Asymptotic limit for rotational quantum compressible Navier–Stokes equations with multiple scales

Accepted Manuscript Asymptotic limit for rotational quantum compressible Navier–Stokes equations with multiple scales

Young-Sam Kwon

PII: DOI: Reference:

S0022-247X(18)30380-9 https://doi.org/10.1016/j.jmaa.2018.04.073 YJMAA 22227

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

17 August 2016

Please cite this article in press as: Y.-S. Kwon, Asymptotic limit for rotational quantum compressible Navier–Stokes equations with multiple scales, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.04.073

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ASYMPTOTIC LIMIT FOR ROTATIONAL QUANTUM COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH MULTIPLE SCALES YOUNG-SAM KWON

Abstract. In this paper we consider the degenerate quantum compressible NavierStokes equations giving rise to a variety of mathematical problems in many areas. We study the asymptotic limit for the rotational compressible Navier-Stokes equations with quantum term and the ill-prepared initial data.

Contents 1. Introduction 2. Main results 3. Proof of Theorem 2.2 3.1. Uniform bounds 3.2. Relative entropy inequality 3.3. Dispersive Estimates 3.4. Convergence of viscosity and velocity terms 3.5. Convergence of pressure terms 3.6. Convergence of quantum potential terms 3.7. Convergence of initial data and conclusion References

1 3 5 5 6 7 8 11 14 14 15

1. Introduction The models of compressible Navier-Stokes equations arise in science and and a variety of engineering in many practical applications such as geophysics, astrophysics, and some engineering problems appearing in plasma confinement, liquid-metal cooling of nuclear reactors, and electromagnetic casting. We here consider the degenerate quantum compressible Navier-Stokes equations with damping om unbounded domain Ω = R2 × T1 where T1 is an one dimensional torus. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a densitydependent viscosity, it reads as: ∂t  + div(u) = 0,

(1.1)

2000 Mathematics Subject Classification. Primary: 35L15 ; Secondary: 35L53. Key words and phrases. Asymptotic incompressible limit, degenerate quantum compressible Navier-Stokes equations. The work of the author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF2013R1A1A2057662). 1

2

YOUNG-SAM KWON

1 ∇γ − μdiv(Du) + f × u (1.2) γ √ Δ  = −αu + κ∇ √  where u is the vector field, γ > 1,  is the density, and Du is defined as follows: 1 Du = [∇u + ∇T u] and f = (0, 0, 1). 2 To begin with, we introduce the scaling limit: ∂t (u) + div(u ⊗ u) +

t → t, u → εu, μ → εμ, α → εα, κ → ε2 .

(1.3)

With the scaling of (1.3), the system (1.1-1.2) read as follows: ∂t ε + div(ε uε ) = 0, ∂t (ε uε ) + div(ε uε ⊗ uε ) +

1 1 ∇γε − με div(ε Duε ) +  f × u γε2 ε  Δ√    = −αε u +  ∇ √ . 

(1.4) (1.5)

We also assume that με −→ 0, αε −→ α > 0, (1.6) as ε tends to 0. Assume that the initial data have the following property at infinity:  (x) → 1,  (x)u (x) → 0 as |x| → ∞.

(1.7)

The existence of global weak solution for the compressible Navier-Stokes equations with density dependent viscosity was also one challenge issues in mathematics community thanks to a variety of applications to ocean physics and the shallow water equations. The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations was first proved by Gamba, J¨ ungel, and Vasseur in [10]. The global existence of weak solutions for the multidimensional quantum Navier–Stokes equations (1.4-1.5) was obtained in the unusual sense of weak solutions by J¨ ungel in [11] such that the density is multiplied to the momentum equation. So, with this solutions, we can not use the relative entropy. Recently, Vasseur and Yu [17, 18] have proved the global-in-time existence solutions for system (1.4 - 1.5) for 3 D degenerate compressible Navier-Stokes equations based on Bresh and Desjardins’ entropy inequality in [1, 2, 3] and the Mellet-Vasseur type inequality [16]. In fact, there is more damping term |u|2 u in [17], which is crucial to show the global existence of 3 D degenerate compressible Navier Stokes equations, but we can omit the term due to the strong convergence of u and u ∈ L2 ((0, T ) × Ω) in [17]. For the incompressible limit problems, there are many recent works by Lions, Masmoudi [13] for isentropic Navier-Stokes equations with constant viscosity and by Feireisl, Novotny [7, 8] for the Navier-Stokes-Fourier systems. They have also worked on the inviscid incompressible limit problems by Masmoudi [14] for isentropic compressible Navier Stokes equations and by Feireisl, Novotny [7] for Navier Stokes Fourier system. In this paper, we study the asymptotic limit for the degenerate quantum compressible Navier-Stokes equations (1.4-1.5) in the whole space when the number ε is very small and we also use the ill-prepared initial data. Our contribution of this paper is physically to derive a rigorous quasi-geostropic equation from compressible quantum Navier-Stokes equations based on the relative entropy method and mathematically to understand the convergence of velocity with the oscillation of initial data. Recently,

ASYMPTOTIC LIMIT FOR NSES

3

Jungel, Lin, and Wu [12] have studied the asymptotic limit for the models with a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity with the well-prepared initial data. In this paper, we deal with the oscillation of the initial velocity and so more a developed result than their result. Our goal is rigorously to investigate the limit  − 1 √ → q,  u → v, (1.8) ε as  tends to 0 in the suitable sense such that the given limits v = [vh (xh ), 0], q = q(xh ) represent a global smooth strong solution of the following system on [0, T ): for q0 ∈ H k (Ω), k ≥ 4 ⎧ vh = ∇⊥ ⎨ h q, (1.9) ⎩ ∂t (Δh q − q) + (∇⊥ h q · ∇h )Δh q + αΔh q = 0, q(0, ·) = q0 where the notations are defined as follows:  ∂g ∂g   ∂g ∂g  x = (xh , x3 ), ∇h g = , ∇⊥ , , − , hg = ∂x1 ∂x2 ∂x2 ∂x1 and divh v =

∂v1 ∂v2 + , Δh = divx ∇h . ∂x1 ∂x2

For the given initial q0 ∈ H k (R2 ), k ≥ 4, the q has the following regularity: q ∈ C([0, T ]); H k (Ω; R2 )) ∩ C 1 ([0, T ]); H k−1 (Ω; R2 )).

(1.10)

We can see the proof of the existence of a global strong solution of (1.9) in Oliver [15] even though the given equation has the diffusion term αΔh q. The outline of this article is as follows: In Section 2 we present the result of global weak solutions for degenerate compressible Navier-Stokes equations (1.4-1.5) and the main result. In Section 3 we give the proof of the asymptotic limit of solutions of degenerate compressible quantum Navier-Stokes equations (1.4-1.5). 2. Main results In this section we introduce the main result of inviscid incompressible limit for degenerate compressible quantum Navier-Stokes equations with damping in the whole space. Before this, we mention the global weak solutions of degenerate compressible quantum Navier-Stokes equations and the result of global existence for the system can be quaranteed Theorem 2.1. (See [17].) Suppose that the initial data (ε,0 , mε,0 ) satisfy: √ ε,0 − 1 ∈ (Lγ ∩ L1 )(Ω), ε,0 ≥ 0, ∇ ε,0 ∈ L2 (Ω), − log− ε,0 ∈ L1 (Ω), |mε,0 |2 mε,0 ∈ L1 (Ω), mε,0 = 0 if ε,0 = 0, ∈ L1 (Ω) ε,0

(2.11)

for fixed ε > 0 where log− g = log min(g, 1). Then, for any γ > 1 and any T > 0, there exists a weak solution of ( 1.4-1.5) in the sense of distribution verifying the following energy inequality and Bresch-Desjardins entropy:  T   με  |Du |2 + αε |u |2 dxdt ≤ Eε,0 (2.12) Eε (t) + 0

Ω

4

YOUNG-SAM KWON

where

   1 1 √  |u |2 + (γ − 1 − γ( − 1)) + |∇  |2 dx, 2 γ(γ − 1) Ω 2    1 1 √ ε,0 |uε,0 |2 + := (γε,0 − 1 − γ(ε,0 − 1)) + |∇ ε,0 |2 dx 2 γ(γ − 1) Ω 2

Eε (t) := Eε,0 and

  1 1 √  |u + ∇ ln  |2 + (γ − 1 − γ( − 1)) + |∇  |2 γ(γ − 1)ε2  Ω 2   −αε log  dx +

T

 Ω

0

γ 1 |∇2 |2 dxdt +  |∇2 log  |2 dxdt 2ε2

(2.13)

≤ Bε,0 + C where Bε,0

  1 1 √ ε,0 |uε,0 |2 + |∇ ε,0 |2 + = (γε,0 − 1 − γ(ε,0 − 1)) 2 2 γ(γ − 1)ε Ω  −αε log− ε,0 dx.

Remark 2.1. We can derive BD entropy inequality by the following control of the Coriolis force term (see Proposition 3.6 in Fanelli [4]):  t    γ 1  t 1   f × u · ∇ log  dxdt ≤ C(t) + 2 |∇2 |2 dxdt  ε 0 Ω 2ε 0 Ω  t  |∇2 log  |2 dxdt + 0

Ω

for any t ∈ [0, T ) where we have used the following fact: ∇2  =  ∇2 log  +

1 ∇ ⊗ ∇ . 

Remark 2.2. As the proof of global existence of the system ( 1.4-1.5) is given in [17] on the periodic domains, we can prove the global weak solution of compressible quantum Navier Stokes equations without the damping term |u|2 u in R2 × T1 . Indeed, let (ur , r ) be weak solution of quantum term with rr |ur |2 ur . Then we can √ show the strong convergence of n un together with rr |ur |2 ur and so we can delete the damping term r |ur |2 ur with the previous result. Theorem 2.2. Let Ω be the unbounded domain Ω = R2 × T1 and ( , u ) be a weak solution to ( 1.4-1.5) verifying the following initial data: √ (1) (1) ε,0 u,0 → u0 in L2 (Ω), ε,0 → 0 in H 1 (Ω) (2.14) where

(1)

(1)

,0 = 1 + εε,0 , 0 ∈ H k−1 (Ω), u0 ∈ H k (Ω; R3 )

(2.15)

for a certain k ≥ 3. Furthermore, we assume that q0 is the unique solution of the following problem:  1  1 (1) curlh [u0 ]h dx3 + 0 dx3 in H 1 (R2 ). −Δh q0 + q0 = 0

0

ASYMPTOTIC LIMIT FOR NSES

Then, one has

⎧  − 1 ⎪ ⎨ → q a.e. on (0, T ) × Ω, ε ⎪ ⎩ √  u → v a.e. on (0, T ) × Ω,

5

(2.16)

such that [q, v] verifies the equation ( 1.9). 3. Proof of Theorem 2.2 In this section we are going to give a rigorous proof of Theorem 2.2. To begin with we derive some uniform estimates from the energy inequality and B-D entropy inequality. 3.1. Uniform bounds. In this section we are going to derive some estimates on the sequence { , u }>0 . From the energy inequality (2.12) and the B-D entropy inequality (2.13), we obtain ⎧ √ ess sup  u (t) L2 (Ω) ≤ C ⎪ ⎪ ⎪ t∈(0,T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

∇

⎪ ⎪



⎪ ⎪ (t) ≤C ess sup

2 √ ⎪ ⎪  L (Ω) ⎪ t∈(0,T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ess sup γ − 1 − γ( − 1)) L1 (Ω) ≤ 2 C (3.17) t∈(0,T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u L2 ((0,T )×Ω) ≤ C ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ∇γ/2 ⎪  L2 ((0,T )×Ω) ≤ ε C ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ √  |∇2 log  | L2 ((0,T )×Ω) ≤ C We consider the properties of convex function γ − 1 − γ( − 1) ≥ C| − 1|2 if γ ≥ 2, γ − 1 − γ( − 1) ≥ C| − 1|2 if γ < 2 and  ≤ R,

(3.18)

γ − 1 − γ( − 1) ≥ C| − 1|γ if γ < 2 and  ≥ R, Let us introduce the set of the essential and residual value g = [g]ess + [g]res where [g]ess = χ( )g, [g]res = (1 − χ( ))g and χ is defined as follows χ(r) = 1 for all r ∈ [1/2, 2] and χ(r) = 0 otherwise. Following the estimate of the third line in (3.17) and the convexity (3.18), we get

  − 1



ess sup

≤C (3.19)

2  ess L (Ω) t∈(0,T ) and

 ess sup t∈(0,T )

Ω

[1 + γ ]res dx ≤ 2 C.

(3.20)

6

YOUNG-SAM KWON

In accordance with (3.19) and (3.20), we obtain  − 1 → (1) weakly − (∗) in L∞ (0, T ; (L2 + Lp )(Ω)), p = min{γ, 2}, ε which also implies that  u → u weakly − (∗) in L∞ (0, T ; (L2 + L2γ/(γ+1 )(Ω; R3 )),

(3.21)

(3.22)

and so we deduce that divu = 0,

(3.23)

f × u + ∇(1) = 0

(3.24)

and in the sense of distribution. 3.2. Relative entropy inequality. In this section we will introduce the relative entropy and it can be derived in the sprite of Feireisl, Jin, and Novotny [9]. Let us set H(, r) = P () − P  (r)( − r) − P (r) p(s) = γ1 sγ and   1 1 √  E(, u|r, U) = |u − U|2 + 2 H(, r) + |∇ |2 dx. ε Ω 2

where P (s) =

1 γ γ(γ−1) s ,

We can derive the relative entropy: for solutiuons ( , u ) verifying (1.4-1.5),   t=τ  τ   E( , u |r, U) με  |Du |2 + αε |u − U|2 dxdt + t=0

 ≤

0

τ

0

Ω

(3.25)

R( , u , r, U)dt,

where the remainder R is given by      ∂t U + u · ∇x U (U − u ) dx R( , u , r, U) = Ω

  + Ω

+

1 ε2

 με  Du : ∇U − αε (u − U) · U dx

  Ω

 (r − )∂t P  (r) + ∇P  (r) · (rU − u) dx

    1 1 τ p() − p(r) divUdx +  f × u dxdt ε2 Ω ε 0    1 √ √ +  ΔdivU − 2(∇  ⊗ ∇  ) : ∇U dx Ω 2 −

where U, r satisfy U ∈ Cc∞ ([0, T ] × Ω; R3 ), and r > 0 in [0, T ] × Ω, r − 1 ∈ Cc∞ ([0, T ] × Ω).

(3.26)

ASYMPTOTIC LIMIT FOR NSES

7

3.3. Dispersive Estimates. Let (s, w) solves the following acoustic equation: ε∂t s + divw = 0,

(3.27)

ε∂t w + f × w + ∇x s = 0,

(3.28)

with the initial data s(0, ·) = s0 , w(0, ·) = w0 . Consider the operator



 divw s → , L: f × w + ∇x s w

which is defined on L2 (Ω) × L2 (Ω; R3 ). Let us introduce the domain of operator D(L) and the null space N (L) as follows: D(L) = {[r, w]|r ∈ H 1 (Ω), w ∈ L2 (Ω; R3 ), divw ∈ L2 (Ω)}, N (L) = {[q, v]|q = q(xh ), q ∈ H 1 (R2 ), v = [vh (xh ), v3 ], divh vh = 0, ∂x3 v3 = 0, f × v + ∇x q = 0} and P : L2 (Ω) × L2 (Ω; R3 ) → N (L), P[r, U] = [q, v]. For  1  1 ˜ h (xh , x3 ) = ˜3 = 0, r(xh , x3 )dx3 , U Uh (xh , x3 )dx3 , U r˜(xh ) = we get

0

0

 −Δh q + q =



1 0

curlh Uh dx3 +

1 0

rdx3

for the minimization problem: ˜ h 2 2 2 2 . [q, v] → q − r˜ 2L2 (R2 ) + vh − U L (R ;R ) In order to derive the dispersive estimates, we first regularize the initial data to remove the interruption of computation of convergence. We consider a family of smooth functions χδ ∈ Cc∞ (0, ∞), 0 ≤ χδ ≤ 1, χδ  1 as δ → 0, and

φδ ∈ Cc∞ (R2 ), 0 ≤ φδ ≤ 1, φδ  1 as δ → 0. (1)

Then we regularize 0 , u0 by

and

    (1) (1) −1 χδ (|ξ|) 0 φδ (ξ, k) exp(−ikx3 ), [0 ]δ (xh , x3 ) = Σ|k|≤1/δ Fξ→x h

(3.29)

   −1  (|ξ|) u φ χ (ξ, k) exp(−ikx3 ). [u0,j ]δ (xh , x3 ) = Σ|k|≤1/δ Fξ→x δ 0,j δ h

(3.30)

Finally, for the initial data:

(1)

[0 ]δ = s0,δ + q0,δ , where

 −Δh q0,δ + q0,δ =



1 0

(3.31)

curlh [[u0 ]h ]δ dx3 +

1 0

(1)

[0 ]δ dx3 ,

and [u0 ]δ = v0,δ + w0,δ , with

v0,δ = ∇⊥ h q0,δ .

(3.32)

8

YOUNG-SAM KWON

We denote (sε,δ , wε,δ ) the solution of the following acoustic equation: ε∂t sε,δ + divwε,δ = 0,

(3.33)

ε∂t wε,δ + f × wε,δ + ∇x sε,δ = 0,

(3.34)

sε,δ (0, ·) = s0,δ , wε,δ (0, ·) = w0,δ .

(3.35)

with the initial data

Then we have the dispersive estimates: Theorem 3.1. (see [5, 6]) Let {sε,δ , vε,δ }ε>0,δ>0 be the solution of system ( 3.33) and ( 3.34) with initial data in ( 3.35). Then, one has wε,δ (t, ·) 2H k (Ω;R3 ) + sε,δ (t, ·) 2H k (Ω) = w0,δ (·) 2H k (Ω;R3 ) + s0,δ (·) 2H k (Ω)

(3.36)

and sε,δ → 0 in Lp (0, T ; W k,q (Ω)), wε,δ → 0 in Lp (0, T ; W k,q (Ω))

(3.37)

as ε tends to 0 for any δ > 0, 1 ≤ p < ∞, 2 < q ≤ ∞, and k = 0, 1, 2, .... 3.4. Convergence of viscosity and velocity terms. In the relative entropy inequality (3.25), we take r = 1 + ε(qδ + sε,δ ), U = vδ + wε,δ . From now on we remove δ to proceed the convenient presentation such that sε = sε,δ , q = qδ , v = vδ , wε = wε,δ . Let us first show that the viscosity term vanishes:   

T 0



  με  Du : ∇U dxdt Ω   T  √ √  = με  (  − 1)  Du : ∇U dxdt + με 0

Ω

T 0



√ Ω

   Du : ∇U dxdt

√ √ ≤ με C  − 1 L∞ (0,T ;L2 (Ω))  Du L2 ((0,T )×Ω) √ +με  Du L2 ((0,T )×Ω) ∇U L2 ((0,T )×Ω) = Λε (τ, δ) (3.38) with lim lim Λε (·, δ) = 0 in C[0, T ]

δ→0 ε→0

where we have here used (1.6), (1.10), (3.17), (3.36), and the following fact (3.39), together with (3.19) and (3.20). √ | x − 1|2 ≤ C|x − 1|k , k ≥ 1.

(3.39)

ASYMPTOTIC LIMIT FOR NSES

9

We next control the velocity terms  τ    (∂t U + u · ∇U)(U − u ) − αε (u − U) · U dxdt Ω

0





τ

= Ω

0



 (U − u ) ⊗ (u − U) : ∇U dxdt

 

τ

  (U − u ) · ∂t U +  (U − u ) ⊗ U : ∇U − αε (u − U) · U dxdt

+ Ω

0

 ≤C 

τ

0

 E( , u |r, U)dt +

 (U − u ) · (∂t v + v · ∇v + αv)dxdt 

Ω

0

 (U − u ) · ∂t wε dxdt +

τ



0



τ

Ω

0

Ω



+ 

Ω

0



τ

+ 



τ

 (U − u ) ⊗ v : ∇wε dxdt +

 (U − u ) ⊗ wε : ∇vdxdt τ

0

 Ω

 (U − u ) · ∇wε · wε dxdt

 

τ

 αε |U|2 − αε u · U −  (U − u ) · αv dxdt.

+ Ω

0

 =C

τ

0

E( , u |r, U)dt +

6 

Ik .

k=1

Using the estimates (3.17), (3.19), and (3.20), together with the regularity (1.10), (3.36), and (3.37) yields that 5 

Ik = Λε (τ, δ).

k=3

On the other hand, from (1.10), (3.17), (3.37), (3.19), and (3.20), one has  τ   αε |U|2 − αε u · U −  (U − u ) · αv dxdt Ω

0



τ

 

= Ω

0

 (αε − α)|U|2 − (αε − α)u · U − (U − u ) · (α( − 1)v − α wε ) dxdt 

τ



+α 0



τ



= Ω

0

 ≤C

τ 0

 ≤C

τ 0

Ω

(U − u ) · (1 −  )wε dxdt

√ √ α   (U − u ) · wε dxdt + Λε (τ, δ) 

E( , u |r, U)dt + C

0

τ

 Ω

E( , u |r, U)dt + Λε (τ, δ)

 |wε |2 dxdt + Λε (τ, δ)

10

YOUNG-SAM KWON

while 

τ 0

 Ω

 |wε |2 dxdt



≤ C  L∞ (0,T ;L3 (Ω)) wε L∞ (0,T ;L2 (Ω))  τ ≤C wε L6 (Ω) dt = Λε (τ, δ),

τ 0

wε L6 (Ω) dt

0

where  ∈ L∞ (0, T ; L3 (Ω))

(3.40)

since the following holds: √ ∇  ∈ L∞ (0, T ; L2 (Ω)) and the Sobolev embedding H 1 (Ω) → L6 (Ω). For I1 , we first observe that 

τ 0

 Ω

 wε · (∂t v + v · ∇v + αv)dxdt.

Employing (3.19), (3.20), and (3.36) gives  0

τ

 Ω

 wε · (∂t v + v · ∇v + αv)dxdt  τ = wε · (∂t v + v · ∇v + αv)dxdt + Λε (τ, δ) = Λε (τ, δ), 0

Ω

while ∂t v and v belong to Lp (0, T ; Lq (Ω)), 1 ≤ p < ∞, 2 < q ≤ ∞ since the following fact holds:   p 1,q (Ω)), 1 ≤ p < ∞, q > 2, ∂t q = (Δh − I)−1 − (∇⊥ h q · ∇h )Δh q − αΔh q ∈ L (0, T ; W together with the interpolation theory and (1.10). Furthermore, by (3.22), it follows that 

  1  (v − u ) · (∂t v + v · ∇v + αv)dx = ∂t |∇⊥ q|2 dx + α |∇⊥ q|2 dx 2  Ω Ω Ω − u · (∂t v + v · ∇v + αv)dx + Λε (τ, δ) Ω

(3.41)

where we have used 3.19), (3.20), and  Ω

v · (v · ∇)vdx = 0.

ASYMPTOTIC LIMIT FOR NSES

11

Thus we get 

E( , u |r, U)

t=τ t=0



τ

 

+ 0

Ω

 με  |Du |2 + αε |u − U|2 dxdt

 τ 1 ⊥ 2 ≤ ∂t |∇ q| dxdt + α |∇⊥ q|2 dxdt 2 0 Ω Ω 0  τ u · (∂t v + v · ∇v + αv)dxdt − 

Ω

0

+

1 ε2



τ



 

τ

Ω

0



 (r −  )∂t P  (r) + ∇x P  (r) · (rU −  u ) dxdt (3.42)

τ  

 p( ) − p(r) divUdxdt

1 ε2 0 Ω    τ 1 τ  (f × u ) · (U − u )dxdt +  (U − u ) · ∂t wε + ε 0 Ω Ω 0  τ   1 √ √  ΔdivU − 2(∇  ⊗ ∇  ) : ∇U dxdt + Ω 2 0 −

+Λε (τ, δ).

3.5. Convergence of pressure terms. In this section we are going to handle the pressure terms:

1 ε2



τ 0

  Ω

(r −  )∂t P  (r) + ∇x P  (r) · (rU −  u ) 1 − 2 ε



τ 0

  Ω



(3.43)

p( ) − p(r) divUdxdt.

The terms in the integration of (3.43) is written by the following form:  1 1 (r −  )∂t P  (r) + ∇x P  (r) · (rU −  u ) − 2 p( ) − p(r) divU 2 ε ε   1 = 2 p(r) − p (r)(r −  ) − p( ) divU ε 1 1 + (r −  )P  (r)(∂t q + div((q + sε )U) + 2  ∇P  (r) · (U − u ) ε ε

(3.44)

12

YOUNG-SAM KWON

where we have used (3.27). On the other hand, we get from (3.19), (3.20), and (3.37), that

1 ε



τ

 Ω

0

(r −  )P  (r)(∂t q + div((q + sε )U)dxdt



τ



=

 1 −  + q + sε )P  (1 + εsε ) ∂t q + ∇q · v ε Ω +∇q · wε + ∇sε · U dxdt (

0



τ



= 0

Ω

(3.45)

(q − (1) )∂t qdxdt + Λε (τ, δ),

while ∇h q · v = ∇h q · ∇⊥ h q = 0. Furthermore, it follows that

1 ε2 =





τ



1 ε2

 ∇P  (r) · (U − u )dxdt

Ω

0

τ

 Ω

0



τ

  ∇ P  (r) − P  (1)(r − 1) − P  (1) · (U − u )dxdt



+ Ω

0



τ

=ε 0



1 2

0

   P (1 + ε(q + sε )) − P  (1)  (q + sε )∇(q + sε ) · (v + wε )dxdt ε(q + sε ) Ω



τ

τ

   P (1 + ε(q + sε )) − P  (1)  ∇(q + sε )2 · ( u )dxdt ε(q + sε ) Ω

0



τ



+ Ω

0

 =−

τ  0

 (∇q + ∇sε )dxdt ε

   − 1 P  (1 + ε(q + sε )) − P  (1)  × (q + sε )∇(q + sε ) · Udxdt ε ε(q + sε ) Ω

+ −

(U − u ) ·

Ω

(U − u ) ·

 (∇q + ∇sε )dxdt ε

  (f × v) · (U − u ) +  (U − u ) · ∂t wε + (U − u ) · (f × wε )dxdt ε ε +Λε (τ, δ). (3.46)

ASYMPTOTIC LIMIT FOR NSES

13

where we have used (3.19), (3.20), (3.22), (3.23), (3.28), and (3.37). Using the equation (1.9) and (3.24), we deduce  −

Ω

(∂t v · u + ∂t q(1) + αu · v + v · ∇v · u)dx  

 [∂t (Δh q − q) + αΔh q](1) + v · ∇v · u dx

= Ω

=− =−

 

 (1) (∇h Δh q · v) + v · ∇v · u dx

Ω

(3.47)

  Ω

 =−

Ω

 (f × u) · vΔh q + v · ∇v · u dx

u · ∇h |v|2 dx = 0.

In virtue of (3.45), (3.46), and (3.47), together with (3.17), we have that 

E( , u |r, U)

t=τ t=0



≤



1 τ ∂t 2 0  τ 

+ Ω

0



τ

+C 0

 ≤

τ

 

+ 0

 Ω

Ω

 με  |Du |2 + αε |u − U|2 dxdt

(|∇h q|2 + q 2 )dxdt + α



τ 0

 Ω

|∇h q|2 dxdt

 1 √ √  ΔdivU − 2(∇  ⊗ ∇  ) : ∇U dxdt 2 (3.48)

E( , u |r, U)dt + Λε (τ, δ).

0

   1 √ √  Δx divU − 2(∇x  ⊗ ∇x  ) : ∇x U dxdt Ω 2



τ

τ

+C 0

E( , u |r, U)dt + Λε (τ, δ),

where the first inequality holds due to (f × u ) · (U − u ) = (f × U) · (U − u ), and the last inequality is given by the following fact: 1 2



τ 0

 ∂t

2

Ω

2

(|∇h q| + q )dxdt + α 

τ

τ 0

 Ω

|∇h q|2 dxdt





= 0



Ω

v · ∇h (Δh q)qdxdt = −

τ 0

 Ω

∇⊥ h q · ∇h q(Δh q)dxdt = 0.

14

YOUNG-SAM KWON

3.6. Convergence of quantum potential terms. In this section we handle the quantum potential term. From (3.17), (3.19), (3.20), and (3.37), we get that 

τ

0

   1 √ √  ΔdivU − 2(∇  ⊗ ∇  ) : ∇U dxdt Ω 2  τ  τ √ √ ≤−  ∇  ∇divwε dxdt + C E( , u |r, U)dt Ω

0

≤

1 2



τ

 |∇divwε |2 dxdt + C

Ω

0

 ≤C

0



τ

0

τ

+C 0



τ

+C 0

0

τ

0

E( , u |r, U)dt

(3.49) E( , u |r, U)dt

√ ≤ C ∇  2L∞ (0,T ;L2 (Ω) +



τ

 L3 (Ω) ∇divwε L2 (Ω) ∇divwε L6 (Ω) dt 

=C





τ 0

∇divwε 2L6 (Ω) dt

E( , u |r, U)dt

E( , u |r, U)dt + Λε (τ, δ),

where we have used the Hollder’s inequality and the Sobolev embedding H 1 (Ω) → L6 (Ω). Thus, we have that



E( , u |r, U)

t=τ t=0

 ≤C

0

τ



τ

 

+ 0

Ω

 με  |Du |2 + αε |u − U|2 dxdt (3.50)

E( , u |r, U)dt + Λε (τ, δ).

3.7. Convergence of initial data and conclusion. Let us apply the Gronwall’s inequality to (3.50) in order to obtain:   E ( , u |r, U)(τ ) ≤ Λε (τ, δ) + E (0,ε , u0ε |r0 , U0 ) (1 + exp(τ C)).

(3.51)

ASYMPTOTIC LIMIT FOR NSES

15

Let us check the convergence of the initial data: E(ε,0 , uε,0 |r0 , U0 ) =

   1 1 √ ε,0 |uε,0 − Uε,0 |2 + 2 H(ε,0 , rε,0 ) + |∇ ε,0 |2 dx ε Ω 2   √ 2 ≤C | ε,0 uε,0 − u0 | dx + C |[u0 ]δ − u0 |2 dx Ω

Ω

(3.52)

  +C Ω

 +C Ω

 (1) (1) (1) (1) (1) |ε,0 − 0 |2 + ε2 |∇ε,0 − ∇0 |2 + ε2 |∇0 |2 dx (1)

(1)

|[0 ]δ − 0 |2 dxdt + Λε (0, δ) = Λε (0, δ).

We are now able to prove the local convergence of  √ |  u − v|2 dx

√  u . Note that

K

 ≤

K

 ≤

Ω

√ √ √ √ |  u −  v −  wε − (1 −  )(v + wε ) + wε |2 dx  |u − U|2 dx +

≤ E( , u |r, U)(τ ) +

 K



|wε |q dx

1/q

1/q

K

|wε |q dx

+ Λε (τ, δ)

(3.53)

+ Λε (τ, δ)

with p > 2 and any compact subset K ⊂ Ω, where we have here used (3.19), (3.20), (3.36), and (3.39). Consequently, using (3.51), (3.52), and (3.53) together with (3.37) and passing to the limit for ε → 0, δ → 0, we prove the second part of (2.16). References [1] D. Bresch, B. Desjardins. Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm. Math. Phys.,238(1-3):211–223 2003. 1 [2] D. Bresch, B. Desjardins. On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J. Math. Pures Appl.,86(4):362–368 2006. 1 [3] D. Bresch, B. Desjardins, and Chi-Kun Lin. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Partial. Differential Equations.,28(3-4):843– 868 2003. 1 [4] F. Fanelli. Highly rotating viscous compressible fluids in presence of capillarity effects. To appear in Mathematische Annalen. 2.1 [5] E. Feireisl and Novotn´ y. Multiple scales and singular limits for compressible rotating fluids with general initial data. Comm. Partial Differential Equations. ,39(6):1104–1127 2014. 3.1 [6] E. Feireisl and Novotn´ y. Scale interactions in compressible rotating fluids. Ann. Mat. Pura Appl.,193(4):1702–1725 2014. 3.1 [7] E. Feireisl and Novotn´ y. Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Comm. Math. Phys. ,321(3):605–628 2013. 1 [8] E. Feireisl and Novotn´ y. The low Mach number limit for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal.,186(1):77–107 2007. 1

16

YOUNG-SAM KWON

[9] E. Feireisl, B. Jin, and Novotn´ y. Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. ,14(4):717–730 2012. 3.2 [10] I. Gamba, A. J¨ ungel, and A. Vasseur. Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations. J. Differ. Equ. 247, 3117–3135 2009. 1 [11] A. J¨ ungel. Global weak solutions to compressible Navier–Stokes equations for quantum fluids. SIAM J. Math. Anal.42, 1025–1045 2010. 1 [12] A. J¨ ungel, C.K, Lin, and K.C. Wu. An Asymptotic Limit of a Navier–Stokes System with Capillary Effects. Commun. Math. Phys. (2)329, 725–744 2014. 1 [13] P.-L. Lions, and N. Masmoudi. Incompressible limit for a viscous compressible fluid J. Math. Pures Appl. (9), 77 585–627, 1998. 1 [14] N. Masmoudi. 1 Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire. (2), 18 199–224, 2001. [15] M. Oliver. 1 Classical solutions for a generalized Euler equation in two dimensions. J. Math. Anal. Appl., 215 471–484, 1997. [16] A. Mellet, A.F., Vasseur. On the barotropic compressible Navier-Stokes equations. Comm. Partial Differential Equations.,32: 1-3, 431–452 2007. 1 [17] A.F. Vasseur, C. Yu. Global weak solutions to compressible quantum Navier-Stokes equations with damping. SIAM J. Math. Anal.(2), 48, 1489–1511, 2016. 1, 2.1, 2.2 [18] A.F. Vasseur, C. Yu. Existence of Global Weak Solutions for 3D Degenerate Compressible Navier-Stokes Equations. To appear in inventiones mathematicae. 1 (Young-Sam Kwon) Department of Mathematics, Dong-A University, Busan 604-714, Korea E-mail address: [email protected]