Asymptotic profile for diffusion wave terms of the compressible Navier-Stokes-Korteweg system

Asymptotic profile for diffusion wave terms of solutions to the compressible Navier-Stokes-Korteweg system is studied on $R^2$. The diffusion wave with time decay estimate is studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002) and Kobayashi and Tsuda (2018) for the compressible Navier-Stokes system and the compressible Navier-Stokes-Korteweg system. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space-time $L^2$ of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by $L^2$ on space, a decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz's energy estimate, and the Fefferman-Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

κ is the capillary constant satisfying κ ≥ 0. If κ = 0 in the Korteweg tensor, the usual compressible Navier-Stokes equation (CNS) appears: P = P(ρ) is pressure assumed to be a smooth function of ρ satisfying P (ρ * ) > 0, where ρ * is a given positive constant and (ρ * , 0) 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 κ = 0, the CNS that describes the motion of one-phase compressible fluid is obtained. Hence, (1) is obtained from adding higher-order derivative terms for ρ, including ∇∆ρ and ∇ρ ⊗ ∇ρ 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 ∇ρ ⊗ ∇ρ. 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 R n (n ≥ 2), by Shibata [10]. Let u = (ρ − ρ * , M) be a solution to CNS and set E := u 0 H s+ ∩L 1 , where u 0 = (ρ 0 − ρ * , M 0 ) = (ρ(0) − ρ * , M(0)), s is an integer part of n/2 and is integer satisfying ≥ 3. 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 L p (R n ) (p > 2, n ≥ 2) of solutions was presented as as t goes to infinity. Here, notation for a positive number C independent of t, similar notation is used hereafter.
is the standard heat kernel and M 0,in is a divergence-free part of M 0 , given by More precisely, it holds that for t > 0, where G = G(t) is the Green function of linearized CNS and ( n−1 4 )( 2 p − 1) ≤ 0 when 2 ≤ p ≤ ∞. K ν (t) * M 0,in and M(t) − K ν (t) * M 0,in are the Stokes flow and potential flow parts of M, respectively, in the Helmholtz decomposition. ρ(t) − ρ * and M(t) − K ν (t) * M 0,in 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 p = 2, the behaviors of both of ρ(t) − ρ * and M(t) − K ν (t) * M 0,in coincide with those of K ν (t) * M 0,in 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 H 1 (defined below), we show some space-time L 2 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 L 2 norm. The precise initial condition given by the Hardy space is shown below. Here, we assume a stronger initial condition by H 1 for density than that by L 1 , 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 ρ(t) − ρ * and M(t) − K ν (t) * M 0,in are clearly different from (3). Measuring by L 2 on space, M(t) − K ν (t) * M 0,in decays slower than the Stokes flow part of M does. By the dependence on κ of constants, the above estimate (4) also holds true for CNS (Theorems 2 and 3). We also obtain a decay rate of L 2 norm of density (Theorem 4). Furthermore, if M 0 ∈ H 1 , space-time L 2 boundedness is obtained for M(t) − K ν (t) * M 0,in , ρ(t) − ρ * and K ν (t) * M 0,in .
The proofs of the main results are based on Morawetz-type energy estimates for a linearized system. The diffusion-wave part of density ρ(t) − ρ * is bounded in space-time L 2 . We rewrite (1) to some linear doubly dispersion equation for ρ 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 H 1 , 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 H 1 and the space of functions of bounded mean oscillation. Another diffusion-wave part M(t) − K ν (t) * M 0,in is shown to grow at the rate of order log t 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 M(t) − K ν (t) * M 0,in . For the Stokes flow part K ν (t) * M 0,in , space-time L 2 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.

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 · X for a given Banach space X. Let 1 p ∞. L p is the usual Lebesgue space of pth powered integrable and essentially bounded functions on R 2 for a finite p and p = ∞, respectively. Let k be a non-negative integer. W k,p and H k are the usual Sobolev spaces of order k, based on L 2 and L p , respectively. As usual, H 0 is defined by H 0 := L 2 .
We also use notation L p to denote the function space of all vector fields w = (w 1 , w 2 ) on R 2 satisfying w j ∈ L p (j = 1, 2), and · L p is norm · (L p ) 2 for brevity if no confusion occurs. Similarly, a function space X is the linear space of all vector fields w = (w 1 , w 2 ) on R 2 satisfying w j ∈ X (j = 1, 2), and · X is norm · X 2 if no confusion occurs. Let In particular, if j = k, we put Let X and Y be given Banach spaces.
More generally, in the case that Y = X 2 , let Symbolsf and F [ f ] stand for the Fourier transform of f with respect to space variable xf Furthermore, the inverse Fourier transform of f is defined by For a non-negative number s, [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 H 1 and BMO space.
for r > 0, and φ is a smooth function on R 2 with compact support in an unit ball with center of the origin, B 1 (0) = {x ∈ R 2 ; |x| < 1}. The definition does not depend on the choice of a function φ.
. We say that f is of bounded mean oscillation, abbreviated as BMO, if where the supremum ranges over all finite balls B ⊂ R 2 , |B| is the 2-dimensional Lebesgue measure of B, and ( f ) B denotes the integral mean of f over B, namely The class of functions of BMO, modulo constants, is a Banach space with norm · BMO defined above.
We also recall the well-known Poincaré inequality.
We denote by C ∞ 0,σ the set of all vector-valued functions φ = (φ 1 , φ 2 ) whose each φ j (j = 1, 2) is C ∞ function having compact support, and satisfying that div The following Hölder type inequality was proved by Amrouche and Nguyen [13].
. Then it holds true that R 2 f (x)dx = 0 and that, for such f and any g ∈ W 1,2 , Lemma 3 also yields the following.
. Then, there holds that R 2 f (x)dx = 0 and, for such f and any

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 L 2 boundedness of the density for the linearized system. First, (1) is reformulated and linearized as follows. Hereafter, we assume that ρ * = 1 without loss of generality. We also set Substituting φ and m into (1), we have system of equations where we use notation By (6), φ satisfies the following doubly dissipative equation: Due to the positivity of ν andν, we may suppose that ν +ν = 1 and γ = 1 without loss of generality. Then, φ satisfies Now, we state the existence of solutions to (7) in the energy class, defined in the following.
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 −∆u tt in a linear system and considered nonlinearity. We apply a similar manner as [14] to the proof without −∆u tt and nonlinearity; thus, we do not obtain in Theorem 1 that ∂ t φ ∈ L ∞ (0, ∞; H 1 ) in contrast to ([14] Theorem 3.1); we omitted the details.
Let φ be a solution to (7). Then, it holds true that for any t > 0, where C is a positive constant independent of t and κ 0 .
In the case that κ 0 = 0, we also have the time-space L 2 estimate for linearized CNS.
Let φ be a solution to (A2) in Appendix B. Then, there holds that for any t > 0, 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 φ ∈ C([0, ∞); L 2 ). We have the following: for any t > 0, 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 m − K ν * m 0,in . System (6) is rewritten as where Let us introduce a semigroup S(t) = e −tA generated by A; for u 0 = (φ 0 , m 0 ) ∈ H s × H s−1 and t ≥ 0. [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.

Remark 2.
In addition of the initial condition in Theorem 2, we assume that m 0 ∈ H 1 ; then, it holds that where C = C(u 0 ) 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 L 2 boundedness for Stokes flow part K ν (t) * M 0,in . Indeed, from (11) together with Theorem A3 in Appendix C and ( [18] Chapter 3, Section 3, Theorem 3), we find that if m 0 ∈ H 1 is added in the assumption of Theorem 2, space-time L 2 boundedness holds true for M(t) − K ν * M 0,in (t), ρ(t) − ρ * and K ν (t) * M 0,in .

Proof of Theorems 2 and 4
In this subsection, we prove Theorems 2 and 4. The proof is performed by modifying Morawetz's energy estimate. For a solution φ to (7), we define function w by where Φ is a solution to the doubly Laplace equation For the existence of a solution to (13), we have such that The proof of Theorem 7 is in Appendix D. By the definition of w, we derive To estimate we take the L 2 inner product of (17) 1 with w; thus, we have where w(0) = −div Φ. Now, we estimate the terms in the right-hand side of (18). Terms ∇w(0) 2 L 2 and (w t (0), w(0)) are directly estimated by (15). A test function w t in (17), being integrated on time interval (0, t) and using (15) and (16), yields estimate On the other hand, from taking the L 2 inner product of −κ 0 ∆w with (17) and integrating on (0, t), we obtain that and thus, Terms having Φ in (20) are estimated by (15) and (16). The fourth and fifth terms on the right-hand side of (20) are also estimated by (19). For term ∇w t (t) 2 L 2 = ∇φ(t) 2 L 2 , we apply the standard energy estimate obtained from (7) with a test function φ t For the estimation of the L 2 −norm on a space of w and the one on the time-space of its spatial gradient, let v = t 0 w(s)ds (22) and proceed to the energy estimates of v. From the direct calculation, v satisfies A test function v t in (23) gives being integrated on (0, t) and yielding Now, the first term of the right-hand side of (25) is of which the first term is evaluated by the use of Lemmas 1 and 2 and Young's inequality as and, the second term is controlled by (15) as Since w(0) = −divΦ, the right-hand side of (25) is bounded by Thus, it follows that Gathering (18), (19), (21), and (27), we obtain Theorem 2.
On the basis of Theorems 2 and 4 is proved as follows. We set a total energy of w as By the proof of Theorem 2 and integration by parts, we find that Since, by integration by parts again, we have This, together with (19), gives the assertion of Theorem 4. The proof is completed.

Proof of Theorem 6
In this section, we show the validity of Theorem 6. By taking the Fourier transform of (6) with respect to space variable x, we have the following ordinary differential equation with a parameter ξ.
If 0 < K < 1 and min 1 2 , where Γ is a closed pass surrounding λ ± and included in set {z ∈ C|Rez ≤ −c 0 }, and c 0 is a positive number satisfying max min 1 2 , Cut-off functions ϕ 1 , ϕ ∞ and ϕ M in C ∞ (R 2 ) are defined by [11] as follows: in the case such that K = 1, ϕ 1 is given by In the case that K = 1, ϕ 1 and ϕ ∞ are We define solution operators E 1 and E ∞ on a low-and high-frequency part of (6), respectively, as follows: where In [11], the solution operator is shown to have an exponential decay in time on the high-frequency part (34). In fact we have for t > 0, k ≥ 0 and |α| ≥ 0, where (δ 1 , δ 2 ) = (1/2, 1) and (1, 3/2) for K = 1 and K = 1, respectively.
Therefore, in order to show Theorem 6, it is enough to consider the low-frequency part. We estimate the Green function. We put for j = 1, ∞.
We set K 1 m 0 is a part of the Green matrix and corresponds to the diffusion-wave part m − K ν * m 0,in . Our claim is the following estimate. Proposition 1. Let m 0 ∈ L 2 ∩ L 1 , |x|m 0 ∈ L 1 andm 0 = 0. Then, it holds that lim sup Proof. By the Plancherel theorem and (31), we see that there exists a positive constant C, such that Hence, we have to estimate e −|ξ| 2 tm 0 (ξ) 2 L 2 (|ξ|≤c 1 ) . It follows from the polar coordinates that where r = |ξ|, ω = ξ/|ξ|, and C ω is some positive constant that appears in the polar coordinates. Changing variables r √ t = s, we have This, together with the fundamental theorem of calculus form 0 , implies that for a positive constant C 2 = C 2 (u 0 ) independent of t.m 0 (0) = 0 by our assumption. Since we estimate t 1 e −|ξ| 2 τm 0 (ξ) 2 L 2 dτ. We set Applying (41) yields that, for t ≥ 1, Let positive constants Q 1 and Q 2 be defined by It obviously follows that, for t ≥ 1, This implies that, for t ≥ 1, and thus, (42) and (43) yield that Therefore, there exists a positive constant Since other parts of diffusion wave m − K ν * m 0,in that appear in the Green matrix on the low-frequency part are estimated similarly to Proposition 1, we obtain the estimation in Theorem 6. The proof is completed.

Conclusions
We studied the asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg system in R 2 . Concerning the linearized system for (1), under some initial conditions given by Hardy space H 1 , we showed some space-time L 2 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 L 2 norm. The asymptotic behaviors in space-time L 2 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 L 2 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 P (ρ) = 0 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.  Acknowledgments: The first author was partly supported by Grants-in-Aid for Scientific Research with grant number 16H03945. The second author was partly supported by Grants-in-Aid for Scientific Research with grant number 18K03375. The third author was partly supported by Grant-in-Aid for JSPS Fellows with grant number A17J047780.

Conflicts of Interest:
The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A
We consider the following Cauchy problem: generally does not hold. This shows a subtle gain of regularity by the Hardy space.
The existence of a unique solution to (A2) is well-known, as follows. For the proof, we can refer to Proposition 2.1 in Ikehata, Todorova, and Yordanov [20] using the Lumer-Phillips theorem.

Appendix C
Let us introduce the following incompressible Stokes system: (A3) The following Helmholtz decomposition is well-known (cf., Simader and Sohr [21]) where G q (R 2 ) denotes the set of all functions of the potential flow part, defined by G q = {∇p ∈ L q (R 2 ); p ∈ L q loc (R 2 )}. Here, we denote by P q the projection operator from L q (R 2 ) to L q σ (R 2 ). On the whole, space P q is given by the Riesz operator P q f = F −1 I 2 − ξ ξ |ξ| 2 f .
The solution to (A4) satisfies u(t) = K ν (t) * M 0,in ; we can thus estimate solution u to (A4) as follows.
Theorem A3. Let M 0,in ∈ L 2 ∩ L 1 and u be a solution to (A4). Then, u satisfies estimate t 0 u(s) 2 L 2 ds ≤ C M 0,in 2 L 1 uniformly for t.
We give the proof of Theorem A3 here.
(A6) and (A7) derive the desired estimate. The proof is completed.