Small BGK waves and nonlinear Landau damping

Transcription

1 Small BGK waes and nonlinear Landau damping hiwu Lin and Chongchun eng School of Mathematics Georgia Institute of Technology Atlanta, GA 333, USA Abstract Consider D Vlaso-poisson system with a ed ion background and periodic condition on the space ariable. First, we show that for general homogeneous equilibria, within any small neighborhood in the Sobole space W p s;p > ; s < + of the steady distribution function, there p eist nontriial traelling wae solutions (BGK waes) with arbitrary minimal period and traeling speed. This implies that nonlinear Landau damping is not true in W s s;p < + space for any homogeneous equilibria and any spatial period. Indeed, in W p s s;p < + neighborhood p of any homogeneous state, the long time dynamics is ery rich, including traelling BGK waes, unstable homogeneous states and their possible inariant manifolds. Second, it is shown that for homogeneous equilibria satisfying Penrose s linear stability condition, there eist no nontriial traelling BGK waes and unstable homogeneous states in some W s;p p > ; s > + neighborhood. Furthermore, when p = ;we proe p that there eist no nontriial inariant structures in the H s s > 3 neighborhood of stable homogeneous states. These results suggest the long time dynamics in the W s s;p > + and particularly, in the H s p s > 3 neighborhoods of a stable homogeneous state might be relatiely simple. We also demonstrate that linear damping holds for initial perturbations in ery rough spaces, for linearly stable homogeneous state. This suggests that the contrasting dynamics in W s;p spaces with the critical power s = + is a trully nonlinear phenomena which can not be traced p back to the linear leel. Introduction Consider a one-dimensional collisionless electron plasma with a ed homogeneous neutralizing ion background. The ed ion background is a good physical approimation since the motion of ions is much slower than electrons. But we

2 consider ed ion mainly to simplify notations and the main results in this paper are also true for electrostatic plasmas with two or more species. The time eolution of such electron plasmas can be modeled by the = = + fd + ; (b) where f(; ; t) is the electron distribution function, E (; t) the electric eld, and is the ion density. The one-dimensional assumption is proper for a high temperature and dilute plasma immersed in a constant magnetic eld oriented in the -direction. For eample, recent discoery by satellites of electrostatic structures near geomagnetic elds can be justi ed by using such Vlaso-Poisson models ([7], [39]). We assume: ) f (; ; t) and E (; t) are T periodic in. ) Neutral condition: T f (; ; ) dd = T. 3) T E (; t) d = ; so E (; t) (; t), where the electric potential (; t) is T periodic in. Since f (; ; t) dd is an inariant, the neutral condition ) is presered for all time. The condition 3) ensures that E (t) is determined uniquely by f (t) from (b) and the system () can be considered to be an eolution equation of f only. It is shown in [] that with condition 3), the system () is equialent to the following one-dimensional @ = = = ; fd + ; f (; ; t) d U; where U is the bulk elocity of the ion background. The system () is nondissipatie and time-reersible. It has in nitely many equilibria, including the homogeneous states (f () ; ) where f () is any nonnegatie function satisfying f () d =. In 946, Landau [9], looking for analytical solutions of the linearized Vlaso- Poisson system around Mawellian e ;, pointed out that the electric eld is subject to time decay een in the absence of collisions. The e ect of this Landau damping, as it is subsequently called, plays a fundamental role in the study of plasma physics. Howeer, Landau s treatment is in the linear regime; that is, only for in nitesimally small initial perturbations. Despite many numerical, theoretical and eperimental e orts, no rigorous justi cation of the Landau damping has been gien in a nonlinear dynamical sense. In the past decade, there has been renewed interest [6] [37] [8] [8] [9] [5] [3] [5] [38] [47] as well as controersy about the Landau damping. In [3] [5], it was shown that there eist certain analytical perturbations for which

3 electric elds decay eponentially in the nonlinear leel. More recently, in [38] nonlinear Landau damping was shown for general analytical perturbations of stable equilibria with linear eponential decay. For non-analytic perturbations, the linear decay rate of electric elds is known to be only algebraic (i.e. [48]) and the nonlinear damping is more di cult to justify if it is true. Moreoer, in the nonlinear regime, it has been known ([4]) that the damping can be preented by particles trapped in the potential well of the wae. Such particle trapping e ect is ignored in Landau s linearized analysis as well as other physically equialent linear theories ([], [49]), which assume that the small amplitude of waes hae a negligible e ect on the eolution of distribution functions. As early as in 949, Bohm and Gross ([8]) already recognized the importance of particle trapping e ects and the possibility of nonlinear traelling waes of small but constant amplitude. In 957, Bernstein, Greene and Kruskal ([6]) formalized the ideas of Bohm and Gross and found a general class of eact nonlinear steady imhomogeneous solutions of the Vlaso-Poisson system. Since then, such steady solutions hae been known as BGK modes, BGK waes or BGK equilibria. The nontriial steady waes of this type are made possible by the eistence of particles trapped foreer within the electrostatic potential wells of the wae. The eistence of such undamped waes in any small neighborhood of an equilibrium will certainly imply that nonlinear damping is not true. Furthermore, numerical simulations [37] [6] [36] [4] [9] indicate that for certain small initial data near a stable homogeneous state including Mawellian, there is no decay of electric elds and the asymptotic state is a BGK wae or superposition of BGK waes which were formally constructed in []. Moreoer, BGK waes also appear as the asymptotic states for the saturation of an unstable homogeneous state ([3]). These suggest that small BGK waes play important role in understanding the long time behaiors of Vlaso-Poisson system, near homogeneous equilibria. In this paper, we proide a sharp characterization of the Sobole spaces in which small BGK waes eist in any small neighborhood of a homogeneous equilibrium. Denote the fractional order Sobole spaces by W s;p () or W; s;p ((; T ) ) with p ; s. These spaces are the interpolation spaces (see [], [46]) of L p space and Sobole space W m;p (m positie integer). Theorem Assume the homogeneous distribution function f () W s;p () p > ; s [; + p ) satis es f () ; f () d = ; f () < +: Fi T > and c. Then for any " >, there eist traelling BGK wae solutions of the form (f " ( ct; ) ; E " ( ct)) to (), such that (f " (; ) ; E " ()) has minimal period T in, f " (; ) ; E " () is not identically zero, and kf " f k L ; + T jf " (; ) f ()j dd + kf " f k W s;p < ": () ; 3

4 The rst two terms in () imply that the BGK wae is close to the homogeneous state (f ; ) in the norms of total mass and energy. When p > ; s = ; the fractional Sobole space is equialent to the usual Sobole space W ;p. The conclusions in Theorem are also true for the Sobole space W; ; by the same proof. Aboe theorem immediately implies that nonlinear Landau damping is not true for perturbations in any W s s;p < + p space, for any homogeneous equilibrium in W s;p and any spatial period. As a corollary of the proof, we show that there eist unstable homogeneous states in W s;p () s < + p neighborhood of any homogeneous equilibrium. Corollary Under the assumption of Theorem, for any ed T >, 9 " >, such that for any < " < ", there eists a homogeneous state (f " () ; ) which is linearly unstable under perturbations of period T, f " () ; f " () d = ; and kf " () f ()k L () + jf " () f ()j d +kf " () f ()k W s;p () < ": Byaboe Corollary and emark following the proof of Theorem, in W; s;p s < + p neighborhood of any homogeneous state there eist lots of unstable homogeneous states and unstable nontriial BGK waes. In a work in progress, we are constructing stable and unstable manifolds near an unstable equilibrium of Vlaso-Poisson system by etending our work ([33]) on inariant manifolds of Euler equations. Such (possible) inariant manifolds might reeal more complicated global inariant structures such as heteroclinic or homoclinic orbits. Moreoer, in some physical reference ([]), small BGK waes are formally shown to follow a nonlinear superposition principle to form time-periodic or quasi-periodic orbits. We note that Mawellian or any homogeneous equilibria f () = with monotonically decreasing, were shown by Newcomb in 95s (see Appendi I, pp. - of [7]) to be nonlinearly stable in the norm kfk L. So our result suggests, in particular, that in any inariant small L neighborhood of Mawellian, the long time dynamical behaiors are ery rich. The following Theorem shows that there eist no nontriial BGK waes near a stable homogeneous state in W; s;p space when p > ; s > + p : Theorem Assume f () W s;p () p > ; s > + p : Let S = f i g l i= be the set of all etrema points of f : Let < T + be de ned by = ma : (3) T ; ma is f () d i Then for any T < T, 9 " (T ) >, such that there eist no nontriial traelling wae solutions (f ( ct; ) ; E ( ct)) to () for any c, satisfying 4

5 that (f (; ) ; E ()) has period T in, E () not identically, T f (; ) dd < ; (assumption of nite energy) and kf f k W s;p ; < " : By Penrose s stability criterion ([4] or Lemma 7) the homogeneous equilibrium (f () ; ) is linearly stable to perturbations of period T < T. Moreoer, in Proposition 3, the linear damping of electrical eld is shown for such stable states in a rough function space. Theorems and imply that for any p > ; s = + p is the critical inde for eistence or non-eistence of small BGK waes in W s;p neighborhood of a stable homogeneous state. In Lemma 5, we show that the stability condition < T < T is in some sense also necessary for the aboe non-eistence result in W s s;p > + p. The following corollaryshows that all homogeneous equilibria in a su ciently small W s;p () s > + p neighborhood of a stable homogeneous state remain linearly stable. With Corollary, it implies that s = + p is also the critical inde for persistence of linear stability of homogeneous states under perturbations in W s;p () space. Corollary Assume f () W s;p () p > ; s > + p : Let S = f i g l i= be the set of all etrema points of f and T be de ned in (3). Then for any T < T, 9 " (T ) > such that any homogeneous state (f () ; ) satisfying kf () f ()k W s;p () < " is linearly stable under perturbations of period T. Theorem and the aboe Corollary suggest that the dynamical structures in small W s s;p > + p neighborhood of a stable homogeneous equilibrium might be relatiely simple, since the only nearby steady structures, including traelling waes, are stable homogeneous states. The physical implication of Theorem is that when the initial perturbation is small in W s s;p > + p, the potential well of the wae is unable to trap particles foreer to form BGK waes. So the particles will get out of the potential well sooner or later and perform free ights, then the linear damping e ect might manifest itself at the nonlinear leel. Furthermore, when p = ; we get a much stronger result that any inariant structure near a stable homogeneous state in H s space s > 3 must be triial, that is, the electric eld is identically zero. Theorem 3 Assume the homogeneous pro le f () H s () s > 3 : For any T < T (de ned by (3)), there eists " >, such that if (f (t) ; E (t)) is a solution to the nonlinear VP equation (a)-(b) and kf (t) f k L H s < " ; for all t ; (4) 5

6 then E (t) for all t : The space L H s is contained in the Sobole space H;. s The aboe theorem ecludes any nontriial inariant structure, such as almost periodic solutions and heteroclinic (homoclinic) orbits, in the H s s > 3 neighborhood of a stable homogeneous state. In Theorem 4, we also show that nonlinear decay of electric eld is true for any positie or negatie inariant structure (see Section 5 for de nition) in the H s s > 3 neighborhood of stable homogeneous states. These results reeal that in contrary to the H s s < 3 case, there are no obstacles in the H s s > 3 neighborhood of stable homogeneous states to preent nonlinear Landau damping: We note that Theorems, and 3 about the contrasting nonlinear dynamics in W s;p spaces with s < + p or s > + p (particularly when p = ), hae no any analogue at the linear leel. Indeed, under Penrose s stability condition, it is shown in Section 4 that the linear decay of electrical elds holds true for ery rough initial data, particularly, no deriaties of f (t = ) is required for linear damping. We refer to Propositions 4 and 3, as well as emark 5 in Section 4 for more details. This shows once again the importance of particle trapping e ects on nonlinear dynamics, which are completely ignored at the linear leel. Finally, we brie y describe main ideas in the proof of Theorems, and 3. For simplicity, we look at steady BGK waes. The rst attempt would be to construct BGK waes near (f () ; ) directly by the bifurcation theory. Howeer, this requires a bifurcation condition: for bifurcation period T > ; = T f () d: (5) For general homogeneous equilibria and period T, the bifurcation condition (5) is not satis ed. For eample, for Mawellian, this condition fails for any T >. Our strategy is to modify f () to get a nearby homogeneous state satisfying (5) and then do bifurcation near this modi ed state. In the modi cation step, we introduce two parameters, one is to to obtain (5) and the other one is to ensure that the modi cation results in a small W s s;p < + p norm change. For the proof of non-eistence of traelling waes in W s s;p > + p, our idea is to get an second order equation for the electrical eld E () from steady Vlaso- Poisson equations and show the integral form of this equation is not compatible when T < T and the perturbation is small in W s s;p > + p. Interestingly, T (de ned by (3)) is eactly the critical period for linear stability by Penrose s criterion, which is also used in the proof of Corollaries and. To proe Theorem 3, we use the integral form of the linear decay estimate (Proposition 4) and the H s s > 3 inariant assumption to obtain similar nonlinear decay estimates in the integral form. From such integral estimates, we can show the homogeneous nature of the inariant structures and the decay of electric eld for semi-inariant solutions. 6

7 Here we are in a position to o er a conceptual eplanation why s = + p appears as the critical Sobole eponent for the eistence of small BKG waes and possibly also in the nonlinear Landau damping. By Penrose stability criterion, the critical spatial period T for linear stability of (f () ; ) is determined in (3) by integrals f c i d, where c i are critical points of f. These integrals are controlled by jjf jj W s;p if s > + p, but not if s < + p. In the latter case, a small homogeneous perturbation to f in W s;p space may dramatically change its stability for any ed spatial period T. Due to this change of stability, bifurcations occur and produce small BKG waes and possibly other complicated structures. In the opposite case when s > + p, small homogeneous perturbations does not change the stability of (f () ; ), therefore the bifurcation of nontriial waes cannot occur and the nonlinear Landau damping may be epected. The result of this paper has also been etended to a related problem of iniscid decay of Couette ow ~ = (y; ) of D Euler equations. The linear decay of ertical elocity near Couette ow was already known by Orr ([4]) in 97. This iniscid decay problem is important to understand the formation of coherent structures in D turbulence. In [34], we are able to obtain similar results near the Couette ow. This paper is organized as follows. In Section, we proe the eistence result in W s s;p < + p. In Section 3, non-eistence of BGK waes in W s s;p > + p is shown. In Section 4, we study the linear damping problem in Sobole spaces. In Section 5, we use the linear decay estimate in Section 4 to show that all inariant structures in H s s > 3 are triial. The appendi is to reformulate Penrose s linear stability criterion used in this paper. 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 waes in W s;p s < + p In this Section, we construct small BGK waes near any homogeneous state in the space W s s;p < + p. Our strategy is to rst construct BGK waes near proper smooth homogeneous states. Then we show that any homogeneous state can be approimated by such smooth states in W s;p. Lemma Assume u () C (), h supp u [ b; b], and u () is een, then pb; p i there eists g C (), supp g b ; such that u () = g. Proof. The proof is essentially gien in [4, P. 394]. We repeat it here for completeness. When k is odd, since u (k) () is odd we hae u (k) () = pb; p. By Theorem..6 in [4], we can choose g C b with the Taylor epansion P u (k) () k = (k)!: Then all deriaties of u () = u () 7

8 g anish at. De ne g () = g () + u ( p ) if > g () if : Then g () satis es all the required properties. In particular, g () is C at = because all deriaties of u () anish there. Proposition Assume f is een near =, and f > ; f () C () \ W ;p () (p > ) ; f () d = ; f () d < : Then for any ed s < + p ; T > ; and any " >, there eist steady BGK solutions of the form (f " (; ) ; E " ()) to (), such that (f " (; ) ; E " ()) has period T in, f " (; ) > ; E " () is not identically zero, and T kf " f k L ; + jf f " j dd + kf " f k W s;p < ": (6) ; Proof. Assume f () is een in [ a; a] (a > ). Let () = (jj) to be the cut-o function such that () C () ; () ; () = when jj ; () = when jj. (7) By Lemma, there eists g () C (), supp g p a; p a ; such that f () = g a : De ne g + () ; g () C () by 8 >< f ( p p ) a + g () if > p a p g () = p g () if a < a >: if p : a Then g+ f () = if > g if : Since f () =, f W ;p () \ C (), we hae f () d <. Indeed, f () d f () jj d + f () jj d! p kf ma jf ()j + jj jj d jj p k L p p = ma jf ()j + jj p kf k L p < : 8

9 We consider three cases. Case : f (). d < T Choose a function F () C () ; such that F W ;p () ; F () is een, F () > ; F () d < ; F () F () d < ; d > : (8) An eample of such functions is gien by! ( ) F () = ep + ep! ( + ) ; where is a large positie constant. Indeed, F () F () F () d = d > ; when is large enough, and other properties in (8) are easy to check. Since F () is een, by Lemma, there eists G () C () such that F () = G. Let ; > be two small parameters to be ed, de ne f ; () = + C f () + F ; (9) where C = F () d >. Note that f ; C ()\W ;p () ; f ; () d = ; and f; () f () d = + C d + F () d : Since f (), d < T there eists < < such that f () < d + F () f () d < < d + F () T d: Thus there eists > small enough, such that f ; < () f; d < < () d; when < <. () T We look for steady BGK waes near the homogeneous states (f ; () ; ). Consider a steady BGK solution f (; ) ; E () = () to (). Denote e = () to be the particle energy. From the steady Vlaso equation, f (; ) is constant along each particle trajectory. So for trapped particles with ma < e < min, f depends only on e, and for free particles with e > min, f depends on e and the sign of the initial elocity. We look for BGK waes near (f ; ; ) of the form f ; (; ) = 8 < : h +C h +C g g + (e) + G (e) + G i e () i e () if > if : () 9

10 For kk L su ciently small, f ; (; ) > and it satis es the steady Vlaso equation, since in particular for trapped particles f ; (; ) = f ; (; ). To satisfy Poisson s equation, we sole the ODE = (; ) d = f ; + C := h ; () : " g + (e) d + > g (e) d + G Then h ; C (). Since f = ; (; ) = f ; (), so h ; () = f ; () d = and ( h ; () = + C g+ d + > f; = () d:! # e () d g! ) d + () G () d Thus when < < ; < < ; we hae h ; () <, which implies that = is a center of the second order ODE = h ; () : () So by the standard bifurcation theory of periodic solutions near a center, for any ed (; ) ; there eists r > (independent of ( ; )), such that for each < r < r, there eists a T (; ; r) periodic solution ;;r to the ODE () with k ;;r k H (;T (;;r)) = r. Moreoer,! T (; ; r) By (), when r is small enough, f; () d, when r! : T (; ; r) < T < T (; ; r) : Since T (; ; r) is continuous in ; for each ; r > small enough, there eists T (; r) ( ; ), such that T (; T ; r) = T. De ne f;r T (; ) = f ; T (; ) from () by setting = ;T ;r and let E ;r () = ; T ;r (). Then f ;r T (; ) ; E ;r () is a nontriial BGK solution to () with period T. For any ed >, let () = lim r! T (; r) [ ; ] :

11 By the dominant conergence theorem, it is easy to show that f T ;r (; ) f ;() () L ; + T f T ;r (; ) + f T ;r (; ) f ;() () W ;p! ; ; f ;() () dd when r = j ;T ;rj H (;T )! : Since s < + p < ; for any > small, there eists r = r (; ") > such that f T ;r (; ) f ;() () L ; + T f T ;r (; ) + f T ;r (; ) f ;() () W s;p < " ; ; f ;() () dd Net, we show that the modi ed homogeneous state f ;() () is arbitrarily close to f () in the sense that f () f ;() () L + T f () f ;() () d+ f () f ;() () W s;p! ; ; when! : Note that the deiation is f () f ;() () = + C Since () [ ; ], when!, () F () d = C f () F () d! ; F : () F d = 4 () F () d! ; () () F = () + p () p kf ()k L p! ; L p d d () F = () p () p kf ()k L p! ; L p and thus f () f ;() () L + T f () f ;() () d+ f () f ;() () W ;p! : ; It remains to check jdjs d d f () f ;() ()! ; when! ; L p

12 where jdj ( > ) is the fractional di erentiation operator with the Fourier symbol jj : By using the scaling equality jdj d () = d jdj ; d where d () = (=d), we hae d jdjs d () F = () L p s+ p () s+ p k(jdj s F ) ()k L p! ; when!, since s < + p. So we can choose = (") > small such that f f L ;() + T f () f () ;() () d+ f f W ;() < " s;p () : Then (f " ; E " ) = f(");r((");") T (; ) ; E (");r((");") () is a steady BGK wae solution satisfying (6). Case : f (). d > T Choose F () = ep ; then F () d <. De ne f ; () as in Case (see (9)). Then there eists < < such that f () < d + F () f () d < < d + F () T d: The rest of the proof is the same as in Case. Case 3: f (). d = T For > ; de ne f () = f : Then f C () \ W ;p () ; f () > ; f () d = ; and f () d = f () d: For any " > small, there eist < (") < < (") such that < f () d < < f () T d and when ( (") ; (")) ; kf () f ()k L () + T jf () f ()j d+kf () f ()k W ;p () < " : For ( (") ; (")) ; we consider bifurcation of steady BGK waes near (f () ; ), which are of the form f (; ) = g + g e e if > if ; e = () ; E =. (3)

13 The eistence of BGK waes is then reduced to sole the ODE = f (; ) d := h () (4) As in Case, for any ( (") ; (")) ; 9 r (") > (independent of ) such that for each < r < r, there eists a T (; r) periodic solution ;r to the ODE (4) with k ;r k H (;T (;r)) = r. Moreoer,! T (; r) f () d, when r! : So when r is small enough, T ( ; r) < T < T ( ; r) and there eists T (r; ") ( (") ; (")) such that T ( T ; r) = T. De ne fr;" T (; ) = f T (r;") (; ) with = T (r;");r in (3) and E r;" () = T (r;");r (). Then f r;" T (; ) ; E r;" () is a nontriial BGK solution to () with period T. Let (") = lim r! T (r; ") [ (") ; (")] As in Case, by dominance conergence theorem, we can choose r = r (") > small enough, such that Then fr(");" T (; ) f (") () + + L ; f T r(");" (f " ; E " ) = (; ) f T is a steady BGK wae solution satisfying kf " f k L ; + T f T r(");" (; ) (") () W ;p ; < " : fr(");" T (; ) ; E r(");" () f (") () dd jf f " j dd + kf " f k W ;p ; < "; which certainly implies (6). This nishes the proof of the Proposition. To nish the proof of Theorem, we need the following approimation result. Lemma Fied p >, s < + p and c. Assume f W s;p () ; f >, f () d =, and f () d <. Then for any " >, there eists f " () C () \ W ;p (), such that f " is een near = c; and kf " f k L () + jf " f j dd + kf " f k W s;p () ": 3

14 Proof. Let () be the standard molli er function, that is, ( C ep () = if jj < if jj ; and () =. De ne f () := ()f () : Then by the properties of molli ers, we hae f C () ; f () > ; f () d = ; and when is small enough f k L () + jf f j dd + kf f k W s;p () " : (5) kf We can assume f () W ;p. Since otherwise, we can modify f () near in nity by cut-o to get f ~ () such that f ~ () W ;p and f f ~ L + f f ~ f dd + ~ W " f () s;p () : Solely to simplify notations, we set c = below: Let () = (jj) to be the cut-o function de ned by (7). Let > be a small number, and de ne f () + f ( ) f ; () = f () + f () f ( ) = f () Then obiously, f ; C () ; f ; () > ; f ; () d = f () d = ; and f ; () is een on the interal [ ; ]. Below, we proe that: when is small enough f k L () + jf f j d + kf f k W s;p () " : (6) kf Since kf f ; k L f () d jj jf f ; j d ( ) jj f () d; kf f ; k L p kf k Lp [ ; ] ; 4

15 (f f f ; ) = () + f ( ) + f () f ( ) ; k@ (f f ; )k L p f Lp [ + ma j ; ] j f () f ( ) f Lp [ + ; ] ma j j f d so when!, kf f ; k L + Net, we show L p f jj g L p f jj g f Lp [ ; ] + ma j j (4 ) p f Lp [ ; ] ( ) p ( + ma j j) f Lp [ ; ] ; jf f ; j d + kf f ; k W ;p! : k@ (f f ; )k W s ;p ()!, when!. This follows from Lemma 3 below, since s f () f ( ) W s ;p () = f (( ) ) d W s ;p () C j j So when!, kf f ; k L + p f L p + j j < p and f (( ) ) d W s ;p () ( p s+) jdj s f L p jf f ; j d + kf f ; k W s;p! : d C kf k W s;p () : Thus by choosing small enough, (6) is satis ed. By setting f " = f ;, the conclusion of the lemma follows from (5) and (6). Lemma 3 Gien f W p ;p () \ L ; g W s;p () p > ; s < p, then for any > ; f g () W s;p () and f g!, when!. (7) W s;p 5

16 Proof. First, we cite a result of Strichartz ([44]): Gien h W p ;p () \ L ; h W s;p () ; then h h W s;p () and kh h k W s;p C kh k ;p W p ; kh k L kh k W s;p. Aboe result immediately implies that f g () W s;p (). To show (7), for any " > ; we pick g C () such that kg g k W s;p < ". Since Lp f + jdj p Lp f = p kf ()kl p + jdj p f ; L p so when, f W p ;p C kfk W p ;p ; for some C independent of. Thus f g W s;p W f (g g ) + f g W s;p s;p W C kfk ;p W p ; kfk L kg g k W s;p + f C kg k ;p s;p W p ; kg k L C kfk ;p W p ; kfk L " + p kf ()kl p + p s jdj p f C Letting!, we get lim f g C W s;p! L p kfk ;p W p ; kfk L ": kg k W p ;p ; kg k L Since " is arbitrarily small, (7) is proed: Proof of Theorem. Fied the period T > and the trael speed c :Then by Lemma, for any " small enough, there eists f () C () \ W ;p (), such that f () is een near = c and kf f k L () + T jf f j d + kf f k W s;p () "=: Our goal is to construct traelling BGK wae solutions of the form near (f () ; ), such that kf " (; ) f ()k L ; + (f " ( ct; ) ; E " ( ct)) jf " (; ) f ()j dd+kf " (; ) f ()k W s;p ; < " : It is equialent to nd steady BGK solutions (f " (; + c) ; E " ()) near (f ( + c) ; ). By Proposition, there eists steady BGK solution (f (; ) ; E ()) near (f ( + c) ; ) such that E () not identically ; kf (; ) f ( + c)k L + jf ; (; ) f ( + c)j dd 6 :

17 Setting + kf (; ) f ( + c)k W s;p ; < " (5 + 4c ) : f " (; ) = f (; c) ; E " () = E () ; then (f " ( ct; ) ; E " ( ct)) is a traelling BGK solution and kf " f ()k L + ; ( c) jf " f ()j dd Since j jjjcj + kf " f ()k W s;p ; < " (5 + 4c ) : cj jj = when jj jcj ;so jf " (; ) f ()j dd jf " (; ) f ()j dd + jjjcj jf " (; ) 4 ( c) jf " (; ) f ()j dd + 4c kf " f k L ; < 4 + 4c " (5 + 4c ) ; and thus kf " f ()k L ; + " < (5 + 4c ) c (5 + 4c ) = " ; So kf " f ()k L ; + and the proof of Theorem is nished. f ()j dd jf " (; ) f ()j dd + kf " (; ) f ()k W s;p ; " jf " (; ) f ()j dd + kf " (; ) f ()k W s;p ; < ": emark For steady BGK waes (f (; ) ; E ()) of the form E () = and f (; ) = + (e) if ; e = (e) if < () ; (8) with + ; C () ; such as constructed in the proof of Theorem, E () has only two zeros in one minimal period. This is because the electric potential satisfying the nd order autonomous ODE = + d + d = h () (9) < 7

18 with h C () : Any periodic solution of minimal period to the ODE (9) has only one minimum and maimum, and therefore E = anishes at only two points. By Theorem, for T >, near any homogeneous equilibria we can construct small BGK waes such that multiple of its minimal period equal T. By [3] and [3], any of such multi-bgk waes are linearly and nonlinearly unstable under perturbations of period T. So far, the eistence of stable BGK wae of minimal period remains open, although some numerical eidences suggest the eistence of such stable BGK wae. For eample, in [5] starting near a unstable multi-bgk wae, numerical simulations shows that the long time asymptotics is to tend to a seemingly stable BGK wae of minimal period. emark In ([] [3]), Dorning and Holloway (see also [], [7]) studied the bifurcation of small traelling BGK waes with speed p near homogeneous equilibria (f () ; ) under the bifurcation condition ( p ) = P f () p d > ; () where P denotes the principal alue integral. It is equialent to nd steady BGK waes near (f ( + p ) ; ). The approach in ([] [3]) is as follows. De ne f e;p () = (f ( + p ) + f ( + p )) ; f o;p () = (f ( + p ) f ( + p )) : Then d d f e;p () d = P f () p d = ( p ) > : So by the bifurcation theory, there eist small BGK waes (f e (; ) ; ) near (f e;p () ; ) with periods close top, and f e (; ) is een in. Net, the (p) odd part f o;p (; ) is de ned by f o;p (; ) = e G o (e) if min G o (e) if < ; where () is the cut-o function as de ned in (7) and G o (e) = f o;p when e >. De ne f (; ) = f e;p (; ) + f o;p (; ) ; p e then (f (; ) ; ) is a steady BGK wae, since for trapped particles with e < min, f o;p (; ) = and f (; ) only depends on e. It can be shown that (f (; ) ; ) is close to (f ( + p ) ; ) in L p ; norm. The periods of the BGK waes constructed aboe are only near. In [] [3], [7], it was suggested p (p) that BGK waes with eact period p and " (p) close to (f ( + p ) ; ) in L p ; 8

19 norm can be constructed by performing aboe bifurcation from (( + (")) f ( + p ) ; ) for proper small parameter. It should be pointed out that this strategy actually does not work to get eact period p. Since to ensure that (( + (")) f e;p () ; ) (p) is a bifurcation point, it is required that ( + (")) f e;p () d = ; and thus (") =, i,e, is not adjustable at all. Second, by Lemma 4 below, d d j ( p )j = f e;p () d kf e;p ()k W ;p = kf ()k W ;p. So by the method in [] [3], one can not get small BGK waes with spatial periods less than = p kf ()k W ;p. By comparison, we construct BGK waes with any minimal period near any homogeneous equilibrium (f () ; ) in any W s s;p < + p neighborhood. It is also claimed in [] [3] that for p such that ( p ) <, there eist no traelling BGK waes with trael speed p, arbitrarily near (f () ; ). For Mawellian f () = e, the critical speed is about p = :35 since ( p ) < when p < :35: Howeer, by our Theorem, BGK waes with arbitrary trael speed eist near (in W s;p space, s < + p ) any homogeneous equilibrium including Mawellian. So the claim of the critical trael speed based on () is not true. Proof of Corollary. From the proof of Theorem and Proposition, it follows that: Fied T > ; for any " >, there eists a homogeneous pro le f " () C () \ W ;p () ; such that f " () ; f " () d = ; and kf " () = k f" () = d with f" ( " ) = ; T " f ()k L () + T De ne f () C () \ W ;p () by f () = f " " + " ; Then f () ; f () d = and k () = jf " () f ()j d+kf " () f ()k W s;p () < " : f () " d = 9 : T ()

20 We consider two cases below. Case : f" ( " ) >. By Lemma 7 and emark 6 thereafter, there eist unstable modes of the linearized VP equation around (f " () ; ) ; for wae numbers k in the internal (k ; k ). Here k is de ned by k f" () = d; c and c is a maimum point f " (). If there is no maimum point c of f " such that f" () d < k c ; then k =. Choose < such that kf " () f ()k L () + T jf " f j d + kf " f k W s;p () < " : () Then again by Lemma 7 and emark 6, there eist unstable modes of the linearized VP equation around (f () ; ) ; for wae numbers k in the internal (k () ; k ()). Since k () > k and k () k ()! k k > when!, we hae k (k () ; k ()) when is close enough to. This implies that (f () ; ) is linearly unstable under perturbations of period T. Moreoer, the inequalities () and () imply that kf () f ()k L () + T jf () f ()j d+kf () f ()k W s;p () < ": Case : f" ( " ) <. Choose > su ciently close to, then by the same argument as in Case, (f () ; ) is linearly unstable under perturbations of period T and kf () f ()k L () + T jf () f ()j d+kf () f ()k W s;p () < ": This nishes the proof of Corollary. 3 Noneistence of BGK waes in W s;p s > + p In this Section, we proe Theorem. The net lemma is a Hardy type inequality. Lemma 4 If u () W s;p () p > ; s > p ; and u () = ; then for some constant C. u () d C kuk W s;p () ;

21 Proof. Since s > p, the space W s;p () is embedded to the Hölder space C ; with ; s p. So ju ()j = ju () u ()j jj kuk C ; C kuk W s;p jj and thus u () d u () d + C kuk W s;p C kuk W s;p (). jj jj u () d + d + jj! p d kuk jj p L p Proof of Theorem. Suppose otherwise, then there eist a sequence " n!, and nontriial traelling wae solutions (f n ( c n t; ) ; E n ( c n t)) to () such that E n () is not identically zero; T E n () d = ; f n (; ) and n () are T periodic in, T f n (; ) dd < and kf n (; ) f ()k W s;p < " n: ; The traelling BGK waes satisfy and ( c n f n E f n = ; = + f n d + : (4) Because f n W; s;p with s > + p > p ; so by Sobole embedding kf n k L ; By a standard estimate in kinetic theory, C kf n k W s;p ; < : n = f n d kf n k 3 L ; 3 f n d (5) and thus n L 3 (; T ). So E n () W ;3 (; T ) which implies that E n () H (; T ) and E n () is absolutely continuous. De ne two sets P n = fe n 6= g and Q n = fe n = g. Then P n is of non-zero measure and En = a.e. on Q n : Thus we hae T T jen ()j d = n () En () d = n () En () d: (6) P n

22 Since s > p, by the trace theorem for fractional Sobole f n j =cn f n j =cn L p (; T ) : So from equation f n j =cn = for a.e. P n. By Lemma 4, for a.e. P n f n d c n () C kf n (; )k W s;p L p (P n ) : From (3), when P n ; n () = and it follows from (6) that T je n ()j f n c n de n () L p (P n ) P f n c n de n () d = : (7) Denote jp n j to be the measure of the set P n. We consider two cases. Case : jp n j! when n!. Since ke n k L (;T ) ke nk L (;T ) p T ke nk L (;T ) ; so from (7), kenk L (;T ) T ke f n L (;T ) P n c n dd T kenk L (;T ) kf n (; )k W s;p d P n T kenk L (;T ) kf n (; ) P n C kf n (; ) T ke nk L (;T ) < ke nk L (;T ) ; f k W s;p d + jp n j kf k W s;p f k W s;p + jp nj kf ; k W s;p when n is large enough. Thus for large n; kenk L (;T ) = and thus E n () ; which is a contradiction. Case : jp n j! d > when n!. When n is large enough, we hae jp n j d. By the trace Theorem, k@ f n (; c n f (c n )k L p (P n) k@ f n (; c n f (c n )k L p (;T ) C kf n f k W s;p C" n : f n (; c n ) = for a.e. P n, so k@ f (c n )k L p (P n) C" n which implies that j@ f (c n )j C" n. d p

23 f (c n )! when n! +. Therefore there eist a subsequence of fc n g, such that either it conerges to one of the critical points of f, say i S or it dierges. We discuss these two cases separately below. To simplify notations, we still denote the subsequence by fc n g. Case.: c n! i S. ewrite (7) as T where V n () = jen ()j d f n d c f d E n () d + V n () E n () d; (8) i P n P f d = (f n (; + c n ) f ( + i )) d: Note (f n (; + c n ) f ( + i )) j = = for P n, so by Lemma 4, we hae jv n ()j d C kf n (; + c n ) f ( + i )k W s;p d P n P n C C T kf n kf n f k W s;p So P n jv n ()j d! when n!. H (; T ) is T periodic; we hae Also by the assumption of Theorem, + kf ( + c n ) f ( + i )k W s;p d f k W s;p ; + kf ( + c n ) f ( + i )k W s;p ke nk L (;T ) T ke nk L (;T ) : a i = Combining aboe, from (8), we f i < Since T E n () d = and E n : T kenk L (;T ) ma fa i; g ke n k L (;T ) P + jv n ()j d ke n k L n T ke nk L (;T ) < ke nk L (;T ) ;! ma fa i ; g + T jv n ()j d P n when n is large enough. A contradiction again. Case.: fc n g dierges. We assume c n! +; and the case when c n! is similar. Again, for a.e. P f n (; c n ) =. Let n () be a cut-o function such that: n ; n () = when c n ; 3cn ; 3 T

24 n () = when = c n ; 3cn + and j n j C M (independent of n). Since W s;p,! W s;p when s > s ; we can assume p < s. Then P n C T f n d c n d P n + k f n k W s ;p P n j c nj cn k (f n f )k W s ;p + k f k f n c n d f n d d c n ( n f n c n + c + p W s ;p n kf n k W ;p C (M) kf n f k W s;p + CT k n@ ; f k + CT c + p W s ;p n kf n k W ;p ;! ; when n!, d d d and this again leads to a contradiction as in Case. In the aboe, we use two estimates: i) ii) k (f n f )k W s ;p C (M) k@ (f n f )k W s ;p k f! ; when n!. (9) kw s ;p We proe them below. Estimate i) follows from the following general estimate: Gien u () C () ; then for any g W ;p () (p > ; ), we hae kugk W ;p () C (kuk C ) kgk W ;p () : (3) This estimate is obious for = and =, and the case (; ) then follows from the interpolation theorem. To show estimate ii), we rst note that for any h C () ; obiously k n C k! ; when n! : hkw s ;p nhkw ;p Then the estimate (9) follows by using the fact that C () is dense in W s ;p and the estimate (3). This nishes the proof of Theorem. In the aboe proof of Theorem, we do not assume that the possible BGK waes to hae the form (8) or the electric eld to anish only at nitely many points. So we can eclude any traeling structures which might hae the form of a nontriial wae pro le plus a homogeneous part. The following Lemma shows that the condition < T < T in Theorem is necessary. Lemma 5 Assume f () C 4 () \ W ;p () (p > ) : Let S = f i g l i= be the set of all etrema points of f and < T < + be de ned by T = ma is f () d = i f () m d (3) : 4

25 Then 9 " >, such that for any < " < " there eist nontriial traelling wae solutions (f " ( m t; ) ; E " ( m t)) to (), such that (f " (; ) ; E " ()) has period T in, E " () not identically zero, and T kf " f k L + jf ; f " j dd + kf " f k W ;p < " (3) ; Proof. To simplify notations, we assume m =. As in the proof of Lemma, for > we de ne f () + f ( ) f () = f () + f () f ( ) = f () ; where () is the cut-o function de ned by (7). Then we hae: f i) f () = ; () f d = () d = ; and ii) f () C 4 () \ W ;p () ; kf f k W ;p ()! ; when! : Property i) follows since () is een. To proe property ii), we only need to show that k@ (f f )k L p ()! when!. Since in the proof of Lemma, it is already shown that kf f k W ;p ()! when!. Note (f f ) = (f () f ( )) + (f () + f ( )) + (f () f ( )) Since and = I + II + III: f () f ( ) = = jf () f ( )j p C ( ) f () d C p f (s) ds = s ( ) f () d + + f () dds ( ) f () d T ( ) f () d; p! p jf ()j p d ( ) p p d + jf ()j p d C p kf k p L p ( ;) ; p! ( + ) p p d 5

26 so jij p Similarly, and d C p jf () f ( )j p d C p C kf k p L p ( : ; ) jiij p d C p C p thus when!, k@ (f f () d p kf k p L p ( ;) p + p kf k p L p ( ;) d f jiiij p d C kf k p L p ( ; ) ; p! () d d d C kf k p L p ( ; ) f )k L p ()! : Choose > such that kf f k W ;p () < "=: Since f C 4 () \ W ;p () and f () d = ; this is eactly the Case 3 treated in the proof of Proposition, so we can construct a nontriial BGK solution (f " ; E " ) near (f () ; ) satisfying T kf " f k L ; + jf " f j dd + kf " f k W ;p < " ; : Thus (f " ; E " ) is a BGK solution satisfying (3). From the proof of Theorem, it is easy to get Corollary. Proof of Corollary. Suppose otherwise, then there eists a sequence " n!, and homogeneous states ff n ()g which are linear unstable with period T and kf n f k W s;p () < " n. By Lemma 7, for each n, there eists a critical point n of f n () such that Since f n () n d > T : T jf ( n )j k@ (f n f )k C() C kf n f k W s;p () C" n; either f n g conerges to one of the critical point of f () ; say, or f n g dierges. As in the proof of Theorem, in the rst case, we hae f n () f d! () d; when n! : n 6

27 This implies that f () d > T ; a contradiction. For the second case, we hae f n () n d! ; when n! ; a contradiction again. 4 Linear damping In this section, we study in details the linear damping problem in Sobole spaces. First, the linear decay estimates deried here are used in Section 5 to show that all inariant structures in H s s > 3 neighborhood of stable homogeneous states are triial. Second, the linear decay holds true for initial data as rough as f (t = ) L, and this suggests that Theorems, and 3 about nonlinear dynamics hae no analogues at the linear leel. We refer to emark 5 for more discussions. The linearized Vlaso-Poisson around a homogeneous state (f () ; ) is the = + f d; (33) where f and E are T periodic in and the neutralizing condition becomes T f dd =. Notice that any (f; E) = (g () ; ) with g () d = is a steady solution of the linear system (33). For a general solution (f; E) of (33); the homogeneous component of f remains steady and does not a ect the eolution of E. So we can consider a function h (; ) which is T periodic in and T h (; ) d =. Denote its Fourier series representation by h (; ) = X e i T k h k () : We de ne the space H s H s by h H s H s 6=k if khk H s H s X k6= T jkj s kh k k H s A < : Proposition Assume f () H s () s > 3 and let < T + be de ned by (3). Let (f (; ; t) ; E (; t)) be a solution of (33) with period T T < T and g (; ) = f (; ; ) T f (; ; ) d. If g Hs H s with js j s ; then kt s E (; t)k 3 L t H +s+s for some constant C : C kgk H s H s C kf (; ; )k H s H s ; (34) 7

28 One may compare this proposition with other smoothing estimates in PDEs. Here based on the most naie estimate, the initial alue g H s H s H; s+s only implies E() H s+ which is much weaker than H 3 +s+s in the aboe proposition. Howeer, this improed regularity of E may blow up as t!. Proof. To simplify notations, we assume T =. Let g (; ) = X e ik g k () ; then by assumption Let then f (; ; t) = X k6= 6=k kgk H s H s = X jkj s kg k k H <. s 6=k e ik h k (; t) ; E (; t) = X k6= e ik E k (t) ; E k (t) = h k (; t) d; E k () = g k () d: ik ik Below we denote C to be a generic constant depending only on f :When k > ; we use the the well-known formula for E k (t) where G k (z) = E k (t) = +i i i + g k () + d; F (z) = z G k ( p=ik) k F ( p=ik) ept dp; (35) f () d, Im z > z and is chosen so that the integrand in (35) has no poles for e p >. The formula (35) was deried in Landau s original 946 paper ([9]) by using Laplace transforms. Here we follow the notations in ([48]). By using the new ariable z = p=ik, we get E k (t) = k i k + i k G k (z) k F (z) e ikzt dz: (36) By assumption k = T > T, so by Penrose s criterion (Lemma 7), there eist no unstable modes to the linearized equation with period =k. Therefore, k F (z) 6= when Im z >. Moreoer, by the proof of Lemma 7, under the condition k > T ; k F ( + i) 6= for any. It is also easy to see that F ( + i)! when!. So there eists c >, such that k F ( + i) c k ; for any and k: (37) 8

29 Note that for z = i + ; when! +; by (5), g k () G k (z)! G k ( + i) = P d + ig k () = Hg k + ig k ; and F (z)! F ( + i) = P f () d + if () = Hf + if ; where H is the Hilbert transform. So letting! +, from (36), we hae E k (t) = k G k ( + i) k F ( + i) e ikt d: (38) Let A k (t) = G k ( + i) k F ( + i) e it d be the Fourier transform of H k () = G k ( + i) k F ( + i) ; then E k (t) = ka k (kt). Since H : H s! H s is bounded for any s, kg k ( + i)k H s By (37) and the inequality C kg k k H s ; jf ( + i)k H s C kf k H s : kf f k H s C s;s kf k H s kf k H s ; if s > ; jsj s ; we hae kh k k H s k kg k ( + i)k H s + F ( + i) k 4 G k ( + i) F ( + i) =k H s C k kg kk H s + F ( + i) k F ( + i) =k C H s k kg kk H s : where C depends on f but not k. In the aboe, the second inequality holds since the estimates F ( + i) =k c and kf ( + i)k H s C kf k H s ; imply F ( + i) F ( + i) =k < C kf k H s (39) H s through direct eri cation where, for < s <, one needs to use the equialent characterization of W s;p ( n ) when < s < ; p > (See [45, Lemma 35.]): p W s;p ( n ) = u L p ( n ju () u (y)j ) j n n j yj n+sp ddy < : 9

30 So and jtj s ja k (t)j dt kh k k H s C k 4 kg kk H s kt s E k (t)k L = jtj s je k (t)j dt = jtj s k ja k (kt)j dt = k s jtj s ja k (t)j dt Ck 3 s kg k k H s : For k <, the same estimate kt s E k (t)k L C jkj 3 s kg k k H s follows by taking the comple conjugate of the k > case. Thus ; kt s E (; t)k L t H 3 +s+s = X jkj 3+s+s kt s E k (t)k L k6= C X k6= jkj s kg k k H s = C kgk H s H s. This nishes the proof. The decay estimate in Proposition 4 is in the integral form. With some additional assumption on the initial data, we can obtain the pointwise decay estimate. Proposition 3 Assume f () H s () s > 3 and let < T + be de ned by (3). Let (f (; ; t) ; E (; t)) be a solution of (33) with period T < T and T g (; ) = f (; ; ) f (; ; ) d: T If g H s, then s H s, g H s H s with s > ; s + s and ma fjs j ; js jg kek H s (t) = o t s +s, when t! ; where 3 s = min + s + s ; + s + s : (4) Corollary 3 Assume f () H s () s > 3 and T < T. (i) If g H 3 L and g H L, then kek L (t)! when t!. (ii) If g; g H k ; with k s ; then kek H k+ (t) = o t k when t!. Proposition 3 and its Corollary shows that linear damping is true for initial data of ery low regularity, een in certain negatie Sobole spaces. It also 3

31 shows that the decay rate is mainly determined by the regularity in, although the regularity in a ects the norm of electrical eld that decays. Proof of Proposition 3. First we derie a formula for E t (t). We notice that (f t ; E t ) satis es the linear system (33) and f t (; ; ) f (; ; ) + E (; ) f () = X e ik ikg k () g k () d f () = ik k6= where 3X ~g j (; ) ; ~g (; ) (g (; )) ; ~g (; ) = X e ik g k () () d f () = X e ik ~g k () ; ik k6= k6= ~g 3 (; ) = X e ik g k () ( ()) d f () = X e ik ~g k 3 () ; ik k6= k6= and () is the cut-o function de ned by (7). Then ~g H s H s ; ~g H s+ H s ; ~g 3 H s + H s, and k~g k s H H s = kgk s H H s j= k~g k H s+ H s = X k X k jkj s jkj s kg k k H s g k () () d f k ()k H s H s kf k H s C kgk H s H s ; k~g 3 k X H s + H s k jkj s kg k k H s Correspondingly, we decompose () H s kf k H s C kgk : H s H s (f t ; E t ) = 3X i= ft i ; Et i with f i t ; E i t being the solution of (33) with initial data f i t (t = ) = ~g i (; ). Then by Proposition we hae t s E L t t H +s +s C kgk s ; H H s and t s E 3 t t s L t H 5 +s +s 3 E t L t H 5 +s+s C kgk s : H H s C kgk H s H s

32 For any t > t su ciently large and s de ned by (4), we hae kek H (t ) kek s H (t ) s t t 3X = he (t) ; E t (t)i H s dt ke (t)k H s E i t (t)! H s dt t t s s + t + t t t t i= kt s E (t)k H s t s E H t +s dt +s t kt s E (t)k H s t s E t t s (s ) t s (s ) kt s E (t)k H s t s E 3 t dt t 5 H +s+s t s s kt s E (; t)k 3 L t (t ;t )H +s+s t s E L t t (t;t)h +s + t s E +s t L 5 t (t;t)h +s+s + t s Et 3 o So nkek H (t) is a Cauchy sequence, thus lim s t! kek H (t) eists and s t must be zero since kt s Ek L < with s t Hs >. By ing t and letting t! in the aboe computation, it follows that kek H (t ) = o t s s s : This nishes the proof. emark 3 The integral decay estimate in Proposition 4 is optimal and the pointwise decay estimate in Proposition 3 is close to be optimal. Intuitiely, the integral estimate (34) suggests that ke (; t)k H 3 +s+s dt = o t (s+ ) : (4) In [48], the single-mode solution e ik (f (; t) ; E (t)) with initial pro le L t (t;t)h 5 +s +s : f (; ) = g () = ( ) e ( ) ; is arbitrary constant, was calculated eplicitly for the linearized problem at Mawellian, and the decay rate for je (t)j was found to be O t 3. Note that g () ; g H and g ; (g) are delta functions which belong to H ( +") for any " > ; and thus g () ; g H 5 ". So Proposition 4 suggests a decay rate o t (3 ") in the integral form 5 and Corollary 3 (ii) yields a pointwise decay rate o t. +" In [, pp ], the authors made a more general claim about the decay rate of single mode 3

33 solutions: for initial pro le g () with (n + ) th deriatie being function like, the decay rate of je (t)j is O t (n+). In such cases, our results gie the decay rates o t (n ") in the integral form and o t (n+ ") pointwise. In Theorem 3, we use the integral estimate (34) to proe that H 3 is the critical regularity for eistence or noneistence of nontriial inariant structures near stable homogeneous states. This again suggests that the decay estimate in Proposition 4 is optimal. emark 4 The linear decay result is also true for initial data in L p space. For simplicity, we consider a single mode solution (f (; ; t) ; E (; t)) = e ik (h (; t) ; E (t)) (4) to (33) with h (; ) = g (). Assume f () L () \ W ;p () (p > ) and < T + be de ned by (3). We hae the following result: If T = k < T and g () L p (p > ) ; g L, then je (t)j! when t! +. We proe it brie y below. Since g L p ; g L, so kgk L () jgj d + jgj d =p kgk L p + g L < ; and jj kgk L q jgj jj L jgj L p g kgk L L p ; for < q < satisfying q = + p. Since q < p, for any < q < q; letting q = q q ; we hae kgk L q () kgk L q (jj) + kgk L q (jj)! C kgk L p + kgk L q < : L q (jj) Since H is bounded L p! L p for any p > and the Fourier transform is bounded L p! L p for any < p, so from (38), ke (t)k L q C kgk L q () < : (43) As in the proof of Proposition 3, (f t ; E t ) satis es (33)with f t (t = ) = e ik ikg () g () d f () = e ik ~g () : ik Since k~g ()k L q by using the estimate for E (t) ; we get C kgk L q + kgk L () kf k L q < ; ke (t)k L q C k~g ()k Lq < : (44) The decay of je (t)j follows from the estimates (43) and (44). 33

34 emark 5 In Proposition 3, we proe that the linear decay of electrical eld E in L norm holds true for initial data as rough as f (t = ) H 3 L ; f (t = ) H L : In particular, it is not necessary to hae any assumption on deriaties of f (t = ) to get linear decay of E. The linear decay result implies that there eist no nontriial inariant structures een in H 3 L space for the linearized problem. So our result on eistence of BGK waes in W s s;p < + p neighborhood (Theorem ) can not be traced back to the linearized leel. Also, the contrasting nonlinear dynamics in W s s;p > + p and particularly in H s s > 3 spaces (Theorems and 3) hae no analogue on the linearized leel. These again are due to the fact that particle trapping e ects are completely ignored on the linear leel, but instead they play an important role on nonlinear dynamics. 5 Inariant structures in H s s > 3 We de ne inariant structures near a homogeneous state (f () ; ) in H s ; (s ) space to be the solutions (f (t) ; E (t)) of nonlinear VP equation (a)- (b), satisfying that for all t ; kf (t) f k Hs ((;T )) < " ; for some constant " >. The aboe de ned inariant structures include the well known structures such as traelling waes, time-periodic, quasi-periodic or almost periodic solutions. In Sections and 3, we proe that W + p ;p is the critical regularity for eistence of nontriial traelling waes near a stable homogeneous state. For p = ; this critical regularity is H 3. In this section, we proe a much stronger result that H 3 is also the critical regularity for eistence of any nontriial inariant structure near a stable homogeneous state. In the proof, we use the linear decay estimate in Proposition 4. Lemma 6 Assume f () H s () s > 3 and let < T + be de ned by (3). Let (f (; ; t) ; E (; t)) be a solution of (a)-(b) with period T < T, satisfying that: For some 3 < s s and su ciently small " ; kf (t) f k L H s ((;T )) < " ; for all t : Then ( + t) s E (; t) L ftg H 3 C" ; (45) for some constant C: Proof. Denote L to be the linearized operator corresponding to the linearized Vlaso-Poisson equation at (f () ; ), and E is the mapping from f (; ) 34