STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA

Transcription

1 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA ROBERT M. STRAIN AND YAN GUO Abstract. The relativistic Landau-Maxwell system is among the most fundamental and complete models for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field. We construct the first global in time classical solutions. Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the Jüttner solution. Contents 1. Collisional Plasma 1. Main Results 4 3. The Relativistic Landau Operator 1 4. Local Solutions Positivity of the Linearized Landau Operator Global Solutions 5 References Collisional Plasma A dilute hot plasma is a collection of fast moving charged particles [7]. Such plasmas appear commonly in such important physical problems as in Nuclear fusion and Tokamaks. Landau, in 1936, introduced the kinetic equation used to model a dilute plasma in which particles interact through binary Coulombic collisions. Landau did not, however, incorporate Einstein s theory of special relativity into his model. When particle velocities are close to the speed of light, denoted by c, relativistic effects become important. The relativistic version of Landau s equation was proposed by Budker and Beliaev in 1956 [1]. It is widely accepted as the most complete model for describing the dynamics of a dilute collisional fully ionized plasma. The relativistic Landau-Maxwell system is given by t F + + c p p + t F + c p p x F + + e + E + p p + x F e E + p p B p F + = CF +, F + + CF +, F B p F = CF, F + CF, F + Comm. Math. Phys. 51 4, no., 63 3.

2 R. M. STRAIN AND Y. GUO with initial condition F ±, x, p = F,± x, p. Here F ± t, x, p are the spatially periodic number density functions for ions + and electrons, at time t, position x = x 1, x, x 3 T 3 [ π, π] 3 and momentum p = p 1, p, p 3. The constants ±e ± and m ± are the magnitude of the particles charges and rest masses respectively. The energy of a particle is given by p ± = m ± c + p. The l.h.s. of the relativistic Landau-Maxwell system models the transport of the particle density functions and the r.h.s. models the effect of collisions between particles on the transport. The heuristic derivation of this equation is total derivative along particle trajectories = rate of change due to collisions, where the total derivative of F ± is given by Newton s laws ẋ = the relativistic velocity = p m± + p /c, ṗ = the Lorentzian force = ±e ± E + p p ± B. The collision between particles is modelled by the relativistic Landau collision operator C in 1 and [1, 9] sometimes called the relativistic Fokker-Plank-Landau collision operator. To completely describe a dilute plasma, the electromagnetic field Et, x and Bt, x is generated by the plasma, coupled with F ± t, x, p through the celebrated Maxwell system: p p t E c x B = 4πJ = 4π e + p + F + e p F dp, t B + c x E =, with constraints x B =, x E = 4πρ = 4π e + F + e F dp, and initial conditions E, x = E x and B, x = B x. The charge density and current density due to all particles are denoted ρ and J respectively. We define relativistic four vectors as P + = p +, p = p+, p 1, p, p 3 and Q = q, q. Let g +p, h p be two number density functions for two types of particles, then the Landau collision operator is defined by 1 Cg +, h p p ΦP +, Q p g + ph q g + p q h q dq. The ordering of the +, in the kernel ΦP +, Q corresponds to the order of the functions in argument of the collision operator Cg +, h p. The collision kernel is given by the 3 3 non-negative matrix ΦP +, Q π p + c e +e L q 1 +, ΛP +, Q SP +, Q, m + c m c where L +, is the Couloumb logarithm for + interactions. The Lorentz inner product with signature + is given by P + Q = p + q p q.

3 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 3 We distinguish between the standard inner product and the Lorentz inner product of relativistic four-vectors by using capital letters P + and Q to denote the fourvectors. Then, for the convenience of future analysis, we define Λ S P+ Q m c m + c P+ m + c P+ + m + c 3/ P+ m + c Q 1, m c p 1 I 3 m + c Q p 1 m c m + c q m c + Q m c q p m c q m c m + c p m + c. q m c This kernel is the relativtistic counterpart of the celebrated classical non-relativistic Landau collision operator. It is well known that the collision kernel Φ is a non-negative matrix satisfying and [8, 9] 3 Φ ij P +, Q i=1 i,j qi q p i p + = 3 Φ ij P +, Q j=1 p Φ ij P +, Q w i w j > if w d p + qj q q q p j p + d R. =, The same is true for each other sign configuration +, +,, +,,. This property represents the physical assumption that so-called grazing collisions dominate, e.g. the change in momentum of the colliding particles is perpendicular to their relative velocity [9] [p. 17]. This is also the key property used to derive the conservation laws and the entropy dissipation below. It formally follows from that for number density functions g + p, h p 1 p p + Ch +, g p + 1 p p Cg, h + p dp =. The same property holds for other sign configurations. By integrating the relativistic Landau-Maxwell system and plugging in this identity, we obtain the conservation of mass, total momentum and total energy for solutions as d m + F + t = d m F t =, dt T 3 R dt 3 T 3 d dt pm + F + t + m F t + 1 Et Bt =, T 3 R 4π 3 T 3 1 m + p + F +t + m p F t + 1 T 3 R 8π 3 d Et + Bt =. dt T 3 The entropy of the relativistic Landau-Maxwell system is defined as Ht F + t, x, p ln F + t, x, p + F t, x, p ln F t, x, p dxdp. T 3 Boltzmann s famous H-Theorem for the relativistic Landau-Maxwell system is d Ht, dt

4 4 R. M. STRAIN AND Y. GUO e.g. the entropy of solutions is non-increasing as time passes. The global relativistic Maxwellian a.k.a. the Jüttner solution is given by exp cp ± J ± p = /k BT 4πe ± m ±ck B T K m ± c /k B T, where K is the Bessel function K z z 3 e zt t 1 3/ dt, T is the temperature and k B is Boltzmann s constant. From the Maxwell system and the periodic boundary condition of Et, x, we see that d dt Bt, xdx. We thus have a T 3 constant B such that 1 3 T 3 Bt, xdx = B. T 3 Let [, ] denote a column vector. We then have the following steady state solution to the relativisitic Landau-Maxwell system 1 [F ± t, x, p, Et, x, Bt, x] = [J ±,, B], which minnimizes the entropy Ht =. It is our purpose to study the effects of collisions in a hot plasma and to construct global in time classical solutions for the relativistic Landau-Maxwell system with initial data close to the relativistic Maxwellian Theorem 1. Our construction implies the asymptotic stability of the relativistic Maxwellian, which is suggested by the H-Theorem.. Main Results We define the standard perturbation f ± t, x, p to J ± as F ± J ± + J ± f ±. We will plug this perturbation into the Landau-Maxwell system of equations to derive a perturbed Landau-Maxwell system for f ± t, x, p, Et, x and Bt, x. The two Landau-Maxwell equations for the perturbation f = [f +, f ] take the form t + c p p ± x ± e ± E + p p ± B p f ± e ±c p J E k B T p ± ± + L ± f 4 = ± e ±c k B T E p p ± f ± + Γ ± f, f, with f, x, p = f x, p = [f,+ x, p, f, x, p]. The linear operator L ± f defined in 1 and the non-linear operator Γ ± f, f defined in 3 are derived from an expansion of the Landau collision operator 1. The coupled Maxwell system takes the form p p t E c x B = 4πJ = 4π e + J p + + f + e J p f dp, 5 t B + c x E =, with constraints 6 x E = 4πρ = 4π e + J + f + e J f dp, x B =, with E, x = E x, B, x = B x. In computing the charge ρ, we have used the normalization J ± pdp = 1 e ±.

5 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 5 Notation: For notational simplicity, we shall use, to denote the standard L inner product in and, to denote the standard L inner product in T 3. We define the collision frequency as the 3 3 matrix 7 σ ij ±, p Φ ij P ±, Q J qdq. These four weights corresponding to signatures +, +, +,,, +,, are used to measure the dissipation of the relativistic Landau collision term. Unless otherwise stated g = [g +, g ] and h = [h +, h ] are functions which map t T 3 R. We define g, h σ σ ij +,+ + σ ij +, σ ij +,+ + σ ij +, σ ij, + σ ij,+ pj g + pi h + + σ ij, + σ ij,+ pj g pi h dp, pi p j p + p + pi p j p p g + h + dp g h dp, where in 8 and the rest of the paper we use the Einstein convention of implicitly summing over i, j 1,, 3 unless otherwise stated. This complicated inner product is motivated by following splitting, which is a crucial element of the energy method used in this paper Lemma 6 and Lemma 8: Lg, h = [L + g, L g], h = g, h σ + a compact term. We will also use the corresponding L norms g σ g, g σ, g σ g, g σ g, g σ dx. T 3 We use to denote the L norm in and to denote the L norm in either T 3 or T 3 depending on whether the function depends on both x, p or only on x. Let the multi-indices γ and β be γ = [γ, γ 1, γ, γ 3 ], β = [β 1, β, β 3 ]. We use the following notation for a high order derivative γ β γ t x γ1 1 x γ x γ3 3 p β1 1 p β p β3 3. If each component of β is not greater than that of β s, we denote by β β; β < β β β means β β, and β < β. We also denote by C β β. Let f t f σt [E, B] t γ + β N γ + β N γ β ft, γ β ft σ, [ γ Et, γ Bt]. It is important to note that our norms include the temporal derivatives. For a function independent of t, we use the same notation but we drop the t. The above norms and their associated spaces are used throughout the paper for arbitrary functions.

6 6 R. M. STRAIN AND Y. GUO We further define the high order energy norm for a solution ft, x, p, Et, x and Bt, x to the relativistic Landau-Maxwell system 4 and 5 as 9 Et 1 f t + [E, B] t + Given initial datum [f x, p, E x, B x], we define t E = 1 f + [E, B ], f σsds. where the temporal derivatives of [f, E, B ] are defined naturally through equations 4 and 5. The high order energy norm is consistent at t = for a smooth solution and Et is continuous Theorem 6. Assume that initially [F, E, B ] has the same mass, total momentum and total energy as the steady state [J ±,, B], then we can rewrite the conservation laws in terms of the perturbation [f, E, B]: m + f + t 1 J + m f t J, T 3 T T 3 p m + f + t J + + m f t J 1 4π T 3 m + p + f +t J + + m p f t J 1 8π T 3 Et Bt, T 3 Et + Bt B. We have used 3 for the normalized energy conservation 1. The effect of this restriction is to guarantee that a solution can only converge to the specific relativistic Maxwellian that we perturb away from if the solution converges to a relativistic Maxwellian. The value of the steady state B is also defined by the initial conditions 3. We are now ready to state our main results: Theorem 1. Fix N, the total number of derivatives in 9, with N 4. Assume that [f, E, B ] satisfies the conservation laws 1, 11, 1 and the constraint 6 initially. Let F,± x, p = J ± + J ± f,± x, p. There exist C > and M > such that if E M, then there exists a unique global solution [ft, x, p, Et, x, Bt, x] to the perturbed Landau-Maxwell system 4, 5 with 6. Moreover, F ± t, x, p = J ± + J ± f ± t, x, p solves the relativistic Landau-Maxwell system and Remarks: sup Es C E. s These solutions are C 1, and in fact C k, for N large enough. Since f σtdt < +, ft, x, p gains one momentum derivative over it s initial data and f σt in a certain sense.

7 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 7 Further, Lemma 5 together with Lemma 13 imply that γ Et + γ Bt B C γ ft σ. 1 Therefore, except for the highest order derivatives, the field also converges. It is an interesting open question to determine the asymptotic behavior of the highest order derivatives of the electromagnetic field. Recently, global in time solutions to the related classical Vlasov-Maxwell-Boltzmann equation were constructed by the second author in [6]. The Boltzmann equation is a widely accepted model for binary interactions in a dilute gas, however it fails to hold for a dilute plasma in which grazing collisions dominate. The following classical Landau collision operator with normalized constants was designed to model such a plasma: C cl F, F + v φv v v F vf + v F v v F + v dv. The non-negative 3 3 matrix is 13 φ ij v = δ ij v iv j 1 v v. Unfortunately, because of the crucial hard sphere assumption, the construction in [6] fails to apply to a non-relativistic Coulombic plasma interacting with it s electromagnetic field. The key problem is that the classical Landau collision operator, which was studied in detail in [5], offers weak dissipation of the form 1+ v 1 f dv. The global existence argument in Section 6 from [6] does not work because of this weak dissipation. Further, the unbounded velocity v, which is inconsistent with Einstein s theory of special relativity, in particular makes it impossible to control a nonlinear term like E v f ± in the classical theory. On the other hand, in the relativistic case our key observation is that the corresponding nonlinear term c E p/p ± f± can be easily controlled by the dissipation because cp/p ± c and the dissipation in the relativisitc Landau operator is f dp Lemma 5 and Lemou [8]. However, it is well-known that the relativity effect can produce severe mathematical difficulties. Even for the related pure relativistic Boltzmann equation, global smooth solutions were only constructed in [3, 4]. The first new difficulty is due to the complexity of the relativistic Landau collision kernel ΦP +, Q. Since P + m + c Q m c 1 1 p c q m + when P + m + c Q m c, the kernel in 1 has a first order singularity. Hence it can not absorb many derivatives in high order estimates Lemma 7 and Theorem 4. The same issue exists for the classical Landau kernel φv v, but the obvious symmetry makes it easy to express v derivatives of φ in terms of v derivatives. It is then possible to integrate by parts and move derivatives off the singular kernel in the estimates of high order derivatives. On the contrary, no apparent symmetry exists beween p and q in the relativistic case. We overcome this severe difficulty with the splitting m pj Φ ij P +, Q = q p + qj Φ ij P +, Q + pj + q p + qj Φ ij P +, Q,

8 8 R. M. STRAIN AND Y. GUO where the operator pj + q p + qj does not increase the order of the singularity mainly because pj + q p + qj P + Q =. This splitting is crucial for performing the integration by parts in all of our estimates Lemma and Theorem 3. We believe that such an integration by parts technique should shed new light on the study of the relativistic Boltzmann equation. As in [5, 6], another key point in our construction is to show that the linearized collision operator L is in fact coercive for solutions of small amplitude to the full nonlinear system 4, 5 and 6: Theorem. Let [ft, x, p, Et, x, Bt, x] be a classical solution to 4 and 5 satisfying 6, 1, 11 and 1. There exists M, δ = δ M > such that if 1 14 γ ft + γ Et + γ Bt M, then L γ ft, γ ft δ γ ft σ. Theorem is proven through a careful study of the macroscopic equations These macroscopic equations come from a careful study of solutions f to the perturbed relativistic Landau-Maxwell system 4, 5 with 6 projected onto the null space N of the linearized collision operator L = [L +, L ] defined in 1. As expected from the H-theorem, L is non-negative and for every fixed t, x the null space of L is given by the six dimensional space 1 i 3 15 N span[ J +, ], [, J ], [p i J +, p i J ], [p + J +, p J ]. This is shown in Lemma 1. We define the orthogonal projection from L p onto the null space N by P. We then decompose ft, x, p as f = Pf + I Pf. We call Pf = [P + f, P f] R the hydrodynamic part of f and I Pf = [I P + f, I P f] is called the microscopic part. By separating its linear and nonlinear part, and using L ± Pf =, we can express the hydrodynamic part of f through the microscopic part up to a higher order term hf: 16 where t + c p p ± x P ± f e ±c k B T E l ± I Pf t + p p ± h ± f e ± E + p p ± ± e ±c E k B T p p ± J ± = l ± I Pf + h ± f, x I P ± f + L ± I Pf, B p f ± p f ± + Γ ± f, f. p ±

9 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 9 We further expand P ± f as a linear combination of the basis in P ± f a ±t, x + 3 b j t, xp j + ct, xp ± J ±. j=1 A precise definition of these coefficients will be given in 94. The relativistic system of macroscopic equations 98-1 are obtained by plugging 19 into 16. These macroscopic equations for the coefficients in 19 enable us to show that there exists a constant C > such that solutions to 4 which satisfy the smallness constraint 14 for M > small enough will also satisfy γ a ± + γ b + γ c CM I P γ ft σ. This implies Theorem since Pf σ is trivially bounded above by the l.h.s. Proposition and L is coercive with respect to I P γ ft Lemma 8. Since our smallness assumption 14 involves no momentum derivatives, in proving the presence of momentum derivatives in the collision operator 1 causes another serious mathematical difficulty. We develop a new estimate Theorem 5 which involves purely spatial derivatives of the linear term 1 and the nonlinear term 3 to overcome this difficulty. To the best of the authors knowledge, until now there were no known solutions for the relativistic Landau-Maxwell system. However in, Lemou [8] studied the linearized relativistic Landau equation with no electromagnetic field. We will use one of his findings Lemma 5 in the present work. For the classical Landau equation, the 199 s have seen the first solutions. In 1994, Zhan [1] proved local existence and uniqueness of classical solutions to the Landau-Poisson equation B with Coulomb potential and a smallness assumption on the initial data at infinity. In the same year, Zhan [11] proved local existence of weak solutions to the Landau-Maxwell equation with Coulomb potential and large initial data. On the other hand, in the absence of an electromagnetic field we have the following results. In, Desvillettes and Villani [] proved global existence and uniqueness of classical solutions for the spatially homogeneous Landau equation for hard potentials and a large class of initial data. In, the second author [5] constructed global in time classical solutions near Maxwellian for a general Landau equation both hard and soft potentials in a periodic box based on a nonlinear energy method. Our paper is organized as follows. In section 3 we establish linear and nonlinear estimates for the relativistic Landau collision operator. In section 4 we construct local in time solutions to the relativistic Landau-Maxwell system. In section 5 we prove Theorem. And in section 6 we extend the solutions to T =. Remark 1. It turns out that the presence of the physical constants do not cause essential mathematical difficulties. Therefore, for notational simplicity, after the proof of Lemma 1 we will normalize all constants in the relativistic Landau-Maxwell system 4, 5 with 6 and in all related quantities to be one.

10 1 R. M. STRAIN AND Y. GUO 3. The Relativistic Landau Operator Our main results in this section include the crucial Theorem 3, which allows us to express p derivatives of ΦP, Q in terms of q derivatives of ΦP, Q. This is vital for establishing the estimates found at the end of the section Lemma 7, Theorem 4 and Theorem 5. Other important results include the equivalence of the norm σ with the standard Sobolev space norm for H 1 Lemma 5 and a weak formulation of compactness for K which is enough to prove coercivity for L away from the null space N Lemma 8. We also compute the sum of second order derivatives of the Landau kernel Lemma 3. We first introduce some notation. Using, we observe that quadratic collision operator 1 satisfies CJ +, J + = CJ +, J = CJ, J + = CJ, J =. Therefore, the linearized collision operator Lg is defined by 1 Lg = [L + g, L g], L ± g A ± g K ± g, where A + g J 1/ + C J + g +, J + + J 1/ + C J + g +, J, A g J 1/ C J g, J + J 1/ C J g, J +, K + g J 1/ + CJ +, J + g + + J 1/ + CJ +, J g, K g J 1/ CJ, J g + J 1/ CJ, J + g +. And the nonlinear part of the collision operator 1 is defined by where 3 Γg, h = [Γ + g, h, Γ g, h], Γ + g, h J 1/ + C J + g +, J + h + + J 1/ + C J + g +, J h, Γ g, h J 1/ C J g, J h + J 1/ C J g, J + h +. We will next derive the null space 15 of the linear operator in the presence of all the physical constants. Lemma 1. Lg, h = Lh, g, Lg, g. And Lg = if and only if g = Pg. Proof. From 1 we split Lg, h, with Lg = [L + g, L g], as C J + g +, J + + CJ +, J + g + dp J+ 4 h + h + C J + g +, J + CJ +, J g dp J+ h C J g, J + + CJ, J + g + dp J h C J g, J + CJ, J g dp. J

11 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 11 We use the fact that qi J q = c q i k B T J q q and pi J 1/ + p = c k B T as well as the null space of Φ in to show that CJ 1/ + g +, J = pi Φ ij P +, Q J qj 1/ + p = pi = pi And similarly Φ ij P +, Q J qj 1/ + p qi q pi p + p i p + p i p + J + p g + p + pj g + p dq g + p + pj g + p dq Φ ij P +, Q J qj + p pj J 1/ + g + pdq CJ, J 1/ + g + = pi Φ ij P, Q + J pj 1/ + q pi p qi q i q + g + q + qj g + q dq = pi Φ ij P, Q + J pj 1/ + q q + g + q + qj g + q dq 5 = pi Φ ij P, Q + J pj + q qj J 1/ + g + qdq. Similar expressions hold by exchanging the + terms and the terms in the appropriate places. For the first term in 4, we integrate by parts over p variables on the first line, then relabel the variables switching p and q on the second line and finally adding them up on the last line to obtain = Φ ij P +, Q + J + pj + q pi h + J 1/ + p = pj g + J 1/ + p qj g + J 1/ + qdpdq Φ ij P +, Q + J + pj + q qi h + J 1/ + q qj g + J 1/ + q pj g + J 1/ + pdpdq = 1 Φ ij P +, Q + J + pj + q pi h + J 1/ + p qi h + J 1/ + q pj g + J 1/ + p qj g + J 1/ + qdpdq. By the first term in 4 is symmetric and if h = g. The fourth term can be treated similarly with + replaced by everywhere. We combine the second and third terms in 4; again we integrate by parts over p variables to compute = Φ ij P +, Q J + pj q pi h + J 1/ + p pj g + J 1/ + p qj g J 1/ qdpdq + Φ ij P, Q + J pj + q pi h J 1/ p pj g J 1/ p qj g + J 1/ + qdpdq.

12 1 R. M. STRAIN AND Y. GUO We switch the role of p and q in the second term to obtain = Φ ij P +, Q J + pj q pi h + J 1/ + p qi h J 1/ q pj g + J 1/ + p qj g J 1/ qdpdq. Again by this piece of the operator is symmetric and if g = h. We therefore conclude that L is a non-negative symmetric operator. We will now determine the null space 15 of the linear operator. Assume Lg =. From Lg, g = we deduce, by, that there are scalar functions ζ l p, q l = ± such that pi g l J 1/ l p qi g l J 1/ l q ζ l p, q pi p l q i q l, i 1,, 3. Setting q =, pi g l J 1/ l p = ζ l p, pi +b p l li. By replacing p by q and subtracting we obtain pi g l J 1/ l p qi g l J 1/ l q = ζ l p, p i = ζ l p, p l pi ζ l q, q i p l q i q l q l + ζ l p, ζ l q, q i q l. We deduce, again by, that ζ l p, ζ l q, = and therefore that ζ l p, c l a constant. We integrate pi g l J 1/ p l p = c i l + b li to obtain g l = a g l + 3 i=1 p l b g li p j + c g l pl J 1/ l. Here a g l, bg lj and cg l are constants with respect to p but could be functions of t and x. Moreover, we deduce from the middle terms in 4 as well as that pi g + J 1/ + p qi g J 1/ q ζp, pi q. Therefore b g +i bg i + cg + pi p + c g qi q p + q i q = ζp, q pi qi q. We conclude b g +i b g i, i = 1,, 3; c g + c g. That means gt, x, p N as in 15, so that g = Pg. Conversely, LPg = by a direct calculation. For notational simplicity, as in Remark 1, we will normalize all the constants to be one. Accordingly, we write = 1 + p, P =, p, and the collision kernel ΦP, Q takes the form 6 where ΦP, Q ΛP, Q SP, Q, q Λ P Q 3/, P Q 1 S P Q 1 I 3 p q p q + P Q 1 p q + q p.

13 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 13 We normalize the relativistic Maxwellian as Jp J + p = J p = e p. We further normalize the collision freqency 7 σ±, p ij = σ ij p = Φ ij P, QJqdq, and the inner product, σ takes the form g, h σ σ ij pj g + pi h + + pj g pi h dp, + 1 σ ij p i p j 8 g + h + dp + g h dp. R p 3 The norms are, as before, naturally built from this normalized inner product. The normalized vector-valued Landau-Maxwell equation for the perturbation f in 4 now takes the form t + p x + ξ 9 E + p B p E = ξ p f E f + Γf, f. p Jξ1 + Lf with f, x, v = f x, v, ξ 1 = [1, 1], and the matrix ξ is diag1, 1. Further, the normalized Maxwell system in 5 and 6 takes the form p 3 t E x B = J = Jf+ f dp, t B + x E =, R p 3 31 x E = ρ = Jf+ f dp, x B =, with E, x = E x, B, x = B x. We have a basic but useful inequality taken from Glassey & Strauss [3]. Proposition 1. Let p, q with P =, p and Q = q, q then 3 p q + p q q P Q 1 1 p q. This will inequality will be used many times for estimating high order derivatives of the the collision kernel. Notice that pi + q qi P Q = pi + q qi q p q qi p i =. q = p i q q i + q This is the key observation which allows us do analysis on the relativistic Landau Operator Lemma. We define the following relativistic differential operator 33 Θ α p, q p3 + q α 3 q3 p + q α q p1 + q α 1 q1. Unless otherwise stated, we omit the p, q dependence and write Θ α = Θ α p, q. Note that the three terms in Θ α do not commute and we choose this order for no special reason.

14 14 R. M. STRAIN AND Y. GUO We will use the following splitting many times in the rest of this section, 34 A = p q + p q [ p + 1]/, B = p q + p q [ p + 1]/. The set A is designed to be away from the first order singularity in collision kernel ΦP, Q Proposition 1. And the set B contains a ΦP, Q singularity 6 and 3 but we will exploit the fact that we can compare the size of p and q. We now develop crucial estimates for Θ α ΦP, Q: Lemma. For any multi-index α, the Lorentz inner product of P and Q is in the null space of Θ α, Θ α P Q =. Further, recalling 6, for p and q on the set A we have the estimate 35 Θ α p, qφp, Q Cp α q 6. And on B, q 6q. Using this inequality, we have the following estimate on B 37 Θ α p, qφp, Q Cq 7 p α p q 1. Proof. Let e i i = 1,, 3 be an element of the standard basis in. We have seen that Θ ei P Q =. And the general case follows from a simple induction over α. By 6 and 33, we can now write S Θ α p, qφ ij ij p, q P, Q = ΛP, QΘ α p, q, q where 38 S ij p, q Θ α q = P Q 1 Θ α δ ij / q +P Q 1Θ α p i q j + p j q i / q Θ α p i q i p j q j / q. We will break up this expression and estimate the different pieces. Using 33, the following estimates are straight forward 39 4 On the other hand, we claim that Θ α δ ij / q Cq 1 p 1 α, Θ α p i q j + p j q i / q Cp α. 41 Θ α p i q i p j q j / q C p q p α. q This last estimate is not so trivial because only a lower order estimate of p q is expected after applying even a first order derivative like Θ ei. The key observation is that pi + q p i q i p j q j = 1 q p j q j, qi and the r.h.s. is again second order. Therefore the operator Θ α can maintain the order of the cancellation.

15 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 15 Proof of claim: To prove 41, it is sufficient to show that for any multi-index α and any i, j, k, l 1,, 3 there exists a smooth function G α,ij kl p, q satisfying 3 4 Θ α p i q i p j q j / q = p k q k p l q l G α,ij kl p, q k,l=1 as well as the decay 43 ν1 ν q G α,ij kl p, q Cq 1 ν which holds for any multi-indices ν 1, ν. induction over α. If α =, we define G,ij kl p, q = δ kiδ lj. q p 1 α ν1, We prove 4 with 43 by a simple The decay 43 for G,ij kl p, q is straight forward to check. And 4 holds trivially for α =. Assume the 4 with 43 holds for α n. To conclude the proof, let α = n + 1 and write Θ α = Θ em Θ α for some multi-index α with m = maxj : α j >. This specification of m is needed because of our chosen ordering of the three differential operators in 33, which don t commute. Recalling 33, Θ em p k q k = δ km 1 q. From the induction assumption and the last display, we have We compute Θ α p i q i p j q j / q 3 = Θ em p k q k p l q l G α,ij p, q, = k,l=1 1 q 3 p k q k + k=1 kl G α,ij mk 3 p k q k p l q l Θ em G α,ij p, q. k,l=1 1 q = q = p q p q p + q = = + q + q p, q + Gα,ij km p, q kl l p l q l p l + q l. + q We plug this display into the one above it to obtain 4 for α with the new coefficients G α,ij G α,ij kl p, q Θ em G α,ij mk p, q + Gα,ij km p, q p l + q l kl p, q +. + q We check that G α,ij kl p, q satisfies 43 using the Leibnitz differentiation formula as well as the induction assumption 43. This establishes the claim 41.

16 16 R. M. STRAIN AND Y. GUO With the estimates 39, 4 and 41 in hand, we return to establishing 35 and 37. We plug the estimates 39, 4 and 41 into Θ α Φ ij P, Q from 38 to obtain that 44 Θα Φ ij P, Q Cp α P Q P Q 1 3/ P Q 1 q +Cp α P Q P Q 1 3/ P Q 1 +Cp α P Q P Q 1 3/ p q q. We will use this estimate twice to get 35 and 37. We first establish 35. On the set A we have p q + p q p q + p q 1 4 p + p 1 4 p. From 3 and the last display we have P Q + 1 P Q q From the Cauchy-Schwartz inequality we also have P Q 1 P Q q + p q q. We plug these last two inequalities one at a time into 44 to obtain Θα Φ ij p, q CP Q P Q 1 3/ α p p q + p q + p q q C q P Q 1 3/ p α q C q 3 P Q 1 3 p α 3 C q 3 p p α. q We move on to establishing 36. If p 1, then q. Assume p 1, using B we compute q q p p q 1 p Therefore, 6q on B. For the other half of 36, q + p q We move on to establishing 37. On the set B we have a first order singularity. Also 3 tells us p q q P Q 1 1 p q.

17 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 17 We plug this into 44 to observe that on B we have Θα Φ ij p, q CP Q P Q 1 3/ α P Q + 1 p q + p q + p q q q CP Q P Q 1 3/ p α p q C q P Q 1 3/ p α p q C q q 3/ p q 1 P Q + 1 3/ p α C q 7/ p q 1 p α. We achieve the last inequality because 3 says P Q 1. Next, let µp, q be an arbitrary smooth scalar function which decay s rapidly at infinity. We consider the following integral Φ ij P, QJ 1/ qµp, qdq. Both the linear term L and the nonlinear term Γ are of this form Lemma 6. We develop a new integration by parts technique. Theorem 3. Given β >, we have 45 = β Φ R ij P, QJ 1/ qµp, qdq 3 Θ β1 Φ ij P, QJ 1/ q q β β3 µp, qϕ β β 1,β,β 3 p, qdq β 1+β +β 3 β where ϕ β β 1,β,β 3 p, q is a smooth function which satisfies 46 ν q 1 ν ϕ β β 1,β,β 3 p, q Cq β ν1 p β1 + β3 β ν, for all multi-indices ν 1 and ν. Proof. We prove 45 by an induction over the number of derivatives β. Assume β = e i i = 1,, 3. We write 47 pi = q qi + pi + q qi = q qi + Θ ei

18 18 R. M. STRAIN AND Y. GUO Instead of hitting Φ ij P, Q with pi, we apply the r.h.s. term above and integrate by parts over q qi to obtain = pi Φ R ij P, QJ 1/ qµp, qdq 3 Φ ij P, QJ 1/ q pi µp, qdq R 3 + Φ ij P, QJ 1/ q q qi µp, qdq R p 3 + Φ ij P, QJ 1/ qi q q i µp, qdq R q 3 + Θ ei Φ ij P, QJ 1/ qµp, qdq. We can write the above in the form 45 with the coefficients given by 48 φ ei,, p, q = q i q i, φ ei e q p i,, p, q = 1, φei,e p, q = q i,, φ ei,,e p i p, q = 1. And define the rest of the coefficients to be zero. Note that these coefficients satisfy the decay 46. This establishes the first step in the induction. Assume the result holds for all β n. Fix an arbitrary β such that β = n+1 and write β = pm β for some multi-index β and m = maxj : β j >. This specification of m is needed because of our chosen ordering of the three differential operators in 33, which don t commute. By the induction assumption β Φ ij P, QJ 1/ qµp, qdq = pm Θ β1 Φ R ij P, QJ 1/ q q β β3 µp, qϕ β β1, β, β p, qdq 3 3 β 1+ β + β 3 β We approach applying the last derivative the same as the β = 1 case above. We obtain = Θ β1 Φ ij P, QJ 1/ qϕ β β1, β, β 3 p, q pm q β β3 µp, qdq + Θ β1 Φ ij P, QJ 1/ qϕ β β1, β, β 3 p, q q qm q β β3 µp, qdq + Θ em Θ β1 Φ ij P, QJ 1/ q q β β3 µp, qϕ β β1, β, β 3 p, qdq + Θ β1 Φ ij P, QJ 1/ q q β β3 µp, q R 3 pm + q qm + q m q m ϕ β β1, q p β, β p, qdq, 3

19 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 19 where the unspecified summations above are over β 1 + β + β 3 β. We collect all the terms above with the same order of differentiation to obtain = Θ β1 Φ ij P, QJ 1/ q q β β3 µp, qϕ β β 1,β,β 3 p, qdq, β 1+β +β 3 β where the functions ϕ β β 1,β,β 3 p, q are defined naturally as the coefficient in front of each term of the form Θ β1 Φ ij P, QJ 1/ q q β β3 µp, q and we recall that β = β + e m. We check 46 by comparing the decay with the order of differentiation in each of the four terms For 49, the order of differentiation is β 1 = β 1, β = β, β 3 = β 3 + e m. And by the induction assumption, ν q 1 ν ϕ β β1, β, β p, q 3 Cq β ν1 This establishes 46 for 49. For 5, the order of differentiation is p β 1 + β 3 β ν, Cq β+em ν1 p β1 + β 3+e m β+e i ν = Cq β ν 1 p β1 + β3 β ν. β 1 = β 1, β = β + e m, β 3 = β 3. And by the induction assumption as well as the Leibnitz rule, q q ν 1 ν ϕ β β1, p β, β p, q Cq β +1 ν1 3 p β 1 + β 3 β 1 ν, This establishes 46 for 5. For 51, the order of differentiation is = Cq β ν 1 p β1 + β3 β ν. β 1 = β 1 + e m, β = β, β 3 = β 3. And by the induction assumption, ν q 1 ν ϕ β β1, β, β p, q 3 Cq β ν1 This establishes 46 for 51. For 5, the order of differentiation is p β 1 + β 3 β ν, Cq β ν 1 p β1 + β3 β ν. β 1 = β 1, β = β, β 3 = β 3. And by the induction assumption as well as the Leibnitz rule, q ν 1 ν pm + q qm + q m q m ϕ β β1, q p β, β p, q 3 Cq β +1 ν1 p β 1 + β 3 β 1 ν, = Cq β ν 1 p β1 + β3 β ν. This establishes 46 for 5 and therefore for all of the coefficients.

20 R. M. STRAIN AND Y. GUO Next, we compute derivatives of the collision kernel in 6 which will be important for showing that solutions F ± to the relativistic Landau-Maxwell system are positive. Lemma 3. We compute a sum of first derivatives in q of 6 as 53 j qj Φ ij ΛP, Q P, Q = P Qp i q i. q This term has a second order singularity at p = q. We further compute a sum of 53 over first derivatives in p as 54 i,j pi qj Φ ij P, Q = 4 P Q q P Q 1 1/. This term has a first order singularity. This result is quite different from the classical theory, it is straightforward compute the derivative of the classical kernel in 13 as vi v j φ ij v v =. On the contrary, the proof of Lemma 3 is quite technical. i,j Proof. Throught this proof, we temporarily suspend our use of the Einstein summation convention. Differentiating 6, we have qj Φ ij P, Q q j ΛP, Q S ij P, Q q ΛP, Q + qj S ij P, Q q j q q S ij P, Q. And 3/ qj ΛP, Q = P Q P Q qj 1 p j q 3P Q 3 5/ P Q qj 1 p j. q Since implies j Sij qj P, Q q p j =, we conclude j j qj ΛP, QS ij P, Q q =. Therefore it remains to evaluate the r.h.s. of 55 qj Φ ij ΛP, Q P, Q = qj S ij P, Q q j q q S ij P, Q. j

21 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 1 We take a derivative of S ij in 6 as qj S ij qj = P Q p j q δ ij + qj p j p i q j + q i p j q + P Q 1 p i + δ ij p j δ ij p i q i qj = P Q p j δ ij + q j p i + p j q j q i p i p j q j q i p j q q q +P Q 1 + δ ij p i 1 + δ ij q i. Next, sum this expression over j to obtain qj S ij qi = P Q p i q j p qp i p q i + 4P Qp i 4q i. + q q p i + p qq i We collect terms which are coefficients of p i and q i respectively qj S ij = q i P Q + p q p p 4 q q 56 j +p i P Q + q p p q + 4P Q q + p i = q i p q P Q 3 3P Q q where the last line follows from plugging p = p p 1 = q q 1 into the first line and plugging q = q = q q q q q into the second line. Turning to the computation of 1 q j q js ij, we plug 6 into the following q j S ij P, Q = P Q 1 q i p i q i q p q j + P Q 1 p qq i + q p i We collect terms which are coefficients of p i and q i respectively to obtain = q i P Q 1 + q p q + p q P Q 1 +p i q p q + q P Q 1 = q i P Q 1 q + p q P Q + p i p q + q P Q = q i p q q p q + p i q P Q p q = q i p q q p q q + pi q P Q P Q p q = q q i P Q q + q p i P Q p. q Divide this expression by q to conclude q j S ij p 57 = q i P Q 1 + p i q j q P Q q,.

22 R. M. STRAIN AND Y. GUO This and 56 are very symmetric expressions. We combine 56 and 57 to obtain qj S ij q j q S ij p = q i P Q 3 + p i 3P Q p j q q p q i P Q 1 p i P Q p q q = P Qp i q i. We note that this term has a first order cancellation at p = q. We plug this last display into 55 to obtain 53. We differentiate 53 to obtain pi qj Φ ij P, Q = pi ΛP, Q P Qp i q i p i j i q 58 ΛP, Q + q i pi p i p P Qp i q i. And we can write the derivative of Λ as 5/ pi ΛP, Q = P Q P Q pi P 1 q q i Q +. We compute pi q q i P Qp i q i = p i i We further add and subtract q P Q to obtain q P Qp i q p i q i p i q i P Q + qi = q P Q p q p q p qp Q + q. = q P Q q P Q q p q p qp Q + q = q P Q q p qp Q + q = q P Q p qp Q 1 = P Q 1. We conclude that pi ΛP, Q 59 q i P Qp i q i = P Q q P Q + 3/. P Q 1 This term has a third order singularity. We will find that the second term in 58 also has a third order singularity, but there is second order cancellation between the two terms in 58. We now evaluate the sum in second term in 58 as i pi p i p P Qp i q i = i P Q + p i p i q q i = 3P Q + p q p q P Q p p P Q p i p + p iq i p + p q p.

23 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 3 We add and subtract q as well as P Q p to obtain = 3P Q q + q p q P Q + P Q p + p q p = 3P Q q + q p = 3P Q. Therefore, plugging in 6, we obtain ΛP, Q pi p i P Q3 3/. q p P Qp i q i = 3 P Q 1 q i Further plugging this and 59 into 58 we obtain 54. In the following Lemma, we will use Lemma 3 to obtain a simplified expression for part of the collision operator 1 which will be used to prove the positvity of our solutions F ± to the relativistic Landau-Maxwell system. Lemma 4. Given a smooth scalar function Gq which decays rapidly at infinity, we have pi Φ R ij P Q 1/ P, Q qj Gqdq = 4 P Q 1 Gqdq p 3 q +κpgp, where κp = 7/ π π 1 + p sin θ 3/ sin θdθ. Proof. We write out pi as in 47 to observe pi Φ ij P, Q qj Gqdq = q qi + Θ ei Φ ij P, Q qj Gqdq = Θ ei Φ ij P, Q qj Gqdq + q qi Φ ij P, Q qj Gqdq. We split these integrals into p q ɛ and p q > ɛ for ɛ >. We note that the integrals over p q ɛ converge to zero as ɛ. We will eventually send ɛ, so we focus on the region p q > ɛ. We rewrite q qi Φ ij P, Q qj Gq = qj q qi Φ ij P, QGq Gq. qj q qi Φ ij P, Q After an integration by parts, the integrals over p q > ɛ are = qj Θei Φ ij P, Q Gqdq p q >ɛ + p q >ɛ p q >ɛ q qj qi Φ ij P, Q Gqdq q qj qi Φ ij P, QGq dq.

24 4 R. M. STRAIN AND Y. GUO By the definition of Θ ei in 33, this is = qj pi Φ ij P, QGqdq + p q >ɛ p q >ɛ q qj qi Φ ij P, QGq dq. We plug 54 into the first term above to obtain the first term on the r.h.s. of this Lemma as ɛ. For the second term above, we apply the divergence theorem to obtain p q >ɛ q qj qi Φ ij q P, QGq dq = qi Φ ij P, Q p j q j p q =ɛ p q GqdS, where ds is given below. By a Taylor expansion, P Q = 1 + O p q. Using this and 53 we have q qi Φ ij P, Q p j q j p q Q = ΛP, p p q + O p q 1. And the integral over p q = ɛ which includes the terms in O p q 1 goes to zero as ɛ. We focus on the main part ΛP, Q p q GqdS. p p q =ɛ We multiply and divide by q + p q + 1 to observe that P Q 1 = p q + p q q + p q + 1. This and 6 imply 3/ Λ = P Q p q + p q + 1 p q + p q 3/. q p q + 1 We change variables as q p q so that the integrand becomes Λ q Gp q and we define q = 1 + p q so that after the change of variables p q Λ = q p + p q + p 3/ p q + 1 q q p + p q 3/. + p q + 1 We choose the angular integration over q = ɛ in such a way that p q = p q cos θ and ds = ɛ sin θdθdφ = ɛ dω with θ π, φ π. Note that as ɛ on q = ɛ Hence, as ɛ, p q =ɛ ɛ 3 Λ 3/ 1 + p sin θ 3/ p 3. Λ q Gp qds = p ɛ 3 ΛGp ɛωdω κpgp, S with κ defined in the statement of this Lemma. Lemma 5. There exists C >, such that 1 6 p g C + g g σ C p g + g. Further, σ ij p is a smooth function satisfying 61 β σ ij p β C.

25 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 5 Proof. The spectrum of σ ij p, 7, consists of a simple eigenvalue λ 1 p > associated with the vector p and a double eigenvalue λ p > associated with p ; there are constants c 1, c > such that, as p, λ 1 p c 1, λ p c. In Lemou [8] there is a full discussion of these eigenvalues. This is enough to prove 6; see [5] for more details on a similar argument. We move on to 61. We combine 7 and 45 with µp, q = J 1/ q to obtain β σ ij p = β Φ R ij P, QJqdq 3 = Θ β1 ΦP, QJ 1/ q q β J 1/ qϕ β β 1,β, p, qdq. β 1+β β By 46 then β σ ij p C β 1+β β p β1 β Θ β1 ΦP, Q J 1/ qdq. Recall 34, we split this integration into the sets A, B. We plug in the estimate 35 to get p β1 β Θ β1 ΦP, Q J 1/ qdq Cp β1 β p β1 = Cp β. A On B we have a first order singularity but q is larger than p, in fact we use 36 to get exponential decay in p over this region. With 37 we obtain Θ β1 ΦP, Q J 1/ qdq CJ 1/16 p p q 1 J 1/4 qdq. B We can now deduce 61. We now write the Landau Operators A, K, Γ in a new form which will be used throughout the rest of the paper. Lemma 6. We have the following representations for A, K, Γ R, which are defined in and 3, 6 Ag = J 1/ pi J 1/ σ ij pj g + p j g = pi σ ij pj g 1 σij p i p j g + pi σ ij p j g, 63 Kg = Jp 1/ pi Jp Φ ij J 1/ q qj gq [1, 1]dq [1, 1] R 3 Jp 1/ pi Jp Φ ij J 1/ q q j gq [1, 1]dq [1, 1], R q 3 where Φ ij = Φ ij P, Q. Further 64 Γg, h = [Γ + g, h, Γ g, h],

26 6 R. M. STRAIN AND Y. GUO where Γ ± g, h = pi p i Φ ij J 1/ q pj g ± p hq [1, 1] dq, pi p i Φ ij J 1/ qg ± p qj hq [1, 1] dq. Proof. For 6 it suffices to consider Jp 1/ CJ 1/ g ±, J: Jp 1/ pi Φ R ij P, Q pj J 1/ g ± p Jq J 1/ g ± p 3 = Jp 1/ pi Φ ij P, QJqJp 1/ qj pj g ± + p j R q 3 = Jp 1/ pi σ ij pjp 1/ pj g ± + p j g ±. g ± dq qj Jq Above, we have used the null space of Φ in. Below, we move some derivatives inside and cancel out one term. = pi σ ij p pj g ± + p j g ± p i σ ij p pj g ± + p j g ± = pi σ ij p pj g ± + pi σ ij p p j g ± 1 σij p p i p j g ±. For K simply plug 5 with normalized constants into. For Γ, we use the null condition to compute Jp 1/ C Jg +, Jh = Jp 1/ pi Φ ij P, QJ 1/ qj 1/ p h q pj g + p qj h qg + p dq +Jp 1/ pi Φ ij P, QJ 1/ qj 1/ qj p p j h qg + pdq q = Jp 1/ pi Φ ij P, QJ 1/ qj 1/ p h q pj g + p qj h qg + p dq = pi p i Φ ij P, QJ 1/ q h q pj g + p qj h qg + p dq. dq Plug four of these type calculations into 3 to obtain 64. We will use these expressions just proven to get the estimates below. Lemma 7. Let β >. For any small η >, there exists C η > such that 65 β Ag, β g β g σ η α g σ C η g, 66 β Kg, β h η β β α β βg σ + C η g βh σ. Proof. We will prove 65 for a real valued function g to make the notation less cumbersome, although the result follows trivially for g = [g +, g ]. We write out

27 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 7 the inner product in 65 using 6 to achieve 67 β Ag, β g = β pi σ ij p i 1 β g σ ij p i p j β gdp. g β g + β [σ ij pj g] pi β g dp σ ij p i α g β gdp = β g σ + Cβ α β α pi α β Cβ α β α σ ij α pj g pi β gdp α<β Cβ α 1 α<β R β α σ ij p i p j p 3 α g β gdp Since α < β in the last two terms below, 61 gives us the following estimate β α pi σ ij p i + β ασ ij + β α σ ij p i p j Cp 1 We bound the second and fourth terms in 67 by C α β p 1 α g β g dp = On the unbounded part we use Cauchy-Schwartz α β p >m p 1 α g β g dp C m βg σ α β p m + p >m α g σ C α g σ m On the compact part we use the compact interpolation of Sobolev-spaces p m α β α g β g dp p m α β η η α = β +1 α β α g + β g dp p m α g σ + C η g α g dp + C η α β p m g dp For the third term in 67, we split into two cases, first suppose α < β 1 and integrate by parts on pi to obtain α < β 1 β α pi σ ij α pj g + β α σ ij pi α pj g β gdp

28 8 R. M. STRAIN AND Y. GUO We bound this term by C α pj g β g dp = α < β p 1 p m p m p m + C m By the compact interpolation of Sobolev spaces α pj g β g dp η α < β p m α β + p >m + C m βg σ α β α < β α g σ α g σ + C η g α g σ Finally, if α = β 1 for the third term in 67, we integrate by parts and use symmetry β α σ ij α pj g pi β gdp = β α σ ij α pj g β α α pi gdp R 3 = β ασ ij α pj g α pi gdp. Because the order of the derivatives on g is now = β, we can again use the compact interpolation of Sobolev spaces and 61 to get the same bounds as for the last case α < β 1. We obtain β Ag, β g β g σ η + C α g σ C η g m, α β This completes the estimate 65. We now consider β Kg, β h and 66. µp, q = Recalling 63, we use 45 with qj + qj q gq [1, 1], the Leibnitz formula as well as an integration by parts to express β Kg, β h as Θ α1 Φ ij P, Q JpJq α q qj + q j gq [1, 1] q β1 hp [1, 1] ϕ β,β1 α 1,α, p, qdqdp, where the sum is α 1 + α β, β β 1 β + 1. And ϕ β,β1 α 1,α, p, q is a collection of the inessential terms, it satisfies the decay estimate 46 independent of the value of β 1. Therefore we can further express β Kg, β h as µ β1β p, qjp 1/4 Jq 1/4 β1 hp [1, 1] q β gq [1, 1]dqdp where the sum is over β β + 1, β β 1 β + 1. And, using 35 and 37, we see that µ β1β p, q is a collection of L functions. Therefore, as in 8, we split β Kg, β h to get ψ ij p, qjp 1/4 Jq 1/4 β1 hp [1, 1] q β gq [1, 1]dqdp + µ β1β ψ ij Jp 1/4 Jq 1/4 β1 hp [1, 1] q β gq [1, 1]dqdp.

29 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 9 In the same fashion as J in 8, for any m >, we estimate the second term above by C m βh σ β β βg σ. Since ψ ij p, q is a smooth function with compact support, we integrate by parts over q repeatedly to bound the first term by β1 ψ ij p, qjq 1/4 Jp 1/4 gq [1, 1] β1 hp [1, 1]dqdp Cm Jq 1/8 Jp 1/4 gq [1, 1] β1 hp [1, 1] dqdp Cm g β1 h Cm g β h σ. where the final sum above is over β β 1 β + 1. And we have used 6 to get the last inequality. We conclude our lemma by choosing m large. We now estimate the nonlinear term Γf, g: Theorem 4. Let γ + β N, then 68 γ β Γf, g, γ β h C γ 1 β 3 f γ γ1 where the summation is over γ 1 γ, β + β 3 β. Further β g σ + γ1 β 3 f σ γ γ1 β g γ β h σ, γ β Γf, g, γ β h C γ β h σ g σ f + f σ g. Proof. Notice γ Γf, g = γ 1 γ Cγ1 γ Γ γ1 f, γ γ1 g; thus it suffices to only consider the p derivatives. From 64, using 45, we can write β Γf, g as pi Θ β1 Φ ij P, Q Jq q β β3 pj f l pg k q ϕ β β 1,β,β 3 p, qdq pi Θ β1 Φ ij P, Q Jq q β β3 fl p qj g k q ϕ β β 1,β,β 3 p, qdq + Θ β1 Φ ij P, Q Jq q β β3 f l p qj g k q p i ϕ β β p 1,β,β 3 p, qdq Θ β1 Φ ij P, Q Jq q β β3 pj f l pg k q p i ϕ β β p 1,β,β 3 p, qdq. Above, we implicitly sum over i, j 1,, 3, β 1 + β + β 3 β and k +,. And l +,. Recall ϕ β β 1,β,β 3 p, q from Theorem 3. Further 46 implies ϕ β β 1,β,β 3 p, q Cq β.

30 3 R. M. STRAIN AND Y. GUO We use the above two displays and integrate by parts over pi in the first two terms, to bound β Γf, g, β h above by C J 1/4 q Θ β1 Φ ij P, Q q 69 β g k q β3 pj f l p pi β h l p dqdp +C J 1/4 q Θ β1 Φ ij P, Q q 7 β qj g k q β3 f l p pi β h l p dqdp +C J 1/4 q Θ β1 Φ ij P, Q q β qj g k q β3 f l p β h l p dqdp +C J 1/4 q Θ β1 Φ ij P, Q q β g k q β3 pj f l p β h l p dqdp. In the above expressions, we add the summation over l +,. It suffices to estimate 69 and 7; the other two terms are similar. While estimating these terms we will repeatedly use the equivalence of the norms in 6 without mention. We start with 69. We next split 69 into the two regions, A and B, defined in 34. We then use 37 and the Cauchy-Schwartz inequality to estimate 69 over B J 1/4 q Θ β1 Φ ij P, Q q β g k q β3 pj f l p pi β h l p dqdp B C p q 1 J 1/8 q q β g k q β3 pj f l p pi β h l p dqdp B C β h σ β3 pj f l p B p q 1 J 1/8 q q β g k q dq dp C β h σ β g β3 pj f l p 1/ p q J 1/4 qdq dp B C β h σ β g β3 f σ, 1/ where use 36 to say that B p q J 1/4 qdq C. This completes the estimate of 69 over B. We estimate 69 over A using 35 J 1/4 q Θ β1 Φ ij P, Q q β g k q β3 pj f l p pi β h l p dqdp A C J 1/8 q q β g k q β3 pj f l p pi β h l p dqdp C β h σ β g β3 f σ. This completes the estimate for 69. Note that the difference between 7 and 69 is that qj hits g in 7 whereas pj hit f in 69. This difference means we will need to use the norm σ to estimate g but we are able to use the smaller norm to estimate f. This is in contrast to the estimate for 69 where the opposite situation held. The main point is that σ includes first order p derivatives. Using the same splitting over A and B as well as the same type calculation, 7 is bounded by β h σ β g σ β3 f. This completes the estimate for 7. The proof of 69 follows from the Sobolev embedding: H T 3 L T 3. Without loss of generality, assume γ 1 N/. This Sobolev embedding grants us

31 STABILITY OF THE RELATIVISTIC MAXWELLIAN IN A COLLISIONAL PLASMA 31 that sup γ1 β fx γ γ1 x γ β f β 3 gx σ + γ γ1 β 3 gx σ + sup x γ1 β fx σ γ γ1 β 3 gx γ β f σ γ γ1 β 3 gx, where summation is over γ γ 1 + N since N 4. We conclude 69 by integrating 68 further over T 3. We next prove the important estimates which are needed to prove Theorem in Section 5. Theorem 5. Let γ N. let gx, p be a smooth vector valued L T 3 x p; R function and hp a smooth vector valued L p; R function, we have 71 γ Γg, g, h C β h γ g γ g σ Moreover, β 7 L γ g, h C γ g β h β Proof. We begin with the linear term. By Lemma 6, 1 and two integrations by parts L γ g, h is given by γ g pj pi hpσ ij + 1 p σij i p j γ g hp dp + pi σ ij p i γ g hp dp qj Φ ij P, Q JqJp γ g l q pi Jp 1/ h k p q Jp γ g l q pi qj Φ ij P, Q Jq dqdp, Jp 1/ h k p dqdp we implicitly sum over i, j 1,, 3 and k, l +,. Using cauchy s inequality, L γ g, h is bounded by γ g pj pi hpσ ij + 1 p σij i p j γ g hp dp dx 73 plus + 74 pi σ ij p i γ g hp dp dx qj Φ ij P, Q JqJp γ g l q pi Jp 1/ h k p q dqdp dx qj Φ ij P, Q Jq Jp γ g l q pi Jp 1/ h k p dqdp. dx We have split 73 and 74 because we will estimate each one separately.

32 3 R. M. STRAIN AND Y. GUO By the Cauchy-Schwartz inequality as well as 61, and that hp is not a function of x, the first and second lines of 73 are bounded by C β h γ gx dx = C γ g β h β β This completes 7 for the first and second lines of 73. By the Cauchy-Schwartz inequality over dp the third line of 73 is bounded by C 1/ β h J 1/ q γ g l q Φ ij P, Q J pdp dq 1/ dx β 1 Recall again the splitting in 34, apply 35 and 37 with α 1 = to obtain Φ ij P, Q J 1/ pdp = + A B Cq 1 + q 14 p q J 1/4 pdp Cq 14 Using the Cauchy-Schwartz inequality again, this implies that the third line of 73 is bounded by C γ g β 1 βh. To establish 7 it remains to estimate 74. From 33, we write qj = q pj + qj + q pj B = q pj + Θ ej q, p where e j is an element of the standard basis in. For the rest of this proof, we write Θ ej = Θ ej q, p for notational simplicity although the reader should note that it is the opposite of the shorthand we were using previously. Further, qj Φ ij P, QJ 1/ q = Φ ij P, Q q j Jq q Jq q pj Φ ij P, Q + JqΘ ej Φ ij P, Q. We plug the above into 74 and integrate by parts for the middle term in 74 to bound 74 by 75 C Jq Jp Φ ij P, Q γ g l q β h k p dqdp dx +C Jq Jp Θ ej Φij P, Q γ g l q β h k p dqdp dx. Above, we add the implicit summation over β. By the same estimates as for 73, the first term in 75 is C γ g β βh. To establish 7, it remains to estimate the second term in 75. To this end, we note that Θ ej q, pp Q =. Further, Θ ej q, pφ ij P, Q satisfies the estimates Θ ej q, pφp, Q Cqq 6 1 on A, Θ ej q, pφp, Q Cqq 7 1 on B, where we recall the sets from 34. This can be shown directly, by repeating the proof of 35 and 37 using the operator Θ ej q, p in place of the operator Θ ej p, q. Plugging these estimates into the second term of 75, we can show that it is