The Vlasov-Poisson-Landau System in a Periodic Box

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1 The Vlasov-Poisson-Landau System in a Periodic Box Yan Guo Division of Applied Mathematics, Brown University March 9, Abstract The classical Vlasov-Poisson-Landau system describes dynamics of a collisional plasma interacting with its own electrostatic eld as well as its grazing collisions. Such grazing collisions are modeled by the famous Landau (Fokker-Planck) collision kernel, proposed by Landau in 936. We construct global uniue solutions to such a system for initial data which have small weighted H norms, but can have large H k (k 3) norms with high velocity moments. Our construction is based on accumulative study on the Landau kernel in the past decade [G] [SG-3], with four extra ingredients to overcome the speci c mathematical di culties present in the Vlasov-Poisson-Landau system: a new exponential weight of electric potential to cancel the growth of the velocity, a new velocity weight to capture the weak velocity di usion in the Landau kernel, a decay of the electric eld to close the energy estimate, and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude. Introduction In the absence of magnetic e ects, the dynamics of charged dilute particles (e.g., electrons and ions) is described by the Vlasov Poisson-Landau t F + + v r x F + + e + E r v F + = Q(F + ; F + ) + Q(F ; F + ); m + t F + v r x F m E r vf = Q(F + ; F ) + Q(F ; F ); () F (; x; v) = F ; (x; v): Here F (t; x; v) are the spatially periodic number density functions for the ions (+) and electrons (-) respectively, at time t, position x = (x ; x ; x 3 ) [ ;] 3 = T 3, velocity v = (v ; v ; v 3 ) R 3, and e ; m the magnitude of their charges and masses, c the speed of light. The collision between charged

2 particles is given by Q(G ; G )(v) = c r v m R 3 (v v ) G (v )r v G (v) G (v)r vg (v ) dv m m () where is the famous Landau (Fokker-Planck) kernel [G]: (u) = u u I juj juj and c = e e ln ; ln = ln( D b ); D = ( n ee ) = being the Debye shielding distance and b = e 3T being a typical distance of closest approach for a thermal particle [H]. The self-consistent electrostatic eld E(t; x) = r, and the electric potential satis es: = = fe + F + R 3 e F gdv; (t; x)dx = : T 3 () T (3) It is well-known that for classical solutions to the Vlasov-Poisson-Landau system, the following conservation laws of mass, total momentum, and total energy hold: d dt d dt d dt m F (t) = ; T 3 R 3 v(m + F + (t) + m F (t)) = ; T 3 R 3 T 3 R 3 jvj (m + F + (t) + m F (t)) + 8 je(t)j = : T 3 Moreover, we also have the following celebrated H-Theorem of Boltzmann d (F + (t) ln F + (t) + F (t) ln F (t)) : (5) dt T 3 R 3 It is our purpose in this article to construct uniue global solutions for the Vlasov-Poisson-Landau system () and () near global Maxwellians: + (v) = n e + ( m + T ) 3= e m+jvj =T ; (v) = n e ( m T ) 3= e m jvj =T : For notational simplicity, we normalize all constants in the Vlasov-(Poisson)- Landau system to be one. Accordingly, we normalized the Maxwellian as (v) + (v) = (v) = e jvj : (6) We de ne the standard perturbation f (t; x; v) to as F = + p f : (7)

3 Let f(t; x; v) = f+(t;x;v) f (t;x;v) ; the Vlasov-Poisson-Landau system for the perturbation now takes the form f@ t + v r x E r v gf fe vg p + L f = fe vgf + (f; f)(8) = pu[f+ f ]dv (9) with R dx = : For any g = T 3 is given by the vector L+ g Lg L g g g ; the linearized collision operator Lg in (8) p Q(; g ) + Q( p fg + g g; ) p Q(; p g ) + Q( p : () fg + g g; ) For g = [g ; g ] and h = [h ; h ]; the nonlinear collision operator (g; h) is given by the vector +(g; h) (g; h) p Q( g ; p h ) + Q( p g ; p h ) p (g; h) Q( p g ; p h ) + Q( p g ; p : () h ) By assuming that initially F has the same mass, total momentum and total energy as the steady state, we can then rewrite the conservation laws in terms of the perturbation f as f + (t) p f (t) p ; () T 3 R 3 T 3 R 3 vff + (t) + f (t)g p ; (3) T 3 R 3 jvj ff + (t) + f (t)g p je(t)j ; () T 3 R 3 T 3 In an attempt [G-] to construct global smooth solution near Maxwellian for the Vlasov-Poisson-Landau system, the author initiated a nonlinear energy method for a general dissipative t g + Lg = N (g): (5) We denote jj jj to be the L norm. Upon taking L inner product with g; we obtain d dt jjgjj + (Lg; g) = (N (g); g): (6) Global solutions with small L norm can be constructed if one can identify a dissipation rate jjj jjj such that the following estimates can be established (Lg; h) & jjjgjjj ; (7) (N (g); g). jjgjj jjjgjjj : (8) 3

4 For jjhjj << we obtain d dt jjhjj + jjjhjjj ; with some > : This implies global uniform bound for jjgjj and hence stability. Several remarks are needed: () Usually higher order Sobolev norms are needed to close the argument. () Such a dissipation rate jjj jjj satisfying both (7) and (8), if exists, usually is uniue up to a constant. The proof of (7) and (8) can be very challenging [G] [GrS-]. (3) Such an approach is designed to treat cases when jjj jjj is weaker, or it can not be compared with jj jj so that standard spectra analysis and semigroup approach is di cult to apply. () In many interesting applications, such (Lg; g) can not control full jjjgjjj ; and one has to control the missing part via further study of the nonlinear euation (5). In particular, in the context of Landau or Boltzmann euations, the linear collision operator L as in () can be shown to be positive de nite ([G], [SG-3]) along the nonlinear dynamics even it has a kernel. Such a exible approach turns out to be robust, leading to constructions of global solutions to several di erent applied PDE [G-5], [GS], [GrS-], [SG- 3], [GT], [Ha]. Unfortunately, despite these advances and many attempts, the original motivation in this program, the stability of Maxwellian for the Vlasov- Poisson-Landau system has remained out of reach, as pointed out in [SG3]. For other work related to the Landau euation from di erent approaches, see [AB], [AV], [CDL], [HY], [L], [V], and [-] among others. Let w(v) be a weight function and jj jj ;w to denote the weighted L norm. We de ne jjfjj ;w jjfjj H w jjhvi fjj;w + jjhvi 3 rv f v jvj jj ;w + jjhvi rv f v jvj jj ;w (9) with hvi = p + jvj : It is well-known that jjfjj jjfjj ; captures the dissipation for the Landau kernel [DL], [G]. There are two intrinsic di culties [G] associated with the Landau kernel: the vanishing factors hvi and hvi 3 make jjfjj soft, and their di erent vanishing rates along di erent directions makes jjfjj non-isotropic for r v f. There are two major mathematical di culties in the study of stability in Vlasov-Poisson-Landau system. The rst di culty is created by the (innocent looking!) nonlinear term E vf in (8). Such a term comes from the factor p in our linearization F = + p f ; which is the only known choice so far capturing the linear dissipation rate of L, a linear analog to the entropy production in the fundamental Boltzmann s H-theorem: Hence, the presence of the term E vf is a basic feature of interaction between the electric eld and the particles in the near Maxwellian context. Upon multiplying f one hopes to bound jvef j () in terms of jjfjj jjfjj (9). The electric eld E = E f behaves nicely, but the extra velocity factor v makes the control by jjfjj (which only controls

5 jjhvi = fjj in (9)) impossible. In a simpler model problem [G], the Landau collision is replaced by the hard-sphere interaction so that R jvjf can be exactly bounded by the dissipation rate of the hard-sphere kernel. In a relativistic Landau collision kernel [SG3] [DL], the relativistic counterpart is R je ^pf j; where the velocity ^p is bounded, and jjfjj is controlled by the relativistic Landau dissipation. These two facts led to the resolution for the relativistic Landau problem [SG3]. As a matter of fact, this extra v factor is the key reason that only hard-sphere like interaction can be treated for the Boltzmann type of euations in the presence if a force term. With even a bit weaker (softer) than hardsphere interaction, () is beyond the control of either the energy or dissipation rate so that even the local in-time solutions can not be constructed within this framework. The second main di culty stems from controlling the velocity v derivative of f: Taking v derivative of (8), we estimate jjr v fjj via the standard energy method. In this process, the free streaming term v r x f produces a uadratic term as: r x f r v f: () It is well-known that r v f can produce growth in time in the kinetic theory. To estimate () by the norm (9) is tricky; again due to the negative weight. In the absence of the electric eld E [G], the author designed a weighted norm to overcome this di culty with more v derivatives associated with more negative velocity weights. The rst step was to take pure x derivative (with no weight and no electric eld E) and to bound jjr x fjj : In the next step, jjhvi r v fjj (instead of jjr v fjj ) was estimated in the v derivative of (8), which contains a dissipation rate of jjr v fjj ;hvi : The weighted mixed terms () could be bounded by (see (9)): hvi jr x f r v fj C " jjr x fjj + " jjr v fjj ;hvi ; which could be closed for " small. Such a weight disparity between x and v derivatives of f has played an important role in treating soft potentials [G3][GrS- ]. Unfortunately, in the presence of E; this strategy fails completely because R t jjr xfjj can not be estimated independently of r v f at the rst step: In fact, taking x derivatives of (8) produces new contribution Er x f r v f () Now this can not be controlled with a norm for r v f with a negative weight hvi : The key to overcome the rst di culty () is to realize that, instead of treating E vf as a second order perturbation in (8), we need to combine or 5

6 cancel it with the linear term streaming term v r x f in (8), which also contains an extra v factor! Upon using the fact that E = r x in this problem, upon multiplying with e ; we can rewrite e [v r x f r x vf ] = v r x fe f g: Indeed, such a perfect derivative leads no contribution in the integration (see (7)). Even though such a spatial weight destroys the exact energy structure from R the Poisson euation, fortunately, new error contributions are of the type rx v p f (e ) which can be controlled if is small. Our observation works for all forces given by a potential. To overcome the second main di culty and to control the v derivative of f; we need to design new weight function in v: In light of (), we need to assign same weight functions for both r x f and r v f, which seems to contradict to the weight disparity for controlling (). We observe crucially that jjfjj contains (weak) v derivative of f: Hence r v f can be also viewed, not as a part of jjr v fjj ; but as a part of jjf jj ;hvi with no v derivative but with an extra stronger weight hvi : In fact, thanks to (9), we can estimated () by r x f r v f jjr x fjj jjfjj ;hvi provided jjfjj ;hvi is controlled at an earlier step. The weight disparity is not between x and v derivatives, but between jjfjj ;hvi and jjr x fjj : That is, less derivatives of f should reuires stronger weight. Since higher spatial derivatives are associated with weaker velocity weight in our norm (6), more careful analysis is needed for spatial Sobolev imbedding to close the energy estimate, especially when we take one derivative (Step in the proof of Proposition 6). Such a cascade of weight takes advantage of the crucial feature of the Landau operator: a weak gain of r v : Because of this reason, our new strategies do not work for a soft potential with Grad s angular cuto, even if it just a bit weaker than the hard-sphere interaction. On the other hand, it is interesting to use such a weight for the full inverse power law without angular cuto [GrS-]. For notational simplicity, we use jj jj p to denote L p norms with weight w(v) in T 3 R 3 or T 3 ; and jj jj p;w for L p norms with weight w(v): Let the multi-indices and be = [ ; ; 3 ]; = [ ; ; 3 ]; and we x 3 3 v 3 : If each component of is not greater than that of s, we denote by ; < means ; and jj < j j: We de ne the velocity weight w(; )(v) e jvj hvi (l jj jj) ; l jj + jj; < : (3) The presence of e jvj would lead to stretched exponential decay. Recall (9). 6

7 Let the instant energy and dissipation rate are: E m;l; (f)(t) jj@ f (t)jj ;w(;) ; () D m;l; (f)(t) We remark that from the de nition, jj+jjm jj+jjm jj@ f (t)jj ;w(;) ; (5) E ;l+m; (f) E ;l+m ; (f) ::: E m;l; (f); D ;l+m; (f) D ;l+m ; (f) ::: D m;l; (f); (6) so that less derivatives of f demands stronger velocity weight. It is important to note that due to the presence of (), all our estimates can only be obtained with (nonlinear) exponential weight of e (+) f (see Es (76) to (86) and Lemma ): But if the electric potential jj f jj is bounded (as we shall prove); such weighted norms are euivalent to E m;l; (f) and D m;l; (f); and we need to use E m;l; (f) and D m;l; (f) without the nonlinear weight to close our continuity argument. Our main result is as follows. Theorem Assume that f satis es the conservation laws (), (3), () with F ; (x; v) = + p f ; (x; v) : There exists a su ciently small M > such that if E ;; (f ) M; then there exists a uniue global solution f(t; x; v) to the Vlasov-Poisson-Laudau system (8) and (9) with F (t; x; v) = + p f (t; x; v) : () If E ;l; (f ) < + for l and ; then there exists C l > ; Furthermore, sup E ;l; (f(s)) + s D ;l; (f(s))ds C l E ;l; (f ): (7) jj@ t (t)jj + jjr x (t)jj + jjf(t)jj C l ( + t) l+ E ;l; (f ); (8) jj@ t (t)jj + jjr x (t)jj + jjf(t)jj C l e C lt =3 E ;l; (f ) for > :(9) () In addition, if E m;l; (f ) < for any l > ; l m ; ; there exists an increasing continuous function P m;l () with P m;l () = such that the uniue solution satis es sup E m;l; (f(t)) + t D m;l; (f(s))ds P m;l (E m;l; (f )): (3) The continuous function P m;l () can be determined inductively on m: We remark that we only reuire E ;; (f ) to be small, but for l > and m ; the high momentum or high Sobolev norm E m;l; (f ) can be arbitrarily large. A H 7

8 type of construction was rst carried out in [GrS-] for the Boltzmann euation with an non-cuto inverse power collision kernel. Note from the Sobolev s imbedding, L is not necessarily bounded by E ;; (f ): We also note that the estimate (3) is uniform in time, so that we verify the bounds in [DV] and more decay can be obtained (see also [SG-]). Such an estimate (3) also gives a natural approximation mechanism to establish the gain of smoothness for t > [CDH]. Throughout the paper, we introduce the notation A. B (A & B and A v B) if A is bounded by B up to a universal constant C which does not depend on either l or m: The introduction of the weight spatial weight function e (+) creates a new analytical di culty: we need to control (see Es. (5) and (6)) [jj@ t (s)jj + jjr x (s)jj ]ds (3) to close the global energy estimate for E ;; (f): Note that (3) is di erent from the dissipation estimate R D ;; (f(s))ds < since jj@ t (s)jj + jjr x (s)jj. D ;; (f(s)); and D ;; (f(s)) is expected to be small most of the time. It is a typical di culty that R p D;; (f(s))ds < can not be derived directly from the energydissipation estimate (7). We need to make an interplay (see (3)) between the decay estimate of (8) (with l = ; = ) and the energy estimate (7). This is di erent from the program in [SG-], in which the energy estimate can be close alone rst and the decay is obtained after. In fact, the proof of (8) is intertwined with (7) with l =, and we are able to close a di erential ineuality () with up to only one spatial derivative. This leads to a decay rate of jj@ t (s)jj + jjr x (s)jj v s in terms of (7) with l = ; which is su cient to close the estimate. The strong decay rate of s is a conseuence of the periodic box T 3 and it remains an open uestion if su cient decay rate can be obtained for the Vlasov-Poisson-Landau system in the whole space R 3 : The last novelty of the paper is the proof of high moments and high regularity (3) with only small E ;; (f ): It is important to note, that we reuire l m; the total number of derivatives in w(; ) (3): This is because that the starting point of our method is to control pure x derivatives of f without any weight, which demands that l m at the highest level of derivatives. It is also important for our analysis to reuire w(; ) : This implies that jjfjj ;l E m;l; (f ); which can not be small for all l! Therefore it is natural to assume E m;l; (f ) is only nite but not small for l > : The hope is to obtain, at the highest order 8

9 derivatives,. jw (; ) (@ f; f)@ fj + E ;; (f)jj@ f(t)jj ;w(;) ; jw (; ) f)@ fj (3) so that the smallness of E ;; (f ) would be su cient. Unfortunately, due to the combination of the non-local feature of the Landau operator as well as the velocity weight w(; ), as in Lemma [SG], a term like E ;l; (f)jj@ f(t)jj jj@ f(t)jj ;w(;) will occur in the upper bound for (3). Since p E ;l; (f) is large for l >, we can not absorb the whole product by the dissipation rate of jj@ f(t)jj ;w(;). In Proposition 6, we are able to move the weight function from p E ;l; (f) to jj@ f(t)jj and the price to pay is an additional contribution of C l D ;l; (f)jj@ f(t)jj ;w(;) : Even though C l D ;l; (f) can be very large, but jj@ f(t)jj ;w(;) belongs to the instant energy not the dissipation rate, therefore we can still control this via the Gronwall lemma and the fact R t D ;l;(f)(s)ds < : A new splitting of the domain needs to designed to achieve this goal. We believe that our method will lead to new estimates for solutions recently constructed in [GrS-] for the inverse power law. Moreover, the presence of E also makes the construction of the local solutions more delicate and we need to employ fractional Sobolev spaces to gain compactness of the approximate solutions. Our paper is organized as follows. In section, new re ned estimates are developed to cope with nonlinear terms in (8). In section 3, local in time solutions are constructed via estimates of the pseudo energy and dissipation rate () and (5). In section, decay estimates (8) and (9) are obtained to bootstrap into global in time solutions. Basic Estimates We use h; i to denote the standard L inner product in R 3 v for a pair of functions g = g+ g and h = h + h and de ne: hf; hi = hf + ; h + i + hf ; h i: (33) We also denote j j ;w and j j ;w to be corresponding w weighted L and H norms (9) in R 3 v: Recall the linear Landau operator L in (). We rst recall a basic property of L (see [H] [G] [SG3] for a proof). 9

10 Lemma We have hlg; hi = hlh; gi; hlg; gi ; and Lg = if and only if g = P g where P being the L v(r 3 ) projection with respect to the vector L inner product (33) onto the null space of L : p span ; p p ; v i ; jvj p ; (3) with i 3: Moreover hlg; gi & jfi P ggj j[fi P gg] j (35) where [fi P gg] are two components of fi P gg: Lemma 3 Let w(v) : Then for any > sup jg(x; )j w. kjgj ;w k 7 x H. jj@ gjj ;w + C jjgjj ;w ; jj= sup jg(x; )j ;w. kjgj ;w k 7 x H. jj@ gjj ;w + C jjgjj ;w ; jj= jjgjj L x (L w ). kjgj ;w k 3 H. jj@ gjj ;w + C jjgjj ;w ; jj= jjgjj L (Hw ). kjgj ;w k 3 H. jj@ gjj ;w + C jjgjj ;w : jj= We note that L (T 3 ) H 7 (T 3 ), L (T 3 ) (T 3 ): Moreover, H H 7, and H compactly, which gives rise to the arbitrary small constant : The Lemma then follows from general functional Sobolev ineuality of H k, see [GrS-]. We also need the following version of the Gronwall Lemma. Lemma Let A(t); B(t); y(t) satisfy y(t) R t A(s)y(s)ds + B(t), then y(t) e R t A(s)ds A(s)B(s)ds + B(t): The following lemma is the key to treat the streaming term r v f r x f via our dissipation rate (5). Denote ei = if e i ; or ei = ; otherwise: (36) Lemma 5 We have w (; ) e i f@ f. jj e i fjj ;w(; ei)jj@ +ei e i f jj ;w(+ei; e i):

11 Proof. Recalling (3), we note w( + e i ; e i )(v) = hvi w(; )(v): We rewrite from e e i ei (9) +ei e i f to get w (; ) e i f@ f jw(; )hvi = e e i )(v) = w(; )(v) and w(; and we use e i f jjw(; )hvi +ei e i f j jjw(; e i )hvi 3= e e i f jj jj@ +ei e i f jj ;w(;) ; where hvi = w(; ) = hvi 3= w(; e i )(v): We now use (9) again e i f to conclude the proof jjw(; e i )hvi 3= e e i f jj. jj e i f jj ;w(; ei): The following proposition is a re ned estimate for the non-linear collision term (g; g) in [G] [SG-]. The key improvement is that the factors in front of the highest order dissipation rate is bounded by only p E ;; (g )+ p E ;; (g ): Proposition 6 () For jj + jj = m ; recall w = w(; ) in (3): We have [g ; g g i. C l j@ g j j@ g j ;w j@ g j ;w + ; [j@ g j j@ g j ;w + j@ g j j@ g j ;w ]j@ g j ;w :(37) Moreover, for any > ; there exists C l; > such that [g ; g g ijdx T 3. (E ;; (g ) + )D ;l; (g ) + C l; j@ g j () For jj + jj = m 3; we have. [g ; g g i C C C l ; j j+j j jj+jj j j+j j jj+jj j j+j j jj+jj C C C C C C j j+j j j@ g j j@ g j ;w j@ g j ;w E ;l; (g ):(38) [j@ g j j@ g j ;w + j@ g j j@ g j ;w ]j@ g j ;w [j@ g j ;w j@ g j + j@ g j ;w j@ g j ]j@ g j ;w C l;;j@ g j j@ g j ;w j@ g j ;w : (39)

12 Furthermore, for any > ; there exists C l;m; > jhw (; )@ [g ; g g ijdx T 3. f E ;; (g ) + E ;; (g )gfjj@ g jj ;w + jj@ g jj ;wg +f E ;; (g ) + E ;; (g ) + g fjj@ g jj ;w + jj@ g jj ;wg +C l;m; j j+j j[ m ] j j+j j=m ( j@ g j w( ; ; ) hvi + j@ g j w( ; ; ) hvi ) fe m;l; (g ) + E m;l; (g )g +C l;m; fe m ;l; (g ) + E m ;l; (g ) + gfd m ;l; (g ) + D m ;l; (g )g: () We remark that from (3), (5) and Lemma 3, j j+j j[ m ] j j+j j ( j@ j@ gj gj ; w( ; ) hvi ).. j j+j j;jj j j+j j[ m ] jj gjj. D ;; (g); () gjj ;w( +; ). D [ m ]+;l; (g); (replacing by H ) which are su cient for our estimates for global solutions. However, for the construction of the local solutions, such sharper estimates are important for compactness of the approximate solutions. Proof. Recall Lemma in [SG] and Theorem 3 in [G]. We rst separate: (l jj i [w ] = fv i w g + + jvj v i w. () We can express [g ; g g i = P C C G ; with G ; as hw f [ g ]g@ g g i (3) ( + )hw f [v i g ]g@ g g i () +hw f [ g ]g@ g g i (5) +( + )hw f [v i g ]g@ g g i (6) (l jj jj) h + jvj w f [v i g ]g@ g g i (7) (l jj jj) +h + jvj w f [ = v g ]g@ g g i: (8) with double summations over i; j 3 ei vi i : Step. Estimate of (7) and (8).

13 We note that there is a (large) factor (l jj jj) in both (7) and (8): We follow exactly as in the proof of Lemma in [SG]. From ij =8 +jvj and the Cauchy-Schwarz ineuality, j [v i g ]j + j [v i g ]j. hvi j@ g j ; and we use the exponential weight v i = to bound j@ g j in (9): Therefore (l jj jj)jh w hvi ij f@ [v i g ]@ g g ij +(l jj jj)jh w hvi ij f@ [v i g ]@ g g ij w C l;m j@ g j hvi 3 (j@ j@ g j + j@ g j)j@ g jdv C l;m j@ g j j@ g j ;w j@ g j ;w ; where we have used jwhvi g j. j@ g j ;w by (9). This concludes the control of (7) and (8) via the rst terms on the right hand side of both (37) and (39). Step. Proof of (37) and (38) for jj + jj = m : In light of step, we only need to bound (3) to (6), which are precisely bounded as in Lemma of [SG] leading to (37). Note that one can eliminate the weight function in the g factor due to exponential decay factor = in the Landau integral. To conclude (38), we need to take x integration of (37) and we need to separate three cases. If jj + jj = ; w = w(; ); we have hw [g ; g ]; g i T 3. C l sup jg j jg j ;w jg j ;w + sup jg j jg j x T 3 x T ;w + sup jg j jg j ;w jg j ;w 3 x T 3. C l kjg j k 7 H D ;l; (g ) E ;l; (g ) + E ;; (g )D ;l; (g ). ( E ;; (g ) + )D ;l; (g ) + C l; kjg j k H 7 E ;l;(g ): We have used from () and (5) and Lemma 3: sup jg j. kjg j k 7 x H and sup jg j. E ;; (g ): (9) x If jj + jj =, either ( ; ) = or ( ; ) =. In the case 3

14 ( ; ) = ; we use (9) to bound similarly C l sup jg j j@ x T g j ;w j@ g j ;w 3 + sup jg j j@ x T g j ;w + sup jg j j@ 3 x T g j ;w j@ g j ;w 3. ( E ;; (g ) + )D ;l; (g ) + C l; kjg j k H 7 E ;l;(g ): On the other hand, if ( ; ) = ; we take L L L and use Lemma 3 to get C l j@ T g j jg j ;w j@ g j ;w + [j@ g j jg j ;w + j@ g j jg j ;w ]j@ g j ;w 3 T 3. C l j@ g j H 3 jj@ g jj ;w jj@ g jj ;w + g jj jj@ g jj ;w jj@ g jj ;w (5) where jj : Note jj@ g jj ;w. g jj. p E ;; (g ): For w = w(; ); D ;l; (g ). D ;l; (g ) from () and (5). Therefore, (5) is further bounded by j@ H C l g j 3 D ;l; (g ) E ;l; (g ) + E ;; (g )D ;l; (g ) j@. ( E ;; (g ) + )D ;l; (g ) + C l; g j E ;l; (g ): We remark that we can not take L x of sup x jg j ;w(;) and sup x jg j ;w(;) in this case, because w(; ) now is associated with jj + jj = ; and stronger than w allowed for second order derivatives from (3): for jj ; w(; ) < w(; ): As a conseuence, P jj jj@ g jj ;w(;) and P jj jj@ g jj ;w(;) can not be bounded by p D ;l; (g ) or p E ;l; (g ): We now consider the third case jj + jj = : We rst consider either = = or = = : (5) We take L x of the term without derivatives and L x of the other two terms. We note that by (3), accordingly, w(; ) w( + ; ) or w(; ) w( + ; ) (5) for jj : Therefore for any g sup x jgj. kjgj k 7 H ; and sup jgj. x E ;; (g) sup jgj ;w. jj@ gjj ;w(;). jj@ gjj ;w(;). x jj jj sup jgj ;w. jj@ gjj ;w(;). jj@ gjj ;w(;). x jj jj D ;l; (g) E ;l; (g): (53)

15 By Lemma 3 and (53), we bound (37) by [g ; g g ij T 3. C l kjg j k 7 H D ;l; (g ) E ;l; (g ) + E ;; (g )D ;l; (g ) + kjg j k E H 7 ;; (g ) D ;l; (g ). C l kjg j k 7 H D ;l; (g ) E ;l; (g ) + E ;; (g )D ;l; (g ): We note kjg j k 7 H D ;l; (g ) E ;l; (g ) D ;l; (g ) + C kjg j k H 7 E ;l;(g ): (5) We conclude (38) if (5) is valid. If ( ; ) 6= and ( ; ) 6= : Now j j + j j = and j j + j j = since jj + jj =. We take L L L to get: [g ; g g i j@. C l g H j 3 g jj ;w jj@ g jj ;w g jj g jj ;w +. C l j@ g j H 3 D ;l; (g ) j@ g H j 3 g jj ;w ]jj@ g jj ;w E ;l; (g ) + E ;; (g )D ;l; (g ); where jj : This concludes step by (5). Step 3. Proof of (39). In the case of j j + jj jj+jj ; the estimate again follows exactly as in Lemma of [SG] for (3) to (6). The case of j j + jj jj+jj is most delicate as we need to avoid a contribution of j@ g j jg j ;w j@ g j ;w in Lemma of [SG]. Our goal is to move the weight w out of jg j ;w to j@ g j : To accomplish this, we following exactly the proof of Lemma of [SG] but with a di erent splitting of the phase space v; v ; depending on l and m : jv j fjwj g; jvj " l;m; jwj and with " l;m < satisfying (l jj + jj from (3)): jv j jvj " l;m; jwj " l;m ( " l;m ) < and ( " l;m) (l jj jj) : (55) The estimate in the rst region fjwj g follows the proof of case in Lemma in [SG] with an upper bound of terms (3) to (6) for j j + j j jj+jj as [j@ g j j@ g j ;w + j@ g j j@ g j ;w ]j@ g j ;w (56). j@ g j ;w j@ g j j@ g j ;w + j@ g j ;w j@ g j j@ g j ;w ; 5

16 where we have used the important fact w and the property w(; ) : We remark even if > and l = in (3): w(; ) = e jvj hvi jj jj ; w c ;! as jj+jj! :A large constant of c ; in (56); when we bound j@ g j by j@ c ; j@ g j ;w jg j j@ g j ;w appears in front of j@ g j ;w j@ g j j@ g j ;w g j ;w : In particular can not be absorbed by the dissipation rate (5). This illustrates the importance of the choice of l jj + jj n to guarantee w : o jv For the second region j " l;m ; jwj we shall swap the weight w(v) by w(v ): Since ( jvj " l;m )jvj jv j ( + " l;m )jvj, we deduce that w(v) w(v ) l jj jj = e jvj jv j + jvj + jv j e jv j [ ( " l;m ) + + jv j jv j ( " l;m ) A l jj jj " l;m ( " e l;m ) jv j ( " l;m ) (l jj jj) e jv j (57) by (55). In the Case of the proof of Lemma of [SG], due to exponential decay of v e jv j in the Landau kernel which dominates e jv j ; by (57) we can move weight function w(v) = w(v ) w(v) w(v ) to g to obtain bounds (w ) for (3) to (6) as: [j@ g j ;w j@ g j + j@ g j ;w j@ g j ]j@ g j ;w : n o jv For the third region of j " l;m ; jwj ; we note jvj jv vj jjv j jvjj " l;m jvj and jvj l;m > : We can repeat the same argument as in Case 3 of the proof for Lemma [SG], upon further integration by parts in v to bring down the v derivative in j@ g j. This process creates a constant depending on (large) constant C l;m and we obtain a desired upper bound of C l;m j@ g j j@ g j ;w j@ g j ;w 6

17 to conclude step 3. Step. Proof of (). We shall separate three cases in (39). The rst case is either = ; = or = ; and = : Indeed we now have the coe cients C C = : Recall (5) and (5) in this case. We note that m 3 so that we can apply L x estimate to terms without derivatives, and then take corresponding L L L for the x integration. By (53), we obtain an upper bound of (39) C l [kjg j k 7 H jj@ g jj ;w + jj@ g jj kjg j k 7 H ] E m;l; (g ) +[ E ;; (g )jj@ g jj ;w + kjg j k E H 7 m;l; (g )]jj@ g jj ;w +[ E m;l; (g ) kjg j k + jj@ g H 7 jj ;w E ;; (g )]jj@ g jj ;w +C l;m E m;l; (g ) kjg j ;w k 7 H jj@ g jj ;w. f E ;; (g ) + E ;; (g ) + gfjj@ g jj ;w + jj@ g jj ;wg +C l;m; fkjg j ;w k H 7 + kjg j ;w k H 7 gfe m;l;(g ) + E m;l; (g )g: We have used the fact w and l ; : We note that.. kjg j ;w k H 7 + kjg j ;w k H 7 j@ g j ;w(;) j j+j j j j+j j[ m ] j@ g j w( ; ; ) hvi + j@ g j ;w(;) + j@ g j w( ; ; ) hvi ; (58) as [ m ] and w(; ) w( ; ) hvi for jj + jj j j + j j + from (3): We thus conclude the rst case. The second case is when j j + jj m and j j + j j m : In this case, we shall take L L L and by Lemma 3 to nd an upper bound of (39): j@ C l;m g H j@ j H 3 g j ;w 3 jj@ g jj ;w +C l;m [ j@ g H j@ j H 3 g j ;w 3 + j@ g H j@ j H 3 g j ;w 3 ]jj@ g jj ;w +C l;m [ j@ g H j@ j H ;w 3 g j 3 + j@ g H j@ j H ;w 3 g j 3 ]jj@ g jj ;w +C l;m j@ g j H 3 j@ g j ;w H 3 jj@ g jj ;w : (59) For the last three terms, we use H and the facts j@ H j@ gj ;w + gj ;w. E H m ;l; (g) j@ H j@ gj ;w + gj ;w. D m ;l; (g) H 7

18 to obtain a desired upper bound of jj@ g jj ;w + C l;m; fe m ;l; (g ) + E m ;l; (g )gfd m ;l; (g ) + D m ;l; (g )g: For the rst term in (59), we combine the factor of smaller total derivatives (less then [ m ]) with jj@ g jj ;w to obtain an upper bound: [ m ]j j+j jm +C l;m; j j+j j=m j j+j j[ m ] f +C l;m; fj@ g j ;w j@ g j ;w + j@ g j ;w + j@ g j ;w g gjj@ g jj ;w fjj@ g jj ;w + jj@ g jj ;wg + fd m ;l; (g ) + D m ;l; (g )g j j+j j[ m ] f j@ g j ; w hvi where w = w( ; ) and w(; ) w ( ; ) hvi + j@ g j ; w hvi gjj@ g jj ;w: (6) for jj + jj j j + j j + by (3): We therefore conclude the second case. We now consider the last case of following possibilities: = j j + jj; j j + jj = m, = j j + j j; or j j + j j = m : Note that in this case the weight function satis es: w(; ) w( + ; ); w(; ) w( + ; ); for jj : We bound (59) by taking L L L as j@ C l;m g H j@ j H 3 g j ;w 3 jj@ g jj ;w +C l;m [ j@ g H j@ j H 3 g j ;w 3 + jj+j j + j@ g H j@ j H 3 g j ;w 3 ]jj@ g jj ;w +C l;m [ j@ g H j@ j H ;w 3 g j 3 + j j+j j + j@ g H j@ j H ;w 3 g j 3 ]jj@ g jj ;w +C l;m j@ g H j@ j H 3 g j ;w 3 jj@ g jj ;w :(6) j j+j j The rst term above can be estimated exactly as (6). Note that from (5) j@ 3. E ;; (g); (6) H jj+j j gj 8

19 by Lemma 3, the second term in (6) is bounded by a desired upper bound: jj@ g jj ;w + C l;m; E ;; (g ) j@ g j ;w + C l;m; j@ g j. jj@ g jj ;w + C l;m; E ;; (g ) +C l;m; jj+j j j@ g j. jj@ g jj ;w + E ;; (g ) +C l;m; E ;; (g )D m ;l; (g ); j j+j j=m j j+j j=m E m;l; (g ) jj+j j j@ g j ;w jj@ g jj ;w( ; ) + C l;m;; E ;; (g )D m ;l; (g ) jj@ g jj ;w( ; ) + C l;m; jj+j j with further small. Similarly, the second and the third terms in (6) are bounded by a desired upper bound of (w = w(; )) 8 < jj@ g jj ;w +C l;m; : j j+j j j@ g j ;w. jj@ g jj ;w + C l;m; + j j+j j=m j j+j j E m;l; (g ) + j@ g j ;w j j+j j E m;l; (g ) jj@ g jj ;w( ; ) + C l;m;e ;; (g )D m ;l; (g ); where we have applied compact imbedding again to j@ g j ;w : By (58), we complete the proof of the proposition. In order to obtain decay for the electric eld, we now treat pure spatial derivatives in more details (f; f)@ f : Lemma 7 () If jj = ; ; ; then h@ (f; f idx. E ;; (f) j jjj jj@ fjj : () If jj = m 3; then for any > there exists C m; > such that jh@ (f; f ijdx. [ E ;; (f) + ]jj@ fjj + [ E ;; (f) + ] j@ g j j j=m +C m; [D ;; (f)e m;l; (f) + E m ;l; (f)d m ;l; (f)]: Proof. Since there is no weight w; we use Theorem 3 in [G] with w = : h@ (f; f idx. j@ fj j@ fj j@ fj dx: (63) C j@ g j ;w jj@ fjj E m;l; (g ) 9 = E ;; (g ) ; 9

20 We now estimate case by case according to jj. Assume jj = : We can take sup x jfj. p E ;; by () and L L L in (63) to obtain an upper bound of jfj jfj jfj dx. p E ;; jjfjj : Assume jj = : We take L L L in (63) if = ; and L L L in (63) if j j = : By Lemma 3 with = and w =, we obtain an upper bound of j@ fj j@ fj j@ fj dx. p E ;; jj@ fjj : jj Assume jj = : If = ; we take L L L ; if j j = we L L L ; if j j = ; we take L L L in (63) respectively. By Lemma 3 with w = and = ; we obtain: j@ fj j@ fj j@ fj dx. p E ;; jj@ fjj : jj Combining with the cases jj = ; ; we conclude part () of the lemma. Assume jj = m 3: We need to be careful with the (large) constant C. We single out the terms with highest order derivatives with either = or p = : By taking L on the term without any derivatives with sup x jfj. D;; ; sup x jfj. p E ;; ; we apply Lemma 3 with w = to nd an upper bound of (63): [jfj j@ fj + j@ fj jfj j@ fj ]dx p E ;; jj@ fjj + p D ;; jj@ fjj jj@ fjj [ p E ;; + ]jj@ fjj + C D ;; E m;l; (f): For the remaining cases of j j jj ; we always take L L L and use Lemma 3 with small (depending on ): By singling out the cases of j j = m and j j = ; and combining the rest of lower order terms together, we obtain C m fjj fjj jj@ fjj + C m; jj@ fjj jj@ fjj jj@ fjj jj= C m [ E ;; (f) jj=;j j=m fjj + D ;; (f) jj=;j j= fjj jj@ fjj ] +C m; E m ;l; (f) D m ;l; (f)jj@ fjj [C m E ;; (f) + ] jj@ fjj + C m; [D ;; (f)e m;l; (f) + E m ;l; (f)d m ;l; (f)]:

21 We thus conclude the lemma by further adjusting (depending on m). Next we estimate the other nonlinear terms with the electric eld. Lemma 8 Let jj + jj = m : For < with w = w(; ) de ned in (3) r x r f j + r [v@ f ]j. jj@ f jj ;w + C jf j w(;) ; jj@ r xjj + C [jjr xjj hvi H + m jjr xjj ]D m ;l; (f): Proof. Note < ; w(; ) = w( ; )hvi jj+jj = w( ; )hvi hvi [jj jj ] ; (6) from (9), we obtain: r x r f jwhvi r x hvi [jj jj ] w( ; )hvi 3= r f j j@ f j ;w j@ r x jj@ f j ; w( ;) hvi [jj j j ] dx (65) We now separate three cases. Case. j j + jj = m or m = : Since j j + jj + j j = m and > ; we have jj j j = in this case: Taking L L L in (65) yields a desired upper bound: jj@ f jj ;w jjr xjj j@ f j ;w(;). jj@ f jj ;w+c jjr xjj D m ;l; (f): For the rest of the cases, we take L L L of (65) to get an upper bound:. jj@ f jj ;w jj@ r x jj 3 H j@ f j w( ; ;) : (66) 3 H hvi [jj j j ] Case. j j = m : We have = = now. Since jj j j now; we further estimate (66) by () jj@ f jj ;w jj@ r x jj 3 H jf j w(;) ; hvi 3 H. jj@ f jj ;w + C jj@ r xjj jf j w(;) ; hvi. jj@ f jj ;w + C jf j w(;) ; jj@ r xjj : hvi

22 Case 3. j j+jj m and j j m : By j j+j j+jj = m; we deduce j j : Hence hvi [jj jj ] hvi : We estimate (66) as jj@ f jj ;w jj@ r x jj 3 H j@ f j w( ; ;) hvi 3 H. jj@ f jj ;w + C jj@ r x jj H j@. jj@ f jj ;w + C jj@ r jj D m ;l; (f). jj@ f jj ;w + C jjr jj H m D m ;l;(f): Here we have used the fact w(;) hvi j@ f j w( ; ;) = hvi H jj f j ; w( ;) hvi H w( + ; ) for jj from (3) so that D m ;l; (f): f j ; w( ;) hvi. f j ;w(+;) jj We now turn to the second term. Similarly, by (6) w(; ) = w( ; e i )hvi hvi [jj jj ] ; we have from (9): r [v@ f ] j r x v@ f ]j + C j r x e i f ]j jwhvi w( ; ) r x hvi hvi 3 [jj jj f j +C r x w( ; e i ) hvi hvi 3 [jj jj] e e i f j C m C m j@ f j ;w(;) j@ j@ f j ;w(;) j@ r x jj@ f j ; w( ;) hvi [jj j j ] r x jj e i f j ; w( ; e i ) hvi [jj j j] : The rst term is estimated exactly as in (66). For the second term, we note that jj so that j j m ; and j j + jj m : Since < j e i f j ; w( ; e i ) hvi [jj j j] j e i f j ; w( ; e i ) hvi ; we thus can apply case 3 above to complete the proof. We also need more precise estimate for purely spatial derivatives of E r v f : Lemma 9 Let = R p [f+ f ]dv with R = :

23 () For jj = ; ; then for < r x r f dv dx r x v@ f dv dx. E ;; (f) jj@ fjj : j jjj () For jj = m 3; then for < ;any > r x r f dv dx r x v@ f dv dx. E ;; (f)jj@ fjj + jj@ fjj + C m; D m ;l; (f)e m ;l; (f): j j=jj Proof. We rst perform integration by part in v to r x r f dv r x v@ f dv (67) = r r f dv dx r x v@ f dv dx [jhvi r v f j + f j]j@ r x [hvi f ]j. j@ f j j@ r x jjhvi f j dx: Here we have used norm (9) f : If jj = ; then since < so = : From the elliptic estimate: jj@ r x jj. jj@ r xjj = jj p@ f jj. jj@ fjj and P jj f jj. p E ;; (f): Since jhvi r v f j. j@ f j in (67), we take L L L to get (67). jj@ fjj jj@ r x jj jjhvi f jj (68) jj. jj@ fjj jjhvi f jj. jj E ;; (f)jj@ fjj : If jj = ; the case = is treated as in (68) and we only need to treat the case of j j = : Note jjhvi f jj. p E ;; (f): We take L L L in (67) to obtain (67). j@ f j j@ r x jjhvi f j dx. jj@ f jj jjr jj jjhvi f jj. jj@ f jj E ;; (f): (69) j jjj 3

24 Here we have used the elliptic estimate (j j = ): jjr jj. jjr xjj. j jjj jj@ fjj : This completes the proof of the rst part of the lemma. If jj = m 3; the case j j = is again treated in (68). When j j = m ; as in (69), by (), jjhvi f jj. E m ;l; (f); and jjr xjj. p D ;; (f): We take L L L in (67) to get (67). jj@ f jj jjr xjj jjhvi f jj. jj@ f jj + C D ;; (f)e m ;l; (f): When j j m ; jj@ r x jj. jj@ r xjj. jj@ fjj. jjhvi + f jj. E m ;l; (f): jj D m ;l; (f); By (69), we obtain by taking L L L in (67): jj@ f jj jj@ r x jj jjhvi + f jj jj. jj@ f jj + C m; D m ;l; (f)e m ;l; (f): This completes the proof of the lemma. 3 Local Solutions Our goal is to construct a uniue local-in time solution to the Vlasov-Poisson- Landau system (8) and (9) if E ;; (f ) is su ciently small. The construction is based on an uniform energy estimate for a seuence of iterating approximate solutions. We rst note, by direct computations P ij ij = 8 and the Landau collision operator has the following non-divergent form of [G]: We start with Q(G ; G ) = f ij G g@ ij G + 8G G F (t; x; v) = or f : (7) To preserve the positivity for F n+, we design the following iterating seuence of F n+ as [SG3]: [@ t + v r x r x n r v ]F n+ = Q(F n ; F n+ ) 8F n (F n+ F n ) +Q(F; n F n+ ) 8F(F n n+ F) n = ij [F n + F]@ n ij F n+ + 8fFg n + 8FF n ; n n+ = (F+ n+ F n+ )dv (7)

25 with R n+ = : We note that F n T 3 implies F n+ from (7): We now rewrite the above iteration in the perturbation form of F n+ = + p f n+ : [@ t + v r x r x n r v ]f n+ Af n+ r x n vf n+ = r x n v p + K f n + (f n ; f n+ ) 8(f n + f) n p (f n+ f) n 6(f n+ f) n n+ = (f+ n+ f n+ )dv: (7) with f n+ j t= = f : Here for g = g+ g ; we denote as in [G] [SG-]: Ag = K g = p Q(; p g ); p Q( p [g + g ]; ): To solve such f n+ ; we can add an arti cial dissipation "fa f n+ + x ( + jvj )f n+ g with A = p Q(; p g ) with (u) = juj I u u juj in (3). This choice makes the problem strongly parabolic in both x and v with strong bound in v which justi es the moments estimates [SG]: We shall construct f n+ as "! with uniform bound in ". The procedure is standard and for notational brevity, we ignore such a regularization. We of the Landau-Poisson system (setting " = ): [@ t + v r x r x n r v ]@ f n+ [r x n v@ f n+ Af n+ = e i f n+ C f@ [r x n v p ] K f n (f n ; f n+ ) 8@ [ p (f n + f)(f n n+ f)] n 6@ [(f n+ r x n r f r x [v@ f n+ ]g f n )]: 5

26 We multiplying with e () n w with w = w(; ) in (3) to get: [e (+)n f n+ ] f[@ t + v r x r x n r v ]@ f n+ [r x n v@ f n+ ]g = d nw (@ dt fe(+) f n+ ) g ( + ) n t e (+)n w (@ f n+ ) nw +v r x f e(+) (@ f n+ ) g ( + )v r x n e (+) n (@ f n+ ) w nw r x n r v f e(+) (@ f n+ ) g r x n v(@ f n+ ) e (+)n w (l jj jj) + jvj r x n ve (+)n w (@ f n+ ) r x n v(@ f n+ ) w e (+)n (73) where we have used (). Our weight function is so designed such that there is an exact cancellation for the high momentum contributions: Therefore, we can rewrite (73) as: ( + )v r x n e (+) n (@ f n+ ) w r x n ve (+)n w (@ f n+ ) r x n v(@ f n+ ) w e (+)n = : (7) d fe(+) nw (@ f n+ dt ) g ( + ) n t e (+)n w (@ f n+ ) nw +v r x f e(+) (@ f n+ ) g (75) nw r x n r v f e(+) (@ f n+ ) g (l jj jj) + jvj r x n ve (+)n w (@ f n+ ) : 6

27 Upon integration over T 3 R 3 ; and combining terms we obtain: = d dt ( e (+) n w (@ f n+ ) ) Af n+ f n+ i (76) e (+)n w e i f f n+ (77) < < C C e (+)n f r x n r f n+ (78) e (+)n f n+ r x [v@ f n+ ] (79) (l jj jj) [ + jvj r x n v ( + ) n t ]e (+) nw (@ f n+ ) (8) + w (e (+)n )@ f Af n+ (8) e (+)n w r [v p ]@ f n+ (8) + w e () K f f n+ (83) + w e (f n ; f n+ )@ f n+ (8) 8 w e [ p (f n + f)(f n n+ f)]@ n f n+ (85) 6 w e [(f n+ f)]@ n f n+ : (86) We now derive estimates for (76) to (86). Lemma Assume for M su ciently small, E ;; (f n ) M: 7

28 () We have E ;l; (f n+ ) + +C l [ D ;l; (f n+ )ds D ;l; (f n )ds + C l [E ;l; (f ) + +C l +C l +C l j j+j j j j+j j j@ f n j w( ; ; ) hvi j@ f n+ j w( ; ; ) hvi jj fjj f n jj + jj f n+ jj g] + jjr x n jj + jj@ t n jj ]E ;l; (f n+ ) E ;l; (f n ) E ;l; (f n+ f n )[ + jjf n jj 7 H ] E ;l; (f n+ ) jjf n+ f n jj 7 H E ;l; (f n ) E ;l; (f n+ ): (87) () For m 3; we have E m;l; (f n+ ; n ) + +C l;m [ D m;l; (f n+ )ds D m;l; (f n )ds + C l E m;l; (f ) + C l;m j j+j j[ m ] ( j@ f n j w( ; ; ) hvi fjj f n jj + jj f n+ jj g + j@ f n+ j w( ; ; ) [E m;l; (f n ) + E m;l; (f n+ )] +C l;m [ + jjf n jj H [ E m ]+ 3 ] m;l; (f n+ f n ) E m;l; (f n+ ) +C l;m jjf n+ f n jj H [ m ]+ 3 E m;l; (f n ) E m;l; (f n+ ) hvi ) + jjr n jj + jj@ t n jj] +C l;m [D m ;l; (f n ) + D m ;l; (f n+ )][E m ;l; (f n ) + E m ;l; (f n+ ) + ]: (88) We remark the exponential factor 8 is chosen for convenience. Proof. First note that n = R p (f n + f n )dv and R n = so that jj n jj. M by the elliptic estimate so that e (+)n. : 8

29 We shall estimate term by term in (76) to (86). In the second term of (76), we apply both Lemma 8 and 9 of [SG]. For = ; hw (; )@ Af n+ f n+ idx & jj@ f n+ jj ;w(;) C m jj@ f n+ jj ; (89) (v) being a general cuto function in v. For 6= ; for any > ; we have hw (; )@ Af n+ f n+ idx & jj@ f n+ jj ;w(;) D m;l; (f n+ ) C l; jj@ fn+ jj ;w(; ) :(9) From Lemma 5, we have for any > and e i < (77). jj@ e i f n+ jj ;w(; ei)jj@ +ei e i f n+ jj ;w(+ei; e i) D m;l; (f n+ ) + C jj e i f n+ jj ;w(; ei): (9) To estimate (78) and (79); from the elliptic estimate and our assumption, jjr n jj. jjf n jj H. p M; jjr n jj H m. jj pf n jj H m. E m ;m ; (f n ): We deduce from Lemma 8 that (78) + (79). jj@ f n+ jj ;w + C jf n+ j ; w(;) hvi In particular, if m = ; we have E m;m; (f n ) M and by (3) n+ jf. jj@ f n+ jj ;w(;). so that j ; w(;) hvi H Next, we easily control jj (78) + (79). ( + C M) jj+jj (9) E m;m; (f n ) + C [E m ;m ; (f n ) + M]D m ;l; (f n+ ): jj+jj jj@ f n+ jj ;w(;) ; jj@ f n+ jj ;w: (93) (8) C l;m fjj n t jj + jjr x n jj gw (@ f n+ ) : (9) To estimate (8); since (e (+)n ). p M; we control from Lemmas 8-9 of [SG]: for any > ; (8). p Mjj@ f n+ jj ;w(;) + C p l;m M jj@ f n+ jj (95) jjm +D m;l; (f n+ ) + C l;m; jj@ fn+ jj ;w(; ) : < 9

30 In (8); since < in (3), via repeated integration by part in v; we can move all the v out f n+ to the [v p w ]; so that (8). C l;m By Lemma 8 in [SG], we have (83). f D m;l; (f n ) + C l;m; 8 [j@ f n+ j + j@ f n j ]dvdx: (96) jjm. [D m;l; (f n ) + D m;l; (f n+ )] + C l;m; jj@ f n jj g D m;l; (f n+ ) (97) jjm jj@ f n jj : We now turn to (85) and (86). If jj + jj ; due to the decay of p and the fact < in (3); by product rule and Sobolev imbedding (Lemma 3) in T 3 : jjw@ [ p (f n + f)(f n n+ f)] n [(f n+ f)]jj n. Cf E ;l; (f n+ f n )[ + jjf n jj 7 H ] + E ;l; (f n )jjf n+ f n jj 7 H g; so that for jj + jj ; we have (85) + (86) (98). Cf E ;l; (f n+ f n )[ + jjf n jj 7 H ] E ;l; (f n+ ) + E ;l; (f n )jjf n+ f n jj 7 H g: For jj + jj = m 3; due to the decay of p ; we have jjw@ [ p (f n + f)(f n n+ f)] n [(f n+ f)]jj n (99). C m f E m;l; (f n+ f n )[ + jjf n jj H [ m ]+ 3 ] + jjf n+ f n jj H [ m ]+ 3 E m;l; (f n )g: We now prove the rst part (). Applying Proposition 6 with g = f n ; g = f n+, we choose small and M further small such that C M <<. By (3) and (5), j@ fj. j@ fj ; w( ; ) for j j + j j : We collect terms to hvi 3

31 get f n+ jj ;w +. C l jj@ f n+ ()jj w + +C l; 8 < +C l +C l : +C l; jj@ f n+ jj ;w jj@ f n jj ;w [jj f n+ jj + jj f n jj ] jj j j+j j j j+j j j j;j j<jj j@ f n j w( ; ; ) hvi j@ f n+ j w( ; ; ) hvi jj@ fn+ jj ;w( ; ) +C m f E ;l; (f n+ f n )[ + jjf n jj 7 H ] + 9 = + jjr x n jj + jj@ t n jj ; E ;l;(f n+ ) E ;l; (f n ) E ;l; (f n )jjf n+ Here we have bounded the cuto function by : As in [G], we can get R t P rid of the contribution of C l; j j;j j<jj jj@ fn+ jj ;w( ; ) which has less v derivatives. Upon an induction starting from jj = ; ; by choosing di erent small value of each time; we conclude that for any > (di erent value of C l ) that. jj+jj jj@ f n+ jj ;w + jj+jj 8 < +C l : +C l; +C l; jj+jjm jj@ f n+ jj ;w + C l; j j+j j j j+j j jj@ f n+ jj ;w jj+jjm j@ f n j w( ; ; ) hvi j@ f n+ j w( ; ; ) hvi jj@ f n+ ()jj ;w f n jj 7 H g E ;l; (f n+ ): 9 = + jjr x n jj + jj@ t n jj ; E ;l;(f n+ ) E ;l; (f n ) [jj f n+ jj + jj f n jj ] jj +C m f E ;l; (f n+ f n )[ + jjf n jj 7 H ] + E ;l; (f n )jjf n+ We sum over jj+jj then choose su ciently small (but xed) to conclude the rst part of our lemma. We remark that M is also small but xed. f n jj 7 H g E ;l; (f n+ ): 3

32 For jj + jj = m 3; we simply put terms like P < jj@ fn+ jj ;w(; ) into D m ;l; (f n+ ): We apply Proposition 6 and collect terms to get. jj@ f n+ jj ;w + jj@ f n+ jj ;w [D m;l; (f n ) + D m;l; (f n+ )] + C l;m; jj@ f n+ ()jj ;w +C l;m; +C l;m [ [jj f n+ jj + jj f n+ jj ] j j+j j[ m ] ( j@ f n j w( ; ; ) hvi + j@ f n+ j w( ; ; ) fe m;l; (f n ) + E m;l; (f n+ )g +C l;m; [ + jjf n jj H [ E m ]+ 3 ] m;l; (f n+ f n ) E m;l; (f n+ ) +C l;m; jjf n+ f n jj H [ m ]+ 3 E m;l; (f n+ ) E m;l; (f n ) +C l;m fe m ;l; (f n ) + E m ;l; (f n+ ) + gfd m ;l; (f n ) + D m ;l; (f n+ )g: We therefore conclude the lemma by setting su ciently small. We are now ready to construct local in time solutions by showing uniform bounds for f n which reuires the use of fractional Sobolev norms. Lemma Assume f C c such that F = + p f > with (7). () There exist small constants < T and M > ; such that if E ;; (f ; ) su ciently small, E ;; (f n+ ) + () ff n g is Cauchy in L ([; T ]; L x;v): (3) There exists C l > such that for t T : E ;l; (f n+ )(t) + hvi D ;; (f n+ )(s)ds M: () D ;l; (f n+ )(s)ds C l E ;l; (): () () Assume () is valid. For m 3; there exists an increasing continuous function P m;l with P m;l () = such that for t T : (5) ff n g: E m;l; (f n+ ) + D m;l; (f n+ )ds P m;l (E m;l; (f )): () ) + jjr n jj + jj@ t n jj] 3

33 Proof. To prove (), we apply (87) with an induction over n: We assume () is valid for k = ; ; ; ::n: Recall n = R p [f n + f n ]dv and j n = R v p [f n + f n ]dv. We now note that from the continuity euation of we have and n t + r x j n = ; r x n = r x n t n = r x j n (3) jj@ t n jj + jjr x n jj. j@ t [f+ n f n ] p dvj + j@ x [f+ n f n ] p dvj. p M: We therefore deduce that for t T () It follows that T fjj n t jj + jjr x n jj gds. p M: fjj f n+ jj + jj f n jj g. [E ;l; (f n ) + E ;l; (f n+ )]: We then summarize from Lemma with l =, by jjfjj 7 H by ():. p E ;; (f n ); and E ;; (f n+ ) + D ;; (f n )ds +CE ;; (f ) + C +C D ;; (f n+ )ds [D ;; (f n+ ) + [D ;; (f n ) + E ;; (f n ) + ]E ;; (f n+ ) E ;; (f n ) + ]E ;; (f n )] M 3 + CfE ;l;(f ) + M 3= T + MT + [M + p M + T ] sup E ;; (f n+ ) + M tt We have used the induction hypothesis for f n : For M and T both small, we have E ;; (f n+ ) + D ;; (f n+ )ds M 5 + CE ;;(f ) < M We thus deduce part () of the lemma for E ;; (f ) su ciently small. We now prove part (). It is standard to prove ff n g is Cauchy in light of strong bound obtained in part (). In particular, sup jvf n j and sup jr v f n j are D ;; (f n+ )g: 33

34 bounded (part of E ;; (f n )). We take di erence in (7) to obtain [@ t + v r x r x n r v ][f n+ f] n A [f n+ f n ] r x n v[f n+ f] n [r x n r x n ]r v f n [r x n r x n ] vf n = [r x n r x n ] v p + K [f n f n ] + [ (f n+ ; f n ) (f n ; f n )] 8[f(f n n+ f) n f n (f n f n )] 6 p ([f n+ f] n [f n f n ]) [ n+ n ] = ([f+ n+ f+] n [f n+ f+])dv; n By multiplying with e n (f n+. jje n (f n+ f n )jj (t) + f n ), (f n+ and f n has the same initial value) jje n (f n+ f n )jj (s)ds + jje n (f n+ f n )jj (s)ds jje n (f n f n )jj (s)ds: Since n is uniformly bounded, by Lemma, we can repeat this process to obtain jjf n+ f n jj (t). jjf n f n jj (s)ds. tn n! : Hence ff n g is Cauchy and we can take the limit as n! to obtain H m solutions for all m: We denote f to be the limit of f n : We remark, however, unlike [G], it is impossible to establish f n is Cauchy with respect to p E ;; ; due to the presence of the electric eld E: We now turn to part (3) for which we have to make use of the sharper estimates of H 3 norms. Collecting terms, we rewrite (87) as where A n (s) E ;l; (f n+ ) + D ;l; (f n+ )ds (5) D ;l; (f n )ds + C l E ;l; (f ) + C l A n (s)[e ;l; (f n+ ) + E ;l; (f n )] j j+j j ( j@ f n j w( ; ; ) hvi + j@ f n+ j w( ; ; ) +jjf n+ jj H 7 + jjf n jj H 7 + jjr x n jj + jj@ t n jj + The key di culty to prove () is to replace A n (s) by a xed, integrable function. First of all, in light of parts () and (), we shall prove T T A n (s)ds = [ j@ fj ; w( ; ) + jjfjj 7 H + jjr x jj + jj@ t jj + ] lim n! T j j+j j hvi A(s)ds < : (6) 3 hvi )

35 In fact, since f n! f in L from part (), we deduce from (): T E ;; (f) + D ;; (f) M: Again from (), sup tt E ;; (f n ) is uniformly bounded so that max tt jjf n fjj H 7! and T jjf n fjj H 7 ds! by compact imbedding. Moreover, by (3) and (), max t jj@ t t jj + jjr x n r x jj. [jjr x j n r x jjj + jjr x n r x jj ]. jjf n fjj H! ; (7) from () and f n! f in L : We separate jvj R and jvj R to get, for any > ; by (3) and (5) T T T j j+j j j j+j j j j+j j jj j@ f n j@ T. [ D ;; (f n ) + +C T f n f n T j@ f n j j+j j T T. [ D ;; (f n ) + +C ;R fj ; w( ; ) fj ; w( ; ) hvi + C fjj ;w( +; ) + C D ;; (f)] + C T R j j+j fj ; w( ; ) (jvjr) hvi j j+j j T j@ j@ j j+j j f f n j@ D ;; (f)] + C T T R [ D ;; (f n ) + D ;; (f)] f n fj ;w( ; fj ; w( ; ) fj ; w( ; ) hvi jj[r v f n r v f] (jvjr) jj H] (8) where is a cuto function. In light of R T D ;;(f n ) + R T D ;;(f) < ; and by (5), R T jj[r vf n r v f] (jvjr) jj H is uniformly bounded. Hence T lim n! jj[r v f n r v f] (jvjr) jj H! 35