Asymptotic stability of boundary layers to the Euler-Poisson equation arising in plasma physics. Based on joint research with
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- ÏἈχαϊκός Μακρής
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1 Asymptotic stability of boundary layers to the Euler-Poisson equation arising in plasma physics (, Based on joint research with ( (
2 Contents 1. Physical background of the problem 2. Mathematical formulation of the problem 3. Review on previous results with emphasis on the unique existence of stationary solution & its stability in 1D 4. Main results Asymptotic stability of sheath in multi-dimensional space 5. Concluding remarks
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6 Bohm s Sheath Criterion For the sheath formation, physical observation requires the Bohm sheath criterion (BSC: u 2 + K + 1, u + < 0, (BSC u + : Ion s velocity component normal to wall around sheath edge K : Const. proportional to abs. temperature (= (Acoustic Velocity 2 (p(ρ = Kρ, K > 0, Isothermal Validate BSC from the mathematical point of view. Remark : (BSC Supersonic condition : u 2 + > K.
7 1. Mathematical formulation of the problem Euler-Poisson equations (dimension N = 1, 2, 3 ρ t + div (ρu = 0, (ρu t + div (ρu u + p(ρ = ρ φ, φ = ρ ρ e. (E.a (E.b (E.c t > 0 : Time variable x = (x 1, x = (x 1, x 2,..., x N : Space variables R N + := (0, RN 1 ρ = ρ(t, x > 0 : Ion density u = u(t, x R N : Ion velocity φ = φ(t, x R : Electrostatic potential ( 1 p(ρ = Kρ (K > 0 : Pressure (Isothermal ρ e = e φ > 0 (Boltzmann relation : Electron density = ( x1,..., xn, (u uij = u i u j, = ( 2 x x N. [Chen, Introduction to plasma physics, 77]
8 Initial data (ρ, u(0, x = (ρ 0, u 0 (x, ρ 0 (x > 0, inf x R N + (I.a lim x 1 (ρ 0, u 0 (x 1, x = (ρ +, u +, 0,..., 0 where ρ + (> 0, u + (< 0 are constants. x R N 1, (I.b Boundary data where φ b is constant. φ(t, 0, x = φ b, x R N 1 (B Reference point of potential lim x 1 φ(t, x 1, x = 0, x R N 1 (R To construct classical solution to (E.c: φ = ρ e φ, it must be that ρ + = 1. (A
9 Stationary problem We define sheath by a planar stationary solution ( ρ, ũ, φ(x 1 to (E independent of t, x 2,..., x N, ( ρũ x1 = 0, ( ρũ 2 + p( ρ = ρ φ x1, x 1 φ x1 x 1 = ρ e φ, (S.a (S.b (S.c with conditions (I.b, (B, (R, (A inf ρ(x 1 > 0, x 1 R + lim x 1 ( ρ, ũ, φ(x 1 = (ρ +, u +, 0, φ(0 = φ b. Question 1. When does a stationary solution exist? 2. Is the stationary solution asymptotically stable? 3. What is the convergence rate?
10 Derivation of the Bohm criterion in physical textbooks In the stationary problem, by integrating (E.a, (E.b and (E.c over (x, in spatial coordinate, we have 1 2 φ (x 2 = u 2 e v u = u +, (GB.a 2 + Kv φ = u2 + 2, (GB.b x φ (ye v(y dy + e φ(x 1. Given φ(x, solve for v(x(< 0 using (GB.a, (GB.b. (GB.c Substitute v(φ in (GB.c. RHS of (GB.c must be positive everywhere. u + < K + 1 is necessary.
11 Derivation of ion acoustic wave speed Rewrite (E.a and (E.b by deviding by ρ. (dimension N = 1 v t + uv x + u x = 0, (E.a u t + uu x + Kv x φ x = 0, (E.b φ xx = e v e φ. (E.c Under quasi-neutrality assumption, drop (E.c, replace φ with v. v t + uv x + u x = 0, (E.a u t + uu x + (K + 1v x = 0. (E.b Characteristic of this system are u ± K + 1. K + 1 is the phase velocity of the wave supported in the linearized system of (E.a and (E.b. This wave is called the ion acoustic wave.
12 2. Previous results on asymptotic analysis (a Results over bounded domain (0, 1 [A. Ambroso, F. Méhats, P.-A. Raviart, AA 01] shows existence of stationary solution under (BSC. [A. Ambroso M3AS 06] numerically shows stability of stationary solution. (b Results over R + [M. Suzuki 10] shows existence of monotone stationary solution over R + and its stability.
13 Results about stationary solution [M.Suzuki] Theorem 1 (Existence of monotone stationary solution i Let u 2 + K or K + 1 = u2 + or K + 1 < u2 +. φ b f( u + / K, V (φ b 0 Monotone stationary sol exists. Moreover, assume monotonicity unique. ii Let K < u 2 + < K + 1. Non-trivial stationary solution does NOT exist. f( ρ := K log ρ + u2 + 2 ρ 2 u2 φ + 2, V ( φ := [f 1 (η e η ] dη. 0 Remark (A: u 2 + > K + 1 & φ b 1 or (B: u 2 + = K + 1 & φ b 0 gives sufficiency for the existence of sheath. ( Sagdeev potential Corollary 2 (Property of sheath (A j x 1 ( ρ ρ + (x 1 + j x 1 (ũ u + (x 1 + j x 1 φ(x 1 C φ b e cx 1
14 Derive conditions for the existence of stationary solution x (S.b/ ρ dx 1, lim x 1 ( ρ, ũ(x 1 = (1, u + φ = f( ρ, f( ρ := K log ρ + u2 + 2 ρ 2 u O M + (1, 0 ρ f(ρ takes its minimum at M + := u + / K. φ b f(m +. M + is the Mach number. Take the inverse by choosing branch containing asymptotics (1,0 ρ = f 1 ( φ Substitute this into (S.c, (S.c φ x1 dx 1 x φ 2 φ x1 = 2V ( φ, V ( φ := [f 1 (η e η ] dη. V (φ b 0. 0 V ( φ is the Sagdeev potential.
15 Graph of V V (φ V (φ φ f(m + φ f(m + φ f(m + V (φ (i u 2 + K (ii K < u2 + < K + 1 (iii u2 + = K + 1 V (φ V (φ f(m + φ f(m + φ (iv K + 1 < u 2 + ω (v ω < u2 + φ x1 2 = 2V ( φ, φ(0 = φ b, lim x 1 φ(x 1 = 0, V is Lipschitz continuous over [0, φb ]. φ b f( u + / K and V (φ b 0 gives sufficient condition for the existence of a monotone stationary solution.
16 Result about asymptotic stability over R + (exponential weight [M.Suzuki] Perturbation (ψ, η, σ(t, x := (log ρ, u, φ(t, x (log ρ, ũ, φ(x 1. Theorem 3 (Asymptotic stability of sheath u + < 0, u 2 + > (K + 1/ 2 2 /K. If (e x 1/2 ψ 0, e x 1/2 η 0 H 2 (R + then ɛ, C, γ > 0(Const. s.t. φ b + (e x 1/2 ψ 0, e x 1/2 η 0 2 ɛ 1 Time global solution (ψ, η, σ e x 1/2 ψ, e x 1/2 η 2 i=0 C i ( [0, ; H 2 i (R +, e x 1/2 σ 2 i=0 C i ( [0, ; H 4 i (R +, and (e x 1/2 ψ, e x 1/2 η(t ex 1/2 σ(t 2 4 C (ex 1/2 ψ 0, e x 1/2 η e γt. i := H i : H i Sobolev norm. Remark: Condition (1 u + < 0, u 2 + > K + 1.
17 3. Asymptotic stability of sheath in multi-dimensions Study asymptotic stability of sheath in multi-dimensional space under exactly (BSC Stationary solution Consider 1D sheath is embedded in multi-dimensional space as ṽ(x = ṽ(x 1, ũ(x = (ũ(x 1, 0,..., 0, φ(x = φ(x 1. (x R N + Hereafter, we simply write (ṽ, ũ, φ by (ṽ, ũ, φ Perturbation (v, ṽ := (log ρ, log ρ, (ψ, η, σ(t, x := (v, u, φ(t, x 1, x (ṽ, ũ, φ(x 1.
18 Perturbation (ψ, η, σ satisfies equations ( ψ η t + N j=1 M j ( ψ η M j (u j := u j I N+1 + x j = η 1 ( ṽ ũ x 1 + σ = e ψ+ṽ eṽ e (σ+ φ + e φ K ĵ + 1 with initial and boundary data to (P < j + 1 ( 0 σ, (P.a ( matrix of size (N+1 (P.b (ψ, η(0, x = (ψ 0, η 0 (x := (log ρ 0 log ρ, u 0 ũ, σ(t, 0, x = 0, lim x 1 (ψ 0, η 0 (x = (0, 0, lim x 1 σ(t, x 1, x = 0, x R N 1. (PI (PB
19 In case perturbation is small ((BSC supersonic In x 1 direction, characteristics of hyperbolic equations (P.a are λ 1 := u 1 K < 0, λ 2 := u 1 + K < 0, λ i := u 1 < 0 (i = 3,..., N + 1. ( u1 = u + + (ũ u + + η 1, (BSC : u + K + 1 For hyperbolic equation (P.a, no boundary condition is necessary. For elliptic equation (P.b, one boundary condition is necessary. x 1 t x 1 = 0 Well-posed with 1 boundary condition (PB, λ i (i = 3,..., N + 1 λ 1 λ 2 σ(t, 0, x = 0, x R N 1.
20 Difficulty to show asymptotic stability System of linearized equations of (P around asymptotic state (ρ, u, φ = (ρ +, u +, 0,..., 0, 0 is ( ψ η t + N j=1 M + j ( ψ η x j = ( 0 σ M + j = M j (u + (j = 1,..., N, σ = ψ + σ. (L Spectrums of (L are given by ( µ(iξ = i ξ 1 u + ± ξ K ξ 2, iξ 1 u + (N-1 multiple ξ R N. Real parts of all spectrums are ZERO. To resolve this difficulty, we employ weighted energy method. All characteristics go into boundary. Decay of (ψ 0, η 0 as x 1 convergence of solution towards stationary solution as t.
21 Introduce new variables (Ψ, H, Σ := (e βx 1/2 ψ, e βx 1/2 η, e βx 1/2 σ. Rewrite systems of equation (P w.r.t. (Ψ, H, Σ (P. Linearize (P around asymptotic state (ρ, u, φ = (ρ +, u +, 0,..., 0, 0 (L. Spectrums of (L are given by µ(iξ = βu ( i ξ 1 u + ± Kζ 1ζ + 1 K, (ζ = 1 + ξ 2 β2 4 + iβξ 1 βu + 2 iξ 1u + (N-1 multiple for ξ R N. Linearly Stable ( sup Re (µ(iξ < 0 u 2 ξ R N + > K β 2 /4. ( sup ξ R N Re (µ(iξ = Re (µ(0 If u 2 + > K + 1, setting β 1 ensures (.
22 Main result (with exponential weight [S.Nishibata, M.O., M.Suzuki] Theorem 4 (Asymptotic stability of sheath (N, m = (1, 2, (2, 3, (3, 3. u + < 0, u 2 + > K + 1, K > 0. If (e λx1/2 ψ 0, e λx1/2 η 0 H m (R N + then δ > 0 s.t. β (0, λ], β + ( φ b + (e βx1/2 ψ 0, e βx1/2 η 0 m /β δ 1 Time global solution (ψ, η, σ m ( e βx1/2 ψ, e βx1/2 η C i [0, ; H m i (R N +, e βx1/2 σ i=0 m i=0 ( C i [0, ; H m+2 i (R N +. C > 0, γ (0, β] s.t. (e βx 1/2 ψ, e βx 1/2 η(t m + e βx 1/2 σ(t m+2 C (e βx 1/2 ψ 0, e βx 1/2 η 0 m e γt.
23 Outline of proof (for N = 2, 3 (Local existence + (A-priori estimate (Global existence Lemma 5 (Local existence (e λx 1/2 ψ 0, e λx 1/2 η 0 H 3 (R N + with sup {η 0 + ũ + K} < 0, sup ψ 0 + φ b 1. x R N + x R N + α, T > 0, s.t., 1 solution (ψ, η, σ s.t. (e αx 1/2 ψ, e αx 1/2 η C([0, T ]; H 3 (R N +, eαx 1/2 σ C([0, T ]; H 5 (R N +. N e αx 1 (T := sup 0 t T (e αx 1/2 ψ, e αx 1/2 η(t 3. Proposition 6 (A-priori estimate β (0, α], β + (N e αx 1 (T + φ b /β 1 C, γ > 0 ( e γt e βx1/2 (ψ, η(t eβx1/2 σ(t t 0 e γτ ( e βx 1/2 (ψ, η(τ eβx 1/2 σ(τ 2 5 s.t. dτ C e βx 1/2 (ψ 0, η 0 2 3
24 Main result (algebraic weight [S.Nishibata, M.O., M.Suzuki] w λ,α := (1 + αx 1 λ for λ > 0, α > 0 Theorem 7 (Asymptotic stability of sheath (N, m = (1, 2, (2, 3, (3, 3. u + < 0, u 2 + > K + 1, K > 0. If (w λ/2,β ψ 0, w λ/2,β η 0 H m (R N + for λ 2, β > 0, then α (0, λ] δ = δ(α > 0 s.t. if β + ( φ b + (w λ/2,β ψ 0, w λ/2,β η 0 m /β δ 1 Time global solution (ψ, η, σ s.t. m w α/2,β ψ, w α/2,β η C i ([0, ; H m i (R N +, w α/2,β σ i=0 m i=0 C i ([0, ; H m+2 i (R N + and C(α > 0 s.t. (w α/2,β ψ, w α/2,β η(t 2 m + w α/2,β σ(t 2 m+2 C (w λ/2,β ψ 0, w λ/2,β η 0 2 m (1 + βt (λ α.
25 Degenerate case : u 2 + = K + 1, u + < 0 Proposition 8 (Decay rate of stationary sol. in the degenerate case Under the degenerate condition, δ 0 > 0, C = C(δ 0 > 0 s.t. x 1 0 and φ b (0, δ 0 ] i x 1 φ(x 1 G(x i+2 c i Cφ b, i x 1 ( ρ(x1 ρ + G(x i+2 + c i Cφ b, i x 1 (ũ(x1 u + /u+ G(x i+2 + c i Cφ b, for i = 0, 1, 2, 3..., where G(x := Γ x 1 + φ 1/2 b, Γ := (K + 1/6 and c 0 := 1, c 1 := 2Γ, c 2 := K + 1, c 3 := 4Γ (K + 1,...
26 Main result (degenerate case [S.Nishibata, M.O., M.Suzuki] Theorem 9 (Asymptotic stability of sheath (N, m = (1, 2, (2, 3, (3, 3. u + < 0, u 2 + = K + 1, K > 0. Let λ 0 R satisfy λ 0 (λ 0 1(λ (λ = 0, λ [4, λ 0. α (0, λ], θ (0, 1], δ = δ(α, θ > 0 s.t. if φ b (0, δ], β/γ φ 1/2 [θ, 1], (w λ/2,β ψ 0, w λ/2,β η 0 H m (R N + and (w λ/2,γ ψ 0, w λ/2,γ η 0 m /β 3 δ, b 1 Time global solution (ψ, η, σ m w α/2,β ψ, w α/2,β η C i ([0, ; H m i (R N +, w α/2,β σ i=0 m i=0 C i ([0, ; H m+2 i (R N + and C(α, θ > 0 s.t. (w α/2,β ψ, w α/2,β η(t 2 m + w α/2,β σ(t 2 m+2 C (w λ/2,β ψ 0, w λ/2,β η 0 2 m (1 + βt (λ α/3.
27 5. Concluding Remarks u 2 + K + 1 (BSC Existence of stationary solution, NOT unique Monotone stationary solution is unique. Monotone stationary solution is time asymptotically stable Spectrum analysis supports { (BSC Linearly stable. Otherwise Linearly unstable. (BSC may be a necessary condition for stability. We call monotone stationary solution with (BSC sheath.
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