Nonlinear instability of periodic BGK waves for Vlasov-Poisson system

Transcription

1 Nonlinear instability of periodic BGK waves for Vlasov-Poisson system Zhiwu Lin Courant Institute Abstract We investigate the nonlinear instability of periodic Bernstein-Greene-Kruskal(BGK waves. Starting from an exponentially growing mode to the linearized equation, we proved nonlinear instability in the L 1 -norm of the electric field. 1 Introduction We consider a one-dimensional electron plasma with a fixed homogeneous neutralizing ion background. In such a plasma collisions are relatively rare; hence we assume no collisions at all. So the time evolution can be modelled by the Vlasov-Poisson system (1.1a (1.1b f t + v f x E f =, E + x = fdv + 1, where f(x, v, t is the electron distribution function, E (x, t = x φ (x, t the electric field (φ is the electric potential, and 1 is the ion density. In 1957, Bernstein, Greene and Kruskal ([2] showed the existence of an infinite family of exact stationary solutions to (1, called BGK waves. Since then, the stability of these BGK waves has been of great interest. In [4], [5], [6], Guo and Strauss initiated a rigorous study of instability of spatially-dependent equilibria in plasmas. Using some delicate perturbation arguments, they proved in particular that inhomogeneous periodic BGK waves with small amplitude are linearly and nonlinearly unstable, if the corresponding homogeneous case has a growing mode according to Penrose s criteria. In [7], we show that an arbitrary periodic BGK wave is linearly unstable under multi-periodic perturbations. In this paper, we finally complete such instability investigation of BGK waves by showing that linear instability always implies nonlinear instability, without the smallness assumption. In particular, combining with the result in [7], we Communications on Pure and Applied Mathematics, Vol., 1 27 (199X c 199X John Wiley & Sons, Inc. CCC 1 364/98/1-27

2 2 ZHIWU LIN prove that any periodic BGK wave is nonlinearly unstable under multiperiodic perturbations. Before stating our main theorem, we introduce some notations as in [7]. A BGK wave is a steady state of the form ( 1 f (x, v = µ 2 v2 β (x, E (x = β x (x, with β a periodic solution of the differential equation + ( 1 (1.2 β xx = µ 2 v2 β (x dv 1. We assume that neutral, that is, (1.3 (i µ (θ is a nonnegative C 2 function on ; (ii µ is + and (iii For some γ > 1 (1.4 ( 1 µ 2 v2 dv 1 = ; µ (θ, µ (θ C 1 + θ γ Let β be any periodic solution of (1.2 and P β be its minimal period. By adjusting the starting point, we can arrange that the solution satisfies: (1.5 β( = β(p β = min x P β β(x, β( P β 2 = max x P β β(x, (1.6 β (x = β (P β x, x [, P β ], [ and β (x is strictly increasing in, P β 2 ]. Notice that by extending β (x to [, kp β ] periodically, we obtain new BGK waves with period kp β. It was proved in [7] that if k 2, then there exists some kp β -periodic growing mode to the linearized equation. That is, there is a solution (e λt f, e λt Ẽ with λ > to (1.7a (1.7b f t + v f x E f = E f, E + x = fdv, where f and Ẽ are kp β-periodic functions in x and Ẽ H1. The main theorem in this paper is

3 NONLINEA INSTABILITY OF BGK WAVES 3 Theorem 1.1 Let (E, f be the above periodic BGK solution to (1 with period P = kp β (k 2 and µ satisfying (1.4 and (3.1. Then there exists positive constants ε, C 1 and a family of solutions [ f δ (t, (t] Ēδ W 1,1 ([, P ] of (1 with period P in x, defined for δ sufficiently small, with f δ (t non-negative, such that (1.8 ( f 1 δ ( µ 2 v2 W β (x + Ēδ ( E δ, 1,1 (,P W 1,1 ([,P ] and L (1.9 sup Ēδ (t E ε. t C 1 ln δ 1 (,P We note that in [4] [5], nonlinear instability is proved in terms of the L 1 norm of the electric field plus the electron distribution function. In our result, nonlinear instability is in the L 1 norm of the electric field only, which is clearer physically. In particular, this implies that the deviation from the unstable BGK waves is macroscopic, not in the microscopic sense. Before sketching the main idea of proving Theorem 1.1, we indicate some difficulties in the nonlinear instability study. The main difficulty is that the nonlinear term of (1, E v f, contains derivative and thus is unbounded in L p space of E and f. This is a typical situation in many equations in continuum physics and kinetic theory, and so far there is no general method to dealt with it. Starting with [4], Guo and Strauss developed a bootstrap technique to overcome this difficulty and prove nonlinear instability. Their basis idea is to derive the growth estimate for the derivative of the perturbation from the growth of the perturbation, in a time period during which the perturbation is exponentially growing while the amplitude is kept small. In [4] [5], they used this technique to bootstrap v f form the estimate of f, under the assumption that the amplitude of the BGK wave is small. In [1], this technique was used to study nonlinear instability of ideal plane flows, with some new arguments. It was showed that if the growth rate of the growing mode exceeds the Liapunov exponent of the steady velocity field, then linear instability implies nonlinear instability. If we use the argument of [1] to the current problem, then similar result can be obtained. That is, nonlinear instabil-

4 4 ZHIWU LIN ity can only be showed if the growth rate exceeds the Liapunov exponent ( β xx ( in this case of the particle equation. To overcome such a difficulty, we introduce some new ideas. Let E (t and f (t be the small perturbations satisfying the nonlinear Vlasov- Poisson system (3. Our method is to study the L 1 norm of E (t by using the following evolution formula for f (t t (1.1 f (t = e tl f ( + e (t sl v (Ef (s ds, where L is the linearized operator. The idea of this coupled approach is to use the regularizing effect of Poisson s equation to get rid of the derivative v in the nonlinear term v (Ef. In this way, the nonlinear term essentially becomes Ef which is easier to handle. To make this thinking formal, we use the following inequality to estimate E (t 1 E (t 1 2 sup f (t, x, v a(xdxdv a W 1,1 (,P a x 1 [,P ] and then use a duality argument and integration by parts to move v in (1.1 to a differentiable test function. Another important point in our proof is that even though the quantity X(s; x, v has pointwise exponential growth, where (X(s; x, v, V(s; x, v satisfies the particle trajectory equation (2, the integral X(s; x, v dv only has linear growth. This is due to the geometric property of the particle trajectory. This observation is crucial in our proof of regularity of growing modes and nonlinear instability. The method of proving Theorem 1.1 can be directly used to show that linear instability implies nonlinear instability of any periodic BGK waves, in the case of two species of particles or relativistic case. Moreover the new ideas we introduced in this paper are quite flexible and could apply to other physical systems. In ([9], we use these new ideas to study general 2D ideal flows and prove nonlinear instability from linear instability, without assumption on the linear growth rate.

5 NONLINEA INSTABILITY OF BGK WAVES 5 2 egularity of growing modes In the following, we denote P = kp β. First, we recall some facts about growing modes proved in [7]. The growing mode found in [7] is of the form (e λt f, e λt E with λ >, E = x φ and (2.1 f(x, v = µ (eφ (x + µ (e λe λs φ(x(s; x, vds. Here φ H 2 per(, P satisfies 2 x 2 φ µ (edvφ (x + µ (e λe λs φ(x(sdsdv = where e = 1 2 v2 β(x, and (X(s; x, v, V(s; x, v is the solution of the characteristic equation (2.2a Ẋ(s = V(s, (2.2b V(s = E (X(s, with initial data X( = x, V ( = v. It was showed in [8] that the growing mode (e λt f, e λt E satisfies (1 weakly and f is almost everywhere differentiable in x, v. Now we show that Theorem 2.1 The function f defined by (2.1 is in W 1,1 ([, P ]. To see the main difficulty in the proof, we take the v-derivative of (2.1 and the following term appears in the right hand side (2.3 I (x, v := µ (e λe λs φ x (X(s X(s ds. If the particle energy e = β min, then X(s x grows exponentially in s with the rate β xx (. Thus if λ < β xx (, the integral in (2.3 might diverge and I (x, v becomes infinite on the critical set (2.4 A cr = {(x, v 1 } 2 v2 β(x = β min. However we can show that I (x, v L 1. This is the following lemma.

6 6 ZHIWU LIN Lemma 2.2 The function I (x, v defined by (2.3 is in L 1 ([, P ]. Proof: We need to show that e λs µ (eφ x (X(s X(s dsdvdx <. [,P ] Let e cr = β min and e < e cr < e 1 < e 2 be three numbers to be determined later. We divide the (x, v space into the following five regions. (D 1 (fast particles e = 1 2 v2 β(x > e 2. (D 2 (mildly free particles e 1 < e < e 2. (D 3 (barely free particles e cr < e < e 1. (D 4 (barely trapped particles e < e < e cr. (D 5 (strongly trapped particles β max e < e. First we claim: For e 2 sufficiently large, if (x, v D 1 then there exists constant C 1 such that (2.5 X(s; x, v C 1e λ 2 s for s +. To show (2.5, we consider the case of positive time and positive velocity only since (2 is time reversible. Differentiating (2.2a with respect to v, we get So (2.6 Ẋ(s X(s = = V(s. s V(u du. For any solution ((X(s; x, v, V(s; x, v of (2, we have ( v2 β(x = 1 2 V(s2 β(x(s. Differentiating above with respect to v, we get (2.8 v = V(s V(s It follows from (2.6 and (2.8 that X(s = s ( v β x (X(s X(s. V(u + β x(x(u V(u X(u du.

7 NONLINEA INSTABILITY OF BGK WAVES 7 It is easy to see that if e 2 is large and (x, v D 1, then we have 1 2 v V(u 3 2, for u +. Thus we have X(s 3 2 s + λ 2 β x (X(u V(u S < λ 2 X(u du and the estimate (2.5 follows from Gronwall s inequality. Since D 2 and D 5 are bounded and the Liapunov exponent of (2 is zero there, there exists some constant C 2 such that X(s C 2e λ 2 s, for s +. Let C 3 = max{c 1, C 2 }, then e λs µ (eφ x (X(s X(s D 1 D 2 D 5 dsdvdx C 3 e λ 2 s µ D (eφ x (X(s dvdxds 1 D 2 D 5 C 3 e λ 2 s µ (eφ x (x dvdx [,P ] C 3 e λ 2 s sup µ (e dv φ x (x dx x [,P ] ( C φ x (x 2 dx [,P ] 1 2, which implies that (2.9 I (x, v dxdv <. D 1 D 2 D 5 Here in the second inequality above, we used the fact that the mapping (x, v ((X(s; x, v, V(s; x, v is 1-1 with Jacobian 1. In D 3 and D 4, X(s might have growth e βxx(s and we need to do more delicate analysis. First we recall some basic facts about the particle

8 8 ZHIWU LIN motion satisfying (2. If e e cr, particles with energy e do periodic motion. Free particles with energy e > e cr travel through the whole interval [, P ] with the period T (e = P 1 2(e + β (y dy. and trapped particles with energy e < e cr go back and forth within one of the intervals [a (e, b (e], [a + P β, b + P β ],, [a + (k 1 P β, b + (k 1 P β ] with period b(e 1 T (e = 2 dy. a(e 2(e + β (y Here a (e, b (e are two points in [, P β ] such that β (x = e. For particles with energy e cr, it takes infinite time to approach the saddle point. We have the following properties of T (e: (I T (e is twice differentiable for ( β max, e cr (e cr,. (II T (e + as e e cr andt (e 2π as e β βxx( max. (III T (e < for e > e cr (IV There exists e ( β max, e cr such that T (e, T (e > for e < e < e cr. Properties (I,(II are standard (e.g. see [3] and (III is obvious. It follows from (I and (II that T (e, T (e tends to + as e e cr +. So if we choose e close to e cr then (IV holds. Now we estimate D 3 I (x, v dxdv. Let J x = {v (x, v D 3 }, J x + = J x {v > } and Jx = J x {v < }. Then I (x, v dxdv D 3 = e λs µ (eφ x (X(s X(s [,P ] J x + Jx dvdxds (2.1 C e λs X(s dvdxds where [,P ] J + x J x C = µ (e φ x. We will prove that ( (2.11 X(s s J x + Jx dv 2 T (e P.

9 NONLINEA INSTABILITY OF BGK WAVES 9 Then it follows from (2.1 and (2.11 that (2.12 D 3 I (x, v dxdv <. We only show (2.11 for positive s. First we notice that J + x X(s dv = J + x X(s X X(s;J dv = dx. x + Here in the second equality, we use the change of variable v X(s and denote X(s; J x + to be the image of J x + under this mapping. Free particle with energy e > e cr always go in one direction. If they start from the same position x with different velocity, they will never meet. The particles in D[ 3 have ] the minimal period T (e 1. So by ( time s they can finish at most s T (e 1 periods of motion and we can use s T (e number of interval [, P ] to cover the set X(s; J x +. So J + x ( X(s s dv T (e P. The estimate on Jx is the same as above. Thus (2.11 is proved. We use the same idea as above to estimate D 4 I (x, v dxdv. But particles in D 4 can go back and forth, we need to do more careful analysis. We use the same notations as above. We will prove that (2.13 ( X(s s J x + Jx dv 4 T (e + 1 P β. Assuming (2.13, then as what we did for D 3 above, we have D 4 I (x, v dxdv <. Combining with (2.9 and (2.12, we have I (x, v L 1 ([, P ]. We only prove (2.13 for positive s and x (, P β since the proof is the same for other cases. First by co-area formula, we have (2.14 J + x X(s dv = dx. X X(s;J x +

10 1 ZHIWU LIN So to show (2.13 we need to understand the set X(s; J [ x +. We ] know s that by time s, each particle in D 4 can finish at most T (e periods of motion. Consider the following question: for particles starting from the same position x and with different energy e (e, e cr, how many of them can collide at x 1 by time ( s? We classify the possible collision in the following way. Denote P j + P j + to be the set of all particles getting to x 1 at time s with positive (negative ( velocity after having exactly finished ( j periods of motion and n + j n j is the number of particles in P j + P j +. [ ] Note that as mentioned above, we have j. For a particle in P + j with energy e, we have x1 (2.15 s = jt (e + x if x 1 > x and (2.16 s = jt (e + T (e s T (e dx 2 (e + β (x x x 1 dx 2 (e + β (x if x 1 < x. For a particle with energy e in P j, we have (2.17 s = jt (e + T (e 2 if P β x 1 < x and (2.18 s = jt (e + T (e 2 x P β x 1 Pβ x 1 + x dx 2 (e + β (x dx 2 (e + β (x if P β x 1 > x. Above equations of s can be obtained by analyzing the phase space of (2 and here the symmetry property of β ((1.5, (1.6 is used. By property (IV of T (e, it is easy to see that right hand sides of (2.15, (2.16, (2.17 and (2.18 are either monotone or convex functions of e. So there are at most two particles with different energy such that one of (2.15, (2.16, (2.17 (2.18 is true. Therefore we have n + j, n j 2, for j [ s T (e 1 ]. So we can use 2 ([ s T (e 1 ] + 1 cover the set X(s; J x +. Thus from (2.14, ( X(s s dv 2 T (e + 1 P β. J + x intervals [, P β ] to Since the estimate for J x is the same, (2.13 is proved.

11 NONLINEA INSTABILITY OF BGK WAVES 11 Now we can prove the regularity of growing modes. Proof of Theorem 2.1: Since f defined by (2.1 is differentiable besides the critical set A cr (defined by (2.4, almost everywhere we have v f = µ (evφ (x + µ (e v λe λs φ(x(s; x, vds + I (x, v. The first two terms in the right hand side above are obviously integrable. So by Lemma 2.2, v f L 1 ([, P ]. Now consider the integrability of x f. Besides the set A cr, f is twice differentiable and satisfy the equation (2.19 λf + v x f E v f = φ x vµ (e. We take x of (2.19 to get (2.2 λf x +v x ( x f E v ( x f = β xx v f φ xx vµ (e+φ x β x vµ (e. From (2.2, we have (2.21 d ( e λs f x (X(s, V(s = e λs ( β xx v f φ xx vµ (e ds + φ x β x vµ (e (X(s, V(s. Integrating (2.21 from to, we get f x (x, v = e λs ( β xx v f φ xx vµ (e + φ x β x vµ (e (X(s, V(sds.

12 12 ZHIWU LIN So = f x (x, v dvdx [,P ] e λs ( β xx v f + φ xx vµ (e [,P ] = 1 λ e λs [,P ] + β x φ x vµ (e (X(s, V(sdsdvdx ( β xx v f + φ xx vµ (e + β x φ x vµ (e (x, vdvdxds ( β xx v f + φ xx vµ (e [,P ] + β x φ x vµ (e dvdx 1 λ ( β xx v f L sup vµ (e dv φ xx 2 P <. x + β x φ x vµ (e L 1 Here in third line, we change the variable (x, v (X(s, V(s. We shall use the following fact: Assume a continuous function a (x, y which is differentiable beyond a closed C 1 curve. If a x, a y (defined almost everywhere are both integrable. Then a x, a y are also the weak derivatives of a in the distribution sense. It is straightforward to prove the above fact by the definition of weak derivative. Using this fact and the integrability of f x, f v just proved, we know that f defined by (2.1 is in W 1,1 ([, P ]. In ([8], we showed that f is a weak solution to (2.19. By Theorem 2.1, f is also the strong solution to (2.19. emark 2.3 The method of proof ( of Theorem 2.1 can be used to get the following regularity result: Let e λt f, e λt E (e λ > be a growing mode to (1 with f L 1 ([, P ], then we have f W 1,1 ([, P ]. 3 Nonlinear instability of Periodic BGK waves By the result in [7], there exists a growing mode to (1. We pick the growing mode (e λt f g, e λt E g with the largest growth rate e λ. By the

13 NONLINEA INSTABILITY OF BGK WAVES 13 regularity result proved in the last section, f g W 1,1 ([, P ] and E g W 1,1 (, P. Consider the more general case that λ is not real. The usual way to construct unstable initial state is to let f δ ( = f + δ Im f g and Ēδ ( = E + δ Im E g. However to be physically meaningful, the distribution function f δ must be nonnegative. So some truncation process is required. This was done in [5] for the relativistic case. Here the construction is almost the same as in [5] [6], so we only state the result and indicate some minor modifications. Lemma 3.1 Let h (s be either s σ or exp (l s ( s = 1 + s 2 with σ, l. Assume for sufficiently large v, we have (3.1 ( ( ( vµ v /2 2 C h ( v µ v 2 /2, µ v 2 /2 h ( v [h ( v ] 2 m for some m >. Then there exists δ, θ, c > such that for < δ < δ, there exists perturbed initial state ( ( f δ (, ( Ēδ = f + δfg δ, E + δeg δ such that f δ ( W 1,1 ([, P ], Ēδ ( W 1,1 (, P, f δ W ( f + Ē δ W ( E = δ, 1,1 ([,P ] 1,1 (,P f δ (, x Eg δ = fg δ dv. Moreover, denote (f L δ (t, Eδ L (t be the solution to the linearized equation ( (1 with (f L δ (, Eδ L ( = δfg δ, δeg δ, then we have (3.2 c δe e λt EL δ (t 3c L δe 1 (,P for t T (here δe e λt = θ. e λt We can use the same arguments in [5] to prove Lemma 3.1 and we refer to [5] for the details. A minor modification is the following. From (2.1 for the growing mode f g, we have f g (x, v 2 φ g µ (e.

14 14 ZHIWU LIN This estimate can be used to prove Lemma 14 in [5] without the assumption that E is small. A minor extension of Lemma 3.1 is to dealt with the case when f = µ (e has compact support. In this case, by assuming the growth condition like (3.1 near the boundary of the support of µ, we can construct corresponding nonnegative f δ ( by the similar truncation technique. We skip these details and move on to the nonlinear instability proof below. ( Denote f δ (t, (t Ēδ to be the solution to (1 with initial data ( f δ (, ( Ēδ as constructed in Lemma 3.1. Denote ( ( f δ (t, E δ (t = f δ (t f, Ēδ (t E. to be the perturbations satisfying the equation (3.3a (3.3b f t f = v x + E f + E f + E f, E + x = fdv, with ( ( f δ (, E δ ( = δfg δ, δeg δ. (( In the following we consider a fixed δ < δ, so we simply denote f δ (t, Ēδ (t ( and f δ (t, E δ (t by (( f (t, Ē (t and (f (t, E (t. Denote D = v x + E v then the linearized operator L corresponding to the linear part of (3 can be written as ( L f = Df + µ (e D xx 1 fdv = Df + Kf. Here we define 1 xx Then we have f (t = e tl f ( + in the following way: ψ = xx 1 ρ is the solution to ψ xx = ρ, with t P ψ (x dx =. e (t sl ( v (Ef (s ds = f L (t + f N (t

15 and correspondingly NONLINEA INSTABILITY OF BGK WAVES 15 E (t = x φ (t ( = x xx 1 ( = x xx 1 f (t dv f L (t dv ( x xx 1 f N (t dv = E L (t + E N (t = x φ L (t x φ N (t. We define the dual operator L of L as ( L f := Df xx 1 µ (e Dfdv = Df + K f. Consider and We have e tl : L 1 ([, P ] L 1 ([, P ] e tl : L ([, P ] L ([, P ]. Lemma 3.2 For any f L 1 ([, P ], f L ([, P ] and t, the following is true ( ( e tl f f dxdv = f e tl f dxdv. [,P ] [,P ] Proof: Since L 1 and L are not dual spaces, we cannot use the abstract theory directly. We sketch the proof below. Let f (t, x, v = e td f = f (X( t; x, v, V( t; x, v t f 1 (t, x, v = f + = f + t e (t sd Kf (s ds (Kf (s, X( (t s ; x, v, V( (t s ; x, v ds t f k+1 (t, x, v = f + (Kf k (s, X( (t s ; x, v, V( (t s ; x, v ds and f (t, x, v = e td f = f (X(t; x, v, V(t; x, v

16 16 ZHIWU LIN t f1 (t, x, v = f + (K f (s, X(t s; x, v, V(t s; x, v ds t fk+1 (t, x, v = f + (Kfk (s, X(t s; x, v, V(t s; x, v ds. Then since K, K are compact operators, by the standard theory we have f k e tl f in L 1 and f k e tl f in L as k. By deduction on k and a tedious computation using integration by parts, we can show that f k f dxdv = ffk dxdv [,P ] [,P ] for any integer k. Letting k in the above, we get the conclusion. We need the following simple lemma. Lemma 3.3 (i If g (x L 1 per (, P and P g (x dx =, then P (3.4 g 1 2 sup a Wper 1, (,P ga x dx. a x 1 Moreover if g x L 1 (, P, then P (3.5 g 1 2 sup a Wper 1, (,P g x a dx. a x 1 (ii If g (x L per (, P and P g (x dx =, then P (3.6 g 2 sup a Wper(,P 1,1 ga x dx. a x 1 1 Moreover if g x L (, P, then P (3.7 g 2 sup a Wper(,P 1,1 g x a dx. a x 1 1

17 NONLINEA INSTABILITY OF BGK WAVES 17 Proof: have We only prove (i since the proof of (ii is the same. We P (3.8 g 1 = sup g (x b (x dx. b L per b =1 Since the function b 1 P P b (x dx has zero integral, there exists some function a W 1, per (, P such that So b 1 P P b (x dx = a x. (3.9 b = 1 P P b (x dx + a x and clearly a x 2 b = 2. From (3.8, (3.9 and the assumption that P gdx =, we have g 1 = P sup g (x a x dx b L per b =1 sup a W 1, per (,P a x 2 P ga x dx. The estimate (3.5 follows from (3.4 by integration by parts. The following lemma will be used later. Lemma 3.4 For any ε <, there exists constant C ε such that: for any function a(x in Wper 1, (, P with a x 1, we have the following estimate ( (3.1 v e tl X(t t a P + C ε e (e λ+εs X(t s ds. Proof: Denote h (t, x, v = e tl a,

18 18 ZHIWU LIN then h is the solution to the equation ( (3.11 t h = Dh xx 1 µ (e Dhdv with g ( = a (x. We have the following estimate: (3.12 h (t C εe (e λ+εt for some constant C ε. To prove (3.12, we notice that: there exists C ε such that e tl L 1 L C 1 ε e (e λ+εt. This is due to the fact that L is compact perturbation of the operator D which generate an isometry group in any L p space (1 p <, and e λ is the maximal growth rate. Now (3.12 follows easily by duality since h (t = e tl a = sup (e tl a k (x, v dxdv k 1 =1 [,P ] = sup k 1 =1 ( a (x e tl k (x, v dxdv (by Lemma 3.2 [,P ] e tl a e tl k P a x L L 1 P C ε e (e λ+εt. From (3.11, we have So t h (t = e td a = a (X(t t v h (t = a x (X(t X(t t ( (3.13 x xx 1 Denoting ( e (t sd xx 1 ( xx 1 µ (e Dhdv µ (e Dh (s dv ds (s, X(t s ds. µ (e Dhdv (s, X(t s ( g (s, x = x xx 1 µ (e Dh (s dv, X(t s ds.

19 NONLINEA INSTABILITY OF BGK WAVES 19 then by (3.7 we have Let g (s 2 sup P b Wper(,P 1,1 b x 1 1 = 2 sup b W 1,1 per(,p b x 1 1 = 2 sup b Wper(,P 1,1 b x 1 1 g x b dx [,P ] [,P ] µ (e Dh (s b (x dxdv h (s D ( µ (e b dxdv = 2 sup h (s µ (e vb x dxdv b Wper(,P 1,1 [,P ] b x h (s sup vµ (e dv b x 1 x 2C ε sup vµ (e dve (e λ+εs (by (3.12. x C ε = 2C ε sup vµ (e dv, x then (3.1 follows from (3.13 and the above estimate. We need the following bootstrap lemma. Lemma 3.5 For < δ < δ, consider the solution (E (t, f (t to (3 with ( (f (, E ( = δfg δ, δeg δ. If (3.14 E (t L 1 (,P 4c e λt δe for t T (here δ, c, T are as in Lemma 3.1, then there exists constants c 1, c 2, c 3 such that f (t 1 c 1 δe e λt, f (t c 2 δe e λt e λt, E (t c 3 δe for t T.

20 2 ZHIWU LIN (3.15 Proof: First we estimate f (t 1. We rewrite (3.3a as f t + v f f Ē (t x = E f, where Ē (t = E + E (t is the perturbed electric field. Denote ( X(s; x, v, V(s; x, v to be the solution of the perturbed characteristic equation (3.16 (3.17 d ds X(s = V(s, d ds V(s = Ē(s, X(s, with initial data X( = x, V ( = v. Then integrating (3.15 along the perturbed trajectory, we have f (t, x, v = f (, X( t, V( t t ( + E f (s, X( (t s, V( (t s ds. So = δ f (t, x, v dxdv [,P ] f (, X( t, V( t dxdv [,P ] t + E f [,P ] t f ( dxdv + [,P ] [,P ] fg δ + sup vµ (e t dv 1 x ( f δ g + 4c 1 e λ sup vµ (e dv x ( s, X( (t s, V( (t s dxdvds E f (s dxdvds E (s 1 ds δe e λt : = c 1 δe e λt. Here in the second equality above we changed the variable by (x, v ( X( (t s, V( (t s

21 NONLINEA INSTABILITY OF BGK WAVES 21 which has Jacobian 1. The estimate of E (t follows easily by letting c 3 = P c 1, since E (t P x E (t 1 P f (t 1 P c 1 δe e λt. Now we estimate f (t. Define the electron current j (t = vf (t dv. First we need the following estimate (3.18 j (t 1 c e λt δe for some constant c and t T. following equation from (3.15 To show (3.18, we obtain the (vf t + v (vf x (vf Ē (t = Ev f Ē (t f. Integrating above along the perturbed trajectory, we have vf (t, x, v = V( tf (, X( t, V( t t ( + Ev f (s, Ēf X( (t s, V( (t s ds. So [,P ] [,P ] t + vf (t dxdv vf ( dxdv [,P ] ( Ev f + ( E + E f (s dxdv. Then by the estimates on f (t 1 and E (t just proved, there exists some constant c such that vf (t 1 c e λt δe for t T, which implies (3.18. Since f t + v f x Ē f =,

22 22 ZHIWU LIN we have f (t, x, v = f (, X( t, V( t and thus f (t = f (t f ( 1 = µ 2 V( t 2 β ( X( t ( 1 µ 2 v2 β (x + δfg δ ( X( t, V( t. If ( X(s; y, u, V(s; y, u is a solution to (3.16 with initial data (y, u then and so ( d 1 ds 2 V(s 2 φ ( s, X(s = φ ( s s, X(s 1 2 V(t; y, u 2 φ ( t, X(t; y, u = 1 2 u2 φ (, y = 1 2 u2 φ (, y t t where φ (t = β + φ (t is the perturbed potential. Letting (y, u = X( t; x, v, V( t; x, v, φ s ( s, X(s; y, u ds φ s ( s, X(s; y, u ds, then we have 1 2 V( t 2 φ (, X( t = 1 2 v2 φ t ( (t, x + φ s s, X(s t; x, v ds. So 1 2 V( t 2 β ( X( t ( 1 2 v2 β (x = φ (, X( t t ( φ (t, x + φ s s, X(s t; x, v ds. Combining above we have (3.19 f (t µ ( φ (t + δ φ δ t g + φ s (s ds. Since e λt φ (t P E (t 1 4c P δe

23 NONLINEA INSTABILITY OF BGK WAVES 23 and φ t (t P x φ t (t 1 = P t E (t 1 2P j (t 1 c P δe e λt, by (3.19 there exists c 2 such that e λt f (t c 2 δe for t T. Here we use the estimate t E (t 1 2 j (t 1. which we prove in the below. Denote the electron density by ρ (t = f (t dv, then from (3 we have So by (3.5 of Lemma 3.3, t E (t 1 2 x E = ρ, t ρ = x j. sup a Wper 1, (,P a x 1 = 2 sup a W 1, per (,P a x 1 = 2 sup a W 1, per (,P a x 1 = 2 sup a W 1, per (,P a x 1 2 j (t 1. This ends the proof of Lemma 3.5. P P P P x ( t E (t a (x dx t ρ (t a (x dx x j (t a (x dx j (t a x (x dx With the preparations above, we can prove Theorem 1.1 now.

24 24 ZHIWU LIN Proof of Theorem 1.1: Denote T to be the maximal time such that E (t 1 4c δe e λt. Notice that (3.2 implies E ( 1 = δeg δ 3c δ, 1 so T >. We claim that T > T, where T is as in Lemmas 3.1, 3.5. Suppose otherwise, we have T T. By (3.5, we have P (3.2 E N (t 1 2 sup a Wper 1, (,P x E N (t a (x dx. a x 1 For any a W 1, per (, P with a x 1, we have = = = = P x E N (t a (x dx [,P ] t [,P ] t t [,P ] [,P ] t f N (t a (x dxdv e (t sl ( v (Ef (s a (x dxdvds v (Ef (s (e (t sl a dxdvds (Ef (s v ( e (t sl a dxdvds Ef (s (P X(t s [,P ] t s + C ε e (e λ+εu X(t s u du dxdvds t t t = ( dxdvds + ( dxdvds + ( dxdvds D 1 D 2 D 5 D 3 D 4 = I 1 (t + I 2 (t + I 3 (t. In the above we used Lemma 3.4 with ε = 1 4 e λ. Now we estimate I 3 (t. As showed in Section 2, there exists constant C 3 such that X(t C 3e 1 2 e λ t.

25 NONLINEA INSTABILITY OF BGK WAVES 25 So I 1 (t t t C 4 E (s f (s 1 (P C 3 e 1 2 e λ(t s t s + C ε e (e λ+εu C 3 e 1 2 e λ(t s u du ds ( c 1 c 3 δe e λs 2 (P C3 e 1 e λ(t s 2 (δe e λt 2 t s + C ε e (e λ+εu C 3 e 1 2 e λ(t s u du ds for some constant C 4 and t T. To estimate I 2 (t, we notice that by (2.11 Thus I 2 (t t t D 3 X(s P dxdv J x + Jx 2 X(s ( s T (e P 2. E (s f (s (2P 3 ( t s T (e c 2 c 3 ( δe e λs 2 (2P 3 C 4 (δe e λt 2 t s + C ε e (e λ+εu 2 ( t s T (e t s + C ε e (e λ+εu 2 ( t s u T (e 1 ( t s u T (e 1 dvdx + 1 P 2 duds + 1 P 2 duds for some constant C 4 and t T. The estimate of I 3 is the same and we have I 3 (t C 4 (δe e λt 2 for some constant C 4 and t T. Let C 4 = 2 (C 4 + C 4 + C 4. Combining these estimates, we have P x E N (t a (x dx 1 2 C 4 (δe e λt 2

26 26 ZHIWU LIN and by (3.2 (3.21 E N (t 1 C 4 ( δe e λt 2 for t T. So we have E (T 1 E L (T 1 + E N (T 1 3c δe e λt + C 4 ( δe e λt 2 (3c + C 4 θ < 3.5c δe e λt, ( δe e λt (by (3.2 and (3.21 if we choose θ to be small such that C 4 θ < 1 2. This is a contradiction to the definition of T. So we must have T < T. Then E (T 1 E L (T 1 E N (T 1 c δe e λt C 4 ( δe e λt 2 = c θ C 4 θ 2 > 1 2 c θ if θ is small such that C 4 θ < 1 2 c. We let θ = 1 2 c θ and the proof of Theorem 1.1 is finished. Acknowledgment. I thanks Walter Strauss and Yan Guo for helpful discussions about this work. Bibliography [1] Bardos, C., Guo, Y., Strauss, W. A. Stable and unstable ideal plane flows. Dedicated to the memory of Jacques-Louis Lions. Chinese Ann. Math. Ser. B 23 (22, no. 2, [2] Bernstein, I., Greene, J., Kruskal, M., Exact nonlinear plasma oscillations. Phys. ev. 18 (1957, no. 3, [3] Chafee, N.; Infante, E. F. A bifurcation problem for a nonlinear partial differential equation of parabolic type. Applicable Anal. 4 (1974/75, [4] Guo, Y.; Strauss, W. A. Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48 (1995, no. 8, [5] Guo, Y.; Strauss, W. A. elativistic unstable periodic BGK waves. Comput. Appl. Math. 18 (1999, no. 1, [6] Guo, Y.; Strauss, W. A. Unstable BGK solitary waves and collisionless shocks. Comm. Math. Phys. 195 (1998, no. 2,

27 NONLINEA INSTABILITY OF BGK WAVES 27 [7] Lin, Z. Instability of periodic BGK waves. Math. es. Lett. 8 (21, no. 4, [8] Lin, Z. Stability and instability of equilibria in collisionless plasmas and ideal plane flows, Doctoral dissertation, Brown University, 23. [9] Lin, Z. Nonlinear instability of ideal plane flows. Preprint, 23. eceived Month 199X.