On the Linearized Balescu Lenard Equation

Transcription

1 Communications in Partial Differential Equations, 3: , 7 Copyright Taylor & Francis Group, LLC ISSN print/ online DOI:.8/ On the Linearized Balescu Lenard Equation ROBERT M. STRAIN Department of Mathematics, Harvard University, Cambridge, Massachusetts, USA The Balescu Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau s fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation. Keywords Balescu Lenard; Collisional kinetic theory; Plasma theory. Mathematics Subject Classification Primary 76P5; Secondary 8B4, 8C4, 8D.. Introduction to the Balescu Lenard Equation The Balescu Lenard equation is a widely accepted kinetic equation which describes the dynamics of a spatially homogeneous plasma. It is { F } t = Bv v v FF F F Fdv () 3 F = Ft v is the velocity distribution, F = Ft v and F = v Ft v. () is a correction to the spatially homogeneous Landau equation (4). But in stark contrast to (4), the kernel B = B ij introduces a strong nonlocal nonlinearity: ( ) B ij v v v n e 4 F m e k k k i k j k v v dk () k 4 k k v F The main advantage of this operator is it s inclusion of the effects of Debye shielding, which we will see can lead to very different behavior. Shielding is encoded Received March, 6; Accepted October, 6 Address correspondence to Robert M. Strain, Department of Mathematics, Harvard University, Science Center 3, One Oxford St., Cambridge, MA 38, USA; strain@math.harvard.edu 55

2 55 Strain into the collision kernel through the plasma dispersion function: k k v F + k ( v e e ) k Fu lim du (3) 3 k v u i In (), (), (3), t and the velocity is v = v v v 3 3. Also, the wavenumber k = k k k 3 3 is cut-off at <k <. The parameter, k, represents the wavenumber beyond which collisions are no longer grazing collisions. The constant n is the average density, e is the charge, e is the Debye length, v e is the thermal speed and m e is the mass. The kinetic equation () was derived in a much more complicated form by Bogolyubov (946, 96). Then, Lenard (96) showed how to write the equation explicitly in terms of the distribution, as (). Lenard argued that this equation formally satisfies the expected physical properties including positivity of F, the standard conservation laws and the H-Theorem. Further, Landau s equation (4) was derived as an approximation to (). Independently and in the same year, Balescu (96) also derived (), which is now commonly known as the Balescu Lenard equation. Due to the nonlocal nonlinearity in the collision kernel, there has always been an air of extreme difficulty surrounding this model. As far as we know there is only one other mathematically oriented paper on the subject (Merchant and Liboff, 973) from the early 97 s. For more information on the physical background and relevance of the Balescu Lenard equation see for instance Balescu (96), Decoster et al. (998), Hazeltine and Waelbroeck (988), Lenard (96), Liboff (998), Lifshitz and Pitaevskiĭ (98), Montgomery and Tidman (964), Nicholson (983), Thompson (96), and Villani (). The Landau equation, proposed by Landau in 936, is one of the most fundamental partial differential equations in plasma physics; see several of the references just cited. The spatially homogenous Landau equation takes the same form as (), save that the kernel Bv v v F is replaced by bv v = b ij with b ij v v = { L v v ij v i vi v j vj } (4) v v Above L is a parameter which is logarithmically divergent. L is proportional to L d which is divergent at both zero and infinity and therefore requires a cut-off resulting in the well known Coulomb logarithm. The truncation at infinity is needed because Landau s equation does not model the effects of wide angle collisions, which is also the rational for k in the Balescu Lenard case. However the cut-off near zero is needed because the Coulomb potential decreases very slowly at large distances. The inclusion of the effects of Debye shielding at large distances causes a rapid decrease. One of the key advantages of the Balescu Lenard collision operator over the Landau collision operator is that it does not require a cut-off at small wavenumbers and can therefore model very precisely an electrically neutral plasma. On the other hand, we will argue that the inclusion of effects at small wave numbers makes the difference between these two operators enormous.

3 On the Linearized Balescu Lenard Equation 553 Let s try to make this difference more precise. Roughly speaking, one can see the Coulomb logarithm in the Balescu Lenard kernel as follows ( ) n e 4 k dk L v F = kk k v F m e Switching to spherical coordinates, k =k, we can write () as B ij v v v F = L v F i j v v d S And if L is a constant this is just the Landau kernel (4) because S i j v v d = { v v ij v i vi v j vj } v v But one of the key results of our analysis shows that the Balescu Lenard kernel () can be very far away from the Landau kernel (4) in the following sense. Consider the normalized steady state Maxwellian v = 3/ e v / We show that up to some lower order decay, the kernel behaves like B ij v v v C e v R v v +v R 3+ Here v R is v in the direction perpendicular to the relative velocity v v : ( ) v v v R =v v v v (5) See Theorem 6 for a precise statement. The main new difficulty in our analysis is contained in this observation that the effect of Debye shielding on the Balescu Lenard kernel, when evaluated at maxwellian, is to create an exponentially growing velocity factor. Further, since () satisfies the H-Theorem, we speculate that this exponentially growing factor will be present for solutions to the Balescu Lenard equation at the nonlinear level for large times. We consider B ij v v v above because this turns out to be the kernel of the linearized Balescu Lenard collision operator. Before stating our main results, we will linearize (). For suitable functions F, G and H define the Balescu Lenard collision operator by { } QF G H = Bv v v HF G F Gdv 3 We linearize this operator around the normalized steady state Maxwellian. To this end, consider the standard perturbation F = + f

4 554 Strain Then we can write the Balescu Lenard equation for the perturbation as f + Lf = Nf t where the linearized Balescu Lenard collision operator takes the form Lf = / Q f + Q f (6) The non-linear part of the Balescu Lenard collision operator is Nf = /{ Q + f + f + f Q f Q f } At first glance, due to the way we have written it, L may seem like a fabricated linear operator. But a Taylor expansion of the kernel Bv v v + f reveals that Nf is a nonlinear function of the perturbation f. The terms subtracted off on the right cancel with the linear terms in the taylor expansion. More precisely, for F satisfying k k v F, by () we have Therefore, Q F = Q + f + f F = Q ff+ Q f F + Q f f F Let us now look at the collision kernel () with (3) for F = + f. From (3) k k v + f = k k v + k ( v e e ) k fu lim 3 k v u i du From here we see the first term in a taylor expansion of Bv v v + f in terms of the rightmost term above is Bv v v and all other terms depend on f. Therefore (6) is the linearized Balescu Lenard collision operator. Then the linearized Balescu Lenard equation is f + Lf = (7) t The presence of physical constants does not create intrinsic mathematical difficulties. Accordingly, to simplify our presentation, we will normalize all constants to one. Because of the null space of L (Lemma 9), a solution to (7) formally satisfies f vvv vdv = ft vvv vdv 3 3 We are interested in the asymptotic properties of solutions to (7). We will prove time decay to maxwellian in weighted energy spaces. Consider the velocity weight ( ) q wl v +v l/ exp 4 +v (8)

5 On the Linearized Balescu Lenard Equation 555 Above l, q>, and. If = we further assume <q<. Depending on our choice of parameters, the velocity in this weight can grow either with an arbitrarily low polynomial power or alternatively almost but not quite as fast as /. Define the following weighted norm g 3 w l gv dv Above i = vi. Let denote the standard L v 3 inner product. We put a zero in the norm to drop the entire weight (8), e.g., if l = = then g = 3 g dv =g g In these norms we can show linear decay: Theorem. Let f v satisfy the conservation of mass, momentum, and energy: f vv = If >, a solution to (7) with initial data fv= f v and f < satisfies where C > and p =. + ft Ce tp f (9) It is our hope that this linear decay will aid in a future in construction of classical solutions with small amplitude to the full Balescu Lenard equation. The only previous result for the Balescu Lenard equation that we know of is the work of Merchant and Liboff (973). They show that the spectrum of the linearized Balescu Lenard operator is continuous from zero to minus infinity. Additionally they obtain analytic expressions for some spherical harmonic eigenfunctions. This lack of a spectral gap in the linear operator makes it difficult to prove time decay. Indeed, the proof of Theorem requires some development because there are many complicated elements of the Balescu Lenard operator even at the linear level. In Section we will analyse the pointwise behavior of the longitudinal permittivity k k v. We write down another formula for in (5). Then we use this formula to determine the asymptotics of and thereby show that the Balescu Lenard kernel B ij v v v is well defined (Lemma 3). Then, in Section 3, we establish an alternate formula for the Balescu Lenard kernel B ij v v v in Lemma 4 via a series of changes of variables. We use this representation to show that the kernel contains an exponentially growing factor in Theorem 6. In the proof of Theorem 6 we split the integration region in order to squeeze a bit of extra decay out of the kernel, which we later need to use to estimate the linear operator. In Section 4, we look at the eigenvalues of the so-called collision frequency, which is defined as ij v B ij v v v v dv ()

6 556 Strain Despite the exponential difference in the growth of the kernel as compared to the Landau kernel (4), the eigenvalues of the collision frequency decay similarly to the Landau case. Using a different series of changes of variables, we find a useful formula for the eigenvalues in and around Lemma 7. Then in Lemma 8 we prove that the eigenvalues decay for large v as v c log +v +v 3 v c +v In the Landau case, the decay is as above if we remove the log +v factor (Degond and Lemou, 997). This study motivates the definition of the following weighted Sobolev norm g ( w l ij v ij 3 i g j g + v i v j g If is absent, this means that we drop the entire weight as follows g =g ( ij v i g j g + v ) i v j ij 3 g dv ) dv () These turn out to be anisotropic spaces which are motivated by related spaces used by Guo in the Landau case (Guo, ) and by Strain and Guo on the relativistic Landav collision operator (Strain and Guo, 4). They help measure the dissipation of (6). We study the full linear operator, L, in Section 5. We write down it s null space in Lemma 9. Then we split the linearized collision operator as Lf f =f Kf f See Lemma. The main estimate in our paper is the following Theorem. For any small > there is < C < such that w i ij v j g g +w Kg g g g + Cg C g C Here w = w l and C v is the indicator function for the set v C. Due to the exponential growth of the collision kernel (Theorem 6) it was difficult to expect that such a result was even true, especially with the possible inclusion of these exponentially growing weights. However the exponential growth of the kernel is only present in the direction perpendicular to the relative velocity and there is some decay remaining in the other directions. We design a splitting to show that this left over decay is just barely enough to prove Theorem. In the rest of Section 5, we use Theorem to deduce coercivity of the linear operator 6 and exponential decay of solutions to the linearized equation (7). We use a standard compactness argument to prove the coercivity of the linear operator. But all other arguments in this paper are constructive, including the estimate of K. To prove Theorem, we use a splitting between velocity and time which was previously used to establish decay rates for soft potential kinetic equations in Caflisch (98) and Strain and Guo (in press).

7 On the Linearized Balescu Lenard Equation 557 To end this section, we remark that the results in each of the following sections build on and crucially make use of the results in previous sections. We begin by analysing the longitudinal permittivity.. The Dispersion Function, k ˆk v The main objective of this section is to determine the pointwise asymptotic behavior of the plasma dispersion function (3) evaluated at maxwellian. Since this appears to be difficult to accomplish from (3) due to the nonlocal operator our first step is to derive an alternate formula. This formula (5) seems to be known but we were unable to find a complete derivation in any one reference. Therefore we briefly derive (5) for completeness. Then we establish the asymptotics of in Lemma 3. Since we focus on the linear operator, which includes the dispersion function only evaluated at Maxwellian k k v, in the rest of this article we will drop the dependence of the maxwellian in our notation, to write only k k v Now we begin our computation of at Maxwellian. From (3) we write k s k s = + k lim k u 3 s k u i du If k, let k = k and k k k 3 be an orthonormal basis for 3. Define an k orthogonal matrix O k such that k O k u =ku as By this change of coordinates, 3 k u s k u i du = 3/ O k = k k k 3 () 3 k ue u/ s k u i du = 3/ ku e u/ 3 s ku i du + = / ku e u / s ku i du In the last step we integrated out the extra variables u and u 3. To go further, we recall the well known Plemelj formula: ( ) lim x y i = P.V. + ix y x y which is a distribution. Here P.V. denotes the Cauchy principle value. Thus, + lim ku e u / + s ku i du = lim u e u / s u k i k du + u = P.V. e u / s u du + i s k k e s k /

8 558 Strain The second term on the r.h.s. is in exactly the form we want. For the first term, further decompose u = ( s u ) k + s to obtain k + P.V. u e u / s u du = k s + k P.V. = s + k P.V. e u / s u du k + e u / s u du k e u / du We collect the last few computations to obtain kx= +k x where = R + i I and + e y / R x = / xp.v. x y dy I x = / xe x (3) The integral part of R is a well studied function in plasma physics (Fried and Conte, 96). We further evaluate R via the well known formula (Thompson, 96, pp , eq. (8..3)): + e y / P.V. x y dy = e x / This yields the following simplified version of (3): R x = xe x / By plugging x = s k = k k v = ˆk v into, we can write x x e y / dy e y / dy (4) k ˆk v = +k ˆk v (5) Notice that above only depends upon the magnitude of v in the direction of k. Now that we have a tractable formula for evaluated at Maxwellian (5), we will study it s pointwise behavior. The key term in the decay of is R : Lemma 3. As x ±, x R x Proof. We first write x R x as a fraction: x R x = x ( xe x / = / x ex x x 3 e x / x ey / dy ) e y / dy

9 On the Linearized Balescu Lenard Equation 559 Next, we apply l Hôpital s rule to obtain as x ± / x ex x / ey du e x 3 x / ( ) / ex x e x / e ( ) = x / 3 3 x x 4 x That s it. Remark 3.. In particular, Lemma 3 shows that R x, given by (4), is eventually negative for large enough x. From (5) we have Rek ˆk v = + Rˆk v k Lemma 3 therefore tells us that the real part will be zero if ˆk v is large enough and k small enough k depending on the size of R ˆk v. There are then lots of places where Re is zero, but if Rek ˆk v = then fortunately Im = I = ˆk v e ˆk v k We conclude that, when evaluated at maxwellian, for all finite v. By Lemma 3 and Remark 3., the kernel of the Balescu Lenard collision operator () evaluated at Maxwellian is well defined. In the next section we will further use Lemma 3 to obtain asymptotic estimates of Bv v v in Theorem The Collision Kernel, Bv v v In this section we consider the Balescu Lenard collision kernel () at Maxwellian. Since we will only consider this case, in the rest of the paper we will write Bv v v = Bv v v (6) We will look progressively more closely at the pointwise behavior of this kernel in a series of Lemmas. The asymptotic analysis of the dispersion function from Section will play an important role. We first record some basic properties of the collision kernel which are also shared with the Landau kernel (4). A proof of the following can be found in Montgomery and Tidman (964): B and u t Bv v v u = iff u = cv v with c The main result of this section (in Theorem 6) is our pointwise estimate of the asymptotic growth rate the collision kernel, which turns out to be exponential. The first step in this direction is to develop a more tractable expression for (6):

10 56 Strain Lemma 4. The Balescu Lenard collision kernel (6) can be expressed as B ij v v v = ij w v R + ij w v R v v Above, ij and ij are non-negative symmetric matricies that are connected to the Landau projection (4) by the formula ( ij + ij = ij v i vi v j vj ) v v Further, ij v i v i = i i ij v i v i = (7) We define w v R, w v R in (8) and = ij, = ij in (). The weights are scalar functions which are given by w v R = w v R = / / sin Jv R cos d cos Jv R cos d (8) Here Js includes the effects of the dispersion function (5) as follows k d Jx = 4 x k 4 3 = d (9) + R x + I x These quantities will help us get all our later estimates. The main idea in the proof of Lemma 4 is to use three changes of coordinates, one at a time. The first one is designed to extract the singularity from the delta function in (). The second coordinate change will give us a useful scalar quantity in the form of v R. And with the final rotation we obtain the integral J which will be evaluated precisely in () below. Proof. Let u 3 = v v = v v and u u u 3 be an orthonormal basis. Define R v v to be the rotation matrix satisfying Rk v v = k 3 v v, e.g., R = u u u 3 Then we rotate the k variable in () with (5), in other words (6), to obtain k B ij v v v i k j k v v k k k 4 k ˆk v dk Rk i Rk j k = 3 v v k k k 4 krˆk v dk

11 On the Linearized Balescu Lenard Equation 56 With the formula ak 3 = a k 3 for a> we have B ij v v v = Rk i Rk j k 3 v v k k k 4 krˆk v dk We will expand Rk i Rk j in order to simplify this expression. Note that Rk i = k u i + k u i + k 3 u 3 i Here u l = u l ul ul 3 t with l 3. We therefore have We can therefore write Rk i Rk j = B ij v v v = 3 l= k l ul i ul j + l m v v k k ij k k l k m u l i um j k 3 krˆk v dk Since terms involving k 3 vanish because of the delta function, we can define ij k { k u i u j + k u i u j + k k ( u i u j + u j u i )} This completes the first change of variable. With each new change of variable, below, we will elect to redefine ij as needed instead of repeatedly introducing a new temporary notation. Next we evaluate the delta function. Consider k = k k t and write Evaluating the delta function yields where we redefine B ij v v v = = k k u + k k u ij v v k k k k v dk ij =k { k u i u j + k u i u j + k k ( u i u j + u j u i )} Before we rotate the k coordinate system again, we look at the following vector v R v u v u t This is the velocity, v, in the direction perpendicular to the relative velocity, v v. Since u u u 3 is an orthonormal basis, we can expand v = v u + v u + v u 3

12 56 Strain Thus, ( ) v R =v v u 3 =v v v v v v This is (5). Also v = k k v R Now we are set up for another change of variables. Let O R be the orthogonal matrix such that O R k v R = k v R e.g., O R = ˆv R ˆv R Here v R u v u v t We apply this rotation to obtain B ij v v v = where ij is redefined as ij v v k k k k v R k /k dk ij k { O R k u i u j + O Rk u i u j + O Rk O R k ( u i u j + u j u i )} To simplify this expression, we have to write out O R k. We expand ( OR k ) = k ) i (ˆv R + k ) i (ˆvR i Hence O R k i O R k j = k ˆv R i (ˆv R )j + k ˆv R i (ˆvR )j + k k {(ˆvR ) (ˆv R )j + (ˆv ) ) R (ˆv j R i} Plugging this into ij yields a long expression. But the factor k k involves the integration of an odd function over an even domain, which is zero. Disregarding terms with this factor, we can write i where ij = k k ij + k k ij ij = u v u v R i u j + u v u v R i u j u vu v( u v R i u j + ) u j u i () ij = u v u v R i u j + u v u v R i u j + u vu v( u v R i u j + ) u j u i This completes our reduced expression after a second change of variables.

13 On the Linearized Balescu Lenard Equation 563 For the third and final change of variables, we will split into an angular integral and a magnitude integral. The magnitude integral is evaluated precisely in (). For now, we choose polar coordinates as k = sin k = cos First changing coordinates and second plugging in (5) yields B ij v v v { k ij sin + ij = cos } d d v v v R cos { k ij sin + ij = cos } d d v v + R + 4( ) R + I = v v { ij sin + ij cos } Jv R cos d 4 where the last line follows from the definition (9). It remains to reduce the integral over to an integral over /. Notice that R x is an even function (4) and I x is odd (3), so that I x is even. In particular, Jx = J x. Since Jx is an even function (9), the reduction follows by first translating on the region and second translating on the region /. Next, it is not hard to evaluate (9) precisely as: ( J = log + k4 + Rk R + I ) + ( ( ) ( )) R tan R k tan + R () I I I Here we clearly see the Logarithmic divergence of the Balescu Lenard kernel when k is sent to infinity. With (), we can determine the asymptotic limits of J: Lemma 5. x 3 e x / Jx ± 8 as x ± Furthermore, if one were to add a cut-off at small wave number in the Balescu Lenard kernel (say k s > ) then it would appear in the first arctangent factor in () as follows. One would replace the factor R by k s+ R. Our proof below implies this I I large exponential growth would then disappear because of cancellation. Proof. We will first examine the behavior of the log term. Notice that the only zero of I x is at x = and R =. From this we see that the argument of the log is bounded away from zero on any compact set. So there are no finite singularities. It is not hard to see that as x ± ( x 3 e x / log + k4 + ) Rxk Rx + I x We therefore only need to look at the term involving the difference of tangents.

14 564 Strain We now consider the second term of Jx in (), which defines the asymptotics. By Lemma 3 and (3) R x I x x R x x I x / x 3 ex Above means the quantities have the same limit. Similarly k + Rx I x x ex / Since tan x ± as x ±, we conclude ( ( ) ( )) lim tan R x k tan + R x =± x ± I x I x Therefore, lim x ± x3 e x / ( ( ) ( )) Rs tan R s k tan + R s =± 8 I s I s I s And the same is true for x 3 e x / Jx. Given u 3, we define the projection Then in matrix form we have P v u ˆv uˆv = I P v v ij = ij v v i v v j v v ( v ) v v u v () We now use Lemmas 4 and 5 to get asymptotic bounds for the kernel. Theorem 6. Consider the Balescu Lenard kernel (6) and the relative velocity (5). For any << there exists C > such that B ij v v v e v R v v C +v R 3 Moreover, u 3 and any <q< there exists C q > such that u t Bv v v u I P q v v u e v R C q v v +v R 3 In this sense, the upper bound is exponentially sharp. Although the bounds above are sufficient for the rest of our analysis, it seems tractable to refine the bounds established in Theorem 6 and make them sharp by evaluating more precisely the integrals in (3) below.

15 On the Linearized Balescu Lenard Equation 565 From Lemma 4, (8) and () we see that B ij v v v 4 w v R + w v R v v Therefore the upper bound in Theorem 6 requires an upper bound on the weights w, w. For the lower bound, we use () to observe u t Bv v v u = w v R v v ut u + w v R v v ut u minw v R w v R ( u t u + u t u ) v v Then using the formula for + in Lemma 4 yields u t u + u t u = u t + u =I P v v u We thereby see that for the lower bound in Theorem 6 it is enough to get a lower bound on minw v R w v R. This is what we prove. Proof. We first establish the upper bound. From (8) and Lemma 5, we have w v R C w v R C / / e v R cos sin +v R cos d 3 e v R cos cos +v R cos d 3 (3) Without loss of generality say v R. We split into v R and v R. Then We will use vr e v R cos +v R cos d e v R 3 v R +v R cosv R 3 cos ( v R )! v R From this lower bound and << we have +v R cosv R +v R v R +v R Here we have utilized v R. Thus, e v R e v R v R +v R cosv R v R This completes the estimate over v R. 3+

16 566 Strain For the second half of the splitting, v R, we have / v R Now we will use the upper bound e v R cos +v R cos d Ce 3 vr cos v R cos ( v R )! v R + 4! v R 4 From here, we get some weak exponential decay as long as <<: e v R cos v R Ce v R e 4 v R e 48 v R Ce v R e 8 v R This is more than enough decay to establish the upper bound. Next we consider a lower bound for minw v R w v R. By Lemma 5, / e v R cos w v R sin C +v R cos d 3 / C +v R 3 sin e v R cos d / w v R cos C +v R cos d 3 / C +v R 3 cos e v R cos d e v R cos This time we consider w and w separately. For w and <q< we have / And similarly for w, / cos q cos q cos e v R cos d q cos qe q v R ( cos sin e v R cos q d ) sin d e q v R > These lower bounds for w and w establish the theorem. Remark 6.. It is a basic but important fact that Bv v v = Bv v v, the delta function yields B ij v v v = = k k k k k i k j k 4 k v v k ˆk v dk k i k j k 4 k v v k ˆk v dk = B ijv v v

17 On the Linearized Balescu Lenard Equation 567 We will use this later to split up the growth of B ij v v v in Theorem 6 between v R and vr. Alternatively, the next identity can be used and additionally will be useful in other contexts below: ( ) v v v R =v v ( ) v v =v v =v v v v v R This is seen by a difference of squares argument: ( ) v v v ( ) v v v =v v v v v v In particular, Theorem 6 says that B ij v v v L loc 3 v 3 v. This completes our estimates for the collision kernel Bv v v. In the next section we consider the collision frequency. 4. The Collision Frequency, v We recall the Balescu Lenard collision frequency (). We will use the reduced form of B from Lemma 4 to do an asymptotic analysis of the collision frequency. The following quantity will also be used: i v { v v } = i j B ij v v v v j v dv (4) In this section, we compute the eigenvalues v and v of v in Lemma 7. Then in Lemma 8 we transform these eigenvalues into a form which is conducive to obtaining precise pointwise information. Somewhat surprisingly, despite the exponential growth of the collision kernel (Theorem 6) in comparision to the Landau operator (4), the collision frequency still decays at large velocities, and in a way which is different but also closely related to the Landau case. We also used the collision frequency to define an anisotropic norm (),, which measures the dissipation of the linear operator L. An equivalent norm will be established in Corollary 8., using all the asymptotics developed in this section. This norm is important for establishing the main results in Section 5. First we look at the eigenvalues: Lemma 7. The matrix v from () has an eigenvalue v with eigenvector v and a double eigenvalue v whose eigenspace is perpendicular to v: v = / v = / k e vk /k k k k 5 kk v/k dk k k k + k 3 k 5 e vk /k kk v/k dk And we can expand ij v =ˆv iˆv j v + ( ij ˆv iˆv j ) v with ˆv = v v

18 568 Strain We remark that the strategy for computing these eigenvalues is to use a series of two rotations in a different way from Lemma 4. Here we first evaluate the v integration which is not present in B ij v v v. Proof. First we translate v v v and second we use (), O k, to obtain k ij v = B ij v v v v dv i k j k v v v = dk dv k k k 4 k ˆk v k i k j k v v v = dv dk k k k 4 k ˆk v k i k j k O = k v v O k v dv dk k k k 4 k ˆk v k i k j v = v O kv dv dk k k k 5 k ˆk v Next, with k = ˆk and k k k 3 an orthonormal basis for 3 as in (), we expand the exponent of the Maxwellian as v O k v = v v k v k v 3 k3 We can further write v = v k k + v k k + v k 3 k 3. Then by orthogonality v Ov = 3/ e {v k v +v k v +v k 3 v 3 } Now we evaluate the delta function and use translation invariance to obtain k ij i k j v v = v O kv dv dk k k k 5 k ˆk v k = / i k j e v ˆk dk (5) k k k 5 k ˆk v We will use this formula to compute the eigenvalues. Now that we have evaluated the v integrations via a rotation in the k direction, we will simplify the k integrations by rotating in the v direction. Let v =ˆv and v v v 3 be an orthonormal basis for 3. Further define the rotation matrix O v = v v v 3 (6) Notice that O v k i = k v i + k v i + k 3 v 3 i We rotate the k variable with O v in (5) to achieve ij v = / = / k k k k k O v k i O v k j k 5 e v O v ˆk ko v ˆk v dk v i v j + k v i v j + k 3 v3 i v3 j e k k v k 5 kk v/k dk

19 On the Linearized Balescu Lenard Equation 569 Above, the cross terms in O v k i O v k j disappear because they give you the integral of an odd function over an even domain. By symmetry, k e k k v k k k 5 m kk v/k dk = k k k 3 k 5 e k k v m kk v/k dk Recall (). By the spectral theorem I P v = P v + P v 3 or in another form v i v j + v3 i v3 j = ij ˆv iˆv j These last three points yield the result. With a sequence of two rotations, we can write these eigenvalues in the form v = v = (ˆk ˆv ) k v v v k k k k ˆk v dk dv k k k k v k v v v v k k ˆk v dk dv This follows by first rotating the v variable with () to obtain v = / v = / k k k k (ˆk ˆv ) e ˆk v k 3 k ˆk v dk k k v e ˆk v v k 3 k ˆk v dk Next rotate the k variable in the direction of v with (6) to get the eigenvalues as written in Lemma 7. These forms of the eigenvalues are similar in form to the eigenvalues found for the collision frequency of the Landau equation in Degond and Lemou (997). We will now transform these eigenvalues into a form which is conducive to asymptotic analysis in Lemma 8 below. We switch to spherical coordinates via k = cos k = sin cos k 3 = sin sin k Then we can write the eigenvalues in Lemma 7 as v = v = d k d k d cos sin d sin3 e v cos v cos e v cos v cos

20 57 Strain Plug (9) into the eigenvalues to obtain v = d 8 cos sin e v cos Jv cos v = d 4 sin 3 e v cos Jv cos We change variables as y =vcos above to obtain v = +v y e v 3 y Jydy 8 v v = +v ) ( y e v 4 v v y Jydy The fact that the integrand s are even functions yields v = v y e v 3 y Jydy v = v ) ( y e v 8 v y Jydy Now we are ready to look at the decay of these eigenvalues: Lemma 8. As v log +v v +v 3 v 8 e y Jydy +v Moreover, for any multiindex with we have D v C log +v +v 3 D v C +v (7) where C and C are positive constants. Remark 8.. As v these eigenvalues converge to a unique finite limit: lim x = lim x = x x 3 J J is clearly finite by (9). The eigenvalues thus have no finite singularites. Proof. By Lemma 5 and l Hôpital s rule, as v + v y e y Jydy v 3 e v Jv 8 logv which implies the decay for in Lemma 8.

21 On the Linearized Balescu Lenard Equation 57 For the decay of, we consider v v = 8 v e y Jydy v 8 v y e y Jydy Since the second term on the r.h.s. converges to zero, as v +we have v v 8 e y Jydy And the integral on the r.h.s. is finite by Lemma 5. This yields the decay of. The decay of in (7) follows from taking the derivative of and then applying Lemma 5 and l Hôpital s rule, exactly as we have done for the no derivative case. The same recipe will establish the decay of the derivative of in (7). Then all the results in this section yield the following lower bound: Corollary 8.. g c c > such that 3 w { log +v P +v 3 v g + I P vg + +v where w = w l v is defined in (8) and P v is defined in (). Proof. By (), Lemma 7 and (), log +v g }dv +v g = 3 w l { v ( P v g +v g ) + vi P v g } dv Plugging in the asymptotics from Lemma 8 yields Corollary 8.. An analogous upper bound can also be established in the same way. This completes our study of the Balescu Lenard collision frequency (). In Section 5 we will use these asymptotics to prove bounds for the Linearized Balescu Lenard Operator. 5. Compactness of K, Coercivity of L and Exponential Decay The main result of this section (Theorem ) is to show that L can be split into as L = A K (Lemma ) where K is compact (in the sense of the inequality in Theorem ). This is a standard Theorem for a linearized collision operator of a kinetic equation, such as the Boltzmann or the Landau equation. However the exponential growth of the kernel (Theorem 6), which is not present in the Boltzmann or Landau kernel, creates new difficulties. As a consequence of Theorem, we deduce coercivity for L in the anisotropic norm () in Corollary.. Then we finish section 5 by proving exponential decay of solutions to the linearized Balescu Lenard equation. First define the projection Pg = { a g + b g v + c g v } / v

22 57 Strain where a g c g, and b g 3 depend on the function gv. Then we have the following standard lemma for the linearized collision operator (6). Lemma 9. Lg g =g Lg, L and Lg g = if and only if g v = Pg. In this case LPg =. For the Landau equation, there is a standard argument used to prove this Lemma, see for instance Guo (, Lemma 4). The proof in the Balescu Lenard case is exactly the same as for the Landau equation because the null space (7) is exactly the same. Now we will split the operator L = A K using () and (4): Lemma. We split L = A K, where { Ag = i ij j g } + { i i} g ij v i g { } Kg = / i { B ij v v v / 3 j g + v j g }dv Here and in the proof we use the convention of summing over repeated indicies. The proof of Lemma is virtually the same as Guo (, Lemma ). This is expected because the Landau kernel and the Balescu Lenard kernel share the same null space. v j Proof. We define Ag = / Q / g Kg = / Q / g Then L = A K by (6). We will simplify A and K. Notice that i = v i ( i / g = / i v ) i g i / g = / ( i + v i ) g We will use these and the null space (7) of () several times below. We compute Ag = / i B ij v v v { } j / g / g j dv 3 {( = / i B ij v v v / j v } j )g + v j 3 g dv ( ) = / i B ij v v v / j + v j gdv 3

23 On the Linearized Balescu Lenard Equation 573 where we used (7) in the last step. By (7) again, and then by () we have ( = / i B ij v v v / j + v ) j gdv 3 { ( = / i ij / j g + v )} j g Next, we take the derivatives on each term to obtain = / i { / ij j g } + / i { / ij v j { = i ij j g } ij v { i jg + v ij j = i { ij j g } + i { ij v j This is the desired expression for A. Next, for K, we have } g ij v i v j g } g } i g + i { ij v j } g ij v i v j g Kg = / i { 3 B ij v v v { / g j j / g } dv } { } = / i { B ij v v v / 3 v j g + v j g j g }dv By (7) this is { = / i { B ij v v v / 3 j g + v } j g }dv { } = / i { B ij v v v / 3 j g + v j g }dv And this is the expression we sought for K. We are ready to prove Theorem. Proof of Theorem. write We first estimate w l Kg g. Recall Lemma and w l Kg g = w l B ij v v v { ( ) }{ vj j g g i g + v } i g dv dv + i w l B ij v v v { ( ) } vj j g g g dv dv where we recall that means that the function in parenthesis is evaluated at v. We will split the integration region several times to obtain the estimate.

24 574 Strain The derivative of the weight function 8 is i w l = w l w v i (8) where w v = {l +v + q +v } In particular w v C<. Then we can write where w Kg g = ij 3 3 w l B ij v v v d j g d i g dv dv (9) d j = j + v j di = i + v i + w vv i (3) We will estimate w l Kg g in this condensed form. Now we outline the main strategy of the proof. A key point is to get sufficient upper bounds for wl vb ij v v v. We first want to control B ij v v v by something which approximates the dissipation ij in our norm. But we also want velocity decay left over to generate a small constant factor as in the statement of Theorem. And further we want to show that this bound has additional velocity decay to allow us to distribute and control the exponential weight (8), which only depends on v. This is done via several splittings. We will now look for a bound for wl v in terms of wl v. To this end, we expand ( ) v v ( ) v v v =v v v v + v v v Then, since, we obtain e q 4 +v ( ( ( ) q v v ) exp +v 4 v v v ) ( ) e q v v 4 v v v If l then we similarly have ( ( ) v v ) l/ ( ( ) v v ) l/ +v l/ C +v v v v + v v v But if l< then +v l/ ( ( ) v v ) l/ +v v v v Then, by the last few inequalities, under any conditions we have shown ( ( ) v v ) l/ wl v Cwl v R + v v v exp ( q 4 ( ) v v ) v v v

25 On the Linearized Balescu Lenard Equation 575 Since v R =vr (Remark 6.) we have ( ( ) v v ) l/ wl v R = wl v R Cwl v + v v v The extra factor on the end is needed only if l<. We have this shown that ( ( ) v v ) l wl v wl v + v v v exp ( q 4 ( ) v v ) v v v (3) This estimate allows us to distribute the exponentially growing velocity weight from the v variable onto the v variable. We have to pay with some extra growth of v in the direction of the relative velocity, but this can be controlled by terms in the upper bound (3) below. The next step is to get bounds for wl vb ij v v v. From Theorem 6 and Remark 6., B ij v v v C exp( ( v v v) ( 4 v v v v v ) ) 4 v v (3) v v +v R 3+ So we have some exponential decay in the direction of the relative velocity, and this is how we control the exponentially growing weight (8) and (3). Since either < and <qor = and <q< there is <q < such that wl vb ij v v v ( ( ) Cwl v exp q v v v v +v R 3+ 4 v v v ( ) v v ) 4 v v v (33) We will use this upper bound several times below. We first split the integration into a compact region and a large region. For large m >, define a smooth function r so that r = for r m and r = for r m. Then, with (9) and (3), we define K so that w K g g = v+v w B ij v v v ( d j ij 3 3 g ) di g dv dv This is really the hardest term to estimate. We will split the integration region a few more times to do it. We first split into two regions where (33) yields solid exponential decay in v and v. Define On S,ifv v then S = v v v v v v v+v 3 v

26 576 Strain And further v v v v v v v v v v v v v v = v v v 3 v 3 v Alternatively, if v v then And similarly v v v+v 3 v v v v v v v vv v v v v v 3 v 3 v In either case, we plug these estimates into (33) to obtain wl vb ij v v v / v / v q wl v Ce 36 v +v (34) v v This is the strongest decay estimate we ll get. Here and below we will define terms like S w K g g to be S w K g g = ij On S we use (34) (and recall (3)) to get S w K g g C By Cauchy Schwartz, this is S v+v w B ij v v v d j g d i g dv dv wl d j v v g wl d i g Ce q e q 44 m 7 v +v wl d j v v g wl d i g dv dv e v+v m q 36 v +v ( ) Ce q e q 44 m 7 v +v wl d j v v g / dv dv ( e q 7 v +v wl d i v v g / dv dv) ( Ce q 44 m e q 7 v ( )( wl d j g ) dv e q v 7 w ( l ) ) d i g dv By (3) and Corollary 8., we conclude S w K g g Ce q 44 m g g Since m > will be chosen large, this yields Theorem for K restricted to S.

27 On the Linearized Balescu Lenard Equation 577 Next fix << and consider the region { } { v v S = v v v v v v v v v v } Also S c = v < v < 4v. Then on S c S, 33 yields wl vb ij v v v / v / v v +v Cwl v e q (35) v v where q = q. Since > we can use exactly the same estimates as in the 4 previous case to establish S c S w l K g g Ce q m g g with q = q >. And this grants Theorem for K restricted to S c S if m > is large enough. It remains to estimate K on S c Sc, where { } { S c = v v v v v v v v v v v v } with <<. Over this region, we do not expect to get anymore general exponential decay in v and v out of (3) (33). However, fortunately, Theorem 6 allows us the possibility of finding polynomial decay in this region. Without loss of generality assume v /, which means v v. Then, using (5), (33) and S c, with << we have wl vb ij v v v Cwl [ ( ) v v v +v v ] 3+/ v v v v Cwl v [ +v v ] 3+/ v v Cwl v v v [ + v ] 3+/ In particular, since v > v on S c Sc we have decay in both variables: wl vbv v v Cwl v +v 3+/4 (36) v v +v 3+/4 And the next step is to use (36) to complete the estimate for K on S c Sc. By (36) and the definition of we have S c S cw l K g g v m /3 wl d j C g wl d i g S c v v Sc +v 3+/ +v 3+/ dv dv

28 578 Strain We next use Fubini and Cauchy Schwartz to obtain wl d i = C g { wl d j g } v m /3 +v 3+/ v S c v v Sc +v dv dv 3+/ ( { wl d j C g } / v m /3 v v +v dv dv) 3+/ ( v m /3 S c Sc w l v di g dv +v3+ ) / (37) We now focus on bounding the term involving wl d j g. The main difficulty with this term is controlling the singular factor v v. The following is designed to control it. First we remark that on S c Sc v v = v v v v v v v +v + v Therefore, S c Sc Sc v v 3v Then v S c Sc +v 3+/ v v wl d j g dv ( ) / ( v v dv v v 3v S c Sc w l v d j g +v dv 3+ ) / Moreover, v v 3v v v dv C +v Combining these last two calculations yields { v m /3 ( C S c Sc S c Sc +v 3+/ ( } wl d j v v g ) dv dv ) v m /3w l v +v d j +v 3+ g / dv dv Since v is comparable to v on S c, this is ( C v S c m /6w l v dj g ) / Sc +v 3+ dv dv

29 On the Linearized Balescu Lenard Equation 579 And by Fubini s theorem, the integral in parenthesis is = v m /6 w l v dj g ( ) dv dv +v 3+ v S c Sc ( ) dv dv v v 3v C w l v dj g v m /6 +v 3+ C w l v dj g v m /6 +v 3+ +v 3 dv Since << are otherwise arbitrary, we can choose 4 to be arbitrarily close to. But this is not needed for the current case, so we merely choose = 3 4 and = 8 to obtain 4 = 4. Then, for m > large, the integral above is C m /4 logm 3 w l v log +v +v 3 d j g dv We plug the result of the last few inequalities back into (37) to see that S c S cw l K g g C( w l v log+v d j 3 +v 3 g dv ) / m /8 logm ( d w i l v g ) / dv v m /3 +v3+ ( w l v log+v d j C 3 +v 3 g ) dv / m / logm ( w log +v l v d i 3 +v 3 g / dv) By (3) and Corollary 8. then S c S cw l K g g Cg g m / logm This completes the estimate for K with m > large. Next define K = K K. In this definition of K we have a smooth cutoff function r = r such that (for some m > large) r = for r m and r = ifr m. Then from (9) and (3) w K g g = ij where w = w l. Theorem 6 implies 3 3 v+v w B ij v v v d j g d i g dv dv ij w l v+v B ij v v v L 3 3

30 58 Strain Therefore, for any given m > we can choose a C c function ij v v such that ij ij L 3 3 m supp ij v +v Cm < We split K into a small part and a compact part as K g g =K c g g +K s g g Here we have used the splitting ij = ij + [ ] ij ij to define K c g g = ij ij v v ( d j g ) di g dv dv K s g g = ij ij ij ( d j g ) di g dv dv We introduce this second smooth cuttoff so that we can integrate by parts. We estimate each of these terms separately. The second term K s g g is bounded by ij ij L 3 3 d j g d i g C L / d j 3 3 m g / d i g C m g g where the last line follows from Corollary 8.. After integrations by parts, the first term is K c g g = ij (d j di ij v v ) g v g vdv dv where d j = vj + v j di = v i + v i + w vv i Since w v C, we therefore have { } / { / K c g g C ij C g dv g dv} v Cm v Cm Combining the last two estimates we conclude that for any m > w l K g g C m g g + Cm g Cm g Cm

31 On the Linearized Balescu Lenard Equation 58 We thus conclude Theorem for the K part by first choosing m > large and then m > is chosen large enough. Finally consider w l i i g g. For m>, we split the integration region into a compact part and a large part: Then by (4) and (7), v m v m v m w l i i g g dv C w log +v l g v m +v 3 g dv C w log +v l g m v m +v g dv And then, using Corollary and Cauchy Schwartz, the integral on the r.h.s. is g g. Thus, v m w l i i g g dv C m g g By Remark, we see that i and i i have no finite singularities. Therefore, on v m, 4 and 7 imply that there is a constant Cm such that i i Cm Therefore, v m w l ( ) / ( / i i g g dv Cm g dv g dv) v m v m We conclude that for any m> w l i i g g C m g g + Cmg Cm g Cm Since m> can be arbitrarily large, this completes the estimate for the last term w l i i g g and thereby finishes the proof of Theorem. Next we will use Theorem to deduce coercivity of the linear operator, L, via a standard compactness argument. Corollary.. > such that Lg g I Pg. Proof. We use Theorem and the method of contradiction. In this case, we have a sequence of functions g n v n satisfying I Pg n > and Lg n g n < n I Pg n

32 58 Strain Without loss of generality suppose Pg n = and g n. Consider the inner product g g = ( ij v ij 3 Then there exists a g v such that i g j g + v i v ) j g g dv g n g In other words, g n h g h. By lower semi continuity, as n for all hv which are bounded in g Equivalently g = lim n g n g lim n g n g =g. From Lemma we can write Lg n g n =g n i i g n g n Kg n g n (38) We claim that lim i i g n g n = i i g g n lim Kg ng n =Kg g n (39) These limits will follow directly from Theorem. It is then a standard application of (39) to prove the coercivity. We first consider the limit of i i g n g n. Splitting i i g n g n i i g g = i i g n g g n + i i g g n g From Theorem then i i g n g n i i g g 8 g n g g n + g + Cg n g C ( gn C +g C ) Then by Corollary 8., g C C g and g g n =. We thus have i i g n g n i i g g + Cg n g C Similarly, Kg n g n Kg g + Cg n g C Furthermore i g n are bounded in L v C because g n = ; the Rellich Kondrachov Compactness Theorem thus yields lim g n g C = n Then (39) follows by first choosing small and second sending n.

33 On the Linearized Balescu Lenard Equation 583 By sending n in (38), using (39), we get We therefore conclude = i i g g Kg g Lg g =g Since L we deduce g =. Equivalently, Lg g =. We thus conclude that g = Pg by Lemma 9. On the other hand, since Pg n =, we also conclude that g =. This contradicts g = and thereby establishes the result. Now that we have established Coercivity of the linearized Balescu Lenard operator, we are ready to prove Theorem. Our arguments in this section are based on techniques developed in Strain and Guo (in press) combined with previous results in this work. Proof of Theorem. We will prove that a solution to the linearized Balescu Lenard equation satisfying the assumptions of Theorem also satisfies: d dt f t +f t (4) Here we use the notation f and f loosely in the sense that we establish the above only up to equivalent norms. Then we will show that (4) implies (9). We prove (4). Multiply 7 by f and integrate over v 3 to obtain d dt f +Lf f = We establish (4) for l = = by Corollary.. Next assume l and. In this case we multiply (7) by w l f and integrate over v 3 to obtain d dt f +w l Lf f = Our goal is to get a lower bound for w l Lf f. By Lemma we can write w l Lg g =g + iw ij j g g w i i g g w Kg g (4) Recall that C is the indicator function of the set v C. Weclaim that w l Lg g q g C C g (4) where q > depends on q from (8) and > chosen small enough. Notice that by plugging (4) into the differential equality at the beginning of this paragraph and adding the result to (4) for the case = l = yields (4) in the general case (at least up to an equivalent norm). To establish (4), it remains to prove (4). We do this first assuming l and <. Later we will handle the case = separately. We estimate each of the

34 584 Strain last three terms on the right side of (4). By Lemma 7 and (8) we can write i w l v ij v = w vw v vv j From the definition of w v in 8, w v C +v. By Lemma 8, vv j C log+v. Thus for any m > +v i w ij j g g w l log +v C +v j g g dv +v For m> large, we split the integral into a bounded part and an unbounded part. We have v>m log +v j gg w dv Cm +v +v v>m Cm g 3 g log +v w +v j ggdv where we have used Cauchy Schwartz, Corollary 8., < and m> chosen large enough. We use Cauchy Schwartz on the bounded part to obtain log +v j g g w v m +v +v dv Cmg m g 3 g + C mg We estimate the last two terms on the right side of (4) in the same way using Theorem. This establishes the claim for <. For =, to prove (4) we will split linear operator in a different way. Specifically, split L = A K (as in Lemma ) and define Mv exp ( ( )) q 4 +v. First we can show that there is q > such that ( w l Ag g q +v l ij v 3 Second we can establish ( +v l ij v 3 C q Cq g i Mg j Mg + v i i Mg j Mg + v i ) v j Mg dv ) v j Mg dv q g C q Cq g where q = q > since <q< and > can be chosen arbitrarily small. This is enough to establish (4) because the K part is controlled by Theorem. Using slightly different notation, this exact result was shown for the linearized Landau collision operator in Strain and Guo (in press, Lemma 9) in equations (64) and (65) of that paper. Since the proof is very much the same in this case, we will not repeat it. Now that we have established our claim and thereby the differential inequality (4), it remains to show that this implies exponential decay. As in other works on

35 On the Linearized Balescu Lenard Equation 585 time decay problems for soft potentials (Caflisch, 98; Strain and Guo, in press), a key point is to split f t into a time dependent low velocity part Then from Corollary 8., we have E = +v + t p f t C + t p f E t where E is just the indicator of the set E. Notice that we have ignored the weak logarithmic factor. Plugging this into the the differential inequality (4) we obtain Thus d dt f t + d dt f t + C f + t E p t C + t p f t C + t p f E c t Define = C p where for now p = p and p > is otherwise arbitrary. Then d ( e +t p f dt t) p + t p e +tp f E c t The integrated form is f t f e +tp t + t + s p e pe +tp +sp f E c tds Since f E c t is on Ec = +v > + t p we have f E c s Ce q +s p f s In the last display we have used the region and ( ) q exp q +v +v The integrated form of (4) implies f t e tp (f t + pf t ( ) q exp +v e q +s p ) + s p e +sp q +s p ds (43) The biggest exponent p that we can allow with this splitting is p = p ; since also p = p + we have p = so that + p = + + = +

36 586 Strain Further choose > large enough so that = C < q p. Therefore the right side of (43) is finite. This completes the proof of decay. Acknowledgments The author would like to express his gratitude to Yan Guo for suggesting that he study this equation. He also thanks Clément Mouhot for several stimulating discussions regarding this work. This work was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. References Balescu, R. (96). Irreversible processes in ionized gases. Phys. Fluids 3:5 63. Bogolyubov, N. N. (946). Problemy dinamičeskoĭteorii v statisti českoĭfizike. [Problems of Dynamical Theory in Statistical Physics.] Moscow-Leningrad, Russian: Gosudarstv. Izdat. Tehn.-Teor. Lit. Bogolyubov, N. N. (96). Problems of a dynamical theory in statistical physics. Studies in Statistical Mechanics : 8. Caflisch, R. E. (98). The Boltzmann equation with a soft potential. I. Linear, spatiallyhomogeneous. Comm. Math. Phys. 74():7 95. Decoster, A., Markowich, P. A., Perthame, B. (998). Modeling of Collisions. Series in Applied Mathematics (Paris). Vol.. Gauthier-Villars Éditions Scientifiques et Médicales Elsevier, Paris 998. With Contributions by I. Gasser, A. Unterreiter and L. Desvillettes; Edited and with a foreword by P. A. Raviart. Degond, P., Lemou, M. (997). Dispersion relations for the linearized Fokker Planck equation. Arch. Rational Mech. Anal. 38(): Fried, B. D., Conte, S. D. (96). The Plasma Dispersion Function. The Hilbert Transform of the Gaussian. New York: Academic Press, pp. v+49. Guo, Y. (). The Landau equation in a periodic box. Comm. Math. Phys. 3(3): Hazeltine, R. D., Waelbroeck, F. L. (988). The Framework of Plasma Physics. Frontiers in Physics. Vol.. Perseus Books. Lenard, A. (96). On Bogoliubov s kinetic equation for a spatially homogeneous plasma. Ann. Physics :39 4. Liboff, R. L. (998). Kinetic Theory: Classical, Quantum, and Relativistic Descriptions. nd ed. New York: John Wiley. Lifshitz, E. M., Pitaevskiĭ, L. P. (98). Course of Theoretical Physics [ Landau-Lifshits ]. Vol.. Pergamon International Library of Science, Technology, Engineering and Social Studies. Oxford: Pergamon Press, pp. xi+45. Translated from the Russian by J. B. Sykes and R. N. Franklin. Merchant, A. H., Liboff, R. L. (973). Spectral properties of the linearized Balescu Lenard operator. J. Mathematical Phys. 4:9 9. Montgomery, D. C., Tidman, D. A. (964). Plasma Kinetic Theory. McGraw-Hill. Nicholson, D. R. (983). Introduction to Plasma Theory. John Wiley & Sons. Strain, R. M., Guo, Y. Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. (in press), strain/research.html Strain, R. M., Guo, Y. (4). Stability of the relativistic Maxwellian in a collisional plasma. Comm. Math. Phys. 5():63 3. Thompson, W. B. (96). An Introduction to Plasma Physics. Oxford: Pergamon Press, pp. viii+56. Villani, C. (). A review of mathematical topics in collisional kinetic theory. Handbook of Mathematical Fluid Dynamics. Vol. I. Amsterdam: North-Holland, pp