Some Applications of an Energy Method in Collisional Kinetic Theory

Transcription

1 Some Applications of an Energy Method in Collisional Kinetic Theory by Robert Mills Strain III B. A., New York University, Sc. M., Brown University, A Dissertation submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Division of Applied Mathematics at Brown University Providence, Rhode Island May 5

2 c Copyright 5 by Robert Mills Strain III

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4 Contents Chapter. Introduction.. Precise statements of the results Chapter. Stability of the Relativistic Maxwellian in a Collisional Plasma 8.. Collisional Plasma 8.. Main Results 3.3. The Relativistic Landau Operator.4. Local Solutions Positivity of the Linearized Landau Operator Global Solutions 75 Chapter 3. Almost Exponential Decay near Maxwellian Introduction Vlasov-Maxwell-Boltzmann Relativistic Landau-Maxwell System Soft Potentials The Classical Landau Equation 93 Chapter 4. Exponential Decay for Soft Potentials near Maxwellian Introduction Introduction Boltzmann Estimates Landau Estimates Energy Estimate and Global Existence Proof of Exponential Decay 54 Chapter 5. On the Relativistic Boltzmann Equation 56 v

5 5.. The Relativistic Boltzmann Equation Examples of relativistic Boltzmann cross-sections Lorentz Transformations Center of Momentum Reduction of the Collision Integrals Glassey-Strauss Reduction of the Collision Integrals Hilbert-Schmidt form Relativistic Vlasov-Maxwell-Boltzmann equation 87 Appendix A. Grad s Reduction of the Linear Boltzmann Collision Operator 89 Bibliography 96 vi

6 Abstract of Some Applications of an Energy Method in Collisional Kinetic Theory by Robert Mills Strain III, Ph.D., Brown University, May 5 The collisional Kinetic Equations we study are all of the t F + v r x F + V (t, x) r v F = Q(F, F). Here F = F (t, x, v) is a probabilistic density function (of time t, space x and velocity v R 3 ) for a particle taken chosen randomly from a gas or plasma. V (t, x) is a field term which usually represents Maxwell s theory of electricity and magnetism, sometimes this term is neglected. Q(F, F) is the collision operator which models the interaction between colliding particles. We consider both the Boltzmann and Landau collision operators. We prove existence, uniqueness and regularity of close to equilibrium solutions to the relativistic Landau-Maxwell system in the first part of this thesis. Our main tool is an energy method. In the second part, we prove arbitrarily high polynomial time decay rates to equilibrium for four kinetic equations. These are cuto soft potential Boltzmann and Landau equations, but also the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system. The main technique used here is interpolation. In the third part, we prove exponential decay for the cuto soft potential Boltzmann and Landau equations. The main point here is to show that exponential decay of the initial data is propagated by a solution. In the fourth and final part of this thesis, we write down a few important calculations in the relativistic Boltzmann theory which are scattered around the literature. We also calculate a few Lorentz transformations which maybe useful in relativistic transport theory. We use these calculations to comment about extending the results in this thesis to the relativistic Boltzmann equation.

7 CHAPTER Introduction In this thesis, we study some mathematical properties of a class of nonlinear partial di erential equations called kinetic equations which arise out of the Kinetic Theory of Gases and Plasmas. We are primarily concerned with stability theorems, e.g. results on existence, uniqueness and decay rates of solutions which are initially close to steady state.... Brief Introduction to Kinetic Theory. A rarefied gas is a large collection of electrically neutral particles (around 3 ) in which collisions form a small part of each particles lifetime. A plasma is a large collection of fast-moving charged particles. The large number of particles makes it di cult to attempt a physical description of a gas or plasma using Classical Mechanics. Boltzmann (87) instead decided to posit the existence of a probability density function F (t, x, v) which roughly speaking tells you the average number of particles a time t, position x and velocity v. He proposed a partial di erential equation which governs the time evolution of F. Boltzmann s Equation is the foundation of kinetic theory [,, 5, 3, 5, 67, 68]. The density function F (t, x, v) contains a great deal of information, it can be a practical tool for determining detailed properties of dilute gases and plasmas as well as calculating many important physical quantities. Kinetic Theory has been among the most active areas in the mathematical study of nonlinear partial di erential equations over the last few decades spectacular progress has been made yet still more is expected. The main goal of this research is to study the existence, uniqueness, regularity and asymptotic properties of solutions F. And the major result thus far is the global renormalized weak solutions of Diperna & Lions [3] to the Boltzmann equation with large initial data.

8 Recently, Yan Guo has developed a flexible nonlinear energy method [39] used to construct global in time smooth solutions to the hard sphere Vlasov-Maxwell- Boltzmann system for initial data close to Maxwellian equilibrium. This system is a generalization of the Boltzmann equation which models two species of particles (electrons and ions) interacting and includes electromagnetic e ects. It is a system of ten partial di erential equations, two coupled Boltzmann equations that represent electrons and ions interacting also coupled with Maxwell s equations for electricity and magnetism. See [35 4] for more applications of such a method. There is also related work near vacuum [4]. In this Ph.D thesis, we apply this energy method a few important problems in kinetic theory... Precise statements of the results In this doctoral thesis, I develop new techniques to study existence, uniqueness, regularity and asymptotic convergence rates of several nonlinear partial di erential equations from collisional Kinetic Theory. In Section.., we discuss the relativistic Landau-Maxwell system. In Section.., we discuss the asymptotic decay of four kinetic equations. And in Section..3, we discuss exponential decay for soft potential Boltzmann and Landau equations.... Relativistic Landau-Maxwell System. Dilute hot plasmas appear commonly in important physical problems such as Nuclear fusion and Tokamaks. The relativistic Landau-Maxwell system [5] is the most fundamental and complete model for describing the dynamics of a dilute collisional plasma in which particles interact through binary Coulombic collisions and through their self-consistent electromagnetic field. It is widely accepted as the most complete model for describing the dynamics of a dilute collisional fully ionized plasma. Yan Guo and I proved existence of global in time classical solutions [6] with initial data near the relativistic Maxwellian. To the best of our knowledge, this is the first existence proof for the relativistic Landau-Maxwell system. In, Lemou [47] studied the linearized relativistic Landau equation with no electromagnetic field. In

9 the non-relativistic situation there are a few results for classical solutions to kinetic equations with the Landau collision operator [7, 38, 7]. We considered the following coupled relativistic Landau-Maxwell system (two t F ± + p r x F ± ± E + p B r p F ± = C(F ±,F ± )+C(F ±,F ) Here F ± (t, x, p) are the spatially periodic number density functions for ions (+) and electrons ( ) at time t, position x = (x,x,x 3 ) T 3 = [, ] 3 and momentum p =(p,p,p 3 ) R 3. The energy of a particle is given by = p + p. The electromagnetic field, E(t, x) and B(t, x), is coupled with F ± (t, x, p) through the celebrated Maxwell t E r x B = 4 with constraints r x B =, r x E =4 R R 3 {F + initial conditions. R 3 p {F + F } t B + r x E =, F } dp and spatially periodic The relativistic Landau collision operator is defined by C(G, H) r p (p, q) {r p G(t, x, p)h(t, x, q) G(t, x, p)r q H(t, x, q)} dq, R 3 where the 3 3 matrix (p, q) is a complicated expression [47, 5, 6]. Steady state solutions are [F ± (t, x, p),e(t, x),b(t, x)] = [e,, B], where B is a constant which is determined initially. We consider the standard perturbation around the relativistic Maxwellian F ± (t, x, p) =e + e f ± (t, x, p). Define the high order energy norm for a solution to the relativistic Landau-Maxwell system as E N (t) [f +,f ] N(t)+ [E,B] N(t)+ t [f +,f ],N(s)ds, where N represents an L sobolev norm in (x, p) (or just in x in the case of [E,B]) but also includes t derivatives. N contains (t, x, p) derivatives of all orders apple N. The dissipation,n represents the same norm, but with a momentum weight and one extra p-derivative. At t = the temporal derivatives are defined naturally through the equations. 3

10 Theorem.. [6]. Fix N 4. Let F,± (x, p) =e + e f,± (x, p). Assume that initially [F,±,E, B ] has the same mass, total momentum and total energy as the steady state [e,, B]. 9C >, > such that if E N () apple then there exists a unique global in time solution [F ± (t, x, p),e(t, x),b(t, x)] to the relativistic Landau-Maxwell system. Moreover, F ± (t, x, p) =e + e f ± (t, x, p) and suapplesapple E N (s) apple CE N (). The proof of this theorem uses the energy method from [39]. However, severe new mathematical di culties were present in terms of the complexity of the relativistic Landau Collision kernel (p, q) and the momentum derivatives in the collision operator C. One important question left open in this result and in the case of the Vlasov-Maxwell-Boltzmann system [39] is the question of a precise time decay rate to equilibrium.... Almost Exponential Decay Near Maxwellian. The key di culty in obtaining time decay for the relativistic Landau-Maxwell system is contained in the following di erential inequality ( N > ): dy N dt (t)+ N [f +,f ],N(t) apple, where the instant energy, y N (t), is equivalent to The di [f +,f ] N(t)+ [E,B B] N (t). culty lies in the fact that the instant energy functional at each time is stronger than the dissipation rate [f +,f ],N (t): [f +,f ] N(t)+ [E,B B] N (t) apple C [f +,f ],N(t). No estimates for the N-th order derivatives of E and B are available. It is therefore di cult to imagine being able to use a Gronwall type of argument to get the time decay rate. Our main observation is that a family of energy estimates have been 4

11 derived in [6]. The instant energy is stronger for fixed N, but it is possible to be bounded by a fractional power of the dissipation rate via simple interpolations with stronger energy norms. Only algebraic decay is possible due to such interpolations, but more regular initial data grants faster decay. Theorem.. [6]. Fix N 4, k and choose initial data [F,±,E,B ] which satisfies the assumptions of the previous Theorem for both N and N + k. Then there exists C N,k > such that [f +,f ] N(t)+ [E,B B] N (t) apple C N,k E N+k () + t k. k And the same decay holds for the Vlasov-Maxwell-Boltzmann system under similar conditions. On a related note, Desvillettes and Villani have recently undertaken an impressive program to study the time-decay rate to Maxwellian of large data solutions to soft potential Boltzmann type equations. Even though their assumptions in general are a priori and impossible to verify at the present time, their method does lead to an almost exponential decay rate for the soft potential Boltzmann and Landau equations for solutions close to a Maxwellian. This surprising new decay result relies crucially on recent energy estimates in [37, 38] as well as other extensive and delicate work [8, 53, 64, 69]. To obtain time decay for these models with very weak collision e ects has been a very challenging open problem even for solutions near a Maxwellian, therefore it is of great interest to find simpler proofs. Using interpolations with higher velocity weights, Yan Guo and I gave a much simpler proof of almost exponential decay for the Boltzmann and Landau equations. The Boltzmann equation is written t F + v r x F = Q(F, F), F(, x, v) =F (x, v). Above F (t, x, v) is the spatially periodic density function for particles at time t, position x =(x,x,x 3 ) T 3 and velocity v =(v,v,v 3 ) R 3. The collision operator 5

12 is Q(F, F) du R 3 d!b( ) u S v {F (t, x, u )F (t, x, v ) F (t, x, u)f (t, x, v)}. Here = (u, v,!) is the scattering angle and [u,v ]=[u (u, v,!),v (u, v,!)] are the post-collisional velocities. The exponent in the collision operator is 3 < < (soft potentials) and we assume B( ) satisfies the Grad angular cuto assumption: <B( ) apple C cos. The Landau Equation is obtained in the so-called grazing collision limit of the Boltzmann equation [5]. We write the perturbation from Maxwellian, µ = e v, as F (t, x, v) =µ(v)+ p µ(v)f(t, x, v). Then define the high order energy norm (.) E`(t) f `(t)+ t f,`(s)ds, where f `(t) is an instantaneous norm in time and an energy norm in the space and velocity variables which includes derivatives in all variables (t, x, v) and of all orders apple N where N 8. Here ` denotes a polynomial velocity weight like ( + v ) `/. At t = we define the temporal derivatives naturally through the equation.,` is the same norm as `, but with another dissipation weight. Theorem.3. [6]. Fix 3 < < ; fix integers ` and k>. Let F (x, v) =µ + p µf (x, v). Assume the initial data F (x, v) has the same mass, momentum and energy as the steady state µ. 9C` >, ` > such that if E`() < ` then there exists a unique global in time solution to the soft potential Boltzmann equation F (t, x, v) =µ + µ / f(t, x, v) with suapplesapple E`(f(s)) apple C`E`(). 6

13 Moreover, if E`+k () < `+k, then the unique global solution satisfies f `(t) apple C`,k E`+k () + t k. k The same result holds for the Landau Equation. We remark that the existence for both the Boltzmann and Landau case were proven in [37, 38] at ` =...3. Exponential Decay Near Maxwellian. These almost exponential decay results suggest strongly that under some conditions exponential decay can be achieved. Indeed, in the case of the Boltzmann equation with a soft potential Caflisch [8, 9] got decay of the form exp( t p ) for some > and <p< but only for < <. Caflisch uses an exponential weight of the for e q v in his norms for <q<. Yan Guo and I extend Caflisch s result to the soft potential 4 Boltzmann equation for the full range 3 < < and to the Landau equation using an exponential weight like e q v # for apple # apple. 7

14 CHAPTER Stability of the Relativistic Maxwellian in a Collisional Plasma Abstract. The relativistic Landau-Maxwell system is the most fundamental and complete model 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. This result has already appeared in a modified form as [6]... Collisional Plasma A dilute hot plasma is a collection of fast moving charged particles [4]. Such plasmas appear commonly in such important physical problems as in Nuclear fusion and Tokamaks. Landau, in 936, 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 e ects become important. The relativistic version of Landau s equation was proposed by Budker and Beliaev in 956 [3 5]. 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 t F + + c p p t F r x F + + e + E + p + c p r x F e p + E + p B r p F + = C(F +,F + )+C(F +,F ) B r p F = C(F,F )+C(F,F + ) 8

15 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,x,x 3 ) T 3 [, ] 3 and momentum p =(p,p,p 3 ) R 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 ± = 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 e ect 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 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 (.) and [3, 4, 5] (sometimes called the relativistic Fokker-Plank-Landau collision operator). To completely describe a dilute plasma, the electromagnetic field E(t, x) and B(t, x) is generated by the plasma, coupled with F ± (t, x, p) through the celebrated Maxwell system: t E cr x B = 4 J = 4 e + R p + F + 3 e p F t B + cr x E =, with constraints r x B =, r x E =4 =4 {e + F + R 3 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 and J respectively. 9

16 We define relativistic four vectors as P + =(p +,p)=(p +,p,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 (.) C(g +,h )(p) r p (P +,Q ) {r p g + (p)h (q) g + (p)r q h (q)} dq. R 3 The ordering of the +, in the kernel (P +,Q ) corresponds to the order of the functions in argument of the collision operator C(g +,h )(p). The collision kernel is given by the 3 3 non-negative matrix (P +,Q ) p + c e +e L q +, (P +,Q )S(P +,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. We distinguish between the standard inner product and the Lorentz inner product of relativistic four-vectors by using capital letters P + and Q Then, for the convenience of future analysis, we define to denote the four-vectors. S ) 3/ Q, m c ( P+ m + c Q P+ m c m + c ( P+ m + c Q p ) I 3 m c m + c P+ + m + c Q p m c m + c q m c q m c + q m c p 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 3 i= ij (P +,Q ) qi q p i p + = 3 j= is a non-negative matrix satisfying ij (P +,Q ) qj q p j p + =,

17 and [47, 5] i,j ij (P +,Q )w i w j > if w 6= d p p + q q 8d 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 [5] [p. 7]. 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) R >< >= B p A C(h +,g )(p)+ B p A C(g,h +)(p) dp =. >: >; p + The same property holds for other sign configurations. By integrating the relativistic Landau-Maxwell system and plugging in this identity, we deduce the local conservation of mass R 3 m t + c p p ± r x F ± (t, x)dp =, the local conservation of total momentum (both kinetic and electromagnetic) R 3 p m t + c p p + r x F + (t, t + c p r x F (t, x) dp + t (E(t) B(t)) = (B r x ) B +(E r x + r x E) E 4 r x B + E, and the local conservation of total energy (both kinetic and electromagnetic) R 3 m + p t + c p p + r x F + (t, t + c p r x F (t, x) dp + t E(t) + B(t) = 4 {E (r x B) B (r x E)}.

18 Integration over the periodic box T 3 yields 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 R 3 d p(m + F + (t)+m F (t)) + E(t) B(t) =, dt T 3 R 4 3 T 3 d (m + p + F + (t)+m p dt F (t)) + E(t) + B(t) =. T 3 R 8 3 T 3 The entropy of the relativistic Landau-Maxwell system is defined as H(t) {F + (t, x, p) ln F + (t, x, p)+f (t, x, p) ln F (t, x, p)} dxd. T 3 R 3 Boltzmann s famous H-Theorem for the relativistic Landau-Maxwell system is d H(t) apple, dt 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 J ± (p) = where K ( ) is the Bessel function K (z) z 3 exp cp ± /(k B T ) 4 e ± m ±ck B TK (m ± c /(k B T )), R e zt (t ) 3/ dt, T is the temperature and k B is Boltzmann s constant. From the Maxwell system and the periodic R boundary condition of E(t, x), we see that d B(t, x)dx. We thus have a con- dt T 3 stant B such that (.3) B(t, x)dx = T 3 B. T 3 Let [, ] denote a column vector. We then have the following steady state solution to the relativisitic Landau-Maxwell system which minnimizes the entropy (H(t) = ). [F ± (t, x, p),e(t, x),b(t, x)] = [J ±,, B], It is our purpose to study the e ects 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.). Our construction

19 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 ± + p 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), E(t, x) and B(t, x). The two Landau-Maxwell equations for the perturbation f =[f +,f ] take the t + c p p ± r x ± e ± E + p p ± B r p f ± e ±c p p E J k B T p ± ± + L ± f (.4) = ± e ±c k B T E with f(, x, p) =f (x, p) =[f,+ (x, p),f, in (.) and the non-linear operator p p ± f ± + ± (f,f), (x, p)]. The linear operator L ± f defined ±(f,f) defined in (.3) are derived from an expansion of the Landau collision operator (.). The coupled Maxwell system takes the form (.5) p t E cr x B = 4 J = 4 e + J R p + + f + t B + cr x E =, with constraints (.6) r x E =4 =4 R 3 n e + p J + f + e p J f e p p J f o dp, r x B =, with E(,x)=E (x), B(,x)=B (x). In computing the charge, we have used the normalization R R 3 J ± (p)dp = e ±. Notation: For notational simplicity, we shall use h, i to denote the standard L inner product in R 3 and (, ) to denote the standard L inner product in T 3 R 3. We define the collision frequency as the 3 3 matrix ij (.7) ±, (p) ij (P ±,Q )J (q)dq. 3 dp,

20 These four weights (corresponding to signatures (+, +), (+, ), (, +), (, )) are used to measure the dissipation of the relativistic Landau collision term. otherwise stated g =[g +,g ] and h =[h +,h ] are functions which map {t T 3 R 3! R. We define Unless } (.8) hg, hi R R 3 R 3 ij +,+ + ij pj g pi h + + ij, + pj pi h dp, ij +,+ + ij +, ij, + ij,+ p i p + p i p j p + g + h + dp p j g h dp, where in (.8) and the rest of the paper we use the Einstein convention of implicitly summing over i, j {,, 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): hlg, hi = h[l + g, L g],hi = hg, hi + a compact term. We will also use the corresponding L norms g hg, gi, kgk (g, g) hg, gi dx. T 3 We use to denote the L norm in R 3 and k k to denote the L norm in either T 3 R 3 or T 3 (depending on whether the function depends on both (x, p) or only on x). Let the multi-indices and be =[,,, 3 ], =[,, 3 ]. We use the following notation for a high 3 x 3 p 3. 4

21 If each component of is not greater than that of s, we denote by apple ; < means apple, and <. We also A by C. Let f (t) f (t) + applen + applen [E,B] (t) f(t), [@ B(t)]. 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. We further define the high order energy norm for a solution f(t, x, p), E(t, x) and B(t, x) to the relativistic Landau-Maxwell system (.4) and (.5) as (.9) E(t) f (t)+ [E,B] (t)+ Given initial datum [f (x, p),e (x),b (x)], we define t E() = f + [E,B ], f (s)ds. 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 E(t) 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) p (.) J + m f (t) p J, T 3 R 3 T 3 R 3 n p m + f + (t) p J + + m f (t) p o J (.) E(t) B(t), T 3 R 4 3 T 3 m + p + f + (t) p J + + m f (t) p J (.) E(t) + B(t) B. T 3 R 8 3 T 3 We have used (.3) for the normalized energy conservation (.). 5

22 The e ect 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.. Fix N, the total number of derivatives in (.9), with N 4. Assume that [f,e,b ] satisfies the conservation laws (.), (.), (.) and the constraint (.6) initially. Let F,± (x, p) =J ± + p J ± f,± (x, p). There exist C > and M> such that if E() apple M, then there exists a unique global solution [f(t, x, p),e(t, x),b(t, x)] to the perturbed Landau-Maxwell system (.4), (.5) with (.6). Moreover, F ± (t, x, p) =J ± + p J ± f ± (t, x, p) solves the relativistic Landau-Maxwell system and Remarks: sup E(s) apple C E(). applesapple These solutions are C, and in fact C k,forn large enough. Since R f (t)dt < +, f(t, x, p) gains one momentum derivative over it s initial data and f (t)! in a certain sense. Further, Lemma.5 together with Lemma.3 imply that E(t) {B(t) B} apple C f(t). 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. 6

23 Recently, global in time solutions to the related classical Vlasov-Maxwell-Boltzmann equation were constructed by the second author in [39]. 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 + ) r v (v v ) {r v F (v)f + (v ) F (v)r v F + (v )} dv. R 3 The non-negative 3 3 matrix is (.3) ij (v) = ij v i v j v v. Unfortunately, because of the crucial hard sphere assumption, the construction in [39] 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 [38], o ers weak dissipation of the form R R 3 ( + v ) f dv. The global existence argument in Section.6 (from [39]) 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 ± apple c and the dissipation in the relativisitc Landau operator is R R 3 f dp (Lemma.5 and Lemou [47]). However, it is well-known that the relativity e ect can produce severe mathematical di culties. Even for the related pure relativistic Boltzmann equation, global smooth solutions were only constructed in [3, 33]. kernel The first new di (P +,Q ). Since culty is due to the complexity of the relativistic Landau collision P + m + c Q m c c p m + q m when P + m + c Q m c, 7

24 the kernel in (.) 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 (v v ), but the obvious symmetry makes it easy to express v derivatives of in terms of v derivatives. It is then possible to integrate by parts and move derivatives o 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 di culty with the ij pj (P +,Q )= q p ij qj (P +,Q pj + q p ij qj (P +,Q ), where the pj + p + qj does not increase the order of the singularity mainly 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 [38,39], 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 [f(t, x, p),e(t, x),b(t, x)] be a classical solution to (.4) and (.5) satisfying (.6), (.), (.) and (.). There exists M, = (M ) > such that if (.4) then f(t) E(t) B(t) apple M, applen (L@ f(t)) f(t). Theorem. is proven through a careful study of the macroscopic equations (.98) - (.). These macroscopic equations come from a careful study of solutions f to the perturbed relativistic Landau-Maxwell system (.4), (.5) with (.6) projected 8

25 onto the null space N of the linearized collision operator L =[L +,L ] defined in (.). 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 ( apple i apple 3) (.5) N span{[ p J +, ], [, p J ], [p i p J +,p i p J ], [p + p J +, p J ]}. This is shown in Lemma.. We define the orthogonal projection from L (R 3 p) onto the null space N by P. We then decompose f(t, x, p) as f = Pf + {I P}f. We call Pf = [P + f,p f] R the hydrodynamic part of f and {I P}f = [{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 h(f): t + c p p ± r x P ± f e ±c k B T E p p ± p J ± = l ± ({I P}f)+h ± (f), where (.7) (.8) l ± ({I t + p p ± h ± (f) e ± E + p ± e ±c k B T r x {I P} ± f + L ± {{I P}f}, p ± E B r p f ± p p ± f ± + ± (f,f). We further expand P ± f as a linear combination of the basis in (.5) ( ) 3 pj (.9) P ± f a ± (t, x)+ b j (t, x)p j + c(t, x)p ± ±. j= A precise definition of these coe cients will be given in (.94). The relativistic system of macroscopic equations (.98) - (.) are obtained by plugging (.9) into (.6). These macroscopic equations for the coe cients in (.9) enable us to show that there exists a constant C>such that solutions to (.4) which satisfy the smallness 9

26 constraint (.4) (for M > small enough) will also satisfy (.) a ± b c } apple C(M ) {I P}@ f(t). applen applen This implies Theorem. since kpfk is trivially bounded above by the l.h.s. (Proposition.) and L is coercive with respect to {I P}@ f(t) (Lemma.8). Since our smallness assumption (.4) involves no momentum derivatives, in proving (.) the presence of momentum derivatives in the collision operator (.) causes another serious mathematical di culty. We develop a new estimate (Theorem.5) which involves purely spatial derivatives of the linear term (.) and the nonlinear term (.3) to overcome this di culty. To the best of the authors knowledge, until now there were no known solutions for the relativistic Landau-Maxwell system. However in, Lemou [47] 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 99 s have seen the first solutions. In 994, han [7] 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, han [73] 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 [7] 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 [38] 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

27 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.. It turns out that the presence of the physical constants do not cause essential mathematical di culties. Therefore, for notational simplicity, after the proof of Lemma. we will normalize all constants in the relativistic Landau-Maxwell system (.4), (.5) with (.6) and in all related quantities to be one..3. The Relativistic Landau Operator Our main results in this section include the crucial Theorem.3, which allows us to express p derivatives of (P, Q) in terms of q derivatives of (P, 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 with the standard Sobolev space norm for H (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 (.) satisfies C(J +,J + )=C(J +,J )=C(J,J + )=C(J,J )=. Therefore, the linearized collision operator Lg is defined by (.) Lg =[L + g, L g], L ± g A ± g K ± g, where A + g J / + C( p J + g +,J + )+J / + C( p J + g +,J ), A g J / C( p J g,j )+J / C( p J g,j + ), (.) K + g J / + C(J +, p J + g + )+J / + C(J +, p J g ), K g J / C(J, p J g )+J / C(J, p J + g + ).

28 And the nonlinear part of the collision operator (.) is defined by (g, h) =[ +(g, h), (g, h)], where (.3) +(g, h) J / + C( p J + g +, p J + h + )+J / + C( p J + g +, p J h ), (g, h) J / C( p J g, p J h )+J / C( p J g, p J + h + ). We will next derive the null space (.5) of the linear operator in the presence of all the physical constants. Lemma.. hlg, hi = hlh, gi, hlg, gi. And Lg =if and only if g = Pg. Proof. From (.) we split hlg, hi, with Lg =[L + g, L g], as (.4) p J+ {C( p J + g +,J + )+C(J +, p J + g + )}dp R 3 h + R 3 R 3 R 3 h+ p J+ {C( p J + g +,J )+C(J +, p J g )} hp J {C(p J g,j + )+C(J, p J + g + )} h pj {C( p J g,j )+C(J, p J g )}dp. dp dp We use the fact qi J (q) = as the null space of c k B T in (.) to show that q i J (q) q pi J / c + (p) = k B T p i p + J + (p) as well C(J / + g +,J ) ij pi (P +,Q )J (q)j / + (p) R 3 pi pi R 3 ij (P +,Q )J (q)j / + (p) qi q pi p + p i p + g + (p)+@ pj g + (p) g + (p)+@ pj g + (p) R 3 ij (P +,Q )J (q)j + (p)@ pj (J / + g + (p))dq dq dq

29 And similarly C(J,J / + g + ) ij pi (P,Q + )J (p)j / + (q) R 3 pi R 3 ij (P,Q + )J (p)j / + (q) pi qi q + q i q + g + (q)+@ qj g + (q) g + (q)+@ qj g + (q) (.5) ij pi (P,Q + )J (p)j + (q)@ qj (J / + g + (q))dq. R 3 dq dq 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 = ij (P +,Q + )J + (p)j + (q)@ pi (h + J / + (p)) = {@ pj (g + J / + qj (g + J / + (q))}dpdq ij (P +,Q + )J + (p)j + (q)@ qi (h + J / + (q)) {@ qj (g + J / + pj (g + J / + (p))}dpdq = ij (P +,Q + )J + (p)j + (q){@ pi (h + J / + qi (h + J / + (q))} {@ pj (g + J / + qj (g + J / + (q))}dpdq. 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 = ij (P +,Q )J + (p)j (q)@ pi (h + J / + (p)) {@ pj (g + J / + qj (g J / (q))}dpdq + ij (P,Q + )J (p)j + (q)@ pi (h J / (p)) {@ pj (g J / qj (g + J / + (q))}dpdq. 3

30 We switch the role of p and q in the second term to obtain = ij (P +,Q )J + (p)j (q){@ pi (h + J / + qi (h J / (q))} {@ pj (g + J / + qj (g J / (q))}dpdq. Again by (.) this piece of the operator is symmetric and conclude that L is a non-negative symmetric operator. if g = h. We therefore We will now determine the null space (.5) of the linear operator. Assume Lg =. From hlg, gi = we deduce, by (.), that there are scalar functions l (p, q) (l = ±) such pi (g l J / l qi (g l J / pi l (q)) l (p, q) p l q i q l,i {,, 3}. Setting q pi (g l J / l (p)) = l (p, ) p i + b p l li. By replacing p by q and subtracting we pi (g l J / l qi (g l J / l (q)) = l (p, ) p i p l pi = l (p, ) p l l (q, ) q i q i q l q l +( l (p, ) l (q, )) q i. q l We deduce, again by (.), that l (p, ) l (q, ) = and therefore that l (p, ) c l (a constant). We pi (g l J / p l (p)) = c i l + b p l li to obtain g l = {a g l + 3 i= b g li p j + c g l pl }J / 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 Therefore b g pi (g + J / + qi (g J / (q)) (p, pi q) b g i + cg + p i p + c g q i q = (p, q) p i b g +i b g i,i=,, 3; c g + c g. 4 p + q i q q i q. We conclude.

31 That means g(t, x, p) N as in (.5), so that g = Pg. Conversely, L{Pg} =bya direct calculation. For notational simplicity, as in Remark., we will normalize all the constants to be one. Accordingly, we write = p + p, P =(,p), and the collision kernel (P, Q) takes the form (.6) where (P, Q) (P, Q) S(P, Q), q (P Q) (P Q) 3/, S (P Q) I 3 (p q) (p q)+{(p Q) } (p q + q p). We normalize the relativistic Maxwellian as J(p) J + (p) =J (p) =e. We further normalize the collision freqency (.7) ij ±, (p) = ij (p) = ij (P, Q)J(q)dq, and the inner product h, i (.8) hg, hi takes the form + R 3 R 3 pj g pi h + pj pi h ij p i p j {g + h + dp + g h } dp. The norms are, as before, naturally built from this normalized inner product. The normalized vector-valued Landau-Maxwell equation for the perturbation f in dp, (.4) now takes the form t + p r x + E + p B r p f = E p f + (f,f). E p p J + Lf 5

32 with f(, x, v) =f (x, v), = [, ], and the matrix is diag(, ). Further, the normalized Maxwell system in (.5) and (.6) takes the form p p t E r x B = J = J(f+ f t B + r x E =, R p 3 p (.3) r x E = = J(f+ f )dp, r x B =, R 3 with E(,x)=E (x), B(,x)=B (x). We have a basic (but useful) inequality taken from Glassey & Strauss [3]. Proposition.. Let p, q R 3 with P =(,p) and Q =(q,q) then (.3) p q + p q q apple P Q apple p q. This will inequality will be used many times for estimating high order derivatives of the the collision kernel. Notice pi + qi P Q pi + qi ( q p q) = p i q q i + q qi q This is the key observation which allows us do analysis on the relativistic Landau p i =. Operator (Lemma.). We define the following relativistic di erential operator (.33) (p, p3 + q p + p + q. Unless otherwise stated, we omit the p, q dependence and write = (p, q). Note that the three terms in do not commute (and we choose this order for no special reason). We will use the following splitting many times in the rest of this section, (.34) A = { p q + p q [ p +]/}, B = { p q + p q apple [ p +]/}. The set A is designed to be away from the first order singularity in collision kernel (P, Q) (Proposition.). And the set B contains a 6 (P, Q) singularity ((.6) and

33 (.3)) but we will exploit the fact that we can compare the size of p and q. We now develop crucial estimates for (P, Q): Lemma.. For any multi-index, the Lorentz inner product of P and Q is in the null space of, (P Q) =. Further, recalling (.6), for p and q on the set A we have the estimate (.35) (p, q) (P, Q) apple Cp q 6. And on B, (.36) 6 q apple apple 6q. Using this inequality, we have the following estimate on B (.37) (p, q) (P, Q) apple Cq 7 p p q. Proof. Let e i (i =,, 3) be an element of the standard basis in R 3. We have seen that ei (P Q) =. And the general case follows from a simple induction over. By (.6) and (.33), we can now write S (p, q) ij ij (p, q) (P, Q) = (P, Q) (p, q), q where (.38) S ij (p, q) q = (P Q) { ij /( q )} +(P Q ) {(p i q j + p j q i )/( q )} {(p i q i )(p j q j )/( q )}. We will break up this expression and estimate the di erent pieces. Using (.33), the following estimates are straight forward (.39) (.4) { ij /( q )} apple Cq p, {(p i q j + p j q i )/( q )} apple Cp. 7

34 On the other hand, we claim that (.4) {(p i q i )(p j q j )/( q )} apple C p q p. q This last estimate is not so trivial because only a lower order estimate of p expected after applying even a first order derivative like ei. The key observation is pi + qi (p i q i )(p j q j )= q (p j q j ), and the r.h.s. is again second order. Therefore the operator can maintain the order of the cancellation. q is Proof of claim: To prove (.4), it is su cient to show that for any multi-index and any i, j, k, l {,, 3} there exists a smooth function G,ij kl (p, q) satisfying (.4) {(p i q i )(p j q j )/( q )} = as well as the decay 3 (p k q k )(p l q l )G,ij kl (p, q) q G,ij kl (p, q) apple Cq p, which holds for any multi-indices,. induction over. If =, we define We prove (.4) with (.43) by a simple G,ij kl (p, q) = ki lj q. The decay (.43) for G,ij kl (p, q) is straight forward to check. And (.4) holds trivially for =. Assume the (.4) with (.43) holds for apple n. = n + and write = em for some multi-index with To conclude the proof, let m = max{j :( ) j > }. 8

35 This specification of m is needed because of our chosen ordering of the three di erential operators in (.33), which don t commute. Recalling (.33), q em (p k q k )= km. From the induction assumption and the last display, we have {(p i q i )(p j q j )/( q )} 3 = em (p k q k )(p l q l )G,ij (p, q), = + k,l= q 3 k= kl (p k q k ) G,ij mk (p, q)+g,ij km (p, q) 3 (p k q k )(p l q l ) em G,ij (p, q). k,l= kl We compute q = q = q ( + q ) = (p q) (p + q) ( + q ) = P l (p l q l )(p l + q l ). ( + q ) We plug this display into the one above it to obtain (.4) for coe cients with the new G,ij kl (p, q) em G,ij kl (p, q)+ G,ij mk (p, q)+g,ij km (p, q) (p l + q l ). ( + q ) We check that G,ij kl (p, q) satisfies (.43) using the Leibnitz di erentiation formula as well as the induction assumption (.43). This establishes the claim (.4). With the estimates (.39), (.4) and (.4) in hand, we return to establishing (.35) and (.37). We plug the estimates (.39), (.4) and (.4) into ij (P, Q) from (.38) to obtain that (.44) ij (P, Q) apple Cp (P Q) (P Q) +Cp (P Q) (P Q) +Cp (P Q) (P Q) 3/ (P Q) q 3/ (P Q ) 3/ p q q. We will use this estimate twice to get (.35) and (.37). 9

36 We first establish (.35). On the set A we have p q + p q ( p q + p q ) 4 + p 4. From (.3) and the last display we have P Q + P Q. 6 q From the Cauchy-Schwartz inequality we also have apple P Q apple P Q apple q + p q apple q. We plug these last two inequalities (one at a time) into (.44) to obtain ij (p, q) apple C(P Q) (P Q) apple C( q ) (P Q) 3/ p p q + p q + p q q 3/ p q apple C( q ) 3 {P Q } 3 p apple C( q ) 3 p q 3 p. We move on to establishing (.36). If p apple, then apple apple q. Assume p, using B we compute q q p p q p 4. Therefore, apple 6q on B. For the other half of (.36), q apple + p q apple 3 + apple. We move on to establishing (.37). On the set B we have a first order singularity. Also (.3) tells us p q q apple P Q apple p q. 3

37 We plug this into (.44) to observe that on B we have ij (p, q) apple C(P Q) (P Q) 3/ p (P Q +) p q + p q + p q q q apple C(P Q) (P Q) 3/ p p q apple C( q ) (P Q) 3/ p p q apple C( q ) ( q ) 3/ p q {P Q +} 3/ p apple C( q ) 7/ p q p. We achieve the last inequality because (.3) says P Q. Next, let µ(p, q) be an arbitrary smooth scalar function which decay s rapidly at infinity. We consider the following integral ij (P, Q)J / (q)µ(p, q)dq. Both the linear term L and the nonlinear term develop a new integration by parts technique. R 3 are of this form (Lemma.6). We (.45) Theorem.3. Given >, we ij (P, Q)J / (q)µ(p, q)dq = apple R 3 R 3 where ', (p, q) is a smooth function which satisfies, 3 ij (P, Q)J / (q)@ 3µ(p, q)', (p, q)dq, 3 ',, 3 (p, q) apple Cq p + 3, for all multi-indices and. Proof. We prove (.45) by an induction over the number of derivatives. Assume = e i (i =,, 3). We write pi = qi pi + qi 3 = qi + ei

38 Instead of hitting ij (P, Q) pi, we apply the r.h.s. term above and integrate by parts over qi to obtain pi R 3 ij (P, Q)J / (q)µ(p, q)dq R 3 ij (P, Q)J / (q)@ pi µ(p, q)dq + ij (P, Q)J / (q) qi µ(p, q)dq R p 3 + ij (P, Q)J / qi q i (q) µ(p, q)dq R q 3 + ij (P, Q)J / (q)µ(p, q)dq. R 3 ei We can write the above in the form (.45) with the coe cients given by (.48) e i,,(p, q) = q i q q i, e i e i,,(p, q) =, e i,e i,(p, q) = q, e i,,e i (p, q) =. And define the rest of the coe cients to be zero. Note that these coe cients satisfy the decay (.46). This establishes the first step in the induction. Assume the result holds for all apple n. Fix an arbitrary such that = n + and for some multi-index and m = max{j :( ) j > }. This specification of m is needed because of our chosen ordering of the three di erential operators in (.33), which don t commute. By the induction assumption R apple ij (P, Q)J / (q)µ(p, pm R 3 ij (P, Q)J / (q)@ 3µ(p, q)', (p, q)dq, 3 3

39 We approach applying the last derivative the same as the = case above. We obtain (.49) = R 3 ij (P, Q)J / (q)',, 3 (p, q)@ p 3µ(p, q)dq (.5) + R 3 ij (P, Q)J / (q)',, 3 (p, q) 3µ(p, q)dq (.5) + R 3 em ij (P, Q)J / (q)@ 3µ(p, q)', (p, q)dq, 3 (.5) + ij (P, Q)J / (q)@ 3µ(p, q) R pm + qm + q m q m ' q p, (p, q)dq,, 3 where the unspecified summations above are over apple the terms above with the same order of di erentiation to obtain = apple R 3. We collect all ij (P, Q)J / (q)@ 3µ(p, q)' (p, q)dq,,, 3 where the functions ' (p, q) are defined naturally as the coe,, 3 cient in front of each term of the form ij (P, Q)J / (q)@ 3µ(p, q) and we recall that = +e m. We check (.46) by comparing the decay with the order of di erentiation in each of the four terms (.49-.5). For (.49), the order of di erentiation is =, =, 3 = 3 + e m. And by the induction ',, 3 (p, q) apple Cq p + 3, This establishes (.46) for (.49). For (.5), the order of di erentiation is apple Cq +em p + 3+e m +e i = Cq p + 3. =, = + e m, 3 = 3. 33

40 And by the induction assumption as well as the Leibnitz q ' p, (p, q) apple Cq +, 3 p + 3, This establishes (.46) for (.5). For (.5), the order of di erentiation is And by the induction assumption, This establishes (.46) for (.5). = Cq p + 3. = + e m, =, 3 = ',, 3 (p, q) apple Cq p + 3, For (.5), the order of di erentiation is apple Cq p + 3. =, =, 3 = 3. And by the induction assumption as well as the Leibnitz q pm + qm + q m q m ' q p, (p, q), 3 apple Cq + p + 3, = Cq p + 3. This establishes (.46) for (.5) and therefore for all of the coe cients. 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. (.53) Lemma.3. We compute a sum of first derivatives in q of (.6) as ij (P, Q) qj (P, Q) = (P Qp i q i ). q 34

41 This term has a second order singularity at p = q. We further compute a sum of (.53) over first derivatives in p as qj i,j ij (P, Q) =4 P Q q (P Q) /. This term has a first order singularity. This result is quite di erent from the classical theory, it is straightforward compute the derivative of the classical kernel in (.3) ij v j (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. Di erentiating (.6), we ij qj (P, q j (P, Q) S ij (P, Q) q (P, Q) + qj S ij (P, Q) q j q S ij (P, Q). qj (P, Q) = (P Q) (P Q) 3(P Q) 3 (P Q) 3/ qj q 5/ qj q p j p j. Since (.) implies P qj j Sij (P, Q) q p j =, we conclude qj (P, Q)S ij (P, Q) q =. Therefore it remains to evaluate the r.h.s. of ij qj (P, Q) = j (P, Q) q j qj S ij (P, Q) q j q S ij (P, Q).