Global existence of the Navier-Stokes-Korteweg equations with a non-decreasing pressure in L p -framework
aa r X i v : . [ m a t h . A P ] D ec Global existence of the Navier-Stokes-Korteweg equations with anon-decreasing pressure in L p -framework Keiichi Watanabe Department of Pure and Applied Mathematics, Waseda University
Abstract
We consider the isentropic Navier-Stokes-Korteweg equations with a non-decreasing pressure on thewhole space R n ( n ≥ p -in-time and L q -in-space framework,especially in the maximal regularity class, by assuming ( p, q ) ∈ (1 , × (1 , ∞ ) or ( p, q ) ∈ { } × (1 , s +1 ,q ( R n ) × H s,q ( R n ) n provided s ≥ n/q if q ≤ n and s ≥ q > n . Our results allow the case when the derivative of thepressure is zero at a given constant state, that is, the critical states that the fluid changes a phase fromvapor to liquid or from liquid to vapor. The arguments in this paper do not require any exact expressionor a priori assumption on the pressure. This paper investigates the motion of isentropic compressible fluids with capillary effects on the wholespace R n ( n ≥ ∂ t ̺ + div m = 0 ,∂ t m + div (cid:18) m ⊗ m ̺ (cid:19) − div T ( ̺, m ) + ∇ P ( ̺ ) = 0 , ( ̺, m ) | t =0 = ( ̺ ( x ) , m ( x )) , (1.1)where ̺ = ̺ ( x, t ) and m = m ( x, t ) = ⊤ ( m ( x, t ) , . . . , m n ( x, t )) stand the unknown density and mo-mentum, respectively, at time t > x ∈ R n ( n ≥ ̺ = ̺ ( x ) and m = m ( x ) = ⊤ ( m , ( x ) , . . . , m ,n ( x )) stand given initial data; T ( m /̺ ) stands the stress tensor de-fined by T ( ̺, m ) = S (cid:18) m ̺ (cid:19) + K ( ̺ ) , S (cid:18) m ̺ (cid:19) = 2 µ D (cid:18) m ̺ (cid:19) + ( ν − µ ) (cid:18) div (cid:18) m ̺ (cid:19)(cid:19) I , K ( ̺ ) = κ ̺ − |∇ ̺ | ) I − κ ∇ ̺ ⊗ ∇ ̺, where D ( m /̺ ) = 2 − ( ∇ ( m /̺ ) + ⊤ ( ∇ ( m /̺ ))) is the deformation tensor; P ( ̺ ) stands the pressure thatis assumed to be a smooth function with respect to ̺ satisfying P ′ ( ρ ∗ ) ≥
0, where ρ ∗ is a given positiveconstant; µ and ν stand given constants denoting the viscosity coefficients and κ stands given constantdenoting the capillary coefficient, which satisfy the conditions: µ > , ν > n − n µ, and κ > . (1.2)Here, the first and second conditions of (1.2) ensure strong ellipticity for the Lam´e operator. We em-phasize that any exact expression or a priori assumption of the pressure P is not imposed in the presentpaper. *0 AMS subject classifications: 35Q30; 76N10 *1 [email protected] he system (1.1) is said to be the Navier-Stokes-Korteweg equations , which were originally introducedby Korteweg [46]. Notice that, when κ = 0, the system (1.1) deduces to the usual compressible Navier-Stokes equations. If κ depends on the density ̺ , it is known that there are applications to shallow watermodels [6, 32, 42] or quantum fluids models [7, 33]. A feature of the Korteweg-type model is that thedensity gradient ∇ ̺ appears in the stress tensor, which is a consequence of the second gradient theory dueto van der Walls [58]. The tensor K containing ∇ ̺ is called the Koteweg tensor formulated by Dunn andSerrin [24] in order to analyze the structure of liquid-vapor mixtures with phase transitions. In particular,they proposed a concept of the interstitial working , which is a mechanical quantity, and determine thecoefficients of the Korteweg tensor such that the Clausius-Duhem inequality holds. Roughly speaking, forthe Korteweg-type fluids, the entropy flux does not obey to the classical Fourier law but the “Korteweg”law—the entropy flux consists of the heat transfer and the interstitial working. For further details on theinterstitial working, the reader may consult the paper by Dunn [23], Dunn and Serrin [24], and referencestherein. Although the formulation due to Dunn and Serrin [24] assumed that κ is a non-negative constant,Heida and M´alek [37] showed that it is allowed to consider κ as a non-negative function of ̺ with a slightmodifications on the Korteweg tensor, where the term ̺κ ′ ( ̺ ) |∇ ̺ | I is added to K . In the presentpaper, however, we suppose that the contribution of ̺ to κ is so small that we can see that κ is aconstant, which is the similar assumption as for the viscosity coefficients. Since the Cauchy problemfor (1.1) in the case when κ ≡ κ is a positive constant.There are two different type of models to interpret liquid-vapor phase interfaces: sharp interfacemodels and diffuse interface models . We here mention that diffuse interface models are also called phasefield models. Sharp interface models treat the interfaces as zero thickness, which were introduced toinvestigate capillary effects occurring at the interface, especially, to consider the boundary conditionssuch as the Young-Laplace law in the early 19th century (cf. Finn [27]). When we adopt sharp interfacemodels on the fluid, we recognize the interface separates the fluid into liquid or vapor without spinodalphases taken into account, so that physical quantities, e.g., the density and momentum, are discontinuous across the interfaces. To the best of our knowledge, sharp interface models have been known as one ofthe well-adopted models for moving boundary problems of two-phase flows [3, 41, 56, 65], but there isan example that this model collapses. Indeed, when we simulate the moving contact line along a rigidsurface, the thickness of interfaces cannot be negligible due to some numerical reasons [4, 49]. Comparedwith sharp interface models, in diffuse interface models, the liquid-vapor phase interfaces are construedas interfacial layers , which are also called transition zone, with extremely small but non-zero thickness.Diffuse-interface models provide smoothness of physical quantities across the interfaces. More precisely,a order parameter varies sharply but smoothly, which enables us to overcome some difficulties arisenin sharp interface models [4, 49]. It is known that the Navier-Stokes-Korteweg equations (1.1) can beunderstood as a diffuse interface model due to the capillary terms, see, e.g., [4, 19, 47, 49]. We remarkthat, although the Navier-Stokes-Cahn-Hilliard equations have also been adopted to describe liquid-vapormixtures (cf. Gurtin et al. [30,31] and Oden et al. [54]), this system can be reduced to the Navier-Stokes-Korteweg system by choosing suitable phase transformation rates and capillary stress tensors, see [28,50].We mention that this result is physically reasonable because the derivation of Cahn-Hilliard equations [8]are based on the Ginzburg-Landau phase transition theory [29], which is essentially owing to van derWaals’s idea [58].The diffuse interface model takes the density ̺ as an order parameter to identify the phases. As faras explaining a background of phase transitions on the fluid, let ̺ be a smooth function and boundedsatisfying 0 < ̺ < b with some constant b >
0. Furthermore, let T ∗ > T ∗ , wedefine the Hemholtz free energy W ∈ C ((0 , b )) by W ( ̺ ) := ̺w ( ̺ ), where w ( · ) ∈ C ((0 , b )) is somegiven function depending only on physical properties of the fluid. If the fluid allows phase transitions,the functional W should satisfy the following conditions:1. The Helmholtz energy W ( ̺ ) is a double-well potential. Namely, there exist constants a , a ∈ (0 , b )such that W ′′ ( ̺ ) > ̺ ∈ (0 , a ) ∪ ( a , b ) and W ′′ ( ̺ ) < ̺ ∈ ( a , a ).2. For all ̺ ∈ (0 , b ), we have W ( ̺ ) ≥ ̺ ↓ W ( ̺ ) = lim ̺ ↑ b W ( ̺ ) = + ∞ .Then, the phase of fluid with the density ̺ can be classified into vapor, spinodal, and liquid phases ifthe value of the density ̺ is contained in the interval (0 , a ], ( a , a ), and [ a , b ), respectively. From theusual thermodynamical theory, it is known that the pressure P can be written by P ( ̺ ) = ̺W ′ ( ̺ ) − W ( ̺ ) for ̺ ∈ (0 , b ) . e notice that the pressure P is still positive but its derivative may be negative, that is, the pressure isa non-monotone function because P ′ ( ̺ ) = ̺W ′′ ( ̺ ) and the sign of W ′′ ( ̺ ) may change. We remark thatthe most typical model describing liquid-vapor phase transitions is derived from the van der Waals law :The pressure P ( ̺ ) is given by P ( ̺ ) := RT ∗ bb − ̺ − a̺ , (1.3)where R is the specific gas constant and a, b are some positive constants independent of ̺ and dependingonly on the physical quantity of the fluid. Defining W ( ̺ ) := ̺ (cid:18)Z ̺ρ ∗ P ( ϑ ) ϑ d ϑ − P ( ρ ∗ ) ρ ∗ (cid:19) , (1.4)we observe that the functional (1.3) and (1.4) satisfy the required properties such that phase transitionsoccur. Notice that W is convex when the pressure P is non-decreasing, which includes the standardisentropic pressure laws of γ -type. Summing up, our assumption P ′ ( ρ ∗ ) ≥ critical case P ′ ( ρ ∗ ) = 0. For moreinformation on the background of phase transitions, we refer to [4, 19, 55, 57] and references therein. Theauthor remark that it is expected to show unstable global-in-time solutions to the system (1.1) in thecase when P ′ ( ρ ∗ ) < P ′ ( ρ ∗ ) <
0, there exists an eigenvalue of the correspondinglinearised equations to (1.1) such that its real part is positive for some frequencies. For the local existenceresult on the case P ′ ( ρ ∗ ) <
0, we refer to Charve [9].Concerning the Cauchy problem (1.1) on the whole space R n ( n ≥ all these results were obtained in the L framework because their proof strongly rely on the energy methodor the Plancherel theorem, which is only applicable within the L framework in general. Hence, anothernew technique is required to show a unique global solvability to the system (1.1) in the L p -in-time andL q -in-space framework. Recently, Murata and Shibata [53] overcame this difficulty by using the maximalregularity theory and proved that the system (1.1) admits a global-in-time solution ( ̺, v ) satisfying ̺ − ρ ∗ ∈ W ,p ((0 , ∞ ); W ,q ( R n )) ∩ L p ((0 , ∞ ); W ,q ( R n )) , v ∈ W ,p ((0 , ∞ ); L q ( R n ) n ) ∩ L p ((0 , ∞ ); W ,q ( R n ) n )with m = ̺ v and some given constant ρ ∗ by assuming 2 < p < ∞ and n < q < ∞ with 2 /p + n/q < maximal regularity class . However, this result does not cover anyL framework result because the assumptions on ( p, q ) guarantee some embedding properties in orderto bound the nonlinear terms in a suitable norm, so that it does not admit to take ( p, q ) = (2 , et al. [12] establish the global existence theorem in the L p framework, but theirresult imposed the L integrability for initial data with respect to space variables on the low frequencies.The aim of this paper is to show a unique global solvability of the system (1.1) in the L p -in-time andL q -in-space framework, especially in the maximal regularity class including the L framework results. Inthis paper, especially, we seek a solution ( ̺, m ) to the system (1.1) around a constant state ( ρ ∗ , ̺ ( x ) = ρ ∗ + π ( x ) with somegiven function π ( x ). To simplify the notation, we normalize ρ ∗ to 1 by an appropriate rescaling. Thefollowing is the main theorem of this paper. Theorem 1.1.
Let µ , ν , κ satisfy (1.2) . Suppose that P is a given smooth function with respect to ̺ such that P ′ (1) ≥ . If P ′ (1) = 0 , we additionally assume that ( µ + ν ) ≥ κ . Let p and q satisfy ( p, q ) = (1 , × (1 , ∞ ) or ( p, q ) = { } × (1 ,
2] (1.5) and s satisfy ( s > n/q if q ≤ n,s ≥ if q > n. (1.6) There exists a positive constant C such that if k ̺ − k H s +1 ,q ( R n ) + k m k H s,q ( R n ) < C holds, then thesystem (1.1) admits a unique global strong solution ( ̺, m ) satisfying ̺ − ∈ W ,p ((0 , ∞ ); H s,q ( R n )) ∩ L p ((0 , ∞ ); H s +2 ,q ( R n )) , m ∈ W ,p ((0 , ∞ ); H s − ,q ( R n ) n ) ∩ L p ((0 , ∞ ); H s +1 ,q ( R n ) n ) . emark 1.2. We shall give some comments on Theorem 1.1.(1) The assumption P ′ (1) ≥ all frequencies. As we mentioned above, thisassumption is reasonable from the view point of the thermodynamics. However, if P ′ (1) = 0, wesuppose ( µ + ν ) ≥ κ . This additional assumption will guarantee the Fourier multiplier of thesolution operator satisfies the Mikhin condition.(2) It suffices to suppose P is a C [ s ]+1 -function, see Lemma 5.2 below.(3) According to the trace method of real interpolation, for 1 < p < ∞ it holdW ,p ((0 , ∞ ); W ,q ( R n )) ∩ L p ((0 , ∞ ); W ,q ( R n )) ֒ → BUC([0 , ∞ ); B s +2 − /pq,p ( R n )) , W ,p ((0 , ∞ ); H s − ,q ( R n ) n ) ∩ L p ((0 , ∞ ); H s +1 ,q ( R n ) n ) ֒ → BUC([0 , ∞ ); B s +1 − /pq,p ( R n )) , (1.7)see Amann [2, Theorem III. 4.10.2] (cf. Tanabe [62, pp. 10]). Hence, Theorem 1.1 yields thatthe system (1.1) is globally well-posed for small initial data belonging to H s +1 ,q ( R n ) × H s,q ( R n ) n assuming that (1.5) and (1.6) because we have H s,q ( R n ) ֒ → B s +1 − /pq,p ( R n ) and H s +1 ,q ( R n ) ֒ → B s +2 − /pq,p ( R n ), see, e.g. , Pr¨uss and Simonett [56, Chapter 4] for further general results.The rest of this paper is constructed as follows: The next section introduces notation and lemmaswhich we use throughout this paper. In Section 3, we focus on the linearized system and give the resultsfor the corresponding eigenvalues. Section 4 is devoted to show the linear estimates, which play animportant role for our discussion on proving Theorem 1.1 in Section 5. In this subsection, we introduce symbols and function spaces we use throughout this paper. The symbols N , R and C is the set of all natural, real, and complex numbers, respectively. We define N := N ∪ { } and R + := (0 , ∞ ).For a Banach space X , let X n be the n -product space of X and its norm be denoted by k · k X forshort. The identity mapping on X is denoted by I when no confusion is possible.The Fourier transform on R n is defined by F [ f ]( ξ ) = b f ( ξ ) := Z R n f ( x ) e − ix · ξ d x and the inverse Fourier transform of f is defined by F − [ f ]( x ) := 1(2 π ) n Z R n f ( x ) e ix · ξ d ξ, where x · ξ is the dot product defined by x · ξ := x ξ + · · · + x n ξ n for x = ( x , . . . , x n ) and ξ = ( ξ , . . . , ξ n ).The symbols L p ( R n ), H s,p ( R n ), and B sq,p ( R n ) stand the usual Lebesgue, Bessel potential, and Besovspaces, respectively. For a Banach space X , we denote L p ( I ; X ) by the standard Bochner-Lebesguespaces, where I ⊂ R + is an open time interval and T >
0. Similarly, functions spaces W ,p ( I ; X )describe the set of all functions u such that u, ∂ t u ∈ L p ( I ; X ). In this subsection, we give auxiliary lemmas we use in below. It is well-known that the negative ofthe Laplace operator admits the maximal regularity , see, e.g. , Hieber and Pr¨uss [38] and Pr¨uss andSimonett [56, Chapter 4].
Lemma 2.1.
Let < p, q < ∞ and f : R + → L p ( R + ; L q ( R n )) be a given function. Consider theinhomogeneous heat equation: ∂ t u − ∆ u = f, u ( x,
0) = u ( x ) (2.1) or u ∈ B − /p ) q,p ( R n ) . Then the problem (2.1) admits a unique strong solution u in the maximalregularity class, u ∈ W ,p ( R + ; L q ( R n )) ∩ L p ( R + ; W ,q ( R n )) . Especially, we have the estimate k u k L p ( R + ;L q ( R n )) + k ∂ t u k L p ( R + ;L q ( R n )) + k ( − ∆) u k L p ( R + ;L q ( R n )) ≤ C (cid:16) k u k B − /p ) q,p ( R d ) + k f k L p ( R + ;L q ( R n )) (cid:17) with some positive constant C . We shall recall the
Bernstein-type inequalities which are convenient when we bound spectrally local-ized functions. The proof can be seen in Bahouri et al. [5, Lemma 2.1].
Lemma 2.2.
For < r < R < ∞ . There exists a positive constant C such that for any k ∈ N , for any ≤ p ≤ q ≤ ∞ and f ∈ L p ( R n ) , and for some λ > , we have supp b f ( ξ ) ⊂ { ξ ∈ R n : | ξ | ≤ λr } = ⇒ sup | a | = k k ∂ ax f k L q ( R n ) ≤ C k λ k +3( p − q ) k f k L p ( R n ) , supp b f ( ξ ) ⊂ { ξ ∈ R n : λr ≤ | ξ | ≤ λR } = ⇒ C − k λ k k f k L p ( R n ) ≤ sup | a | = k k ∂ ax f k L p ( R n ) ≤ C k λ k k f k L p ( R n ) , where s ∈ N n is a multi-index. Here, the constant C is independent of λ and f . In this section, we introduce the linearised system associated with (1.1). Noting ̺ = 1 + π , we see thatthe system (1.1) can be reformulated in the form of ∂ t π + div m = 0 ,∂ t m − (cid:0) µ ∆ m + ν ∇ div m − γ ∇ π + κ ∇ ∆ π (cid:1) = F ( π, m ) , ( π, m ) | t =0 = ( π , m ) , (3.1)with F ( π, m ) = − div (cid:18)
11 + π m ⊗ m (cid:19) − µ ∆ (cid:18) π π m (cid:19) − ν ∇ div (cid:18) π π m (cid:19) − ( G ( π ) + P ′′ (1) π ) ∇ π − κ div (cid:26)(cid:18) ( π ∆ π ) − |∇ π | (cid:19) I − ∇ π ⊗ ∇ π (cid:27) , where γ = P ′ (1) and G ( θ ) = P ′ (1 + θ ) − γ − P ′′ (1) θ for θ ∈ R . We remark that G (0) = G ′ (0) = 0 and G is independent of t . We then obtain the linearized systemof (3.1): ∂ t (cid:18) φ u (cid:19) − (cid:18) − div − γ ∇ + κ ∇ ∆ µ ∆ I + ν ∇ div (cid:19) (cid:18) φ u (cid:19) = 0 , ( φ, u ) | t =0 = ( φ , u ) , (3.2)where u ( x, t ) = ⊤ ( u ( x, t ) , . . . , u n ( x, t )) and u ( x ) = ⊤ ( u , ( x ) , . . . , u ,n ( x )). We mention that it isfail to adopt a general theory for symmetric hyperbolic-parabolic systems established by Shizuta andKawashima [60] due to the capillary term. Applying the Fourier transform with respect to x implies ∂ t (cid:18) b φ b u (cid:19) − (cid:18) − i ⊤ ξ − ( γ + κ | ξ | )i ξ − µ | ξ | I − νξ ⊤ ξ (cid:19) (cid:18) b φ b u (cid:19) = 0 , ( b φ, b u ) | t =0 = ( b φ , b u ) . (3.3) et det( λ ) be the polynomial of λ ∈ C defined bydet( λ ) := λ + ( µ + ν ) | ξ | λ + ( γ + κ | ξ | ) | ξ | . (3.4)The equation det( λ ) = 0 has two roots: λ ± ( ξ ) = − µ + ν | ξ | ± µ + ν s | ξ | − γ + κ | ξ | ) | ξ | ( µ + ν ) = − µ + ν | ξ | ± µ + ν s(cid:18) − κ ( µ + ν ) (cid:19) | ξ | − γ | ξ | ( µ + ν ) =: − A | ξ | ± A r(cid:16) − κA (cid:17) | ξ | − γA | ξ | , (3.5)where we have set A := − ( µ + ν ) / λ ± ( ξ ) are classified into the followingcases: Case 1: A > κ and γ > Case 2: A < κ and γ > Case 3: A = κ and γ > Case 4: A > κ and γ = 0; Case 5: A < κ and γ = 0; Case 6: A = κ and γ = 0 . In this paper, we exclude Case 5.Let us focus on Case 1. We see that λ + ( ξ ) = λ − ( ξ ) holds if ξ satisfies | ξ | = r γA − κ =: B. Noting (3.5), we observe that λ ± ( ξ ) = − A | ξ | ± i √ γ | ξ | r − | ξ | B for 0 < | ξ | < B, − A | ξ | ± √ γB | ξ | s − B | ξ | for | ξ | > B. (3.6)Especially, in this case, the eigenvalues λ ± ( ξ ) have the following asymptotic behaviors: λ ± ( ξ ) = ± i √ γ | ξ | − A | ξ | ∓ i √ γB | ξ | + i O ( | ξ | ) as | ξ | → , − A (cid:18) ∓ r − κA (cid:19) | ξ | ∓ √ γ B + O (cid:18) | ξ | (cid:19) as | ξ | → ∞ and the solution to (3.3) is represented by b φ ( ξ, t ) = λ + ( ξ ) e λ − ( ξ ) t − λ − ( ξ ) e λ + ( ξ ) t λ + ( ξ ) − λ − ( ξ ) ! b φ ( ξ ) − e λ + ( ξ ) t − e λ − ( ξ ) t λ + ( ξ ) − λ − ( ξ ) ! (i ξ · b u ( ξ )) , b u ( ξ, t ) = − e λ + ( ξ ) t − e λ − ( ξ ) t λ + ( ξ ) − λ − ( ξ ) ! ( γ + κ | ξ | )i ξ b φ ( ξ ) + e − µ | ξ | t b u ( ξ )+ λ + ( ξ ) e λ + ( ξ ) t − λ − ( ξ ) e λ − ( ξ ) t λ + ( ξ ) − λ − ( ξ ) − e − µ | ξ | t ! ξ ( ξ · b u ( ξ )) | ξ | (3.7)for ξ ∈ R n \ { } . In our analysis, it is convenient to rewrite as b φ ( ξ, t ) = X ℓ = ± e λ ℓ ( ξ ) t G ( ℓ )1 , ( ξ ) b φ ( ξ ) + n X k =1 X ℓ = ± e λ ℓ ( ξ ) t G ( ℓ )1 ,k +1 ( ξ ) b u ,j ( ξ ) , b u j ( ξ, t ) = X ℓ = ± e λ ℓ ( ξ ) t G ( ℓ ) j +1 , ( ξ ) b φ ( ξ ) + e − µ | ξ | t b u ,j ( ξ )+ n X k =1 (cid:18) X ℓ = ± e λ ℓ ( ξ ) t G ( ℓ ) j +1 ,k +1 ( ξ ) + e − µ | ξ | t ξ j ξ k | ξ | (cid:19)b u ,k ( ξ ) (3.8) or j = 1 , . . . , n , where G ( ℓ ) JK ( ξ ), J, K = 1 , . . . , n + 1, are defined by G ( ± )1 , = ∓ λ ∓ ( ξ ) λ + ( ξ ) − λ − ( ξ ) , G ( ± )1 ,k +1 = ∓ i ξ k λ ± ( ξ ) − λ − ( ξ ) ,G ( ± ) j +1 , = ∓ λ ± ( ξ )( γ + κ | ξ | )i ξ j λ + ( ξ ) − λ − ( ξ ) , G ( ± ) j +1 ,k +1 = ± λ ± ( ξ ) ξ j ξ k | ξ | ( λ + ( ξ ) − λ − ( ξ )) . (3.9)We notice that, for Case 1, these formulas hold if | ξ | 6 = B . On the other hand, if B/ < | ξ | < B , wewrite the solution to (3.3) in the form of b φ ( ξ, t ) = (cid:18) π i I Γ ( z + 2 A | ξ | ) e zt det( z ) d z (cid:19) b φ ( ξ ) + (cid:18) π I Γ e zt det( z ) d z (cid:19) ξ · b u ( ξ ) , b u ( ξ, t ) = − (cid:18) π I Γ e zt det( z ) d z (cid:19) ( γ + κ | ξ | ) b φ ( ξ ) + e − µ | ξ | t b u ( ξ )+ (cid:18) π i I Γ ze zt det( z ) d z − e − µ | ξ | t (cid:19) ξ ( ξ · b u ) | ξ | , (3.10)where Γ ⊂ { z ∈ C : Re z ≤ − c } denotes a closed pass including the zero points of det( z ) with somepositive constant c independent of t and ξ such thatmax B/ ≤| ξ |≤ B Re λ ± ( ξ ) ≤ − c . (3.11)Here, the formulas (3.7)–(3.10) can be derived in the same manner as in Hoff and Zumbrun [39, Section 3]and Kobayashi and Shibata [44, Section 2], see next section. We, in addition, remark that we have usedCauchy’s integral expression to derive (3.10).We next treat the other cases. For Case 2, we have λ ± ( ξ ) = − A | ξ | ± i A | ξ | r(cid:16) κA − (cid:17) | ξ | + γA , which have the asymptotic behaviors: λ ± ( ξ ) = ± i γ | ξ | − A | ξ | ± i κ − A A | ξ | + i O ( | ξ | ) as | ξ | → , − A | ξ | ± i p κ − A | ξ | ± i Aγ √ κ − A + i O (cid:18) | ξ | (cid:19) as | ξ | → ∞ . For the rest cases, the eigenvalues can be written by λ ± ( ξ ) = ± i √ γA | ξ | − A | ξ | for Case 3 , − A (cid:18) ∓ r − κA (cid:19) | ξ | for Case 4 , − A | ξ | ± i (cid:18)r κA − (cid:19) | ξ | for Case 5 , − A | ξ | for Case 6 , where we have a real double root for Case 6. For Case 2–4, the solution to (3.3) is expressed in the formof (3.7)–(3.10). As for Case 6, the solution to (3.3) is denoted by b φ ( ξ, t ) = (1 + µ | ξ | t ) e − µ | ξ | b φ − te − µ | ξ | t ( iξ · b u ) , b u ( ξ, t ) = A | ξ | te − A | ξ | t i ξ b φ + e − µ | ξ | t b u + (cid:18) ( A | ξ | t − e − A | ξ | t − e − µ | ξ | t (cid:19) ξ ( ξ · b u ) | ξ | , (3.12)see the appendix for derivations. According to the above analysis, we emphasize that we may expect aparabolic smoothing for the solution to (1.1) for all frequencies, which enables us to prove the globalsolvability to (1.1) via the Banach fixed point iteration. Estimates for the solution operator
We define the solution operator T ( t ) = ⊤ ( T φ ( t ) , T u ( t )) such that T ( t ) U ( x ) = F − h b U ( ξ, t ) i ( x ) = U ( x, t )for every t >
0, where U := ⊤ ( π, u ) and U := ⊤ ( φ , u ) are a solution and an initial data of thesystem (3.2), respectively. In this section, we prove the following lemmas. Lemma 4.1.
Let ≤ q ≤ r ≤ ∞ . Let γ ≥ and A ≥ κ if γ = 0 . There exists positive constants c and C independent of t and U such that k ∂ ax T ( t ) U k L r ( R n ) ≤ C (cid:18) t − n ( q − r ) + e − ct + e − ct t − n ( q − r ) − | a | (cid:19) k U k L q ( R n ) holds for any a ∈ N n , t > , and U ∈ L q ( R n ) n +1 . Lemma 4.2.
Let ( p, q ) satisfy (1.5) . Assume that γ ≥ and A ≥ κ if γ = 0 . Then, for any U ∈ H ,q ( R n ) , we have the estimates k T ( t ) U k W ,p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) , k ( − ∆) T ( t ) U k L p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) , where C is some positive constant independent of t . Lemma 4.3.
Let < p, q < ∞ satisfy (1.5) and let γ ≥ . If γ = 0 , we additionally suppose A ≥ κ .There exists a positive constant C independent of t such that the estimates (cid:13)(cid:13)(cid:13)(cid:13)Z t T ( t − τ ) f ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) W ,p ( R + ;L q ( R n )) ≤ C k f k L p ( R + ;L q ( R n )) , (cid:13)(cid:13)(cid:13)(cid:13)Z t ( − ∆) T ( t − τ ) f ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;L q ( R n )) ≤ C k f k L p ( R + ;L q ( R n )) are valid for any f ∈ L p ( R + ; L q ( R n )) . We first consider Case 1. To establish the estimates for T ( t ), we decompose the operator T ( t ) asfollows: Let ϕ , ϕ ∞ , ϕ M ∈ C ∞ ( R n ) be cut-off functions defined by ϕ ( ξ ) = ( | ξ | ≤ B/ , | ξ | ≥ B/ √ ,ϕ ∞ ( ξ ) = ( | ξ | ≥ B, | ξ | ≤ max (1 , √ B ) , and ϕ M ( ξ ) = 1 − ϕ ( ξ ) − ϕ ∞ ( ξ ), respectively. Using these cut-off functions, we decompose T ( t ) into thelow, medium, and high frequencies in the Fourier space: T ( t ) = T ( t ) + T M ( t ) + T ∞ ( t ) , T ( t ) = ⊤ ( T φ, ( t ) , T u , ( t )) ,T M ( t ) = ⊤ ( T φ,M ( t ) , T u ,M ( t )) , T ∞ ( t ) = ⊤ ( T φ, ∞ ( t ) , T u , ∞ ( t )) ,T φ, ( t ) (cid:18) φ ( x ) u ( x ) (cid:19) = F − [ ϕ ( ξ ) b φ ( ξ, t )]( x ) ,T u , ( t ) (cid:18) φ ( x ) u ( x ) (cid:19) = F − [ ϕ ( ξ ) b u ( ξ, t )]( x ) ,T φ,M ( t ) (cid:18) φ ( x ) u ( x ) (cid:19) = F − [ ϕ M ( ξ ) b φ ( ξ, t )]( x ) ,T u ,M ( t ) (cid:18) φ ( x ) u ( x ) (cid:19) = F − [ ϕ M ( ξ ) b u ( ξ, t )]( x ) ,T φ, ∞ ( t ) (cid:18) φ ( x ) u ( x ) (cid:19) = F − [ ϕ ∞ ( ξ ) b φ ( ξ, t )]( x ) ,T u , ∞ ( t ) (cid:18) φ ( x ) u ( x ) (cid:19) = F − [ ϕ ∞ ( ξ ) b u ( ξ, t )]( x ) . (4.1) o show the estimates for the operator T ℓ ( t ) ( ℓ = 1 , ∞ ), we set g ( ± )1 ( θ ) = ± i √ γθ s − A − κγ θ ! for 0 < θ < B,g ( ± ) ∞ ( θ ) = (cid:18)p A − κ ± r A − κ − γθ (cid:19) θ for θ > B ,so that the eigenvalues λ ± ( ξ ) given in (3.6) can be written by λ ± ( ξ ) = − K | ξ | + g ( ± ) ℓ ( | ξ | )for ℓ = 1 , ∞ , where we have set K := A (cid:18) − r − κA (cid:19) > . We emphasize that K does not vanish for all ξ ∈ R n \ { } such that | ξ | 6 = B . We first give the L q − L r estimate for T ℓ ( t ). Lemma 4.4.
Let q and r satisfy ≤ q ≤ r ≤ ∞ . For all a ∈ N n , t > , and U ∈ L q ( R n ) n +1 , theestimates k ∂ ax T ( t ) U k L r ( R n ) ≤ Ct − n ( q − r ) k U k L q ( R n ) , k ∂ ax T ∞ ( t ) U k L r ( R n ) ≤ Ce − ct t − n ( q − r ) − | a | k U k L q ( R n ) hold with some positive constants c and C independent of t and U .Proof. Let T ( ± ) ℓ ( t ) and T ( µ ) ℓ ( t ) be the operators defined by T ( ± ) ℓ ( t ) U ( x ) = F − h e λ ± t ϕ ℓ ( ξ ) G ( ± ) ( ξ ) b U ( ξ ) i ( x ) ,T ( µ ) ℓ ( t ) U ( x ) = F − h e − µ | ξ | t ϕ ℓ ( ξ ) G ( µ ) ( ξ ) b U ( ξ ) i ( x ) (4.2)for ℓ = 1 , ∞ with function G independent of t and U such that | ξ a ∂ aξ G ( ξ ) | ≤ C for any multi-index a ∈ N n . According to the L q − L r estimate for the heat semigroup and the Miklin-type Fourier multipliertheorem, we have (cid:13)(cid:13)(cid:13) ∂ ax T ( ± ) ∞ ( t ) U (cid:13)(cid:13)(cid:13) L r ( R n ) = (cid:13)(cid:13)(cid:13)(cid:13) ∂ ax e K t ∆ F − (cid:20) e t (cid:16) − K | ξ | + g ( ± ) ∞ ( | ξ | ) (cid:17) ϕ ∞ ( ξ ) G ( ± ) ( ξ ) b U ( ξ ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) L r ( R n ) ≤ Ce − ct t − n ( q − r ) − | a | k U k L q ( R n ) for all t >
0, 1 ≤ q ≤ r ≤ ∞ , a ∈ N n , and U ∈ L q ( R n ), where c is some positive constant independentof x , t , and U . Analogously, we also obtain (cid:13)(cid:13)(cid:13) ∂ ax T ( µ ) ∞ ( t ) U (cid:13)(cid:13)(cid:13) L r ( R n ) ≤ Ce − ct t − n ( q − r ) − | a | k U k L q ( R n ) for all t >
0, 1 ≤ q ≤ r ≤ ∞ , and U ∈ L q ( R n ). If | ξ | 6 = B , from (3.9), we observe that | G ( ± ) J,K ( ξ ) | ≤ C B for J, K = 1 , . . . , n + 1. Hence, combined with (3.8), the required estimate has been shown. Here, to derivethe estimate for T ( t ), we have used same argument above and additionally employed the Bernstein-typeinequality.We next show the space-time estimates for T ℓ ( t ), ℓ = { , ∞} . Lemma 4.5. If p and q satisfy (1.5) , there exists a positive constant C independent of t such that k T ( t ) U k W ,p ( R + ;L q ( R n )) ≤ C k U k L q ( R n ) , k ( − ∆) T ( t ) U k L p ( R + ;L q ( R n )) ≤ C k U k L q ( R n ) , k T ∞ ( t ) U k W ,p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) , k ( − ∆) T ∞ ( t ) U k L p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) for any U ∈ H ,q ( R n ) . roof. Let T ( ± ) ℓ and T ( µ ) ℓ be the operators defined in (4.2). Using Lemma 2.1 with f = 0, we have (cid:13)(cid:13)(cid:13) T ( ± ) ℓ ( t ) U (cid:13)(cid:13)(cid:13) W ,p ( R + ;L q ( R n )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) F − (cid:20) e t (cid:16) − K | ξ | + g ( ± ) ℓ ( | ξ | ) (cid:17) ϕ ℓ ( ξ ) G ( ± ) ( ξ ) b U ( ξ ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) B − /p ) q,p ( R n ) Since we have the continuous embedding H ,q ( R n ) ֒ → B − /p ) q,p ( R n ) under the condition (1.5), by theMiklin-type Fourier multiplier theorem, we deduce that (cid:13)(cid:13)(cid:13)(cid:13) F − (cid:20) e t (cid:16) − K | ξ | + g ( ± ) ℓ ( | ξ | ) (cid:17) ϕ ℓ ( ξ ) G ( ± ) ( ξ ) b U ( ξ ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) B − /p ) q,p ( R n ) ≤ C k U k H ,q ( R n ) , which yields that (cid:13)(cid:13)(cid:13) T ( ± ) ℓ ( t ) U (cid:13)(cid:13)(cid:13) W ,p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) . Similarly, we have (cid:13)(cid:13)(cid:13) T ( ν ) ℓ ( t ) U (cid:13)(cid:13)(cid:13) W ,p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) . Noting (3.8) and using Lemma 2.1 and the Bernstein-type inequality, the proof has been finished.The following lemma will be used when we estimate the nonlinear terms.
Lemma 4.6.
Let < p, q < ∞ satisfy (1.5) and let ℓ = 1 , ∞ . Then, there exists a positive constant C independent of t such that the estimates (cid:13)(cid:13)(cid:13)(cid:13)Z t T ℓ ( t − τ ) f ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) W ,p ( R + ;L q ( R n )) ≤ C k f k L p ( R + ;L q ( R n )) , (cid:13)(cid:13)(cid:13)(cid:13)Z t ( − ∆) T ℓ ( t − τ ) f ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;L q ( R n )) ≤ C k f k L p ( R + ;L q ( R n )) hold for all f ∈ L p ( R + ; L q ( R n )) .Proof. Applying Lemma 2.1 with u = 0 and the similar argument as in the proof of Lemma 4.5, theproof is completed.We finally focus on T M ( t ). The following is the L q − L r mapping property for T M ( t ). Lemma 4.7.
Let ≤ q ≤ r ≤ ∞ and a ∈ N n . For any t > and U ∈ L q ( R n ) , we have k ∂ ax T M ( t ) U k L r ( R n ) ≤ Ce − ct k U k L q ( R n ) , where C and c are some positive constants independent of t .Proof. The required estimate immediately follows from the L q − L q decay estimates for T M ( t ) with1 ≤ q ≤ ∞ . Indeed, since F [ T M ( t ) f ] is a compactly supported function, we can adopt the Bernstein-type inequality and the inclusion L q ( R n ) ֒ → L r ( R n ) for 1 ≤ r ≤ q ≤ ∞ . As for this embedding property,we refer to Wang et al . [64, Proposition 1.16].Set N , ( x, t ) = 12 π i F − (cid:20)(cid:18)I Γ ( z + 2 A | ξ | ) e zt det( z ) d z (cid:19) ϕ M ( ξ ) (cid:21) ,N ,k +1 ( x, t ) = 12 π i F − (cid:20) i ξ k (cid:18)I Γ e zt det( z ) d z (cid:19) ϕ M ( ξ ) (cid:21) ,N j +1 , ( x, t ) = − π F − (cid:20) ( γ + κ | ξ | ) (cid:18)I Γ e zt det( z ) d z (cid:19) ϕ M ( ξ ) (cid:21) ,N j +1 ,k +1 ( x, t ) = 12 π i F − (cid:20) ξ j ξ k | ξ | (cid:18)I Γ ze zt det( z ) d z (cid:19) ϕ M ( ξ ) (cid:21) , e N ( x, t ) = F − h e − µ | ξ | t ϕ M ( ξ ) i , e N j,k ( x, t ) = F − (cid:20) ξ j ξ k | ξ | e − µ | ξ | t ϕ M ( ξ ) (cid:21) . or j, k = 1 , . . . , n . Recalling (3.10), we have the following formulas: F − h ϕ M ( ξ ) b φ i ( x, t ) = N , ( · , t ) ∗ φ + n X k =1 N ,k +1 ( t, · ) ∗ u ,k , F − h ϕ M ( ξ ) b u j i ( x, t ) = N j +1 , ( · , t ) ∗ φ + e N ( · , t ) ∗ u ,j + n X k =1 (cid:16) N j +1 ,k +1 ( · , t ) + e N j,k ( · , t ) (cid:17) ∗ u ,k (4.3)for j = 1 , . . . , n . Notice that the Fourier transforms of N J,K ( x, t ) ( J, K = 1 , . . . , n + 1), e N ( x, t ), and e N j,k ( x, t ) are compactly supported functions due to the cut-off function ϕ M ( ξ ). Thus, from the residuetheorem and the condition (3.11), we easily see that | N J,K ( x, t ) | ≤ Ce − c t holds for J, K = 1 , . . . , n + 1 with some positive constant C independent of x and t , where c is the sameconstant given in (3.11). On the other hand, we also have | e N j,k ( x, t ) | ≤ Ce − c t for j, k = 1 , . . . , n and | e N ( x, t ) | ≤ Ce − c t , where C is some positive constant independent of t and x and c is some positive constant dependingonly on µ and B . Hence, applying the Young inequality to (4.3), we obtain k T φ,M ( t ) U k L q ( R n ) ≤ C A,B,n,q e − ct k U k L q ( R n ) , k T u ,M ( t ) U k L q ( R n ) ≤ C A,B,n,q e − ct k U k L q ( R n ) for 1 ≤ q ≤ ∞ and U ∈ L q ( R n ) with c := min( c , c ). Namely, we obtain k T M ( t ) U k L q ( R n ) ≤ C A,B,n,q e − ct k U k L q ( R n ) for all 1 ≤ q ≤ ∞ and U ∈ L q ( R n ).The following lemmas can be proved by employing the same arguments as in the proof of Lemmas 4.5and 4.6, so that we may omit the proofs. Lemma 4.8.
Let ( p, q ) satisfy (1.5) . Then the estimates k T M ( t ) U k W ,p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) , k ( − ∆) T M ( t ) U k L p ( R + ;L q ( R n )) ≤ C k U k H ,q ( R n ) , hold for all U ∈ H ,q ( R n ) with some positive constant C independent of t . Lemma 4.9.
Let p and q satisfy (1.5) . There exists a positive constant C independent of t such thatthe estimates (cid:13)(cid:13)(cid:13)(cid:13)Z t T M ( t − τ ) f ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) W ,p ( R + ;L q ( R n )) ≤ C k f k L p ( R + ;L q ( R n )) , (cid:13)(cid:13)(cid:13)(cid:13)Z t ( − ∆) T M ( t − τ ) f ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;L q ( R n )) ≤ C k f k L p ( R + ;L q ( R n )) hold true for all f ∈ L p ( R + ; L q ( R n )) . Recalling that T ( t ) = T ( t ) + T M ( t ) + T ∞ ( t ), we see that Lemmas 4.4–4.9 imply Lemmas 4.1–4.3, sothat we may omit the proof of Lemmas 4.1–4.3 for Case 1.We next deal with Case 2,3,4,6. To simplify the notation, we introduce functions g ( ± ) m ( θ ), m =2 , , ,
6, defined by g ( ± )2 ( θ ) = ± i A | ξ | s(cid:18) κA − (cid:19) | ξ | + γA for Case 2 ,g ( ± )3 ( θ ) = ± i √ γA | ξ | for Case 3 ,g (+)4 ( θ ) = 0 for Case 4 ,g ( − )4 ( θ ) = − A r − κA | ξ | for Case 4 ,g ( ± )6 ( θ ) = 0 for Case 6 , here 0 < θ < ∞ , and constants K m defined by K = A for Case 2 ,K = A for Case 3 ,K = A (cid:18) − r − κA (cid:19) for Case 4 ,K = A for Case 6 . We see that λ ± ( ξ ) can be written in the form of λ ± ( ξ ) = − K m | ξ | + g ( ± ) m ( | ξ | ) for each Case m = 2 , , , K m does not vanish for all ξ ∈ R n \ { } . Accordingly, applying the argument for the proof ofLemmas 4.4–4.6, we immediately have the estimates of T ( t ) for Case 2,3,4,6. Summing up, we obtainLemmas 4.1–4.3 for Case 1–6 exclude Case 5. Finally, we prove Theorem 1.1. As we mentioned before, we can prove the existence of global-in-timesolution via the standard Picard fixed point iteration because we have the parabolic smoothing not onlyfor the momentum but for the density, which is different from the case for the “usual” compressibleNavier-Stokes equation—this fact is a consequence of the capillary effects on the fluids.From the Duhamel principle, the equations (3.1) can be transformed to the integral equation: U ( t ) = T ( t ) U − Z t T ( t − τ ) N ( τ ) d τ (5.1)for t ≥
0, where N ( τ ) := ⊤ (0 , F ( π ( x, τ ) , m ( x, τ ))). In what follows, let p , q , and s satisfy (1.5) and (1.6).Then we define the underlying space X p,q,s by X p,q,s := (cid:8) U ∈ Y p,q,s | k U k X p,q,s ≤ C k U k H s +1 ,q ( R n ) × H s,q ( R n ) (cid:9) endowed with the norm k U k X p,q,s := k π k W ,p ( R + ;H s,q ( R n )) + k π k L p ( R + ;H s +2 ,q ( R n )) + k m k W ,p ( R + ;H s − ,q ( R n )) + k m k L p ( R + ;H s +1 ,q ( R n )) . Here, we have set U ∈ Y p,q,s ⇐⇒ ( π ∈ W ,p ( R + ; H s,q ( R n )) ∩ L p ( R + ; H s +2 ,q ( R n )) , m ∈ W ,p ( R + ; H s − ,q ( R n )) ∩ L p ( R + ; H s +1 ,q ( R n )) , k U k H s +1 ,q ( R n ) × H s,q ( R n ) := k π k H s +1 ,q ( R n ) + k m k H s,q ( R n ) . Using the norm k · k X p,q,s , from Lemma 4.2, there exists a positive constant C independent of t and U such that k T ( t ) U k X p,q,s ≤ C k U k H s +1 ,q ( R n ) × H s,q ( R n ) (5.2)for any U ∈ H s +1 ,q ( R n ) × H s,q ( R n ) n . To construct a solution to (5.1), we have to estimate the nonlinearterms. Lemma 5.1.
Let p , q , and s satisfy (1.5) and (1.6) . Define N ( τ ) and N ( τ ) by N ( τ ) := ⊤ (0 , F ( π ( x, τ ) , m ( x, τ ))) , N ( τ ) := ⊤ (0 , F ( π ( x, τ ) , m ( x, τ ))) for τ ∈ [0 , t ] with t > . Assume that C k U k H s +1 ,q ( R n ) × H s,q ( R n ) ≤ , (5.3) where C is a constant satisfying (5.2) . Then, for all t > and U , U , U ∈ X p,q,s , we have the estimates (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) N ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) X p,q,s ≤ C k U k s +1 ,q ( R n ) × H s,q ( R n ) , (5.4) (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ )( N ( τ ) − N ( τ )) d τ (cid:13)(cid:13)(cid:13)(cid:13) X p,q,s ≤ C k U k H s +1 ,q ( R n ) × H s,q ( R n ) k U − U k X p,q,s , (5.5) with some positive constants C and C depending on C , where U := ⊤ ( π , m ) and U := ⊤ ( π , m ) . e here introduce the estimates for composition operators to bound the nonlinear terms includingpressure term. Lemma 5.2.
Let < q < ∞ and ∈ I ⊂ R . Let s satisfy (1.6) . Furthermore, let f : I → R be asmooth (at least C [ s ]+1 -class) function such that f (0) = 0 . The following assertions hold true.(1) If u ∈ H s,q ( R n ) , then the composition operator f ( u ) belongs to H s,q ( R n ) bounded by k f ( u ) k H s,q ( R n ) ≤ C (cid:16) k u k H s,q ( R n ) + k u k s H s,q ( R n ) (cid:17) with some positive constant C depending only on f ′ , . . . , f ([ s ]) , n , q , and s .(2) If, in addition, f satisfies f ′ (0) = 0 , then for any u, v ∈ H s,q ( R n ) , the difference f ( u ) − f ( v ) is theelement of H s,q ( R n ) possessing the estimate: k f ( u ) − f ( v ) k H s,q ( R n ) ≤ C (cid:16) k u − v k H s,q ( R n ) sup τ ∈ [0 , k u + τ ( v − u ) k H s,q ( R n ) + k u − v k H s,q ( R n ) sup τ ∈ [0 , k u + τ ( v − u ) k s H s,q ( R n ) (cid:17) , where C is some constant depending only on f ′′ , . . . , f ([ s ]+1) , n , q , and s . Remark 5.3.
When 0 < s ≤
1, it holds k f ( u ) k H s,q ( R n ) ≤ CK k u k H s,q ( R n ) with a positive constant C , where 1 < q < ∞ , f (0) = 0, and | f ′ | ≤ K . One can find the proof in Christand Weinstein [18] and Taylor [63, Chapter 2]. Proof of Lemma 5.2.
We first extend the domain of f to R by zero. According to Adams and Frazier [1],we have the estimate k f ( u ) k H s,q ( R n ) ≤ C max k ∈{ ,..., [ s ] } sup θ ∈ R (cid:12)(cid:12)(cid:12) f ( k ) ( θ ) (cid:12)(cid:12)(cid:12) (cid:16) k u k H s,q ( R n ) + k u k s ˙H ,sq ( R n ) (cid:17) ≤ C max k ∈{ ,..., [ s ] } sup θ ∈ R (cid:12)(cid:12)(cid:12) f ( k ) ( θ ) (cid:12)(cid:12)(cid:12) (cid:16) k u k H s,q ( R n ) + k u k s H s,q ( R n ) (cid:17) , where we have employed the embedding H s,q ( R n ) ֒ → H ,sq ( R n ). Notice that this embedding holds forany q ∈ (1 , ∞ ) whenever s ≥
1. Here, the constant C is independent of f and u . In addition, we knowthe identity f ( u ) − f ( v ) = ( v − u ) Z t f ′ ( u + τ ( v − u )) d τ, which concludes the proof.We also need the estimates for composition operators with the Besov norm. The proof of the followinglemma can be found in the book of Bahouri et al . [5], see Theorem 2.87 and Corollary 2.91. Lemma 5.4.
Let ≤ p, q ≤ ∞ , s > , and ∈ I ⊂ R . Furthermore, let f : I → R be a smooth (at least C [ s ]+1 -class) function such that f (0) = 0 . The following statements are valid.(1) If u ∈ B sq,p ( R n ) ∩ L ∞ ( R n ) , then we have f ( u ) ∈ B sq,p ( R n ) ∩ L ∞ ( R n ) bounded by k f ( u ) k B sq,p ( R n ) ≤ C k u k B sq,p ( R n ) with some positive constant C depending only on f ′ , s , and k u k L ∞ ( R n ) .(2) If, furthermore, f satisfies f ′ (0) = 0 , then for any u, v ∈ B sq,p ( R n ) ∩ L ∞ ( R n ) , the difference f ( u ) − f ( v ) also belongs to B sq,p ( R n ) ∩ L ∞ ( R n ) with the estimate: k f ( u ) − f ( v ) k H s,q ( R n ) ≤ C (cid:16) k u − v k B sq,p ( R n ) sup τ ∈ [0 , k u + τ ( v − u ) k L ∞ ( R n ) + k u − v k L ∞ ( R n ) sup τ ∈ [0 , k u + τ ( v − u ) k B sq,p ( R n ) (cid:17) , where C is some constant depending on f ′′ , k u k L ∞ ( R n ) , and k v k L ∞ ( R n ) . o estimate the nonlinear terms, we will use the Kato-Ponce inequality [43], see also Sawano [59,Theorem 4.44].
Lemma 5.5.
Let < q < ∞ and s > . For f, g ∈ H s,q ( R n ) ∩ L ∞ ( R n ) , we have k fg k H s,q ( R n ) ≤ C (cid:16) k f k H s,q ( R n ) k g k L ∞ ( R n ) + k f k L ∞ ( R n ) k g k H s,q ( R n ) (cid:17) with a positive constant C .Proof of Lemma 5.1. To show (5.4), it suffices to prove the bounds (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) F ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C k U k s +1 ,q ( R n ) × H s,q ( R n ) , (5.6) (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) F ( τ ) d τ (cid:13)(cid:13)(cid:13)(cid:13) W ,p ( R + ;H s − ,q ( R n )) ≤ C k U k s +1 ,q ( R n ) × H s,q ( R n ) . (5.7)Using the embedding property (1.7) and the assumption (5.3), for U ∈ X p,q,s we have the boundsup t ∈ R + k π k H s,q ( R n ) ≤ C sup t ∈ R + k π k B s +2 − /pq,p ( R n ) ≤ C (cid:16) k π k W ,p ( R + ;H s,q ( R n )) + k π k L p ( R + ;H s +2 ,q ( R n )) (cid:17) ≤ CC k U k H s +1 ,q ( R n ) × H s,q ( R n ) ≤ C < ∞ , (5.8)which, combined with Lemma 5.2, yields that k G ( π ) k H s,q ( R n ) ≤ C (cid:0) k π k H s,q ( R n ) + k π k s H s,q ( R n ) (cid:1) . Besides, employing Lemma 5.4, we observe k G ( π ) k B s +2 − /pq,p ( R n ) ≤ C k π k B s +2 − /pq,p ( R n ) , where the constant may depend on C . Using Lemma 5.5, we obtain k G ( π ) ∇ π k H s − ,q ( R n ) ≤ C (cid:16) k G ( π ) k H s − ,q ( R n ) k∇ π k L ∞ ( R n ) + k G ( π ) k L ∞ ( R n ) k∇ π k H s − ,q ( R n ) (cid:17) . Then, from Lemma 4.3, we have the estimate (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) (cid:16) G ( π ) ∇ π (cid:17) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C k G ( π ) ∇ π k L p ( R + ;H s − ,q ( R n )) ≤ C (cid:16) k G ( π ) k L p ( R + ;H s − ,q ( R n )) k∇ π k L ∞ ( R + ;L ∞ ( R n )) + k G ( π ) k L ∞ ( R + ;L ∞ ( R n )) k∇ π k L p ( R + ;H s − ,q ( R n )) (cid:17) ≤ C (cid:16) k G ( π ) k L p ( R + ;H s − ,q ( R n )) k∇ π k L ∞ ( R + ;B s +1 − /pq,p ( R n )) + k G ( π ) k L ∞ ( R + ;B s +2 − /pq,p ( R n )) k∇ π k L p ( R + ;H s − ,q ( R n )) (cid:17) ≤ C n(cid:16) k π k L p ( R + ;H s,q ( R n )) + k π k s L p ( R + ;H s,q ( R n )) (cid:17) · (cid:16) k π k W ,p ( R + ;H s,q ( R n )) + k π k L p ( R + ;H s +2 ,q ( R n )) (cid:17) + (cid:16) k π k L p ( R + ;H s,q ( R n )) + k π k s L p ( R + ;H s,q ( R n )) (cid:17) k π k L p ( R + ;H s,q ( R n )) o ≤ C k U k s +1 ,q ( R n ) × H s,q ( R n ) for U ∈ X p,q,s , where we have used the embedding properties:B s +2 q,p ( R n ) ֒ → B s +1 q,p ( R n ) ֒ → L ∞ ( R n ) , W ,p ( R + ; H s,q ( R n )) ∩ L p ( R + ; H s +2 ,q ( R n )) ֒ → L ∞ ( R + ; B s +2 − /pq,p ( R n )) nd the estimate k π k L p ( R + ;H s,q ( R n )) + k π k s L p ( R + ;H s,q ( R n )) ≤ (cid:16) k π k s − p ( R + ;H s,q ( R n )) (cid:17) k π k L p ( R + ;H s,q ( R n )) ≤ (cid:16) k U k s − X p,q,s (cid:17) k π k L p ( R + ;H s,q ( R n )) ≤ (cid:16) C s − k U k s − s +1 ,q ( R n ) × H s,q ( R n ) (cid:17) k π k L p ( R + ;H s,q ( R n )) ≤ (cid:18) s − (cid:19) k π k L p ( R + ;H s,q ( R n )) . (5.9)Similarly, we also obtain (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) (cid:16) P ′′ (1) π ∇ π (cid:17) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C k U k s +1 ,q ( R n ) × H s,q ( R n ) . Since we see that (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +2 ,q ( R n )) ≤ − C k U k H s +1 ,q ( R n ) × H s,q ( R n ) ≤ , (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R + ;L ∞ ( R n )) ≤ − C k U k H s +1 ,q ( R n ) × H s,q ( R n ) ≤ U ∈ X p,q,s supposing (5.3), the Sobolev embedding theorem and Lemma 4.3 implies (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ )div (cid:18)
11 + π ( τ ) m ( τ ) ⊗ m ( τ ) (cid:19) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)
11 + π m ⊗ m (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s − ,q ( R n )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s − ,q ( R n )) k m k ∞ ( R + ;L ∞ ( R n )) +2 (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R + ;L ∞ ( R n )) k m k L p ( R + ;H s − ,q ( R n )) k m k L ∞ ( R + ;L ∞ ( R n )) ! ≤ C (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s − ,q ( R n )) k m k ∞ ( R + ;B s +1 ,qq,p ( R n )) +2 (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R + ;B s +2 − /pq,p ( R n )) k m k L p ( R + ;H s − ,q ( R n )) k m k L ∞ ( R + ;B s +1 − /pq,p ( R n )) ! ≤ C (cid:18)(cid:16) k m k W ,p ( R + ;H s − ,q ( R n )) + k m k L p ( R + ;H s +1 ,q ( R n )) (cid:17) + k m k L p ( R + ;H s − ,q ( R n )) (cid:16) k m k W ,p ( R + ;H s − ,q ( R n )) + k m k L p ( R + ;H s +1 ,q ( R n )) (cid:17)(cid:17) ≤ CC k U k s +1 ,q ( R n ) × H s,q ( R n ) . Using the same argument, we also arrive at (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ )∆ (cid:18) π ( τ )1 + π ( τ ) m ( τ ) (cid:19) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) π π m (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) k π k L ∞ ( R + ;L ∞ ( R n )) k m k L ∞ ( R + ;L ∞ ( R n )) + (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R + ;L ∞ ( R n )) k π k L p ( R + ;H s +1 ,q ( R n )) k m k L ∞ ( R + ;L ∞ ( R n )) (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R + ;L ∞ ( R n )) k π k L ∞ ( R + ;L ∞ ( R n )) k m k L p ( R + ;H s +1 ,q ( R n )) ! ≤ CC k U k s +1 ,q ( R n ) × H s,q ( R n ) (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) ν ∇ div (cid:18) π ( τ )1 + π ( τ ) m ( τ ) (cid:19) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) π π m (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) k π k L ∞ ( R + ;L ∞ ( R n )) k m k L ∞ ( R + ;L ∞ ( R n )) + (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R + ;L ∞ ( R n )) k π k L p ( R + ;H s +1 ,q ( R n )) k m k L ∞ ( R + ;L ∞ ( R n )) + (cid:13)(cid:13)(cid:13)(cid:13)
11 + π (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R + ;L ∞ ( R n )) k π k L ∞ ( R + ;L ∞ ( R n )) k m k L p ( R + ;H s +1 ,q ( R n )) ! ≤ CC k U k s +1 ,q ( R n ) × H s,q ( R n ) (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) κ div ( π ∆ π ) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C k π ∆ π k L p ( R + ;H s,q ( R n )) = C (cid:13)(cid:13)(cid:13)(cid:13) n X j =1 ∂ j ( ∂ j π ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s,q ( R n )) ≤ C (cid:13)(cid:13) π (cid:13)(cid:13) L p ( R + ;H s +2 ,q ( R n )) ≤ C k π k L p ( R + ;H s +2 ,q ( R n )) k π k L ∞ ( R + ;L ∞ ( R n )) ≤ CC k U k s +1 ,q ( R n ) × H s,q ( R n ) (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) κ div (cid:26)(cid:18) − |∇ π | (cid:19) I − ∇ π ⊗ ∇ π (cid:27) (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C k ( ∇ π ) k L p ( R + ;H s,q ( R n )) ≤ C k∇ π k L p ( R + ;H s,q ( R n )) k∇ π k L ∞ ( R + ;L ∞ ( R n )) ≤ CC k U k s +1 ,q ( R n ) × H s,q ( R n ) . Combining the estimates above, we obtain (5.6). Similarly, we have (5.7), which implies (5.4).We now turn to prove (5.5). To this end, it is enough to show the estimates (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ )( F ( τ ) − F ( τ )) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C k U − U k X p,q,s , (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ )( F ( τ ) − F ( τ )) d τ (cid:13)(cid:13)(cid:13)(cid:13) W ,p ( R + ;H s − ,q ( R n )) ≤ C k U − U k X p,q,s (5.10)hold for U , U ∈ X p,q,s , where we have set F ( τ ) := F ( π ( x, τ ) , m ( x, τ )) and F ( τ ) := F ( π ( x, τ ) , m ( x, τ )).Since we have π , π ∈ H s,q ( R n ), see (5.8), Lemmas 5.2, 5.4 and the condition (5.3) imply k G ( π ) − G ( π ) k H s,q ( R n ) ≤ C k π − π k H s,q ( R n ) (cid:0) k π k H s,q ( R n ) + k π k H s,q ( R n ) (cid:1) + C k π − π k H s,q ( R n ) (cid:0) k π k H s,q ( R n ) + k π k H s,q ( R n ) (cid:1) s ≤ C k π − π k H s,q ( R n ) (cid:16) k π k L ∞ ( R + ;B s +2 − /pq,p ( R n )) + k π k L ∞ ( R + ;B s +2 − /pq,p ( R n )) (cid:17) + C k π − π k H s,q ( R n ) (cid:16) k π k L ∞ ( R + ;B s +2 − /pq,p ( R n )) + k π k L ∞ ( R + ;B s +2 − /pq,p ( R n )) (cid:17) s ≤ C k π − π k H s,q ( R n ) , k G ( π ) − G ( π ) k B s +1 − /pq,p ( R n ) C k π − π k B s +1 − /pq,p ( R n ) (cid:16) k π k B s +1 − /pq,p ( R n ) + k π k B s +1 − /pq,p ( R n ) (cid:17) ≤ C k π − π k B s +1 − /pq,p ( R n ) · (cid:16) k π k L ∞ ( R + ;B s +2 − /pq,p ( R n )) + k π k L ∞ ( R + ;B s +2 − /pq,p ( R n )) (cid:17) ≤ C k π − π k B s +1 − /pq,p ( R n ) because we have the embedding B s +2 − /pq,p ( R n ) ֒ → B s +1 − /pq,p ( R n ) ֒ → H s,q ( R n ) ֒ → L ∞ ( R n ) and the esti-mate similar to (5.9). Thus, we observe that (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ ) (cid:16) G ( π ) ∇ π − G ( π ) ∇ π (cid:17) d τ (cid:13)(cid:13)(cid:13)(cid:13) L p ( R + ;H s +1 ,q ( R n )) ≤ C k ( G ( π ) − G ( π )) ∇ π + G ( π ) ∇ ( π − π ) k L p ( R + ;H s − ,q ( R n )) ≤ C (cid:16) k G ( π ) − G ( π ) k L p ( R + ;H s,q ( R n )) k∇ π k L ∞ ( R + ;L ∞ ( R n )) + k G ( π ) − G ( π ) k L ∞ ( R + ;L ∞ ( R n )) k∇ π k L p ( R + ;H s − ,q ( R n )) + k G ( π ) k L p ( R + ;H s − ,q ( R n )) k∇ ( π − π ) k L ∞ ( R + ;L ∞ ( R n )) + k G ( π ) k L ∞ ( R + ;L ∞ ( R n )) k∇ ( π − π ) k L p ( R + ;H s − ,q ( R n )) (cid:17) ≤ C (cid:16) k π − π k L p ( R + ;H s,q ( R n )) k∇ π k L ∞ ( R + ;B s +1 − /pq,p ( R n )) + k π − π k L ∞ ( R + ;B s +1 − /pq,p ( R n )) k∇ π k L p ( R + ;H s − ,q ( R n )) + k π k L p ( R + ;H s − ,q ( R n )) k∇ ( π − π ) k L ∞ ( R + ;B s +1 − /pq,p ( R n )) + k π k L ∞ ( R + ;B s +2 − /pq,p ( R n )) k∇ ( π − π ) k L p ( R + ;H s − ,q ( R n )) (cid:17) ≤ C k U k H s +1 ,q ( R n ) × H s,q ( R n ) k U − U k X p,q,s . The rest terms can be shown by employing the similar arguments above, which yields (5.10).
Proof of Theorem 1.1.
Define the map Φ byΦ( U )( t ) = T ( t ) U − Z t T ( t − τ ) N ( τ ) d τ. According to the estimate (5.2) and Lemma 5.1, for all U , we see that k Φ( U ) k X p,q,s ≤ C k U k H s +1 ,q ( R n ) × H s,q ( R n ) + C k U k s +1 ,q ( R n ) × H s,q ( R n ) ≤ ( C + C k U k H s +1 ,q ( R n ) × H s,q ( R n ) ) k U k H s +1 ,q ( R n ) × H s,q ( R n ) On the other hand, we observe that k Φ( U ) − Φ( U ) k X p,q,s ≤ (cid:13)(cid:13)(cid:13)(cid:13) Z t T ( t − τ )( N ( τ ) − N ( τ )) d τ (cid:13)(cid:13)(cid:13)(cid:13) X p,q,s ≤ C C k U k H s +1 ,q ( R n ) × H s,q ( R n ) k U − U k X p,q,s for all U , U ∈ X p,q,s . Taking the initial data U ∈ H s +1 ,q ( R n ) × H s,q ( R n ) n such that k U k H s +1 ,q ( R n ) × H s,q ( R n ) ≤ min (cid:18) C , C C , C C (cid:19) we deduce that k Φ( U ) k X p,q,s ≤ C k U k H s +1 ,q ( R n ) × H s,q ( R n ) , k Φ( U ) − Φ( U ) k X p,q,s ≤ k U − U k X p,q,s assuming that U , U , U ∈ X p,q,s . Namely, Φ is the contraction mapping on X p,q,s , so that there existsa unique solution U ∈ X p,q,s satisfying (1.1) with ̺ = 1 + π . This concludes the proof. Derivation of the Green matrix
We here give a derivation of the Green matrix. According to (3.3), we have the initial-value problem for b φ ( ξ, · ): ( ∂ t b φ + ( µ + ν ) | ξ | ∂ t b φ + ( γ + κ | ξ | ) | ξ | b φ = 0 , ( ∂ t b φ ( ξ, , b φ ( ξ, − i ξ · b u ( ξ ) , b φ ( ξ )) . (A.1)We then see that the characteristic equation corresponding to (A.1) is given by (3.4). In the following,let λ ± ( ξ ) be the roots of (3.4).We first consider the case λ + ( ξ ) = λ − ( ξ ), that is, Case 1 with | ξ | 6 = B and Case 2–5. In these cases, b φ is denoted by b φ = − λ − b φ + i ξ · b u λ + − λ − e λ + t + λ + b φ + i ξ · b u λ + − λ − e λ − t = λ + e λ − t − λ − e λ + t λ + − λ − ! b φ − e λ + t − e λ − t λ + − λ − ! (i ξ · b u ) . (A.2)On the other hand, by (3.3), we also obtain ∂ t b u = − ( γ + κ | ξ | )i ξ b φ + ( − µ | ξ | b u − νξ ( ξ · b u )) . We decompose b u into components parallel to and orthogonal to ξ : b u ( ξ, t ) = a ( ξ, t ) ξ | ξ | + b ( ξ, t ) , where a := ( ξ · b u ) / | ξ | is the scalar and b is the vector perpendicular to ξ . We see that a and b enjoy ∂ t a = − ( µ + ν ) | ξ | a − i( γ + κ | ξ | ) | ξ | b φ, ∂ t b = − µ | ξ | b , respectively. Hence, we obtain a ( ξ, t ) = e − ( µ + ν ) | ξ | t a ( ξ, e − ( µ + ν ) | ξ | t ( γ + κ | ξ | ) | ξ | Z t e ( µ + ν ) | ξ | τ b φ ( ξ, τ ) d τ, b ( ξ, t ) = e − µ | ξ | t (cid:18) I − ξ ⊤ ξ | ξ | (cid:19)b u . (A.3)Recalling (A.2) and noting λ ± + ( µ + ν ) | ξ | = − λ ∓ and λ + λ − = ( γ + κ | ξ | ) | ξ | , we arrive at (3.7).We next focus on Case 6: λ + ( ξ ) ≡ λ − ( ξ ) for any ξ ∈ R n \ { } . Since λ + = λ − = − A | ξ | is the doubleroot of (3.4), b φ can be written by b φ = αe − A | ξ | t + βte − A | ξ | t with some scalar functions α = α ( ξ ) and β = β ( ξ ). The initial conditions imply α = b φ , β = A | ξ | b φ − i ξ · b u , so that we obtain b φ = (1 + A | ξ | t ) e − A | ξ | b φ − te − A | ξ | t (i ξ · b u ) . (A.4)Noting λ ± + ( µ + ν ) | ξ | = − λ ∓ and λ + λ − = A | ξ | and using (A.3) and (A.4), we have (3.12). Acknowledgments
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