (t; Γ a j )d, S 2 A0 (t; Γ a j )f introduced in Subsection 3.3. At this point, we introduce several lemmas used in the following argumentation. From [7, Lemmas 4.3 and 4.4], we have the following two lemmas.
(2) For f ∈ L p (0, ∞) and τ > 0, set Then there exists a positive constant C, independent of f , such that for any τ > 0 provided that b 2 r > 1.
Next, we introduce time-decay estimates arising in the study of an evolution equation with the fractional Laplacian. Lemma 5.3. Let 1 ≤ p ≤ q ≤ ∞, θ > 0, and ν > 0. Then the following assertions hold.
(1) For any τ > 0 and ϕ ∈ L p (R N −1 ) with a positive constant C independent of τ and ϕ. (2) If it is assumed that 1 ≤ p ≤ 2 additionally, then for any τ > 0 and ϕ ∈ L p (R N −1 ) (2) The desired estimate follows from (1) and Parseval's identity immediately, so that the detailed proof may be omitted.
Let L p (R n , X) be the X-valued Lebesgue spaces on R n , n ∈ N, for 1 ≤ p ≤ ∞. The following lemma is proved in [9, Theorem 2.3].
Lemma 5.4. Let X be a Banach space and · X its norm. Suppose that L and n be a non-negative integer and positive integer, respectively. Let 0 < σ ≤ 1 and s = L + σ − n. Let f (ξ) be a C ∞ -function on R n \ {0} with value X and satisfy the following two conditions: (1) ∂ γ ξ f ∈ L 1 (R n , X) for any multi-index γ ∈ N n 0 with |γ| ≤ L. (2) For any multi-index γ ∈ N n 0 , there exists a positive constant M γ such that . Then there exists a positive constant C n,s such that

5.1.
Analysis for Γ ± Res . In this subsection, we prove Theorem 5.5. Let 1 ≤ p < 2 ≤ q ≤ ∞ and t = t + 1. Then there exists a constant A 0 ∈ (0, A 3 ) such that for any t > 0 and (d, where C is a positive constant independent of t, d, and f .
Recalling A = |ξ ′ | and λ ± given in Proposition 4.4, we define Then we immediately obtain Lemma 5.6. There exists a constant A 4 ∈ (0, A 3 ) such that for any A ∈ (0, A 4 ) with positive constants C 1 and C 2 independent of ξ ′ , and also with a positive constant C independent of ξ ′ , where a, b ∈ {+, −} and j, m = 1, . . . , N .
Let A ∈ (0, A 4 ). Then, by Lemma 5.6, we have the following estimates for the symbols of the representation formulas given in Subsection 3.3: for the height function, for the fluid velocity, To prove Theorem 5.5, we introduce some technical lemma. Let us define the following operators: 4) and also for ±x N > 0 Here it is assumed that the symbols are infinitely many times differentiable with respect to ξ ′ ∈ R N −1 \ {0} and holomorphic with respect to λ ∈ C \ (−∞, −z 0 |ξ ′ | 2 ]. Then we have and that there exists a constant A 5 ∈ (0, A 3 ) such that for any A ∈ (0, A 5 ) with some positive constant C independent of ξ ′ . Then there exists a constant A 0 ∈ (0, A 5 ) such that the following assertions hold.
(1) For any t > 0 and (d, , with some positive constant C independent of t, d, and f . Case 1: K A0 (t; Γ + Res ). By the residue theorem, we have Recalling ℜζ ± = − √ 2α 1/4 βA 5/4 , we write this formula as Combining this formula with Lemma 5.3 yields We choose a sufficiently small A 0 so that and thus we have by Parseval's identity, Proposition 4.4, and the assumption for k(ξ ′ , λ + ) Since 0 ≤ ϕ A0 ≤ 1, this implies Applying Lemma 5.3 to the right-hand side of the last inequality furnishes the desired estimate for K A0 (t; Γ + Res ). This completes the proof of Case 1. Case 2: K +,+ A0,M (t; Γ + Res ). In the same way as we have obtained (5.6), we obtain We choose a sufficiently small A 0 so that for positive constant C and c. Then which, combined with Lemmas 5.1 and 5.3, implies Since 1 ≤ p < 2, applying Lemma 5.2 to the right-hand side of the last inequality shows that the desired estimate for K +,+ A0,M (t; Γ + Res ) holds. This completes the proof of Case 2.
Case 3: L + A0,M (t; Γ + Res ). In the same way as we have obtained (5.6), we obtain By (5.7), we see that When q = 2, it follows from (5.8) that . Combining this with Lemma 5.3 yields the desired estimate of L + A0,M (t; Γ + Res ) for q = 2. When q > 2, it follows from (5.8) and Lemmas 5.1 and 5.3 that In this inequality, taking L q norm of both sides with respect to x N ∈ (0, ∞) furnishes the desired estimate of L + A0,M (t; Γ + Res ) for q > 2. This completes the proof of Case 3.

5.2.
Analysis for Γ ± 1 . In this subsection, we prove Theorem 5.8. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞ and t = t + 1. Then there exists a constant A 0 ∈ (0, A 3 ) such that for any t > 0 and (d, where C is a positive constant independent of t, d, and f .
To prove Theorem 5.8, we start with the following lemma.
Proof. See [7, Lemma 4.9] for B ± . Then the desired estimates for Φ a,b j , Ψ a,b j , I m± , and J m follow from the estimates of B ± immediately.
Note that |L A (λ)| ≥ CA for λ ∈ Γ + 1 ∪ Γ − 1 when A is small enough as seen in Case 1 of the proof of Lemma 4.1, and thus it follows from Lemma 5.9 that |L(A, λ)| ≥ CA 2 . By this inequality and Lemma 5.9, we have the following estimates for the symbols of the representation formulas given in Subsection 3.3: for the height function, (5.12) for the velocity Lemma 5.10. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞, t = t+1, and a, b ∈ {+, −}. Suppose that there exists a constant A 7 ∈ (0, A 3 ) such that for any A ∈ (0, A 7 ) and λ ∈ Γ with some positive constant C independent of ξ ′ and λ. Then there exists a constant A 0 ∈ (0, A 7 ) such that the following assertions hold.
(1) For any t > 0 and (d, , with some positive constant C independent of t, d, and f .
, with some positive constant C independent of t, d, and f .
(1) For any t > 0 and (d, , with some positive constant C independent of t, d, and f . are proved in [7,Lemma 4.13], and K a,b A0,B (t; Γ ± 4 ) can be proved similarly to the case of L ± A0,B (t; Γ).
It thus holds that by Lemma 5.3, Parseval's identity, and the assumption for k(ξ ′ , λ) We choose a sufficiently small A 0 ∈ (0, 1) so that e z0A 2 t /8 e (ℜλ)t ≤ Ce −z0A 2 t /8 e −cs t on supp ϕ A0 for positive constants C and c. Then ds.
Theorem 5.13. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞. Then there exist constants A 0 ∈ (0, A 3 ) and c 0 > 0 such that for any t ≥ 1 and (d, f ) where C is a positive constant independent of t, d, and f .

Time-decay estimates for high frequency part
This section proves Theorem 2.5. Suppose ρ − > ρ + > 0 throughout this section. Let us denote the points of intersection between λ = −1 + is (s ∈ R) and Γ ± 0 given in (2.3) by z ± 4 , and let A 0 be the positive constant given in Theorem 2.3. We define A ∞ = A high (1, ℑz + 4 ) for the positive constant A high given in Proposition 4.6. In addition, we set M 1 = A 0 /2 and M 2 = 3A ∞ in Proposition 4.7. Then we have a 0 ∈ (0, 1) from Proposition 4.7 and denote the points of intersection between λ = −a 0 + is (s ∈ R) and Γ ± 0 by z ± 5 . Note that ℑz − 5 = −ℑz + 5 . Now we define integral paths Γ 6 and Γ 7 as follows: Then Propositions 4.6 and 4.7 yield the following lemma.
where C and c are positive constants independent of t, d, and f . Theorem 6.2 yields Theorem 2.5 immediately. This completes the proof of Theorem 2.5. A.
Applying ∂ N to these formulas yields
Proof. The first and third formulas follow from the residue theorem. Differentiating the first formula with respect to a, we have the second formula. Analogously the fourth and fifth formulas follow from the third formula. This completes the proof of Lemma A.1.
Together with the above formulas of ψ k± , ψ N ± , ∂ N ψ k± , and ∂ N ψ N ± , we obtain by Lemmas A.1 and A.2 This completes the proof of the appendix.