Inviscid limit of the inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in \mathbb{R}^3
aa r X i v : . [ m a t h . A P ] F e b Inviscid limit of the inhomogeneous incompressible Navier-Stokes equationsunder the weak Kolmogorov hypothesis in R Dixi Wang a , Cheng Yu b , Xinhua Zhao c, ∗ a Department of Mathematics, University of Florida, Gainesville, FL 32611, United States of America b Department of Mathematics, University of Florida, Gainesville, FL 32611, United States of America c School of Mathematics, South China University of Technology, Guangzhou 510641, China
Abstract
In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equationsunder the weak Kolmogorov hypothesis in R . In particular, we first deduce the Kolmogorov-type hy-pothesis in R , which yields the uniform bounds of α th -order fractional derivatives of √ ρ µ u µ in L x forsome α >
0, independent of the viscosity. The uniform bounds can provide strong convergence of √ ρ µ u µ in L space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations. Keywords:
Inviscid limit, Kolmogorov hypothesis, inhomogeneous Navier-Stokes equations, Eulerequations.
Contents1 Introduction 12 The Kolmogorov hypotheses for the inhomogeneous incompressible fluids 53 Compactness of weak solutions 74 Vanishing viscosity limit 115 Appendix 13References 141. Introduction
The main purpose of this paper is to study the vanishing viscosity limit of inhomogeneous Navier-Stokes equations in R under the well known hypothesis by Kolmogorov [9, 10]. In particular, a weakerversion Assumption (KHw) of Assumption (KH) which was derived in [4, 5] in a periodic domain. Thiscan provide the convergence of weak solutions of the Navier-Stokes equations through a subsequenceto a solution of the Euler equations. We are particularly interested in extending these results to theinhomogeneous fluids in the whole space, because of the special structure of inhomogeneous equationsand di ff erent formulation of Kolmogorov hypothesis in R . ∗ Corresponding author
URL: [email protected] (Dixi Wang), [email protected] (Cheng Yu), [email protected] (Xinhua Zhao)
Preprint submitted to Elsevier February 5, 2021 e are particularly interested in the following Navier-Stokes equations for the inhomogeneous fluids: ρ t + div( ρ u ) = , ( ρ u ) t + div( ρ u ⊗ u ) + ∇ p − µ ∆ u = ρ f , div u = , (1.1)with initial condition ( ρ, ρ u ) | t = = ( ρ , m )( x ) , x ∈ R , (1.2)and boundary condition u → u ∞ , as | x | → + ∞ , f or all t ≥ , (1.3)where u ∞ is fixed vector in R . Here ρ, u , P stand for the density, velocity and pressure of the fluidrespectively, µ > ffi cient, and f = f ( x , t ) = ( f , f , f )( x , t ) denotes a given externalforce.There are many literatures studying the existence and uniqueness of solutions to inhomogeneous in-compressible Navier-Stokes equations, see Refs. [1, 8, 17] and the references therein. Ladyˇzenskaja andSolonnikov [11] first addressed the question of unique solvability of (1.1). The global existence of weaksolutions to (1.1) with the large initial data was first proved by Lions [14]. For clarity of presentation, weassume that u ∞ = T > f ∈ L (0 , T ; L ( R )), and theinitial data satisfies the following conditions: ρ ≥ a . e . in R , ρ ∈ L ∞ ( R ) , m ∈ L ( R ) , m = a . e . on { ρ = } , m ρ ∈ L ( R ) , (1.4)(1 / √ ρ )1 ( ρ <δ ) ∈ L ( R ) f or some positive constant δ . (1.5)If the initial data verifies the above conditions, Lions [14] proved the global existence of weak solutionsin the following sense: Definition 1.1 ([14]) . For any T > , ( ρ µ , u µ )( t , x ) is a weak solution on [0 , T ] of (1.1) - (1.3) if • √ ρ µ u µ ∈ L ∞ (0 , T ; L ( R )) , ∇ u µ ∈ L ((0 , T ) × R ) , ρ µ ∈ C ([0 , T ]; L p ( R )) f or ≤ p < ∞ , • For any φ ∈ C ∞ ( R + × R ; R ) such that div φ = , Z T Z R (cid:0) ρ µ u µ · φ t + ( ρ µ u µ ⊗ u µ ) : ∇ φ + ρ µ f · φ (cid:1) d x d t + Z R m ( x ) · φ (0 , x ) d x = Z T Z R µ ∇ u µ · ∇ φ d x d t . (1.6) • For any φ ∈ C ∞ ( R + × R ; R ) Z T Z R ( ρ µ φ t + ρ µ u µ · ∇ φ ) d x d t + Z R ρ µ φ (0 , x ) d x = . (1.7) • The energy inequality holds for any t ∈ [0 , T ] Z R ρ µ | u µ | x + Z t Z R µ |∇ u µ | d x d s ≤ Z R | m | ρ d x + Z t Z R ρ µ u µ · f d x d s . (1.8)2owever the uniqueness of the weak solutions in three-dimensional space is still an interesting openquestion, which is linked to the global regularity.The inviscid limit for Navier-Stokes equations has also been extensively studied (see Refs.[2, 3, 15]for instance). The goal of this paper is to show that the weak solutions of (1.1)-(1.3) converge to a weaksolution of the following Euler equations, if they are under the Kolmogorov hypothesis: ρ t + div( ρ u ) = , ( ρ u ) t + div( ρ u ⊗ u ) + ∇ P = ρ f , div u = , (1.9)with the same initial data (1.2). The definition of global weak solution to (1.9) and (1.2) is given asfollows: Definition 1.2.
For any T > , ( ρ, u )( t , x ) is called a global weak solution on [0 , T ] of (1.9) with initialdata ( ρ , m ) if ( ρ, u ) satisfies: • ( ρ, u ) solves the system (1.9) in the sense of distributions in [0 , T ] × R , i.e.(a). For any φ ∈ C ∞ ( R + × R ; R ) such that div φ = , Z T Z R ( √ ρ √ ρ u · φ t + ( √ ρ u ⊗ √ ρ u ) : ∇ φ + ρ f · φ ) d x d t + Z R m ( x ) · φ (0 , x ) d x = , (1.10) (b). For any φ ∈ C ∞ ( R + × R ; R ) Z T Z R ( ρφ t + √ ρ √ ρ u · ∇ φ ) d x d t + Z R ρ φ (0 , x ) d x = . (1.11) • (1.2) holds in D ′ ( R ) , • the energy inequality holds for any t ∈ (0 , ∞ ) Z R ρ | u | x ≤ Z R | m | ρ d x + Z t Z R ρ u · f d x d t . (1.12)In this paper, we aim to investigate such limits of global weak solutions from the inhomogeneous in-compressible Navier-Stokes equations to the corresponding Euler equations under the weaker Kolmogorov-type hypothesis, which was particularly motivated by [4] for incompressible fluids and [5] for com-pressible fluids. Compared to these two models, the special features of inhomogeneous incompressibleNavier-Stokes equations bring new di ffi culties to the mathematical analysis. Specifically, on one hand,the pressure-density relation for the compressible flows does not hold for the inhomogeneous flow any-more. On the other hand, the term ∂ t ( ρ u ) is nonlinear in ( ρ, u ) thus requires additional argument to handlethe regularity in time, which is not necessary for homogeneous incompressible Euler system.In fact, the same issue arises in the mathematical study of the energy equality for the weak solutionsof compressible or inhomogeneous flows, see reference [6, 7, 12, 18] and their references. To circumventthis time regularity assumption, one approach is to formulate the equations in terms of the density andenergy equality for the momentum m = ρ u and obtain the energy equality by multiplying the momentumequation by ( ρ u ) ε /ρ ε instead of u ε , where ε is a suitable regulation. However, ( ρ u ) ε /ρ ε cannot keepthe divergence free structure, thus requires a commutator estimate involving the pressure and additionalregularity of the pressure is needed. In [6, 18], the authors use ( ϕ ( t ) u ε ) ε where ϕ ( t ) is su ffi cient nicefunction. This function is divergence free so it does not require any additional regularity of the pressure.It succeeded in removing the di ffi culty related to the regularity of pressure in these works. Such ideascarried into this current paper as well. 3n addition to the special feature of inhomogeneous incompressible Navier-Stokes equations, we aremainly interested in the inviscid limit problem in the whole space. We adopt the Fourier integrals toexpress the Kolmogorov-type hypothesis in R , which is di ff erent from the case in T P = [ − P , P ] , P > , in [4, 5]. Our goal is to show that the same conclusion in [4, 5] holds in R . To this end, we introducethe Kolmogorov-type hypothesis for inhomogeneous incompressible fluids in R using physical notionin [16]. The details can be founded in section 2. Accordingly, it is interesting to investigate the inviscidlimit under the weaker version of Kolmogorov-type hypothesis in R . We present our main result asfollows. Theorem 1.3.
If the weak solutions ( ρ µ , u µ ) of (1.1) - (1.3) as in Definition 1.1 are under Assumption(RICKHw) (2.26) , there exists a subsequence (still denoted) ( ρ µ , u µ ) and a function ( ρ, u ) such that as µ → , ρ µ → ρ weakly in L p ((0 , T ) × R ) , p ρ µ u µ → √ ρ u in L ((0 , T ) × R ) , (1.13) where < p < ∞ , and ( ρ, u ) is a weak solution of (1.9) with initial data ( ρ , m ) . Remark 1.4.
The inviscid limit for compressible Navier-Stokes equations in R for any γ > can beobtained by the same method as this paper. However, for homogeneous incompressible Navier-Stokesequations, the total energy E ( t ) per unit mass vanishes as defined in Kolmogorov hypothesis in [16]. Meanwhile, we can establish a similar result in T P = [ − P , P ] , P > . The proof will be given inAppendix at the end of this paper.
Theorem 1.5.
If we consider the fluid in a domain with period T P = [ − P / , P / ⊂ R , P > , one candeduce the same result as Theorem 1.3. However, in this case, the Kolmogorov hypothesis is deduced asfollows: sup k ≥ k ∗ (cid:16) | k | + β Z T | d √ ρ u ( t , k ) | d t (cid:17) ≤ C T , f or some β > di ff erent from Assumption (RICKHw). In fact, for the domain T P , the total energy E ( t ) per unit masssatisfies E ( t ) = R T P ρ µ d x Z T P ρ µ | u µ | x = X k ≥ E ( t , k ) = X k ≥ π q ( t , k ) k , (1.15) and the weighted velocity √ ρ u ( t , x ) can be expanded to Fourier series, i.e., √ ρ u ( t , x ) = X k d √ ρ u ( t , k ) e i k · x . (1.16) The proof highly relies on (1.14) and (1.16) , see Appendix 5.
In general, the global existence theory of weak solutions to Euler equations in three dimensionalspace has not been established yet. We make our attempt to find a way of obtaining existence result ofthe inhomogeneous Euler equations under special conditions. In general, the vanishing viscosity limit ofNavier-Stokes equations is not a solution to the corresponding Euler equations, since the uniform boundscannot guarantee the convergence of the nonlinear term ρ µ u µ ⊗ u µ .In order to pass the limit of the nonlinear term ρ µ u µ ⊗ u µ , if we follow the techniques in [5] toshow strong convergene of ρ µ u µ , the additional regularity on the pressure is required. It seems noteasy to obtain such a regularity for the weak solutions. However, we do not have a pressure law forinhomogeneous fluids, and the velocity is not good enough to be a test function even after smoothing. Ourkey idea is to show the strong convergence of √ ρ µ u µ in L t , x , this yields the weak limit of a subsequenceis a weak solution to the Euler equations. We emphasize the fact that our argument does not need anyadditional regularity on the pressure, since we find a suitable divergence free test function. This yieldsour Lemma 3.5 so as to show the strong convergence of √ ρ µ u µ . ffi cient weaker version of Assumption(RICKHw) for this current project. In section 3, we derive the compactness of weak solutions of theNavier-Stokes equations when the viscosity coe ffi cient vanishes, which is crucial to obtain the weaksolution to the Euler equations. In Section 4, we prove our main result based on the compactness resultsin Section 3. We give an outline of the proof to Theorem 1.5 in Section 5 as an appendix.
2. The Kolmogorov hypotheses for the inhomogeneous incompressible fluids
In this section, we introduce the Kolmogorov hypotheses for the inhomogeneous incompressible flu-ids in R and the corresponding Kolmogorov-type hypothesis (RICKH) in mathematical terms. Note that,two fundamental assumptions for the isotropic incompressible turbulence were proposed by Kolmogorov[9, 10]:(i) At su ffi ciently high wavenumbers, the energy spectrum E ( t , k ) can depend only on the fluid vis-cosity µ, the dissipation rate ε and the wavenumber k itself.(ii) E ( t , k ) should be independent of the viscosity as the Reynolds number tends to infinity: E ( t , k ) ≈ αε / k − / in the limit of infinite Reynolds number, where α may depend on t, but is independent of k , ε. Under the above Kolmogorov’s two hypotheses, Chen-Glimm [4, 5] interpreted in mathematicalterms for the incompressible and compressible Kolmogorov-type hypothesis in T P . We can generalizethem to R n for the fluid equations as follows. Assumption (RICKH).
For any T > , there exists C T > and k ∗ (su ffi ciently large) depending on ρ , m and f but independent of the viscosity µ such that, for k = | k | ≥ k ∗ , Z T E ( t , k )d t ≤ C T k − . (2.17)On the one hand, McComb [16] defined the spectral tensor Q αβ ( t , k ), and stated the relationshipbetween the trace of Q αβ ( t , k ) and the energy E ( t ) per unit mass of fluid at time t, i.e.2 E ( t ) = trQ αβ ( t , r ) | r = = tr Z ∞ q ( t , k ) k d k Z D αβ ( k )d Ω k , (2.18)where D αβ ( k ), q ( t , k ) and d Ω k denote projection operator, spectral density and the elementary solid anglein wavenumber space respectively. Using (2.18), Leslie [13] obtained the following crucial result E ( t ) = π tr δ αβ Z ∞ q ( t , k ) k d k = Z ∞ π k q ( t , k )d k = Z ∞ E ( t , k )d k . (2.19)On the other hand, from the energy inequality (1.8) of the weak solutions ( ρ µ , u µ ), the Gronwallinequality yields to Z R ρ µ | u µ | x + Z t Z R µ |∇ u µ | d x d t ≤ M T , (2.20)5here M T is a positive constant that depends on intial data, f and T, but independent of µ . By (2.19) and(2.20), the total energy E ( t ) per unit mass at time t for the inhomogeneous turbulence is: E ( t ) = R R ρ µ d x Z R ρ µ | u µ | x = Z ∞ E ( t , k )d k = Z ∞ π q ( t , k ) k d k . (2.21)Furthermore, when we consider the domain T P = [ − P / , P / ⊂ R , P > , the wavevector k = ( k , k , k ) = π P ( n , n , n ) ∈ R , with n j = , ± , ± , · · · and j = , , , is discrete. When the settingis in R , in some sense, we can view that it is the large box limit as P → ∞ . Since the wavevector iscontinuous, we introduce the Fourier transform for the weighted velocity √ ρ u ( t , x ) in the x -variable asfollows d √ ρ u ( t , k ) = π ) Z R √ ρ u ( t , x ) e − i k · x d x , (2.22)thus √ ρ u ( t , x ) = Z R d √ ρ u ( t , k ) e i k · x d k . (2.23)By Parseval identity, we have k p ρ µ u µ k L x = k [ p ρ µ u µ k L k . (2.24)Clearly, it is usually more convenient to use the spherical coordinates ( k , θ, ϕ ), where k = | k | , ≤ θ ≤ π, ≤ ϕ ≤ π. Note that the facts [ p ρ µ u µ ( t , k ) = [ p ρ µ u µ ( t , k , θ, ϕ ) , and (2.24), we derive that Z ∞ E ( t , k ) d k = R R ρ µ d x Z R ρ µ | u µ | x = R R ρ d x Z R | [ p ρ µ u µ ( t , k ) | d k = R R ρ d x Z π Z π Z ∞ | [ p ρ µ u µ ( t , k , θ, ϕ ) | k sin ϕ d k d θ d ϕ, (2.25)where we used Z R ρ µ ( x , t ) d x = Z R ρ ( x ) d x , which can be derived from the mass equation in (1.1).Thanks to Assumption (RICKH), E ( t , k ) = E ( t , k , θ, ϕ ) and the equality (2.25), we are able to obtainthe following weaker version of Assumption (RICKHw). This can provide the uniform bound for ourcompactness analysis in this current paper. Assumption (RICKHw).
For any T > , there exists C T > and k ∗ (su ffi ciently large) depending on ρ , m and f but independent of the viscosity µ such that, for k = | k | ≥ k ∗ , sup k ≥ k ∗ (cid:16) | k | + β Z T Z π Z π | [ p ρ µ u µ ( t , k , θ, ϕ ) | k sin ϕ d θ d ϕ d t (cid:17) ≤ C T , f or some β > . (2.26)6 . Compactness of weak solutions We develop the compactness of weak solutions when the viscosity coe ffi cient vanishes in this section.The following lemma is crucial to show the compactness of weak solutions. Lemma 3.1.
Under Assumption (RICKHw), for any T ∈ (0 , + ∞ ) , there exists C > , independent of µ > , such that Z T Z R | D α x ( p ρ µ u µ ) | d x d t ≤ C , (3.27) where α ∈ (0 , + β ) , C the generic positive constants depending on ρ , m , f , α, k ∗ and T > , butindependent of µ > . Remark 3.2.
A similar version for periodic domain case were given in [4, 5], where the proof relied onthe definition of fractional derivatives via Fourier series.Proof.
Note that the definition of fractional derivatives via Fourier transform, the Parseval identity, thespherical coordinates and Assumption (RICKHw), we have Z T Z R | D α x ( p ρ µ u µ )( t , x ) | d x d t = Z T Z R | [ D α x ( p ρ µ u µ )( t , k ) | d k d t = Z T Z R | k | α | [ p ρ µ u µ ( t , k ) | d k d t = Z T Z π Z π Z ∞ k α | [ p ρ µ u µ ( t , k , θ, ϕ ) | k sin ϕ d k d θ d ϕ d t = Z T Z π Z π Z k ∗ k α | [ p ρ µ u µ | k sin ϕ d k d θ d ϕ d t + Z T Z π Z π Z ∞ k ∗ k α | [ p ρ µ u µ | k sin ϕ d k d θ d ϕ d t ≤ Ck α ∗ Z T Z R | p ρ µ u µ | d x d t + C Z ∞ k ∗ k α − − β d k ≤ C , (3.28)where α < + β . (cid:3) With Lemma 3.1 at hand, we are able to have the following uniform bound of √ ρ µ u µ in µ. Corollary 3.3.
Under
Assumption (RICKHw) , for any T > , there exists C > , independent of µ > , such that for q = q ( β ) > , k p ρ µ u µ k L q ((0 , T ) × R ) ≤ C . (3.29) Proof.
In view of Gagliardo-Nirenberg interpolation inequality and Lemma 3.1, we have k p ρ µ u µ k L q ≤ k p ρ µ u µ k α L k p ρ µ u µ k − α H α , (3.30)where q = α + (cid:0) − α (cid:1) (1 − α ) for 0 < α < . Note that √ ρ µ u µ is uniformly bounded in L (0 , T ; H α ( R )) , we get q = + α > . (cid:3) Now it turns to show that √ ρ µ u µ is equicontinuous with respect to the space variable x , which isnecessary for our argument to get its convergence. 7 emma 3.4. Under
Assumption(RICKHw) , for any T ∈ (0 , + ∞ ) , √ ρ µ u µ is equicontinuous with respectto the space variable x in L ((0 , T ) × R ) , independent of µ, i.e., Z T Z R | p ρ µ u µ ( t , x + ∆ x ) − p ρ µ u µ ( t , x ) | d x d t → , as ∆ x → . (3.31) Proof.
Using Lemma 3.1 and Parseval identity, we obtain Z T Z R | p ρ µ u µ ( t , x + ∆ x ) − p ρ µ u µ ( t , x ) | d x d t = Z T Z R (cid:12)(cid:12)(cid:12)(cid:12) Z R [ p ρ µ u µ ( t , k ) e i k · x ( e i k · ∆ x −
1) d k (cid:12)(cid:12)(cid:12)(cid:12) d x d t ≤ C Z T Z R Z R e i k · x d k Z R | [ p ρ µ u µ ( t , k ) | ( e i k · ∆ x − d k d x d t ≤ C | ∆ x | α Z R δ (2 x ) d x Z T Z R | k | α | [ p ρ µ u µ ( t , k ) | d k d t ≤ C | ∆ x | α Z T Z R | D α x ( p ρ µ u µ ) | d x d t ≤ C | ∆ x | α , where δ ( x ) is the Dirac delta function. (cid:3) To show the strong convergence of √ ρ µ u µ in L space, it is crucial to have the following lemma. Lemma 3.5.
Under
Assumption (RICKHw) , for any T > , √ ρ µ u µ is equicontinuous with respect tothe time variable t in L ((0 , T − ∆ t ) × R ) , independent of µ, i.e. Z T − ∆ t Z R | p ρ µ u µ ( t + ∆ t , x ) − p ρ µ u µ ( t , x ) | d x d t → , as ∆ t → . (3.32) Proof.
For simplicity, we drop the superscript µ of √ ρ µ u µ in the following proof. For any ϕ ( t ) ∈D (0 , + ∞ ) , we have Z T − ∆ t Z R | √ ρ u ( t + ∆ t , x ) − √ ρ u ( t , x ) | d x d t = Z T − ∆ t Z R [ ρ u ( t + ∆ t , x ) − ρ u ( t , x )][ u ( t + ∆ t , x ) − u ( t , x )] d x d t + Z T − ∆ t Z R √ ρ u ( t + ∆ t , x ) u ( t , x )[ √ ρ ( t + ∆ t , x ) − √ ρ ( t , x )] d x d t + Z T − ∆ t Z R √ ρ u ( t , x ) u ( t + ∆ t , x )[ √ ρ ( t , x ) − √ ρ ( t + ∆ t , x )] d x d t = Z T − ∆ t Z R [ ρ u ( t + ∆ t , x ) − ρ u ( t , x )] (cid:8) [ u ( t + ∆ t , x ) − ϕ ( t ) u ǫ ( t + ∆ t , x )] − [ u ( t , x ) − ϕ ( t ) u ǫ ( t , x )] (cid:9) d x d t + Z T − ∆ t Z R [ ρ u ( t + ∆ t , x ) − ρ u ( t , x )][ ϕ ( t ) u ǫ ( t + ∆ t , x ) − ϕ ( t ) u ǫ ( t , x )] d x d t + Z T − ∆ t Z R √ ρ u ( t + ∆ t , x ) u ( t , x )[ √ ρ ( t + ∆ t , x ) − √ ρ ( t , x )] d x d t + Z T − ∆ t Z R √ ρ u ( t , x ) u ( t + ∆ t , x )[ √ ρ ( t , x ) − √ ρ ( t + ∆ t , x )] d x d t = I + I + I + I , (3.33)8here u ǫ ( t , x ) = ( j ǫ ∗ u )( t , x ) = Z | t − s |≤ ǫ Z | x − y |≤ ǫ j ǫ ( t − s , x − y ) u ( s , y ) d y d s , and j ǫ is a standard mollifier. Note that ϕ ( t ) u ǫ ( t + ∆ t , x ), ϕ ( t ) u ǫ ( t , x ) are well defined in D (0 , + ∞ ) . We divide the equality (3.33) into four terms, and each term will be considered separately. First, wecan rewrite I as following I = Z T − ∆ t Z R √ ρ u ( t + ∆ t , x ) u ( t , x )[( √ ρ ( t + ∆ t , x ) − p ρ ǫ ( t + ∆ t , x )) − ( √ ρ ( t , x ) − p ρ ǫ ( t , x ))] d x d t + Z T − ∆ t Z R √ ρ u ( t + ∆ t , x ) u ( t , x )[ p ρ ǫ ( t + ∆ t , x ) − p ρ ǫ ( t , x )] d x d t = Z T − ∆ t Z R √ ρ ( u ( t + ∆ t , x ) u ( t , x )( √ ρ ( t + ∆ t , x ) − p ρ ǫ ( t + ∆ t , x )) d x d t + Z T − ∆ t Z R √ ρ u ( t + ∆ t , x ) u ( t , x )( p ρ ǫ ( t , x ) − √ ρ ( t , x )) d x d t + Z T − ∆ t Z R √ ρ u ( t + ∆ t , x ) u ( t , x )( p ρ ǫ ( t + ∆ t , x ) − p ρ ǫ ( t , x )) d x d t = I + I + I . (3.34)Here we decompose u into u = u { ρ<δ } + u { ρ ≥ δ } = u + u , then in view of √ ρ { ρ <δ } ∈ L ( R ) , wecan get k u { ρ<δ } k L l ≤ (cid:13)(cid:13)(cid:13) √ ρ { ρ<δ } (cid:13)(cid:13)(cid:13) L k √ ρ u k L q , l = qq + . (3.35)Thus u ∈ L l ((0 , T ) × R ). As for u , since √ ρ u ∈ L ∞ (0 , T ; L ( R )), we have u ∈ L ∞ (0 , T ; L ( R )) . Moreover, it holds that as ǫ → , k u − u ǫ k L l (0 , T ; L l ( R )) → , k u − u ǫ k L ∞ (0 , T ; L ( R )) → . (3.36)For I + I , using √ ρ ǫ → √ ρ in C ([0 , T ] , L p ( R )) , ≤ p < ∞ , as ǫ = ǫ ( ∆ t ) →
0, we have I + I ≤ Z T − ∆ t Z R √ ρ u u ( p ρ ǫ − √ ρ ) d x d t + Z T − ∆ t Z R √ ρ u u ( p ρ ǫ − √ ρ ) d x d t ≤ C k √ ρ u k L q (0 , T ; L q ( R )) k u k L l (0 , T ; L l ( R )) k p ρ ǫ − √ ρ k L ∞ (0 , T ; L p ( R )) + C k √ ρ u k L q (0 , T ; L q ( R )) k u k L ∞ (0 , T ; L ( R )) k p ρ ǫ − √ ρ k L ∞ (0 , T ; L p ( R )) → , as ∆ t → , (3.37)where q > p and p satisfy q + l + p = q + + p = I , using H ¨older inequality, we obtain I ≤ C k √ ρ u k L q (0 , T ; L q ( R )) k u k L l (0 , T ; L l ( R ) k p ρ ǫ ( t + ∆ t , x ) − p ρ ǫ ( t , x ) k L qq − ([0 , T ] × R ) + C k √ ρ u k L ∞ (0 , T ; L ( R )) k u k L ∞ (0 , T ; L ( R ) ) k p ρ ǫ ( t + ∆ t , x ) − p ρ ǫ ( t , x ) k L ∞ ([0 , T ] × R ) → , as ∆ t → , (3.38)where q > . Similarly, we have as ∆ t → , I → , independent of µ. I , from div u ǫ = I = Z T − ∆ t Z R Z t +∆ tt ( ρ u ) s ( s , x ) · [ ϕ ( t ) u ǫ ( t + ∆ t , x ) − ϕ ( t ) u ǫ ( t , x )] d s d x d t = Z T − ∆ t Z R Z t +∆ tt ( ρ u ⊗ u )( s , x ) : ∇ [ ϕ ( t ) u ǫ ( t + ∆ t , x ) − ϕ ( t ) u ǫ ( t , x )] d s d x d t − Z T − ∆ t Z R Z t +∆ tt µ ∇ u ( s , x ) · ∇ [ ϕ ( t ) u ǫ ( t + ∆ t , x ) − ϕ ( t ) u ǫ ( t , x )] d s d x d t + Z T − ∆ t Z R Z t +∆ tt ( ρ f )( s , x ) · [ ϕ ( t ) u ǫ ( t + ∆ t , x ) − ϕ ( t ) u ǫ ( t , x )] d s d x d t = I + I + I . (3.39)Notice that, for any x ∈ R , | ϕ ( t ) u ǫ ( t , x ) | ≤ C ǫ (cid:12)(cid:12)(cid:12)(cid:12) Z | t − s |≤ ǫ Z | x − y |≤ ǫ j ( t − s ǫ , x − y ǫ ) u ( s , y ) d y d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ǫ k u k L l (0 , T ; L l ( R ) (cid:16) Z | τ |≤ Z | z |≤ | j ( τ, z ) | l ′ ǫ d z d τ (cid:17) / l ′ + C ǫ k u k L ∞ (0 , T ; L ( R ) (cid:16) Z | τ |≤ Z | z |≤ | j ( τ, z ) | ǫ d z d τ (cid:17) / ≤ C ǫ q + / q , (3.40)and | ϕ ( t ) ∇ u ǫ ( t , x ) | ≤ C ǫ (cid:12)(cid:12)(cid:12)(cid:12) Z | t − s |≤ ǫ Z | x − y |≤ ǫ ∂ j ( t − s ǫ , x − y ǫ ) u ( s , y ) d y d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ǫ k u k L l (0 , T ; L l ( R ) (cid:16) Z | τ |≤ Z | z |≤ | ∂ z j ( τ, z ) | l ′ ǫ d z d τ (cid:17) / l ′ + C ǫ k u k L ∞ (0 , T ; L ( R ) (cid:16) Z | τ |≤ Z | z |≤ | ∂ z j ( τ, z ) | ǫ d z d τ (cid:17) / ≤ C ǫ (3 q + / q . (3.41)For I + I , by (3.40),(3.41) and H ¨older inequality, we have I + I ≤ C ∆ t k √ ρ u k L | ϕ ∇ u ǫ | + C ∆ t k ρ k L k f k L | ϕ u ǫ |≤ C ∆ t ǫ (3 q + / q . (3.42)For I , in view of H ¨older inequality and the energy inequality in Definition 1.1, we obtain I ≤ C (cid:12)(cid:12)(cid:12)(cid:12) Z T − ∆ t Z R Z t +∆ tt √ µ ∇ u ( s , x ) d s · [ √ µ ∇ u ǫ ( t + ∆ t , x ) − √ µ ∇ u ǫ ( t , x )] d x d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C h Z T − ∆ t Z R (cid:16) Z t +∆ tt √ µ ∇ u ( s , x ) d s (cid:17) d x d t i h Z T − ∆ t Z R µ |∇ u ǫ | d x d t i ≤ C ( ∆ t ) h Z T − ∆ t Z R Z t +∆ tt µ |∇ u | ( s , x ) d s d x d t i ≤ C ∆ t . (3.43)10lugging (3.42) and (3.43) into (3.39) yields I ≤ C ∆ t ǫ (3 q + / q + C ∆ t ≤ C ∆ t ǫ (3 q + / q . (3.44)By choosing ǫ = ( C ∆ t ) q q + , we deduce that I ≤ ∆ t → , as ∆ t → . (3.45)To handle the term I , we choose a C ∞ nonnegative cut-o ff function as the following ϕ ( t ) = , t ≤ ∆ t , ∆ t ≤ t ≤ T − ∆ t , t ≥ T − ∆ t , where ∆ t > | ϕ ′ ( t ) | ≤ ∆ t . Notice that ϕ ( t ) ∈ D (0 , + ∞ ), and ϕ ( t ) converges to 1 almosteverywhere as ∆ t → . By using H ¨older inequality, Corollary 3.3 and (3.36), as ∆ t →
0, we have I = Z T − ∆ t Z R [ ρ u ( t + ∆ t , x ) − ρ u ( t , x )] (cid:2)(cid:0) u ( t + ∆ t , x ) − ϕ ( t ) u ǫ ( t + ∆ t , x ) (cid:1) − (cid:0) u ( t , x ) − ϕ ( t ) u ǫ ( t , x ) (cid:1)(cid:3) d x d t ≤ C Z T − ∆ t k √ ρ u k L q (cid:2) k u − u ǫ k L l k √ ρ k L p + k (1 − ϕ ( t )) u ǫ k L l k √ ρ k L p (cid:3) d t + C Z T − ∆ t k √ ρ u k L (cid:2) k u − u ǫ k L + k (1 − ϕ ( t )) u ǫ k L (cid:3) d t ≤ C k √ ρ u k L q (0 , T ; L q ( R )) (cid:2) k u − u ǫ k L l (0 , T ; L l ( R )) + k u ǫ k L l (0 , T ; L l ( R )) k − ϕ k L p (0 , T ) (cid:3) + C k √ ρ u k L ∞ (0 , T ; L ( R )) (cid:2) k u − u ǫ k L ∞ (0 , T ; L ( R )) + k u ǫ k L ∞ (0 , T ; L ( R )) k − ϕ k L ∞ (0 , T ) (cid:3) → , (3.46)where q + l + p = , q > . Combining (3.34),(3.45) and (3.46), we complete the proof. (cid:3)
Thanks to Lemmas 3.4-3.5, we have L -equicontinuity of √ ρ µ u µ . Thus, the following propositionfollows. Proposition 3.6.
Under
Assumption(RICKHw) , for any T > , there exists a subsequence (still denoted) √ ρ µ u µ and a function √ ρ u ∈ L ((0 , T ) × R ) such that √ ρ µ u µ → √ ρ u in L ((0 , T ) × R ) as µ → . (3.47)
4. Vanishing viscosity limit
In this section, we aim to prove our main result by the compactness argument as µ tends to zero.Note that ρ µ and √ ρ µ are uniformly bounded in L ∞ ([0 , T ] × R ) ∩ L ∞ (0 , T ; L p ( R )) , where 1 ≤ p < ∞ ,thus we have ρ µ → ρ weakly in L p ([0 , T ] × R ) , < p < ∞ , p ρ µ → √ ρ weakly in L p ([0 , T ] × R ) , < p < ∞ , (4.48)as µ → . With the help of Proposition 3.6, for any test function φ ∈ C ∞ ( R + × R ; R ), we have Z T Z R p ρ µ p ρ µ u µ · φ d x d t → Z T Z R √ ρ √ ρ u · φ d x d t . (4.49)11eanwhile, Proposition 3.6 yields that Z T Z R (cid:16) p ρ µ u µ ⊗ p ρ µ u µ (cid:17) : ∇ φ d x d t → Z T Z R (cid:0) √ ρ u ⊗ √ ρ u (cid:1) : ∇ φ d x d t . (4.50)The viscous term in the Navier-Stokes vanishes by letting µ tend to zero, in the following way (cid:12)(cid:12)(cid:12)(cid:12) Z T Z R µ ∇ u µ · ∇ φ d x d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ µ k √ µ ∇ u µ k L (0 , T ; L ( R )) → . (4.51)Note that in Definition 1.1, the weak solutions ( ρ µ , u µ ) of the Navier-Stokes equations satisfy thefollowing weak formulation Z T Z R ( ρ µ u µ · φ t + ( ρ µ u µ ⊗ u µ ) : ∇ φ + ρ µ f · φ ) d x d t + Z R m ( x ) · φ (0 , x ) d x = Z T Z R µ ∇ u µ · ∇ φ d x d t , (4.52)and Z T Z R ρ µ φ t + ρ µ u µ · ∇ φ d x d t + Z R ρ µ φ (0 , x ) d x = . (4.53)With (4.48)-(4.51) at hand, letting µ → Z T Z R ( √ ρ √ ρ u · φ t + ( √ ρ u ⊗ √ ρ u ) : ∇ φ + ρ f · φ ) d x d t + Z R m ( x ) · φ (0 , x ) d x = , (4.54)and Z T Z R ρφ t + √ ρ √ ρ u · ∇ φ d x d t + Z R ρ φ (0 , x ) d x = . (4.55)Since R t R R µ |∇ u µ | d x d t ≥
0, from (1.8) we obtain Z R ρ µ | u µ | x ≤ Z R | m | ρ d x + Z t Z R ρ µ u µ · f d x d t . (4.56)By Proposition 3.6 and (4.48), we have Z t Z R ρ µ u µ · f d x d t → Z t Z R ρ u · f d x d t . Thus, by letting µ → Z R ρ | u | x ≤ Z R | m | ρ d x + Z t Z R ρ u · f d x d t . (4.57)It is clear that (4.55), (4.54) and (4.57) meet with Definition 1.2. Therefore we conclude that ( ρ, u ) is aweak solution of (1.9) with initial data ( ρ , m ). 12 . Appendix In this section, we sketch the proof of Theorem 1.5. In the domain with period T P , k = ( k , k , k ) = π P ( n , n , n ) ∈ R , with n j = , ± , ± , · · · , and j = , , , is the discrete wavevector. Thus the totalenergy E ( t ) per unit mass at time t for the inhomogeneous turbulence in T P is: E ( t ) = R T P ρ d x Z T P ρ µ | u µ | x = X k ≥ E ( t , k ) = X k ≥ π q ( t , k ) k . (5.58)And the Fourier transform of the weighted velocity √ ρ u ( t , x ) in the x -variable is d √ ρ u ( t , k ) = | T P | Z T P √ ρ u ( t , x ) e − i k · x d x , then we have √ ρ u ( t , x ) = X k d √ ρ u ( t , k ) e i k · x . We have the following weaker version of Assumption (ICKHw) in T P : Assumption (ICKHw).
For any T > , there exists C T > and k ∗ (su ffi ciently large) depending on ρ , m , P and f but independent of the viscosity µ such that, for k = | k | ≥ k ∗ , sup k ≥ k ∗ (cid:16) | k | + β Z T | d √ ρ u ( t , k ) | d t (cid:17) ≤ C T , f or some β > . (5.59)By the Assumption (ICKHw) and the Fourier transform, we can get the following uniform bound of √ ρ µ u µ in L (0 , T ; H α ( T P )) for α ∈ (0 , + β ) . For completeness, we present the proof which is similar to[4, 5].
Lemma 5.1.
Under Assumption (ICKHw), for any T ∈ (0 , + ∞ ) , there exists C > , independent of µ > , such that Z T Z T P | D α x ( p ρ µ u µ ) | d x d t ≤ C , (5.60) where α ∈ (0 , + β ) . Proof.
Note that the definition of fractional derivatives via the Fourier transform, the Parseval identityand Assumption (ICKHw) imply that Z T Z T P | D α x ( p ρ µ u µ ) | d x d t ≤ C Z T X k | [ D α x ( p ρ µ u µ ) | d t ≤ C Z T X k | k | α | [ p ρ µ u µ | d t ≤ C Z T X ≤| k |≤ k ∗ | k | α | [ p ρ µ u µ | d t + C Z T X | k |≥ k ∗ | k | α | [ p ρ µ u µ | d t ≤ Ck α ∗ Z T Z T P | p ρ µ u µ | d x d t + C X | k |≥ k ∗ | k | α − − β ≤ C , (5.61)where α < + β . (cid:3) √ ρ µ u µ , i.e., k p ρ µ u µ k L q ((0 , T ) × T P ) ≤ C , f or q = q ( β, r ) > . It also gives us the L -equicontinuity of √ ρ µ u µ in space variable x in L ((0 , T ) × T P ) , independent of µ. Lemma 5.2.
Under
Assumption (ICKHw) , for any T ∈ (0 , + ∞ ) , we get the equicontinuity of √ ρ µ u µ with respect to the space variable x in L ((0 , T ) × T P ) , independent of µ, i.e., Z T Z T P | p ρ µ u µ ( t , x + ∆ x ) − p ρ µ u µ ( t , x ) | d x d t → , as ∆ x → . (5.62) Proof.
Using Lemma 5.1 and Parseval identity, we obtain Z T Z T P | p ρ µ u µ ( t , x + ∆ x ) − p ρ µ u µ ( t , x ) | d x d t = Z T X k | [ p ρ µ u µ | ( e i k · ∆ x − d t ≤ C | ∆ x | α Z T X k | k | α | [ p ρ µ u µ | d t ≤ C | ∆ x | α Z T Z T P | D α x ( p ρ µ u µ ) | d x d t ≤ C | ∆ x | α . (cid:3) For L − equicontinuity of √ ρ µ u µ with respect to t , the proof is similar to Lemma 3.5 with some slightmodification. Since the domain considered is a domain with period T P = [ − P / , P / ⊂ R , P > . Hence, it need not to estimate I i ( i = , , ,
4) by dividing u into u and u . We here only state thefollowing lemma. Lemma 5.3.
Under
Assumption (ICKHw) , for any T > , we have the equicontinuity of √ ρ µ u µ withrespect to the time variable t in L ((0 , T − ∆ t ) × T P ) , independent of µ, i.e. Z T − ∆ t Z T P | p ρ µ u µ ( t + ∆ t , x ) − p ρ µ u µ ( t , x ) | d x d t → , as ∆ t → . (5.63)With the L − equicontinuity of √ ρ µ u µ with respect to x and t , we directly deduce that there exists asubsequence (still denoted) √ ρ µ u µ and a function √ ρ u ∈ L ((0 , T ) × T P ) such that √ ρ µ u µ → √ ρ u in L ((0 , T ) × T P ) as µ → . Finally we can get a same theorem as Theorem 1.3 as follows:
Theorem 5.4.
Under Assumption (ICKHw) (5.59) , for the weak solution ( ρ µ , u µ ) of (1.1) - (1.2) as inDefinition 1.1, there exists a subsequence (still denote) ( ρ µ , u µ ) and a function ( ρ, u ) such that as µ → ,ρ µ → ρ weakly in L p ((0 , T ) × T P ) , p ρ µ u µ → √ ρ u , in L ((0 , T ) × T P ) , (5.64) where < p < ∞ , and ( ρ, u ) is a weak solution of (1.9) with the initial data ( ρ , m ) . Acknowledgments
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