Analysis of Oscillations and Defect Measures in Plasma Physics - PowerPoint PPT Presentation

Analysis of Oscillations and Defect Measures in Plasma Physics Donatella Donatelli (joint work with P. Marcati) Dipartimento di Matematica Pura ed Applicata Universit` a degli Studi dell’Aquila 67100 L’Aquila, Italy donatell@univaq.it

Simplified Model for Plasma Compressible Navier Stokes Poisson System ∂ t ρ λ + div( ρ λ u λ ) = 0 ( ρ λ ) γ = µ ∆ u λ +( µ + ν ) ∇ div u λ + ρ λ ∇ V λ ∂ t ( ρ λ u λ )+div( ρ λ u λ ⊗ u λ ) + ∇ λ 2 ∆ V λ = ρ λ − 1 , x ∈ R 3 , t ≥ 0 ρ λ ( x, t ) is the negative charge density m λ ( x, t ) = ρ λ ( x, t ) u λ ( x, t ) is the current density u λ ( x, t ) is the velocity vector density V λ ( x, t ) is the electrostatic potential µ is the shear viscosity and ν is the bulk viscosity λ = λ D /L , λ D is the Debye length 2

Quasineutral limit ∂ t ρ λ + div( ρ λ u λ ) = 0 ( ρ λ ) γ = µ ∆ u λ +( µ + ν ) ∇ div u λ + ρ λ ∇ V λ ∂ t ( ρ λ u λ )+div( ρ λ u λ ⊗ u λ ) + ∇ λ 2 ∆ V λ = ρ λ − 1 , x ∈ R 3 , t ≥ 0 Study the limit λ → 0 ⇓ 3

Quasineutral limit ∂ t ρ λ + div( ρ λ u λ ) = 0 ( ρ λ ) γ = µ ∆ u λ +( µ + ν ) ∇ div u λ + ρ λ ∇ V λ ∂ t ( ρ λ u λ )+div( ρ λ u λ ⊗ u λ ) + ∇ λ 2 ∆ V λ = ρ λ − 1 , x ∈ R 3 , t ≥ 0 Study the limit λ → 0 ⇓ Formally yields to an Incompressible Dynamics ρ = 1 div u = 0 3

Main Issues This formal limit will not be in general true Control of Acoustic waves Control of Space localized, high frequency in time Wave Packets 4

What is a plasma ? A Plasma is a fluid which contains ions and electrons, such that charge neutrality is mantained A gas heated up to sufficiently high temperatures so that the atoms ionise Sun’s core. The plasma at the center of the sun, where fusion of hydrogen to form helium generates the suns heat. Solar wind. The wind of plasma that blows off the sun and outward through the region between the planets. Interstellar medium. The plasma, in our Galaxy, that fills the region between the stars. 5

a charged particle inside a plasma attracts particles with opposite charge and repels those with the same charge ⇓ creation of a net cloud of opposite charge around itself ⇓ the cloud shields the particle’s own charge from external view how large is this cloud? 6

V = V ( r ) , n 0 =mean density of electrons and protons n e =electron density= n 0 e eV/kT n i =ion density= n 0 e − eV/kT ∆ V = − 1 ( n i − n e ) = − n 0 e ( e − eV/kT − e eV/kT ) ǫ 0 ǫ 0 potential energy eV ≪ kinetic energy kT � � � 2 n 0 e 2 � ∆ V = − n 0 e 1 − eV kT − 1 − eV = V ǫ 0 kT ǫ 0 kT � ǫ 0 kT λ D = 2 n 0 e 2 = Debye lenght ∆ V − 1 V = 0 Debye law λ 2 D 7

� ǫ 0 kT V ( r ) = q re − r/λ D λ D = 2 n 0 e 2 ⇒ the electric field dies off on distance greater than λ D ⇒ this is the screening effect due to the polarization cloud which screens the field of charge for distances larger than λ D ⇒ charge fluctuation may occur over distances smaller than λ D ⇒ the plasma is quasineutral for a distance L >> λ D (we can define as a parameter the “plasma density” ) Plasma T ( K ) λ D ( m ) 10 4 10 − 4 Gas discharge 10 7 10 − 11 Sun’s core 10 5 Solar wind 10 10 4 Interstellar medium 10 8

Simplified Model for Plasma Compressible Navier Stokes Poisson System ∂ t ρ λ + div( ρ λ u λ ) = 0 ( ρ λ ) γ = µ ∆ u λ +( µ + ν ) ∇ div u λ + ρ λ ∇ V λ ∂ t ( ρ λ u λ )+div( ρ λ u λ ⊗ u λ ) + ∇ λ 2 ∆ V λ = ρ λ − 1 , x ∈ R 3 , t ≥ 0 ρ λ ( x, t ) is the negative charge density m λ ( x, t ) = ρ λ ( x, t ) u λ ( x, t ) is the current density u λ ( x, t ) is the velocity vector density V λ ( x, t ) is the electrostatic potential µ is the shear viscosity and ν is the bulk viscosity λ = λ D /L , λ D is the Debye length 9

Plasma Oscillation 1 − D linearized system Fourier Transform   � ˆ � � ˆ � σ t + u x = 0 0 iξ σ t σ   u t + c 2 σ x = V x + i = 0 iξc 2 + u t ˆ 0 ˆ u λ 2 V xx = σ λ 2 ξ 10

Plasma Oscillation 1 − D linearized system Fourier Transform   � ˆ � � ˆ � σ t + u x = 0 0 iξ σ t σ   u t + c 2 σ x = V x + i = 0 iξc 2 + u t ˆ 0 u ˆ λ 2 V xx = σ λ 2 ξ Solutions � ˆ � � ˆ � � ˆ � σ ( ξ, t ) σ + ( ξ ) σ − ( ξ ) e iθ ( ξ ) t + e − iθ ( ξ ) t = u ( ξ, t ) ˆ u + ( ξ ) ˆ u − ( ξ ) ˆ � � 1 / 2 c 2 ξ 2 + 1 θ ( ξ ) = λ 2 10

Mathematical difficulties - 1 u = P [ u ] + Q [ u ] ���� ���� solenoidal part gradient part div P [ u ] = 0 Q [ u ] = ∇ Ψ ∂ t σ + div u = 0 ∂ t σ + ∆Ψ = 0 ∂ t u + ∇ σ = ∇ V ∂ t ∇ Ψ + ∇ σ = 1 λ 2 ∇ ∆ − 1 σ λ 2 ∆ V = σ Problem: “weak compactness” 11

Mathematical difficulties - 1 u = P [ u ] + Q [ u ] ���� ���� solenoidal part gradient part div P [ u ] = 0 Q [ u ] = ∇ Ψ ∂ t σ + div u = 0 ∂ t σ + ∆Ψ = 0 ∂ t u + ∇ σ = ∇ V ∂ t ∇ Ψ + ∇ σ = 1 λ 2 ∇ ∆ − 1 σ λ 2 ∆ V = σ Problem: “weak compactness” div( ρu ⊗ u ) ≈ div( u ⊗ u ) = div( u ⊗ P [ u ]) + div( P [ u ] ⊗ ∇ Ψ) + 1 2 ∇|∇ Ψ | 2 + ∆Ψ ∇ Ψ � ∂ t ( σ ∇ Ψ) − 1 2 ∇ σ 2 + 1 λ 2 σ ∇ ∆ − 1 σ 11

Mathematical difficulties -2 ∂ t ρ λ + div( ρ λ u λ ) = 0 ( ρ λ ) γ = µ ∆ u λ +( µ + ν ) ∇ div u λ + ρ λ ∇ V λ ∂ t ( ρ λ u λ )+div( ρ λ u λ ⊗ u λ ) + ∇ λ 2 ∆ V λ = ρ λ − 1 λ ∇ V λ = λE λ ρ λ ∇ V λ ∼ λE λ ⊗ λE λ L ∞ t L 2 bound on x 12

Mathematical difficulties -2 ∂ t ρ λ + div( ρ λ u λ ) = 0 ( ρ λ ) γ = µ ∆ u λ +( µ + ν ) ∇ div u λ + ρ λ ∇ V λ ∂ t ( ρ λ u λ )+div( ρ λ u λ ⊗ u λ ) + ∇ λ 2 ∆ V λ = ρ λ − 1 λ ∇ V λ = λE λ ρ λ ∇ V λ ∼ λE λ ⊗ λE λ L ∞ t L 2 bound on x simplified example: space independent case λ 2 ∂ tt E λ + E λ = F Fourier transform in time 1 λ ˆ 1 − λ 2 τ 2 ˆ E λ ( τ ) = λ F ( τ ) E λ concentrates around τ = 1 the L 2 mass of λ ˆ λ as λ → 0 = ⇒ corrector analysis 12

State of Art - References Quasineutral limit for Euler Poisson system in 1D weak solutions, Gasser and Marcati (’01)(’03) Quasineutral limit for Euler Poisson system in H s Cordier and Grenier (’00), Grenier (’96), Cordier, Degond, Markowich and Schmeiser (’96), Loeper (’05), Peng, Wang and Yong (’06) Combined quasineutral limit and inviscid limit in T d smooth solutions, well- prepared initial data ,Wang (’04) weak solutions, general initial data, Jiang and Wang (’06) Quasineutral limit for Navier Stokes Poisson system regular solutions, ill-prepared data, Ju,Li and Li (’09) weak solutions, well-prepared data, Ju, Li and Wang (’08) weak solutions, D. Donatelli P.Marcati, A quasineutral type limit for the Navier Stokes Poisson system with large data, Nonlinearity, 21, (2008), 135-148. Singular Limits E. Feireisl, A. Novotny, Singular Limits in Thermodynamics of Viscous Fluids , Birkh¨ auser Verlag, 2009. L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation , Springer, 2009. 13

Existence for Navier Stokes Poisson ∂ t ρ λ + div( ρ λ u λ ) = 0 ( ρ λ ) γ = µ ∆ u λ +( µ + ν ) ∇ div u λ + ρ λ ∇ V λ ∂ t ( ρ λ u λ )+div( ρ λ u λ ⊗ u λ ) + ∇ λ 2 ∆ V λ = ρ λ − 1 Renormalized pressure: Total Energy: � � � π λ = ( ρ λ ) γ − 1 − γ ( ρ λ − 1) ρ λ | u λ | 2 + π λ + λ 2 2 |∇ V λ | 2 E ( t ) = dx γ ( γ − 1) 2 R 3 Initial conditions: � � � π λ | t =0 + | m ε 0 | 2 + λ 2 | V λ 0 | 2 ρ λ u λ | t =0 = m λ dx ≤ C 0 , where 0 . 2 ρ λ R 3 0 Existence of global weak solution (B. Ducomet, E. Feireisl, H. Petzeltov´ a, and I. Straˇ skraba, 2004) � t � � µ |∇ u λ | 2 +( ν + µ ) | div u λ | 2 � E ( t ) + dxds ≤ E (0) . R 3 0 14

Strategy Uniform bounds Strong convergence for P u Recover Acoustic equation and control oscillations Strichartz estimates Strong convergence for Q u Compactness for λ ∇ V λ introduction of correctors construction of microlocal defect measure 15

Uniform bounds � � � + ( ρ λ ) γ − 1 − γ ( ρ λ − 1) ρ λ | u λ | 2 + λ 2 |∇ V λ | 2 dx 2 γ ( γ − 1) R 3 � t � � µ |∇ u λ | 2 +( ν + µ ) | div u λ | 2 � + dxds ≤ C 0 . R 3 0 the convexity of z → z γ − 1 − γ ( z − 1) ⇓ density fluctuation = σ λ = ρ λ − 1 ∈ L ∞ t L k 2 , k = min { 2 , γ } = ⇒ ∇ u λ is bounded in L 2 λ ∇ V λ is bounded in L ∞ t L 2 t,x , x , u λ is bounded in L 2 t,x ∩ L 2 t L 6 σ λ u λ is bounded in L 2 t H − 1 x x 16

Leray Projectors u = P [ u ] + Q [ u ] ���� ���� soleinodal part gradient part div P [ u ] = 0 Q [ u ] = ∇ Ψ where Q [ u ] = ∇ ∆ − 1 div[ u ] P [ u ] = I − Q [ u ] 17

Convergence of P u λ convolution techniques L p compactness � P u λ ( t + h ) − P u λ ( t ) � L 2 ([0 ,T ] × R 3 ) ≤ C T h 1 / 5 ⇓ P u λ − strongly in L 2 (0 , T ; L 2 loc ( R 3 )) → P u, 18

Strategy Uniform bounds Strong convergence for P u Recover Acoustic equation and control oscillations Strichartz estimates Strong convergence for Q u Compactness for λ ∇ V λ introduction of correctors construction of microlocal defect measure 19