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Weak-strong uniqueness for the Navier–Stokes–Poisson equations
Journal Pre-proof Weak-strong uniqueness for the Navier–Stokes–Poisson equations Lianhua He, Zhong Tan
PII: DOI: Reference:
S0893-9659(19)30467-7 https://doi.org/10.1016/j.aml.2019.106143 AML 106143
To appear in:
Applied Mathematics Letters
Received date : 25 September 2019 Revised date : 11 November 2019 Accepted date : 11 November 2019 Please cite this article as: L. He and Z. Tan, Weak-strong uniqueness for the Navier–Stokes–Poisson equations, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106143. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Journal Pre-proof revision--weak-strong uniqueness of nsp system.tex Click here to view linked References
WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS LIANHUA HE AND ZHONG TAN Abstract. This paper is devoted to study the weak-strong uniqueness property of compressible NavierStokes-Poisson system. By means of relative entropy method, we prove the result that the weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists.
Keywords: relative entropy inequality; weak-strong uniqueness; Navier-Stokes-Poisson equations.
1. Introduction Let T > 0 and Ω = T3 be the 3-dimensional torus. In this paper, we consider the following compressible Navier-Stokes-Poisson(NSP) system
na lP repr oo f
ρt + div(ρu) = 0 (ρu)t + div(ρu ⊗ u) − µ∆u − (λ + µ)∇divu + ∇p(ρ) = ρ∇Φ + ρ f Z 1 ∆Φ = 4πg ρ − ρdx |Ω| Ω
(1.1) (1.2) (1.3)
in (0, T ) × Ω, where ρ ∈ R, u ∈ R3 and Φ ∈ R denote the electron density, electron velocity, and the electrostatic potential, respectively. The pressure function p(ρ) satisfies p(ρ) = aργ with a > 0 and γ > 1. µ, λ are the constant viscosity coefficients satisfying the physical requirements µ > 0, λ + 23 µ ≥ 0. f is the external force. The initial conditions are imposed as follows: ρ(0, ·) = ρ0 (·),
u(0, ·) = u0 (·),
(ρu)(0, ·) = (ρ0 u0 )(·)
(1.4)
Jo
ur
The Navier-Stokes-Poisson system (1.1)-(1.3) can be used to describe the transportation of charge particles in electronic devices. More details about its background are introduced in [1]. Many researchers have been devoted to many topics of the compressible NSP system. Zhang and Tan [2] established the local existence of unique strong solution to the isentropic compressible Navier-Stokes-Poisson system (1.1)-(1.3) by means of Schauder fixed point theorem. The existence and asymptotic behavior of global solution are established by Li et al.[3] and Shi et al.[4]. The global well-posedness for NSP system in some Besov spaces have been investigated recently(see [5], [6]). T.Kobayashi[7] proved the existence of finite-energy weak solutions of isentropic compressible NSP equation with the pressure p(ρ) = ργ (γ > 3 2 ). By use of Orlicz space theory, Zhang and Tan[8] got the existence of finite-energy weak solutions to NSP system with the pressure function p(ρ) = ρ logd ρ (d > 1). Other properties of NSP system, for example, time delay rate and long-time behavior of solution, are investigated(see[9], [10]). However, to best of our knowledge, there are no results about the weak-strong uniqueness of Naiver-Stokes-Poisson equations. In this paper, we consider the weak-strong uniqueness principle of NSP system (1.1)-(1.3) which means that the weak solution must coincide with a strong solution emanating from the same initial data as long as the latter exists. The relative entropy method is an important method to study partial differential equations. Carrillo et al.[11] used entropy dissipation method to consider the large-time asymptotic of quasilinear degenerate parabolic problems and proved the generalized Sobolev-inequalities. The relative entropy method is devoted to study the incompressible Euler limit of the Boltzmann equation in [12]. J.Glesselmann et al.[13] applied the modified relative entropy approach to derive the weak-strong stability of NaiverStokes-Korteweg system. The weak-strong uniqueness property of the Navier-Stokes-Fourier system in bounded domain or unbounded domain is proved by E.Feiresisl in [14] and Jessle et al. in [15]. In 2010 Mathematics Subject Classification. 35D30, 35E15, 76N10. Corresponding author: Zhong Tan, [email protected]. 1
Journal Pre-proof 2
LIANHUA HE AND ZHONG TAN
particular, the relative entropy method is the most important method to research weak-strong uniqueness of compressible Navier-Stokes equation with monotone pressure in [18][19] [20] and non-monotone pressure in [21]. In [22], Kwon applied the refine relative entropy method to prove the convergence of the weak solution of degenerate compressible quantum NSP system to the strong solution of the incompressible Euler equation. Unfortunately, Kwon didn’t mention weak-strong uniqueness of NavierStokes-Poisson equations in [22]. In this paper, by the relative entropy method, we shall show weakstrong uniqueness property of Navier-Stokes-Poisson system for the first time. The paper is organized as follows. In Section 2, we recall the definition of finite-energy weak solution for NSP equations (1.1)-(1.3) and state the main results. In Section 3, we derive the relative entropy inequality to (1.1)-(1.3). In last section, we prove the weak-strong uniqueness property of (1.1)-(1.3). 2. Main results Definition 2.1. We call that (ρ, u, Φ) is a finite-energy weak solution to the Navier-Stokes-Poisson system (1.1)-(1.4) if i) ρ ≥ 0, ρ ∈ L∞ [0, T ); Lγ (Ω) , u ∈ L2 [0, T ); W 1,2 (Ω) , Φ ∈ L∞ [0, T ); W 2,γ (Ω) . (2.1) ii) The energy inequality
′
Z
2
|∇u| dx + (λ + µ)
Z
2
|divu| dx ≤
na lP repr oo f
dE(t) +µ dt
Ω
Ω
Z
Ω
ρ f · udx
holds in D ((0, T )) with the energy ! Z 1 a γ 1 2 2 E(t) = ρ|u| + ρ + |∇Φ| (t, ·)dx < ∞ γ−1 8πg Ω 2
for t ∈ [0, T ). iii) For any τ ∈ (0, T ) and any test fuction φ ∈ C ∞ ([0, T ) × Ω), it hlods Z Z Z τZ ρ(τ, ·)φ(τ, ·)dx − ρ0 φ(0, ·)dx = ρ∂t φ + ρu · ∇φdxdt. Ω
0
Ω
(2.2)
(2.3)
(2.4)
Ω
iv) For any τ ∈ (0, T ) and any test fuction ϕ ∈ C ∞ ([0, T ) × Ω), ϕ|∂Ω = 0, it hlods Z Z ρu(τ, ·)ϕ(τ, ·)dx − ρ0 u0 ϕ(0, ·)dx Ω ZΩτ Z = ρu∂t ϕ + (ρu ⊗ u) : ∇ϕ − µ∇u : ∇ϕ − (λ + µ)divudivϕ + p(ρ)divϕ + ρϕ · ∇Φ + ρ f · ϕdxdt. 0
Ω
(2.5)
v)The equation (1.1) is satisfied in the sense of renormalized solution, i,e. (b(ρ))t + div (b(ρ)u) + (b′ (ρ)ρ − b(ρ)) divu = 0
(2.6)
ur
for any b ∈ C ′ (R) such that b′ (z) = constant, for any z large enough, sayz ≥ M. Lemma 2.1. ([23],Proposition 2.2)
q20 ρ0
3 2.
Assume that the initial data ρ0 , q0 = ρ0 u0 satisfy
L1 (Ω).
∈ ρ0 ∈ ≥ 0, q0 = 0 whenever ρ0 = 0. one finite energy weak solution (ρ, u, Φ) in (0, T ) × Ω.
Jo
Lγ (Ω),ρ0
Let γ >
We define H(s) =
asγ γ−1 .
Then the problem (1.1) − (1.4) admits at least
Then the following equalities hold
H ′ (s)s − H(s) = p(s), H ′′ (s)s = p′ (s). T Theorem 2.1. Suppose that f ∈ L∞ 0, T ; L1 (Ω) L∞ (Ω) . Let (ρ, u, Φ) be a finite energy weak solution to the NSP system (1.1) − (1.4) in the sense of Definition2.1. Let r > 0, U, Ψ ∈ C0∞ ([0, T ) × Ω) and satisfy Z 1 ∂t r + div(rU) = 0, ∆Ψ = 4πg r − rdx . (2.7) |Ω| Ω
Journal Pre-proof WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS
Then the following relative entropy inequality holds for a.e τ ∈ (0, T ): Z τZ 2 2 µ ∇(u − U) + (λ + µ) div(u − U) dxdt ε(ρ, u, Φ r, U, Ψ)(τ) + 0 Ω Z τ R(t)dt, ≤ε ρ0 , u0 , Φ0 r(0, ·), U(0, ·), Ψ(0, ·) + 0
where
3
(2.8)
Z Z 2 1 ε(ρ, u, Φ r, U, Ψ)(τ) = ρ u − U (τ, ·)dx + H(ρ) − H ′ (r)(ρ − r) − H(r) (τ, ·)dx Ω 2 Ω Z 2 1 ∇(Φ − Ψ) (τ, ·)dx, + Ω 8πg
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and the remainder R(t) is defined as Z µ∇U : ∇(U − u) + (λ + µ)divUdiv(U − u)dx R(t) = ZΩ Z + ρ ∂t U + u · ∇U (U − u)dx + ρ f (u − U)dx Ω Ω Z Z + (r − ρ)∂t (H ′ (r)) + (rU − ρu)∇(H ′ (r))dx + (p(r) − p(ρ))divUdx Ω ZΩ + ρ(u − U) · ∇Φ + (rU − ρu) · ∇(Φ − Ψ)dx.
(2.9)
Ω
For convenience, we abbreviate ε(ρ, u, Φ r, U, Ψ)(t) by ε(t). 2γ
Theorem 2.2. Let γ > 2 and f ∈ L2 (0, T ; L γ−1 (Ω)). Suppose that (ρ, u, Φ) is a finite energy weak solution of the NSP system (1.1)-(1.4) in (0, T ) × Ω in the sense of Definition 2.1. Assume that (r, U, Ψ) is a strong solution of the same problem satisfying 0 < inf r ≤ r(t, x) ≤ sup r < ∞, (0,T )×Ω
∇r ∈ L2 (0, T ; Lq (Ω)),
(0,T )×Ω
∇2 U ∈ L2 (0, T ; Lq (Ω)),
2γ }) with the same initial data. Then (q > max{3, γ−1
ρ = r,
u = U,
Φ=Ψ
∇Ψ ∈ L2 (0, T ; Lq (Ω))
in (0, T ) × Ω.
3. relative entropy inequality
3.1. Proof of Theorem2.1.
ur
Proof. Taking 21 |U|2 as a test function in (2.4), we can get Z Z Z τZ 1 1 ρU · ∂t U + ρu · ∇U · Udxdt. ρ|U|2 (τ, ·)dx = ρ0 |U(0, ·)|2 dx + Ω 2 Ω 2 0 Ω
Jo
Similarly, substituting ϕ for U as a test function, we can obtain Z Z Z τZ ρuU(τ, ·)dx = ρ0 u0 U(0, ·)dx + ρu∂t U + (ρu ⊗ u) : ∇Udxdt Ω Ω 0 Ω Z τZ −µ∇u : ∇U − (λ + µ)divu divU + p(ρ)divU + ρU · ∇Φ + ρ f · Udxdt. +
(3.1)
0
(3.2)
Ω
By virtue of Φ, Ψ satisfying (1.3)(2.7), we can get ! Z Z 1 2 |∇(Φ − Ψ)| dx = (rU − ρu) · ∇(Φ − Ψ)dx. 8πg Ω Ω t
(3.3)
Journal Pre-proof 4
LIANHUA HE AND ZHONG TAN
From (2.2), we can deduce Z Z τZ 2 1 2 2 ρ|u| (τ, ·) + H(ρ)(τ, ·)dx + µ ∇u + (λ + µ) divu dxdt Ω 2 Z τ Z0 Ω Z 1 ρ|u0 |2 + H(ρ0 )dx + ρu · ∇Φ + ρ f · udxdt. (3.4) ≤ 0 Ω Ω 2 Summing up relations (3.1)-(3.4), we have ! Z 1 1 2 2 ρ|u − U| + H(ρ) + |∇(Φ − Ψ)| (τ, ·)dx 8πg Ω 2 Z τZ + µ∇u : ∇(u − U) + (λ + µ)divu div(u − U)dxdt Z τZ Z0 Ω 1 1 2 2 ρ(u − U) · ∇Φ + ρ f · (u − U)dxdt ρ|u0 − U(0, ·)| + H(ρ0 ) + |∇(Φ(ρ0 ) − Ψ(ρ0 ))| dx + ≤ 8πg 0 Ω Ω 2 Z τZ ρ(U − u)∂t U + ρu · ∇U · (U − u)dxdt + 0 Ω Z τZ Z τZ + (rU − ρu) · ∇(Φ − Ψ)dxdt − p(ρ)divUdxdt. (3.5) 0
Ω
Note that
Z h
Z h i i − H(r) − H (r)(ρ − r) (τ, ·)dx − − H(r(0, ·)) − H ′ (r(0, ·))(ρ0 − r(0, ·)) dx Ω ZΩτ Z ∂t (p(r)) − ∂t (H ′ (r)ρ)dxdt, = 0
and
Ω
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0
′
Ω
Z τZ 0
We may rewrite (3.5) as Z ε(ρ, u, Φ r, U, Ψ)(τ) +
Ω
rU · ∇(H ′ (r)) + p(r)divUdxdt = 0.
τZ
2 2 µ ∇(u − U) + (λ + µ) div(u − U) dxdt 0 Ω Z τZ ≤ε ρ0 , u0 , Φ0 r(0, ·), U(0, ·), Ψ(0, ·) + µ∇U : ∇(U − u) + (λ + µ)divUdiv(U − u)dx 0 Ω Z τZ Z τZ + ρ ∂t U + u · ∇U (U − u)dx + ρ f (u − U)dx 0 Ω 0 Ω Z τZ Z ′ ′ + (r − ρ)∂t (H (r)) + (rU − ρu)∇(H (r))dx + (p(r) − p(ρ))divUdx 0 Ω ZΩτ Z Z τ + ρ(u − U) · ∇Φ + (rU − ρu) · ∇(Φ − Ψ)dx = ε(0) + R(t)dt. 0
Ω
0
ur
The proof of Theorem 2.1 is completed.
Jo
3.2. Extending the admissible class of test function. By means of density argument, we can extend considerably the class of test function (r, U, Ψ) appearing in the relative entropy inequality (2.8), (2.9). For the left hand side of (2.8) to be well defined, the function (r, U, Ψ) must belong at least to the class: 2γ (3.6) r ∈ Cweak ([0, T ]; Lγ (Ω)), U ∈ Cweak ([0, T ]; L γ−1 (Ω)); ∇Ψ ∈ L∞ ([0, T ]; L2 (Ω)). Similarly, a short inspection of the integrals in (2.9) yields
(3.7)
6γ
2γ
∂t U ∈ L1 ((0, T ); L γ−1 (Ω)) + L2 ((0, T ); L 5γ−6 (Ω));
(3.8)
∇U ∈ L∞ ((0, T ); L
(3.9)
3γ 2γ−3
1
(Ω)) + L2 ((0, T ); L ∞
divU ∈ L ((0, T ); L (Ω));
6γ 2γ−3
(Ω));
(3.10)
Journal Pre-proof WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS 2γ
6γ
2γ γ−1
6γ 5γ−6
5
∇2 U ∈ L1 ((0, T ); L γ−1 (Ω)) + L2 ((0, T ); L 5γ−6 (Ω));
(3.11)
(Ω)).
(3.12)
∇Ψ ∈ L1 ((0, T ); L
(Ω)) + L2 ((0, T ); L
Moreover, the function r must be bounded away from zero, and γ
∂t (H ′ (r)) ∈ L1 ((0, T ); L γ−1 (Ω)); 6γ
(3.13) 2γ
∇(H ′ (r)) ∈ L2 ((0, T ); L 5γ−6 (Ω)) + L1 ((0, T ); L γ−1 (Ω)).
(3.14)
It is easy to prove that the relative entropy inequality (2.8)(2.9) can be extended to (r, U, Ψ) satisfying (3.6)-(3.14) by density argument. 4. weak-strong uniqueness In order to prove the Theorem2.2 , we firstly rewrite the expression of R(t).
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Lemma 4.1. For r > 0, the remainder R(t) can read as Z R(t) = ρ(u − U) · ∇U · (U − u)dx Ω Z µr−1 (ρ − r)∆U(U − u) + (µ + λ)r−1 (ρ − r)(U − u) · ∇divUdx + Z ZΩ p(r) − p′ (r)(r − ρ) − p(ρ) divUdx. + (r − ρ)U · ∇(Φ − Ψ)dx + Ω
(4.1)
Ω
Proof. For r > 0, there exists a0 > 0 so that 0 < a0 ≤ r < ∞. Since (r, U, Ψ) is the strong solution of the system (1.1)-(1.3), the following equality holds ∂t U + U · ∇U = µr−1 ∆U + (λ + µ)r−1 ∇divU − r−1 ∇p(r) + ∇Ψ + f.
Substituting (4.2) into (2.9), one has Z µ∇U : ∇(U − u) + (λ + µ)divUdiv(U − u)dx R(t) = ZΩ µρr−1 ∆U(U − u) + (µ + λ)ρr−1 ∇divU(U − u)dx + Ω Z Z + ρ(u − U) · ∇U · (U − u)dx + ρ(u − U) · ∇(H ′ (r))dx Ω Ω Z Z p(r) − p(ρ) divUdx + (r − ρ)∂t (H ′ (r)) + (rU − ρu)∇(H ′ (r))dx + Ω ZΩ + ρ(u − U) · ∇(Φ − Ψ) + (rU − ρu) · ∇(Φ − Ψ)dx.
(4.2)
(4.3)
Ω
Note that
ur
(r − ρ)∂t (H ′ (r)) + (rU − ρu)∇(H ′ (r)) + ρ(u − U) · ∇(H ′ (r)) = −(r − ρ)p′ (r)divU.
Then we can rewrite R(t) as (4.1). The proof of Lemma 4.1 is completed.
Jo
In order to estimate the remainder R(t), we can deduce the following lemma: Lemma 4.2. Let ρ ≥ 0 and 0 < a0 ≤ r ≤ b0 < ∞. There exist a1 ∈ (0, a0 ), M >> 1 and a constant c > 0 so that c(ρ − r)2 if a1 ≤ ρ ≤ Mr; p(r) (4.4) H(ρ) − H ′ (r)(ρ − r) − H(r) ≥ if 0 ≤ ρ < a1 ; cρ2 γ if ρ > Mr.
Proof. By virtue of Taylor’s formula and the definition of H(r), it is easy to infer the inequality (4.4). Hence, we omit to prove this Lemma. Finally, we prove Theorem 2.2.
Journal Pre-proof 6
LIANHUA HE AND ZHONG TAN
Proof. Step1: we estimate the remainder R(t). For ∇U, divU ∈ L1 (0, T ; L∞ (Ω)), it is easy to infer that ! Z ′ I1 = ρ(u − U) · ∇U · (U − u) − p(ρ) − p (r)(ρ − r) − p(r) divU dx ≤ k∇UkL∞ (Ω) ε(t) ≤ η(t)ε(t). (4.5) Ω
The second term of R(t) can be rewrote as Z I2 = µr−1 (ρ − r)∆U(U − u) + (µ + λ)r−1 (ρ − r)(U − u) · ∇divUdx Ω ! Z Z Z + r−1 (ρ − r) µ∆U(U − u) + (µ + λ)(U − u) · ∇divU dx + = {a1 ≤ρ≤Mr}
:= I21 + I22 + I23
{ρ>Mr}
{0≤ρ
By virtue of H¨older’s inequality, Sobolev’s inequality and Lemma 4.2, one has I21 ≤ Ck∇2 UkL3 (Ω) kρ − rkL2 ({a1 ≤ρ≤Mr}) kU − ukL6 (Ω) Z Z ≤ Ck∇2 Uk2L3 (Ω) (ρ − r)2 dx + δ |∇(U − u)|2 dx {a ≤ρ≤Mr} Ω Z 1 ≤ η(t)ε(t) + δ |∇(U − u)|2 dx
(4.6)
and
2
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Ω
I22 ≤ k∇ UkL3 (Ω) 2
≤ Ck∇
Z
{0≤ρ
Uk2L3 (Ω)
≤ η(t)ε(t) + δ
Z
Z
Ω
1 1dx 2 kU − u)kL6 (Ω)
{0≤ρ
′
H(ρ) − H (r)(ρ − r) − H(r)dx + δ
Z
Ω
|∇(U − u)|2 dx
|∇(U − u)|2 dx.
(4.7)
with δ > 0 sufficient small. From Lemma 4.2, we can get
1
kρkLγ ({ρ>Mr}) ≤ c[ε(t)] γ
and
1
γ
kρ 2 kL2 ({ρ>Mr}) ≤ c[ε(t)] 2 .
(4.8)
Due to γ > 2 and by using of Sobolev’s inequality, H¨older’s inequality and (4.8), we can deduce Z γ ρ−r γ | ρ 2 |U − u| |∇2 U| dx ≤ Ck∇2 UkL3 (Ω) kρ 2 kL2 ({ρ>Mr}) kU − ukL6 (Ω) I23 ≤ C | {ρ>Mr} ρr ≤ Ck∇2 Uk2L3 (Ω) ε(t) + δk∇(U − u)k2L2 (Ω)
≤ η(t)ε(t) + δk∇(U − u)k2L2 (Ω) .
Ω
ur
The third term of R(t) is denoted as Z Z I3 = (r − ρ)U · ∇(Φ − Ψ)dx =
+
{a1 ≤ρ≤Mr}
Z
:= I31 + I32 + I33 .
+
{0≤ρ
Z
{ρ>Mr}
(4.9)
(r − ρ)U · ∇(Φ − Ψ)dx
and
Jo
By Lemma 4.2 and H¨oder’s inequality, we can obtain 12 Z Z 1 2 (r − ρ) dx I31 ≤ CkUkL∞ (Ω) |∇(Φ − Ψ)|2 dx 2 ≤ η(t)ε(t). {a1 ≤ρ≤Mr}
I32 ≤ CkUkL∞ (Ω) ≤ CkUkL∞ (Ω) ≤ η(t)ε(t).
Z
{0≤ρ
Z
{0≤ρ
(4.10)
Ω
12 Z 1 1dx |∇(Φ − Ψ)|2 dx 2 Ω
1 1 H(ρ) − H ′ (r)(ρ − r) − H(r)dx 2 ε 2 (t)
(4.11)
Journal Pre-proof WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS
7
In view of (4.8) and Lemma 4.2, we have Z γ ρ 2 |U| |∇(Φ − Ψ)|dx I33 ≤ C {ρ>Mr}
γ
≤ CkUkL∞ (Ω) kρ 2 kL2 ({ρ>Mr}) k∇(Φ − Ψ)kL2 (Ω) ≤ CkUkL∞ (Ω) ε(t) ≤ η(t)ε(t)
(4.12)
with γ > 2. Step2: Basing on the estimate of R(t), we prove the result of Theorem 2.2. Plugging relations(4.5)-(4.12) to (2.8), we can deduce Z τZ Z τ 2 2 µ ∇(u − U) + (λ + µ) div(u − U) dxdt ≤ ε(τ) + η(t)ε(t)dt 0
0
Ω
L1 (0, T ).
with η(t) ∈ By using of Gronwall’s inequality, we infer ε(t) = 0 in (0, T ). This implies ρ = r, u = U, Φ = Ψ. The proof of Theorem 2.2 is completed.
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Acknowledgments The work of L.H He and Z.Tan was supported by the National Natural Science Foundation of China (11531010, 11726023). References
Jo
ur
[1] P. Degond, Mathematical modelling of microelectronics semiconductor devices, some current topics on nonlinear conservation laws. in: AMS/IP Stud. Adv. Math., vol. 15, Amer. Math. Sco., Providence, RI, 2007: 77-110. [2] Y.H. Zhang, Z. Tan. Strong solutions of the coupled Navier-Stokes-Poisson equations for isentropic compressible fluids. Acta Mathematica Scientia(B),30(2010): 1280-1290 [3] H.L. Li, A. Matsumura, G. Zhang. Optimal Decay Rate of the Compressible Navier-Stokes-Poisson System in R3 . Arch. Rational Mech. Anal. 196 (2010): 681-713. [4] W.X. Shi, J. Xu. A sharp time-weighted inequality for the compressible Navier-Stokes-Poisson system in the critical L p framework. J. Differ. Equ. 266(2019): 6426-6458. [5] Xiaoxin Zheng. Global well-posedness for the compressible Navier-Stokes-Poisson system in the L p framwork. Nonlinear Analysis, 75(2012): 4156-4175. [6] C. Hao, H.L. Li, Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differ. Equ. 246 (2009): 4791-4812. [7] T. Kobayashi, T. Suzuki. Weak solution to the Navier-Stokes-Poisson equation. Adv.Math.Sci.Appl.,18(2008):141-168. [8] Y.H. Zhang, Z. Tan. On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow. Math. Meth. Appl. Sci. (30)2007:305-329. [9] Y.J. Wang. Decay of the Navier-Stokes-Poisson equations. J.Differ.Equ.253(2012):273-297. [10] Q. Bie, Q. Wang, Z. Yao. Optimal decay rate for the compressible Navier-Stokes-Poisson system in the critical L p framework. J. Differ. Equ. 263(2017):8391-8417. [11] J. Carrillo, A. J¨ungle, P.A. Markowich, G. Toscani, A. Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized sobolev inequalities. Monatshefte Math 133(2001):1-82. [12] L.S. Raymond. Hydrodynamic limits: some improvements of the relative entropy method. Ann. I. H. Poincar-AN 26 (2009): 705-744. [13] Jan Giesselmann, Corrado Lattanzio, Athanasios E. Tzavaras. Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics. Arch. Rational Mech. Anal. 223(2017): 1427-1484. [14] E. Feireisl, A. Novoton´y. Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch.Rational Mech.Anal. 204(2012): 683-706. [15] D. Jessle, B. Jin, A. Novoton´y. Navier-Stokes-Fourier System on Unbounded Domains: Weak Solutions, Relative Entropies, Weak-Strong Uniqueness. SIAM J. Math. Anal. 45(2013): 1907-1951. [16] P. Germian. Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations. J.Differ.Equ. 226(2006): 373-428. [17] Q. Liu. On weak-strong uniqueness of solutions to the generalized incompressible Navier-Stokes equations. Computers and Mathematics with Applications 72 (2016): 675-686 [18] P. Germain. Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J.Math.Fluid Mech. 13(2010):137-146. [19] E. Feireisl, B. Jin, A. Novotny. Relative entropis, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J.Math.Fluid Mech.14(2012):717-730. [20] E. Feireisl, A. Novotn´y, Y. Sun. Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60(2011): 611-631
Journal Pre-proof 8
LIANHUA HE AND ZHONG TAN
[21] E. Feireisl. On weak-strong uniqueness for the compressible Navier-Stokes system with non-monotone pressure law. Comm. Partial Differential Equations 44 (2019):271-278. [22] Young-sam Kwon. From the degenerate quantum compressible Navier-Stokes-Poisson system to incompressible Euler equations. J. Math. Phys. 59 (2018), no. 12, 123101, 13 pp. [23] S. Wang, S. Jiang. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun.Partial Differ.Equ., 31(2006):571-591. School of Mathematical Sciences, Xiamen University , Xiamen, 361005, China. And School of Mathematics Science, Guizhou Normal University, Guiyang, 550001, China E-mail address, L. He: [email protected]
Jo
ur
na lP repr oo f
School of Mathematical Sciences, Xiamen University , Xiamen, 361005, China. E-mail address, Z. Tan: [email protected]
*Author Contributions Section
Journal Pre-proof
Author
Contributions
Jo
ur
na lP repr oo f
There are many researchers to investigate the weak-strong uniqueness of Navier -Stokes equations with various pressure functions . However, in this paper, we study the weak-strong uniqueness of N-S-P system for the first time. We apply relative entropy method to obtain the weak-strong uniqueness property of N-S-P equations. Many tricks and the methods applied in this paper are relative novel. In the writing of this paper, He is responsible for writing the paper, and Tan is responsible for revising the pape. We togerher discuss and solve the problems in the paper.
PII: DOI: Reference:
S0893-9659(19)30467-7 https://doi.org/10.1016/j.aml.2019.106143 AML 106143
To appear in:
Applied Mathematics Letters
Received date : 25 September 2019 Revised date : 11 November 2019 Accepted date : 11 November 2019 Please cite this article as: L. He and Z. Tan, Weak-strong uniqueness for the Navier–Stokes–Poisson equations, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106143. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Journal Pre-proof revision--weak-strong uniqueness of nsp system.tex Click here to view linked References
WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS LIANHUA HE AND ZHONG TAN Abstract. This paper is devoted to study the weak-strong uniqueness property of compressible NavierStokes-Poisson system. By means of relative entropy method, we prove the result that the weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists.
Keywords: relative entropy inequality; weak-strong uniqueness; Navier-Stokes-Poisson equations.
1. Introduction Let T > 0 and Ω = T3 be the 3-dimensional torus. In this paper, we consider the following compressible Navier-Stokes-Poisson(NSP) system
na lP repr oo f
ρt + div(ρu) = 0 (ρu)t + div(ρu ⊗ u) − µ∆u − (λ + µ)∇divu + ∇p(ρ) = ρ∇Φ + ρ f Z 1 ∆Φ = 4πg ρ − ρdx |Ω| Ω
(1.1) (1.2) (1.3)
in (0, T ) × Ω, where ρ ∈ R, u ∈ R3 and Φ ∈ R denote the electron density, electron velocity, and the electrostatic potential, respectively. The pressure function p(ρ) satisfies p(ρ) = aργ with a > 0 and γ > 1. µ, λ are the constant viscosity coefficients satisfying the physical requirements µ > 0, λ + 23 µ ≥ 0. f is the external force. The initial conditions are imposed as follows: ρ(0, ·) = ρ0 (·),
u(0, ·) = u0 (·),
(ρu)(0, ·) = (ρ0 u0 )(·)
(1.4)
Jo
ur
The Navier-Stokes-Poisson system (1.1)-(1.3) can be used to describe the transportation of charge particles in electronic devices. More details about its background are introduced in [1]. Many researchers have been devoted to many topics of the compressible NSP system. Zhang and Tan [2] established the local existence of unique strong solution to the isentropic compressible Navier-Stokes-Poisson system (1.1)-(1.3) by means of Schauder fixed point theorem. The existence and asymptotic behavior of global solution are established by Li et al.[3] and Shi et al.[4]. The global well-posedness for NSP system in some Besov spaces have been investigated recently(see [5], [6]). T.Kobayashi[7] proved the existence of finite-energy weak solutions of isentropic compressible NSP equation with the pressure p(ρ) = ργ (γ > 3 2 ). By use of Orlicz space theory, Zhang and Tan[8] got the existence of finite-energy weak solutions to NSP system with the pressure function p(ρ) = ρ logd ρ (d > 1). Other properties of NSP system, for example, time delay rate and long-time behavior of solution, are investigated(see[9], [10]). However, to best of our knowledge, there are no results about the weak-strong uniqueness of Naiver-Stokes-Poisson equations. In this paper, we consider the weak-strong uniqueness principle of NSP system (1.1)-(1.3) which means that the weak solution must coincide with a strong solution emanating from the same initial data as long as the latter exists. The relative entropy method is an important method to study partial differential equations. Carrillo et al.[11] used entropy dissipation method to consider the large-time asymptotic of quasilinear degenerate parabolic problems and proved the generalized Sobolev-inequalities. The relative entropy method is devoted to study the incompressible Euler limit of the Boltzmann equation in [12]. J.Glesselmann et al.[13] applied the modified relative entropy approach to derive the weak-strong stability of NaiverStokes-Korteweg system. The weak-strong uniqueness property of the Navier-Stokes-Fourier system in bounded domain or unbounded domain is proved by E.Feiresisl in [14] and Jessle et al. in [15]. In 2010 Mathematics Subject Classification. 35D30, 35E15, 76N10. Corresponding author: Zhong Tan, [email protected]. 1
Journal Pre-proof 2
LIANHUA HE AND ZHONG TAN
particular, the relative entropy method is the most important method to research weak-strong uniqueness of compressible Navier-Stokes equation with monotone pressure in [18][19] [20] and non-monotone pressure in [21]. In [22], Kwon applied the refine relative entropy method to prove the convergence of the weak solution of degenerate compressible quantum NSP system to the strong solution of the incompressible Euler equation. Unfortunately, Kwon didn’t mention weak-strong uniqueness of NavierStokes-Poisson equations in [22]. In this paper, by the relative entropy method, we shall show weakstrong uniqueness property of Navier-Stokes-Poisson system for the first time. The paper is organized as follows. In Section 2, we recall the definition of finite-energy weak solution for NSP equations (1.1)-(1.3) and state the main results. In Section 3, we derive the relative entropy inequality to (1.1)-(1.3). In last section, we prove the weak-strong uniqueness property of (1.1)-(1.3). 2. Main results Definition 2.1. We call that (ρ, u, Φ) is a finite-energy weak solution to the Navier-Stokes-Poisson system (1.1)-(1.4) if i) ρ ≥ 0, ρ ∈ L∞ [0, T ); Lγ (Ω) , u ∈ L2 [0, T ); W 1,2 (Ω) , Φ ∈ L∞ [0, T ); W 2,γ (Ω) . (2.1) ii) The energy inequality
′
Z
2
|∇u| dx + (λ + µ)
Z
2
|divu| dx ≤
na lP repr oo f
dE(t) +µ dt
Ω
Ω
Z
Ω
ρ f · udx
holds in D ((0, T )) with the energy ! Z 1 a γ 1 2 2 E(t) = ρ|u| + ρ + |∇Φ| (t, ·)dx < ∞ γ−1 8πg Ω 2
for t ∈ [0, T ). iii) For any τ ∈ (0, T ) and any test fuction φ ∈ C ∞ ([0, T ) × Ω), it hlods Z Z Z τZ ρ(τ, ·)φ(τ, ·)dx − ρ0 φ(0, ·)dx = ρ∂t φ + ρu · ∇φdxdt. Ω
0
Ω
(2.2)
(2.3)
(2.4)
Ω
iv) For any τ ∈ (0, T ) and any test fuction ϕ ∈ C ∞ ([0, T ) × Ω), ϕ|∂Ω = 0, it hlods Z Z ρu(τ, ·)ϕ(τ, ·)dx − ρ0 u0 ϕ(0, ·)dx Ω ZΩτ Z = ρu∂t ϕ + (ρu ⊗ u) : ∇ϕ − µ∇u : ∇ϕ − (λ + µ)divudivϕ + p(ρ)divϕ + ρϕ · ∇Φ + ρ f · ϕdxdt. 0
Ω
(2.5)
v)The equation (1.1) is satisfied in the sense of renormalized solution, i,e. (b(ρ))t + div (b(ρ)u) + (b′ (ρ)ρ − b(ρ)) divu = 0
(2.6)
ur
for any b ∈ C ′ (R) such that b′ (z) = constant, for any z large enough, sayz ≥ M. Lemma 2.1. ([23],Proposition 2.2)
q20 ρ0
3 2.
Assume that the initial data ρ0 , q0 = ρ0 u0 satisfy
L1 (Ω).
∈ ρ0 ∈ ≥ 0, q0 = 0 whenever ρ0 = 0. one finite energy weak solution (ρ, u, Φ) in (0, T ) × Ω.
Jo
Lγ (Ω),ρ0
Let γ >
We define H(s) =
asγ γ−1 .
Then the problem (1.1) − (1.4) admits at least
Then the following equalities hold
H ′ (s)s − H(s) = p(s), H ′′ (s)s = p′ (s). T Theorem 2.1. Suppose that f ∈ L∞ 0, T ; L1 (Ω) L∞ (Ω) . Let (ρ, u, Φ) be a finite energy weak solution to the NSP system (1.1) − (1.4) in the sense of Definition2.1. Let r > 0, U, Ψ ∈ C0∞ ([0, T ) × Ω) and satisfy Z 1 ∂t r + div(rU) = 0, ∆Ψ = 4πg r − rdx . (2.7) |Ω| Ω
Journal Pre-proof WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS
Then the following relative entropy inequality holds for a.e τ ∈ (0, T ): Z τZ 2 2 µ ∇(u − U) + (λ + µ) div(u − U) dxdt ε(ρ, u, Φ r, U, Ψ)(τ) + 0 Ω Z τ R(t)dt, ≤ε ρ0 , u0 , Φ0 r(0, ·), U(0, ·), Ψ(0, ·) + 0
where
3
(2.8)
Z Z 2 1 ε(ρ, u, Φ r, U, Ψ)(τ) = ρ u − U (τ, ·)dx + H(ρ) − H ′ (r)(ρ − r) − H(r) (τ, ·)dx Ω 2 Ω Z 2 1 ∇(Φ − Ψ) (τ, ·)dx, + Ω 8πg
na lP repr oo f
and the remainder R(t) is defined as Z µ∇U : ∇(U − u) + (λ + µ)divUdiv(U − u)dx R(t) = ZΩ Z + ρ ∂t U + u · ∇U (U − u)dx + ρ f (u − U)dx Ω Ω Z Z + (r − ρ)∂t (H ′ (r)) + (rU − ρu)∇(H ′ (r))dx + (p(r) − p(ρ))divUdx Ω ZΩ + ρ(u − U) · ∇Φ + (rU − ρu) · ∇(Φ − Ψ)dx.
(2.9)
Ω
For convenience, we abbreviate ε(ρ, u, Φ r, U, Ψ)(t) by ε(t). 2γ
Theorem 2.2. Let γ > 2 and f ∈ L2 (0, T ; L γ−1 (Ω)). Suppose that (ρ, u, Φ) is a finite energy weak solution of the NSP system (1.1)-(1.4) in (0, T ) × Ω in the sense of Definition 2.1. Assume that (r, U, Ψ) is a strong solution of the same problem satisfying 0 < inf r ≤ r(t, x) ≤ sup r < ∞, (0,T )×Ω
∇r ∈ L2 (0, T ; Lq (Ω)),
(0,T )×Ω
∇2 U ∈ L2 (0, T ; Lq (Ω)),
2γ }) with the same initial data. Then (q > max{3, γ−1
ρ = r,
u = U,
Φ=Ψ
∇Ψ ∈ L2 (0, T ; Lq (Ω))
in (0, T ) × Ω.
3. relative entropy inequality
3.1. Proof of Theorem2.1.
ur
Proof. Taking 21 |U|2 as a test function in (2.4), we can get Z Z Z τZ 1 1 ρU · ∂t U + ρu · ∇U · Udxdt. ρ|U|2 (τ, ·)dx = ρ0 |U(0, ·)|2 dx + Ω 2 Ω 2 0 Ω
Jo
Similarly, substituting ϕ for U as a test function, we can obtain Z Z Z τZ ρuU(τ, ·)dx = ρ0 u0 U(0, ·)dx + ρu∂t U + (ρu ⊗ u) : ∇Udxdt Ω Ω 0 Ω Z τZ −µ∇u : ∇U − (λ + µ)divu divU + p(ρ)divU + ρU · ∇Φ + ρ f · Udxdt. +
(3.1)
0
(3.2)
Ω
By virtue of Φ, Ψ satisfying (1.3)(2.7), we can get ! Z Z 1 2 |∇(Φ − Ψ)| dx = (rU − ρu) · ∇(Φ − Ψ)dx. 8πg Ω Ω t
(3.3)
Journal Pre-proof 4
LIANHUA HE AND ZHONG TAN
From (2.2), we can deduce Z Z τZ 2 1 2 2 ρ|u| (τ, ·) + H(ρ)(τ, ·)dx + µ ∇u + (λ + µ) divu dxdt Ω 2 Z τ Z0 Ω Z 1 ρ|u0 |2 + H(ρ0 )dx + ρu · ∇Φ + ρ f · udxdt. (3.4) ≤ 0 Ω Ω 2 Summing up relations (3.1)-(3.4), we have ! Z 1 1 2 2 ρ|u − U| + H(ρ) + |∇(Φ − Ψ)| (τ, ·)dx 8πg Ω 2 Z τZ + µ∇u : ∇(u − U) + (λ + µ)divu div(u − U)dxdt Z τZ Z0 Ω 1 1 2 2 ρ(u − U) · ∇Φ + ρ f · (u − U)dxdt ρ|u0 − U(0, ·)| + H(ρ0 ) + |∇(Φ(ρ0 ) − Ψ(ρ0 ))| dx + ≤ 8πg 0 Ω Ω 2 Z τZ ρ(U − u)∂t U + ρu · ∇U · (U − u)dxdt + 0 Ω Z τZ Z τZ + (rU − ρu) · ∇(Φ − Ψ)dxdt − p(ρ)divUdxdt. (3.5) 0
Ω
Note that
Z h
Z h i i − H(r) − H (r)(ρ − r) (τ, ·)dx − − H(r(0, ·)) − H ′ (r(0, ·))(ρ0 − r(0, ·)) dx Ω ZΩτ Z ∂t (p(r)) − ∂t (H ′ (r)ρ)dxdt, = 0
and
Ω
na lP repr oo f
0
′
Ω
Z τZ 0
We may rewrite (3.5) as Z ε(ρ, u, Φ r, U, Ψ)(τ) +
Ω
rU · ∇(H ′ (r)) + p(r)divUdxdt = 0.
τZ
2 2 µ ∇(u − U) + (λ + µ) div(u − U) dxdt 0 Ω Z τZ ≤ε ρ0 , u0 , Φ0 r(0, ·), U(0, ·), Ψ(0, ·) + µ∇U : ∇(U − u) + (λ + µ)divUdiv(U − u)dx 0 Ω Z τZ Z τZ + ρ ∂t U + u · ∇U (U − u)dx + ρ f (u − U)dx 0 Ω 0 Ω Z τZ Z ′ ′ + (r − ρ)∂t (H (r)) + (rU − ρu)∇(H (r))dx + (p(r) − p(ρ))divUdx 0 Ω ZΩτ Z Z τ + ρ(u − U) · ∇Φ + (rU − ρu) · ∇(Φ − Ψ)dx = ε(0) + R(t)dt. 0
Ω
0
ur
The proof of Theorem 2.1 is completed.
Jo
3.2. Extending the admissible class of test function. By means of density argument, we can extend considerably the class of test function (r, U, Ψ) appearing in the relative entropy inequality (2.8), (2.9). For the left hand side of (2.8) to be well defined, the function (r, U, Ψ) must belong at least to the class: 2γ (3.6) r ∈ Cweak ([0, T ]; Lγ (Ω)), U ∈ Cweak ([0, T ]; L γ−1 (Ω)); ∇Ψ ∈ L∞ ([0, T ]; L2 (Ω)). Similarly, a short inspection of the integrals in (2.9) yields
(3.7)
6γ
2γ
∂t U ∈ L1 ((0, T ); L γ−1 (Ω)) + L2 ((0, T ); L 5γ−6 (Ω));
(3.8)
∇U ∈ L∞ ((0, T ); L
(3.9)
3γ 2γ−3
1
(Ω)) + L2 ((0, T ); L ∞
divU ∈ L ((0, T ); L (Ω));
6γ 2γ−3
(Ω));
(3.10)
Journal Pre-proof WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS 2γ
6γ
2γ γ−1
6γ 5γ−6
5
∇2 U ∈ L1 ((0, T ); L γ−1 (Ω)) + L2 ((0, T ); L 5γ−6 (Ω));
(3.11)
(Ω)).
(3.12)
∇Ψ ∈ L1 ((0, T ); L
(Ω)) + L2 ((0, T ); L
Moreover, the function r must be bounded away from zero, and γ
∂t (H ′ (r)) ∈ L1 ((0, T ); L γ−1 (Ω)); 6γ
(3.13) 2γ
∇(H ′ (r)) ∈ L2 ((0, T ); L 5γ−6 (Ω)) + L1 ((0, T ); L γ−1 (Ω)).
(3.14)
It is easy to prove that the relative entropy inequality (2.8)(2.9) can be extended to (r, U, Ψ) satisfying (3.6)-(3.14) by density argument. 4. weak-strong uniqueness In order to prove the Theorem2.2 , we firstly rewrite the expression of R(t).
na lP repr oo f
Lemma 4.1. For r > 0, the remainder R(t) can read as Z R(t) = ρ(u − U) · ∇U · (U − u)dx Ω Z µr−1 (ρ − r)∆U(U − u) + (µ + λ)r−1 (ρ − r)(U − u) · ∇divUdx + Z ZΩ p(r) − p′ (r)(r − ρ) − p(ρ) divUdx. + (r − ρ)U · ∇(Φ − Ψ)dx + Ω
(4.1)
Ω
Proof. For r > 0, there exists a0 > 0 so that 0 < a0 ≤ r < ∞. Since (r, U, Ψ) is the strong solution of the system (1.1)-(1.3), the following equality holds ∂t U + U · ∇U = µr−1 ∆U + (λ + µ)r−1 ∇divU − r−1 ∇p(r) + ∇Ψ + f.
Substituting (4.2) into (2.9), one has Z µ∇U : ∇(U − u) + (λ + µ)divUdiv(U − u)dx R(t) = ZΩ µρr−1 ∆U(U − u) + (µ + λ)ρr−1 ∇divU(U − u)dx + Ω Z Z + ρ(u − U) · ∇U · (U − u)dx + ρ(u − U) · ∇(H ′ (r))dx Ω Ω Z Z p(r) − p(ρ) divUdx + (r − ρ)∂t (H ′ (r)) + (rU − ρu)∇(H ′ (r))dx + Ω ZΩ + ρ(u − U) · ∇(Φ − Ψ) + (rU − ρu) · ∇(Φ − Ψ)dx.
(4.2)
(4.3)
Ω
Note that
ur
(r − ρ)∂t (H ′ (r)) + (rU − ρu)∇(H ′ (r)) + ρ(u − U) · ∇(H ′ (r)) = −(r − ρ)p′ (r)divU.
Then we can rewrite R(t) as (4.1). The proof of Lemma 4.1 is completed.
Jo
In order to estimate the remainder R(t), we can deduce the following lemma: Lemma 4.2. Let ρ ≥ 0 and 0 < a0 ≤ r ≤ b0 < ∞. There exist a1 ∈ (0, a0 ), M >> 1 and a constant c > 0 so that c(ρ − r)2 if a1 ≤ ρ ≤ Mr; p(r) (4.4) H(ρ) − H ′ (r)(ρ − r) − H(r) ≥ if 0 ≤ ρ < a1 ; cρ2 γ if ρ > Mr.
Proof. By virtue of Taylor’s formula and the definition of H(r), it is easy to infer the inequality (4.4). Hence, we omit to prove this Lemma. Finally, we prove Theorem 2.2.
Journal Pre-proof 6
LIANHUA HE AND ZHONG TAN
Proof. Step1: we estimate the remainder R(t). For ∇U, divU ∈ L1 (0, T ; L∞ (Ω)), it is easy to infer that ! Z ′ I1 = ρ(u − U) · ∇U · (U − u) − p(ρ) − p (r)(ρ − r) − p(r) divU dx ≤ k∇UkL∞ (Ω) ε(t) ≤ η(t)ε(t). (4.5) Ω
The second term of R(t) can be rewrote as Z I2 = µr−1 (ρ − r)∆U(U − u) + (µ + λ)r−1 (ρ − r)(U − u) · ∇divUdx Ω ! Z Z Z + r−1 (ρ − r) µ∆U(U − u) + (µ + λ)(U − u) · ∇divU dx + = {a1 ≤ρ≤Mr}
:= I21 + I22 + I23
{ρ>Mr}
{0≤ρ
By virtue of H¨older’s inequality, Sobolev’s inequality and Lemma 4.2, one has I21 ≤ Ck∇2 UkL3 (Ω) kρ − rkL2 ({a1 ≤ρ≤Mr}) kU − ukL6 (Ω) Z Z ≤ Ck∇2 Uk2L3 (Ω) (ρ − r)2 dx + δ |∇(U − u)|2 dx {a ≤ρ≤Mr} Ω Z 1 ≤ η(t)ε(t) + δ |∇(U − u)|2 dx
(4.6)
and
2
na lP repr oo f
Ω
I22 ≤ k∇ UkL3 (Ω) 2
≤ Ck∇
Z
{0≤ρ
Uk2L3 (Ω)
≤ η(t)ε(t) + δ
Z
Z
Ω
1 1dx 2 kU − u)kL6 (Ω)
{0≤ρ
′
H(ρ) − H (r)(ρ − r) − H(r)dx + δ
Z
Ω
|∇(U − u)|2 dx
|∇(U − u)|2 dx.
(4.7)
with δ > 0 sufficient small. From Lemma 4.2, we can get
1
kρkLγ ({ρ>Mr}) ≤ c[ε(t)] γ
and
1
γ
kρ 2 kL2 ({ρ>Mr}) ≤ c[ε(t)] 2 .
(4.8)
Due to γ > 2 and by using of Sobolev’s inequality, H¨older’s inequality and (4.8), we can deduce Z γ ρ−r γ | ρ 2 |U − u| |∇2 U| dx ≤ Ck∇2 UkL3 (Ω) kρ 2 kL2 ({ρ>Mr}) kU − ukL6 (Ω) I23 ≤ C | {ρ>Mr} ρr ≤ Ck∇2 Uk2L3 (Ω) ε(t) + δk∇(U − u)k2L2 (Ω)
≤ η(t)ε(t) + δk∇(U − u)k2L2 (Ω) .
Ω
ur
The third term of R(t) is denoted as Z Z I3 = (r − ρ)U · ∇(Φ − Ψ)dx =
+
{a1 ≤ρ≤Mr}
Z
:= I31 + I32 + I33 .
+
{0≤ρ
Z
{ρ>Mr}
(4.9)
(r − ρ)U · ∇(Φ − Ψ)dx
and
Jo
By Lemma 4.2 and H¨oder’s inequality, we can obtain 12 Z Z 1 2 (r − ρ) dx I31 ≤ CkUkL∞ (Ω) |∇(Φ − Ψ)|2 dx 2 ≤ η(t)ε(t). {a1 ≤ρ≤Mr}
I32 ≤ CkUkL∞ (Ω) ≤ CkUkL∞ (Ω) ≤ η(t)ε(t).
Z
{0≤ρ
Z
{0≤ρ
(4.10)
Ω
12 Z 1 1dx |∇(Φ − Ψ)|2 dx 2 Ω
1 1 H(ρ) − H ′ (r)(ρ − r) − H(r)dx 2 ε 2 (t)
(4.11)
Journal Pre-proof WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES-POISSON EQUATIONS
7
In view of (4.8) and Lemma 4.2, we have Z γ ρ 2 |U| |∇(Φ − Ψ)|dx I33 ≤ C {ρ>Mr}
γ
≤ CkUkL∞ (Ω) kρ 2 kL2 ({ρ>Mr}) k∇(Φ − Ψ)kL2 (Ω) ≤ CkUkL∞ (Ω) ε(t) ≤ η(t)ε(t)
(4.12)
with γ > 2. Step2: Basing on the estimate of R(t), we prove the result of Theorem 2.2. Plugging relations(4.5)-(4.12) to (2.8), we can deduce Z τZ Z τ 2 2 µ ∇(u − U) + (λ + µ) div(u − U) dxdt ≤ ε(τ) + η(t)ε(t)dt 0
0
Ω
L1 (0, T ).
with η(t) ∈ By using of Gronwall’s inequality, we infer ε(t) = 0 in (0, T ). This implies ρ = r, u = U, Φ = Ψ. The proof of Theorem 2.2 is completed.
na lP repr oo f
Acknowledgments The work of L.H He and Z.Tan was supported by the National Natural Science Foundation of China (11531010, 11726023). References
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[1] P. Degond, Mathematical modelling of microelectronics semiconductor devices, some current topics on nonlinear conservation laws. in: AMS/IP Stud. Adv. Math., vol. 15, Amer. Math. Sco., Providence, RI, 2007: 77-110. [2] Y.H. Zhang, Z. Tan. Strong solutions of the coupled Navier-Stokes-Poisson equations for isentropic compressible fluids. Acta Mathematica Scientia(B),30(2010): 1280-1290 [3] H.L. Li, A. Matsumura, G. Zhang. Optimal Decay Rate of the Compressible Navier-Stokes-Poisson System in R3 . Arch. Rational Mech. Anal. 196 (2010): 681-713. [4] W.X. Shi, J. Xu. A sharp time-weighted inequality for the compressible Navier-Stokes-Poisson system in the critical L p framework. J. Differ. Equ. 266(2019): 6426-6458. [5] Xiaoxin Zheng. Global well-posedness for the compressible Navier-Stokes-Poisson system in the L p framwork. Nonlinear Analysis, 75(2012): 4156-4175. [6] C. Hao, H.L. Li, Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differ. Equ. 246 (2009): 4791-4812. [7] T. Kobayashi, T. Suzuki. Weak solution to the Navier-Stokes-Poisson equation. Adv.Math.Sci.Appl.,18(2008):141-168. [8] Y.H. Zhang, Z. Tan. On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow. Math. Meth. Appl. Sci. (30)2007:305-329. [9] Y.J. Wang. Decay of the Navier-Stokes-Poisson equations. J.Differ.Equ.253(2012):273-297. [10] Q. Bie, Q. Wang, Z. Yao. Optimal decay rate for the compressible Navier-Stokes-Poisson system in the critical L p framework. J. Differ. Equ. 263(2017):8391-8417. [11] J. Carrillo, A. J¨ungle, P.A. Markowich, G. Toscani, A. Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized sobolev inequalities. Monatshefte Math 133(2001):1-82. [12] L.S. Raymond. Hydrodynamic limits: some improvements of the relative entropy method. Ann. I. H. Poincar-AN 26 (2009): 705-744. [13] Jan Giesselmann, Corrado Lattanzio, Athanasios E. Tzavaras. Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics. Arch. Rational Mech. Anal. 223(2017): 1427-1484. [14] E. Feireisl, A. Novoton´y. Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch.Rational Mech.Anal. 204(2012): 683-706. [15] D. Jessle, B. Jin, A. Novoton´y. Navier-Stokes-Fourier System on Unbounded Domains: Weak Solutions, Relative Entropies, Weak-Strong Uniqueness. SIAM J. Math. Anal. 45(2013): 1907-1951. [16] P. Germian. Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations. J.Differ.Equ. 226(2006): 373-428. [17] Q. Liu. On weak-strong uniqueness of solutions to the generalized incompressible Navier-Stokes equations. Computers and Mathematics with Applications 72 (2016): 675-686 [18] P. Germain. Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J.Math.Fluid Mech. 13(2010):137-146. [19] E. Feireisl, B. Jin, A. Novotny. Relative entropis, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J.Math.Fluid Mech.14(2012):717-730. [20] E. Feireisl, A. Novotn´y, Y. Sun. Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60(2011): 611-631
Journal Pre-proof 8
LIANHUA HE AND ZHONG TAN
[21] E. Feireisl. On weak-strong uniqueness for the compressible Navier-Stokes system with non-monotone pressure law. Comm. Partial Differential Equations 44 (2019):271-278. [22] Young-sam Kwon. From the degenerate quantum compressible Navier-Stokes-Poisson system to incompressible Euler equations. J. Math. Phys. 59 (2018), no. 12, 123101, 13 pp. [23] S. Wang, S. Jiang. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun.Partial Differ.Equ., 31(2006):571-591. School of Mathematical Sciences, Xiamen University , Xiamen, 361005, China. And School of Mathematics Science, Guizhou Normal University, Guiyang, 550001, China E-mail address, L. He: [email protected]
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School of Mathematical Sciences, Xiamen University , Xiamen, 361005, China. E-mail address, Z. Tan: [email protected]
*Author Contributions Section
Journal Pre-proof
Author
Contributions
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na lP repr oo f
There are many researchers to investigate the weak-strong uniqueness of Navier -Stokes equations with various pressure functions . However, in this paper, we study the weak-strong uniqueness of N-S-P system for the first time. We apply relative entropy method to obtain the weak-strong uniqueness property of N-S-P equations. Many tricks and the methods applied in this paper are relative novel. In the writing of this paper, He is responsible for writing the paper, and Tan is responsible for revising the pape. We togerher discuss and solve the problems in the paper.