Small BGK waves and nonlinear Landau damping (higher dimensions)

Transcription

1 Small BGK waves and nonlinear Landau damping higher dimensions Zhiwu Lin and Chongchun Zeng School of Mathematics Georgia Institute of Technology Atlanta, GA, USA Abstract Consider Vlasov-Poisson system with a fied ion background and periodic boundary conditions on the space variables, in dimension d =,. First, we show that for general homogeneous equilibrium and any periodic bo, within any small neighborhood in the Sobolev space W,v s,p p >, s < + of the steady distribution function, there eist nontrivial travelling wave solutions BGK waves with arbitrary trav- p eling speed. This implies that nonlinear Landau damping is not true in W s s,p < + space for any homogeneous equilibria and in any period p bo. The BGK waves constructed are one dimensional, that is, depending only on one space variable. Higher dimensional BGK waves are shown to not eist. Second, we prove that there eist no nontrivial invariant structures in some neighborhood of stable homogeneous equilibria, in the + v b -weighted H s,v b > d, s > 4 space. Since arbitrarilly small BGK waves can also be constructed near any homogeneous equilibria in such weighted H,v s s < norm, this shows that s = is the critical regularity for the eistence of nontrivial invariant structures near stable homogeneous equilibria. These generalize our previous results in the one dimensional case. Introduction Consider a collisionless electron plasma with a fied homogeneous neutralizing ion background. The Vlasov-Poisson system in d dimension is t f + v f E v f =, E = φ, φ = f dv +, d a b where f t,, v is the distribution function, E, t is the electrical field and φ, t is the electrical potential. We consider the Vlasov-Poisson system

2 in a periodic bo, with periods T i in i. In 946, Landau [6], looking for analytical solutions of the linearized Vlasov-Poisson system around Mawellian e v,, pointed out that the electric field is subject to time decay even in the absence of collisions. The effect of this Landau damping, as it is subsequently called, plays a fundamental role in the study of plasma physics. However, Landau s treatment is in the linear regime; that is, only for infinitesimally small initial perturbations. ecently, nonlinear Landau damping was shown [] for analytical perturbations of stable equilibria with linear eponential decay. For general perturbations in Sobolev spaces, the proof of nonlinear damping remains open. We refer to [9] [] for more discussions and references on this topic. In [9], the following results were obtained for D Vlasov-Poisson system. First, we show that for general homogeneous equilibria, within any small neighborhood in the Sobolev space W p s,p >, s < + p of the steady distribution function, there eist nontrivial travelling wave solutions BGK waves with arbitrary minimal period and traveling speed. This implies that nonlinear Landau damping is not true in W s s,p < + p space for any homogeneous equilibria and any spatial period. Second, it is shown that for homogeneous equilibria satisfying Penrose s linear stability condition, there eist no nontrivial travelling BGK waves and unstable homogeneous states in some W p s,p >, s > + p neighborhood. Furthermore, we prove that there eist no nontrivial invariant structures in the H s s > neighborhood of stable homogeneous states. In particular, these results suggest the contrasting long time dynamics in the H s s > and H s s < neighborhoods of a stable homogeneous state. In this paper, we generalize above results to higher dimensions d =,. Denote the fractional order Sobolev spaces by W s,p d or W s,p,v, T d with p >, s. These spaces are the comple interpolation of of L p space and Sobolev spaces W m,p m positive integer. Our first result is to construct D BGK waves in W s,p,v s < + p spaces. Theorem. Assume the homogeneous distribution function f v W s,p d d, p >, s [, + p satisfies f v, f v dv =, v f v dv < +. Fi T > and c. Then for any ε >, there eist travelling BGK wave solutions of the form f = f ε ct, v, E = Eε ct e to??, such that f ε, v, E ε has minimal period T in, f ε, v, E ε is not identically zero, and f ε f L,v + T d v f ε f d dv + f ε f W s,p,v < ε.

3 In Proposition., we show that there eist no D and D BGK waves. Therefore, the form of D BGK waves in Theorem. is in some sense necessary. For any b > d 4, we denote Hs,b v d to be the + v b weighted H s space, that is, Hv s,b d { = f + v } b s f <. L d and f H s,b v = + v b s f L d. Let T d be a periodic bo with periods T i in i i =,, d, and { π Z d = j,, π } j d j,, j d are integers. T T d We define the space H s Hv sv,b T d d by h = e i k h k v H s Hv sv,b if h H s Hv k Z d = h Hv + k Z d k s h k Hv <. Given a homogeneous profile f v H s,b d d, s >, b > d 4, we have the following generalization of Penrose s stability condition in higher dimensions k ma v i S k/ k f, k/ k α α v i dα >, for any k Z d, 4 where f, k/ k α is the projection of f v to k direction, as defined by and S k/ k is the set of all critical points of f, k/ k α. By Corollary., one only need to check the condition 4 for finitely many k Z d with k C d, s, b f H s,b d. In particular, for a single humped profile f v = µ v with µ e <, the stability condition 4 is satisfied for any period set T,, T d. The following Theorem ecludes any nontrivial invariant structures steady, time periodic, quasi-periodic etc. near stable homogeneous equilibria satisfying 4, in the H s Hv sv,b spaces of high v regularity.

4 Theorem. Consider a homogeneous profile f v H s,b d d, s >, b > d, 5 4 with f v and f v dv =. Let T d be a periodic bo with periods T i in i i =,, d. Assume that f v satisfies the Penrose stability condition 4 for T,, T d. Let f, v, t, E, v, t be the solution of?? in T d. For any s, s v satisfying there eists ε >, such that if s, s > d, and < s v s, 6 f t f H s then E t for all t. Hv < ε, for all t, 7 In the above Theorem, the assumption s > d, s is to make H s an algebra which is needed in the proof of Lemma.. The use of weighted Sobolev space Hv s,b in Theorem. is rather natural in higher dimensions. Indeed, even to state the Penrose s stability condition 4, we need to assume that the homogeneous equilibrium f v H s,b d with s, b satisfying 5. This is because that linear instability stability of homogeneous equilibria of Vlasov- Poisson is longitudinal along the wave direction of perturbation. The weighted Sobolev space 5 is needed to ensure that the projected steady distribution function in any wave direction is in H s s > which is necessary to get the D Penrose stability criterion. Moreover, in Theorem. we also construct D BGK waves arbitrarily near any homogeneous equilibrium in H s Hv sv,b d, b > d 4 for any s > and s v <. Combined with Theorem., this shows that for weighted Sobolev spaces H s Hv sv,b, the critical v regularity for the eistence of nontrivial invariant structures near a stable homogeneous equilibrium is s v =. This generalizes our D results in [9] to higher dimensions. We note that the critical regularity s v = does not depend on the dimension. This illustrates again the longitudinal D nature of Landau damping, which is obvious in the linear regime. We briefly mention some differences of the long time behaviors of Vlasov- Poisson in D and higher dimensions. For the D case, numerical simulations e.g. [4] [] indicated that for certain small initial data near a stable homogeneous state including Mawellian, there is no decay of electric fields and the asymptotic state is a BGK wave or the superposition of BGK waves. Moreover, BGK waves also appear as the asymptotic states for the saturation of an unstable homogeneous state [] in D. These suggest that small BGK waves play an important role in understanding the long time behaviors of D Vlasov-Poisson system. However, for D and D Vlasov-Poisson system, numerical simulations [] [] suggested that when starting near a homogeneous state both stable 4

5 and unstable, the electric fields decay eventually. Our Theorems. and. on eistence of D BGK waves show that such decay of electric field is not true for general initial data near homogeneous states. But the numerical simulations seem to suggest that these D BGK waves do not appear in the long time dynamics in D and D cases. To eplain these phenomena, it will be important to understand the transversal instability of D BGK waves. This paper is organized as follows. In Section, we prove the eistence of D BGK waves in W s s,p < + p neighborhoods of homogeneous states. In Section, we use the linear decay estimate to show that all invariant structures near stable homogeneous equilibria in H s Hv sv,b spaces satisfying 6 are trivial. Throughout this paper, we use C to denote a generic constant in the estimates and the dependence of C is indicated only when it matters in the proof. Eistence of BGK waves in W s,p s < + p In this Section, we construct nontrivial steady states BGK waves near any homogeneous state in the space W,v s,p s < + p. We consider d = only, since the proof is almost the same for d =. The BGK waves we construct are one-dimensional, that is, the steady distribution f = f, v, v and the electric field E = E e. We will show that such a restriction is necessary by ecluding D and D BGK waves. Proof of Theorem.. We adapt the line of proof in [9] to construct BGK wave solutions for D Vlasov-Poisson equations. First, we modify f v to a smooth function f v with some additional properties. Let η v v be the standard mollifier function. For > define f v = η v f v, where η v = η and when is small enough v. Then by the properties of mollifiers, we have f C, f v, f v dv =, f f L + v f f dv + f f W s,p ε 6. Modifying f v near infinity by cut-off, we can assume in addition that f v H,b defined in. In the second step, let σ = σ be the D cut-off function. Let > be a small number, and define v f v, v + f v, v v f, v, v = f v, v σ + σ f v, v f v, v v = f v, v σ. 5

6 Then, f, C, f, v >, f, v dv = f v dv =, and f, v, v is even in v when v [, ]. We show that: when is small enough f, f L + v f, f dv+ f, f W s,p ε 6. 8 Fist, a minor modification of the proof of Lemma. in [9] yields that: when, f f, L + v f f, dv + f f, W,p. It remains to show that f f, W s,p, when. 9 We have v f v, v v f v, v v f f, = σ v, and v f v, v + v f v, v v f f, = σ + σ v v f v, v f v, v. v v By a scaling argument as in the proof of Lemma. of [9], f v, v f v, v v C f W s,p. W s,p So 9 follows from Lemma. below. Thus for any fied ε >, by choosing, small enough, we get f, f L + v f, f dv + f, f W s,p ε. We set f v, v = f, v, v, then f v >, f v C H,b, f v dv =, f v is even for v in [, ] and within ε distance of f v in the norm of. Below, we denote a = /. 6

7 Fi the period T >, we only consider the travel speed c = since the construction for any c follows by the Galilean transform as in [9]. Our strategy is to construct BGK wave solutions of the form f ε, v, v, E ε e by bifurcation at a modified homogeneous profile near f v, v. Denote σ = σ to be the cut-off function such that σ C, σ ; σ = when ; σ = when. Similar to Lemma. in [9], there eists g, C, g = when 4a, such that v f v, v σ = g v a, v. Define g +,, g, C by f ±, σ a g ±, = Then + g, if > a g, if 4a < a. if 4a { g+ v f v, v =, v g v, v if v > if v. Since v f, v =, f C H,b, we have v f v dv v <. Indeed, let f v = f v, v dv, then since f =, by Corollary., v f v dv v = f v dv v C f H,b. We consider three cases. Case : F v = ep v f v v dv < v v π T. Let + ep v + v = G v, and F v = e v, where v is a large positive constant such that F v dv >. v Let γ, > be two small parameters to be fied, define [ f γ, v, v = f v, v + γ F + C γ 7 v γ ] F v,

8 where C = F v F v dv >. The rest of the proof is similar to the proof of Proposition. in [9]. We sketch it below. There eists < < such that for γ > small enough < v f γ, v, v v dv < π < T v f γ, v, v v dv, when < γ < γ. Let β be a T periodic function and denote e = v β. We look for D BGK wave solution f = f β γ,, v, E = E e near f γ,,, where f β γ, +C, v = γ [g + e, v + γ G +C γ [g e, v + γ G e γ e γ ] F v ] F v if v > if v and E = β. The steady Vlasov-Poisson equation is reduced to the ODE β = f β γ,, v dv 4 = + C γ { g + e, v dv + g e, v dv v > v γ + G e γ F v dv} := h γ, β. Since and h γ, = f γ, v dv = d = h γ, + C γ { v > g + v, v dv + + γ γ G v v γ g v, v dv F v dv } v f γ, v, v = dv <, when < γ < γ, < <, v so β = is a center for the ODE 4 and there eist bifurcation of periodic solutions. More precisely, for any fied γ, γ, there eists r > independent 8

9 of,, such that for each < r < r, there eists a T γ, ; r periodic solution β γ,;r to the ODE 4 with β γ,;r H,T γ,;r = r. Moreover, π v f γ, v, v dv, when r. T γ, ; r v To get a solution with the given period T, we adjust [, ] by using the inequality and the fact that T γ, ; r is continuous to. So for each γ, r > small enough, there eists T γ, r,, such that T γ, T ; r = T. Define f γ,r, v = f β γ, T, v, β γ,r = β γ,t ;r and let E γ,r = β γ,r e. Then f γ,r, v, E γ,r is a nontrivial steady solution to?? with period T. For any fied γ >, let γ = lim r T γ, r [, ]. By the dominant convergence theorem, it is easy to show that fγ,r, v f γ,γ v T L + v fγ,r, v f γ,γ v d dv,v + fγ,r, v f γ,γ v W,p,,v when r = β γ,r H,T. So for any γ > and ε >, there eists r = r γ, ε > such that fγ,r, v f γ,γ v L,v + Since f v f γ,γ v = T v fγ,r, v f γ,γ v d dv + fγ,r, v f γ,γ v W,p < ε,v. [ + C γ C γ f v γ F and γ [, ], by using Lemma., for s < + p, f v f γ,γ v W, when γ. s,p ] v F v. γ It is also easy to show that f v f γ,γ v L + v f v f γ,γ v dv, when γ. Thus we can choose γ > small enough such that T f v f γ,γ v L + T v f v f γ,γ v dv 9

10 + f v f γ,γ v < ε W s,p. So the nontrivial steady solution f γ,r, v, E γ,r is within ε distance of the homogeneous state f v, in the norm of. Case : v f v. π v dv < T Choose F v = ep v and F v is the same as before. Define f γ, v as in Case see. Then there eists < < such that v f γ, < v, v π v f γ, dv < < v, v dv. v T v The rest of the proof is the same as in Case. v f v π T. For >, define f v, v = f Case : v dv = For any ε >, there eist < ε < < ε such that < and when ε, ε, v f v v dv < T f v f v L + T π < T v f v dv, v v f v f v dv + f v f v W s,p < ε. We construct steady BGK waves near f v,, which are of the form { f β, v = g + g e e, v, v v, v. if v > if v, e = v β, 5 and E = β e. The eistence of BGK waves is then reduced to solve the ODE β = f β, v dv := h β. 6 As in Case, for any ε, ε, r ε > independent of such that for each < r < r, there eists a T ; r periodic solution β ;r to the ODE 6, satisfying β ;r H,T ;r = r and π v f v dv, when r. T ; r v For r small enough, again there eists T r, ε ε, ε such that T T ; r = T. Define f r,ε, v = f β T, v and E r,ε = β T ;r e.

11 Then f r,ε, v, E r,ε is a nontrivial steady solution to?? with period T. As in Cases and, by choosing r small enough, f r,ε, v is within ε distance of the homogeneous state f v, in the norm of. This finishes the proof of the Theorem.. Lemma. i Given f W p,p L,and Then for >, f g W s,p p >, s < p v ii Given f, g W s,p f v. g v, v, when. 7 W s,p p >, s < p. Then for >, g v, when. W s,p Proof. Proof of i: First we consider g C for W s,p norm see [5], we have v f g v, v W s,p v C f g v, v v + f L p v W s,p v. By Fubini Theorem g v, v W s,p v. L p v By the estimates in the proof of Theorem. of [5], for any p >, s < p, when h W s,p, h W p,p L, we have h h W s,p C h W s,p h W p,p + h L. So f v C f C f C f g v, v v v W s,p v L p v + g L v W s,p g p,p Wv W s,p g Wv,p L p v W g s,p W,p, v L p v

12 when.since f v W s,p under the assumption s < p see [9] for a proof. By the trace Theorem, we also have v v Lp f g v, v W s,p v C f g L p v W,p, when. This proves 7 for g C. When g W s,p, 7 can be proved by using C functions as approimations. Proof of ii: By Fubini Theorem for W s,p norm, v f g v W s,p f v W v L C g v s,p L p v v + g v W s,p v f p v, when. By the similar proof of Theorem., we can get the following. Theorem. Assume the homogeneous distribution function satisfies f v H sv,b d d, b > d 4, s v [, f v, f v dv =, v f v dv < +. Fi T > and c. Then for any ε >, s, there eist travelling wave solutions of the form f = f ε ct, v, E = E ε ct e to??, such that f ε, v, E ε has minimal period T in, f ε, v, E ε is not identically zero, and T f ε f + L v f ε, v f v d dv+ f ε f,v H s H < ε. d v 8 Proof. The construction of BGK waves follows the same line of the proof of Theorem.. First, we modify f v to a smooth profile f v. Then by adding proper perturbations in a scaling form to f v, we get the modified profile f γ, v. The BGK waves f ε, v, E ε are obtained by bifurcation near f γ, v,. To show the estimate 8, we need to control three deviations in the norm of 8: i f ε, v f γ, v ; ii f γ, v f v ; and iii f v f v. For the estimate of i, we choose integers s s, s v s v, and b b and it is easy to show that f ε, v f γ, v H s Hv C f ε, v f γ, v H s sv, b Hv and the right hand side can be made arbitrarily small by using the dominant convergence Theorem. For estimates of ii and iii, we use the following analogue of Lemma..

13 Lemma. i Given f v H L, g v, v H s,b s <, b > 4. For >, f v g v, v, when. H s,b ii Given f, g H s,b s <, b > 4. For >, v f g v, when. H s,b Proof. First, we show that for any function h H s,b d d, s, b > 4, the norm h H s,b d defined by is equivalent to both + v b H f 9 s d and + v b + + v d b f H s d. We only need to prove the equivalence of the norms and 9 for s = and s =, since then for < s < it follows from interpolation. For s =, it is trivial. For s =, by choosing a > small enough, we have + a v b f + a v b L f d = f + a v b + f + a v b L + a v b L f. Thus + a v b L f + a v b L f + a v b L f. The equivalence of and can be proved in the same way. Proof of i: By Lemma. i, v f g v, v C H + v b v f g v, v s,b, Hs when. Since f v H, L and + v b g v, v C g H s,b <. H s

14 Proof of ii: By using the equivalent norm and Lemma. ii, v f g v H s,b C + v b + v b v f g v Hs v C{ f g v + b v b v f g v H s Hs v + f v b g v } H s, when. In the following, we show that there eist no truly D or D BGK solutions. Therefore, the D BGK form of solutions constructed in Theorem. is in some sense necessary. Proposition. i d = Assume µ C L +, µ. If f, v = µ v β, E = β is a solution of the Vlasov-Poisson system, then E. ii d = Assume µ C L +, µ. If f,, v, v, v = µ v + v β,, v, E, = β, β, is a solution of the Vlasov-Poisson system, then E. iii d = Assume µ C, µ, µ r r L +. If f, v = µ v β, E = β is a solution of the Vlasov-Poisson system, then E. Proof. We only prove i since the proof of ii and iii is similar. The electric potential β satisfies β = v β dv + = g β. µ By the assumptions on µ, we have g β C and g β = v β dv = π µ µ s β ds = πµ β, 4

15 Taking derivative of and integrating with β, we have β T d = g β β d. T So T β d = and β is a constant C. By the periodic assumption of β, C = and thus β. Similarly, β. emark. In D and D, the function g β defined in always satisfies g β and thus the elliptic problem only has trivial solutions. For D, the function g β can change signs and thus the eistence of D BGK waves is possible. We note that Proposition. does not eclude steady travelling wave solutions not of BGK types. It would be interesting to construct or eclude nontrivial steady solutions not of BGK types. Non-eistence of Invariant structures In this section, we prove that there eist no nontrivial invariant structures near stable equilibria in spaces H s Hv sv,b sv >. First, we derive the optimal linear decay estimate in a space-time norm under the Penrose stability condition 4. By assuming the uniform in time bound close to the stable homogeneous state above the regularity threshold s v >, it is shown that the solutions of nonlinear equation is dominated by the linearized equation and a decay estimate in space-time is obtained for the nonlinear solution. By the time translation property of Vlasov-Poisson system, such a space-time decay implies that the invariant solution must be trivial, that is, homogeneous. Lemma. Given f v H s,b d d, s, b > d 4. For any unit vector e d, let v = α e + w where v d and w e. Define f e α = f α e + w dw. d Then for some constant C independent of e. f e α H s C f H s,b d Proof. To simplify notations, we only consider e =,,,. Then α = v and f e v = f v dv dv d. d 5

16 Let ξ = ξ, ξ d be the Fourier variable. Then f e v H s = + ξ s ˆf e ξ L = + ξ s ˆf ξ,,, L C + ξ s H ˆf ξ b d = C + v b s f = C f H s,b d. L d Here, the first inequality above is due to the trace theorem and that b > d. Corollary. Given f v H s,b d d, s >, b > d 4. For any unit vector e d, we have i If α is a critical point of f e α, then f e α α α dα C d, s, b f H s,b d. ii For any α, P f e α α α dα C d, s, b f H s,b d, where P is the principal value integral. Proof. i follows from Lemma. and the following Hardy inequality see Lemma. of [9]: If u v W s,p p >, s > p, and u =, then u v v dv C u W s,p, for some constant C. Proof of ii: Since P f e α α α dα = d dα f e α + α + f e α + α α by Hardy inequality P f e α α α dα f C e α + α Hs + f e α + α Hs C f e α H s C f H s,b d dα, by Lemma.. 6

17 The net lemma generalizes the one dimensional result in [9] about linear decay estimates in a space-time norm. The linearized Vlasov-Poisson system at an homogeneous state f, E = f v, is t f + v f E v f =, E = φ, φ = f dv, d 4a 4b Lemma. Assume f v H s,b d d, s >, b > d 4 satisfies the Penrose stability condition 4 for period tuple T,, T d. Let f, v, t, E, t be a solution of the linearized Vlasov-Poisson system 4a-4b with period tuple T,, T d. If f, v, H s Hv sv,b with s v s, then t sv E, t C f, v, H s, 5 L t H +s+sv Hv Proof. First, we reduce the linearized problem to the one dimensional case. Since the homogeneous component of f, v, t remains steady for the linearized equation and therefore has no effect on E, t, we assume that f has no homogeneous component. Let f, v, t = e i k f k v, t k Z d and the electric potential Then E, t = φ = φ, t = k Z d e i k φ k t. k Z d i kφ k t e i k = where E k t = i kφ k t. Denote e = k/ k, then k Z d E k t e i k, E k t = i kφ k t = Ẽ k t e where Ẽ k t = i k φ k t. Let v = α e + w where α, w e, and f k α, t = f k, e α, t = f k α e + w, t dw d 7

18 The linearized Vlasov equation implies that = t f k + v i k f k E k v f = t f k + iα k f k Ẽk α f An integration of the w variable on above equation yields t f k α, t + iα k f k α, t Ẽkf, e α =. 6 The Poisson equation implies k φ k t = f k v, t dv, d and thus i k Ẽ k t = f k α, t dα. 7 Equations 6 and 7 imply that f k α, t, Ẽk t e i k solves the linearized D Vlasov-Poisson equations at the homogeneous profile f, e α. Thus by the D representation formula in [9] and the Penrose stability condition 4, we have k G k y + i Ẽ k t = π k e i k yt dy. F e y + i Here, and G k y + i = P F e y + i = P f k α, α y dα + iπ f k y, f, e α α y dα + iπf, e y. By the Penrose stability condition 4 and P f, e α α y dα C d, s, b f H s,b d Corollary., there eists c > independent of k, such that k F e y + i c k. Then by the same proof of Proposition 4. in [9], t svẽ k t C k sv f k α, L Hv sv C k sv f k v,. H s,b d 8

19 So t sv E, t = C k Z d k Z d L t H +s+sv k +sv+s t s v Ẽ k t C f, v, H s L k s f k v, H s,b d Hv. Lemma. Assume f v H s,b d d, s >, b > d 4 and the Penrose stability condition 4 is satisfied for period tuple T,, T d. Let f, v, t, E, t be a solution of the Vlasov-Poisson system a-b with period tuple T,, T d. For any s, s v satisfying 6, there eists ε >, such that if then f t f H s Hv + t s v E, t L < ε, for all t, {t } H +s Cε. 8 Proof. Denote L to be the linearized operator corresponding to the linearized Vlasov-Poisson equation at f v,, and E is the mapping from f, v to E by the Poisson equation 4b.It follows from Lemma. that: For any s v s, if h, v H s,then + t sv E e tl h L t H C h, v +s H s Denote f t = f t f, then Thus f t = e tl f + and correspondingly t Hv sv,b t f = L f + E v f. Hv. 9 e t ul E v f u du = f lin t + f non t, E t = E f lin t + E f non t = E lin t + E non t. By the linear estimate 9, + t sv Elin, t L {t } H +s = + t sv E e tl f L t H C f H s Hv, +s 9

20 and + t sv Enon, t = C = C C C t L {t } H +s + t sv Enon, t t + t sv E H +s dt [e t ul E v H f u] du dt +s t + t sv + t u sv + u sv du t + u sv + t u sv E [ e t ul E v f u ] + u sv + t u sv E [ e t ul E v f u + u sv u + u sv E v f u + u sv E u Cε + t sv E, t + t u sv E [ e t ul E v f u H +s L {t } H +s H s In the above estimate, we use the fact that t du sv,b Hv f u H s. Hv + t u sv + u sv du C + t sv du H +s ] dudt H +s because of the assumption that s v >. The assumption 6 ensures that the following inequality is true E v f u H s Thus + t sv E L, t + t sv Elin, t C f H s Hv sv,b Hv {t } H +s L {t } H +s C E u H +s f u H s + + t sv Enon, t + Cε + t s v E L, t {t } H. +s By taking ε = C, we get the estimate 8. Hv. ] L {t } H +s H +s dudt dtdu

21 Theorem. follows from Lemma. and the time translation symmetry of the Vlasov-Poisson equation. Since the arguments are eactly the same as in the D case [9], we skip the details. As a corollary of Theorem., we get the following nonlinear instability result. Corollary. Assume f v H s,b d d, s >, b > d 4 and the Penrose stability condition 4 is satisfied for the period tuple T,, T d. For any s, s v satisfying 6, there eists ε > such that for any solution f, v, t, E, t of the Vlasov-Poisson system?? with period tuple T,, T d and E, not identically zero, the following is true: f T f H s Hv ε, for some T. We can also study the positive negative invariant structures near f v,, which are solutions f t, E t of nonlinear Vlasov-Poisson equation satisfying the conditions 7 for all t t. The net theorem shows that the electric field of these semi-invariant structures must decay when t + t. Theorem. Given a homogeneous profile f v H s,b d d, s >, b > d. 4 Assume that f v satisfies the Penrose stability condition 4 for T,, T d. Let f, v, t, E, v, t be a solution of?? in T d. For any s, s v satisfying 6, there eists ε >, such that if f t f H s Hv < ε, for all t or t, with f L,v <, v f,, v dvd <, T d d then E t, when t + or t. L Proof. By energy conservation, v f, v, t dvd + E, t T d d L = v f, v, dvd + E, L < C. T d d

22 Let j = vf dv. When d =,we have j t = vf t dv v dv f t L,v + v fdv v A A v A C f L,v A + v fdv C f 4L,v v 4 fdv, A by choosing A = v fdv/ f L,v 4. Thus j, t L 4 C v f dvd C. Since d E dt, t = L T d j, t E, t d j, t L 4 E, t L 4 C E, t H, and by Lemma. E, t H dt Cε, + t s dt + t s E, t dt H thus lim t E L, t eists and equals zero. When d =, the proof is very similar. The estimates become j, t 5 L 4 C, and d E dt, t j, t 5 L L 4 E, t L 5 C E, t H +s. The rest is the same. Acknowledgement This work is supported partly by the NSF grants DMS-9875 Lin and DMS-89 Zeng.

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