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 compressible) models from compressible models when the Mach number goes to zero. Ma = v Mach number: c v : relative velocity of the source to the medium c : speed of sound in the medium 2 / 37

The isentropic compressible Euler equations � ∂ t ρ + div( ρu ) = 0 , ∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ P = 0 . P = aρ γ , γ > 1 . Denote ǫ the Mach number, we introduce ρ ( x, t ) = ρ ǫ ( x, ǫt ) , u ( x, t ) = ǫu ǫ ( x, ǫt ) . 3 / 37

The original Euler equations become  ∂ t ρ ǫ + div( ρ ǫ u ǫ ) = 0 ,  ∂ t ( ρ ǫ u ǫ ) + div( ρ ǫ u ǫ ⊗ u ǫ ) + a ∇ ( ρ ǫ ) γ = 0 .  ǫ 2 Let ǫ → 0 + , we formally obtain ρ ǫ ( x, t ) → ρ 0 ( t ) . The initial datum ρ ǫ ( x ) → ¯ ρ 0 ⇒ ρ ǫ ( t ) → ¯ ρ 0 . Taking ¯ ρ 0 ≡ 1 ⇒ div v = 0 (assume that u ǫ → v as ǫ → 0 + ). The limiting equations (incompressible Euler) read ∂ t v + v · ∇ v + ∇ π = 0 , div v = 0 . 4 / 37

For well-prepared initial data � � ρ ǫ �� 0 − ¯ ρ 0 � , u ǫ � 0 − v 0 ( x ) = O ( ǫ ) � � ǫ � � H s (Ω) � ρ ǫ − ¯ � ρ 0 − ǫπ � � , u ǫ − v � � 0 ≤ t < T ∗ = ⇒ ( t ) ≤ Kǫ, � � ǫ ψ 0 � � H s (Ω) s > d Ω = T d or R d , � ψ 0 = P ′ (¯ ρ 0 ) , 2 + 1 . T ∗ : the maximal existing time of the smooth solutions to the incompressible Euler equations. 5 / 37

In this talk : The low Mach number limit to the full compressible magnetohydrodynamics equations Model 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 objects: liquid metals, astrophysics, geophysics, plasma physics, cosmic plasmas, et. al. 6 / 37

Full MHD equations: ρ t + div( ρ u ) = 0 , ( ρ u ) t + div ( ρ u ⊗ u ) + ∇ P = ( ∇ × H ) × H + divΨ , u ( E ′ + P ) � � E t + div � � = div ( u × H ) × H + ν H × ( ∇ × H ) + u Ψ + κ ∇ θ , H t − ∇ × ( u × H ) = −∇ × ( ν ∇ × H ) , div H = 0 . ρ ≥ 0 : the density u ∈ R 3 : the velocity H ∈ R 3 : the magnetic field 7 / 37

θ : the temperature Ψ : the viscous stress tensor given by Ψ = µ ( ∇ u + ∇ u T ) + λ div u I d E : the total energy given by � � � � e + 1 + 1 e + 1 2 | H | 2 and E ′ = ρ 2 | u | 2 2 | u | 2 E = ρ e : the internal energy 1 2 ρ | u | 2 : the kinetic energy 8 / 37

1 2 | H | 2 : the magnetic energy P = P ( ρ, θ ) , e = e ( ρ, θ ) satisfy the equations of state P = ρ 2 ∂e ∂ρ + θ∂P ∂θ I : the 3 × 3 identity matrix ∇ u T : the transpose of the matrix ∇ u λ, µ : the viscosity coefficients of the flow satisfying 2 µ + 3 λ > 0 , µ > 0 ν > 0 : the magnetic diffusivity κ > 0 : the heat conductivity 9 / 37

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. 10 / 37

Two categories on studying the low Mach number limit to the full compressible MHD equations 1. Small variations on density and temperature 2. Large variations on density and temperature 11 / 37

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. Remark: 1. For the low Mach number limit to the full compressible MHD equations in the framework of weak solutions, see : P. Kuku ˇ c ka, 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). 12 / 37

CASE I: Small variations on density and temperature We shall focus our efforts on the ionized fluid obeying the perfect gas relations P = R ρθ, e = c V θ, (1) R > 0 : the gas constant c V > 0 : the heat capacity at constant volume 13 / 37

We rewrite the full MHD equations as follows ∂ t ρ + div( ρ u ) = 0 , (2) ρ ( ∂ t u + u · ∇ u ) + ∇ ( ρθ ) = ( ∇ × H ) × H + divΨ , (3) ǫ 2 ∂ t H − ∇ × ( u × H ) = −∇ × ( ν ∇ × H ) , div H = 0 , (4) ρ ( ∂ t θ + u · ∇ θ ) + ( γ − 1) ρθ div u = ǫ 2 ν |∇ × H | 2 + ǫ 2 Ψ : ∇ u + κ ∆ θ, (5) ǫ : the Mach number µ, λ, ν , κ : the scaled parameters γ = 1 + R /c V : the ratio of specific heats 14 / 37

We further restrict ourselves to the small density and temperature variations, i.e. ρ = 1 + ǫq ǫ , θ = 1 + ǫφ ǫ , u = u ǫ , H = H ǫ . (6) Putting (6) and (1) into the system (2)–(5), and using the identities curl curl H = ∇ div H − ∆ H , ∇ ( | H | 2 ) = 2 H · ∇ H + 2 H × curl H , curl ( u × H ) = u (div H ) − H (div u ) + H · ∇ u − u · ∇ H , 15 / 37

we can rewrite (2)–(5) as ∂ t q ǫ + u ǫ · ∇ q ǫ + 1 ǫ (1 + ǫq ǫ )div u ǫ = 0 , (7) (1 + ǫq ǫ )( ∂ t u ǫ + u ǫ · ∇ u ǫ ) + 1 (1 + ǫq ǫ ) ∇ φ ǫ + (1 + ǫφ ǫ ) ∇ q ǫ � � ǫ − H ǫ · ∇ H ǫ + 1 2 ∇ ( | H ǫ | 2 ) = 2 µ div( D ( u ǫ )) + λ ∇ ( tr D ( u ǫ )) , (8) (1 + ǫq ǫ )( ∂ t φ ǫ + u ǫ · ∇ φ ǫ ) + γ − 1 (1 + ǫq ǫ )(1 + ǫφ ǫ )div u ǫ ǫ = κ ∆ φ ǫ + ǫ 2 µ | D ( u ǫ ) | 2 + λ ( tr D ( u ǫ )) 2 } + ǫν |∇ × H ǫ | 2 , � (9) ∂ t H ǫ + u ǫ · ∇ H ǫ + div u ǫ H ǫ − H ǫ · ∇ u ǫ = ν ∆ H ǫ , div H ǫ = 0 . (10) 16 / 37

Therefore, the formal limit as ǫ → 0 + of (7)–(10) is the following incompressible MHD equations (suppose that the limits u ǫ → w and H ǫ → B exist.) ∂ t w + w · ∇ w + ∇ π + 1 2 ∇ ( | B | 2 ) − B · ∇ B = µ ∆ w , (11) ∂ t B + w · ∇ B − B · ∇ w = ν ∆ B , (12) div w = 0 , div B = 0 . (13) Consider the system (7)-(10) in the Torus T 3 or the whole space R 3 . 17 / 37

Proposition (Local existence of the limiting system, Duvaut-Lions (1972), Sermange-Temam (1983) ) Let s > 3 / 2 + 2 . The initial data ( w , B ) | t =0 = ( w 0 , B 0 ) satisfy w 0 ∈ H s , B 0 ∈ H s , div w 0 = 0 , div B 0 = 0 . T ∗ ∈ (0 , ∞ ] and a unique solution Then, there exist a ˆ ( w , B ) ∈ L ∞ (0 , ˆ T ∗ ; H s ) to (11) – (13) satisfying div w = 0 and div B = 0 , and for any 0 < T < ˆ T ∗ , � || ( w , B )( t ) || H s + || ( ∂ t w , ∂ t B )( t ) || H s − 2 + ||∇ π ( t ) || H s − 2 � sup ≤ C. 0 ≤ t ≤ T 18 / 37

Theorem (Jiang-Ju-L, Nonlinearity(2012) ) Let s > 3 / 2 + 2 . Suppose that ( q ǫ 0 ( x ) , u ǫ 0 ( x ) , H ǫ 0 ( x ) , φ ǫ 0 ( x ))) satisfy � q ǫ 0 ( x ) , u ǫ 0 ( x ) − w 0 ( x ) , H ǫ 0 ( x ) − B 0 ( x ) , φ ǫ 0 ( x ) � H s = O ( ǫ ) . Let ( w , B , π ) be a smooth solution to (11) – (13) obtained in the above proposition satisfying ( w , π ) ∈ C ([0 , T ∗ ] , H s +2 ) ∩ C 1 ([0 , T ∗ ] , H s ) , T ∗ > 0 finite. Then ∃ ǫ 0 > 0 , for all ǫ ≤ ǫ 0 , the system (7) - (10) with initial data ( q ǫ 0 ( x ) , u ǫ 0 ( x ) , H ǫ 0 ( x ) , φ ǫ 0 ( x ))) has a unique smooth solution ( q ǫ , u ǫ , H ǫ , φ ǫ ) ∈ C ([0 , T ∗ ] , H s ) . Moreover, ∃ K > 0 , independent of ǫ , for all ǫ ≤ ǫ 0 , � ǫ 2 π, w , B , ǫ � � � � � ( q ǫ , u ǫ , H ǫ , φ ǫ ) − sup 2 π ( t ) H s ≤ Kǫ. � � � � t ∈ [0 ,T ∗ ] 19 / 37

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. 20 / 37

CASE II: Large variations on density and temperature Full MHD equations: ρ t + div( ρ u ) = 0 , ( ρ u ) t + div ( ρ u ⊗ u ) + ∇ p = ( ∇ × H ) × H + divΨ , u ( E ′ + p ) � � E t + div � � = div ( u × H ) × H + ν H × ( ∇ × H ) + u Ψ + κ ∇ θ , H t − ∇ × ( u × H ) = −∇ × ( ν ∇ × H ) , div H = 0 . with Ψ = µ ( ∇ u + ∇ u T ) + λ div u I 3 , � � � � e + 1 + 1 e + 1 2 | H | 2 and E ′ = ρ 2 | u | 2 2 | u | 2 E = ρ . 21 / 37

As before, we shall focus our efforts on the ionized fluid obeying the perfect gas relations p = R ρθ, e = c V θ, (14) R > 0 : the gas constant c V > 0 : the heat capacity at constant volume 22 / 37

Let ǫ be the Mach number. Consider the full MHD system in the physical regime: P ∼ P 0 + O ( ǫ ) , u ∼ O ( ǫ ) , H ∼ O ( ǫ ) , ∇ θ ∼ O (1) , where P 0 > 0 is a certain given constant which is normalized to be P 0 = 1 . 23 / 37