Low Mach number limit for the compressible viscous MHD equations - - PowerPoint PPT Presentation
Low Mach number limit for the compressible viscous MHD equations Fucai LI Nanjing University Based on joint works with S. Jiang, Q.-C. Ju, and Z.-P. Xin 25-6-2012 1 / 37 The goal of Low Mach number limit: derive incompressible (slightly
Based on joint works with S. Jiang, Q.-C. Ju, and Z.-P. Xin
25-6-2012
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The isentropic compressible Euler equations
∂t(ρu) + div(ρu ⊗ u) + ∇P = 0. P = aργ, γ > 1. Denote ǫ the Mach number, we introduce ρ(x, t) = ρǫ(x, ǫt), u(x, t) = ǫuǫ(x, ǫt).
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The original Euler equations become ∂tρǫ + div(ρǫuǫ) = 0, ∂t(ρǫuǫ) + div(ρǫuǫ ⊗ uǫ) + a∇(ρǫ)γ ǫ2 = 0. Let ǫ → 0+, we formally obtain ρǫ(x, t) → ρ0(t). The initial datum ρǫ(x) → ¯ ρ0 ⇒ ρǫ(t) → ¯ ρ0. Taking ¯ ρ0 ≡ 1 ⇒ divv = 0 (assume that uǫ → v as ǫ → 0+). The limiting equations (incompressible Euler) read ∂tv + v · ∇v + ∇π = 0, divv = 0.
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For well-prepared initial data
0 − ¯
ρ0 ǫ , uǫ
0 − v0(x)
= O(ǫ) = ⇒
ρ0 ǫ − ǫπ ψ0 , uǫ − v
≤ Kǫ, 0 ≤ t < T ∗ Ω = Td or Rd, ψ0 =
ρ0), s > d 2 + 1. T ∗: the maximal existing time of the smooth solutions to the incompressible Euler equations.
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Magnetohydrodynamics (MHD) studies the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. The applications of MHD cover a very wide range of physical
astrophysics, geophysics, plasma physics, cosmic plasmas, et. al.
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ρt + div(ρu) = 0, (ρu)t + div (ρu ⊗ u) + ∇P = (∇ × H) × H + divΨ, Et + div
Ht − ∇ × (u × H) = −∇ × (ν∇ × H), divH = 0. ρ ≥ 0: the density u ∈ R3: the velocity H ∈ R3: the magnetic field
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θ: the temperature Ψ: the viscous stress tensor given by Ψ = µ(∇u + ∇uT) + λ divu Id E: the total energy given by E = ρ
2|u|2
2|H|2 and E′ = ρ
2|u|2
the internal energy
1 2ρ|u|2:
the kinetic energy
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1 2|H|2: the magnetic energy P = P(ρ, θ), e = e(ρ, θ) satisfy the equations of state P = ρ2 ∂e ∂ρ + θ∂P ∂θ I: the 3 × 3 identity matrix ∇uT: the transpose of the matrix ∇u λ, µ: the viscosity coefficients of the flow satisfying 2µ + 3λ > 0, µ > 0 ν > 0: the magnetic diffusivity κ > 0: the heat conductivity
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The compressible MHD equations can be derived from the complete equations describing an electromagnetic dynamics [ compressible Navier–Stokes system coupled with Maxwell system ] as the dielectric constant tends to zero. This is the so-called magnetohydrodynamic approximation. Remark: Although the electric field E does not appear in the MHD equations, it is indeed induced according to the following relation E = ν∇ × H − u × H by the moving conductive flow in the magnetic field.
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Two categories on studying the low Mach number limit to the full compressible MHD equations
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Here we study the low Mach number limit to the full compressible MHD equations in the framework of local smooth solutions and consider the three-dimensional case only.
1. For the low Mach number limit to the full compressible MHD equations in the framework of weak solutions, see :
cka, J. Math. Fluid Mech. (2011); A. Novotny, et. al., M3AS (2011); Y.-S. Kwon, K. Trivisa, JDE (2011). 2. For low Mach number limit to the isentropic MHD equations, see: Hu-Wang (SIAM JMA 2009), Jiang-Ju-L (SIAM JMA 2010, CMP 2010).
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We shall focus our efforts on the ionized fluid obeying the perfect gas relations P = Rρθ, e = cV θ, (1) R > 0: the gas constant cV >0: the heat capacity at constant volume
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We rewrite the full MHD equations as follows ∂tρ + div(ρu) = 0, (2) ρ(∂tu + u · ∇u) + ∇(ρθ) ǫ2 = (∇ × H) × H + divΨ, (3) ∂tH − ∇ × (u × H) = −∇ × (ν∇ × H), divH = 0, (4) ρ(∂tθ + u · ∇θ) + (γ − 1)ρθdivu = ǫ2ν|∇ × H|2 + ǫ2Ψ : ∇u + κ∆θ, (5) ǫ: the Mach number µ, λ, ν, κ: the scaled parameters γ = 1 + R/cV : the ratio of specific heats
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We further restrict ourselves to the small density and temperature variations, i.e. ρ = 1 + ǫqǫ, θ = 1 + ǫφǫ, u = uǫ, H = Hǫ. (6) Putting (6) and (1) into the system (2)–(5), and using the identities curl curl H = ∇ divH − ∆H, ∇(|H|2) = 2H · ∇H + 2H × curl H, curl (u × H) = u(divH) − H(divu) + H · ∇u − u · ∇H,
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we can rewrite (2)–(5) as ∂tqǫ + uǫ · ∇qǫ + 1 ǫ (1 + ǫqǫ)divuǫ = 0, (7) (1 + ǫqǫ)(∂tuǫ + uǫ · ∇uǫ) + 1 ǫ
− Hǫ · ∇Hǫ + 1 2∇(|Hǫ|2) = 2µdiv(D(uǫ)) + λ∇(trD(uǫ)), (8) (1 + ǫqǫ)(∂tφǫ + uǫ · ∇φǫ) + γ − 1 ǫ (1 + ǫqǫ)(1 + ǫφǫ)divuǫ = κ∆φǫ + ǫ
(9) ∂tHǫ + uǫ · ∇Hǫ + divuǫHǫ − Hǫ · ∇uǫ = ν∆Hǫ, divHǫ = 0. (10)
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Therefore, the formal limit as ǫ → 0+ of (7)–(10) is the following incompressible MHD equations (suppose that the limits uǫ → w and Hǫ → B exist.) ∂tw + w · ∇w + ∇π + 1 2∇(|B|2) − B · ∇B = µ∆w, (11) ∂tB + w · ∇B − B · ∇w = ν∆B, (12) divw = 0, divB = 0. (13) Consider the system (7)-(10) in the Torus T3 or the whole space R3.
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Duvaut-Lions (1972), Sermange-Temam (1983))
Let s > 3/2 + 2. The initial data (w, B)|t=0 = (w0, B0) satisfy w0 ∈ Hs, B0 ∈ Hs, div w0 = 0, divB0 = 0. Then, there exist a ˆ T ∗ ∈ (0, ∞] and a unique solution (w, B) ∈ L∞(0, ˆ T ∗; Hs) to (11)–(13) satisfying div w = 0 and divB = 0, and for any 0 < T < ˆ T ∗, sup
0≤t≤T
≤ C.
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Let s > 3/2 + 2. Suppose that (qǫ
0(x), uǫ 0(x), Hǫ 0(x), φǫ 0(x))) satisfy
qǫ
0(x), uǫ 0(x) − w0(x), Hǫ 0(x) − B0(x), φǫ 0(x)Hs = O(ǫ).
Let (w, B, π) be a smooth solution to (11)–(13) obtained in the above proposition satisfying (w, π) ∈ C([0, T ∗], Hs+2) ∩ C1([0, T ∗], Hs), T ∗ > 0 finite. Then ∃ ǫ0 > 0 , for all ǫ ≤ ǫ0, the system (7)-(10) with initial data (qǫ
0(x), uǫ 0(x), Hǫ 0(x), φǫ 0(x))) has a unique smooth solution
(qǫ, uǫ, Hǫ, φǫ) ∈ C([0, T ∗], Hs). Moreover, ∃ K > 0, independent of ǫ, for all ǫ ≤ ǫ0, sup
t∈[0,T ∗]
ǫ 2π, w, B, ǫ 2π (t)
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Remark: From Theorem above, we know that for sufficiently small ǫ and well-prepared initial data, the full MHD equations (2)–(5) admits a unique smooth solution on the same time interval where the smooth solution of the incompressible MHD equations exists. Remark: The KEY points in the proof: energy estimates + compact arguments + convergence-stability lemma. Remark: The approach is still valid for the ideal non-isentropic compressible MHD equations.
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Full MHD equations: ρt + div(ρu) = 0, (ρu)t + div (ρu ⊗ u) + ∇p = (∇ × H) × H + divΨ, Et + div
Ht − ∇ × (u × H) = −∇ × (ν∇ × H), divH = 0. with Ψ = µ(∇u + ∇uT) + λ divu I3, E = ρ
2|u|2
2|H|2 and E′ = ρ
2|u|2
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As before, we shall focus our efforts on the ionized fluid obeying the perfect gas relations p = Rρθ, e = cV θ, (14) R > 0: the gas constant cV >0: the heat capacity at constant volume
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Let ǫ be the Mach number. Consider the full MHD system in the physical regime: P ∼ P0 + O(ǫ), u ∼ O(ǫ), H ∼ O(ǫ), ∇θ ∼ O(1), where P0 > 0 is a certain given constant which is normalized to be P0 = 1.
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Consider the case the pressure P is a small perturbation of the given constant state 1 while the temperature θ has a finite variation. We introduce the following transformation to ensure the positivity of P and θ P(x, t) = eǫpǫ(x,ǫt), θ(x, t) = eθǫ(x,ǫt). (15) Note that (14) and (15) imply that ρ(x, t) = eǫpǫ(x,ǫt)−θǫ(x,ǫt) since we take R ≡ cV ≡ 1.
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Set H(x, t) = ǫHǫ(x, ǫt), u(x, t) = ǫuǫ(x, ǫt), (16) and µ = ǫµǫ, λ = ǫλǫ, ν = ǫνǫ, κ = ǫκǫ. (17)
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The MHD system with (14) takes the following equivalent form: ∂tpǫ + (uǫ · ∇)pǫ + 1 ǫ div(2uǫ − κǫe−ǫpǫ+θǫ∇θǫ) = ǫe−ǫpǫ[νǫ|curl Hǫ|2 + Ψ(uǫ) : ∇uǫ] + κǫe−ǫpǫ+θǫ∇pǫ · ∇θǫ, (18) e−θǫ[∂tuǫ + (uǫ · ∇)uǫ] + ∇pǫ ǫ = e−ǫpǫ[(curl Hǫ) × Hǫ + divΨǫ(uǫ)], (19) ∂tHǫ − curl (uǫ × Hǫ) − νǫ∆Hǫ = 0, divHǫ = 0, (20) ∂tθǫ + (uǫ · ∇)θǫ + divuǫ = ǫ2e−ǫpǫ[νǫ|curl Hǫ|2 + Ψǫ(uǫ) : ∇uǫ] + κǫe−ǫpǫdiv(eθǫ∇θǫ), (21) with Ψǫ(uǫ) = 2µǫD(uǫ) + λǫdivuǫ I3, Ψ(uǫ) : ∇uǫ = 2µǫ|D(uǫ)|2 + λǫ|trD(uǫ)|2.
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Formally, as ǫ goes to zero, suppose that (uǫ, Hǫ, θǫ) → (w, B, ϑ) in some sense, and (µǫ, λǫ, νǫ, κǫ) → (¯ µ, ¯ λ, ¯ ν, ¯ κ), then taking the limit to (18)–(21), we have div(2w − ¯ κ eϑ∇ϑ) = 0, (22) e−ϑ[∂tw + (w · ∇)w] + ∇π = (curl B) × B + divΦ(w), (23) ∂tB − curl (w × B) − ¯ ν∆B = 0, divB = 0, (24) ∂tϑ + (w · ∇)ϑ + divw = ¯ κ div(eϑ∇ϑ), (25) with some function π, where Φ(w) = 2¯ µD(w) + ¯ λdivw I3.
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We supplement the system (18)–(21) with the following initial conditions (pǫ, uǫ, Hǫ, θǫ)|t=0 = (pǫ
in(x), uǫ in(x), Hǫ in(x), θǫ in(x)),
x ∈ R3. (26) For simplicity, we also assume that µǫ ≡ ¯ µ > 0, νǫ ≡ ¯ ν > 0, κǫ ≡ ¯ κ > 0, λǫ ≡ ¯ λ.
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We denote vHσ
η := vHσ−1 + ηvHσ
for any σ ∈ R and η ≥ 0. For each ǫ > 0, t ≥ 0 and s ≥ 0, we will also use the following norm: (pǫ, uǫ, Hǫ, θǫ − ¯ θ)(t)s,ǫ := sup
τ∈[0,t]
θ)(τ)Hs+2
ǫ
t [∇(pǫ, uǫ, Hǫ)2
Hs + ∇(ǫuǫ, ǫHǫ, θǫ)2 Hs+2
ǫ
](τ)dτ 1/2 .
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Let s ≥ 4. Assume that the initial data (pǫ
in, uǫ in, Hǫ in, θǫ in) satisfy
(pǫ
in, uǫ in, Hǫ in, θǫ in − ¯
θ)(t)s,ǫ ≤ L0 (27) for all ǫ ∈ (0, 1] and two given positive constants ¯ θ and L0. Then there exist positive constants T0 and ǫ0 < 1, depending only
θ, such that the Cauchy problem (18)–(21), (26) has a unique solution (pǫ, uǫ, Hǫ, θǫ) satisfying (pǫ, uǫ, Hǫ, θǫ − ¯ θ)(t)s,ǫ ≤ L, ∀ t ∈ [0, T0], ∀ ǫ ∈ (0, ǫ0], (28) where L depends only on L0, ¯ θ and T0.
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Assume further that the initial data satisfy the following conditions |θǫ
0(x) − ¯
θ| ≤ N0|x|−1−ζ, |∇θǫ
0(x)| ≤ N0|x|−2−ζ,
∀ǫ ∈ (0, 1], (29)
in, curl (e−θǫ
inuǫ
in), Hǫ in, θǫ in
(30) in Hs(R3) as ǫ → 0+, where N0 and ζ are fixed positive constants. Then the solution sequence (pǫ, uǫ, Hǫ, θǫ) converges weakly in L∞(0, T0; Hs(R3)) and strongly in L2(0, T0; Hs2
loc(R3)) for all 0 ≤ s2 < s
to the limit (0, w, B, ϑ), where (w, B, ϑ) satisfies the system (22)–(25) with initial data (w, B, ϑ)|t=0 = (w0, B0, ϑ0).
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Two parts in the proofs:
◮ Hs
estimates on (Hǫ, θǫ) and (ǫpǫ, ǫuǫ)
◮ Hs+1 estimates on (ǫuǫ, ǫpǫ, ǫHǫ, θǫ) ◮ Hs−1 estimates on (divuǫ, ∇pǫ) ◮ Hs−1 estimate on curl uǫ
the description of the oscillations in ill-prepared initial data case. (applying a result of G. M´ etivier & S. Schochet,(ARMA,2001))
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Main features:
◮ The propagation of oscillations is described by the wave equations
with unknown variable coefficients
◮ Strong coupling of the hydrodynamic motion and the magnetic fields
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◮ The effect of fluid and magnetic diffusions, and heat conductivity ◮ Div-Curl decomposition of the velocity ◮ Refined energy estimates ◮ Weak compact arguments ◮ Detailed analysis of the oscillation equations
Here we use some ideas from: M´ etivier-Schochet (ARMA,2001), Alazard(ARMA,2006), Levermore-Sun-Trivisa (SIAM JMA, 2012).
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Remark: The fluid diffusion term, heat conductivity term, and magnetic diffusivity term in the system (18)-(21) play a crucial role in
(in order to control some undesirable higher-order terms) Remark: For the two cases–
◮ non-isentropic MHD equations with zero magnetic diffusivity ◮ non-isentropic MHD equations with zero fluid diffusion and heat
conductivity coefficients Some new ideas are needed to deal with the low Mach number limit, see Jiang-Ju-L (arXiv:1111.2926v1).
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Low Mach number limit to the full MHD equations in two cases:
data in the torus or the whole space
in the whole space
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