Global existence for the ion dynamics in the Euler-Poisson equations

Transcription

1 Global existence for the ion dynamics in the Euler-Poisson equations Yan Guo (Brown U), Benoît Pausader (Brown U). FRG Meeting May 2010

2 Abstract We prove global existence for solutions of the Euler-Poisson/Ion dynamics equation t n + (nv) = 0 n ( t v + v v) = n n φ φ = 4π (n 0 exp (φ) n). in R R 3 which are small perturbation of a constant background. This is a joint work with Y. Guo.

3 Modeling a plasma In order to model a plasma composed with 2 species of ions and electrons of charge ±e, with collision parameter κ, For κ large, the ions and electrons form only one fluid and one considers the Magneto-hydrodynamics equation. At medium κ, the dynamics of the ions and electrons decouple due to their very different masses, and one uses a 2 fluids (compressible Euler-Maxwell isothermal) model. At small κ, one switches to a kinetic model, and, neglecting the collisions, one studies the Vlasov-Maxwell equations. We focus now on the intermediate case.

4 The 2 fluids model The 2 fluid equations are t n ± + (n ± v ± ) = 0 n ± m ± ( t v ± + v ± v ± ) + T ± n ± = en ± φ φ = 4πe(n + n ). (1) Here n ±, v ±, m ±, T ± are the ion (+) and electron density, velocity, masses and effective temperatures. Besides, E = φ is the electric field e is the charge of an electron. Note that we have neglected the magnetic field (possible if the initial data are irrotational, or if the field is small).

5 Observations The Zakharov equations describe (at first order) the behavior of some particular solutions of the above Euler-Maxwell system. In general, solutions of the compressible Euler equations will develop shocks (even with small smooth initial data, cf Sideris, 85). However it is possible that the presence of a feed-back due to coupling with a field might stabilize the system.

6 The EP/electron equation The electrons have much smaller mass, hence, as a first approximation, one can look at the fast-time dynamics and treat the ion as a fixed background. One then obtains the EP/electron equation t n + (n v ) = 0 n m ( t v + v v ) + T n = en φ φ = 4πe(n n 0 ). Global existence for small perturbation (Guo, 98) Irrotational initial data (n, v ) = (n 0 + ερ, εv) lead to global solutions. Hence in this case, when one turns on an electric field, the system is stabilized.

7 Key observation: in the case of pure Euler, the linearization around a fixed background satisfies the pure wave equation: tt ρ ρ = 0. For this, linear solutions decay like t 1 and have many quadratic resonances. whereas for the electron equation, the linearization yields a Klein Gordon equation: tt ρ ρ + ωeρ 2 4πe = 0, ω e = 2 n 0 m where ω e is the electron plasma frequency. For KG, solutions decay faster t 3 2 and have no quadratic resonances (hence a normal form transformation eliminates the quadratic terms).

8 The ion equation In this work, we focus on the other extreme dynamics: the long time dynamics. In this case, the electron move so fast that they are constantly at the Boltzman equilibrium state: n = n 0 exp( eφ T ) (also obtained by formally setting m /m + = 0 in the 2 fluid equation). In this case, the system is defined by the ion equation interacting with their self-consistent field (EP/ion) t n + + (n + v + ) = 0 n + m + ( t v + + v + v + ) = T + n + n + e φ ( ( ) ) eφ φ = 4πe n 0 exp n +. T

9 Previous results This equation was studied in several context before. Cordier and Grenier, Peng and Wang, studied the quasi-neutral limit. Feldman, Ha, Slemrod studied the plasma sheath interface. Guo and Tahvildar-Zadeh prove formation of shocks for large amplitude solutions. Other works: Holm,Johnson and Lonngren, Liu and Tadmore, Peng and Wang, Perthame, Texier, Wang and Wang...

10 Main result Global existence for the ion equation Suppose that curlv 0 = 0 and that (ρ, v) is small enough in W 8, 10 9 H 10, then the solution with initial data exists globally and scatters in L 2. (n, v ) = (1 + ρ, v) Remark: the regularity is not optimal. Here again, the electric field stabilize the system. Now, the two extreme dynamics for the 2-fluids system are controlled.

11 Main difficulties-1 Here, the linearized operator yields ( tt ρ + 1 ) ρ = 0 which is really close to the wave equation: the dispersion relation is 2 + k ω(k) = ± k k 2 = ±p(k). This has 2 stationary points where ω = 0: 1 < r 0 < 3 and. Fortunately, this still retains some curvature in the radial direction and the linear solutions decay like t 4 3, i.e. faster than t 1 and are integrable in time.

12 Main difficulties-2 This is strongly resonant (especially around 0 where it is almost the wave equation). We overcome this by careful estimate of the resonance region {(ξ, η) : Φ = ω(ξ) ± ω(ξ η) ± ω(η) = 0} = { ξ ξ η η = 0} and Φ vanishes at 0 at second order. Normal form transformation produces bilinear terms with singular multipliers (not in L ξ H 3 2 η, in particular, not of Coifman-Meyer type), and we need to deal with singular operators with singular kernels.

13 Outline of the proof 1 we first study the linear flow (to get good decay rate) The we turn to the nonlinear term and we bound the norm } ρ X = sup { 1 ρ(t) H 2k+1 + (1 + t) ρ(t) W k,10. t 2 we control L 2 -scale bounds using standard energy estimates In order to control L 10 -like norm, 3 we perform a normal form transformation (as in Shatah, Gustafson, Nakanishi and Tsai, Germain, Masmoudi and Shatah) 4 we study the singular multipliers coming from the transformation.

14 The linear equation-1 To study the linear equation, we need to estimate the half-wave propagator e itω( ) 2 + ξ, ω(ξ) = ξ ξ 2 This operator has a wave-like degeneracy at 0, and 2 stationary points (ω = 0): ξ stat = and. We can counter the degeneracies at 0 and by paying some derivatives: 1 2 e itω( ) P 0 δ L (1 + t ) 3 2 e itω( ) P N δ L N 5 2 t 3 2, N > 10

15 The linear equation-2 We need to have a decay faster than t 1. Since our phase ω is radial, the stationary phase one the sphere automatically gives us a decay like t 1. To get better, we exploit the curvature in the radial direction. At the finite stationary point, we use the fact that ω (3) ( ξ stat ) 0 and only get a slower decay: Finally, we get decay estimates e itω( ) P stat δ L (1 + t ) e itω( ) f L 10 (1 + t ) f W 12 5, 10 9.

16 Reduction of the system-1 We want to consider the system t ρ + div(v) + div(ρv) = 0 ] t v + ρ + φ + (v )v ρ2 = [ln(1 + ρ) ρ + ρ2 2 2 ] ρ = (1 )φ + φ2 + [e φ 1 φ φ The last line defines an operator ρ φ(ρ) with φ(ρ) = (1 ) 1 ρ 1 2 (1 ) 1 [ (1 ) 1 ρ ] 2 + O(ρ 3 ). and since v = 0, the second equation reduces to a scalar equation.

17 Reduction of the system-2 Diagonalizing the linear part and introducing the eigenvector α = ρ i 2 + ξ q( ) R 1 v, q(ξ) = ξ 2 we are left with the equation where ( t iω( ))α = Q(α) + N Q = div(ρv) i { (1 ) 1 [(1 ) 1 ρ] 2 ρ 2 v 2} 2q( ) N = i ] [ln(1 + ρ) ρ + ρ2 q( ) 2 R(ρ).

18 The normal form transformation-1 It suffices to consider the quadratic terms. To bound them, we use a normal form transformation. First, we conjugate by the linear flow: consider β(t) = e itω( ) α(t) Then, if one only takes into account only the linear terms, we get t β = 0

19 The normal form transformation-1 It suffices to consider the quadratic terms. To bound them, we use a normal form transformation. First, we conjugate by the linear flow: consider β(t) = e itω( ) α(t) including the nonlinear terms, we get t β = F(β), F(β) = e itω( ) F (e itω( ) β) key point: no linear point, deriving in time gains nonlinear terms: t β = O(β 2 ).

20 The normal form transformation-2 Using that We write the Duhamel formula t t β = e iω( ) Q(α), β(t) = β(0) + F(β(t s))ds 0 t = β(0) + F 1 ξ e isφ m(ξ, η) ˆβ(ξ η) ˆβ(η)dη 0 η R 3 for m(ξ, η) ξ n 1 (ξ)n 2 (ξ η)n 3 (η) a multiplier and the phase Φ = ω(ξ) ± ω(ξ η) ± ω(η). Integrating by parts in s, we obtain β(t) B[β, β] = β 0 B[β, β](0) + 2F 1 ξ t η R3 e isφ And now the integrand is cubic. 0 iφ m(ξ, η) ˆβ(ξ η) t ˆβ(η)dη.

21 The normal form transformation-3 Finally, going back to α, we get ˆα(t) + F ξ B[α(t), α(t)] = e itω(ξ) (ˆα 0 + F ξ B[α 0, α 0 ]) t + e i(t s)ω(ξ) m 1 (ξ, η)m 2 (η, ζ) ˆα(s, ξ η)ˆα(s, η ζ)ˆα(s, ζ) 0 R 6 Φ(ξ, η) dζdηds where m(ξ, η) FB[α, α] = ˆα(ξ η)ˆα(η)dη. R 3 Φ(ξ, η) is a bilinear form with singular multiplier m Φ and m(ξ, η) ξ.

22 The H 1 -norm Around 0, we have that Φ = ω(ξ) ω(ξ η) ω(η) ξ ξ η η, so in the cubic term, the multiplier (m(ξ, η) ξ ) m 1 (ξ, η)m 2 (η, ζ) Φ(ξ, η) is not even bounded. To account for that, we rewrite m 1 (ξ, η)m 2 (η, ζ) ˆα(s, ξ η)ˆα(s, η ζ)ˆα(s, ζ) Φ(ξ, η) ξ η ξ η ˆα(s, ξ η) ˆα(s, η ζ)ˆα(s, ζ) Φ(ξ, η) ξ η and look for a control of α in Ḣ 1 (possible since in the nonlinearity, all the terms have derivative- null structure).

23 Boundedness for bilinear operator with singular multiplier In order to control the various bilinear pseudo product operators, we use the following result from Gustafson, Nakanishi and Tsai. Consider the operator B[f, g] = F 1 ξ m(ξ, η)ˆf (ξ η)ĝ(η)dη R 3 The Coifman-Meyer theorem is too restrictive for our multiplier because they are not smooth it requires an homogeneous behavior (control on α ξ β η m ξ α η β is not enough (Grafakos and Kalton))

24 Boundedness for bilinear operator with singular multiplier In order to control the various bilinear pseudo product operators, we use the following result from Gustafson, Nakanishi and Tsai. Consider the operator B[f, g] = F 1 ξ m(ξ, η)ˆf (ξ η)ĝ(η)dη R 3 and the norm m M s ξ,η = N 2 Z P η N m(ξ, η) L ξ Ḣs η Boundedness for multiplier operators Assume 0 s 3/2, 2 p, r 2n/(n 2s) B[f, g] L r m M s η,ξ f L 2 g L p.

25 It remains to estimate the phases Φ 1 (ξ, ξ η, η) = p(ξ) p(ξ η) p(η) Φ 2 (ξ, ξ η, η) = p(ξ) + p(ξ η) + p(η) Φ 3 (ξ, ξ η, η) = p(ξ) p(ξ η) + p(η) Φ 4 (ξ, ξ η, η) = p(ξ) + p(ξ η) p(η). We remark that if H = max( ξ, ξ η, η ) L is the min, and γ is the angle between H and L, then [ H 2 ] Φ L (1 cos γ) + (1 + H 2 )(1 + L 2 ) and we get that M = ξ ξ η η Φ ξ η 9 4 η 9 4 M 5 4 ε η,ξ M 5 4 ε ξ,η

26 Control on the cubic term Using this, we can control the quadratic and singular cubic terms in the equation. F 1 e i(t s)ω(ξ) m 1(ξ, η)m 2 (η, ζ) ˆα(ξ η)ˆα(η ζ)ˆα(ζ) Φ W k,10 t 0 F 1 ξ s,η,ζ 1 (1 + t s) m 1 (ξ, η)m 2 (η, ζ) R 6 Φ α 3 16 X (1 + t) 15 ˆα(s, ξ η)ˆα(η ζ)ˆα(ζ)dηdζ W k+ 12 5, 10 9 ds

27 Control on the quadratic term The quadratic/integrated term leads to the big loss of derivatives B[α, α] W k,10 F R 3 ξ ξ η η ξ η r η r Φ ξ η r ˆα(ξ η) ξ η (1 )2 M M 3 α 2 ε W k+ 11 α 5 +,p L q ξ,η α X α W k,10 η r ˆα(η) dη η W k+2,2 with 1 p + 1 q = ε. This is more problematic if the first term is at high frequencies and the second at low frequencies.

28 The 2 fluid system-linear part-1 The linearization of the system yields with ε = m e /m i and T = T i /T e : t n n h ρ + ε (T + ( ) 1 ) 0 1 ε 0 h ρ g 1 0 (1 + ( ) 1 ) 0 g ( ) We set y 1 = ε(t +( ) 1 1 ( ) 4 n 1+( ) 1 e i ε R 1 v T +( ) 1 e ), z 1 = ρ i i R 1 v 1+( ) 1 i, y 2 = w 1, z 2 = z 1,

29 The 2 fluid system-linear part-2 writing A = (y 1, y 2, z 1, z 2 ), we need to investigate the following equation t A + ima = 0 with H ε 0 L L M = 0 H ε L L L L H 1 0 L L 0 H 1 with T + ( ) 1 1 T H ε = = ε ε H 1 = 1 + ( ) 1 = L = =. [ε(t + ( ) 1 )(1 + ( ) 1 )] 1 4 [ε(1 T )(1 )] 1 4

30 The 2 fluid system-linear part-2 The eigenvalues are now given by ( ) ε (1 + T ε Λ 1 = ) + 1 ε ((1 ε) (T ε) ) 2 + 4ε 2 = 1 ( )) ε 1 T (1 + O ε 1 ( ) ε (1 + T ε Λ 3 = ) 1 ε ((1 ε) (T ε) ) 2 + 4ε 2 = ( ( )) T + 1 T ε 1 + O 1 T 1 Λ 2 = Λ 1 and Λ 4 = Λ 3