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Weak–strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows
Nonlinear Analysis 85 (2013) 23–30
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Weak–strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows Yong-Fu Yang a,b,∗ , Changsheng Dou b , Qiangchang Ju b a
Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, Jiangsu Province, PR China
b
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, PR China
article
info
Article history: Received 18 September 2012 Accepted 12 February 2013 Communicated by Enzo Mitidieri Keywords: Three-dimensional magnetohydrodynamics (MHD) equations Compressible flows Relative entropy Weak–strong uniqueness
abstract By means of the concept of relative entropy, we establish the weak–strong uniqueness property in the class of finite-energy weak solutions to the magnetohydrodynamic equations of three-dimensional compressible isentropic flows with the adiabatic exponent γ > 1 and constant viscosity coefficients, under the assumption that weak solutions exist. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids and the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. The applications of magnetohydrodynamics cover a very wide range of physical areas from liquid metals to cosmic plasmas: for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics, and plasma physics. Due to their practical relevance, magnetohydrodynamic problems have long been the subject of intense cross-disciplinary research, but, except for relatively simplified special cases, the rigorous mathematical analysis of such problems remains open. In magnetohydrodynamic flows, magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself. There is a complex interaction between the magnetic and fluid dynamic phenomena, and both hydrodynamic and electrodynamic effects have to be considered. The set of equations which describe compressible viscous magnetohydrodynamics is a combination of the compressible Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. In this paper, we consider the MHD equations of threedimensional compressible flows in the isentropic case [1–4]:
ρ + div (ρ u) = 0, t (ρ u)t + div (ρ u ⊗ u) + ∇ p = (∇ × H) × H + div S(∇ u), div H = 0, Ht − ∇ × (u × tH) = −∇ × (ν∇ × H), S = µ(∇ u + ∇ u) + λdiv uI,
(1.1)
∗ Corresponding author at: Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, Jiangsu Province, PR China. Tel.: +86 18061705228. E-mail addresses: [email protected] (Y.-F. Yang), [email protected] (C. Dou), [email protected] (Q. Ju). 0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.02.015
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Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
where ρ denotes the density, u ∈ R3 the velocity, H ∈ R3 the magnetic field, p(ρ) = aρ γ the pressure, with constant a > 0 and the adiabatic exponent γ > 1, and
S = µ(∇ u + ∇ t u) + λdiv uI the stress, with viscosity coefficients µ, λ satisfying 2µ + 3λ > 0 and µ > 0; ν > 0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, and all these kinetic coefficients and the magnetic diffusivity are independent of the magnitude and direction of the magnetic field. The symbol ⊗ denotes the Kronecker tensor product. Usually, we refer to (1.1)1 as the continuity equation and (1.1)2 as the momentum balance equations. It is well known that the electromagnetic fields are governed by the Maxwell equations. In MHD, the displacement current can be neglected [3,4]. As a consequence, Eqs. (1.1)3 are called the induction equations, and the electric field E can be written in terms of the magnetic field H and the velocity u, E = ν∇ × H − u × H. Although the electric field E does not appear in (1.1), it is indeed induced according to the above relation by the moving conductive flow in the magnetic field. In this paper, we are interested in the weak–strong uniqueness property of solutions to the three-dimensional MHD equations (1.1) in a bounded domain Ω ∈ R3 with the following initial-boundary conditions:
ρ(x, 0) = ρ0 (x) ∈ Lγ (Ω ), ρ0 (x) ≥ 0, ρ(x, 0)u(x, 0) = m0 (x) ∈ L1 (Ω ), m0 = 0 if ρ0 = 0, 2 ′ H(x, 0) = H0 (x) ∈ L (Ω ), div H0 = 0 in D (Ω ), u|∂ Ω = 0, H|∂ Ω = 0.
|m0 |2 ∈ L1 (Ω ), ρ0
(1.2)
In the recent years, there have been a lot of studies on MHD by physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenges; see [2,4–13] and the references cited therein. In particular, the one-dimensional problem has been studied in many papers; see, for example, [5,6,8,10,13–15] and the references cited therein. However, a number of fundamental problems for MHD are still open; for example, even in the one-dimensional case, the global existence of classical solutions to the full perfect MHD equations with large data remains unsolved when all the viscosity, heat conductivity, and diffusivity coefficients are constants, although the corresponding problem for the Navier–Stokes case was solved in [16] a long time ago. The reason is that the presence of the magnetic field and its interaction with the hydrodynamic motion in MHD flows of large oscillation cause serious difficulties. Hu and Wang, in [2], obtained the global existence and large-time behavior of weak solutions to the multi-dimensional isentropic problem (1.1)–(1.2) with γ > 3/2, where all the viscosity coefficients µ, λ, ν are constants. Since the global existence of the weak solutions has been established, it is natural to study their uniqueness property. To our best knowledge, so far there are very few results concerning the uniqueness of weak solutions to the initial-boundary value problem (1.1)–(1.2), while, for incompressible magnetohydrodynamic flows, Li and Wang, in [17], established its weak–strong uniqueness principle. When there is no electromagnetic field, system (1.1) reduces to the compressible Navier–Stokes equations. There have been a number of papers in the literature on the multi-dimensional Navier–Stokes equations (see [18–20] and the references therein).In fact, the known results about the compressible Navier–Stokes equations are mainly related to the global existence of weak solutions with large initial data. All the efforts in these papers are to relax the requirement on the adiabatic exponent γ so as to include more physically relevant cases. It is well known that the uniqueness of weak solutions is another fundamental and important topic in mathematical theory of the compressible Navier–Stokes equations. Contrasting with the existence theory of weak solutions, it seems that the uniqueness issue is more difficult and there are only a few results dealing with uniqueness. With the aid of the concept of relative entropy, Germain [21] introduced a class of (weak) solutions to the compressible Navier–Stokes system satisfying a relative entropy inequality with respect to a (hypothetical) strong solution of the same problem, and established the weak–strong uniqueness property within this class (see also [22]). Unfortunately, the existence of weak solutions belonging to this class, where, in particular, the density possesses a spatial gradient in a suitable Lebesgue space, is not known. Recently, Feireisl et al. [23] introduced the concept of a suitable weak solution for the compressible Navier–Stokes system, satisfying a general relative entropy inequality with respect to any sufficiently regular pair of functions. They showed the global-intime existence of suitable weak solutions for any finite-energy initial data. Moreover, the relative entropy inequality can be used to show that suitable weak solutions comply with the weak–strong uniqueness principle, meaning that a weak and a strong solution emanating from the same initial data coincide as long as the latter exists. More recently, Feireisl et al. [24] further showed that any finite-energy weak solution satisfies a relative entropy inequality with respect to any couple of smooth functions satisfying relevant boundary conditions. Based on the relative entropy inequality, they succeeded in proving the weak–strong uniqueness and in providing a satisfactory answer to the weak–strong uniqueness problem initially related to the fundamental questions of the well-posedness for the compressible Navier–Stokes system. In addition, Feireisl and Novotný [25] also established a similar weak–strong uniqueness property of the variational solutions for a compressible Navier–Stokes–Fourier system. We should remark that the weak–strong uniqueness property for the standard
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
25
incompressible Navier–Stokes system was established in the seminal papers by Prodi [26] and Serrin [27]. The situation is a bit more delicate in the case of a compressible fluid. Our goal in this paper is to establish the weak–strong uniqueness property for compressible MHD flows in the spirit of Feireisl et al. [24]. Our method is based on the relative entropy, the relative entropy inequality, and a Gronwall-type argument. Compared with the result in [24], where Feireisl et al. essentially showed the weak–strong uniqueness property for the adiabatic exponent γ ≥ 6/5, we are going to use a new technique to estimate the remainder R (for the definition see (3.8)), so as to establish the weak–strong uniqueness property for an improved lower bound for any adiabatic exponent γ > 1. We should point out here that, due to the presence of the magnetic field, the main difficulty lies in the interaction ˜ ∥L∞ (0,T ;L2 (Ω )) , where u˜ is the corresponding strong solution, is between the velocity and the magnetic field. Since ∥u − u not included in the relative entropy, some new techniques are required to decouple the velocity u and the magnetic field H when we deal with the terms involving the interaction between u and H. The paper is organized as follows. In Section 2, we recall the definitions of weak and strong solutions to the compressible MHD flows, introduce the relative entropy, and state the main result. Section 3 is devoted to the derivation of the relative entropy inequality. Finally, the proof of the main result is presented in Section 4. 2. Main result
Motivated by the concept of relative entropy in [24], we similarly define the relative entropy E = E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] , with respect to {ρ, ˜ u˜ , H˜ }, as
E =
1 2
Ω
1
ρ|u − u˜ |2 + Π (ρ) − Π ′ (ρ)(ρ ˜ − ρ) ˜ − Π (ρ) ˜ + |H − H˜ |2
2
dx,
(2.1)
where
Π (ρ) =
a
γ −1
ργ .
(2.2)
Noticing that
∇ × (∇ × H) = ∇(div H) − △H, Eqs. (1.1)3 become Ht − ∇ × (u × H) = ν△H.
(2.3)
We say that a triple {ρ, u, H} is a finite-energy weak solution to the initial-boundary value problem (1.1)–(1.2) if the following hold.
• The density ρ is a non-negative function satisfying ρ ∈ C ([0, T ]; L1 (Ω )) ∩ L∞ (0, T ; Lγ (Ω )),
ρ(x, 0) = ρ0 ,
and the momentum ρ u satisfies 2γ
ρ u ∈ C ([0, T ]; Lwγ +eak1 (Ω )). • The velocity u and the magnetic field H satisfy the following: u ∈ L2 (0, T ; H01 (Ω )),
H ∈ L2 (0, T ; H01 (Ω )) ∩ C ([0, T ]; L2weak (Ω )),
ρ u ⊗ u, ∇ × (u × H), and (∇ × H) × H are integrable on (0, T ) × Ω , and ρ u(x, 0) = m0 ,
H(x, 0) = H0 ,
div H = 0
in D ′ (Ω ).
• The first equation in (1.1) verifies a family of integral identities τ ρ(τ , ·)ϕ(τ , ·) dx − ρ0 ϕ(0, ·) dx = (ρ∂t ϕ + ρ u · ∇ϕ) dxdt Ω
Ω
0
(2.4)
Ω
¯ ), and any τ ∈ [0, T ]. for any ϕ ∈ C 1 ([0, T ] × Ω • The momentum equations (1.1)2 are satisfied in the sense of distributions; specifically, ρ u(τ , ·)ϕ(τ , ·) dx − ρ0 u0 · ϕ(0, ·) dx Ω Ω τ = (ρ u · ∂t ϕ + ρ u ⊗ u : ∇ϕ + p(ρ)div ϕ − S(∇ u) : ∇ϕ) dxdt Ω τ
0
+ 0
Ω
(ϕ · (∇ H)H − H · (∇ H)ϕ) dxdt
¯ ), ϕ|∂ Ω = 0, and any τ ∈ [0, T ]. for any ϕ ∈ C 1 ([0, T ] × Ω
(2.5)
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Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
Here and hereafter A · (∇ B)C = ((C · ∇)B) · A, in which A, B, and C are vectors. • Eqs. (1.1)3 (i.e., (2.3)) are replaced by a family of integral identities H0 · ϕ(0, ·) dx H(τ , ·) · ϕ(τ , ·) dx − Ω
Ω
τ
= Ω
0
(H · ∂t ϕ − ν(∇ H) : (∇ϕ) + ϕ · (∇ u)H − ϕ · (∇ H)u − (H · ϕ)div u) dxdt
(2.6)
¯ ), and any τ ∈ [0, T ]. for any ϕ ∈ C 1 ([0, T ] × Ω • The energy inequality E (t ) +
τ
0
Ω
µ|∇ u|2 + (λ + µ)(div u)2 + ν|∇ H|2 dxdt ≤ E (0)
(2.7)
holds for a.e. τ ∈ [0, T ], where E (t ) =
Ω
1 2
ρ|u|2 +
a
γ −1
1
ρ γ + |H|2
2
dx,
and E (0) =
Ω
1 |m0 |2 2
ρ0
+
a
γ −1
γ
1
ρ0 + |H0 |
2
2
dx.
Remark 2.1. Multiplying the continuity equation (1.1)1 by b′ (ρ), we obtain the renormalized continuity equation: b(ρ)t + div (b(ρ)u) + (b′ (ρ)ρ − b(ρ))div u = 0
(2.8)
for some suitable function b ∈ C 1 (R+ ). If (1.1)1 is satisfied in the sense of renormalized solutions, that is, (2.8) holds in D ′ (Ω × (0, T )) for any b ∈ C 1 (R+ ) satisfying b′ ( z ) = 0
for all z ∈ R+ large enough, say, z ≥ z0 ,
where the constant z0 depends on the choice of function b, then the existence of global-in-time renormalized finite-energy weak solutions to the MHD system with the adiabatic exponent γ > 23 was established in [2].
{ρ, ˜ u˜ , H˜ } is called a classical (strong) solutions to the MHD system in (0, T ) × Ω if ¯ ), ρ( ρ˜ ∈ C 1 ([0, T ] × Ω ˜ t , x) ≥ ρ > 0 for all (t , x) ∈ (0, T ) × Ω , ˜ , ∂t H˜ , ∇ 2 H˜ ∈ C ([0, T ] × Ω ¯ ), ¯) ˜u, ∂t u˜ , ∇ 2 u˜ ∈ C ([0, T ] × Ω H
(2.9)
and ρ, ˜ u˜ , H˜ satisfy Eq. (1.1), together with the boundary conditions (1.2)4 . Observe that hypothesis (2.9) requires the following regularity properties of the initial data:
¯ ), ρ0 ≥ ρ > 0, ρ( ˜ 0, ·) = ρ( ˜ 0) = ρ0 ∈ C 1 (Ω ˜ (0, ·) = H˜ 0 = H0 ∈ C 2 (Ω ¯ ), ¯ ), ˜ (0, ·) = u˜ 0 = u0 ∈ C 2 (Ω u H
div H0 = 0.
(2.10)
We are now ready to state the main result of this paper. Theorem 2.1. Let Ω ∈ R3 be a bounded domain with a boundary of class C 2+κ , κ > 0, and γ > 1. Suppose that {ρ, u, H} is a finite-energy weak solution to the MHD system in (0, T ) × Ω in the sense specified above, and suppose that {ρ, ˜ u˜ , H˜ } is a strong solution emanating from the same initial data (2.10). Then
ρ ≡ ρ, ˜
˜, u≡u
˜. H≡H
The rest of the paper is devoted to the proof of Theorem 2.1. 3. Relative entropy inequality We deduce a relative entropy inequality satisfied by any weak solution to the MHD system. To this end, consider a triple
¯ , u˜ |∂ Ω = 0, and H˜ |∂ Ω = 0. In addition, u˜ and H˜ solve {ρ, ˜ u˜ , H˜ } of smooth functions, ρ˜ bounded away from zero in [0, T ] × Ω (2.3).
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
27
˜ as a test function in the momentum equation (2.5) to obtain Take u Ω
ρ u · u˜ (τ , ·) dx −
Ω
τ
(3.1)
|u˜ |2 as a test function in the continuity equation (2.4) to get τ 1 ρ u˜ · ∂t u˜ + ρ u˜ · (∇ u˜ )u dxdt , ρ0 |u˜ |2 (0, ·) dx +
(3.2)
Next, we can use the scalar quantity ϕ =
2
ρ|u˜ |2 (τ , ·) dx =
1 2
2
Ω
Ω
0
1
Ω
ρ u · ∂t u˜ + ρ u ⊗ u : ∇ u˜
˜ · (∇ H)H − H · (∇ H)u˜ dxdt . u
+ p(ρ)div u˜ − S(∇ u) : ∇ u˜ dxdt +
Ω
0
τ
ρ0 u0 · u˜ (0, ·) dx =
Ω
0
˜ · (∇ u˜ )u = ((u · ∇)u˜ ) · u. ˜ Similarly, the choice ϕ = Π ′ (ρ) where u ˜ in (2.4) gives rise to Ω
ρ Π ′ (ρ)(τ ˜ , ·) dx =
Ω
τ
ρ0 Π (ρ)( ˜ 0, ·) dx +
Ω
0
ρ∂t Π ′ (ρ) ˜ + ρ u · ∇ Π ′ (ρ) ˜ dxdt .
(3.3)
˜ to get We test (2.6) on H Ω
˜ (τ , ·) dx = H·H
˜ (0, ·) dx + H·H
Ω
τ
Ω
0
τ
+
˜ − ν(∇ H) · (∇ H˜ ) H · ∂t H
Ω
0
˜ · (∇ u)H − H˜ · (∇ H)u − (H · H˜ )div u H
dxdt
dxdt .
(3.4)
˜ and u˜ solve (1.1)3 , we use H˜ |∂ Ω = 0 to infer that Finally, recalling that the smooth functions H 1
2
Ω
1 |H˜ |2 (τ , ·) dx =
2
Ω
|H˜ |2 (0, ·) dx − ν
τ
0
|∇ H˜ |2 dxdt + Ω
τ
0
Ω
˜ · (∇ H˜ )u˜ − u˜ · (∇ H˜ )H˜ H
dxdt .
(3.5)
Summing up the relations (3.1)–(3.5) with the energy inequality (2.7), we infer that
2 ˜ ρ|u − u˜ | + Π (ρ) − ρ Π (ρ) ˜ + |H − H| (τ , ·) dx 2 2 Ω τ τ (S(∇ u) − S(∇ u˜ )) : (∇ u − ∇ u˜ ) dxdt + ν |∇ H − ∇ H˜ |2 dxdt + 0 Ω 0 Ω 1 1 2 ′ ≤ ρ0 |u0 − u˜ (0, ·)| + Π (ρ0 ) − ρ0 Π (ρ( ˜ 0, ·)) + |H0 − H˜ (0, ·)|2 dx 2 2 Ω τ τ + ρ(∂t u˜ + u · ∇ u˜ ) · (u˜ − u) dxdt + S(∇ u˜ ) : ∇(u˜ − u) dxdt 0 τ Ω 0 τ Ω ˜ dxdt − (ρ∂t Π ′ (ρ) ˜ + ρ u · ∇ Π ′ (ρ)) ˜ dxdt − p(ρ)div u 0 Ω 0 Ω τ ˜ · (∇ H)H − H · (∇ H)u˜ − H˜ · (∇ H˜ )u˜ + u˜ · (∇ H˜ )H˜ dxdt − u
1
Ω
0
τ
− 0
1
′
2
Ω
˜ + ν(∇ H) · (∇ H˜ ) + H˜ · (∇ u)H − H˜ · (∇ H)u − (H · H˜ )div u H · ∂t H
dxdt .
(3.6)
By virtue of the definition (2.2) of Π , it is easy to see that
Π ′ (ρ) ˜ ρ˜ − Π (ρ) ˜ = p(ρ) ˜ and
Ω
ρ∂ ˜ t Π ′ (ρ) ˜ + ρ∇ ˜ Π ′ (ρ) ˜ · u˜ + p(ρ) ˜ div u˜ dx =
Ω
∂t p(ρ) ˜ dx.
As a consequence, we deduce, from the identities
Ω
p(ρ)(τ ˜ , ·) dx −
Ω
p(ρ)( ˜ 0, ·) dx =
τ
0
Ω
∂t p(ρ) ˜ dxdt
and (3.6), the desired entropy inequality:
E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] (τ ) +
τ
0
Ω
(S(∇ u) − S(∇ u˜ )) : (∇ u − ∇ u˜ ) dxdt + ν
τ
0
τ
≤ E [ρ0 , u0 , H0 ]|[ρ( ˜ 0, ·), u˜ (0, ·), H˜ (0, ·)] + 0
R(ρ, u, H, ρ, ˜ u˜ , H˜ ) dt ,
|∇ H − ∇ H˜ |2 dxdt Ω
(3.7)
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Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
where
R = R(ρ, u, H, ρ, ˜ u˜ , H˜ ) := Rd + Rm ,
Rd :=
ρ(u˜ − u) · ∂t u˜ + (∇ u˜ )u dx +
Ω
+ Ω
(3.8)
Ω
S(∇ u˜ ) : ∇(u˜ − u) dx
(ρ˜ − ρ)∂t Π ′ (ρ) ˜ + ∇ Π ′ (ρ) ˜ · (ρ˜ u˜ − ρ u) dx −
Ω
˜ (p(ρ) − p(ρ)) div u ˜ dx,
(3.9)
and
Rm :=
Ω
−
˜ − u˜ · (∇ H)H + H˜ · (∇ H˜ )u˜ − u˜ · (∇ H˜ )H˜ H · (∇ H)u
Ω
dx
˜ + ν(∇ × H) · (∇ × H˜ ) + H˜ · (∇ u)H − H˜ · (∇ H)u − (H · H˜ )div u H · ∂t H
dx.
(3.10)
4. Weak–strong uniqueness In this section, we shall finish the proof of Theorem 2.1 by applying the relative entropy inequality (3.7) to {ρ, ˜ u˜ , H˜ }, ˜ where {ρ, ˜ u˜ , H} is a classical (smooth) solution of the MHD system (1.1), (1.2), such that
ρ( ˜ 0, ·) = ρ0 ,
˜ (0, ·) = u0 , u
˜ (0, ·) = H0 . H
Accordingly, the integrals depending on the initial data on the right-hand side of (3.7) vanish, and we apply a Gronwalltype argument to deduce the desired result, namely,
ρ ≡ ρ, ˜
˜, u≡u
˜. H≡H
Our purpose is to examine all terms in the remainder (3.8) and to show that they can be ‘‘absorbed’’ by the left-hand side of (3.7). Compared with [24], we should remark that the main difficulty comes from the interaction between the velocity and the magnetic field. Moreover, in the context of the weak–strong uniqueness, we only assume the adiabatic exponent as γ > 1, not γ ≥ 6/5 as in [24]. Similar to the proof of Theorem 4.1 in [24], we use (3.9) to find that
1 ρ(u˜ − u) · (∇ u˜ )(u − u˜ ) dx + (ρ − ρ) ˜ div S(∇ u˜ ) · (u˜ − u) dx ˜ Ω Ω ρ ρ ′ ˜ p(ρ) − p (ρ)(ρ (∇ × H˜ ) × H˜ · (u˜ − u) dx − div u ˜ − ρ) ˜ − p(ρ) ˜ dx + ˜ Ω ρ Ω ˜ p(ρ) − p′ (ρ)(ρ = ρ(u˜ − u) · (∇ u˜ )(u − u˜ ) dx − div u ˜ − ρ) ˜ − p(ρ) ˜ dx Ω Ω 1 + (ρ − ρ) ˜ div S(∇ u˜ ) + (∇ × H˜ ) × H˜ · (u˜ − u) dx + (∇ × H˜ ) × H˜ · (u˜ − u) dx ˜ Ω ρ Ω =: Rd + (∇ × H˜ ) × H˜ · (u˜ − u) dx.
Rd =
(4.1)
Ω
If we combine the identity
Ω
˜ dx = H · ∂t H
Ω
˜ ˜ ˜ ˜ ˜ ˜ H · (∇ u)H − H · (∇ H)u − (H · H)div u dx − ν (∇ H˜ ) · (∇ H) dxdt Ω
with (3.10), we find after a tedious but straightforward computation that
Rm := Rm +
(∇ × H˜ ) × H˜ · (u˜ − u) dx
Ω = (H − H˜ ) · (∇ u˜ )(H − H˜ ) + (u − u˜ ) · (∇ H˜ )(H − H˜ ) Ω
+ (H˜ − H) · (∇ H˜ )(u − u˜ ) + (H˜ − H) · ((∇ H˜ − ∇ H)u˜ ) dx,
(4.2)
˜ = 0. Consequently, it follows from where we have used integration by parts several times and the fact that div H = div H the definitions of Rd and Rm that R(ρ, u, H, ρ, ˜ u˜ , H˜ ) = Rd + Rm .
(4.3)
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
29
In order to prove the weak–strong uniqueness property, we have to estimate the remainder R. As for Rd , we mainly follow [24] with some modifications to relax the requirement on the adiabatic exponent γ . In what follows, we are going to focus on the estimation of Rm . From (2.10), it is clear to see that
′ ˜ ˜ ˜ ˜ ρ( u − u ) · (∇ u )( u − u ) dx − div u p (ρ) − p ( ρ)(ρ ˜ − ρ) ˜ − p ( ρ) ˜ dx Ω Ω ≤ C ∥∇ u˜ ∥L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] .
(4.4)
Here and hereafter C stands for a generic constant, which may change from line to line.
˜ , H˜ , ∇ H˜ ) = div S(∇ u˜ ) + (∇ × H˜ ) × H˜ . Obviously, we have Let ˜f = ˜f(∇ u
1 Ω
ρ˜
(ρ − ρ) ˜ ˜f · (u˜ − u) dx =
1 ρ˜
{ 2 <ρ<2ρ} ˜
+ {ρ≥2ρ} ˜
ρ˜ 1
ρ˜
(ρ − ρ) ˜ ˜f · (u˜ − u) dx +
1 ρ˜
{0≤ρ< 2 }
ρ˜
(ρ − ρ) ˜ ˜f · (u˜ − u) dx
(ρ − ρ) ˜ ˜f · (u˜ − u) dx.
(4.5)
Similarly to [24], we make use of Hölder’s inequality, Sobolev’s inequality, and a Korn-type inequality to show that
2 ˜f 1 ˜ + (ρ − ρ) ˜ f · (u˜ − u) dx ≤ C (δ) ρ˜ ρ˜ 3 { ρ˜ <ρ<2ρ} ρ ˜ ˜ {0≤ρ< 2 } 2
E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ]
L (Ω )
+ δ ∥S(∇ u − ∇ u˜ )∥2L2 (Ω )
(4.6)
for any δ > 0. On the other hand, noticing that
Π (ρ) − Π ′ (ρ)(ρ ˜ − ρ) ˜ − Π (ρ) ˜ ≥ C ργ ,
as ρ ≥ 2ρ˜ ≥ 2ρ
and
ρ − ρ˜ 1 − γ ρ ρ˜ ρ 2 2 ≤ C ,
as ρ ≥ 2ρ˜ ≥ 2ρ and γ > 1,
we conclude from the definition of relative entropy E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] that
{ρ≥2ρ} ˜
ρ − ρ˜ 1 ˜ 1 ρ ρ˜ ρ 2 f · ρ 2 |u˜ − u| dx ρ − ρ˜ 1 − γ γ ρ 2 2 ρ 2 ˜f · ρ 12 |u˜ − u| dx ρ ρ˜ {ρ≥2ρ} ˜ 12 21 ρ ≤ C ∥˜f∥L∞ (Ω ) |u − u˜ |2 dx ρ γ dx
˜ (ρ − ρ) ˜ f · (u˜ − u) dx = ρ˜ {ρ≥2ρ} ˜ = 1
Ω
Ω
2
≤ C ∥˜f∥L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] .
(4.7)
Next, we continue to estimate Rm . It is easy to see that
(H − H˜ ) · (∇ u˜ )(H − H˜ ) dx ≤ ∥∇ u˜ ∥L∞ (Ω ) ∥H − H˜ ∥22 L (Ω ) Ω ≤ ∥∇ u˜ ∥L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] .
(4.8)
We deduce from Hölder’s inequality and the Korn-type inequality that
˜ ˜ ˜ ˜ (u − u˜ ) · (∇ H)(H − H) + (H − H) · (∇ H)(u − u˜ ) dx Ω
≤ ∥∇ H˜ ∥L∞ (Ω ) ∥u − u˜ ∥L2 (Ω ) ∥H − H˜ ∥L2 (Ω ) ≤ δ ∥S(∇ u − ∇ u˜ )∥2L2 (Ω ) + C (δ) ∥∇ H˜ ∥2L∞ (Ω ) ∥H − H˜ ∥2L2 (Ω ) ≤ δ ∥S(∇ u − ∇ u˜ )∥2L2 (Ω ) + C (δ) ∥∇ H˜ ∥2L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ]
(4.9)
30
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
for any δ > 0. Finally,
((H˜ − H) · (∇ H˜ − ∇ H)u˜ ) dx ≤ ∥u˜ ∥L∞ (Ω ) ∥H − H˜ ∥L2 (Ω ) ∥∇ H − ∇ H˜ ∥L2 (Ω ) Ω
≤ δ ∥∇ H − ∇ H˜ ∥2L2 (Ω ) + C (δ) ∥u˜ ∥2L∞ (Ω ) ∥H − H˜ ∥2L2 (Ω ) ˜ u˜ , H˜ ] . ≤ δ ∥∇ H − ∇ H˜ ∥2L2 (Ω ) + C (δ) ∥u˜ ∥2L∞ (Ω ) E [ρ, u, H]|[ρ,
(4.10)
Summing up relations (4.1)–(4.10), we conclude that the relative entropy inequality yields the desired conclusion:
˜ E [ρ, u, H]|[ρ, ˜ u˜ , H] (τ ) ≤
τ
h(t )E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] (t ) dt
with some h ∈ L1 (0, T ).
0
Thus, Theorem 2.1 immediately follows from Gronwall’s inequality. Remark 4.1. It should be pointed out that the weak–strong uniqueness property also holds for isentropic compressible Navier–Stokes equations with the adiabatic exponent γ > 1, which improves the result in [24], where γ ≥ 6/5 is required. Acknowledgments The authors would like to thank Prof. Song Jiang for helpful discussions and valuable suggestions. Yong-Fu Yang’s research was partially supported by NSFC (Grant No. 11201115) and China Postdoctoral Science Foundation (Grant No. 2012M510365). Changsheng Dou’s research was in part supported by China Postdoctoral Science Foundation (Grant No. 2012M520205). Qiangchang Ju’s research was partially supported by NSFC (Grant No. 11171035). References [1] H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, 1970. [2] X. Hu, D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197 (2010) 203–238. [3] A.G. Kulikovskiy, G.A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965. [4] L.D. Laudau, E.M. Lifshitz, Electrodynamics of Continuous Media, second ed., Pergamon, New York, 1984. [5] G.-Q. Chen, D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations 182 (2002) 344–376. [6] G.-Q. Chen, D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54 (2003) 608–632. [7] B. Ducoment, E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 226 (2006) 595–629. [8] J. Fan, S. Jiang, G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys. 270 (2007) 691–708. [9] H. Freistühler, P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal. 26 (1995) 112–128. [10] D. Hoff, E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. 56 (2005) 791–804. [11] X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. 197 (2010) 203–238. [12] S. Jiang, Q.C. Ju, F.C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys. 297 (2010) 371–400. [13] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math. 63 (2003) 1424–1441. [14] S. Kawashima, M. Okada, Smooth global solutions for the one-dimensional equation in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982) 384–387. [15] T.-P. Liu, Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc. 599 (1997). [16] V. Kazhikhov, V.V. Shelukhin, Unique global solution with respect to time of initial-boundary-value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. 41 (1977) 273–282. [17] X. Li, D. Wang, Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows, J. Differential Equations 251 (2011) 1580–1615. [18] E. Feireisl, Dynamicals of Viscous Compressible Fluids, in: Oxford Lecture Series in Mathematics and its Applications, vol. 26, Oxford University Press, Oxford, 2004. [19] P.-L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. [20] A. Novotný, I. Straškraba, Introduction to the Theory of Compressible Flow, Oxford University Press, Oxford, 2004. [21] P. Germain, Weak–strong uniqueness for the isentropic compressible Navier–Stokes system, J. Math. Fluid Mech. 13 (2011) 137–146. [22] P. Germain, Multipliers, paramultipliers, and weak–strong uniqueness for the Navier–Stokes equations, J. Differential Equations 226 (2006) 373–428. [23] E. Feireisl, A. Novotný, Y. Sun, Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids, Indiana Univ. Math. J. 60 (2011) 611–631. [24] E. Feireisl, B.J. Jin, A. Novotný, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system, J. Math. Fluid Mech. 14 (2012) 717–730. [25] E. Feireisl, A. Novotný, Weak–strong uniqueness property for the full Navier–Stokes system, Arch. Ration. Mech. Anal. 204 (2012) 683–706. [26] G. Prodi, Un teorema di unicità per le equazioni di Navier–Stokes, Ann. Mat. Pura Appl. 48 (1959) 173–182. [27] J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962) 187–195.
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Weak–strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows Yong-Fu Yang a,b,∗ , Changsheng Dou b , Qiangchang Ju b a
Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, Jiangsu Province, PR China
b
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, PR China
article
info
Article history: Received 18 September 2012 Accepted 12 February 2013 Communicated by Enzo Mitidieri Keywords: Three-dimensional magnetohydrodynamics (MHD) equations Compressible flows Relative entropy Weak–strong uniqueness
abstract By means of the concept of relative entropy, we establish the weak–strong uniqueness property in the class of finite-energy weak solutions to the magnetohydrodynamic equations of three-dimensional compressible isentropic flows with the adiabatic exponent γ > 1 and constant viscosity coefficients, under the assumption that weak solutions exist. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids and the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. The applications of magnetohydrodynamics cover a very wide range of physical areas from liquid metals to cosmic plasmas: for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics, and plasma physics. Due to their practical relevance, magnetohydrodynamic problems have long been the subject of intense cross-disciplinary research, but, except for relatively simplified special cases, the rigorous mathematical analysis of such problems remains open. In magnetohydrodynamic flows, magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself. There is a complex interaction between the magnetic and fluid dynamic phenomena, and both hydrodynamic and electrodynamic effects have to be considered. The set of equations which describe compressible viscous magnetohydrodynamics is a combination of the compressible Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. In this paper, we consider the MHD equations of threedimensional compressible flows in the isentropic case [1–4]:
ρ + div (ρ u) = 0, t (ρ u)t + div (ρ u ⊗ u) + ∇ p = (∇ × H) × H + div S(∇ u), div H = 0, Ht − ∇ × (u × tH) = −∇ × (ν∇ × H), S = µ(∇ u + ∇ u) + λdiv uI,
(1.1)
∗ Corresponding author at: Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, Jiangsu Province, PR China. Tel.: +86 18061705228. E-mail addresses: [email protected] (Y.-F. Yang), [email protected] (C. Dou), [email protected] (Q. Ju). 0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.02.015
24
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
where ρ denotes the density, u ∈ R3 the velocity, H ∈ R3 the magnetic field, p(ρ) = aρ γ the pressure, with constant a > 0 and the adiabatic exponent γ > 1, and
S = µ(∇ u + ∇ t u) + λdiv uI the stress, with viscosity coefficients µ, λ satisfying 2µ + 3λ > 0 and µ > 0; ν > 0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, and all these kinetic coefficients and the magnetic diffusivity are independent of the magnitude and direction of the magnetic field. The symbol ⊗ denotes the Kronecker tensor product. Usually, we refer to (1.1)1 as the continuity equation and (1.1)2 as the momentum balance equations. It is well known that the electromagnetic fields are governed by the Maxwell equations. In MHD, the displacement current can be neglected [3,4]. As a consequence, Eqs. (1.1)3 are called the induction equations, and the electric field E can be written in terms of the magnetic field H and the velocity u, E = ν∇ × H − u × H. Although the electric field E does not appear in (1.1), it is indeed induced according to the above relation by the moving conductive flow in the magnetic field. In this paper, we are interested in the weak–strong uniqueness property of solutions to the three-dimensional MHD equations (1.1) in a bounded domain Ω ∈ R3 with the following initial-boundary conditions:
ρ(x, 0) = ρ0 (x) ∈ Lγ (Ω ), ρ0 (x) ≥ 0, ρ(x, 0)u(x, 0) = m0 (x) ∈ L1 (Ω ), m0 = 0 if ρ0 = 0, 2 ′ H(x, 0) = H0 (x) ∈ L (Ω ), div H0 = 0 in D (Ω ), u|∂ Ω = 0, H|∂ Ω = 0.
|m0 |2 ∈ L1 (Ω ), ρ0
(1.2)
In the recent years, there have been a lot of studies on MHD by physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenges; see [2,4–13] and the references cited therein. In particular, the one-dimensional problem has been studied in many papers; see, for example, [5,6,8,10,13–15] and the references cited therein. However, a number of fundamental problems for MHD are still open; for example, even in the one-dimensional case, the global existence of classical solutions to the full perfect MHD equations with large data remains unsolved when all the viscosity, heat conductivity, and diffusivity coefficients are constants, although the corresponding problem for the Navier–Stokes case was solved in [16] a long time ago. The reason is that the presence of the magnetic field and its interaction with the hydrodynamic motion in MHD flows of large oscillation cause serious difficulties. Hu and Wang, in [2], obtained the global existence and large-time behavior of weak solutions to the multi-dimensional isentropic problem (1.1)–(1.2) with γ > 3/2, where all the viscosity coefficients µ, λ, ν are constants. Since the global existence of the weak solutions has been established, it is natural to study their uniqueness property. To our best knowledge, so far there are very few results concerning the uniqueness of weak solutions to the initial-boundary value problem (1.1)–(1.2), while, for incompressible magnetohydrodynamic flows, Li and Wang, in [17], established its weak–strong uniqueness principle. When there is no electromagnetic field, system (1.1) reduces to the compressible Navier–Stokes equations. There have been a number of papers in the literature on the multi-dimensional Navier–Stokes equations (see [18–20] and the references therein).In fact, the known results about the compressible Navier–Stokes equations are mainly related to the global existence of weak solutions with large initial data. All the efforts in these papers are to relax the requirement on the adiabatic exponent γ so as to include more physically relevant cases. It is well known that the uniqueness of weak solutions is another fundamental and important topic in mathematical theory of the compressible Navier–Stokes equations. Contrasting with the existence theory of weak solutions, it seems that the uniqueness issue is more difficult and there are only a few results dealing with uniqueness. With the aid of the concept of relative entropy, Germain [21] introduced a class of (weak) solutions to the compressible Navier–Stokes system satisfying a relative entropy inequality with respect to a (hypothetical) strong solution of the same problem, and established the weak–strong uniqueness property within this class (see also [22]). Unfortunately, the existence of weak solutions belonging to this class, where, in particular, the density possesses a spatial gradient in a suitable Lebesgue space, is not known. Recently, Feireisl et al. [23] introduced the concept of a suitable weak solution for the compressible Navier–Stokes system, satisfying a general relative entropy inequality with respect to any sufficiently regular pair of functions. They showed the global-intime existence of suitable weak solutions for any finite-energy initial data. Moreover, the relative entropy inequality can be used to show that suitable weak solutions comply with the weak–strong uniqueness principle, meaning that a weak and a strong solution emanating from the same initial data coincide as long as the latter exists. More recently, Feireisl et al. [24] further showed that any finite-energy weak solution satisfies a relative entropy inequality with respect to any couple of smooth functions satisfying relevant boundary conditions. Based on the relative entropy inequality, they succeeded in proving the weak–strong uniqueness and in providing a satisfactory answer to the weak–strong uniqueness problem initially related to the fundamental questions of the well-posedness for the compressible Navier–Stokes system. In addition, Feireisl and Novotný [25] also established a similar weak–strong uniqueness property of the variational solutions for a compressible Navier–Stokes–Fourier system. We should remark that the weak–strong uniqueness property for the standard
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
25
incompressible Navier–Stokes system was established in the seminal papers by Prodi [26] and Serrin [27]. The situation is a bit more delicate in the case of a compressible fluid. Our goal in this paper is to establish the weak–strong uniqueness property for compressible MHD flows in the spirit of Feireisl et al. [24]. Our method is based on the relative entropy, the relative entropy inequality, and a Gronwall-type argument. Compared with the result in [24], where Feireisl et al. essentially showed the weak–strong uniqueness property for the adiabatic exponent γ ≥ 6/5, we are going to use a new technique to estimate the remainder R (for the definition see (3.8)), so as to establish the weak–strong uniqueness property for an improved lower bound for any adiabatic exponent γ > 1. We should point out here that, due to the presence of the magnetic field, the main difficulty lies in the interaction ˜ ∥L∞ (0,T ;L2 (Ω )) , where u˜ is the corresponding strong solution, is between the velocity and the magnetic field. Since ∥u − u not included in the relative entropy, some new techniques are required to decouple the velocity u and the magnetic field H when we deal with the terms involving the interaction between u and H. The paper is organized as follows. In Section 2, we recall the definitions of weak and strong solutions to the compressible MHD flows, introduce the relative entropy, and state the main result. Section 3 is devoted to the derivation of the relative entropy inequality. Finally, the proof of the main result is presented in Section 4. 2. Main result
Motivated by the concept of relative entropy in [24], we similarly define the relative entropy E = E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] , with respect to {ρ, ˜ u˜ , H˜ }, as
E =
1 2
Ω
1
ρ|u − u˜ |2 + Π (ρ) − Π ′ (ρ)(ρ ˜ − ρ) ˜ − Π (ρ) ˜ + |H − H˜ |2
2
dx,
(2.1)
where
Π (ρ) =
a
γ −1
ργ .
(2.2)
Noticing that
∇ × (∇ × H) = ∇(div H) − △H, Eqs. (1.1)3 become Ht − ∇ × (u × H) = ν△H.
(2.3)
We say that a triple {ρ, u, H} is a finite-energy weak solution to the initial-boundary value problem (1.1)–(1.2) if the following hold.
• The density ρ is a non-negative function satisfying ρ ∈ C ([0, T ]; L1 (Ω )) ∩ L∞ (0, T ; Lγ (Ω )),
ρ(x, 0) = ρ0 ,
and the momentum ρ u satisfies 2γ
ρ u ∈ C ([0, T ]; Lwγ +eak1 (Ω )). • The velocity u and the magnetic field H satisfy the following: u ∈ L2 (0, T ; H01 (Ω )),
H ∈ L2 (0, T ; H01 (Ω )) ∩ C ([0, T ]; L2weak (Ω )),
ρ u ⊗ u, ∇ × (u × H), and (∇ × H) × H are integrable on (0, T ) × Ω , and ρ u(x, 0) = m0 ,
H(x, 0) = H0 ,
div H = 0
in D ′ (Ω ).
• The first equation in (1.1) verifies a family of integral identities τ ρ(τ , ·)ϕ(τ , ·) dx − ρ0 ϕ(0, ·) dx = (ρ∂t ϕ + ρ u · ∇ϕ) dxdt Ω
Ω
0
(2.4)
Ω
¯ ), and any τ ∈ [0, T ]. for any ϕ ∈ C 1 ([0, T ] × Ω • The momentum equations (1.1)2 are satisfied in the sense of distributions; specifically, ρ u(τ , ·)ϕ(τ , ·) dx − ρ0 u0 · ϕ(0, ·) dx Ω Ω τ = (ρ u · ∂t ϕ + ρ u ⊗ u : ∇ϕ + p(ρ)div ϕ − S(∇ u) : ∇ϕ) dxdt Ω τ
0
+ 0
Ω
(ϕ · (∇ H)H − H · (∇ H)ϕ) dxdt
¯ ), ϕ|∂ Ω = 0, and any τ ∈ [0, T ]. for any ϕ ∈ C 1 ([0, T ] × Ω
(2.5)
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Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
Here and hereafter A · (∇ B)C = ((C · ∇)B) · A, in which A, B, and C are vectors. • Eqs. (1.1)3 (i.e., (2.3)) are replaced by a family of integral identities H0 · ϕ(0, ·) dx H(τ , ·) · ϕ(τ , ·) dx − Ω
Ω
τ
= Ω
0
(H · ∂t ϕ − ν(∇ H) : (∇ϕ) + ϕ · (∇ u)H − ϕ · (∇ H)u − (H · ϕ)div u) dxdt
(2.6)
¯ ), and any τ ∈ [0, T ]. for any ϕ ∈ C 1 ([0, T ] × Ω • The energy inequality E (t ) +
τ
0
Ω
µ|∇ u|2 + (λ + µ)(div u)2 + ν|∇ H|2 dxdt ≤ E (0)
(2.7)
holds for a.e. τ ∈ [0, T ], where E (t ) =
Ω
1 2
ρ|u|2 +
a
γ −1
1
ρ γ + |H|2
2
dx,
and E (0) =
Ω
1 |m0 |2 2
ρ0
+
a
γ −1
γ
1
ρ0 + |H0 |
2
2
dx.
Remark 2.1. Multiplying the continuity equation (1.1)1 by b′ (ρ), we obtain the renormalized continuity equation: b(ρ)t + div (b(ρ)u) + (b′ (ρ)ρ − b(ρ))div u = 0
(2.8)
for some suitable function b ∈ C 1 (R+ ). If (1.1)1 is satisfied in the sense of renormalized solutions, that is, (2.8) holds in D ′ (Ω × (0, T )) for any b ∈ C 1 (R+ ) satisfying b′ ( z ) = 0
for all z ∈ R+ large enough, say, z ≥ z0 ,
where the constant z0 depends on the choice of function b, then the existence of global-in-time renormalized finite-energy weak solutions to the MHD system with the adiabatic exponent γ > 23 was established in [2].
{ρ, ˜ u˜ , H˜ } is called a classical (strong) solutions to the MHD system in (0, T ) × Ω if ¯ ), ρ( ρ˜ ∈ C 1 ([0, T ] × Ω ˜ t , x) ≥ ρ > 0 for all (t , x) ∈ (0, T ) × Ω , ˜ , ∂t H˜ , ∇ 2 H˜ ∈ C ([0, T ] × Ω ¯ ), ¯) ˜u, ∂t u˜ , ∇ 2 u˜ ∈ C ([0, T ] × Ω H
(2.9)
and ρ, ˜ u˜ , H˜ satisfy Eq. (1.1), together with the boundary conditions (1.2)4 . Observe that hypothesis (2.9) requires the following regularity properties of the initial data:
¯ ), ρ0 ≥ ρ > 0, ρ( ˜ 0, ·) = ρ( ˜ 0) = ρ0 ∈ C 1 (Ω ˜ (0, ·) = H˜ 0 = H0 ∈ C 2 (Ω ¯ ), ¯ ), ˜ (0, ·) = u˜ 0 = u0 ∈ C 2 (Ω u H
div H0 = 0.
(2.10)
We are now ready to state the main result of this paper. Theorem 2.1. Let Ω ∈ R3 be a bounded domain with a boundary of class C 2+κ , κ > 0, and γ > 1. Suppose that {ρ, u, H} is a finite-energy weak solution to the MHD system in (0, T ) × Ω in the sense specified above, and suppose that {ρ, ˜ u˜ , H˜ } is a strong solution emanating from the same initial data (2.10). Then
ρ ≡ ρ, ˜
˜, u≡u
˜. H≡H
The rest of the paper is devoted to the proof of Theorem 2.1. 3. Relative entropy inequality We deduce a relative entropy inequality satisfied by any weak solution to the MHD system. To this end, consider a triple
¯ , u˜ |∂ Ω = 0, and H˜ |∂ Ω = 0. In addition, u˜ and H˜ solve {ρ, ˜ u˜ , H˜ } of smooth functions, ρ˜ bounded away from zero in [0, T ] × Ω (2.3).
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
27
˜ as a test function in the momentum equation (2.5) to obtain Take u Ω
ρ u · u˜ (τ , ·) dx −
Ω
τ
(3.1)
|u˜ |2 as a test function in the continuity equation (2.4) to get τ 1 ρ u˜ · ∂t u˜ + ρ u˜ · (∇ u˜ )u dxdt , ρ0 |u˜ |2 (0, ·) dx +
(3.2)
Next, we can use the scalar quantity ϕ =
2
ρ|u˜ |2 (τ , ·) dx =
1 2
2
Ω
Ω
0
1
Ω
ρ u · ∂t u˜ + ρ u ⊗ u : ∇ u˜
˜ · (∇ H)H − H · (∇ H)u˜ dxdt . u
+ p(ρ)div u˜ − S(∇ u) : ∇ u˜ dxdt +
Ω
0
τ
ρ0 u0 · u˜ (0, ·) dx =
Ω
0
˜ · (∇ u˜ )u = ((u · ∇)u˜ ) · u. ˜ Similarly, the choice ϕ = Π ′ (ρ) where u ˜ in (2.4) gives rise to Ω
ρ Π ′ (ρ)(τ ˜ , ·) dx =
Ω
τ
ρ0 Π (ρ)( ˜ 0, ·) dx +
Ω
0
ρ∂t Π ′ (ρ) ˜ + ρ u · ∇ Π ′ (ρ) ˜ dxdt .
(3.3)
˜ to get We test (2.6) on H Ω
˜ (τ , ·) dx = H·H
˜ (0, ·) dx + H·H
Ω
τ
Ω
0
τ
+
˜ − ν(∇ H) · (∇ H˜ ) H · ∂t H
Ω
0
˜ · (∇ u)H − H˜ · (∇ H)u − (H · H˜ )div u H
dxdt
dxdt .
(3.4)
˜ and u˜ solve (1.1)3 , we use H˜ |∂ Ω = 0 to infer that Finally, recalling that the smooth functions H 1
2
Ω
1 |H˜ |2 (τ , ·) dx =
2
Ω
|H˜ |2 (0, ·) dx − ν
τ
0
|∇ H˜ |2 dxdt + Ω
τ
0
Ω
˜ · (∇ H˜ )u˜ − u˜ · (∇ H˜ )H˜ H
dxdt .
(3.5)
Summing up the relations (3.1)–(3.5) with the energy inequality (2.7), we infer that
2 ˜ ρ|u − u˜ | + Π (ρ) − ρ Π (ρ) ˜ + |H − H| (τ , ·) dx 2 2 Ω τ τ (S(∇ u) − S(∇ u˜ )) : (∇ u − ∇ u˜ ) dxdt + ν |∇ H − ∇ H˜ |2 dxdt + 0 Ω 0 Ω 1 1 2 ′ ≤ ρ0 |u0 − u˜ (0, ·)| + Π (ρ0 ) − ρ0 Π (ρ( ˜ 0, ·)) + |H0 − H˜ (0, ·)|2 dx 2 2 Ω τ τ + ρ(∂t u˜ + u · ∇ u˜ ) · (u˜ − u) dxdt + S(∇ u˜ ) : ∇(u˜ − u) dxdt 0 τ Ω 0 τ Ω ˜ dxdt − (ρ∂t Π ′ (ρ) ˜ + ρ u · ∇ Π ′ (ρ)) ˜ dxdt − p(ρ)div u 0 Ω 0 Ω τ ˜ · (∇ H)H − H · (∇ H)u˜ − H˜ · (∇ H˜ )u˜ + u˜ · (∇ H˜ )H˜ dxdt − u
1
Ω
0
τ
− 0
1
′
2
Ω
˜ + ν(∇ H) · (∇ H˜ ) + H˜ · (∇ u)H − H˜ · (∇ H)u − (H · H˜ )div u H · ∂t H
dxdt .
(3.6)
By virtue of the definition (2.2) of Π , it is easy to see that
Π ′ (ρ) ˜ ρ˜ − Π (ρ) ˜ = p(ρ) ˜ and
Ω
ρ∂ ˜ t Π ′ (ρ) ˜ + ρ∇ ˜ Π ′ (ρ) ˜ · u˜ + p(ρ) ˜ div u˜ dx =
Ω
∂t p(ρ) ˜ dx.
As a consequence, we deduce, from the identities
Ω
p(ρ)(τ ˜ , ·) dx −
Ω
p(ρ)( ˜ 0, ·) dx =
τ
0
Ω
∂t p(ρ) ˜ dxdt
and (3.6), the desired entropy inequality:
E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] (τ ) +
τ
0
Ω
(S(∇ u) − S(∇ u˜ )) : (∇ u − ∇ u˜ ) dxdt + ν
τ
0
τ
≤ E [ρ0 , u0 , H0 ]|[ρ( ˜ 0, ·), u˜ (0, ·), H˜ (0, ·)] + 0
R(ρ, u, H, ρ, ˜ u˜ , H˜ ) dt ,
|∇ H − ∇ H˜ |2 dxdt Ω
(3.7)
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Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
where
R = R(ρ, u, H, ρ, ˜ u˜ , H˜ ) := Rd + Rm ,
Rd :=
ρ(u˜ − u) · ∂t u˜ + (∇ u˜ )u dx +
Ω
+ Ω
(3.8)
Ω
S(∇ u˜ ) : ∇(u˜ − u) dx
(ρ˜ − ρ)∂t Π ′ (ρ) ˜ + ∇ Π ′ (ρ) ˜ · (ρ˜ u˜ − ρ u) dx −
Ω
˜ (p(ρ) − p(ρ)) div u ˜ dx,
(3.9)
and
Rm :=
Ω
−
˜ − u˜ · (∇ H)H + H˜ · (∇ H˜ )u˜ − u˜ · (∇ H˜ )H˜ H · (∇ H)u
Ω
dx
˜ + ν(∇ × H) · (∇ × H˜ ) + H˜ · (∇ u)H − H˜ · (∇ H)u − (H · H˜ )div u H · ∂t H
dx.
(3.10)
4. Weak–strong uniqueness In this section, we shall finish the proof of Theorem 2.1 by applying the relative entropy inequality (3.7) to {ρ, ˜ u˜ , H˜ }, ˜ where {ρ, ˜ u˜ , H} is a classical (smooth) solution of the MHD system (1.1), (1.2), such that
ρ( ˜ 0, ·) = ρ0 ,
˜ (0, ·) = u0 , u
˜ (0, ·) = H0 . H
Accordingly, the integrals depending on the initial data on the right-hand side of (3.7) vanish, and we apply a Gronwalltype argument to deduce the desired result, namely,
ρ ≡ ρ, ˜
˜, u≡u
˜. H≡H
Our purpose is to examine all terms in the remainder (3.8) and to show that they can be ‘‘absorbed’’ by the left-hand side of (3.7). Compared with [24], we should remark that the main difficulty comes from the interaction between the velocity and the magnetic field. Moreover, in the context of the weak–strong uniqueness, we only assume the adiabatic exponent as γ > 1, not γ ≥ 6/5 as in [24]. Similar to the proof of Theorem 4.1 in [24], we use (3.9) to find that
1 ρ(u˜ − u) · (∇ u˜ )(u − u˜ ) dx + (ρ − ρ) ˜ div S(∇ u˜ ) · (u˜ − u) dx ˜ Ω Ω ρ ρ ′ ˜ p(ρ) − p (ρ)(ρ (∇ × H˜ ) × H˜ · (u˜ − u) dx − div u ˜ − ρ) ˜ − p(ρ) ˜ dx + ˜ Ω ρ Ω ˜ p(ρ) − p′ (ρ)(ρ = ρ(u˜ − u) · (∇ u˜ )(u − u˜ ) dx − div u ˜ − ρ) ˜ − p(ρ) ˜ dx Ω Ω 1 + (ρ − ρ) ˜ div S(∇ u˜ ) + (∇ × H˜ ) × H˜ · (u˜ − u) dx + (∇ × H˜ ) × H˜ · (u˜ − u) dx ˜ Ω ρ Ω =: Rd + (∇ × H˜ ) × H˜ · (u˜ − u) dx.
Rd =
(4.1)
Ω
If we combine the identity
Ω
˜ dx = H · ∂t H
Ω
˜ ˜ ˜ ˜ ˜ ˜ H · (∇ u)H − H · (∇ H)u − (H · H)div u dx − ν (∇ H˜ ) · (∇ H) dxdt Ω
with (3.10), we find after a tedious but straightforward computation that
Rm := Rm +
(∇ × H˜ ) × H˜ · (u˜ − u) dx
Ω = (H − H˜ ) · (∇ u˜ )(H − H˜ ) + (u − u˜ ) · (∇ H˜ )(H − H˜ ) Ω
+ (H˜ − H) · (∇ H˜ )(u − u˜ ) + (H˜ − H) · ((∇ H˜ − ∇ H)u˜ ) dx,
(4.2)
˜ = 0. Consequently, it follows from where we have used integration by parts several times and the fact that div H = div H the definitions of Rd and Rm that R(ρ, u, H, ρ, ˜ u˜ , H˜ ) = Rd + Rm .
(4.3)
Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
29
In order to prove the weak–strong uniqueness property, we have to estimate the remainder R. As for Rd , we mainly follow [24] with some modifications to relax the requirement on the adiabatic exponent γ . In what follows, we are going to focus on the estimation of Rm . From (2.10), it is clear to see that
′ ˜ ˜ ˜ ˜ ρ( u − u ) · (∇ u )( u − u ) dx − div u p (ρ) − p ( ρ)(ρ ˜ − ρ) ˜ − p ( ρ) ˜ dx Ω Ω ≤ C ∥∇ u˜ ∥L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] .
(4.4)
Here and hereafter C stands for a generic constant, which may change from line to line.
˜ , H˜ , ∇ H˜ ) = div S(∇ u˜ ) + (∇ × H˜ ) × H˜ . Obviously, we have Let ˜f = ˜f(∇ u
1 Ω
ρ˜
(ρ − ρ) ˜ ˜f · (u˜ − u) dx =
1 ρ˜
{ 2 <ρ<2ρ} ˜
+ {ρ≥2ρ} ˜
ρ˜ 1
ρ˜
(ρ − ρ) ˜ ˜f · (u˜ − u) dx +
1 ρ˜
{0≤ρ< 2 }
ρ˜
(ρ − ρ) ˜ ˜f · (u˜ − u) dx
(ρ − ρ) ˜ ˜f · (u˜ − u) dx.
(4.5)
Similarly to [24], we make use of Hölder’s inequality, Sobolev’s inequality, and a Korn-type inequality to show that
2 ˜f 1 ˜ + (ρ − ρ) ˜ f · (u˜ − u) dx ≤ C (δ) ρ˜ ρ˜ 3 { ρ˜ <ρ<2ρ} ρ ˜ ˜ {0≤ρ< 2 } 2
E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ]
L (Ω )
+ δ ∥S(∇ u − ∇ u˜ )∥2L2 (Ω )
(4.6)
for any δ > 0. On the other hand, noticing that
Π (ρ) − Π ′ (ρ)(ρ ˜ − ρ) ˜ − Π (ρ) ˜ ≥ C ργ ,
as ρ ≥ 2ρ˜ ≥ 2ρ
and
ρ − ρ˜ 1 − γ ρ ρ˜ ρ 2 2 ≤ C ,
as ρ ≥ 2ρ˜ ≥ 2ρ and γ > 1,
we conclude from the definition of relative entropy E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] that
{ρ≥2ρ} ˜
ρ − ρ˜ 1 ˜ 1 ρ ρ˜ ρ 2 f · ρ 2 |u˜ − u| dx ρ − ρ˜ 1 − γ γ ρ 2 2 ρ 2 ˜f · ρ 12 |u˜ − u| dx ρ ρ˜ {ρ≥2ρ} ˜ 12 21 ρ ≤ C ∥˜f∥L∞ (Ω ) |u − u˜ |2 dx ρ γ dx
˜ (ρ − ρ) ˜ f · (u˜ − u) dx = ρ˜ {ρ≥2ρ} ˜ = 1
Ω
Ω
2
≤ C ∥˜f∥L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] .
(4.7)
Next, we continue to estimate Rm . It is easy to see that
(H − H˜ ) · (∇ u˜ )(H − H˜ ) dx ≤ ∥∇ u˜ ∥L∞ (Ω ) ∥H − H˜ ∥22 L (Ω ) Ω ≤ ∥∇ u˜ ∥L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] .
(4.8)
We deduce from Hölder’s inequality and the Korn-type inequality that
˜ ˜ ˜ ˜ (u − u˜ ) · (∇ H)(H − H) + (H − H) · (∇ H)(u − u˜ ) dx Ω
≤ ∥∇ H˜ ∥L∞ (Ω ) ∥u − u˜ ∥L2 (Ω ) ∥H − H˜ ∥L2 (Ω ) ≤ δ ∥S(∇ u − ∇ u˜ )∥2L2 (Ω ) + C (δ) ∥∇ H˜ ∥2L∞ (Ω ) ∥H − H˜ ∥2L2 (Ω ) ≤ δ ∥S(∇ u − ∇ u˜ )∥2L2 (Ω ) + C (δ) ∥∇ H˜ ∥2L∞ (Ω ) E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ]
(4.9)
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Y.-F. Yang et al. / Nonlinear Analysis 85 (2013) 23–30
for any δ > 0. Finally,
((H˜ − H) · (∇ H˜ − ∇ H)u˜ ) dx ≤ ∥u˜ ∥L∞ (Ω ) ∥H − H˜ ∥L2 (Ω ) ∥∇ H − ∇ H˜ ∥L2 (Ω ) Ω
≤ δ ∥∇ H − ∇ H˜ ∥2L2 (Ω ) + C (δ) ∥u˜ ∥2L∞ (Ω ) ∥H − H˜ ∥2L2 (Ω ) ˜ u˜ , H˜ ] . ≤ δ ∥∇ H − ∇ H˜ ∥2L2 (Ω ) + C (δ) ∥u˜ ∥2L∞ (Ω ) E [ρ, u, H]|[ρ,
(4.10)
Summing up relations (4.1)–(4.10), we conclude that the relative entropy inequality yields the desired conclusion:
˜ E [ρ, u, H]|[ρ, ˜ u˜ , H] (τ ) ≤
τ
h(t )E [ρ, u, H]|[ρ, ˜ u˜ , H˜ ] (t ) dt
with some h ∈ L1 (0, T ).
0
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