Time periodic solutions to Navier-Stokes-Korteweg system with friction

In this paper, the compressible Navier-Stokes-Korteweg system with friction is considered in $\mathbb{R}^3$. Via the linear analysis, we show the existence, uniqueness and time-asymptotic stability of the time periodic solution when a time periodic external force is taken into account. Our proof is based on a combination of the energy method and the contraction mapping theorem. In particular, this is the first paper that a time periodic solution can be obtained in the whole space $\mathbb{R}^3$ only under the suitable smallness condition of $H^{N-1}\cap L^1$--norm$(N\geq5)$ of time periodic external force.

In this paper, we consider the problem (1) for (ρ, u) around a constant state (ρ ∞ , 0) in R 3 , where ρ ∞ is a positive constant. Meanwhile P = P (ρ) is smooth in a neighborhood of ρ ∞ satisfying P (ρ ∞ ) > 0, and f is time periodic with period T > 0.
The aim of this paper is to show that the problem (1) admits a time periodic solution around the constant state (ρ ∞ , 0) in the whole space R 3 which has the same period as f . With the energy method and the optimal decay estimates of the solution to the linearized system, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem.
Precisely, let N ≥ 5 be a positive integer, and define the solution space by ∇ρ(t, x) ∈ L 2 (0, T ; H N +1 (R 3 )), for some positive constant and with the norm ||| · ||| given by Then the main results of the present paper are the following. Moreover, the periodic solution is unique in the following sense: if there is another time periodic solution (ρ per 1 , u per 1 ) satisfying (1) with the same f (t, x), and (ρ per 1 − ρ ∞ , u per 1 ) ∈ X 0 (0, T ), then (ρ per 1 , u per 1 ) = (ρ per , u per ). To study the stability of the time periodic solution (ρ per , u per ) obtained in Theorem 1.1, we consider the following initial value problem Here the initial data (ρ 0 , u 0 ) is a small perturbation of the time periodic solution (ρ per , u per ). And we have the following stability result.
Furthermore, there exists a constant C 0 > 0 such that for any t ≥ 0 and The work in this paper is motivated by some similar results which have been obtained for the compressible Navier-Stokes equations, the Boltzmann equation, the compressible Korteweg system and magnetohydrodynamic equations, cf. [4,6,14,16,18,19]. While their studies need the space dimension n ≥ 5. Compared to these results, the important difference in the paper is that we can investigate the similar problem in dimension three due to the good effect of the friction term. In Section 4, it is observed that the presence of friction speeds up the decay rate of the velocity of the NSK system with the factor 1 2 compared to the NS system. There are extensive studies on the existence, stability and convergence rates of solutions to the isentropic or non-isentropic compressible Navier-Stokes-Korteweg system. Here we only mention some of them related to our study. For the compressible Navier-Stokes-Korteweg system without the external force, the results obtained are rich. Hattori and Li [9,10] proved the local existence and the global existence of smooth solutions in Sobolev space for the small initial data. Danchin and Desjardins [5] studied the existence of suitably smooth solutions in critical Besov space. Bresch, Desjardins and Lin [2] considered the global existence of weak solution, then Haspot improved their results in [7]. The local existence of strong solutions was proven in [12]. Recently, Wang and Tan [20] established the optimal decay rates of global smooth solutions. Zhang and Tan [21] discussed the global existence and the optimal L 2 decay rates of solutions on the non-isentropic case. However, for the system (NSK) with potential external force, the related study is very limited so far. Li [13] discussed the global existence and optimal L 2 -decay rates of smooth solutions. And Haspot investigated the existence of global strong solution of Navier-Stokes-Korteweg system with friction in R 2 for the viscosity coefficients and capillarity coefficient depending on density in [8].
The rest of the paper is arranged as follows. We will reformulate the problem and give some preliminaries for later use in Section 2. In Section 3, we give the energy estimates for the linearized system. And in Section 4, we apply the spectral analysis to the semigroup and establish the linear L 2 time-decay estimates. The proof of Theorem 1.1 and Theorem 1.2 is given in the last two sections respectively.
Notations. Throughout this paper, for simplicity, we will omit the variables t, x of functions if it does not cause any confusion. C denotes a generic positive constant which may vary in different estimates. ·, · is the inner product in L 2 (R 3 ). The norm in the usual Sobolev Space H s (R 3 ) is denoted by · s for s ≥ 0. When s=0, we will simply use · . Andf is the Fourier transform of f . Moreover, we denote · H s + · L 1 by · H s ∩L 1 . ∇ = (∂ 1 , ∂ 2 , ∂ 3 ) with ∂ i = ∂ xi , i = 1, 2, 3 and for any integer l ≥ 0, ∇ l g denotes all x derivatives of order l of the function g. Finally, for multi-index α = (α 1 , α 2 , α 3 ), it is standard that

2.
Reformulated system and preliminaries. In order to simplify the coming calculation, we set then the system (1) is reformulated as where Notice that Q 1 and Q 2 have the following properties: Here ∼ means that two sides are of same order. Set U = ( , v), Q = (Q 1 , Q 2 ), F = (0, ν 2 f ) and define then the system (5) takes the form We first consider the linearized system of (5) for any given functionsŨ = (˜ ,ṽ) satisfying Notice that the system (8) can be written as By the Duhamel's principle, the solution to the system (8) can be written in the mild form as where S(t, s) is the corresponding linearized solution operator defined by In the next section, we will establish the energy estimates on ( , v), and in Section 4, the decay rates of the solution operator S(t, s) will be obtained by the spectral analysis.
3. Energy estimates. Throughout this section, we assume that To start with, we recall some known elementary inequalities which will be used frequently later, cf. [1,17].
for any > 0. Here and hereafter, C denotes a positive constant depending only on .
In what follows, two lemmas on the energy estimates are given. Firstly, the lower order energy estimate of ( , v) is obtained in the following lemma. Lemma 3.3. There exist two suitably small constants δ 0 > 0 and 0 > 0 such that where C depends only on ρ ∞ , µ, ν, a and κ.
Proof. Multiplying (8) 1 and (8) 2 by and v, respectively, and integrating them over R 3 , we have by integrating that 1 2 From (6) and Lemma 3.2, we have and

HONG CAI, ZHONG TAN AND QIUJU XU
For I 2 , integrating by parts and using (8) 1 , (6) and Lemma 3.2, we deduce that For I 3 , Cauchy-Schwartz inequality and Lemma 3.1 give Substituting (13)- (16) into (12) yields provided that is small enough, where C depends only on ρ ∞ , µ, ν, a and κ. Next, we estimate ∇ 2 . Taking the L 2 inner product with ∇ on both sides of (8) 2 and then integrating by parts, we have Similar to (15), the term I 4 can be controlled by Integrating by parts and using the Cauchy-Schwartz inequality, it is easy to get Finally, (6) and the Cauchy-Schwartz inequality imply that Combining (18)-(22), we obtain where the constant C depends only on ρ ∞ , µ, ν, a and κ. Multiplying (23) with a small constant δ 0 > 0 and then adding the resultant equation to (17), one can get (11) immediately by the smallness of δ 0 and . This completes the proof of Lemma 3.3.
Next, in the following lemma, we derive the energy estimate of the high order derivatives for ( , v).
Proof. For each multi-index α with 1 ≤ |α| ≤ N , applying ∂ α x to (8) 1 and (8) 2 and then taking the L 2 inner product with ∂ α x and ∂ α x v on the two resultant equations respectively, we have by integrating that 1 2 Now, we estimate J 0 -J 3 term by term. For J 0 , we deduce from (6) and the Cauchy-Schwartz inequality that By using Leibniz's formula and Minkowski's inequality, we get Here C α β denotes the binomial coefficients corresponding to multi-indices. For J 4 , Lemma 3.1 gives For the terms J 5 and J 6 , notice that for any β < α with |β| ≤ N − 2, and for any β < α with |β| > N − 2, Hence, we deduce from Lemma 3.1 that and Putting (28)-(30) into (27), we arrive at Similarly, it holds (32) Combining (26), (31) and (32) yields For the term J 1 , let α 0 ≤ α with |α 0 | = 1, then Similar to the estimate of (31), we have Thus, it follows from (34) and (35) that Notice that (31) and (32) imply Therefore, we derive from (8) 1 , (37) and the Cauchy-Schwartz inequality that Moreover, it holds that where α 0 is defined in (34). Combining (25), (33), (36), (38) and (39), if is small enough, we have where C depends only on ρ ∞ , µ, ν, a and κ. Now we turn to estimate ∂ α x ∆ 2 for 1 ≤ |α| ≤ N . As we did for the first order derivative estimate, applying ∂ α x to (8) 2 and then taking the L 2 inner product with ∂ α x ∇ on the resultant equation, we have by integrating that The first term J 7 is controlled by Here, in the last inequality of (42), we have used (37). By integrating by parts and using the Cauchy-Schwartz inequality, the other terms J 8 -J 12 can be estimated as follows where α 0 is given in (34). Combining (41)-(46), we obtain Multiplying (47) with a suitably small constant δ 1 > 0 and then adding the resultant equation to (40) gives provided that δ 1 and are small enough, where C depends only on ρ ∞ , µ, ν, a and κ. Summing up α with 1 ≤ |α| ≤ N in (48), then (24) follows immediately by the smallness of . This completes the proof of Lemma 3.4.

4.
Linear decay estimates. Consider the Cauchy problem for the corresponding homogeneous linear system to (5) In terms of the semigroup theory for evolutionary system, the solution ( , v) t of the Cauchy problem (49) can be expressed as where G(t, x) is the Green's matrix for the system (49). Applying the Fourier transform to the system (49), we make use of the similar argument of Lemma 2.1 in [20], then the explicit expression for the Fourier transform G(ξ, t) is obtained by a direct computation.
With the help of the formula (50) for Green's matrix in Fourier space and the asymptotical analysis on its elements, we are able to establish the the following linear optimal decay estimates as [3,20]. Lemma 4.2. Let l ≥ 0 be an integer. Assume that ( , v) is the solution of the problem (49) with the initial data s ∈ H l+1 ∩ L 1 and v s ∈ H l ∩ L 1 , then where k is an integer satisfying 0 ≤ k ≤ l.
Let R > 0 be a fixed constant as before. By the pointwise estimates (58) and (59), together with the Parsevel theorem and Hausdroff-Young's inequality, we have the L 2 -decay rates for ( , v) and the derivatives of ( , v) as for 0 ≤ k ≤ l. The proof of Lemma 4.2 is completed.

5.
Existence of time periodic solution. Now, we are ready to prove Theorem 1.1 as follows.
Proof of Theorem 1.1. The proof is divided into two steps.
Similar to the proof of Lemma 4.2, we get The higher-order derivatives of S 1 (t, τ ), S 2 (t, τ ) can be bounded similarly as the estimate (63). Thus, for 0 ≤ k ≤ N − 1, we have This in turn gives Since (64) guarantees the convergence of the integral in (60), letting k → ∞ in (60), we obtain Then (65) shows that U per is a fixed point of ϕ[U ]. Conversely, suppose that ϕ has a unique fixed point, denoted by U 1 (t) = ( 1 , v 1 ) (t). We show that U 1 (t) is time periodic with period T . To this end, set U 2 (t) = U 1 (t + T ). Since the period of f is T , the period of F is T too. Thus, we have where we have used S(t + T, s + T ) = S(t, s). Then by uniqueness, U 2 = U 1 , which proves the periodicity of U 1 (t). Since U 1 (t) is differentiable with respect to t, it is the desired periodic solution of the system (5).
Step 2. Throughout this step, we will show that if sup 0≤t≤T f (t) H N −1 ∩L 1 is suitably small, then ϕ has a unique fixed point in the space X 0 (0, T ) for some appropriate constant 0 > 0. The proof is divided into three parts.
(3) LetŨ 1 = (˜ 1 ,ṽ 1 ) andŨ 2 = (˜ 2 ,ṽ 2 ) be time periodic functions with period T in the space X 0 (0, T ), where 0 > 0 will be determined below. Then similar to (1) and (2), we can get where C 3 is a positive constant depending only on ρ ∞ , µ, ν, a, κ and T . Choose 0 > 0 and a sufficiently small constant h > 0 such that Notice Then there exists a constant h 0 > 0 depending only on ρ ∞ , µ, ν, a, κ and T such that if 0 < h ≤ h 0 , the set of 0 satisfying (75) is not empty. For 0 < h ≤ h 0 , when 6. Stability of time periodic solution. Now, we are in a position to prove the stability of the obtained time periodic solution of Theorem 1.2. We first consider the global existence of the solution to the Cauchy problem (2). Let (ρ per , u per ) be the time periodic solution constructed in Theorem 1.1 and (ρ, u) be the solution of (2). Set .
Let us define the solution space and the solution norm of the Cauchy problem (76) bȳ for any 0 ≤ t 1 ≤ t 2 ≤ ∞. Notice that ( per , v per ) ∈X(0, T ). Then, by the standard argument of the contracting map theorem on general hyperbolic-parabolic system as [11,15], one can obtain the local existence of a strong solution. The details are omitted.
where C 4 is a positive constant independent of (¯ 0 ,v 0 ) ∦.
As usual, to extend the local solution to a global in time solution, we need to establish the following priori estimate. Lemma 6.2. (A priori estimate) Suppose that (¯ 0 ,v 0 ) ∈ H N −1 (R 3 ) × H N −2 (R 3 ), and assume that the Cauchy problem (76) has a unique classical solution (¯ ,v) ∈ X(0, T 1 ) for some positive constant T 1 , satisfying sup 0≤t≤T1 (¯ ,v)(t) ∦≤ ζ, for a small constant ζ > 0. Then there exists a constant C 5 > 0 which is independent of T 1 such that for any t ∈ [0, T 1 ], it holds that Proof. Noticing that some smallness condition can be imposed on ( per , v per ), without loss of generality, we may assume |||( per , v per )||| ≤ with > 0 being sufficiently small. Then by the similar argument as in the proof of Lemma 3.3-3.4, we can obtain d dt and d dt where δ 2 > 0 and δ 3 > 0 are some suitably small constants, and C is a constant depending only on ρ ∞ , µ, ν, a and κ. Adding (78) to (79), it holds d dt provided that is sufficiently small. Integrating (80) in t over (0, t), one can immediately get (77), since by the smallness of δ 2 and δ 3 . This completes the proof of Lemma 6.2.
Proof of Theorem 1.2. The proof of Theorem 1.2 is based on Lemma 6.1-6.2 and the continuity argument, then the Cauchy problem (76) admits a unique solution (¯ ,v) globally in time, which satisfies (3) and (4). Then all the statements in Theorem 1.2 follow immediately. This completes the proof of Theorem 1.2.