THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE

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1 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE ROBERT M. STRAIN, FEBRUARY 13, 2006 Abstract. The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians. 1. The Vlasov-Maxwell-Boltzmann System The Vlasov-Maxwell-Boltzmann system describes the dynamics of dilute charged particles e.g. electrons and ions): tf + + v xf + + e+ E + v m + c B vf + = Q ++ F +, F +) + Q + F +, F ), 1) tf + v xf e E + v m c B vf = Q F, F ) + Q + F, F +). The initial conditions are F ± 0, x, v) = F 0,± x, v). Here F ± t, x, v) 0 are number density functions for ions +) and electrons -) respectively at time t 0, position x = x 1, x 2, x 3 ) and velocity v = v 1, v 2, v 3 ). The constants e ± and m ± are the magnitude of their charges and masses, and c is the speed of light. The self-consistent electromagnetic field [Et, x), Bt, x)] in 1) is coupled with F t, x, v) through the celebrated Maxwell system: t E c x B = 4πJ = 4π v{e + F + e F }dv, t B + c x E = 0, with constraints x B = 0, x E = 4πρ = 4π {e + F + e F }dv, and initial conditions E0, x) = E 0 x), B0, x) = B 0 x). Let g A v), g B v) be two number density functions for two types of particles A and B with masses m i and diameters σ i i {A, B}). In this article we consider the Boltzmann collision operator with hard-sphere interactions [3]: Z Q AB σa + σb)2 2) g A, g B) = u v) ω {g Av )g Bu ) g Av)g Bu)}dudω. 4 S 2 Here ω S 2 and the post-collisional velocities are 3) v = v 2m B [v u) ω]ω, u = u + 2m A [v u) ω]ω. m A + m B m A + m B Then the pre-collisional velocities are v, u and vice versa Mathematics Subject Classification. Primary: 76P05; Secondary: 82B40, 82C40, 82D05.

2 2 ROBERT M. STRAIN, FEBRUARY 13, 2006 Recently global in time smooth solutions to the Vlasov-Maxwell-Boltzmann system near Maxwellians were constructed by Guo [11] in the spatially periodic case. Further, arbitrarily high polynomial time decay for the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system [17] was shown in the periodic box by Strain and Guo [18]. A simpler model can be formally obtained when the speed of light is sent to infinity. This is the Vlasov-Poisson-Boltzmann system, where Bt, x) 0 and Et, x) = x φt, x) in 1). Here the self-consistent electric potential, φ, satisfies a Poisson equation. There are several results for this model. Time decay of solutions to the linearized Vlasov-Poisson-Boltzmann system near Maxwellian was studied by Glassey and Strauss [7, 8]. The existence of spatially periodic smooth solutions to the Vlasov-Poisson-Boltzmann system with near Maxwellian initial data was shown by Guo [10]. Renormalized solutions were constructed by Lions [13], and this was generalized to the case with boundary by Mischler [16]. The long time behavior of weak solutions with additional regularity assumptions are studied in Desvillettes and Dolbeault [4]. For some references on the Boltzmann equation with or without forces see [2, 3, 6, 21]. Next we discuss global in time classical solutions to the Cauchy problem for the Vlasov-Poisson-Boltzmann system. The case of near vacuum data was solved by Guo [9] for some soft collision kernels. Recently Duan, Yang and Zhu [5] have resolved the near vacuum data case for the hard sphere model and some cut-off potentials. The near Maxwellian case was proven by Yang, Yu and Zhao [19] with restrictions on either the size of the mean free path or the size of the background charge density. After the completion of the results in this paper we learned that Yang and Yu [20] have removed those restrictions and further shown a time decay rate to Maxwellian of Ot 1/2 ). However, to the author s knowledge, until now no solutions have been constructed for the full Vlasov-Maxwell-Boltzmann system in the whole space. It is our purpose in this article to establish the existence of global in time classical solutions to the Cauchy Problem for the Vlasov-Maxwell-Boltzmann system near global Maxwellians: µ + v) = n ) 3/2 0 m+ e m+ v 2 /2κT 0, e + 2πκT 0 µ v) = n ) 3/2 0 m e m v 2 /2κT 0. e 2πκT 0 Accordingly, in the rest of this section we will reformulate the problem 1) as a perturbation of the equilibria. Since the presence of the physical constants do not create essential mathematical difficulties, for notational simplicity, we normalize all constants in the Vlasov- Maxwell-Boltzmann system to one. We normalize the Maxwellian µv) µ + v) = µ v) = 2π) 3/2 e v 2 /2. We further normalize the collision operator 2) as Q AB g A, g B ) = Qg A, g B ) = u v) ω {g A v )g B u ) g A v)g B u)}dudω. S 2

3 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 3 And we define the standard perturbation f ± t, x, v) to µ as F ± = µ + µf ±. Let [, ] denote a column vector so that F t, x, v) = [F +, F ]. We then study the normalized vector-valued Vlasov equation for the perturbation which now takes the form ft, x, v) = [f + t, x, v), f t, x, v)], 4) { t + v x + qe + v B) v }f {E v} µq 1 + Lf = q {E v}f + Γf, f), 2 with f0, x, v) = f 0 x, v), q 1 = [1, 1] and q = diag1, 1). For any given g = [g +, g ], the linearized collision operator in 4) is where Lg = [L + g, L g], 5) L ± g = 2µ 1/2 Q µg ±, µ) µ 1/2 Qµ, µ{g ± + g }). We split L in the standard way [6] as Lg = νv)g Kg. The collision frequency is 6) νv) v u µu)dudω. S 2 For g = [g +, g ] and h = [h +, h ], the nonlinear collision operator is with Γg, h) = [Γ + g, h), Γ g, h)], Γ ± g, h) = µ 1/2 Q µg ±, µh ± ) + µ 1/2 Q µg ±, µh ). Further, the coupled Maxwell system now takes the form 7) t E x B = J = v µf + f )dv, t B + x E = 0, R 3 8) x E = ρ = µf+ f )dv, x B = 0, with the same initial data E0, x) = E 0 x), B0, x) = B 0 x). It is well known that the the linearized collision operator L is non-negative. For fixed t, x), the null space of L is given by Lemma 1): 9) N = span{[ µ, 0], [0, µ], [v i µ, vi µ], [ v 2 µ, v 2 µ]} 1 i 3). We define P as the orthogonal projection in L 2 v) to the null space N. With t, x) fixed, we decompose any function gt, x, v) = [g + t, x, v), g t, x, v)] as gt, x, v) = Pgt, x, v) + I P)gt, x, v). Then Pg is call the hydrodynamic part of g and I P)g the microscopic part.

4 4 ROBERT M. STRAIN, FEBRUARY 13, Notation and Main Results We shall use, to denote the standard L 2 inner product in v for a pair of functions g 1 v), g 2 v) L 2 v; R 2 ). The corresponding norm is written g = g 1, g 1. We also define a weighted L 2 inner product in v as g 1, g 2 ν νv)g 1, g 2. And we use ν for its corresponding L 2 norm. We note that the weight 6) satisfies v ) νv) c1 + v ). c We also use to denote L 2 norms in either x v or x. In other words we use to denote an L 2 norm in x v if the function depends on x, v). But if the function only depends upon x x then will denote the norm of L 2 x). The L 2 x v) inner product is written, ). Further, g 2 ν νg, g). Let multi-indices α and β be We define a high order derivative as α = [α 0, α 1, α 2, α 3 ], β = [β 1, β 2, β 3 ]. β α t α0 x α1 1 x α2 2 x α3 3 v β1 1 v β2 2 v β3 3. If each component of α is not greater than that of ᾱ s, we write α ᾱ; α < ᾱ ᾱ means α ᾱ and α < ᾱ. We also denote by C α) α. ᾱ We next define an Instant Energy functional for a solution to the Vlasov- Maxwell-Boltzmann system, [ft, x, v), Et, x), Bt, x)], to be any functional Et) which satisfies the following for an absolute constant C > 0: 1 C Et) β α ft) 2 + [ α Et), α Bt)] 2 CEt). α + β N α N The temporal derivatives of [f 0, E 0, B 0 ] are defined through equations 4) and 7). We will always use C to denote a positive absolute constant which may change from line to line. We additionally define the Dissipation rate for a solution as Dt) Et) 2 + α ft) 2 ν + β α I P)ft) 2 ν. 0< α N α + β N Notice that we include the electric field Et, x) in the dissipation and further we do not take any velocity derivatives of the hydrodynamic part, Pf. Throughout this article we assume N 4. Our main result is as follows. Theorem 1. Assume that [f 0, E 0, B 0 ] satisfies the constraint 8) initially. Let F 0,± x, v) = µ + µf 0,± x, v) 0. There exists an instant energy functional and ɛ 0 > 0 such that if E0) ɛ 0, then there is a unique global solution [ft, x, v), Et, x), Bt, x)] to the Vlasov Maxwell-Boltzmann system 4) and 7) with 8) satisfying Et) + t 0 Ds)ds E0).

5 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 5 And moreover F ± t, x, v) = µ + µf ± t, x, v) 0. For spatially periodic data, Guo [11] designed an energy method to establish global existence of classical solutions to the Vlasov-Maxwell-Boltzmann system near Maxwellians. Additionally, Guo has given another construction of global in time near Maxwellian solutions to the pure Boltzmann equation with no force terms) in the whole space [12], but without regularity in the velocity variables. The proof of Theorem 1 uses techniques developed in [11, 12]. At the same time several new difficulties arise which are fundamental to the Vlasov-Maxwell- Boltzmann system in the whole space. In particular, we use the basic estimates from [10,11] Lemma 2 and Lemma 3) but we use new arguments described below) in order to remove the hydrodynamic part, Pf, from the estimates Lemma 5, Lemma 6, Lemma 7 and Theorem 3) and thereby control the weaker whole space dissipation. We additionally show how to include the velocity derivatives in this energy method in the whole space, which we believe will be useful for working on other kinetic equations with force terms in the whole space. In particular, these techniques can be used to construct solutions to the relativistic Landau-Maxwell system [17] in the whole space. Recently, Yang, H. Yu and Zhao [19] and Yang and Zhao [20] have solved the Cauchy problem for the Vlasov-Poisson-Boltzmann system with smooth near Maxwellian data. Their proof makes use of techniques from Liu and S.-H. Yu [14] and Liu, Yang and S.-H. Yu [15] as well as the study in conservation laws. Our proof of Theorem 1 supplies another construction of solutions to the Cauchy problem for the Vlasov-Poisson-Boltzmann system near Maxwellian. As in [11, 12], a key point in our construction is to show that the linearized collision operator 5) is effectively coercive for solutions of small amplitude to the Vlasov-Maxwell-Boltzmann system, 4) and 7) with 8): Theorem 2. Let [ft, x, v), Et, x), Bt, x)] be a classical solution to 4) and 7) satisfying 8). There exists M 0 > 0 such that if { 10) α ft) 2 + α Et) 2 + α Bt) 2} M 0, α N then there are constants δ 0 = δ 0 M 0 ) > 0 and C 0 > 0 so that α N L α ft), α ft)) δ 0 I P)ft) 2 ν + δ 0 0< α N α ft) 2 dgt) ν C 0, dt where Gt) = x ba + + a )dx and a ±, b are defined in 15) below. To prove Theorem 2, we split the solution f to the Vlasov-Maxwell-Boltzmann system 4) into it s hydrodynamic and microscopic parts as 11) f = Pf + I P)f. Separating into linear and nonlinear parts, and using L{Pf} = 0 Lemma 1), we can express the hydrodynamic part, Pf, through the microscopic part, I P)f, up to a second order term: 12) { t + v x }Pf {E v} µq 1 = li P)f) + hf),

6 6 ROBERT M. STRAIN, FEBRUARY 13, 2006 where 13) 14) li P)f) { t + v x + L}I P)f, hf) qe + v B) v f + q {E v}f + Γf, f). 2 Expanding Pf = [P + f, P f] as a linear combination of the basis in 9) yields { } 3 15) P ± f a ± t, x) + b i t, x)v i + ct, x) v 2 µ 1/2 v). i=1 A system of macroscopic equations for the coefficients at, x) = [a + t, x), a t, x)], b i t, x) and ct, x) can be derived from an expansion of the left side of 12) in the velocity variables, using 15) to obtain [11]: 16) 17) 18) x c = l c + h c, t c + i b i = l i + h i, i b j + j b i = l ij + h ij, i j, 19) t b i + i a ± E i = l ± bi + h± bi, 20) t a ± = l a ± + h ± a, Here j = xj. The terms on the r.h.s. of the macroscopic equations 16)-20) are obtained by expanding the r.h.s. of 12) with respect to the same velocity variables and comparing the coefficients on the two sides. Here l c t, x), l i t, x), l ij t, x), l ± bi t, x) and l± a t, x) are the coefficients of this expansion of the linear term 13), and h c t, x), h i t, x), h ij t, x), h ± bi t, x) and h± a t, x) are the coefficients of the same expansion of the second order term 14). These terms are defined precisely in Section 4. Theorem 2 is proven via energy estimates of these macroscopic equations 16)- 20), which are used to estimate derivatives of a, b and c. In the periodic case [11], the Poincaré inequality was used to estimate a, b and c in terms of their derivatives and the conserved quantities. Further, it was observed in [12] that a, b and c themselves can not be estimated directly from 16)-20). This is why we remove Pf from the dissipation rate in the whole space. Additionally, due to the velocity gradient in the equation 4), it is important to include the velocity derivatives in our norms. On the other hand, in our global energy estimates in Section 5) β Pf behaves almost the same as Pf, which is not part of the dissipation. We therefore observe that the dissipation rate for the Vlasov-Maxwell-Boltzmann system in the whole space should include the velocity derivatives only on the microscopic part of the solution. This lack of control over the hydrodynamic part and it s velocity derivatives causes new difficulties. For this reason, we use Sobolev type inequalities to bound the nonlinear terms by derivatives of f in Lemma 5. This is an important step for bounding the macroscopic equations by our weak dissipation in the proof of Theorem 2. Lemma 5 is also used to include the electric field in the dissipation in Lemma 6. Further, in the periodic case [11], the Poincaré inequality was used to estimate the low order temporal derivatives of b i. Instead, we notice that there is a cancellation of the most dangerous terms in the macroscopic equations which allows us to estimate the low order temporal derivatives of b i in the whole space directly in the proof of Theorem 2.

7 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 7 Further, we give a new, more precise estimate in Lemma 5 using Sobolev s inequality to remove the zero th order derivatives of hf) and thereby include the electric field into this whole space dissipation in Lemma 6. Both of these observations are important for closing the estimates in Theorem 3. In addition, Lemma 5 is used to control the macroscopic equations in the proof of Theorem 2. Moreover, we refine the estimate in Lemma 7 to control the non-linear collision operator by the weaker whole space dissipation with velocity derivatives. In particular we use the splitting 11) extensively to get all of these estimates. This article is organized as follows: In section 3 we recall some basic estimates for the Boltzmann collision operator. In section 4 we establish positivity of 5) for solutions to the Vlasov-Maxwell-Boltzmann system with small amplitude. And in Section 5 we prove global existence Theorem 3). 3. Local Solutions We will make use of the following basic estimates from [1, 10, 11]: Lemma 1 [1, 11]). Lg, h = g, Lh, Lg, g 0. And Lg = 0 if and only if g = Pg, where P was defined in 15). Furthermore there is a δ > 0 such that Lg, g δ I P)g 2 ν. From 5) it is well known that we can write L = ν K where ν is the multiplication operator defined in 6) and K satisfies the following. Lemma 2 [10]). If β 0, β νv) is uniformly bounded. For gv) H β ; R 2 ) and for any η > 0, there is C η > 0 such that β [Kg] 2 2 η β g C η g 2 2. β = β Above and below H β ; R 2 ) denotes the standard L 2 ; R 2 ) Sobolev space of order β. Next up are estimates for the nonlinear part of the collision operator. Lemma 3 [10]). Say gv), pv), rv) Cc ; R 2 ). Then β Γg, p), r C [ β1 g ν β2 p 2 + β1 g 2 β2 p ν ] r ν. β 1+β 2 β Moreover, [ Γg, p)rdv + Γp, g)rdv C sup ν 3 r sup x,v x gx, v) 2 dv] 1/2 p. Using these estimates, it is by now standard to prove existence and uniqueness of local-in time positive classical solutions to the Vlasov-Maxwell-Boltzmann system 4) and 7) with 8) and to establish the continuity of the high order norms for a solution as in [11, Theorem 4]. Therefore, we will not repeat the argument. 4. Positivity of the Linearized Collision Operator In this section, we establish the positivity of 5) for any small solution [f, E, B] to the Vlasov-Maxwell-Boltzmann system 4) and 7) via a series of Lemma s.

8 8 ROBERT M. STRAIN, FEBRUARY 13, 2006 First we simplify the Maxwell system from 7) and 8). By 15) [v µ, v µ] Pfdv = 0, [ µ, µ] fdv = a + a. We can therefore write the Maxwell system from 7) and 8) as t E x B = J = [ v µ, v µ] I P)fdv, R 21) 3 t B + x E = 0, x E = a + a, x B = 0. It is in this form that we will estimate the Maxwell system. Next we discuss the derivation of the macroscopic equations 16)-20). We expand the r.h.s. of 12) in the velocity variables using the projection 15) to obtain v i i c v 2 + { t c + i b i }vi 2 + { i b j + j b i }v i v j + { t b i + i a ± E i }v i µ i<j + t a ± µ, The above represents two equations. Here j = xj. For fixed t, x), this is an expansion of l.h.s. of 12) with respect to the following basis 1 i < j 3): 22) [v i v 2 µ, v i v 2 µ], [v 2 i µ, v 2 i µ], [vi v j µ, vi v j µ], [v i µ, 0], [0, vi µ], [ µ, 0], [0, µ]. By expanding the r.h.s. of 12) with respect to the same velocity variables and comparing the coefficients of each basis element in 22) we obtain the macroscopic equations 16)-20). Given the full basis in 22), for fixed t, x), l c t, x), l i t, x), l ij t, x), l ± bi t, x) and l± a t, x), from 16)-20), take the form 23) li P)f) ɛ n v)dv. Here li P)f) is defined in 13) and the {ɛ n v)}, which only depend on v, are linear combinations of the elements of 22). Similarly h c, h i, h ij, h ± bi and h± a from 16)-20) are also of the form 24) hf) ɛ n v)dv, where hf) is defined in 14). In the next two lemmas, we will estimate the right side of the macroscopic equations using 21), 23) and 24). Lemma 4. Let α = [α 0, α 1, α 2, α 3 ], α N 1, then for any 1 i, j 3, α l c + α l i + α l ij + α l ± bi + α l a ± + α J C I P) ᾱ α f. ᾱ 1 This Lemma is similar to [11, Lemma 7], but we prove it for completeness. Proof. The estimate for J follows from 21): α J C v µ I P) α f dv C I P) α f 2.

9 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 9 Square both sides and integrate over x to get the estimate in Lemma 4 for J. For the other terms in Lemma 4, it suffices to estimate the representation 23) with 13). Let α N 1. Since νv) C1 + v ) and K is bounded from L 2 ; R 2 ) to itself. We have by 13) 2 2 α li P)f) ɛ n v)dv) = { t + v x + L}I P) α f) ɛ n v)dv), C ɛ n v) dv ɛ n v) I P) t α f 2 + v 2 I P) x α f 2 )dv R 3 +C ɛ n v) dv ɛ n v) ν K)I P) α f 2 dv C{ I P) t α f I P) x α f I P) α f 2 2}. Here we have used α I P)f = I P) α f, and the exponential decay of ɛ n v). The estimate in Lemma 4 follows by further integrating over x. Next, we estimate coefficients of the higher order term hf) on the right side of the macroscopic equations 16)-20). Lemma 5. Let 10) hold for M 0 > 0. Then for 1 i j 3, { α h c + α h i + α h ij + α h ± bi + α h ± a } C M 0 α N 0< α N α f. Notice that the left side of this estimate contains zero th order derivatives but the right side does not. We have to tighten the estimate in this way because Pf with no derivatives) is not part of the dissipation rate. We use Sobolev s inequality on the second order terms to remove the terms without derivatives. Proof. Notice that it suffices to estimate 24) with 14). For the first term of hf) in 14), we integrate by parts to get α {qe + v B) v f)} ɛ n v)dv = α {qe + v B) j vj f)} ɛ n v)dv = Cα α 1 vj { α1 E + v α1 B) j q α α1 f} ɛ n v)dv = Cα α 1 α1 E + v α1 B) j {q α α1 f} vj ɛ n v)dv. Take the square of the above and further integrate over x, the result is C { } 25) { α1 E 2 + α1 B 2 } α α1 f 2 dv dx. This follows from Cauchy-Schwartz and the exponential decay of v ɛ n v). We will estimate 25) in two steps. First assume α α 1 N/2. In this case we use the embedding W 1,6 ) L ) combined with the L 6 ) Sobolev inequality for gradients to obtain sup x α α1 fx, v) 2 dv C α α1 f 2 W 1,6 ) dv, R 3 C x α α1 f 2 W 1,2 ) dv. j

10 10 ROBERT M. STRAIN, FEBRUARY 13, 2006 Therefore, when α α 1 N/2 and N 4 we see that 25) C{ α 1 Et) + α1 Bt) } 0< ᾱ 2 ᾱ α α1 ft). The small amplitude assumption 10) completes the estimate in this case. Alternatively if α α 1 > N/2, then α 1 N/2 and we use the embedding H 2 ) L ) to obtain sup α1 E + α1 B ) C { ᾱ α1 Et) + ᾱ α1 Bt) }. x In this case 25) C α α 1 f ᾱ 2 ᾱ 2 { ᾱ α1 Et) + ᾱ α1 Bt) }. Again, the small amplitude assumption 10) completes the estimate. For the second term of hf) in 14) we have q { q E v) 2 α1 α α1 f 2 α {E v)f} ɛ n v)dv = Cα α 1 C { 1/2 α1 E α α1 f dv} 2. } ɛ n v)dv This term is therefore easily treated by the last argument. For the third term of hf), we have α Γf, f) ɛ n v)dv Cα α 1 Γ α1 f, α α1 f) ɛ n v)dv. Without loss of generality assume α 1 N/2. From Lemma 3, the last line is { 1/2 C sup α1 ft, x, v) dv} 2 α α1 ft). x Again, we first use the embedding W 1,6 ) L ) and second the L 6 ) Sobolev inequality to see that the above is C α α1 ft) ᾱ α1 ft) C M 0 ᾱ α1 ft). 0< ᾱ 2 0< ᾱ 2 The last inequality follows from 10). This completes the estimate of hf). Next we estimate the electric field Et, x) in terms of ft, x, v) through the macroscopic equation 19) using Lemma 5. Lemma 6. Say [f, E, B] is a classical solution to 4) and 7) with 8). Let the small amplitude assumption 10) be valid for some M 0 1. C > 0 such that 1 C α N 1 α Et) I P)f + 0< α N α ft). Lemma 5 is used to remove the hydrodynamic part, Pf, from this upper bound.

11 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 11 Proof. We use the plus part of the macroscopic equation 19): 2 α E i = α l + bi + α h + bi α t b i α i a +. By 15), α t b i + α i a + C{ P α t f + P α i f } C{ α t f + α i f }. Since M 0 1 in assumption 10), applying Lemma 4 to α l + bi α h + bi we deduce Lemma 6. and Lemma 5 to We now prove the positivity of 5) for a classical solution of small amplitude to the full Vlasov-Maxwell-Boltzmann system, [ft, x, v), Et, x), Bt, x)]. The proof makes use of the estimates established in this section Lemma 4, 5, 6) in addition to an exact cancellation property of the macroscopic equations. Proof of Theorem 2. From Lemma 1 we have L α f, α f δ I P) α f 2 ν. It thus suffices to show that if 10) is valid for some small M 0 > 0, then there are constants C 1, C 2 > 0 such that 0< α N P α ft) 2 ν C 1 α N I P) α ft) 2 dgt) ν + C 2. dt Here Gt) is defined in Theorem 2. By 15), we clearly have 1 C P α ft) 2 ν α [a +, a ] 2 + α b 2 + α c 2. The rest of the proof is therefore devoted to establishing 26) 0< α N { α [a +, a ] 2 + α b 2 + α c 2 } C α N +CM 0 I P) α ft) 2 ν 0< α N dgt) +C 0, dt α ft) 2 which implies Theorem 2 when M 0 is sufficiently small. This is because the last term on the right can be neglected for M 0 sufficiently small: α ft) 2 = P α ft) 2 + I P) α ft) 2 { α [a +, a ] + α b + α c } 2 + I P) α ft) 2. To prove 26), we estimate the macroscopic equations 16) through 20).

12 12 ROBERT M. STRAIN, FEBRUARY 13, 2006 The estimate for bt, x) with at least one spatial derivative. We first estimate x α b with α N 1. We take j of 17) and 18) to get x α b i = jj α b i = jj α b i + ii α b i j j i = j i{ ij α b j + j α l ij + j α h ij } + { i α l i + i α h i t i α c} = { t i α c i α l j i α h j } + j i j i{ j α l ij + j α h ij } + i α l i + i α h i t i α c. Since j i t i α c = 2 t i α c, we get = t i α c + j i{ i α l j i α h j + j α l ij + j α h ij } + i α l i + i α h i = ii α b i + j i{ i α l j i α h j + j α l ij + j α h ij } + 2{ i α l i + i α h i }. Therefore, multiplying with α b i yields: 27) x α b i C { α l j + α h j + α l ij + α h ij + α l i + α h i }. This is bounded by the first two terms on the r.h.s. of 26) by Lemma 4 and Lemma 5. We treat the pure temporal derivatives of bt, x) at the end the proof. The estimate for ct, x). From 16) and 17), for α N 1, we have x α c α l c + α h c, t α c i α b i + α l i + α h i. By 27), Lemma 4 and 5 both t α c 2 and x α c 2 are bounded by the first two terms on the r.h.s. of 26). The estimate for [a + t, x), a t, x)]. By 20), for α N 1, we get t α [a +, a ] α l ± a + α h ± a. By Lemma 4 and 5, t α [a +, a ] is thus bounded by the first two terms on the r.h.s. of 26). We now turn to purely spatial derivatives of a ± t, x). Let α N 1 and α = [0, α 1, α 2, α 3 ] 0. By taking i of 19) and summing over i, we get 28) x α a ± ± x α E = x t α b i i α {l ± bi + h± bi }. From the Maxwell system in 21) we have x α E = α a + α a.

13 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 13 We plug this into 28), then multiply 28) with α a ± respectively and integrate over x. Adding the two resulting equations yields x α a x α a 2 + α a + α a 2 = i α { a ± t α b i + α l ± bi + α h ± bi} dx. i,± We therefore conclude that x α a + + x α a t α b + i,± α {l ± bi + h± bi }. Since α is purely spatial, this is bounded by the first two terms on the right side of 26) by 27), Lemma 4 and 5. Now we consider the case α = 0. The same procedure yields 1 2 xa xa 2 x t ba + + a ) + C {l ± bi + h± bi } 2. i,± We redistribute the derivative in t, as in [12], to estimate the first term by x t ba + + a )dx = d x ba + + a )dx x b t a + + a ) dt d x ba + + a )dx + C x b 2 + C t a ± 2. dt ± The first term on the right is dg/dt and the last two terms have already been estimated by the other terms on the right side of 26). The estimate for bt, x) with purely temporal derivatives. Lastly, we consider t α b i t, x) with α = [α 0, 0, 0, 0] and α 0 N 1. In [11], Guo used a procedure which involved the Poincaré inequality and Maxwell s equations 21) to estimate α t b i from the macroscopic equation 19), which includes the electric field. But the Poincaré inequality does not hold in the whole space case to estimate the low order pure temporal derivatives of b. On the other hand, there are two equations for t b i in 19). We can add them together to cancel the most dangerous term, E i, and obtain 2 t b i + i a + + i a = l + bi + h+ bi + l bi + h bi. Without this exact cancellation, our estimates for E i are not sufficient to conclude the required estimates for t α b i t, x). From this equation we can estimate the temporal derivatives of b directly: t α b i t, x) 2 C i α a ± 2 + α {l ± bi + h± bi } 2). ± If α 0 > 0 then the terms on the right have already been estimated above by the first two terms terms on the right side of 26). Alternatively if α 0 = 0, we use the previous estimate for i a ± to obtain t b i t, x) 2 C d x ba + + a )dx + C x b 2 dt + C ± t a ± 2 + {l ± bi + h± bi } 2). From Lemma 4 and 5, we finish Theorem 2 with G = x ba + + a )dx.

14 14 ROBERT M. STRAIN, FEBRUARY 13, Global Existence The main goal of this section is to establish global existence of classical solutions to the Vlasov-Maxwell-Boltzmann system, 4) and 7) with 8). Our proof uses Theorem 2 and energy estimates Theorem 3) developed in this section. New difficulties arise in these energy estimates for both the terms with velocity derivatives and those without because we can not control the hydrodynamic part of a solution by the linear operator in the whole space. For this reason we have already removed the hydrodynamic part and it s velocity derivatives from the dissipation. As a consequence, we develop different procedures to prove the energy estimates with weaker dissipation. In particular, we estimate the nonlinear part of the collision operator by the weaker dissipation in Lemma 7. Further, including the electric field in the dissipation rate by Lemma 6) enables us to control other nonlinearities without velocity derivatives through Sobolev inequalities. Additionally, we split the solution as 11) in several places because this allows us to absorb velocity derivatives and exploit the exponential velocity decay of the hydrodynamic part 15) in our estimates. We have the following global existence theorem: Theorem 3. Let [ft, x, v), Et, x), Bt, x)] be a classical solution to to the Vlasov Maxwell-Boltzmann system 4) and 7) with 8) satisfying 10). Then there exists an instant energy functional satisfying d dt Et) + Dt) Et)Dt). With 10), this yields global existence from a standard continuity argument. First we give a nonlinear estimate which will be used in the proof Theorem 3: Lemma 7. Let α + β N. Then there is an instant energy functional such that α β Γf, f), α β I P)f ) E 1/2 t)dt). The main improvement of this estimate over Lemma 3 is the presence of the dissipation, Dt). Clearly E 2 is not needed in the dissipation norm in Lemma 7. Proof. First assume β = 0. We have α Γf, f) = α 1 α Furthermore see for instance [6, p.59-60]): C α α 1 Γ α α1 f, α1 f), 29) Γg, p), r = Γg, p), I P)r. Plugging these last two observations into Lemma 3 yields α Γf, f), α f C α α 1 ) f ν α1 f 2 + α α1 f 2 α1 f ν I P) α f ν. α 1 α Further integrate over x to obtain C α 1 α α Γf, f), α I P)f) α α 1 f ν α1 f 2 + α α1 f 2 α1 f ν ) I P) α f ν dx.

15 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 15 By Cauchy-Schwartz and symmetry the above is C I P) α f ν Our goal is to show that α 1 α { } α α 1 ) 1/2 2 f ν α1 f 2 dx. { } α α 1 ) 1/2 2 f ν α1 f 2 dx E 1/2 t)d 1/2 t). This will imply Lemma 7 in the case β = 0. We will use the embedding W 1,6 ) L ) followed by the L 6 ) Sobolev inequality to one of the terms inside the x integration. But, because the dissipation does not contain Pf, it is not enough to just take the supremum in x over the term with the smaller number of total derivatives. Instead, if α α 1 N/2 then we always take the sup on the ν norm: { } α α 1 ) 1/2 2 f ν α1 f 2 dx α1 f sup α α1 fx) ν x { } 1/2 C α1 f νv) α α1 fv) 2 W 1,6 )dv x C α1 f { } 1/2 νv) α α1 ᾱ x f 2 dxdv E 1/2 t)d 1/2 t). ᾱ 1 Alternatively, if α α 1 > N/2 then α 1 N/2 and α α1 f ν is part of the dissipation so that it is ok to take the supremum on the other norm to obtain { } α α 1 ) 1/2 2 f ν α1 f 2 dx α α1 f ν sup α1 fx) 2 x { } 1/2 C α α1 f ν α1 ᾱ x f 2 dxdv D 1/2 t)e 1/2 t). ᾱ 1 This completes the proof of Lemma 7 when β = 0. If β > 0 then we expand f as in 11) to obtain Γf, f) = ΓPf, Pf) + ΓPf, I P)f) + ΓI P)f, Pf) + ΓI P)f, I P)f). We will estimate each of these terms using similar Sobolev embedding arguments to above. However, we use this splitting because β Pf is not part of the dissipation. The exponential velocity decay of Pf is used to control the hard sphere weight 6). Alternatively I P)f is part of the dissipation. We only estimate the first two terms below. The other two terms can be treated similarly. Lemma 3 and the definition 15) yield α β ΓPf, Pf), β α I P)f C [ α α 1 ] Pf 2 α1 Pf 2 + α α1 Pf 2 α1 Pf 2 α β I P)f ν. α 1 α

16 16 ROBERT M. STRAIN, FEBRUARY 13, 2006 Further integrate the above over x, use Cauchy-Schwartz and symmetry to obtain β α ΓPf, Pf), β α I P)f ) { } 1/2 C β α I P)f ν α α1 Pf 2 2 α1 Pf 2 2dx. α 1 α Without loss of generality assume α 1 N/2. Then by the same embedding ) α α1 Pf 2 2 α1 Pf 2 2dx sup α1 fx) 2 2 α α1 Pf 2 x R 3 C α1 ᾱ x fx) 2 α α1 Pf 2 Dt)Et). ᾱ 1 This yields the estimate in Lemma 7 for the β α ΓPf, Pf) term. Finally, we estimate the β α ΓPf, I P)f) term. In this case Lemma 3 yields β α ΓPf, I P)f), β α I P)f ) { ) } 2 1/2 C β α I P)f ν α α1 β 1 Pf ν α1 β 2 I P)f 2 dx { ) } 2 1/2 +C β α I P)f ν α α1 β 1 Pf 2 α1 β 2 I P)f ν dx. Since I P)f is always part of the dissipation, we can take the supremum in x on the term with the least number of total derivatives and use the embedding H 2 ) L ) to establish Lemma 7 for this term. And the last two terms can be estimated exactly the same as this last one. We will now introduce a bit more notation. For 0 m N, a reduced order instant energy functional satisfies 1 C E mt) β m, α + β N α β ft) 2 + α N Similarly the reduced order dissipation rate is given by D m t) Et) 2 + α ft) 2 + 0< α N [ α Et), α Bt)] 2 CE m t). β m, α + β N α β {I P}ft) 2. Note that, E N t) = Et) and D N t) = Dt). With this notation in hand, we can now prove the main global existence theorem of this last section by induction. Proof of Theorem 3. We will show that for any integer m with 0 m N we have 30) d dt E mt) + D m t) Et)Dt). Theorem 3 is a special case of 30) with m = N. We will prove 30) by an induction over m, the order of the v derivatives. We may use different equivalent instant energy functionals on different lines without mention.

17 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 17 31) For m = 0 and α N, by taking the pure α derivatives of 4) we obtain { t + v x + qe + v B) v } α f { α E v} µq 1 + L{ α f} + = α 1 0 C α α 1 q α1 E + v α1 B) v α α1 f α 1 α { q } Cα α 1 E v) 2 α1 α α1 f + Γ α1 f, α α1 f). We will take the L 2 x v) inner product of this with α f. The first term is clearly 1 d 2 dt α ft) 2. From the Maxwell system 7) we have α E v µq 1, α f) = α E α J = 1 d α Et) 2 + α Bt) 2). 2 dt Next up, since νv) c1 + v ), we use two applications of Cauchy-Schwartz to get q α 1 E + v α1 B) v α α1 f α f dxdv { α1 E + α1 B ) νv) v α α1 f } 1/2 { } 1/2 2 dv νv) α f 2 dv dx R { { 3 α f ν α1 E + α1 B ) 2 νv) v α α1 f } 1/2 2 dv dx}. If α = 0 this term is non-existent. Therefore, for this term, 0 < α N and α 1 > 0 so that either α 1 N/2 or α α 1 +1 N/2. Without loss of generality say α 1 N/2; we use the embedding H 2 ) L ) to obtain sup α1 E + α1 B ) C ᾱ α1 Et) + ᾱ α1 Bt) ) Et). x ᾱ 2 We can put the last two estimates together and use α > 0 to obtain q α 1 E + v α1 B) v α α1 f α f dxdv α f ν Et) v α α1 f ν Et)Dt). It remains to estimate the last two nonlinear terms on the right in 31). First we estimate q 2 E v) α1 α α1 f. Consider α = 0 which implies α 1 = 0. We bound for E by the dissipation to estimate this case. We split f as in 11) so that the exponential velocity decay of the hydrodynamic part controls the extra v growth: q ) 2 {E v}f, f C E v Pf 2 + I P)f 2) dxdv. By the Sobolev embedding H 2 ) L ) we have ) Et, x) v I P)f 2 dxdv C sup Et, x) I P)f 2 ν x R 3 C α Et) I P)f 2 ν Et)Dt). α 2

18 18 ROBERT M. STRAIN, FEBRUARY 13, 2006 For the term with Pf, we obtain E v Pf 2 dxdv C By the Cauchy-Schwartz inequality ) sup Pfx) 2 E Pf 2 dx. x E Pf 2 dx C Et) Pf D 1/2 t)e 1/2 t). We just controlled Et) by the dissipation, D 1/2 t). For the other term, from 15), the embedding W 1,6 ) L ) and the Sobolev inequality we obtain ) sup Pfx) 2 C [a, b, c] W 1,6 ) C 32) x α f CD 1/2 t). x α 1 Adding together the last few estimates we obtain q ) 2 {E v}f, f C E v Pf 2 + I P)f 2) dxdv Et)Dt). This completes the estimate when α = 0. Alternatively if α > 0, we use νv) c1 + v ) and Cauchy-Schwartz twice as q ) E v) 2 α1 α α1 f, α f C α1 E νv) α α 1 f α f dvdx { C α1 E νv) α α1 f } 1/2 { } 1/2 2 dv νv) α f 2 dv dx R { { 3 C α f ν α1 E 2 νv) α α1 f } 1/2 2 dv dx}. Here α f ν is controlled by the dissipation since α > 0. But we could also have α = α 1 = N and then α E would not be controlled by the dissipation. Therefore this estimate requires more care. If α α 1 N/2, then to ensure that we have at least one derivative on f we use the embedding W 1,6 ) L ) followed by Sobolev s inequality to obtain sup x νv) α α1 fx, v) 2 dv C νv) α α1 fv) 2 W 1,6 )dv x C x α α1 ᾱf 2 ν CDt). ᾱ 1 Alternatively, if α α 1 > N/2 then α 1 N/2 and 33) sup α1 Et, x) 2 C α1 E 2 W 1,6 x x ) C x α1 ᾱe 2 CEt). ᾱ 1 In either case the last two estimates establish { α1 E 2 νv) α α 1 f } 2 dv dx Et)Dt). Combining these last few estimates, since α > 0, we obtain q E v) 2 α1 α α1 f, f) α α f ν Et)D 1/2 t) Et)Dt). It remains to estimate the nonlinear collision operator in 31).

19 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 19 But from Lemma 7 and the basic invariant property 29) we have Γ α α1 f, α1 f), α f ) Et)Dt). Adding up all the estimates for the terms in 31) and summing over α N yields 34) 1 d α f 2 + α [E, B] 2 2 dt + L α ft), α ft)) Et)Dt). α N α N By Theorem 2 and Lemma 6 for some small δ 0 > 0 we see that 1 d α f 2 + α [E, B] 2 2C 2 dt 0 Gt) + δ 0D 0 t) Et)Dt). α N We add 34) multiplied by a large constant, C 0 > 0, to the last line to obtain d 35) dt E 0t) + D 0 t) Et)Dt), We have used L 0 from Lemma 1. Above, for C 0 large enough and from the definition of Gt) in Theorem 2, the following is a reduced instant energy functional: E 0 t) = C 0δ 0 + 1) 2δ 0 α f 2 + [ α E, α B] 2 C 0 δ 0 Gt). α N We conclude 30) for β = 0. Say Theorem 2 holds for β = m > 0. For β = m + 1, take α β 36) α 1 =0 { t + v x + qe + v B) v } α β I P)f α E β {v µ}q 1 + β {L α I P)f} + β 1 0 C β β 1 β1 v x α f = Cα α 1 C β q β 1 E 2 { α1 β1 v} α α1 f + β α Γf, f) f C α α 1 q α1 E v α α1 β α 1 + β 1 =0 { t + v x + qe + v B) v } α β Pf. of 4) to obtain: C α α 1 C β β 1 q β1 v α1 B v α α1 f Notice that we have split the equation in terms of microscopic and hydrodynamic parts in a different way from 12). We do this to estimate the linear part of the collision operator properly in terms of our weak whole space dissipation. Standard estimates for β L α f, β α f), e.g. Lemma 2, are too strong in this case. We take the inner product of 36) with β αi P)f over R3 x v. We estimate each term from left to right. The first inner product on the left gives 1 d 2 dt α β I P)ft) 2. Since β = m + 1 > 0, α N m 1. Then by Lemma 6 the second inner product on the left is bounded by α E β {v µ}q 1, α β I P)f ) C α E α β I P)f C η α E 2 η α β I P)f 2 C η D 0 t) η α β I P)f 2.

20 20 ROBERT M. STRAIN, FEBRUARY 13, 2006 For the linear operator, since β1 νv) is bounded for β 1 0 Lemma 2) we have β {ν α I P)f}, α β I P)f ) = α β I P)f 2 ν + β 1 0 C β β 1 β1 ν α I P)f, α β I P)f α β I P)f 2 ν C α β I P)f β 1 0 α I P)f. Then together with Lemma 2, since L = ν K, we deduce that for any η > 0 there is a constant C η > 0 such that the third term on the left side of 36) is bounded from below as β {L α I P)f}, β α I P)f ) β α I P)f 2 ν η β α I P)f 2 ν C η β < β ᾱ β I P)f 2. β = β We can further choose C η > 0 such that the inner product of the last term on left side of 36) is bounded by η α β I P)ft) 2 + C η β 1 =1 We split the last term above as in 11) to obtain η α β I P)ft) 2 + C η β 1 =1 x α Pf 2 + C η η α β I P)ft) 2 + CD 0 t) + C η β 1 =1 x α f 2 β 1 =1 x α I P)f 2 x α I P)f 2. This splitting was used to get rid of the velocity derivatives of the hydrodynamic part, which are not in the dissipation. Next we estimate all the terms on the right side of 36). Let us first consider the first term on the right, using 1 + v ) cνv) we have 37) C q E 2 { α1 β1 v} α α1 f, β α I P)f) α1 E νv) α α1 f β α I P)f dxdv. We use Cauchy-Schwartz twice to see that the last line is { C α1 E νv) C α β I P)f ν { α α1 f } 2 1/2 { dv νv) β α I P)f } 1/2 2 dv dx } νv) α α1 f 2 1/2 dv dx}. { α1 E 2

21 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM IN THE WHOLE SPACE 21 We split f into it s microscopic and hydrodynamic parts 11) so that we can remove the velocity derivatives from the hydrodynamic part: { { } } C β α I P)f ν α1 E 2 νv) α α1 Pf 2 1/2 dv dx R { { 3 } +C β α I P)f ν α1 E 2 νv) α α1 I P)f 2 1/2 dv dx}. Since the total number of derivatives is at most N, either α 1 N/2 or α α 1 + β β 1 N/2. So we take the supremum in x of the term has the least total derivatives, for the microscopic term, using the embedding H 2 ) L ) to observe { } α1 E 2 νv) α α1 I P)f 2 dv dx Et)Dt). On the other hand, for the hydrodynamic variables we have { } α1 E 2 νv) α α1 Pf 2 dv dx C α1 E 2 α α1 Pf 2 2dx. If α α 1 N/2, then as in 32) we see that α1 E 2 α α1 Pf 2 2dx C α1 E 2 Dt) CEt)Dt). If alternatively α α 1 > N/2 and α 1 N/2 then we use 33) to get α1 E 2 α α1 Pf 2 2dx CEt) α α1 f 2 ν CEt)Dt). In any case, we conclude q E 2 { α1 β1 v} α α1 f, β α I P)f) Et)Dt). This is the estimate for the first term on the right side of 36). For the nonlinear collision operator in 36), by Lemma 7 we have α β Γf, f), α β I P)f ) Et)Dt). For the next two terms on the right side of 36) we have q α1 E v α α1 β f + q β1 v α1 B v α α1 f, β α I P)f) C νv) α1 E v α α1 β f + α1 B v α α1 α f ) β I P)f dvdx. Notice that these are in the form 37). They are therefore estimated the same way. Furthermore the same procedure as in the estimate for 37)) yields qe + v B) v } α β Pf, α β I P)f ) Et)Dt). It remains to estimate { t + v x } α β Pf, α β I P)f ).

22 22 ROBERT M. STRAIN, FEBRUARY 13, 2006 Since β > 0, we have α N 1 and the above is C νv) t β α Pf + x β α Pf ) α β I P)f dxdv C β α I P)f ν t β α Pf ν + x β α ) Pf ν C β α I P)f ν t α Pf + x α Pf ) η β α I P)f 2 ν + C η ᾱ α f 2. ᾱ =1 This completes all the estimates for the terms in 36). By collecting all the estimates for the terms in 36) and summing over β = m+1 and α + β N we obtain { } 1 d 2 dt α β I P)ft) 2 + β α I P)ft) 2 ν C η D 0 t) β =m+1, α + β N Cη β α I P)ft) 2 ν +C η β =m+1, α + β N β m, α + β N α β I P)ft) 2 + Et)Dt). Here C is a large constant which does not depend on η. Choosing η = 1 2C { } 1 d 2 dt α β I P)ft) 2 + β α I P)ft) 2 ν 38) β =m+1, α + β N CD m t) + Et)Dt). we have Now add the inequality from 30) for β = m multiplied by a suitably large constant, C m, to 38) to obtain d dt C me m t) + 1 β α I P)ft) C m C) D m t) β =m+1, α + β N β =m+1, α + β N α β I P)ft) 2 ν Et)Dt). Here C m is chosen so that C m C > 0. We further define E m+1 t) = C m E m t) β =m+1, α + β N α β I P)ft) 2. Since β Pf C Pf, E m+1 t) is an instant energy functional. Acknowledgements. The author is very grateful to Yan Guo for bringing his attention to this problem. We thank both referees for several helpful comments concerning the presentation of this paper, and for pointing out the reference [1]. We also thank Tong Yang for sending the works [5, 20].

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