On a singular limit for the compressible rotating Euler system
aa r X i v : . [ m a t h . A P ] J a n On a singular limit for the compressible rotatingEuler system ˇS´arka Neˇcasov´a ∗† Tong Tang , ‡§
1. Institute of Mathematics of the Academy of Sciences of the Czech Republic,ˇZitn´a 25, 11567, Praha 1, Czech Republic2. Department of Mathematics, College of Sciences,Hohai University, Nanjing 210098, P.R. China
Abstract
The work addresses a singular limit for a rotating compressible Euler system in the lowMach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in the case ofill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acousticwaves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in theasymptotic limit is eliminated.
Key words: compressible Euler equations, singular limit, low Mach number, low Rossbynumber, dissipative measure-valued solutions. : 35Q30.
Earth’s graceful rotation is an unignorable factor at geophysical fluids models. These modelsplay an important role in the analysis of complex Earth phenomena in meteorology, geophysicaland astrophysics. In order to describe the effect of rotation, people introduce two factors: Coriolisacceleration and centrifugal acceleration. In many real world applications, the action of centrifugalforce is neglected, as it is in equilibrium with stratification caused by the gravity of the Earth.Under the above assumptions, we consider the following scaled Euler equations in an infinite slapΩ = R × (0 , (cid:26) ∂ t ρ + div( ρ u ) = 0 ,∂ t ( ρ u ) + div( ρ u ⊗ u ) + Ma ∇ x p ( ρ ) + Ro ρ ( ω × u ) = 0 , (1.1)where the unknown fields ρ = ρ ( t, x ) and u = u ( t, x ) represent the density and the velocity of aninviscid compressible fluid, ω = (0 , ,
1) is the rotation axis. The Mach number Ma, proportional ∗ The research of ˇS.N. leading to these results has received funding from the Czech Sciences Foundation (GAˇCR),P201-16-032308 and RVO 67985840. Final version of the paper was made under support the Czech SciencesFoundation (GAˇCR), GA19-04243S. † Email: [email protected] ‡ The research of T.T. is supported by the NSFC Grant No. 11801138. § Email: [email protected]
1o the characteristic velocity field divided by the sound speed, and the Rossby number Ro, definedas the ratio of the displacement due to Coriolis forces, play the role of singular (small) parameters.The symbol p = p ( ρ ) denotes the barotropic pressure (assumptions on the pressure see (3.1)).The system is supplemented by the far field conditions u → , ρ → ρ, as | x | → ∞ , where ρ > , (1.2)and boundary condition u · n | ∂ Ω = 0 , (1.3)where n is outer normal vector to ∂ Ω.From modeling of geophysical fluids, the value of Mach number and Rossby number can beconsidered very small. It is well known that the compressible fluid flow becomes incompressible inthe low Mach number limit, as the density distribution is constant and the velocity field becomessolenoidal. On the other hand, low Rossby number corresponds to fast rotation and the fastrotating fluids will lead to the so-called Taylor-Proudman columns phenomena. Therefore, it isinteresting to observe the phenomenon if the two effects take place simultaneously. In this paper,we address the problem of the double limit for
M a = Ro = ǫ . Let ρ = ρ ǫ , u = u ǫ , the system(1.1) takes the form (cid:26) ∂ t ρ ǫ + div( ρ ǫ u ǫ ) = 0 ,∂ t ( ρ ǫ u ǫ ) + div( ρ ǫ u ǫ ⊗ u ǫ ) + ǫ ∇ x p ( ρ ǫ ) + ǫ ρ ǫ ( ω × u ǫ ) = 0 . (1.4)Our goal is to study the singular limit ǫ → ill–prepared initial data for thescaled system (1.4). The definition of ill–prepared initial data will be introduced in Section 3.3.Supposing we know that in the corresponding spaces, ρ (1) ǫ = ρ ǫ − ρǫ → q, u ǫ → v , we can find that q and v satisfy the following equations: ω × v + p ′ ( ρ ) ρ ∇ x q = 0 , (1.5) ∂ t (∆ h q − p ′ ( ρ ) q ) + ∇ ⊥ h q · ∇ h (∆ h q ) = 0 . (1.6)Equations (1.5)(1.6) can be interpreted as a kind of stream function, according to physicists,named as quasi-physical flows [32]. For non-rotating compressible Euler fluids, a great number ofwell-posedness results have been obtained. However, some classical literatures show that smoothsolutions of the Euler system will exhibit blow-up phenomena in a finite time no matter howsmooth or small the initial data are. Therefore, it seems more appropriate to consider a suitableclass of admissible weak solutions to (1.4). By admissible we mean that solutions will satisfysome form of the energy balance. The need for global admissible solutions of the Euler systemleads to the concept of more general dissipative measure–valued (DMV) solutions introduced inthe context of the full Euler system in [3, 4].The measure-valued solutions to hyperbolic conservations laws were introduced by DiPerna[10]. He used Young measures to pass to the artificial viscosity limit. In the case of the incom-pressible Euler equations, DiPerna and Majda [11] proved the global existence of measure-valued2olutions for any initial data with finite energy. They introduced generalized Young measuresto take into account oscillations and concentrations. Further, the existences of measure-valuedsolutions were shown for further models of fluids, e.g. compressible Euler and Navier-Stokes equa-tions [26, 23]. The measure-valued solution to the non-Newtonian case was proved by Novotn´yand Neˇcasov´a [25]. The generalization was given by Alibert and Bouchitt´e [1]. The weak-stronguniqueness for generalized measure-valued solutions of isentropic Newtonian Euler equations wereproved in [21]. Inspired by previous results, the concept of dissipative measure-valued solutionwas finally applied to the barotropic compressible Navier-Stokes system [19].The reader may consult [17, 18, 23, 24, 26] for applications of the theory of (DMV) solutionsin fluid mechanics or their counterparts [8, 9] in other areas of mathematical physics.Let us discuss the main differences between weak solutions and (DMV) solutions. First im-portant advanatge of (DMV) solution is that DMV solutions to the compressible Euler systemexist globally in time . Secondly (DMV) solutions convergence to the limit system holds for anyill-prepared initial data , which in both case are not valid for weak solutions.Due to the above fascinate advantage, there are some new results concerning singular limits inthe context of measure–valued solutions. The low Mach number limit was studied in [18], whereit is shown that (DMV) solutions approach the smooth solutions of incompressible Euler systemboth for well-prepared and ill-prepared data. Moreover, the singular limit of compressible Eulersystem in the low Mach number and strong stratification regime for the ill-prepared data wasidentified, see [20]. However, to the best of our knowledge, compared with non-rotating case, thereis a few results concerning on the singular limit of rotating compressible Euler system no matterweak solutions or strong solutions. Nilasis [28] proved the singular limit of a rotating compressibleEuler system with stratification at the case of well-prepared initial data. Our goal is to considerthe asymptotic limit of (DMV) solutions to the compressible Euler equations with ill-preparedinitial data. We prove it converges to the strong solutions of quasi-physical flows. Moreover,we should emphasize that boundary conditions in this paper can be replaced by the periodicalconditions in x direction with a few changes. All the choices of boundary conditions prevent theflow from creating a viscous boundary layer. The periodic domain with well-prepared initial datawas considered by Feireisl et al., [18]. If the whole domain is torus T , it is difficult to obtain theanalysis of acoustic waves at the case of ill-prepared initial data. It seems interesting to comparethe results of the present paper with those obtained in [18]. The analysis in [18] leans that the (DMV) solutions of Euler system will converge to incompressible Euler system. Moreover, thereis obvious difference about acoustic wave analysis between rotating and non-rotating case. Theextension of the results of [18] to the rotating Euler system is therefore not straightforward. Lastbut not least, we should emphasis that there are huge results about rotating Navier-Stokes systemsuch as [5, 6, 14, 15, 16].The paper is organized as follows. In Section 2, we introduce the dissipative measure solutions,relative energy and the other necessary material. In Section 3, we state our main theorem. Section4 is devoted to deriving uniform bounds of the Euler system independent of ǫ . In Section 5, weperform the necessary analysis of the acoustic waves. The proof of the main theorem is completedin Section 6. 3 Preliminaries
First let us observe that it is more convenient to rewrite the Euler system in terms of theconservative variables ρ , m = ρ u . Let Q = { [ ρ, m ] | ρ ∈ [0 , ∞ ) , m ∈ R } be the natural phasespace associated to solutions [ ρ, m ] = [ ρ, ρ u ]. A dissipative measure-valued (DMV) solution of the Euler system (1.1) is a parameterizedfamily of probability measures { Y t,x } t ∈ [0 ,T ] ,x ∈ Ω , ( t, x ) Y t,x ∈ L ∞ weak − ( ⋆ ) ((0 , T ) × Ω; P ( Q )) , (2.1)satisfying • the continuity equation Z T Z Ω [ h Y t,x ; ρ i ∂ t ϕ + h Y t,x ; m i∇ x ϕ ] dxdt = − Z Ω h Y ,x ; ρ i ϕ (0) dx, (2.2)for all ϕ ∈ C ∞ c ([0 , T ) × Ω); • the momentum equation Z T Z Ω [ h Y t,x ; m i ∂ t ϕ + h Y t,x ; m ⊗ m ρ i : ∇ x ϕ ] dxdt + Z T Z Ω h Y t,x ; p ( ρ ) i div ϕdxdt + Z T Z Ω h Y t,x ; ω × m i ϕdxdt = − Z Ω h Y ,x ; m i ϕ (0) dx − Z T Z Ω ∇ x ϕ : dµ c , (2.3)for all ϕ ∈ C ∞ c ([0 , T ) × Ω), where µ c ∈ M ([0 , T ] × Ω) is the so–called momentum concentrationmeasure; • the energy inequality Z Ω [ h Y τ,x ; 12 | m | ρ + (cid:0) P ( ρ ) − P ′ ( ρ )( ρ − ρ ) − P ( ρ ) (cid:1) i dx + D ( τ ) ≤ Z Ω h Y ,x ; 12 | m | ρ + (cid:0) P ( ρ ) − P ′ ( ρ )( ρ − ρ ) − P ( ρ ) (cid:1) i dx, (2.4)for a.a τ ∈ (0 , T ), where P ( ρ ) = ρ Z ρρ p ( z ) z dz, (2.5)and D is a non-negative function D ∈ L ∞ (0 , T ), satisfying the compatibility condition Z τ Z Ω | µ c | dxdt ≤ C Z τ ξ ( t ) D ( t ) dt, for some ξ ∈ L (0 , T ) . (2.6) Remark 2.1.
The notion of (DMV) solutions can be founded in many works as it was alreadymention in the Introduction, see e.g. [3, 4, 8, 9, 19]. For convenience of readers, we give moredetails.Let L ∞ weak⋆ ((0 , T ) × Ω; P ( Q )) be the space of essentially bounded weakly ⋆ − measure maps Y : (0 , T ) × Ω → P ( Q ) , ( t, x ) Y t,x . By virtue of fundamental theorem on Young measures (see { ρ ǫ , m ǫ } ǫ> and parameterized family of probability measures { Y t,x } ( t,x ) ∈ (0 ,T ) × Ω [( t, x ) Y t,x ] ∈ L ∞ weak⋆ ((0 , T ) × Ω; P ( Q )) , such that a.a. ( t, x ) ∈ (0 , T ) × Ω h Y t,x ; G ( ρ, m ) i = G ( ρ, m )( t, x ) f or any G ∈ C c ( Q ) , whenever G ( ρ ǫ , m ǫ ) → G ( ρ, m )( t, x ) weakly ⋆ in L ∞ ((0 , T ) × Ω) . Moreover, if G ∈ C ( Q ) is such that Z T Z Ω | G ( ρ ǫ , m ǫ ) | dx ≤ C, then G is Y t,x integrable for almost all ( t, x ) ∈ (0 , T ) × Ω and [( t, x )
7→ h Y t,x ; G ( ρ, m ) i ] ∈ L ((0 , T ) × Ω) , and G ( ρ ǫ , m ǫ ) → G ( ρ, m )( t, x ) weakly ⋆ in M ((0 , T ) × Ω) . The difference µ G ≡ G ( ρ, m ) − [( t, x )
7→ h Y t,x ; G ( ρ, m ) i ] ∈ M ((0 , T ) × Ω) , is called concentration defect measure.For more details, please see [7]. Remark 2.2. • The measure Y ,x plays the role of initial conditions . • The proof of an existence of (DMV) solutions of Euler system was done in the pioneerwork by Neustupa, [26]. Recently, see [27], the authors proved the local strong solutionsof rotating compressible Euler system in R . Feireisl et al. [3, 4] proved the existence of (DMV) solutions to the non-rotating full Euler system. As the rotating term does not bringany trouble in the proof of existence, the existence of (DMV) solutions to (1.4) can beobtained by analogous methods as in [4]. Remark 2.3.
We need to define the function [ ρ, m ]
7→ | m | ρ on the vacuum set as [ ρ, m ] → | m | ρ = ∞ , if ρ = 0 and m = 0 , | m | ρ , if ρ > , , otherwise . (2.7) Accordingly, it follows from the energy inequality (2.4) that
Supp[ Y t,x ] ∩ { [ ρ, m ] ∈ Q| ρ = 0 , m = 0] } = ∅ for a.a. ( t, x ) . (2.8)5 .2 Relative entropy inequality Motivated by [12, 13, 4], we introduce the relative energy functional E ( ρ, m | r, U ) = Z Ω h Y t,x ; 12 ρ | m ρ − U ( t, x ) | + ( P ( ρ ) − P ′ ( r )( ρ − r ) − P ( r )) i dx, (2.9)where r > U are smooth “test” functions, r − ρ , U compactly supported in Ω.As shown in [4], any (DMV) solution of (1.1) satisfies the relative entropy inequality E ( ρ, m | r, U ) | t = τt =0 + D ( τ ) ≤ Z τ Z Ω h Y t,x ; ( ∂ t U + m ρ ∇ x U )( ρ U − m ) i dxdt + Z τ Z Ω h Y t,x ; ( r − ρ ) ∂ t P ′ ( r ) + ( r U − m ) ∇ x P ′ ( r ) i dxdt + Z τ Z Ω h Y t,x ; ω × m ρ i ( ρ U − m ) dxdt − Z τ Z Ω h Y t,x ; p ( ρ ) − p ( r ) i div U dxdt + Z τ Z Ω ∇ x U : dµ c . (2.10)for a.a. τ ∈ [0 , T ], and any r, U ∈ C ([0 , T ] × Ω), r − ρ , U compactly supported in Ω. Before stating our main result, we introduce some notations and collect several mostly techni-cal hypotheses and known facts concerning the limit system. x = ( x h , x ) with x h ∈ R denotingits horizontal component. For a vector field b = [ b , b , b ], we introduce the horizontal compo-nent b h = [ b , b ] writing b = [ b h , b ]. Similarly, we use the symbols ∇ h , div h to denote thedifferential operators acting on the horizontal variables. The following assumptions and resultswill be used in the proof. We suppose the pressure p is a continuously differentiable function of the density such thatfor some γ > p ∈ C [0 , ∞ ) ∩ C ∞ (0 , ∞ ) , p (0) = 0 , p ′ ( ρ ) > ρ > , lim ρ →∞ p ′ ( ρ ) ρ γ − = p ∞ > . (3.1) Remark 3.1.
Similarly to [18], we deduce that p ( ρ ) − p ′ ( r )( ρ − r ) − p ( r ) is dominated by P ( ρ ) − P ′ ( r )( ρ − r ) − P ( r ) , specifically, | ρ − r | ≤ c ( δ )( P ( ρ ) − P ′ ( r )( ρ − r ) − P ( r )) when < δ ≤ ρ, r ≤ δ , δ > , | ρ − r | + P ( ρ ) ≤ c ( δ )( P ( ρ ) − P ′ ( r )( ρ − r ) − P ( r )) if < δ < r < δ ,ρ ∈ [0 , δ ) ∪ ( 1 δ , ∞ ) , δ > . The expected limit problem reads ω × v + p ′ ( ρ ) ρ ∇ x q = 0 , v = [ v h ( x h ) , , q = q ( x h ) , (3.2)6 t (∆ h q − p ′ ( ρ ) q ) + v h · ∇ h (∆ h q ) = 0 . (3.3)supplement with the initial condition q | t =0 = q . As shown by Oliver [29], the problem (3 . − (3 .
3) possesses a unique classical solution q ∈ C ([0 , T ]; W m, ( R )) ∩ C ([0 , T ]; W m − , ( R )) , m ≥ , (3.4)for any initial solution q ∈ W m, ( R ) . (3.5) The ill–prepared initial data for the scaled system (1.4) take the form ρ ǫ (0 , · ) = ρ ,ǫ = ρ + ǫs ,ǫ , u ǫ (0 , · ) = u ,ǫ , (3.6)where s ,ǫ → s in W k, (Ω) ∩ W k, (Ω) , u ,ǫ → u in W k, (Ω) ∩ W k, (Ω) , ( k > , (3.7) u = v + ∇ x Φ . (3.8) For simplicity, we assume ρ = p ′ ( ρ ) = P ′′ ( ρ ) = 1. Now, we are ready to state our main result. Theorem 3.1.
Let { Y ǫt,x } ( t,x ) ∈ [0 ,T ] × Ω be a family of (DMV) solutions to the scaled Euler system (1.4) satisfying the compatibility condition (2.6) with a function ξ independent of ǫ . Let the initialdata { Y ǫ ,x } x ∈ Ω be ill-prepared, namely Z Ω h Y ǫ ,x ; 12 ρ | m ρ − u ,ǫ ( x ) | + 1 ǫ ( P ( ρ ) − P ′ ( ρ ,ǫ )( ρ − ρ ,ǫ ) − P ( ρ ,ǫ )) i dx → , where ρ ,ǫ , u ,ǫ are ill prepared data introduced in Section 3.3.Then D ǫ → in L ∞ (0 , T ) ,Y ǫt,x → δ [ q, v ] in L p (0 , T ; L (Ω; M + ( Q ) weak − ( ⋆ ) )) for any finite p ≥ , where q and v is the unique solution of problem (3.2)-(3.3) starting from the initial data q andwhere q ∈ W k +1 , ( R ) ∩ W k +1 , ( R ) is the unique solution of the elliptic problem − ∆ h q + q = Z curl h [ u ] h dx + Z s dx . (3.9)The rest of the paper is devoted to the proof of Theorem 3.1.7 Energy bounds
We start by deriving uniform bounds on solutions to (1.4) independent of ǫ . Similarly to [18],we introduce the decomposition h ( ρ, m ) = [ h ] ess ( ρ, m ) + [ h ] res ( ρ, m ) , [ h ] ess = ψ ( ρ ) h ( ρ, m ) , [ h ] res = (1 − ψ ( ρ )) h ( ρ, m ) , where ψ ∈ C ∞ c (0 , ∞ ) , ≤ ψ ( ρ ) ≤ , ψ ( ρ ) = 1 on an open interval containing ρ = 1 . As the initial data are ill–prepared, the expression on the right–hand side of the energyinequality (2.4) remains bounded uniformly for ǫ →
0. Consequently, we deduce the followingbound: ess sup t ∈ (0 ,T ) Z Ω h Y ǫt,x ; 12 | m ǫ | ρ + 1 ǫ ( P ( ρ ǫ ) − P ′ (1)( ρ ǫ − − P (1)) i dx ≤ C. (4.1)Thus, exactly as in [18], we use the structural properties of the function p to deduce ess sup t ∈ (0 ,T ) Z Ω h Y ǫt,x ; | [ ρ ǫ − ǫ ] ess | i + h Y ǫt,x ; [ P ( ρ ǫ ) + 1 ǫ ] ess i dx ≤ C ;( t, x )
7→ h Y ǫt,x ; m ǫ i bounded in L ∞ (0 , T ; L (Ω) + L γγ +1 (Ω));( t, x )
7→ h Y ǫt,x ; [ ρ ǫ − ǫ ] ess i bounded in L ∞ (0 , T ; L (Ω));( t, x ) ǫ − γ h Y ǫt,x ; [ ρ ǫ ] res i bounded in L ∞ (0 , T ; L γ (Ω)) . (4.2)Using the same argument in [17], there exist functions ρ (1) ∈ L ∞ (0 , T ; L (Ω)) and m ∈ L ∞ (0 , T ; L q (Ω)) for some q > h Y ǫt,x ; m ǫ i → m weakly in L ∞ (0 , T ; L q (Ω); h Y ǫt,x ; ρ ǫ − ǫ i → ρ (1) weakly in L ∞ (0 , T ; L (Ω)) . Recalling (2.2) and (2.3), we deduce Z T Z Ω m · ∇ x ϕdxdt = 0 , Z T Z Ω [( ω × m ) · ϕ + ρ (1) div ϕ ] dxdt = 0 , for ϕ ∈ C ([0 , T ] × Ω). In other words,div x m = 0 , ω × m + ∇ x ρ (1) = 0 , (4.3)in the sense of distribution.It is easy to check that ρ (1) = ρ (1) ( x h ) , m = ( m h , , div x m = div h m h = 0 . Moreover, the detail of derivation of (3.9) can be seen in [14, 15, 16].8
Acoustic waves
It is well-known that ill-prepared data give rise to rapidly oscillating acoustic waves. Similarlyto [15], the relevant acoustic equation reads (cid:26) ǫ∂ t s ǫ + div( ∇ x Φ ǫ ) = 0 ,ǫ∂ t ∇ x Φ ǫ + ω × ∇ x Φ ǫ + ∇ x s ǫ = 0 , (5.1)supplemented with the initial data s ǫ (0 , · ) = s , ∇ x Φ ǫ (0 , · ) = ∇ x Φ , where s , ∇ x Φ have been introduced in Section 3.3.As a matter of fact, the initial data must be smoothed and cut-off via suitable regularizationoperators, namely s ǫ (0 , · ) = s ,δ = [ s ] δ ; ∇ x Φ ǫ (0 , · ) = ∇ x Φ ,δ = ∇ x [Φ ] δ , where [ · ] δ denotes the regularization introduced in [15].Denoting the corresponding solutions s ǫ,δ , Φ ǫ,δ we report the following energy and dispersiveestimates proved in [15, Section 6]:sup t ∈ [0 ,T ] [ k Φ ǫ,δ ( t, · ) k W m, + k s ǫ,δ ( t, · ) k W m, ] = [ k∇ x Φ ,δ k L + k s ,δ k L ] , (5.2)and Z T [ k Φ ǫ,δ ( t, · ) k W m, ∞ + k s ǫ,δ ( t, · ) k W m, ∞ ] ≤ ω ( ǫ, m, δ )[ k∇ x Φ ,δ k L + k r ,δ k L ] , (5.3)where ω ( ǫ, m, δ ) → ǫ → m ≥ δ >
0. More details about Strichartzestimates and acoustic waves, readers can refer to [30, 31].
The proof of convergence is based on the ansatz r ǫ = 1 + ǫ ( q + s ǫ,δ ) , U ǫ = v + ∇ x Φ ǫ,δ , (6.1)in the relative energy inequality (2.10). The [ s ǫ,δ , ∇ x Φ ǫ,δ ] are solutions of the acoustic system(5.1), and [ q, v ] is solution of the target problem ω × v + ∇ x q = 0 ,∂ t (∆ h q − q ) + ∇ ⊥ h q · ∇ h (∆ h q ) = 0 . (6.2)In addition, to avoid technicalities, we shall assume that s and Φ are sufficiently regular so thatthe δ − regularization is not needed in (5.1–5.3). Accordingly, we have s ǫ,δ = s ǫ , Φ ǫ,δ = Φ ǫ . Thegeneral case may be handled as in [15].First note that the relative energy for the scaled system reads E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ ) = Z Ω h Y t,x ; 12 ρ ǫ | m ǫ ρ ǫ − U ǫ | + 1 ǫ ( P ( ρ ǫ ) − P ′ ( r ǫ )( ρ ǫ − r ǫ ) − P ( r ǫ )) i dx, (6.3)9ith the corresponding relative energy inequality: E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ ) | t = τt =0 + D ǫ ( τ ) ≤ Z τ Z Ω h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( ∂ t U ǫ + m ǫ ρ ǫ ∇ x U ǫ ) dxdt + 1 ǫ Z τ Z Ω [ h Y ǫt,x ; r ǫ − ρ ǫ i ∂ t P ′ ( r ǫ ) + h Y ǫt,x ; r ǫ U ǫ − m ǫ i∇ x P ′ ( r ǫ )] dxdt + 1 ǫ Z τ Z Ω h Y t,x ; ω × m ǫ ρ ǫ i ( ρ ǫ U ǫ − m ǫ ) dxdt − ǫ Z τ Z Ω h Y t,x ; p ( ρ ǫ ) − p ( r ǫ ) i div U ǫ dxdt + Z τ Z Ω ∇ x U ǫ : dµ c . (6.4)Our goal is to show that, with the ansatz (6.1), the relative energy E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ ) tendsto zero for ǫ → t ∈ [0 , T ]. In view of the dispersive estimates (5 . − (5 . ǫ →
0. This programme will be carried over by means of several steps.
First, we compute Z τ Z Ω [ h Y ǫt,x ; r ǫ − ρ ǫ i ∂ t P ′ ( r ǫ ) + h Y ǫt,x ; r ǫ U ǫ − m ǫ i∇ x P ′ ( r ǫ ) − h Y ǫt,x ; p ( ρ ǫ ) − p ( r ǫ ) i div U ǫ ] dxdt = Z τ Z Ω [ h Y ǫt,x ; p ( r ǫ ) − p ′ ( r ǫ )( r ǫ − ρ ǫ ) − p ( ρ ǫ ) i div U ǫ + h Y ǫt,x ; r ǫ − ρ ǫ i ∂ t P ′ ( r ǫ )+ h Y ǫt,x ; ( r ǫ − ρ ǫ ) p ′ ( r ǫ ) i div U ǫ + h Y ǫt,x ; ( r ǫ − ρ ǫ ) ∇ x P ′ ( r ǫ ) i U ǫ + h Y ǫt,x ; ( ρ ǫ U ǫ − m ǫ ) ∇ x P ′ ( r ǫ ) i ] dxdt = Z τ Z Ω [ h Y ǫt,x ; p ( r ǫ ) − p ′ ( r ǫ )( r ǫ − ρ ǫ ) − p ( ρ ǫ ) i div U ǫ + h Y ǫt,x ; ∂ t r ǫ + div x ( r ǫ U ǫ ) i ( r ǫ − ρ ǫ ) P ′′ ( r ǫ )+ h Y ǫt,x ; ( ρ ǫ U ǫ − m ǫ ) ∇ x P ′ ( r ǫ ) i ] dxdt. Note that, in view of (6.2), ∂ t r ǫ + div x ( r ǫ U ǫ ) = ǫ∂ t q + ∂ t s ǫ + div( r ǫ ( v + ∇ x Φ ǫ ))= ǫ∂ t q + ǫ div(( q + s ǫ ) U ǫ )) . Next, by virtue of (5.1) and (6.1), ∇ x P ′ ( r ǫ )( ρ ǫ U ǫ − m ǫ ) = ∇ x ( P ′ ( r ǫ ) − P ′′ (1)( r ǫ − − P ′ (1))( ρ ǫ U ǫ − m ǫ ) + ǫ ∇ x q · ( ρ ǫ U ǫ − m ǫ )+ ǫ ∇ x s ǫ · ( ρ ǫ U ǫ − m ǫ )= ∇ x ( P ′ ( r ǫ ) − P ′′ (1)( r ǫ − − P ′ (1))( ρ ǫ U ǫ − m ǫ ) + ǫ ∇ x q · ( ρ ǫ U ǫ − m ǫ ) − ǫ ( ρ ǫ U ǫ − m ǫ ) · ∂ t ∇ Φ ǫ − ǫ ( ρ ǫ U ǫ − m ǫ )( ω × ∇ Φ ǫ ) . Furthermore, by virtue of the compatibility condition (2.6), we can control the concentrationmeasure, Z τ Z Ω ∇ x U : dµ c ≤ k∇ x U k L ∞ Z τ ξ ( t ) D ǫ ( t ) dt. E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ )(0) → ǫ → . Thus we may conclude that E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ )( τ ) + D ǫ ( τ ) ≤ Z τ Z Ω h Y ǫt,x ; ( ∂ t v + m ǫ ρ ǫ ∇ x U ǫ )( ρ ǫ U ǫ − m ǫ ) i dxdt + 1 ǫ Z τ Z Ω h Y ǫt,x ; ω × v i ( ρ ǫ U ǫ − m ǫ ) dxdt + 1 ǫ Z τ Z Ω h Y ǫt,x ; ∇ x (cid:0) P ′ ( r ǫ ) − P ′′ (1)( r ǫ − − P ′ (1) (cid:1) i ( ρ ǫ U ǫ − m ǫ ) dxdt − ǫ Z τ Z Ω h Y ǫt,x ; p ( ρ ǫ ) − p ( r ǫ ) − p ′ ( r ǫ )( ρ ǫ − r ǫ ) i div U ǫ dxdt + 1 ǫ Z τ Z Ω h Y ǫt,x ; ∂ t q + div(( q + s ǫ ) U ǫ )) i ( r ǫ − ρ ǫ ) P ′′ ( r ǫ ) dxdt + 1 ǫ Z τ Z Ω h Y ǫt,x ; ∇ x q i ( ρ ǫ U ǫ − m ǫ ) dxdt + c Z τ ξ ( t ) D ǫ ( t ) dt + ω ( ǫ ) , where ω ( ǫ ) denotes a generic quantity satisfying ω ( ǫ ) → L (0 , T ) as ǫ → . Using (6.2), we get the following conclusion: E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ )( τ ) + D ǫ ( τ ) ≤ Z τ Z Ω h Y ǫt,x ; ( ∂ t v + m ǫ ρ ǫ ∇ x U ǫ )( ρ ǫ U ǫ − m ǫ ) i dxdt + 1 ǫ Z τ Z Ω h Y ǫt,x ; ∇ x (cid:0) P ′ ( r ǫ ) − P ′′ (1)( r ǫ − − P ′ (1) (cid:1) i ( ρ ǫ U ǫ − m ǫ ) dxdt − ǫ Z τ Z Ω h Y ǫt,x ; p ( ρ ǫ ) − p ( r ǫ ) − p ′ ( r ǫ )( ρ ǫ − r ǫ ) i div U ǫ dxdt + 1 ǫ Z τ Z Ω h Y ǫt,x ; ∂ t q + div(( q + s ǫ ) U ǫ )) i ( r ǫ − ρ ǫ ) P ′′ ( r ǫ ) dxdt + c Z τ ξ ( t ) D ǫ ( t ) dt + ω ( ǫ ) , We write Z τ Z Ω [ h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( ∂ t v + m ǫ ρ ǫ ∇ x U ǫ )] dxdt = Z τ Z Ω h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( ∂ t v + v · ∇ x v ) dxdt + Z τ Z Ω h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( v · ∇ x ∇ x Φ ǫ + ∇ x Φ ǫ ∇ x U ǫ ) dxdt + Z τ Z Ω h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( m ǫ ρ ǫ − U ǫ ) ∇ x U ǫ dxdt = I + I + I . I into their essential and residualparts obtaining (cid:12)(cid:12)(cid:12)(cid:12)Z R h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( v · ∇ x ∇ x Φ ǫ + ∇ x Φ ǫ ∇ x U ǫ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ x Φ ǫ k W , ∞ ( k v k W , + k∇ x U ǫ k W , ) + c E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ ) , where the first term on the right–hand side can be controlled by means of the dispersive estimate(5.2) and (5.3).Summing up the previous observations, we may infer that the relative energy inequality withthe ansatz (6.1) reduces to E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ )( τ ) + D ǫ ( τ ) ≤ Z τ Z Ω h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( ∂ t v + v · ∇ x v ) dxdt + 1 ǫ Z τ Z Ω h Y ǫt,x ; ∇ x (cid:0) P ′ ( r ǫ ) − P ′′ (1)( r ǫ − − P ′ (1) (cid:1) i ( ρ ǫ U ǫ − m ǫ ) dxdt − ǫ Z τ Z Ω h Y ǫt,x ; p ( ρ ǫ ) − p ( r ǫ ) − p ′ ( r ǫ )( ρ ǫ − r ǫ ) i div U ǫ dxdt − ǫ Z τ Z Ω h Y ǫt,x ; ( ρ ǫ − r ǫ ) P ′′ ( r ǫ ) i ( ∂ t q + div(( q + s ǫ ) U ǫ )) dxdt + C Z τ E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ ) dt + C Z τ ξ ( t ) D ǫ ( t ) dt + ω ( ǫ ) . Now, we will deal with pressure term and corresponding term. First, using direct calculation,the Taylor formula and dispersive estimates (5.2-5.3), we deduce that1 ǫ |∇ x (cid:0) P ′ ( r ǫ ) − P ′ (1) − P ′′ (1)( r ǫ − (cid:1) | = 1 ǫ | (cid:0) P ′′ ( r ǫ ) − P ′′ (1) (cid:1) ∇ x ( q + s ǫ ) |→ P ′′′ (1) q ∇ x q as ǫ → . Therefore, combining the previous energy bounds and convergence, we get1 ǫ Z τ h Y ǫt,x ; ∇ x (cid:0) P ′ ( r ǫ ) − P ′ (1) − P ′′ (1)( r ǫ − (cid:1) i ( ρ ǫ U ǫ − m ǫ ) dt → ǫ → . The remaining pressure term is | ǫ Z τ h Y ǫt,x ; p ( ρ ǫ ) − p ( r ǫ ) − p ′ ( r ǫ )( ρ ǫ − r ǫ ) i div U ǫ dt | = | ǫ Z τ h Y ǫt,x ; p ( ρ ǫ ) − p ( r ǫ ) − p ′ ( r ǫ )( ρ ǫ − r ǫ ) i (div v + ∆Φ ǫ ) dt |≤ c | ǫ Z τ h Y ǫt,x ; P ( ρ ǫ ) − P ( r ǫ ) − P ′ ( r ǫ )( ρ ǫ − r ǫ ) i (div v + ∆Φ ǫ ) dt |≤ C Z τ E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ ) dt, E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ )( τ ) + D ǫ ( τ ) ≤ ω ( ǫ ) + C Z τ E ( ρ ǫ , m ǫ | r ǫ , U ǫ ) dt + c Z τ ξ ( t ) D ǫ ( t ) dt + Z τ Z Ω h Y ǫt,x ; ρ ǫ U ǫ − m ǫ i ( ∂ t v + v · ∇ x v ) dxdt − ǫ Z τ Z Ω h Y ǫt,x ; ( ρ ǫ − r ǫ ) P ′′ ( r ǫ ) i ( ∂ t q + div(( q + s ǫ ) U ǫ )) dxdt. Finally, we deal with the remaining pressure terms. Similar to the previous analysis, we obtain E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ )( τ ) + D ǫ ( τ ) ≤ ω ( ǫ ) + C Z τ E ( ρ ǫ , m ǫ | r ǫ , U ǫ ) dt + c Z τ ξ ( t ) D ǫ ( t ) dt + Z τ Z Ω h Y ǫt,x ; v − m i ( ∂ t v + v · ∇ x v ) dxdt + Z τ Z Ω h Y ǫt,x ; q − ρ (1) i ( ∂ t q + div( q v )) dxdt, where Z τ Z Ω h Y ǫt,x ; v − m i ( ∂ t v + v · ∇ x v ) dxdt + Z τ Z Ω h Y ǫt,x ; q − ρ (1) i ( ∂ t q + div( q v )) dxdt = 12 Z τ Z Ω h Y ǫt,x ; ∂ t | v | + ∂ t | q | i dxdt − Z τ Z Ω h Y ǫt,x ; ∂ t v · m + ∂ t q · ρ (1) i dxdt − Z τ Z Ω h Y ǫt,x ; v · ∇ x v · m + ρ (1) div( q v ) i dxdt By virtue of (4.3) and (6.2), we havediv x ( q v ) = ∇ x q · v = ∇ x q · ∇ ⊥ x q = 0and − Z τ Z Ω h Y ǫt,x ; ∂ t v · m + ∂ t qρ (1) i dxdt = − Z τ Z Ω h Y ǫt,x ; ω × m i v ∆ h qdxdt. Moreover, it is easy to check that v · ∇ x v · m + ( ω × m ) · v ∆ h q = m · ∇ h | v | . So we deduce that12 Z τ Z Ω h Y ǫt,x ; ∂ t | v | + ∂ t | q | i dxdt = 12 Z τ Z Ω h Y ǫt,x ; ∂ t |∇ ⊥ h q | + ∂ t | q | i dxdt = Z τ Z Ω h Y ǫt,x ; v · ∇ h (∆ h q ) q i dxdt = − Z τ Z Ω h Y ǫt,x ; v · ∇ h q ∆ h q i dxdt = 0 , where the last equality due to v = ∇ ⊥ q . 13utting together Step 1 to Step 4, we conclude that E ǫ ( ρ ǫ , m ǫ | r ǫ , U ǫ ) + D ǫ ( τ ) ≤ ω ( ǫ ) + Z τ (1 + ξ ( t ))[ E ( ρ ǫ , m ǫ | r ǫ , U ǫ ) + D ǫ ( t )] dt, where r ǫ , U ǫ are given by (6.1). Letting ǫ → Acknowledgements
The authors are grateful to the referee and the editor whose comments and suggestions greatlyimproved the presentation of this paper. The paper was written when Tong Tang was visitingthe Institute of Mathematics of the Czech Academy of Sciences which hospitality and support isgladly acknowledged.
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