LARGE TIME BEHAVIOR OF THE RELATIVISTIC VLASOV MAXWELL SYSTEM IN LOW SPACE DIMENSION

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1 Differential Integral Equations Volume 3, Numbers 1- (1), LARGE TIME BEHAVIOR OF THE RELATIVISTIC VLASOV MAXWELL SYSTEM IN LOW SPACE DIMENSION Robert Glassey Department of Mathematics, Indiana University Bloomington, IN 4745, USA Stephen Pankavich Department of Mathematics, University of Texas at Arlington Arlington, TX 7619, USA Jack Schaeffer Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA 1513, USA (Submitted by: Reza Aftabizadeh) Abstract. When particle speeds are large the motion of a collisionless plasma is modeled by the relativistic Vlasov Maxwell system. Large time behavior of solutions which depend on one position variable two momentum variables is considered. In the case of a single species of charge it is shown that there are solutions for which the charge density (ρ R fdv) does not decay in time. This is in marked contrast to results for the non-relativistic Vlasov Poisson system in one space dimension. The case when two oppositely charged species are present the net total charge is zero is also considered. In this case, it is shown that the support in the first component of momentum can grow at most as t Introduction Consider the relativistic Vlasov-Maxwell system: t f + ˆv 1 x f + e (E 1 + ˆv B) v1 f + e (E ˆv 1 B) v f, ρ(t, x) e f (t, x, v)dv, j(t, x) e f (t, x, v)ˆv dv, x E 1 (t, x) 1 ρ(t, y)dy 1 ρ(t, y)dy, x t E + x B j, t B + x E, (1.1) Accepted for publication: May 9. AMS Subject Classifications: 35L6, 35Q99, 8C1, 8C, 8D1. 61

2 6 Robert Glassey, Stephen Pankavich, Jack Schaeffer for 1,..., N. Here, t is time, x R is the first component of position, v (v 1, v ) R contains the first two components of momentum. Hence, dv dv dv 1 the v integrals are understood to be over R. f gives the number density in phase space of particles of mass m charge e. Velocity is given by ˆv v (m ) + v, where the speed of light has been normalized to one. The effects of collisions are neglected. The initial conditions { f (, x, v) f (x, v), 1,..., N, E (, x) E (x), B(, x) B (x), are given where it is assumed throughout the paper that f C1 (R3 ) is nonnegative compactly supported that E, B C 1 (R) are compactly supported. When the neutrality condition, e f dv dx, holds, we will refer to this as the neutral case. A major goal of this paper is to compare the neutral case with the monocharge case, which may be obtained from (1.1) by setting N 1. In the monocharge case we will drop write, for example, f f f 1 take e e 1 1, m m 1 1. Choose C such that f, E, B vanish (for all ) if x C. The letter C will denote a positive generic constant which may depend on the initial data (but not t, x, v) may change from line to line, whereas a numbered constant (such as C ) has a fixed value. We also define the characteristics, (X (s, t, x, v), V (s, t, x, v)), of f by dx ds ˆV 1, X (t, t, x, v) x dv1 ( ds e E 1 (x, X ) + ˆV ) B(s, X ), V1 (t, t, x, v) v 1 (1.) dv ds e ( E (s, X ) ˆV 1 B(s, X ) ), V (t, t, x, v) v. Theorem 1.1. In the neutral case there is a constant, C, such that [ C E1 + (E B) + f ( )dv] (m ) + v (τ,x t+τ) v 1 dτ

3 + [ E 1 + (E + B) + Large time behavior for Vlasov Maxwell 63 f ( )dv] (m ) + v (τ,x+t τ) + v 1 dτ, (1.3) for all t, x R. In the monocharge case there is a constant, C, such that C(C + t x) [(E B) + f ( 1 + v v 1 ) (τ,x t+τ) dv] dτ (1.4) 1 + v for x < C + t, C(C + t + x) for C t < x. [(E + B) + f ( 1 + v + v 1 ) (τ,x+t τ) dv] dτ (1.5) 1 + v The proof of this theorem relies on conservation of energy of momentum (in the monocharge case) is contained in Section. Theorem 1.. In the neutral case there is a constant, C, such that v C + C t x + C, on the support of f for every. In the monocharge case there is a positive constant, C, such that on the support of f. v C + C (t + C ) x, The proof of Theorem 1. is in Section 3. Theorem 1.3. There are solutions of the monocharge problem for which there exist x R C > such that x +t ρ(t, x)dx > C, (1.6) for all t. Furthermore, there exists C > such that for all t p [1, ]. ρ(t, ) L p (R) > C, The second assertion of Theorem 1.3 follows from (1.6) by using Hölder s inequality: C < C +t x +t ρ(t, x)dx ρ(t, ) L p (R)(C x ) 1 1 p.

4 64 Robert Glassey, Stephen Pankavich, Jack Schaeffer The proof of Theorem 1.3 is contained in Section 4. In [8] an analogous, but more detailed, result is obtained for the relativistic Vlasov Poisson system (which may be obtained from (1.1) by setting E B ). Theorem 1.4. For the neutral problem there is a constant, C, such that on the support of f for every. v 1 C + Ct 1 (t x + C ) 1 4, (1.7) The proof is in Section 5. A similar, but different, estimate is obtained in [8] for the relativistic Vlasov Poisson system. Also, note that (1.7) rules out an estimate like (1.6). If (1.6) held, then there would be characteristics for which f X (t,, x, v) x + t, (1.8) for all t. Then by (1.7) (1.8) so C ln(1 + t) 1 ˆV 1 (t,, x, v) V 1 (t,, x, v) C + Ct 1, 1 + (V ) 1 + V ( 1 + V + V 1 (1 + V ) C 1 + t, ( 1 ˆV ) 1 (s,, x, v) ds t X (t,, x, v) + x x x for all t. Finally, Section 6 contains the proof of the following. Theorem 1.5. In both the neutral monocharge cases there are no nontrivial steady solutions with f, E B compactly supported. The global existence in time of smooth solutions to (1.1) is shown in [9] when a neutralizing background density is included. Adaptation of the essential estimate from [9] to the current situation is briefly discussed in Section. Global existence has been shown in two dimensions, [11], two one-half dimensions, [1], but is open for large data in three dimensions. Some time decay is known for the classical Vlasov Poisson system in three dimensions ([1], [14], [15]). Additionally there are time decay results for the classical Vlasov Poisson system in one dimension ([1], [], [7], [17]). For decay results on the relativistic Vlasov Poisson system, see [7] [13]. References 1 )

5 Large time behavior for Vlasov Maxwell 65 [3], [4], [5] are also mentioned since they deal with time dependent rescalings time decay for other kinetic equations. We also cite [6] [16] as general references on mathematical kinetic theory. A main point to this article is that the non-decay stated in Theorem 1.3 is in marked contrast to the decay found in [1], [], [17]. In [1], [] [17] the problem studied is non-relativistic. Hence, there is no a priori upper bound on particle speed this leads to dispersion. In this paper ( also [8]), particle speeds are bounded by the speed of light this limits the dispersion. Define. Conservation Laws e f (m ) + v dv + 1 E + 1 B, m f v 1 dv + E B, l f v 1ˆv 1 dv 1 E E + 1 B. A short computation reveals that Using (.1), the divergence theorem yields x+t τ x t+τ t e + x m, (.1) t m + x l. (.) ( τ e + y m) dy dτ (.3) (e + m) dτ + (τ,x+t τ) (e m) dτ (τ,x t+τ) x+t x t e(, y)dy. Note that ( ) e ± m f (m ) + v ± v 1 dv + 1 E (E ± B), that j f v ( (m ) + v dv C ) f (m ) + v ± v 1 dv.

6 66 Robert Glassey, Stephen Pankavich, Jack Schaeffer In the neutral case (.3) yields x+t C e(, y)dy (e+m) dτ + (τ,x+t τ) x t (e m) dτ. (.4) (τ,x t+τ) In the monocharge case, since E 1 is not compactly supported, (.3) only yields Ct (e + m) dτ + (e m) dτ. (τ,x+t τ) (τ,x t+τ) It follows that E + B C + C j (τ, x + t τ) dτ + C j (τ, x t + τ) dτ C + Ct p, (.5) where p in the neutral case p 1 in the monocharge case. Global existence of smooth solutions follows in both cases as in [9]. Consider the monocharge case now. Bounds independent of t may be obtained by also using (.). For x < C the divergence theorem yields C +τ x +τ [ τ (e m) + y (m l)] dy dτ C +t [e m + l] dτ + (e m) dy (τ,x +τ) (t,y) x +t C [e m + l] dτ (e m) dy. (τ,c +τ) x (,y) Note that e m + l f ( (m ) + v v 1 ) dv + (E B) (m ) + v is nonnegative vanishes on y C + τ (since E 1 canceled). Hence, C C(C x ) (e m) dy (.6) x (,y) C +t (e m + l) dτ + (e m) dy. (τ,x +τ) (t,y) x +t Similarly, e + m + l f ( (m ) + v + v 1 ) dv + (E + B) (m ) + v

7 Large time behavior for Vlasov Maxwell 67 is nonnegative vanishes on y C τ this leads to C(x + C ) x τ x C τ [ τ (e + m) + y (m + l)] dy dτ; (e + m) dy (.7) C (,y) x t (e + m + l) dτ + (e + m) dy, (τ,x τ) (t,y) C t for x > C. Theorem 1.1 now follows from (.4), (.6), (.7). Define note that 3. Bounds on v Support A(t, x) x B(t, y)dy, t A + x A (E B), so t A x A (E + B), A(t, x) A(, x t) A(, x + t) (E B) dτ (τ,x t+τ) (E + B) dτ. (τ,x+t τ) (3.1) For x C + t, A(t, x) A(, x t) C, so consider x < C + t. Then (3.1) becomes A(t, x) A(, x t) (E B) dτ (τ,x t+τ) max(, t x C ) A(, x + t) (E + B) dτ. max(, x+t C ) (τ,x+t τ) In the neutral case, (1.3) the Cauchy Schwartz inequality yield A(t, x) C + t max(, t x C ) C C + C t + x + C,

8 68 Robert Glassey, Stephen Pankavich, Jack Schaeffer A(t, x) C + t max(, x + t C ) C C + C t x + C. Hence, A(t, x) C + C t x + C (3.) follows. For the monocharge case, (1.4) is used in place of (1.3) to obtain A(t, x) C + t max(, t x C ) C(C + t x) (3.3) C + C (t + C ) x. From (1.) we have f (s, X (s, t, x, v), V (s, t, x, v)) f (t, x, v) V (s, t, x, v) + e A(s, X (s, t, x, v)) v + e A(t, x), for all s, t, x, v. If f (t, x, v), then v + e A(t, x) V (, t, x, v) + e A(, X (, t, x, v)) C. In the neutral case, (3.) yields In the monocharge case, (3.3) yields v C + C t x + C. (3.4) v C + C (t + C ) x (3.5) on the support of f. Theorem 1. follows from (3.4) (3.5) but we make one further observation. On the support of f v + e A(t, x) C so v ( e A(t, x) C, e A(t, x) + C). Thus, the v support has bounded measure. 4. Non-decay of ρ in the Monocharge Case In this section only the monocharge case is considered. Let M ρ(t, x)dx, note that x E 1 1 ρ dy 1 For some x ( C, C ) define µ(t) C +t x +t ρ(t, y)dy E(t) x ρ dy 1 C +t M ρ dy. C +t x +t x (e m) dy. (t,y)

9 Large time behavior for Vlasov Maxwell 69 Then µ (t) j 1 (t, x + t) ρ(t, x + t), by (.1) (.) C E +t (t) y (m l)dy + (e m) (e m) (t,x x +t (t,c +t) +t) (e m + l) (e m + l) (t,x (t,c +t) +t) (e m + l). (t,x +t) Suppose that Then, for y x + t, 1 M µ(), (4.1) 1 (C x )( 1 M) > E(). (4.) so hence E 1 (t, y) E 1 (t, x + t) 1 M µ(t) 1 M µ(), E() E(t) 1 C +t x +t C +t x +t E 1dy 1 (C x )( 1 M µ(t)), E() 1 C x M µ(t), ρ dy µ(t) 1 M E() >. C x Hence, Theorem 1.3 follows from (4.1) (4.). To see that there are initial conditions for which (4.1) (4.) hold consider the following: Let f L, f R C1 (R3 ) be nonnegative compactly supported with 1 f L (x, v) if x 1, f R (x, v) if x / ( 1, ), f L dv dx f R dv dx >.

10 7 Robert Glassey, Stephen Pankavich, Jack Schaeffer Let C sup { x : f L (x, v) for some v }, f(, x, v) f L (x, v) + f R (x C, v 1 W, v ), for W > 1. Taking x 1 we have µ() f R dv dx 1 M, which is (4.1). Taking E (, y) B(, y) ( using x 1), we have C [ E() f( 1 + v v 1 )dv + 1 ] E 1 (,y) dy x C x [ f R 1 + v (y C, v 1 W, v ) dv v + v 1 E 1 C 1 C C W + 1 ( 1 ) dy x M µ() 1 + ( 1 C 1 M) dy C W + C 1 x ( M ) 4 Mµ() + µ () M C W + C ( M ) 4 Mµ() + µ () x 8 M C W + C x M C µ()(m µ()). 8 Now, taking W sufficiently large yields (4.) completing the proof. 5. Bounds on v 1 Support in the Neutral Case In this section we consider only the neutral case. Define k f (m ) + v dv, Then (.1) yields ( ) σ ± f (m ) + v ± v 1 dv. kdx edx e(, x)dx C. ] dy

11 Also (1.3) yields Large time behavior for Vlasov Maxwell 71 [σ (τ, x t + τ) + σ + (τ, x + t τ)] dτ C. These bounds are used in the following. Lemma 5.1. For all t x R f dv C kσ, (5.1) f dv C kσ +. (5.) Proof. We will show (5.1); the proof of (5.) is similar. For any R f dv f dv + Ck. 1 + R For v R, (m ) + v v 1 so f dv v R v R (m ) + v (m ) + v + v 1 (m ) (m ) + v C 1 + R f C ( ) 1 + R (m ) + v v 1 dv + Ck 1 + R C 1 + R σ + Ck 1 + R. If < σ k, taking R k σ 1 leads to (5.1). If k < σ, then f dv < Ck < C kσ, if σ, then f dv kσ. In all cases (5.1) holds so the proof is complete.

12 7 Robert Glassey, Stephen Pankavich, Jack Schaeffer Consider a characteristic (X(s), V (s)) (X (s,, x, v), V (s,, x, v)), of f (defined in (1.)) along which f (s, X(s), V (s)). The idea of the following estimate is that as long as V 1 is large, the integration V 1 (t) V 1 (t ) + e ( E 1 + ˆV (s)b ) (s,x(s)) ds is nearly integration on a light cone (1.3) can be used to obtain an improved estimate. Define { C 1 sup v 1 : t [, 1], x R, v R with } f (t, x, v), suppose that t > V 1 (t) > C 1. Define { sup τ (, t] : V 1 (s) 1 } V 1(t) for all s [t τ, t]. Note that so t > 1 V 1 (t ) 1 V 1(t) > C 1, V 1 (t ) 1 V 1(t) (5.3) follows. Define X C (s) X(t) + s t. Using Theorem 1., we have d ds (X C(s) X(s)) 1 ˆV 1 (s) (m ) + V (s) (m ) + V (s) ( (m ) + V (s) + V 1 (s)) C + C (s X(s) + C ) V1 (s). Since s X(s) is increasing, for t s t we have d ds (X C(s) X(s)) C + C (t X(t) + C ) ( 1 V 1(t) ), hence X C (s) X(s) C (t X(t) + C ) V 1 (t). (5.4)

13 Large time behavior for Vlasov Maxwell 73 By (1.3), the Cauchy Schwartz inequality, (5.1), we have E 1 (s, X(s))ds Now letting C + (5.4) (1.3) yield X(s) E 1 (s, X C (s))ds + X C (s) X(s) X(s) X C (s) C kσ dx ds. X(s) X C (s) kdx ds C, S(t) C (t X(t) + C ) V1 (t), σ dx ds X C (s) X(t) t+s(t) X(t) t X(t) t+s(t) X(t) t e f dv dx ds XC (s)+s(t) X C (s) σ (s, y + s)dy ds Hence, the Cauchy Schwartz inequality yields hence Next consider X(s) X C (s) σ dx ds σ (s, y + s)ds dy CS(t). C kσ dx ds C S(t), E 1 (s, X(s))ds C + C S(t). (5.5) ˆV (s)b(s, X(s))ds. Using Theorem 1. we have, for t s t, ˆV (s) C V (s) V 1 (s) C + C s X(s) + C V 1 (s) C t X(t) + C 1 V. 1(t)

14 74 Robert Glassey, Stephen Pankavich, Jack Schaeffer Hence, by (.5) ˆV (s)b(s, X(s)) ds C t X(t) + C. (5.6) V 1 (t) Collecting (5.5) (5.6) yields V 1 (t) V 1 (t ) + with (5.3) this becomes Hence, V e ( E 1 + ˆV (s)b) (s,x(s)) V 1 (t ) + C + C t X(t) + C, V 1 (t) V 1 (t) C + C t X(t) + C. V 1 (t) 1 (t) C V 1 (t) C t X(t) + C, ( V 1 (t) C ) (C t X(t) + C + C ), 4 V 1 (t) C + (C t X(t) + C + C ) 4 C ( ) 1 + (t X(t) + C ) 1 4 Ct 1 (t X(t) + C ) 1 4. Similar estimates may be derived if V 1 (t) < C 1, thus in all cases. Theorem 1.4 follows. V 1 (t) C 1 + Ct 1 (t X(t) + C ) Nonexistence of Steady States Consider the monocharge case first. The dilation identity is d ( ) fxv 1 dv dx + xe Bdx f (v 1ˆv 1 + x (E 1 + ˆv B)) dv dx dt + x [( x B j ) B + E ( x E )] dx fv 1ˆv 1 dv dx + x (ρe 1 + j B) dx [ x x ( B + E ds ) ] + j B dx

15 fv 1ˆv 1 dv dx + Large time behavior for Vlasov Maxwell 75 xρe 1 dx + 1 (B + E ) dx. Let M fdv dx, then for R > C + t, we have M E 1 (t, R) E 1 (t, x) E 1 (t, R) M, for all x. Hence xρe 1 dx 1 R x x E 1dx 1 ( ( M ) ( M ) R R ( R) R R R ) ) E 1 dx. 1 R (( M Hence, for f not identically zero, d ( ) fxv 1 dv dx + xe Bdx dt fv 1ˆv 1 dv dx >, ) E1dx f cannot be a steady solution. Next, consider a steady solution in the neutral case. Note that from (1.1) we have x E so E for all x follows. Next note that d ( f v 1ˆv 1 dv 1 dx E B) v 1 ˆv 1 x f dv ρe 1 j B v 1 e [(E 1 + ˆv B) v1 f + (E ˆv 1 B) v f ] dv ρe 1 j B e f (E 1 + ˆv B) dv ρe 1 j B, hence f v 1ˆv 1 dv E1 B, (6.1) for all x. If E 1 (x) for some x then, since f, f v 1ˆv 1 dv (6.)

16 76 Robert Glassey, Stephen Pankavich, Jack Schaeffer follows thus B(x) f (x, v) for all v. Suppose E 1 (x ) for some x. A contradiction will be derived from this the proof will be complete. Choose a < x b > x such that E 1 (x) on (a, b) E 1 (a) E 1 (b). Consider E 1 (x) > on (a, b). Choose d (a, b) with < E 1(d) e f (d, v)dv. Choose {1,..., N} w R such that f (d, w) > e >. By continuity we may take w 1. Let If w 1 > define (X(s), V (s)) (X (s,, d, w), V (s,, d, w)). T sup {t > : V 1 (s) X(s) b for all s [, t]}. On [, T ), X(s) [a, b] so E 1 (X(s)). From (6.1) it follows that hence that B(X(s)) E 1 (X(s)), V 1 (s) e (E 1 (X(s)) + ˆV (s)b(x(s))), V 1 (s) w 1 >. It follows that T is finite that X(T ) b. Hence, f (b, V (T )) f (d, w) >, which contradicts (6.). If w 1 < define T inf {t < : V 1 (s) X(s) b for all s [t, ]}. It may be shown that T is finite that X(T ) b, which again contradicts (6.). A contradiction may be reached in a similar manner if E 1 < on (a, b) so the proof is complete. References [1] J. Batt, M. Kunze, G. Rein, On the asymptotic behavior of a one-dimensional, monocharged plasma a rescaling method, Advances in Differential Equations, 3 (1998), [] J.R. Burgan, M.R. Feix, E. Fijalkow, A. Munier, Self-Similar asymptotic solutions for a one-dimensional Vlasov beam, J. Plasma Physics, 9 (1983), [3] L. Desvillettes J. Dolbeault, On long time asymptotics of the Vlasov-Poisson- Boltzmann equation, Comm. Partial Differential Equations, 16 (1991),

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