1. Introduction
We study the asymptotic behavior of solutions to the following compressible Navier–Stokes–Korteweg system in
, called CNSK:
Here,
and
are unknown density and momentum, respectively, at time
and position
;
and
are given initial data;
and
denote the viscous stress tensor and Korteweg stress tensor, respectively, given by
where
;
and
are the viscosity coefficients, supposed to be constants satisfying
is the capillary constant satisfying
. If
in the Korteweg tensor, the usual compressible Navier–Stokes equation (CNS) appears:
is pressure assumed to be a smooth function of
satisfying
, where
is a given positive constant and
is a given constant state for (1). We consider solutions to (1) around the constant state.
(1) is the system of equations of motion of liquid–vapor type two-phase flow with phase transition in a compressible fluid, similarly as in [
1]. To describe the phase transition, this model uses the diffusive interface. Hence, the phase boundary is regarded as a narrow transition layer and change of the density prescribes fluid state. Due to the diffusive interface, it is enough to consider one set of equations and a single spatial domain and difficulty of topological change of interface do not occur. If we assume that
, the CNS that describes the motion of one-phase compressible fluid is obtained. Hence, (1) is obtained from adding higher-order derivative terms for
, including
and
to CNS.
For the derivation of (1), Van der Waals [
2] suggested that a phase-transition boundary be regarded as a thin transition zone, i.e., a diffusive interface caused by a steep gradient of density. On the basis of his idea, Korteweg [
3] modified the stress tensor of the Navier–Stokes equation to that including term
. Dunn and Serrin [
4] generalized Korteweg’s work and strictly provided System (1) with (2). In their recent works, Heida and Málek [
5] derived (1) by the entropy production method.
We focus on the diffusion wave that stems from hyperbolic and parabolic aspects of the system. The diffusion wave is given by convolution between heat kernel and the fundamental solution to the wave equation. The importance of the diffusion wave for problems in one-dimensional cases was first recognized by Liu [
6] for the study of stability of shock waves for viscous conservation laws. The multidimensional diffusion wave with a time-decay estimate of solutions was studied for CNS by Hoff and Zumbrun [
7,
8], and Kobayashi and Shibata [
9]; for the viscoelastic equation on
, by Shibata [
10]. Let
be a solution to CNS and set
, where
,
s is an integer part of
and
ℓ is integer satisfying
. Then, the authors in [
7,
8,
9] showed that the linear parts decay faster than nonlinear parts do in the Duhamel formula, and the asymptotic behavior in
of solutions was presented as
as
t goes to infinity. Here, notation
in
is defined as
for a positive number
C independent of
t, similar notation is used hereafter.
is the standard heat kernel and
is a divergence-free part of
, given by
More precisely, it holds that
and
for
, where
is the Green function of linearized CNS and
when
.
and
are the Stokes flow and potential flow parts of
M, respectively, in the Helmholtz decomposition.
and
are given by the Green matrix of the linearized system, which consists of the convolution with the Green functions of the diffusion and the wave equations and are called the diffusion-wave part. In addition, when
, the behaviors of both of
and
coincide with those of
as the parabolic-type decay rate. Kobayashi and Tsuda studied the diffusion-wave property for (1) in [
11].
In this paper, we consider the linearized system for (1). Under some initial conditions given by the Hardy space
(defined below), we show some space–time
estimates for the density and the Stokes flow part of the momentum. The potential flow part of the momentum is also shown to grow at the rate of logarithmic order in spatial-time
norm. The precise initial condition given by the Hardy space is shown below. Here, we assume a stronger initial condition by
for density than that by
, in contrast to [
11]; thus, our results may show a gain of regularity by the Hardy space in the decay estimates. Such a gain is also obtained for heat equations (see
Appendix A). Nonlinear terms are expected to decay faster than the linear terms in the Duhamel formula do, as in [
7,
11]. As a consequence, the leading terms of the asymptotic expansion of solution
u for (1) are given by
Precisely, the following estimates hold true for solutions to the linearized CNSK:
The above behaviors of the diffusion-wave parts and are clearly different from (3). Measuring by on space, decays slower than the Stokes flow part of M does. By the dependence on of constants, the above estimate (4) also holds true for CNS (Theorems 2 and 3). We also obtain a decay rate of norm of density (Theorem 4). Furthermore, if , space–time boundedness is obtained for , and .
The proofs of the main results are based on Morawetz-type energy estimates for a linearized system. The diffusion-wave part of density
is bounded in space–time
. We rewrite (1) to some linear doubly dispersion equation for
and apply a modified version of Morawetz’s energy estimate. A preliminary function is introduced in the Morawetz estimate (see (12) below), which is defined by use of a doubly Laplace-type equation. The existence of solution to the linear doubly Laplace-type equation is shown by use of the linear theory on
, which may be of its own interest. Through the preliminary function, we perform Morawetz-type energy estimates utilizing the Fefferman–Stein inequality on the duality between
and the space of functions of bounded mean oscillation. Another diffusion-wave part
is shown to grow at the rate of order
as
t goes to infinity. Here, we use fundamental solutions for the linearized system given in [
11]. Since a high-frequency part of the solutions exponentially decays, a low-frequency part only has to be estimated here. By direct computation with the explicit form of the Green matrix, we obtain the growth order for
. For the Stokes flow part
, space–time
boundedness is derived in Theorem A3 bellow. These estimates are combined for a diffusion wave, and the Stokes flow parts yields asymptotic expansion (4).
This paper is organized as follows. In
Section 2 some notations and lemmas are given. In
Section 3, the main results are presented. In
Section 4, the proofs of the estimates for the diffusion wave parts are demonstrated.
2. Preliminaries
In this section, we introduce notations such as function spaces that are used in this paper. We also present lemmas needed in the proof of the main result.
The norm on X is denoted by for a given Banach space X.
Let is the usual Lebesgue space of th powered integrable and essentially bounded functions on for a finite p and , respectively. Let k be a non-negative integer. and are the usual Sobolev spaces of order k, based on and , respectively. As usual, is defined by .
We also use notation to denote the function space of all vector fields on satisfying , and is norm for brevity if no confusion occurs. Similarly, a function space X is the linear space of all vector fields on satisfying , and is norm if no confusion occurs.
Let
with
and
. Then, norm
is defined as that of
u on
In particular, if
, we put
Let X and Y be given Banach spaces. For
with
, we similarly set
More generally, in the case that
, let
Symbols
and
stand for the Fourier transform of
f with respect to space variable
xFurthermore, the inverse Fourier transform of
f is defined by
For a non-negative number s, is the Gaussian symbol that denotes the integer part of s. Symbol denotes the convolution on space variable x.
Now, we prepare Hardy space and BMO space.
Definition 1. Hardy space consists of integrable functions on , such thatis finite, where for , and ϕ is a smooth function on with compact support in an unit ball with center of the origin, . The definition does not depend on the choice of a function ϕ. Definition 2. Let f be locally integrable in , . We say that f is of bounded mean oscillation, abbreviated as , ifwhere the supremum ranges over all finite balls B⊂, is the 2-dimensional Lebesgue measure of B, and denotes the integral mean of f over B, namely . The class of functions of , modulo constants, is a Banach space with norm defined above.
We crucially use the decisive Fefferman–Stein inequality, which means the duality between
and
, i.e.,
. For the proof, see [
12].
Lemma 1. (Fefferman-Stein inequality) There is a positive constant C, such that, if and , then We also recall the well-known Poincaré inequality.
Lemma 2. for .
We denote by the set of all vector-valued functions whose each is function having compact support, and satisfying that . For , is the closure of with respect to the norm.
A spatial weighted function space
is defined by
where
is a spatial weight defined by
.
The following Hölder type inequality was proved by Amrouche and Nguyen [
13].
Lemma 3. ([13] Corollary 2.10) Let . Then it holds true that and that, for such f and any , Since , Lemma 3 also yields the following.
Corollary 1. Let . Then, there holds that and, for such f and any , 3. Main Results
In this section, we consider the linearized system corresponding to (1) and present some decay estimates for its solution. akey estimate to show (4) is space–time
boundedness of the density for the linearized system. First, (1) is reformulated and linearized as follows. Hereafter, we assume that
without loss of generality. We also set
Substituting
and
m into (1), we have system of equations
where we use notation
and put
Therefore, (1) is linearized as
By (6),
satisfies the following doubly dissipative equation:
Due to the positivity of
and
, we may suppose that
and
without loss of generality. Then,
satisfies
Now, we state the existence of solutions to (7) in the energy class, defined in the following.
Definition 3. A function ϕ defined on is called to be a solution to (7) if ϕ belongs to with and satisfies (7) in the distribution sense, i.e., satisfies the following conditions:
(i) For each , and a.e. (ii) and .
Theorem 1. For each , there exists a unique solution with to (7) such thatholds for any . Theorem 1 is valid by the standard Galerkin method based on energy inequality (7) in a similar manner to the proof of Theorem 3.1 in Huafei and Yadong [
14]. In ([
14] Theorem 3.1), they added
in a linear system and considered nonlinearity. We apply a similar manner as [
14] to the proof without
and nonlinearity; thus, we do not obtain in Theorem 1 that
in contrast to ([
14] Theorem 3.1); we omitted the details.
In for a solution to (7).
Theorem 2. Suppose that , and . Set Let ϕ be a solution to (7). Then, it holds true thatfor any , where C is a positive constant independent of t and . In the case that , we also have the time–space estimate for linearized CNS.
Theorem 3. Let , and . Set Let ϕ be a solution to (A2) in Appendix B. Then, there holds thatfor any , where C is a positive constant independent of t. Next, we have a time-decay estimate of the solution in the energy class to (7). By Theorem 1 and the Sobolev inequality . We have the following:
Theorem 4. Under the assumption of Theorem 2, it holds thatfor any , where C is a positive constant independent of t. We now recall the existence of solutions to linear system (6) in the energy class in order to consider another diffusion-wave part
. System (6) is rewritten as
where
Let us introduce a semigroup
generated by
A;
where
Theorem 5. ([15] Proposition 3.3) Let s be a non-negative integer satisfying . Then, is a contraction semigroup for (8) on . In addition, for each and all , satisfiesand there holds the estimatefor and . Remark 1. Proposition 3.3 in [15] is stated on the three-dimensional case. However, the proof is based on the standard energy estimate for the resolvent problem in the Fourier space, and it can also be applied to our two-dimensional case. Lastly, another diffusion-wave part is shown to grow in at the rate of logarithmic order.
Theorem 6. Let and u be a solution of (6), , as in Theorem 5. Suppose that , and . Then, it holds true thatprecisely,where is a positive constant independent of t. Remark 2. In addition of the initial condition in Theorem 2, we assume that ; then, it holds thatwhere is a positive constant independent of t. This shows a gain of regularity by the membership in the Hardy space of data, similarly as in the decay estimates for density in Theorems 2 and 3. a similar phenomenon was already observed in [16,17] for dissipative wave equations. The proof is given by direct computations based on the explicit form of fundamental solution (40) below and a similar argument as in Kobayashi and Misawa [16,17]. We omitted the details here. We state the space–time
boundedness for Stokes flow part
. Indeed, from (11) together with Theorem A3 in
Appendix C and ([
18] Chapter 3, Section 3, Theorem 3), we find that if
is added in the assumption of Theorem 2, space–time
boundedness holds true for
,
and
.
5. Conclusions
We studied the asymptotic behavior of solutions to the compressible Navier–Stokes–Korteweg system in
. Concerning the linearized system for (1), under some initial conditions given by Hardy space
, we showed some space–time
estimates for the density and Stokes flow parts of the momentum. The potential flow part of the momentum was also shown to grow at the rate of logarithmic order in space–time
norm. The asymptotic behaviors in space–time
of the diffusion-wave parts were shown to be essentially different between density and the potential flow part of the momentum. Nonlinear terms are expected to decay faster than the linear terms in the Duhamel formula do, as in [
7,
11]. As a consequence, the leading terms of the asymptotic expansion of solution
u for (1) were given by
Analysis for asymptotic behavior in the two-dimensional case is difficult because the time decay of solutions to the linear system in two-dimensional cases is slower than that in higher-dimensional cases. To overcome this difficulty, we used a gain of regularity by the Hardy space; by the energy estimate of the Morawetz type, we succeeded to derive the asymptotic behavior (44).
Concerning future works, it is important to consider how the pressure term has an effect on the asymptotic behavior of (1). As shown in our paper [
19], since (1) governs the motion of two-phase fluids, pressure is a nonmonotone function. When pressure decreases, solutions are expected to be unstable due to positive eigenvalues in linear systems. Hence, we will study the asymptotic behavior of solutions with relation to a critical value, such that
holds or an initial condition of pressure. Furthermore, from the point of view of engineering, analysis of two-phase fluids in bounded or unbounded domains with boundaries is more important. On the basis of our analysis of the Caucy problem, we will study the asymptotic behavior of solutions to CNSK under boundary conditions.