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On the ℛ-boundedness for the two phase problem: compressible-incompressible model problem
Boundary Value Problems volume 2014, Article number: 141 (2014)
Abstract
The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal - regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).
MSC: 35Q35, 76T10.
1 Introduction
This paper is concerned with the evolution of compressible and incompressible viscous fluids separated by a sharp interface. Typical examples of the physical interpretation of our problem are the evolution of a bubble in an incompressible fluid flow, or a drop in a volume of gas. The problem is formulated as follows: Let be two domains. The region is occupied by a compressible barotropic viscous fluid and the region by an incompressible viscous fluid. Let and be the boundaries of such that . We assume that and . We may assume that one of is an empty set, or that both of are empty sets. Let , , and be the time evolutions of , , and , respectively, where t is the time variable. We assume that the two fluids are immiscible, so that for any . Moreover, we assume that no phase transitions occur and we do not consider the surface tension at the interface and the free boundary for mathematical simplicity. Thus, the motion of the fluids is governed by the following system of equations:
for , subject to the initial conditions
Here, , is a positive constant denoting the mass density of the reference domain , P a pressure function, and (), and are unknown velocities, scalar mass density and scalar pressure, respectively. Moreover, are stress tensors defined by
where denotes the doubled strain tensor whose components are with and we set and for any vector of functions . And also, for any matrix field K with components , the quantity DivK is an N-vector with components . Finally, I stands for the identity matrix, the unit normal to pointed from to , the unit outward normal to , and and are first and second viscosity coefficients, respectively, which are assumed to be constant and satisfy the condition
and and are defined by
Aside from the dynamical system (1.1), further kinematic conditions on and are satisfied, which give
Here, is the solution to the Cauchy problem:
with for and for . This expresses the fact that the interface and the free boundary consist of the same particles for all , which do not leave them and are not incident from . In particular, we exclude the mass transportation through the interface , because we assume that the two fluids are immiscible.
Denisova [1] studied a local in time unique existence theorem to problem (1.1) with surface tension on under the assumption that and with some positive constant and that is bounded and . Here, is a positive constant describing the mass density of the reference body . Thus, in [1], both of are empty sets. The purpose of our study is to prove local in time unique existence theorem in a general uniform domain under the assumption (1.4). Especially, the assumption on the viscosity coefficients is improved compared with Denisova [1] and widely accepted in the study of fluid dynamics.
As related topics about the two phase problem for the viscous fluid flows, the incompressible-incompressible case has been studied by [2]–[11] and the compressible-compressible case by [12], [13] as far as the authors know.
To prove a local in time existence theorem for (1.1), we transform (1.1) to the equations in fixed domains by using the Lagrange transform (cf. Denisova [1]), so that the key step is to prove the maximal regularity for the linearized problem
for any , subject to the initial conditions (1.2), where for . Here, is a positive constant and () are functions defined on such that
for with some positive constants and and with some exponent , and is a positive number describing the mass density of the flow occupied in . Our strategy of obtaining the maximal - result for (1.6) is to show the existence of ℛ-bounded solution operator to the corresponding generalized resolvent problem:
Here, denotes the Laplace transform of f with respect to t. In fact, solutions and are represented by
so that roughly speaking, we can represent the solutions to the non-stationary problem (1.6) by
with Laplace inverse transform . Thus, we get the maximal - regularity result:
for some positive constants γ and C with help of the Weis operator valued Fourier multiplier theorem [14]. To construct an ℛ-bounded solution operator to (1.7), problem (1.7) is reduced locally to the model problems in a neighborhood of an interface point as well as an interior point or a boundary point by using the localization technique and the partition of unity. The model problems for the interior point and boundary point have been studied, but the model problem for the interface point was studied only by Denisova [1] under some restriction on the viscosity coefficients. Moreover, she studied the problem in framework, so that the Plancherel formula is applicable. But our final goal is to treat the nonlinear problem (1.1) under (1.4) and (1.5) in the maximal - regularity class, so that we need different ideas. Especially, the core of our approach is to construct an ℛ-bounded solution operator to (1.7). Thus, we construct the ℛ-bounded solution operator to (1.7) for the model problem in this paper, and in the forthcoming paper [15] we construct an ℛ-bounded solution operator to (1.7) in a domain. Moreover, in [15] the maximal - regularity in a domain is derived automatically with the help of the Weis’ operator valued Fourier multiplier theorem, so that a local in time unique existence theorem is proved by using the usual contraction mapping principle based on the maximal - regularity.
Now we formulate our problem studied in this paper and state the main results. Let , , and be the upper half-space, lower half-space and their boundary defined by
In this paper, we consider the following model problem:
Throughout the paper, , , , and are fixed positive constants and the condition (1.4) holds. Substituting the relation into the equations in (1.8), we have
Thus, and being renamed and h, respectively, and defining by
mainly we consider the following problem:
Here, δ is not only but also chosen as some complex number. More precisely, we consider the following three cases for δ and λ:
(C1), .
(C2) with , with and .
(C3) with , with and .
Here, with , and
We define by
The case (C1) is used to prove the existence of ℛ-bounded solution operator to (1.8) and the cases (C2) and (C3) are used for some homotopic argument in proving the exponential stability of analytic semigroup in a bounded domain. Such homotopic argument already appeared in [16] and [17] in the non-slip condition case. In (C2), we note that when with .
In case (C1), . On the other hand, in cases of (C2) and (C3), we assume that for some . Thus, we assume that
We may include the case where in (1.9), which is corresponding to the Lamé system. We may also consider the case where in (1.8) under the condition that and with some . In fact, first we solve the equation in , which transfers the problem to the case where (cf. Shibata [18], Section 3]). Thus, we only consider the case where in this paper for the sake of simplicity.
Before stating our main results, we introduce several symbols and functional spaces used throughout the paper. For the differentiations of scalar f and N-vector , we use the following symbols:
For any Banach space X with norm , denotes the d-product space of X, while its norm is denoted by instead of for the sake of simplicity. For any domain D, , and denote the usual Lebesgue space and Sobolev space, while and denote their norms, respectively. We set . For any two Banach spaces X and Y, denotes the set of all bounded linear operators from X into Y. denotes the set of all X-valued holomorphic functions defined on U. The letter C denotes generic constants and the constant depends on . The values of constants C and may change from line to line. ℕ and ℂ denote the set of all natural numbers and complex numbers, respectively, and we set . For any multi-index , we set .
We introduce the definition of ℛ-boundedness.
Definition 1.1
A family of operators is called ℛ-bounded on , if there exist constants and such that for any , , and sequences of independent, symmetric, -valued random variables on we have the inequality
The smallest such C is called ℛ-bound of , which is denoted by .
The following theorem is our main result in this paper.
Theorem 1.2
Let, and. Letbe the sets defined in (1.12). Letandbe the sets defined by
Then there exist operator families
such thatandsolve problem (1.10) uniquely for anyand, where.
Moreover, there exists a constant C depending on ϵ, q, and N such that
withand, whereis an operator defined by.
Setting in (1.8), we have the following theorem concerning problem (1.8) immediately with the help of Theorem 1.2.
Theorem 1.3
Let, and. Letbe the sets defined in (1.12). Set
Then there exist operator families
such that for anyand,
solve problem (1.8) uniquely, where.
Moreover, there exists a constant C depending on ϵ, , q, and N such that
2 Solution formulas for the model problem
To prove Theorem 1.2, first we consider problem (1.10) with in this section as a model problem, that is, we consider the following equations:
Let denote the partial Fourier transform with respect to the tangential variable with defined by . Using the formulas
and applying the partial Fourier transform to (2.1), we transfer problem (2.1) to the ordinary differential equations
subject to the boundary conditions
where and for . Here and in the following, j and J run from 1 through and N, respectively. Applying the divergence to the first and second equations in (2.1), we have in and in , so that
Thus, the characteristic roots of (2.2) are
To state our solution formulas of problem: (2.2)-(2.3), we introduce some classes of multipliers.
Definition 2.1
Let s be a real number and let be the set defined in (1.12). Set
Let be a function defined on .
-
(1)
is called a multiplier of order s with type 1 if for any multi-index and there exists a constant depending on , , ϵ, , , , and () such that we have the estimates
(2.5)
-
(2)
is called a multiplier of order s with type 2 if for any multi-index and there exists a constant depending on , , ϵ, , , , and () such that we have the estimates
(2.6)
Let be the set of all multipliers of order s with type i ().
Obviously, are vector spaces on ℂ. Moreover, by the fact and the Leibniz rule, we have the following lemma immediately.
Lemma 2.2
Let, be two real numbers. Then the following three assertions hold.
-
(1)
Given (), we have .
-
(2)
Given (), we have .
-
(3)
Given (), we have .
Remark 2.3
We see easily that (), , and . Especially, . Moreover, for any .
In this section we show the following solution formulas for problem (2.2)-(2.3):
with
Here and in the following, denote the Stokes kernels defined by
From now on, we prove (2.7). We find solutions to problem (2.2)-(2.3) of the forms
Using the symbols , we write (2.2) as follows:
Substituting the formulas of in (2.10) and (2.11) and equating the coefficients of , , and , we have
First, we represent , and by and . Namely, it follows from (2.12) that
Substituting the relations
into (2.3), we have
Using (2.14) and (2.13), we have
with
As is seen in Section 4, we have
Noting the relation , and setting
we have
with
By Lemma 2.2 and (2.16), we see that
The most important fact of this paper is that for any and
This fact is proved in Section 5, which is the highlight of this paper. Since
we have
Writing and using the relations , by (2.21), we have
with
for . By Lemma 2.2, (2.16), (2.19), and (2.20), we have
By (2.13) we have
so that setting and , we have the formula of in (2.7).
By (2.12), we have
Since as follows from (2.13), by (2.22) we have
for . By (2.24) we have
Since , setting
for and , we have and in (2.7). As is seen in Section 4 below, we have
which, combined with (2.23), furnishes , , and .
Analogously, in view of (2.24) we set
for , and , we have in (2.7). By (2.23) and (2.25), we have and .
Using (2.21), we represent by
with
for . By Lemma 2.2, (2.16), (2.19), and (2.20), we have
In particular, noting that and setting , (), , and , we have the and in (2.7), and by (2.27) , , , and for .
From (2.14) it follows that
Noting that , we have
which, combined with (2.24) and (2.26), furnishes
Thus, we set
so that we have the and in (2.7). Moreover, as is seen in Section 4, we have
so that by (2.23), (2.25), (2.27), and (2.29) we have , , , and . This completes the proof of (2.7).
To construct our solution operator from the solution formulas in (2.7), first of all we observe that the following formulas due to Volevich hold:
where . Using the identity , we write
Let denote the partial Fourier inverse transform with respect to variable and let and be corresponding variables to and . If we define by
then we have
Analogously, using the identity , we write
Let , and be the corresponding variables to λk, and . If we define by
then we have
Let us define (), () and by () and , respectively. Setting , and , by (2.7) we see that and satisfy (2.1). According to the formulas (2.30), (2.31), (2.32), and (2.33), we define our solution operators (), () and of problem (2.1) such that
as follows: Note that
where we have set . Let , , , and be the corresponding variables to , , , and , respectively. Then we define the operators , , , and by
Obviously, by (2.31) and (2.33), we have (2.34).
If we define operators by
with , respectively, by (2.34) we have
Moreover, if we set
then, using Lemma 3.1 and Lemma 3.2 in Section 3, we have
The estimates (2.38) are proved in Section 6 below.
3 Technical lemmas
To prove the ℛ-boundedness of solution operators, we use the following two lemmas. The first lemma is used to show the ℛ-boundedness of the compressible part and the second one to show that of the incompressible part.
Lemma 3.1
Letandbe multipliers belonging toand, respectively. Let () be operators defined by
Then there exists a constant C such that
Lemma 3.2
Let, , andbe multipliers belonging to, and, respectively. Let () be operators defined by
Then there exists a constant C such that
The assertions for and in Lemma 3.1 immediately follows from the following lemma.
Lemma 3.3
Letbe a multiplier belonging to. Letandbe operators defined by
Then we have
Proof
Set with and . As was seen in Shibata and Shimizu [19], Proof of Lemma 5.4], the lemma follows from the fact that
Thus, we prove (3.2). Using the following Bell formula for the derivatives of the composite function of and :
with and suitable coefficients , we have
with some positive constant c independent of . Thus, we have
To prove the estimate
using the identity , we write
Since , by (2.6) and (3.4) we have
Thus, by Theorem 2.2 due to Shibata and Shimizu [20], we have
from which we have (3.5).
On the other hand, by (2.6) with and (3.4) with , we have
Thus, using the change of variables , we have , which, combined with (3.5), furnishes (3.2). Analogously, we have , which completes the proof of Lemma 3.3. □
The assertions for and in Lemma 3.1 immediately follow from the following lemma.
Lemma 3.4
Letbe a multiplier belonging to. Letandbe operators defined by
Then we have
Proof
Set with and . As was stated in the proof of Lemma 3.3, the lemma follows from the fact that
First, we prove that
By (2.5), (3.4), and the Leibniz rule, we have
so that by Theorem 2.2 in Shibata and Shimizu [20] we have
Thus, employing the same argumentation as in the proof of Lemma 3.3, we have (3.7).
On the other hand, we have
Thus, we have (3.6) with . Analogously, we have (3.6) with , which completes the proof of Lemma 3.4. □
The assertions for in Lemma 3.2 follows from the same observation as in the proof of Lemma 3.1 for . The assertion for , and in Lemma 3.2 follows from the following lemma due to Shibata and Shimizu [19], Lemma 5.4].
Lemma 3.5
Letbe a multiplier belonging to. Let () be operators defined by
Then we have
4 Some estimates of several multipliers
In this section, we estimate several multipliers. For this purpose, we start with the following lemma.
Lemma 4.1
Let, , and.
-
(1)
For any , and , we have .
-
(2)
There exists a number depending on s, , , , , , and ϵ such that
-
(3)
There exist constants and depending on s, , , , , , and ϵ such that
for any.
Remark 4.2
Lemma 4.1 was proved in Götz and Shibata [21], Lemma 3.1], so that we may omit its proof.
First we estimate , , and . For this purpose, we use the estimates
for any with some positive constants c and , which immediately follows from Lemma 4.1. Here and in the following, c and denote some positive constants essentially depending on , , , , , ϵ, and . In particular, by (4.1) we have
for any . As was shown in Enomoto and Shibata [17], Lemma 4.3], using (4.1), (4.2), and the Bell formula (3.3), we see that
Especially, we have (2.29).
Second, we estimate . For this purpose, we write
By Lemma 4.1, (3.3), and (4.3) we have
so that by Lemma 2.2 we have
Since , by (4.4) and (4.6), we have , which, combined with (4.3), furnishes (2.25).
Applying (4.4) to the formula in (2.15), we have
Noting that , by Lemma 2.2, (2.15), (4.3), (4.6), and (4.7), we have , , , and . In addition, since and , by Lemma 2.2 we have and . Summing up, we have proved (2.16).
5 Analysis of Lopatinski determinant
In this section, we show the following lemma, which implies (2.20).
Lemma 5.1
Let L be the matrix defined in (2.17). Then there exists a positive constant ω depending solely on, , ϵ, , , , , andsuch that
for any.
Moreover, we have
for any multi-indexand. Namely, .
Proof
Recalling (1.13) and setting , we have and . Moreover, by Lemma 4.1
with . To prove (5.1), first we consider the case with large . Let P be the function defined in (4.4). By (4.4) we see easily that , that , and that when and with very small positive number . Thus, by (4.7) we have
On the other hand, we have , so that by (2.15) we have
Summing up, we have
so that we have
Since as , we have
with some large number depending on and . On the other hand, when and , we write
Since and and , by Lemma 4.1(1)
On the other hand, , so that we have
provided that with some constant depending on , , , , , ϵ, and , which furnishes (5.1) when with any constant ω satisfying
Secondly, we consider the case with large . In this case, we have
when and with some very small positive number . By (2.15)
Thus, we have
provided that with some constant depending on , , , , , ϵ, , and , which shows that (5.1) holds when with any constant ω satisfying
Thirdly, we consider the case . Set
If satisfies the condition , then . We define by replacing , A, and by , , and in (2.15), respectively. Setting , we have
First, we prove that provided that , and with some small by contradiction. Suppose that . By (5.6) , so that in view of (2.18) there exist and with such that , and and satisfy (2.2) and (2.3) with , , and , that is, they satisfy the following homogeneous equations:
Set , , and . Multiplying the equations in (5.7) by and using integration by parts and the jump conditions in (5.7), we have
Taking the real part and the imaginary part in (5.8), using the inequality
and setting and for short, we have
First, we consider the case . When or and , we have , that is, . When and , it follows from that . Choosing in such a way that and , by (5.8) with and and (5.9) we have
which furnishes () and . Since as (), we have , which contradicts . Thus, we have when , which implies that
Since , there exists a such that
which, combined with (5.6), implies that
with some positive number provide that and with .
Finally, we consider the case where . First, we consider the case (C1), that is, . In this case, it follows from that , so that we prove that directly provided that . Since and , by (5.10) we have
When , we have , so that , that is, . If , by (5.12) , which, inserted into the second formula in (5.12), furnishes
Since when , we have , which implies that , that is, . Summing up, we have obtained , which contradicts , and therefore we have when . Thus, we have
which, combined with (5.11), furnishes
with some positive constant provided that and .
Secondly, we consider the case where , , , and . Note that this case includes (C2) and . We prove that provided that and by contradiction. Suppose that , and then by (5.6), . Thus, we have (5.10). When and , we have , so that . When , we have . When , by (5.8) with and , we have and . Since as , we have . Thus, we have when and . When , and , we have by the first formula of (5.10). When , and , we have by the first formula of (5.10), so that it follows from the second formula of (5.10) that . Since and since , as follows from , we have , which furnishes . Thus, . Summing up, we have proved that , that is, . But this contradicts , and therefore . In particular,
which, combined with (5.11) and (5.6), implies that
with some positive constant provided that and , , and .
Analogously, we have
with some positive constant provided that and , , and . Therefore, we have proved (5.1).
Since
by (2.19), the Leibniz rule, the Bell formula (3.3) with , , and (5.1), we have
which shows (5.2) with . Analogously, we have (5.2) with , which completes the proof of Lemma 5.1. □
6 Proofs of main results
In this section, we prove Theorem 1.2. For this purpose, first of all we prove (2.38). For the multipliers appearing in of (2.36), by Lemma 2.2, (2.8), and (4.3) we have
so that by Lemma 3.1 with and and Lemma 3.2 with and we have
For the multipliers appearing in of (2.36), by Lemma 2.2, (2.8), and (4.3) we have
so that by Lemma 3.1 with and Lemma 3.2 with we have
For the multipliers appearing in of (2.36), by Lemma 2.2, (2.8), and (4.3) we have
so that by Lemma 3.1 with and Lemma 3.2 with we have
For the multipliers appearing in of (2.36), by Lemma 2.2, (2.8), and (4.3) we have
so that by Lemma 3.1 with and we have
Finally, for the multipliers appearing in of (2.36), by Lemma 2.2, (2.8), and (4.3) we have
so that by Lemma 3.5 with we have
Summing up, we have proved (2.38).
To transfer the problem (1.10) to (2.1), we use the solutions and to the following equations:
respectively. We know the following two theorems. The first theorem is due to Götz and Shibata [21], Theorem 2.5] and the second one is due to Shibata [18], Theorem 3.4].
Theorem 6.1
Let, , and. Letbe the set defined in (1.12). Then there exists an operator familysuch that for anyand, is a unique solution to problem (6.1), andsatisfies the following estimates:
with some constant C depending on ϵ, , , , , , , , q and N.
Theorem 6.2
Letand. Then there exist operator families
such that for anyand, andare unique solutions to problem (6.2), andandsatisfy the following estimates:
with some constant C depending on ϵ, , , q and N.
The composite operator of two ℛ-bounded operators is ℛ-bounded and the sum of two ℛ-bounded operators is also ℛ-bounded. Extending the operator to and the operators and to by the PL Lions method, respectively, we see that the resulting operators also ℛ-bounded, so that combining Theorem 6.1 and Theorem 6.2 with (2.38), we have Theorem 1.2. This completes the proof of Theorem 1.2.
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Acknowledgements
Second author was partially supported by JST CREST, JSPS Grant-in-aid for Scientific Research (S)#24224004, and the JSPS Japanese-German Graduate Externship.
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YS proposed the main idea of this paper and prepared the manuscript. TK and KS performed all the steps of the proofs in this paper. All authors read and approved the final manuscript.
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Kubo, T., Shibata, Y. & Soga, K. On the ℛ-boundedness for the two phase problem: compressible-incompressible model problem. Bound Value Probl 2014, 141 (2014). https://doi.org/10.1186/s13661-014-0141-3
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Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-014-0141-3