A new approach to study the Vlasov-Maxwell system

We give a new proof based on Fourier Transform of the classical 
Glassey and Strauss [6] global existence result for the 3D relativistic Vlasov-Maxwell system, under the assumption of compactly supported particle densities. 
Though our proof is not substantially shorter than that of [6], we believe 
it adds a new perspective to the problem. In particular the proof is based on 
three main observations, see Facts 1-3 following the statement of Theorem 1.4, 
which are of independent interest.


Introduction. We write the Vlasov-Maxwell system as
where (x, v) ∈ R 3 × R 3 , f (t, x, v) denotes the particle density 1 , j(t, x) = vf dv the current density, ρ = fdv the charge density andv = v (1+|v|) 1/2 the relativistic velocity. Consider then the initial value problem (IVP) given by (1) and The standard regularity for the data in (2) and (3) is f 0 ∈ C 1 0 and E 0 , B 0 ∈ C 2 . The question of global well-posedness has been considered by many authors. The reduced Vlasov-Poisson system has been tackled successfully, for large data, by Pfaffelmoser [8], Lions-Parthame [9], Schaeffer [10]. The outstanding result for the full 3D Maxwell-Vlasov system remains that of Glassey-Strauss [6], who were able to prove a global existence result under the hypothesis of compactly supported (for all time !) particle density, (see also [4] and [5]). Later Glassey-Shaeffer [7] were able to remove the additional support hypothesis for the, so called 2 , 2 + 1/2 dimensional system.
We recall below the main result of Glassey and Strauss: Theorem 1.1 (Glassey and Strauss). Assume the above conditions on the initial data. Assume there exists a continuous function β(t) such that for all x ∈ R 3 Then there exists a unique C 1 solution of the system for all t.
S.K. was supported by N.S.F. Grant DMS 215-6322. G.S. was supported in part by N.S.F. Grant DMS 9800879 and grants from Hewlett and Packard and Sloan Foundation. 1 Here we assume that we are dealing with only one specie, for a more detailed introduction of the system we refer to [3]. 2 In which case f = f (t, x, v) is a function of time t ∈ R, x ∈ R 2 and v ∈ R 3 .

SERGIU KLAINERMAN AND GIGLIOLA STAFFILANI
The Glassey-Strauss proof relies on showing uniform bounds in time for the L ∞ norms of E, B, f as well as of all their first derivatives. They start by rewriting (1), (2) and (3) as follows 3 : Then they represent the fields E and B using the explicit form of the fundamental solution of = ∂ 2 t − ∆ in physical space. For example for E they write where E 0 is a solution of the homogeneous equation . The presence of the derivative fields ∇ +v∂ t in the integrand seems to create great difficulties 4 . The main new idea of Glassey and Strauss was to decompose ∇ +v∂ t into fields |y−x| , tangent to the backward characteristic cone 5 passing through (t, x), and the vector field S = ∂ t +v · ∇. Indeed one has, and similarly for ∂ t . The crucial observation is that Sf can be reexpressed by using the transport equation, i.e., One can get rid of the v derivative of f by integrating by parts with respect to the v-integration in (6). All this works, and thus allows the authors to estimate the L ∞ norm of E and B in terms of the L ∞ norm of f , as long as the denominator 1 +v · ω is bounded away from zero. This important fact is guaranteed by the a priori assumption (4). A similar, much more delicate argument, is needed to obtain the estimates for the derivatives of E and B. Glassey-Strauss accomplish this goal by proving in fact that for any t in a fixed interval of time [0, T ], where 6 ∇ x denotes the space-time derivatives in x and D = (∇ x , ∇ v ). On the other hand, using the transport equation, they show by a straightforward argument that: 3 We get rid of the constants c and π by taking c = 4π = 1 in (1). 4 Recall that we need to estimates the first derivatives of E and B. According to the above formula this seems to require two derivatives of f !. 5 Which disappear by simple integration in the formula (6). 6 Note that log + z := log(2 + z) for nonnegative z.
The combination of (7) and (8) followed by a straightforward application of Gronwall's inequality gives a bound for all the quantities involved. In proving (7) and (8) Glassey and Strauss had to deal with denominators of type ( 1 +v · ω) n , n ≤ 4 whose possible singularities are avoided by (4). The last step to complete the proof of Theorem 1.1 is based on a standard recursive method.
In this paper we are taking a different approach based on the Fourier representation of E and B. Observe that, For convenience we shall write the system (5) in the form: where with M (v) and N (v) matrices depending only on v. In the first part of our proof we don't need the explicit representation of α(v), M (v) and N (v); shall only make use of the fact that all their components, and their derivatives with respect to v, are bounded for all v ∈ R 3 . When convenient we will make use of the explicit representation (5).
In this paper we prove the following version of Theorem 1.1: Then the system (9) admits a unique Remark 1.3. Observe that we have substituted the support assumption (4) with the boundedness assumption (13). The two assumptions are, essentially, equivalent. In fact (4) implies (13) through a straightforward application of the Glassey-Strauss decomposition idea mentioned above. On the other hand, (13) together with the assumption of compact support in v for f 0 (t, x, v) implies the Glassey-Strauss assumption (4). This follows easily from the properties of characteristics, see Lemma 2.1. As a consequence of (13) we also have |v| ≤ δ < 1 on the support of f (t, x, v) for all (t, x) ∈ [0, T ] × R 3 , fact on which we shall rely heavily in our proof.
Our strategy is to prove directly that the characteristics associated to the transport equation in (9) are Lipshitz. To do this we look at the problem not just in the physical space, but also in the frequency space.
Denote, as in [6], the characteristics 7 , (X(s, t, x, y), V (s, t, x, y)), s,t ∈ R, x, y ∈ R 3 of the transport equation in (9). They are solutions of the system of ordinary differential equations in s, with initial data at s = t, Our main theorem can easily be reduced to the following: Consider the IVP (9) on a fixed interval [0, T ] and assume that [sup sup sup sup The proof is based on three main observations.
• Fact 1: When we solve Φ = J by the Fourier method we are led to integrals of the form These are smoother than may be apparent at first glance as can be seen integrating first by parts in σ, making use of the transport equation (9) for f and then integrating by parts once more in v. This fact, which follows easily from Lemma 2.2 below 8 , seems to be the Fourier counterpart of the Glassey-Strauss idea of decomposition of general derivatives into S and T , integration by parts and use of the transport equation mentioned above.
• Fact 2: The time integral of a wave solution Φ along characteristics is smoother than Φ itself . More precisely if Φ is solution of the inhomogeneous wave equation and (s, X(s, x)) is a time-like characteristic curve 9 , then t 0 Φ(s, X(s, x)) ds is one derivative smoother than Φ. This fact can be easily seen in Fourier space, i.e.
Thus we are led to integrals of the form J = t 0 e i ±s|ξ|+X(s,x)·ξ H(s)ds. Integrating by parts and using the characteristic equations (14) we find, and therefore, and the gain of differentiability is obvious in view ofV ≤ δ < 1. Observe also that the matrices β(V ), α(V ) are bounded.
• Fact 3: In the process of estimating the sup-norms of our main quantities, expressed as Fourier integrals, we need an extension of the well known Beale-Kato-Majda lemma in [1] to Fourier integral operators. This is the content of the Lemma 2.5 below. To understand its usefulness consider the standard initial value problem in R 3+1 It is well known, from the explicit form of the fundamental solution in physical space, that φ(t) L ∞ t g L ∞ . As discussed above, in the context of the inhomogeneous wave equation, this fact plays a fundamental role in the proof of Glassey-Strauss. Unfortunately this estimate seems very unstable. Indeed consider the Fourier representation of the solution: Can we still bound the L ∞ norm of φ + (t, ·) in terms of g L ∞ ? The answer is no; the best we can hope for is an estimate of the form with t ∈ [0, T ] and p > 1. We prove a more general estimate of this type in Lemma 2.5, for Fourier integral operators of the form with m 0 homogeneous of degree zero in ξ.
Throughout the paper we will denote withf the Fourier transform 10 of the function f :ĝ We will also write for 1 ≤ p < ∞ If the function g has two variables, then for 1 ≤ p, q < ∞ we write 11 and if p = q we simplify the notation by writing Very often we will use the notation B ≈ C, if there exists β ∈ C, β = 0, such that B = βC. We will also write B C, if there exists m > 0 such that B ≤ mC.
2. Proof of Theorem 1.4. During the process of proving Theorem 1.4 we will introduce few lemmas, but in order not to distract the reader from the main flow of ideas, we postpone their proofs in the Appendix. We will also always assume that From the definition of characteristics we have We first observe the following support property for f .
Then if f solves the transport equation in (9) and Φ satisfies (13), then for any t ∈ [0, T ] where 10 Allˆquantities refer to Fourier transform with the exception ofv = v (1+|v|) 1/2 which denotes the relativistic velocity! 11 The definition involving the L ∞ norm is the obvious one.
The proof follows directly from (22) and (23). Since Φ solves the wave equation (9), we write where Φ 0 is the solution of the homogeneous problem Φ 0 = 0 and In what follows we neglect Φ 0 , which contributes only a trivial term in the estimates below. As Φ + and Φ − are treated in the same way we may assume that Φ = Φ + . Therefore, ignoring constants, The first integral can be written in the form, At this point we use Fact 1 listed at the end of Section 1. The precise statement is contained in the following lemma: Lemma 2.2. Assume f solves the equation in (9). Then for any fixed v and ξ It is now recognizable in the left hand side of this equality the smoothing effect represented by the factor 1 |ξ| . We also observe that condition (24) guarantees that 1 ±v ·ξ |ξ| ≈ 1. We postpone the proof of this lemma to the appendix. Returning to I we write

SERGIU KLAINERMAN AND GIGLIOLA STAFFILANI
Therefore, back to (29), ignoring constants and integrating by parts in v, Finally, differentiating in x, It is not hard to show that M 0 ∈ S −1 and M 1 , M 2 ∈ S 0 , uniformly with respect to v, where S m is the class of usual symbols 12 of order m.
We now start the estimates for the derivatives of the characteristics. We first assume that all characteristics start at time t = 0. We begin with ∇ x V . From (23) where, as defined in (10) and (11), Φ = (E, B) and α(v) = (1,v×). Thus where X(σ, 0, x, v))dσ, Because α (V ) is uniformly bounded the estimate for I 0 is trivial. We first estimate I 1 . Using (30) we write where The integral I 11 can be directly estimated since its symbol K 11 (w, ξ) = M 0 (w, ξ)ξ ∈ S 0 . Before estimating I 12 and I 13 we need to do some more work. To get additional smoothing we need in fact to appeal to Fact 2 mentioned at the end of Section 1.
In view of (21) we can write, I 12 = I 121 + I 122 + I 123 + I 124 , In a similar fashion we also have, In a similar manner one can express also I 2 . In fact the only difference will be the presence ofV and ∂ tV . But both functions are uniformly bounded thanks to (13) and (24).
Remark 2.3. We have intentionally suppressed the dependence of the symbols K on σ, x, v. This dependence, throughV , is trivial in view of the fact thatV < 1. We also observe that K 122 and K 133 belong uniformly to S −1 , while all the remaining K ijk belong uniformly to the symbol class S 0 . They are in fact all homogeneous in ξ ∈ R 3 .
To estimate the terms involving these multipliers we use Fact 3 introduced at the end of Section 1. We formulate it explicitly in the following lemmas: Lemma 2.4. Assume that m k = m k (x, λ; ξ) ∈ S k , uniformly with respect to x ∈ R 3 and some family of parameters λ. Let P k be the associated pseudodifferential operator, (48) ii.) For any p > 3 and 1 < q < ∞, where we denote log + x = log(2 + x).
We should need a similar lemma for Fourier integral operators.

Lemma 2.5.
Assume that m 0 = m 0 (t, x, λ; ξ) ∈ S 0 uniformly with respect to t ∈ R, x ∈ R 3 and a family of parameters λ. Assume also that m 0 is homogeneous of degree zero in ξ. Let T = T m0 be the associated Fourier integral operator, defined by for g ∈ S(R 3 ). Then for any p ∈ (1, ∞] The proof of these lemmas can be found in the appendix. We now use (48)-(51) to estimate each of the terms I ijk . For example, follows from the fact that the symbol K 11 = ξM 0 ∈ S 0 and hence (50) can be applied for any p > 3. Using (48) we estimate

SERGIU KLAINERMAN AND GIGLIOLA STAFFILANI
To estimate the remaining I ijk terms, we introduce: whereĤ(τ, ξ) = R 3 e iy·ξ H(τ, y)dy and K ijk ∈ S 0 uniformly. Clearly the Fourier integral T [H] can be estimated by using (51). Using this notation we now write where f (0,w) (y) = f 0 (y, w). Consequently, Similarly, and using the boundedness of β on the time interval [0, T ], and similarly, thanks to the boundedness of α, β and Φ again for p ∈ (1, ∞]. We use a similar argument to bound I 13j , with j = 1, 2, 4.
We now summarize the above estimates for ∂ x V , where, In a similar manner one also obtains On the other hand, from (22), we simply have and We need two more ingredients to be able to complete the proof. The first is the following lemma: Lemma 2.6. Assume that Φ and f are solutions for (9) and that Φ satisfies the uniform boundedness condition (13). Then for any t ∈ [0, T ] Also the proof of this lemma can be found in the appendix 13 . Next we define We then go back to (53), (54) (55) and (56). Making use of Lemmas 2.6 and 2.1 as well as (13) and (24), we obtain: The last important ingredient of the proof is the fact that the derivatives of f can be estimated in terms of the derivatives of the forward characteristics X(s, 0, x, v) and V (s, 0, x, v), for s ∈ [0, T ]. In fact, using the backward characteristics X(0, s, x, v) and V (0, s, x, v), we write therefore , According to Lemma 3.1 we can invert the flow and derive, We can now substitute (62) in (60) and (61) and finally derive It only remains to invoke Lemma 3.2, in the appendix, to conclude that for all and Theorem 1.4 is proved for t = 0. To prove the theorem in full generality we first observe that if we insert (63) into (62) we obtain the uniform bound Then we repeat the argument presented above replacing X(s, 0, It is easy to see that (53) becomes that (55) gives and similarly for (56). Thanks to (64), Lemma 2.6 and 2.1, (13) and (24) we have Then if we set and we insert (68) into (65) and (66) , we obtain and again by Lemma 3.2 for all s, t ∈ [0, T ]. This concludes the proof of the theorem.

Appendix. Proof of Lemma 2.2
The proof is a simple consequence of integration by parts and the transport equation in (9). In fact The transport equation allows us to substitute ∂ σf and continue the chain of equalities with (70) Then the lemma is obtained by moving the second term of (70) to the left hand side of the equality.

SERGIU KLAINERMAN AND GIGLIOLA STAFFILANI
The Lemma is well known; we present its proof here for the sake of completeness and as an introduction to the more difficult proof of Lemma 2.5. For simplicity we shall pretend that our symbols m k depend only on ξ. In view of the uniform boundedness of m k (t, x, λ; ξ) relative to the parameters t, x, λ the general case does not present any additional difficulties. We start by proving (48). First recall [11] that ifK Now write where B 1 (x) is the ball in R 3 centered at x and of radius 1, and B 1 (x) c is its complement. We then use (71) to obtain, for any p ∈ [1, 3), The inequality (49) is a classic result of Harmonic Analysis; we refer to [11]. We now sketch the proof of (50). using a well known argument; see for example [1]. Fix x ∈ R 3 , > 0 and write g = g 1 + g 2 , where g 1 (y) = g(y)χ B (x) (y). Then if we can write for any p > 3 and 1 < q < ∞. Then pick −1/p +2 = ∇g L p and (50) is proved.

Proof of Lemma 2.5
As in the proof of the previous lemma we drop the dependence of m 0 on t, x, λ. Thus m 0 = m 0 (ξ) is an homogeneous symbol of degree zero in ξ. We first observe that it suffices to prove (51) for t = 1. In fact, by a simple change of variable and the homogeneity of m 0 , we have where g t (y) = g(ty). Thus, assuming the estimate to be true for t = 1 we obtain T g(t) ∞ = t (T g t )(1) ∞ g ∞ log + ( ∇g L p ) + g L p + 1.
From now on we write T g(x) = T g (1, x). For simplicity we may also assume x = 0. Then By a simple limiting argument 14 we can replace K(y) in (74) by the kernel with, χ an arbitrary test function on [0, ∞) verifying 0 ≤ χ ≤ 1 and |χ (u)|du ≤ 10.
If m 0 where radially symmetric, then a sharp estimate for K h (y) was given by Stein 15 Notice that because m 0 ∈ S 0 is homogeneous it will not depend on u. We now estimate the inner integral in θ. By integration by parts π 0 e i|y|u cos θ m 0 (θ, w) sin θ dθ = e i|y|u cos θ m 0 (θ, w) i|y|u π 0 − π 0 e i|y|u cos θ ∂ θ m 0 (θ, w) i|y|u dθ.
One can then use the method of stationary phase to estimate the asymptotic behavior of the integral in the right hand side (see for example [11] page 334 and 1 − ψ (y) K(y)g(y) dy, K(y)g(y)dy, with ψ (y) a smooth test function supported in the region 1 − |y| ≤ 2 equal to 1 on 1 − |y| ≤ and such that |∇ y ψ(y)| 1 .
We start by estimating I 3 , which is the easiest. In fact from (81) we obtain by a straightforward estimate that |K(y)| 1 |y| 3/2 for |y| ≤ 1/2.
Then clearly, Therefore,