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DOE/ER/03077-181

MF-100

Courant Institute of

Mathematical Sciences

Magneto-Fluid Dynamics Division

Existence and Uniqueness

Theory of the Vlasov Equation

Stephen Wollman

i

XI U.S. Department of Energy Report

â€¢< Plasma Physics

s~. October 1982

NEW YORK UNIVERSITY

New York University

Courant Institute of Mathematical Sciences

Magneto-Fiuid Dynamics Division

Existence and Uniqueness Theory of the Vlasov Equation

Stephen Wollman

October, 1982

U.S. Department of Energy

Contract No. DE-AC02-76ER03077

-1-

Abstract

The purpose of this report is to communicate recent results on the

existence and uniqueness of solutions to the Vlasov equation. Both the

Vlasov-Poisson system and the Vlasov-Maxwell systems are considered.

Chapter 1 is devoted to the Vlasov-Poisson system. A theorem is stated

which gives conditions for the solvability of the system for general

once dif ferentiable initial data having compact support. As an example

of the application of the general theorem the case where the system has

spherical symmetry is considered. A global-in-time existence and

uniqueness theorem is proved for this case. Chapter 2 deals with the

Vlasov-Maxwell system. Here a local-in-time existence and uniqueness

theorem is proved for a general class of initial data having compact

support.

t

A

Table of Contents

Page

Abstract i

Chapter 1 The Vlasov-Poisson System 1

Chapter 2 The Vlasov-Maxwell System 20

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Chapter 1: The Viasov-Poisson system

I. Introduction

The Viasov-Poisson system of equations describes the motion of a

collection of particles in the presence of its own electrostatic (or

gravitational) field. The system under consideration is

a) |4 + W x f +V^^^ =

(1.1) f(x,v,0) = g(x,v)

b) At = -4tt 2 / (f-b)dv

(x,v) - a point in 6-D phase space

f(x,v,t) - phase space distribution function

b(x,v) - fixed background distribution

We assume g,b Ci(R^), i.e., continuously dif f erentiable functions with

bounded support. The support of a function is the region of phase

space outside of which the function is zero. A function with bounded

support goes to zero outside a bounded region of phase space. In this

paper a general theorem is stated which gives conditions for the

solvability of system (1.1). The proof of the theorem is given in

[14]. The bibliography at the end of this chapter contains a set of

references for the classical existence theory of (1.1). An application

of the general theorem is made to the case where the initial data and

hence the solution have spherical symmetry. The result is a

global-in-time existence and uniqueness theorem for the system with

spherical symmetry.

-2-

II. General theorem

The following notation is used

R^ - six dimensional phase space

Cq(R^) - the space of continuous functions with continuous

bounded derivatives on Rr and which have bounded support in R/-.

C (R^x[0,T]) - The space of continuous functions with continuous

bounded derivatives on R/-x[0,T ] ,T-time.

The term "solution" refers to a function that is of class C ,

satisfies (1.1), and which is integrable over phase space in such a way

that appropriate conservation laws (like mass, momentum, energy) can

hold.

Theorem: Let g,b be functions in C^(R 6 ). The solution to (l.la.b)

exists and is unique as an element of C (R^x[0,T]) if and only if the

system admits an a priori bound on the support of C* solutions for t in

the interval [0,T] .

Proof: see [14].

This theorem gives a criterion to determine whether the solution

to (l.la,b) exists and is unique for a particular initial

configuration. To use the theorem let us make the assumption that a C

solution to the problem exists on the interval of time [0,T]. The

theorem tells us that a C 1 solution to the problem necessarily has

bounded support in phase space. We then try to compute a priori what

the bound on the support of such a solution must be as a function of

time. If it is possible to compute such an a priori bound for a

-3-

solution, then the theorem teiis us that the C 1 solution to the system

in fact exists, is unique, and of course has the bound on support

already computed. The important bound to compute is on velocity.

For initial data with bounded support the statement that the

support of the solution remains bounded is equivalent to the statement

that the electric field remains bounded. Therefore, an alternative way

of stating the general theorem is as follows: a C distribution

function that is a solution to (1.1) cannot give rise to an unbounded

electric field in a finite time. To prove existence and uniqueness of

the solution to (1.1) of class CVR^xfO.T]) it is sufficient to show "a

priori" that the electric field remains bounded for t e [0,T].

III. The spherically symmetric Vlasov-Poisson system

An existence and uniqueness theorem for the spherically symmetric

case was first proved by Batt in [2], The proof in [2] is valid for

the gravitational case alone where the spherically symmetric field

always points inward along a radius. In [13] a different proof is

given which makes no use of an inward directed field and is thus valid

for both the gravitational and electrostatic case. The proof in [13]

builds, however, on some of the development in [2], In this paper we

give an independent development of the result in [13].

Let

r = (xf + *l + xÂ§) 1/2

u = (v 2 + v\ + vÂ§) 1/2

C = xjv^ + XÂ£V2 + X3V3

-4-

Let the functions g,b in (l.la,b) be of ciass Cq(R^) and have the form

(3.1) g(x,v) = G(r,u,C) , b(x,v) =H(r,u,?)

Initial and background distributions of this form are termed

spherically symmetric, and solutions to (1.1) with data of this form

reduce to functions in the variables r,u,5, and t and are termed

spherically symmetric. Spherically symmetric solutions are

characterized by the fact that the field remains a function only of

radius and time. Also if (x(t), v(t)) is a trajectory of (1.1) then

x(t) x v (t) = c .

These facts will become evident as we proceed.

Let

n = (x lVl + x 2 v 2 + x 3 v 3 )/ ru = C/ ru .

For points (x,v) e R^ such that |n | t 1 (i.e. , the velocity vector

does not point along the radius) spherically symmetric solutions to

the Vlasov equation can be written in terms of the variables

r = (x^ + x| + x^) 1/2

u = {v\ + v\ + vÂ§) 1/2

a = cos ((xjv. + x 2 v 2 + x-jv 3 )/ru)

Here a is the angle between the position vector and the velocity vector

-5-

and takes on values between and it. In terms of these variables the

Vlasov-Poisson system reduces to the form

N 3F , 9F n , , 3F

a) - â€” + cos a u - â€” - cos a 0(t,r) - â€”

3 1 3 r 3u

3 F

+ (0/u - u/r) sina â€” =

3a

(3.2) F(0) = G

b) 0(t,r) = - ^- j' h(t,F) ? 2 dr

r

h(t,r) = 2tt / / (F-H) sina da u 2 du

where

H(r,u,a) is a fixed background distribution

h(t,r) the charge density,

-0 the radial electric field

The characteristic equations for (3.2) are

r = u cosa

(3.3) u = - 0(t,r) cosa

a = (0/u - u/r) sina

An integral of (3.3) is

(3. A) r(t)u(t) sin(a(t)) = r(0)u(0) sin(a(0)) = c .

Thus if a trajectory of (3.2) (solution to (3.3)) has its initial point

in the region n * 1 then the trajectory remains in the region n * 1.

-6-

That is n M implies r(0)u(0) sin(ct(0)) = c * 0. From (3.4) it follows

that r(t)u(t) sin (a(t)) * for all t and hence the trajectory remains

in the region n * 1 for ail t. In order to prove the result of giobai

existence and uniqueness for the spherically symmetric Viasov-Poisson

system we analyze trajectories in the region n * 1 and compute bounds

on the trajectories in this region. By continuity results for the

region n t 1 are extended to the lines n = 1. By this means an a priori

bound on velocities in the system is computed.

IV. Bounds on the field

Let the initial data G to (3.2a,b) have the bound

< G < D

and G and H have bounds

8tt 2 / / J Gsinada u 2 du r 2 dr Â£

The continuous anti-derivative of g is

G(r) = 1/2 A r 2 -3/2 A 1/3 M 2/3 r < I

G(r) = -M/r r > I

The bound on is

IÂ© | < G'(r)

V. Existence and uniqueness theorem for the spherically symmetric case

Theorem: Let the initial and background distributions g and b to

(l.la.b) be of class Cq(R 6 ) and be of the form (3.1) (i.e., spherically

symmetric). For such spherically symmetric data the solution to

(l.la.b) of class C (R^ x [0,T]) exists and is unique for ail T.

Proof:

We state without proof that for initial and background data of

form (3.1) the solutions to (l.la,b) satisfy a system of the form

(3.2a,b) in the variables r,u,a, and t in the region n * 1. As has been

shown trajectories that originate in the region n * 1 remain in this

region. An a priori bound on the support of the solution to (l.la,b)

is obtained by analyzing the trajectories (solutions to (3.3)) in the

region n t 1. The results are extended by continuity to the lines n =

1.

Let us assume the existence of a solution F to (3.2a,b) on the

interval of time [0,T] which has bounded support in the variables r,u.

We wish to compute a priori what this bound must be. Let

Q(t) = supp F(t) - support of F in r,u,ct space for each t.

p = (r,u,a) - a point in r,u,a space.

E(p) = u - the projection of p on the u axis.

fl(T) = Ufi(t), t e [0,T]

P(T) = su2 E(p) - a bound on the support

pen(T)

in velocity space up to time T.

-9-

Let the constant A(T) satisfy

|h| < 2tt / | F â€” H | sina da u 2 du < A(T) for t e [0,T]

and set

A = A(T) = 4/3ir ff(T) .

We are assuming

8tt 2 / / / |F - Hi sina da u 2 du r 2 dr < M

J

Let

G(r) = 1/2 Ar 2 - 3/2 A 1/3 M 2/3 r < I = (M/A) 1/3

G(r) = -M/r r > I

The bound on the field for t e [0,T] is

|0| < G'(r)

For any two points r^,r2 tne function G(r) satisfies

(5.1) |G( ri ) - G(r 2 )| < 3/2 M 2/3 A 1/3

Let p(t) = (r(t), u(t), a(t)) be a trajectory for t e [0,T] such

that p(0) is in the region t\ t 1. The proof now proceeds through a

sequence of steps to get the estimate

-10-

u 2 (t) < u 2 (0) + 6M 2/3 A 1/3

.) If r) or r < on [t 1 ,t 2 ]C [0,T]

then

|u 2 (t 2 )'- u 2 (t 1 )| < 2 |G(r(t 2 )) - GCrCt^)

as follows:

A. (1/2 u 2 ) = u u = - (t,r) u cosa

dt

= -0 r

Hence

|u 2 (t 2 ) - u 2 (t 1 )| < 2 / |0r|dt

t l

However

if V > |0r| < 4- G(r(t))

at

if r < |0r| < -A G(r(t))

dt

Thus

-11-

.2/a.x _ .2

u z (t 2 ) - u z (t 1 )| s 2 |G(r(t 2 )) - G(r( tl ))

b) At a local maximum of r(t)

u 2 (t) < M 2 / 3 A 1 / 3

as follows:

At a local maximum

r = u cosa = + a = tt /2

sin(a) = 1

r = u cosa - u sina a = - u a <

->â– a >

Hence from

a = [0 /u - u/r] sina we have

0/u - u/r > + u 2 i.e. the field - < 0. From the bounds on |0

u 2 I

The maximum occurs at r = I and

u 2 (t) < M 2 / 3 A 1/3

c) At a local minimum of r(t)

u 2 (t) < u 2 (0) + 3M 2/3 A 1/3

as follows: Let t > be a local minimum and suppose r(t) = 0. From

(3.3) we have that

r = u coscc - u sin(a)a

= - cos 2 a - u sina [ /u - u/r] sina

2

9 9 U 9

4> cos^a - sin^a + â€” sin a

r

2

-i- u -2

+ â€” sin a

At a minimum a = it /2 and assuming r = then

u 2 = I in which

|G(r(t)) - G(r(t Q ))| < M 2/3 A l/3

and from parts a,b and (5.1)

-14-

.2(t) < .*(*â€ž> + *Â»3 A l/3 Â« 3M2/3 A l/3

li) r(0 < t, r(t ) > i in Â»hich from (5.1) and parts a,b

At)Â« u2(t ) + 3M^ A l/3< rf"*" 1

We further refine this estimate.

(5.2) u 2 (t) < u 2 (t ) + 2|G(r(t)) - G(r(t ))|

there

exists tg < t such that either tg = or tg is a local extremum of r(t)

and r does not change sign on [tg,t]. Thus from (5.1) and parts a-c

5.4) u 2 (t) < u 2 (tg) + 2|G(r(t)) - G(r(tg))|

< u 2 (0) + 6M 2/3 A 1/3

-16-

Estimate (5.4) is now used to get an a priori bound on the support

of F in velocity space. In terms of the bound on velocity

P = P(T) for t e [0,T]

the charge density h has the bound

00 "FT o

|h| < 2tt / ( IF - Hi sina da u z du

;

< !Z. D P 3 (T) + B < !?L (D + B)P 3 (T) < ^- CP 3 (T)

(assuming P(T) > 1). Here D = sup|F| ,

oo it

2tt / j |H | sina da u 2 du < B , C = D + B

o o

Therefore let

X(T) = ^L C P 3 (T)

and set

A = A(T) = 4/3 tt A(T) = (-^-) 2 C P 3 (T)

= (^-) 2 c P 3

From (5.4) it then follows that

(5.5) u 2 (t) < u 2 + 6M 2 / 3 {^-) 2 ^ C 1 ' 3 P

-17-

Thls Inequality Is satisfied for any solution to (3.3). In particular

it Is satisfied for p(0) e fl(0), the support of the Initial data G.

Since

P - sup u(t), t e [0,T]

p(0)en(0)

It follows from (5.5) that an inequality for P is

P 2 < P 2 + 6M 2 ' 3 (^-) 2 / 3 C 1 ' 3 P

P Q = supE(p)

pefi(O)

Thus an inequality for P is of the form

P 2 < Pq + B M 2/3 C 1/3 P

where M and C are constants known a priori from the initial and

background distributions. A bound for P is

P = P(T) < P Q + B 2 M 2/3 C 1/3

The bound is independent of T and is therefore uniform for all T. The

uniform bound on support in velocity space leads to a bound on support

in position space. From the general theorem we thus get the

global-in-time existence and uniqueness of the classical solution to

(l.la.b) with data of the form (3.1).

-18-

Ref erences

1. ARSENEV, A. A. , Existence and uniqueness in the small for the

classical solution of a system of Vlasov equations, Akad. Nauk

SSSR Dokl. 28 (1974), 11-12.

2. BATT , J., Global symmetric solution of the initial value problem

of stellar dynamics, J. Differential Equations 25 (1977),

342-364.

3. BATT, J., Recent developments in the mathematical investigation

of the initial value problem of stellar dynamics and plasma,

Ann. Nuclear Energy, to appear.

4. HORST , E. , "Zur Existenz Globaler Klassischer Losungen des

Anf angswertproblems der Stellardynamik", Disseration, Munchen,

1979.

5. HORST, E. , On the existence of global classical solution of

stellar dynamics, in "Mathematical Problems in the Kinetic Theory

of Gases" (D.C. Pack and H. Neunzert, Eds.), Frankurt, 1980.

6. HORST, E. , On the classical solution of the initial value problem

for the unmodified non-linear Vlasov equation I, Math. Meth. in

the Appi. Sci. 3 (1981), 229-248.

7. HORST, E. , On the classical solution of the initial value problem

for the unmodified non-linear Vlasov equation II, Math. Meth. in

the Appl. Sci. 4 (1982), 19-32.

-19-

8. IORDANSKII, S.V. , The cauchy problem for the kinetic equation of

plasma, Amer. Math. Soc. Transl. Ser. 2 35 (1964), 351-363.

9. KURTH, R. , Das anf angswertproblem der stellar dynamik, Z. Astrophys,

30 (1952), 213-229.

10. UKAI, S. and OKABE , T. , On classical solution in the iarge in time

of two-dimensional Vlasov's equation, Osaka J. Math. 15 (1978),

245-261.

11. WOLLMAN, S., "Classical Solutions to the Viasov-Poisson System of

Equations", Technical Report No. 329, Department of Mathematics,

University of New Mexico, 1977.

12. WOLLMAN, S. , Global-in-time solutions of the two-dimensional

Viasov-Poisson system, Comm. Pure Appl. Math. 33 (1980),

173-197.

13. WOLLMAN, S. , The spherically symmetric Viasov-Poisson system,

J. Differential Equations 35 (1) (1980), 30-35.

14. WOLLMAN, S. , Existence and uniqueness theory of the Viasov-Poisson

system with application to the problem with cylindrical symmetry,

J. Math. Anal. Appl. 89 (1982), to appear.

-20-

Chapter 2: The Vlasov-Maxwell system

I. Introduction

The purpose of the present paper is to give a local exis-

tence and uniqueness theorem for the Vlasov-Maxwell system cf

equations. The Vlasov-Maxwell system is one of the fundamental

equations underlying the kinetic theory of plasma. It is a

non-linear integro-dif f erential system which describes the

motion of a collection of charged particles in the presence of

its own self generated electro-magnetic field. Under discussion

is the following initial value problem

a) |t+W f- (E + vx b)Â«V f = 0, f (0) = g

3t x v

(1.1)

b) || - V x B = 4tt

vf dv , ~ + V x e =

o t

where

E(0) = E . B(0) = B^

o o

c) V'E = - 4tt

o

gdv , V â€¢ B =0

o

E = (E 1 ,E 2 ,E 3 ) , B = (B 1# B 2 ,B )

x = (x 1 ,x 2 ,x 3 ) , v = ( v 1 ' v 2 ' v 3 )

The function, f, gives the number density of particles in phase

space. The functions E, B are the electric and magnetic fields

produced by the distribution, f. The system is internally con-

sistent in that the distribution, f, evolves in time under the

-21-

influence of the fields E, B, which are in turn being generated

by f. Given initial condition (1.1c) it is easy to see that

the solution (1.1) satisfies the additional equations

VÂ«E = - 4tt fdv , VÂ«B = for t > .

In the present context f represents a single species of charged

particles, say electrons. Normally, a plasma is comprised of

multiple species each satisfying a transport equation of the

type (1.1a) and each contributing to the charge and current

terms on the right side of equation (l.lb,c). The analysis

given in this paper can be modified to deal with such an expanded

multiple species system of equations. For the sake of simplicity

we deal, however, with the single species system given by (l.la,b,c)

The existence and uniqueness theorem for (1.1) is produced

by generalizing a theorem of Kato [ 3, theorem II, p. 195] . In [ 3]

Kato proves a local existence and uniqueness theorem for quasi-

linear symmetric hyperbolic systems of equations. In the proof

the author actually generalizes somewhat the type of systems being

considered. The coefficients in the equations are not merely

point wise operators on functions of x and t but are instead

operators which are point wise in t but which are non-local in

x â‚¬ R mapping functions of R into functions on R . In the pre-

n n n r

sent paper we further generalize the result in [ 3] so that the co-

efficients are non-local in both x and t mapping functions of

R x[ 0,T] into functions on R x[ 0,T] . The starting point of

this analysis is the linear theorem [3, Theorem I, p. 189] for

linear symmetric hyperbolic systems. This theorem is used as

-22-

in [ 3] to prove local existence and uniqueness for a general sys-

tem with operator coefficients of the type mentioned above.

Solutions are obtained as continuous mappings of t into classes of

Sobolev spaces on R . The proof given is in fact a modification

of that in [ 3] . Having generalized Kato's work we can then give

a proof for local existence and uniqueness for the Vlasov-Maxwell

system. The proof we give is for an initial distribution with

compact support. For this class of data the Vlasov-Maxwell sys-

tem can be put into a form so that the general theorem applies.

As far as we know the existence and uniqueness theorem pre-

sented in this paper is the first such result obtained for the

fully non-linear three-dimensional Vlasov-Maxwell system. Pre-

viously some results have been obtained for the system in lower

dimensions and with various additional modifications or restric-

tions, [1], [ 2] , [ 4] .

-23-

II. Function spaces and notation

We are adopting as much as possible the notation of [3], Sore chanaes

in notation, however, are made.

Rj^ - m dimensional Euclidean space

P - A real or complex Hilbert space which in the present context

will be identified with ^ for some value of m.

m

For x 6 IV , |x| = y^Xi 2 ) 1 / 2

u(R m ) - a real P valued functions defined on ^

L^R^P) - the linear space of P valued functions on R^ such that

u|lo,m= ( 1 |u| 2 dx)l/ 2 <

H s (R m ,p) - the space of P value functions on ^ for which the

distributions derivatives of order

MF-100

Courant Institute of

Mathematical Sciences

Magneto-Fluid Dynamics Division

Existence and Uniqueness

Theory of the Vlasov Equation

Stephen Wollman

i

XI U.S. Department of Energy Report

â€¢< Plasma Physics

s~. October 1982

NEW YORK UNIVERSITY

New York University

Courant Institute of Mathematical Sciences

Magneto-Fiuid Dynamics Division

Existence and Uniqueness Theory of the Vlasov Equation

Stephen Wollman

October, 1982

U.S. Department of Energy

Contract No. DE-AC02-76ER03077

-1-

Abstract

The purpose of this report is to communicate recent results on the

existence and uniqueness of solutions to the Vlasov equation. Both the

Vlasov-Poisson system and the Vlasov-Maxwell systems are considered.

Chapter 1 is devoted to the Vlasov-Poisson system. A theorem is stated

which gives conditions for the solvability of the system for general

once dif ferentiable initial data having compact support. As an example

of the application of the general theorem the case where the system has

spherical symmetry is considered. A global-in-time existence and

uniqueness theorem is proved for this case. Chapter 2 deals with the

Vlasov-Maxwell system. Here a local-in-time existence and uniqueness

theorem is proved for a general class of initial data having compact

support.

t

A

Table of Contents

Page

Abstract i

Chapter 1 The Vlasov-Poisson System 1

Chapter 2 The Vlasov-Maxwell System 20

-1-

Chapter 1: The Viasov-Poisson system

I. Introduction

The Viasov-Poisson system of equations describes the motion of a

collection of particles in the presence of its own electrostatic (or

gravitational) field. The system under consideration is

a) |4 + W x f +V^^^ =

(1.1) f(x,v,0) = g(x,v)

b) At = -4tt 2 / (f-b)dv

(x,v) - a point in 6-D phase space

f(x,v,t) - phase space distribution function

b(x,v) - fixed background distribution

We assume g,b Ci(R^), i.e., continuously dif f erentiable functions with

bounded support. The support of a function is the region of phase

space outside of which the function is zero. A function with bounded

support goes to zero outside a bounded region of phase space. In this

paper a general theorem is stated which gives conditions for the

solvability of system (1.1). The proof of the theorem is given in

[14]. The bibliography at the end of this chapter contains a set of

references for the classical existence theory of (1.1). An application

of the general theorem is made to the case where the initial data and

hence the solution have spherical symmetry. The result is a

global-in-time existence and uniqueness theorem for the system with

spherical symmetry.

-2-

II. General theorem

The following notation is used

R^ - six dimensional phase space

Cq(R^) - the space of continuous functions with continuous

bounded derivatives on Rr and which have bounded support in R/-.

C (R^x[0,T]) - The space of continuous functions with continuous

bounded derivatives on R/-x[0,T ] ,T-time.

The term "solution" refers to a function that is of class C ,

satisfies (1.1), and which is integrable over phase space in such a way

that appropriate conservation laws (like mass, momentum, energy) can

hold.

Theorem: Let g,b be functions in C^(R 6 ). The solution to (l.la.b)

exists and is unique as an element of C (R^x[0,T]) if and only if the

system admits an a priori bound on the support of C* solutions for t in

the interval [0,T] .

Proof: see [14].

This theorem gives a criterion to determine whether the solution

to (l.la,b) exists and is unique for a particular initial

configuration. To use the theorem let us make the assumption that a C

solution to the problem exists on the interval of time [0,T]. The

theorem tells us that a C 1 solution to the problem necessarily has

bounded support in phase space. We then try to compute a priori what

the bound on the support of such a solution must be as a function of

time. If it is possible to compute such an a priori bound for a

-3-

solution, then the theorem teiis us that the C 1 solution to the system

in fact exists, is unique, and of course has the bound on support

already computed. The important bound to compute is on velocity.

For initial data with bounded support the statement that the

support of the solution remains bounded is equivalent to the statement

that the electric field remains bounded. Therefore, an alternative way

of stating the general theorem is as follows: a C distribution

function that is a solution to (1.1) cannot give rise to an unbounded

electric field in a finite time. To prove existence and uniqueness of

the solution to (1.1) of class CVR^xfO.T]) it is sufficient to show "a

priori" that the electric field remains bounded for t e [0,T].

III. The spherically symmetric Vlasov-Poisson system

An existence and uniqueness theorem for the spherically symmetric

case was first proved by Batt in [2], The proof in [2] is valid for

the gravitational case alone where the spherically symmetric field

always points inward along a radius. In [13] a different proof is

given which makes no use of an inward directed field and is thus valid

for both the gravitational and electrostatic case. The proof in [13]

builds, however, on some of the development in [2], In this paper we

give an independent development of the result in [13].

Let

r = (xf + *l + xÂ§) 1/2

u = (v 2 + v\ + vÂ§) 1/2

C = xjv^ + XÂ£V2 + X3V3

-4-

Let the functions g,b in (l.la,b) be of ciass Cq(R^) and have the form

(3.1) g(x,v) = G(r,u,C) , b(x,v) =H(r,u,?)

Initial and background distributions of this form are termed

spherically symmetric, and solutions to (1.1) with data of this form

reduce to functions in the variables r,u,5, and t and are termed

spherically symmetric. Spherically symmetric solutions are

characterized by the fact that the field remains a function only of

radius and time. Also if (x(t), v(t)) is a trajectory of (1.1) then

x(t) x v (t) = c .

These facts will become evident as we proceed.

Let

n = (x lVl + x 2 v 2 + x 3 v 3 )/ ru = C/ ru .

For points (x,v) e R^ such that |n | t 1 (i.e. , the velocity vector

does not point along the radius) spherically symmetric solutions to

the Vlasov equation can be written in terms of the variables

r = (x^ + x| + x^) 1/2

u = {v\ + v\ + vÂ§) 1/2

a = cos ((xjv. + x 2 v 2 + x-jv 3 )/ru)

Here a is the angle between the position vector and the velocity vector

-5-

and takes on values between and it. In terms of these variables the

Vlasov-Poisson system reduces to the form

N 3F , 9F n , , 3F

a) - â€” + cos a u - â€” - cos a 0(t,r) - â€”

3 1 3 r 3u

3 F

+ (0/u - u/r) sina â€” =

3a

(3.2) F(0) = G

b) 0(t,r) = - ^- j' h(t,F) ? 2 dr

r

h(t,r) = 2tt / / (F-H) sina da u 2 du

where

H(r,u,a) is a fixed background distribution

h(t,r) the charge density,

-0 the radial electric field

The characteristic equations for (3.2) are

r = u cosa

(3.3) u = - 0(t,r) cosa

a = (0/u - u/r) sina

An integral of (3.3) is

(3. A) r(t)u(t) sin(a(t)) = r(0)u(0) sin(a(0)) = c .

Thus if a trajectory of (3.2) (solution to (3.3)) has its initial point

in the region n * 1 then the trajectory remains in the region n * 1.

-6-

That is n M implies r(0)u(0) sin(ct(0)) = c * 0. From (3.4) it follows

that r(t)u(t) sin (a(t)) * for all t and hence the trajectory remains

in the region n * 1 for ail t. In order to prove the result of giobai

existence and uniqueness for the spherically symmetric Viasov-Poisson

system we analyze trajectories in the region n * 1 and compute bounds

on the trajectories in this region. By continuity results for the

region n t 1 are extended to the lines n = 1. By this means an a priori

bound on velocities in the system is computed.

IV. Bounds on the field

Let the initial data G to (3.2a,b) have the bound

< G < D

and G and H have bounds

8tt 2 / / J Gsinada u 2 du r 2 dr Â£

The continuous anti-derivative of g is

G(r) = 1/2 A r 2 -3/2 A 1/3 M 2/3 r < I

G(r) = -M/r r > I

The bound on is

IÂ© | < G'(r)

V. Existence and uniqueness theorem for the spherically symmetric case

Theorem: Let the initial and background distributions g and b to

(l.la.b) be of class Cq(R 6 ) and be of the form (3.1) (i.e., spherically

symmetric). For such spherically symmetric data the solution to

(l.la.b) of class C (R^ x [0,T]) exists and is unique for ail T.

Proof:

We state without proof that for initial and background data of

form (3.1) the solutions to (l.la,b) satisfy a system of the form

(3.2a,b) in the variables r,u,a, and t in the region n * 1. As has been

shown trajectories that originate in the region n * 1 remain in this

region. An a priori bound on the support of the solution to (l.la,b)

is obtained by analyzing the trajectories (solutions to (3.3)) in the

region n t 1. The results are extended by continuity to the lines n =

1.

Let us assume the existence of a solution F to (3.2a,b) on the

interval of time [0,T] which has bounded support in the variables r,u.

We wish to compute a priori what this bound must be. Let

Q(t) = supp F(t) - support of F in r,u,ct space for each t.

p = (r,u,a) - a point in r,u,a space.

E(p) = u - the projection of p on the u axis.

fl(T) = Ufi(t), t e [0,T]

P(T) = su2 E(p) - a bound on the support

pen(T)

in velocity space up to time T.

-9-

Let the constant A(T) satisfy

|h| < 2tt / | F â€” H | sina da u 2 du < A(T) for t e [0,T]

and set

A = A(T) = 4/3ir ff(T) .

We are assuming

8tt 2 / / / |F - Hi sina da u 2 du r 2 dr < M

J

Let

G(r) = 1/2 Ar 2 - 3/2 A 1/3 M 2/3 r < I = (M/A) 1/3

G(r) = -M/r r > I

The bound on the field for t e [0,T] is

|0| < G'(r)

For any two points r^,r2 tne function G(r) satisfies

(5.1) |G( ri ) - G(r 2 )| < 3/2 M 2/3 A 1/3

Let p(t) = (r(t), u(t), a(t)) be a trajectory for t e [0,T] such

that p(0) is in the region t\ t 1. The proof now proceeds through a

sequence of steps to get the estimate

-10-

u 2 (t) < u 2 (0) + 6M 2/3 A 1/3

.) If r) or r < on [t 1 ,t 2 ]C [0,T]

then

|u 2 (t 2 )'- u 2 (t 1 )| < 2 |G(r(t 2 )) - GCrCt^)

as follows:

A. (1/2 u 2 ) = u u = - (t,r) u cosa

dt

= -0 r

Hence

|u 2 (t 2 ) - u 2 (t 1 )| < 2 / |0r|dt

t l

However

if V > |0r| < 4- G(r(t))

at

if r < |0r| < -A G(r(t))

dt

Thus

-11-

.2/a.x _ .2

u z (t 2 ) - u z (t 1 )| s 2 |G(r(t 2 )) - G(r( tl ))

b) At a local maximum of r(t)

u 2 (t) < M 2 / 3 A 1 / 3

as follows:

At a local maximum

r = u cosa = + a = tt /2

sin(a) = 1

r = u cosa - u sina a = - u a <

->â– a >

Hence from

a = [0 /u - u/r] sina we have

0/u - u/r > + u 2 i.e. the field - < 0. From the bounds on |0

u 2 I

The maximum occurs at r = I and

u 2 (t) < M 2 / 3 A 1/3

c) At a local minimum of r(t)

u 2 (t) < u 2 (0) + 3M 2/3 A 1/3

as follows: Let t > be a local minimum and suppose r(t) = 0. From

(3.3) we have that

r = u coscc - u sin(a)a

= - cos 2 a - u sina [ /u - u/r] sina

2

9 9 U 9

4> cos^a - sin^a + â€” sin a

r

2

-i- u -2

+ â€” sin a

At a minimum a = it /2 and assuming r = then

u 2 = I in which

|G(r(t)) - G(r(t Q ))| < M 2/3 A l/3

and from parts a,b and (5.1)

-14-

.2(t) < .*(*â€ž> + *Â»3 A l/3 Â« 3M2/3 A l/3

li) r(0 < t, r(t ) > i in Â»hich from (5.1) and parts a,b

At)Â« u2(t ) + 3M^ A l/3< rf"*" 1

We further refine this estimate.

(5.2) u 2 (t) < u 2 (t ) + 2|G(r(t)) - G(r(t ))|

there

exists tg < t such that either tg = or tg is a local extremum of r(t)

and r does not change sign on [tg,t]. Thus from (5.1) and parts a-c

5.4) u 2 (t) < u 2 (tg) + 2|G(r(t)) - G(r(tg))|

< u 2 (0) + 6M 2/3 A 1/3

-16-

Estimate (5.4) is now used to get an a priori bound on the support

of F in velocity space. In terms of the bound on velocity

P = P(T) for t e [0,T]

the charge density h has the bound

00 "FT o

|h| < 2tt / ( IF - Hi sina da u z du

;

< !Z. D P 3 (T) + B < !?L (D + B)P 3 (T) < ^- CP 3 (T)

(assuming P(T) > 1). Here D = sup|F| ,

oo it

2tt / j |H | sina da u 2 du < B , C = D + B

o o

Therefore let

X(T) = ^L C P 3 (T)

and set

A = A(T) = 4/3 tt A(T) = (-^-) 2 C P 3 (T)

= (^-) 2 c P 3

From (5.4) it then follows that

(5.5) u 2 (t) < u 2 + 6M 2 / 3 {^-) 2 ^ C 1 ' 3 P

-17-

Thls Inequality Is satisfied for any solution to (3.3). In particular

it Is satisfied for p(0) e fl(0), the support of the Initial data G.

Since

P - sup u(t), t e [0,T]

p(0)en(0)

It follows from (5.5) that an inequality for P is

P 2 < P 2 + 6M 2 ' 3 (^-) 2 / 3 C 1 ' 3 P

P Q = supE(p)

pefi(O)

Thus an inequality for P is of the form

P 2 < Pq + B M 2/3 C 1/3 P

where M and C are constants known a priori from the initial and

background distributions. A bound for P is

P = P(T) < P Q + B 2 M 2/3 C 1/3

The bound is independent of T and is therefore uniform for all T. The

uniform bound on support in velocity space leads to a bound on support

in position space. From the general theorem we thus get the

global-in-time existence and uniqueness of the classical solution to

(l.la.b) with data of the form (3.1).

-18-

Ref erences

1. ARSENEV, A. A. , Existence and uniqueness in the small for the

classical solution of a system of Vlasov equations, Akad. Nauk

SSSR Dokl. 28 (1974), 11-12.

2. BATT , J., Global symmetric solution of the initial value problem

of stellar dynamics, J. Differential Equations 25 (1977),

342-364.

3. BATT, J., Recent developments in the mathematical investigation

of the initial value problem of stellar dynamics and plasma,

Ann. Nuclear Energy, to appear.

4. HORST , E. , "Zur Existenz Globaler Klassischer Losungen des

Anf angswertproblems der Stellardynamik", Disseration, Munchen,

1979.

5. HORST, E. , On the existence of global classical solution of

stellar dynamics, in "Mathematical Problems in the Kinetic Theory

of Gases" (D.C. Pack and H. Neunzert, Eds.), Frankurt, 1980.

6. HORST, E. , On the classical solution of the initial value problem

for the unmodified non-linear Vlasov equation I, Math. Meth. in

the Appi. Sci. 3 (1981), 229-248.

7. HORST, E. , On the classical solution of the initial value problem

for the unmodified non-linear Vlasov equation II, Math. Meth. in

the Appl. Sci. 4 (1982), 19-32.

-19-

8. IORDANSKII, S.V. , The cauchy problem for the kinetic equation of

plasma, Amer. Math. Soc. Transl. Ser. 2 35 (1964), 351-363.

9. KURTH, R. , Das anf angswertproblem der stellar dynamik, Z. Astrophys,

30 (1952), 213-229.

10. UKAI, S. and OKABE , T. , On classical solution in the iarge in time

of two-dimensional Vlasov's equation, Osaka J. Math. 15 (1978),

245-261.

11. WOLLMAN, S., "Classical Solutions to the Viasov-Poisson System of

Equations", Technical Report No. 329, Department of Mathematics,

University of New Mexico, 1977.

12. WOLLMAN, S. , Global-in-time solutions of the two-dimensional

Viasov-Poisson system, Comm. Pure Appl. Math. 33 (1980),

173-197.

13. WOLLMAN, S. , The spherically symmetric Viasov-Poisson system,

J. Differential Equations 35 (1) (1980), 30-35.

14. WOLLMAN, S. , Existence and uniqueness theory of the Viasov-Poisson

system with application to the problem with cylindrical symmetry,

J. Math. Anal. Appl. 89 (1982), to appear.

-20-

Chapter 2: The Vlasov-Maxwell system

I. Introduction

The purpose of the present paper is to give a local exis-

tence and uniqueness theorem for the Vlasov-Maxwell system cf

equations. The Vlasov-Maxwell system is one of the fundamental

equations underlying the kinetic theory of plasma. It is a

non-linear integro-dif f erential system which describes the

motion of a collection of charged particles in the presence of

its own self generated electro-magnetic field. Under discussion

is the following initial value problem

a) |t+W f- (E + vx b)Â«V f = 0, f (0) = g

3t x v

(1.1)

b) || - V x B = 4tt

vf dv , ~ + V x e =

o t

where

E(0) = E . B(0) = B^

o o

c) V'E = - 4tt

o

gdv , V â€¢ B =0

o

E = (E 1 ,E 2 ,E 3 ) , B = (B 1# B 2 ,B )

x = (x 1 ,x 2 ,x 3 ) , v = ( v 1 ' v 2 ' v 3 )

The function, f, gives the number density of particles in phase

space. The functions E, B are the electric and magnetic fields

produced by the distribution, f. The system is internally con-

sistent in that the distribution, f, evolves in time under the

-21-

influence of the fields E, B, which are in turn being generated

by f. Given initial condition (1.1c) it is easy to see that

the solution (1.1) satisfies the additional equations

VÂ«E = - 4tt fdv , VÂ«B = for t > .

In the present context f represents a single species of charged

particles, say electrons. Normally, a plasma is comprised of

multiple species each satisfying a transport equation of the

type (1.1a) and each contributing to the charge and current

terms on the right side of equation (l.lb,c). The analysis

given in this paper can be modified to deal with such an expanded

multiple species system of equations. For the sake of simplicity

we deal, however, with the single species system given by (l.la,b,c)

The existence and uniqueness theorem for (1.1) is produced

by generalizing a theorem of Kato [ 3, theorem II, p. 195] . In [ 3]

Kato proves a local existence and uniqueness theorem for quasi-

linear symmetric hyperbolic systems of equations. In the proof

the author actually generalizes somewhat the type of systems being

considered. The coefficients in the equations are not merely

point wise operators on functions of x and t but are instead

operators which are point wise in t but which are non-local in

x â‚¬ R mapping functions of R into functions on R . In the pre-

n n n r

sent paper we further generalize the result in [ 3] so that the co-

efficients are non-local in both x and t mapping functions of

R x[ 0,T] into functions on R x[ 0,T] . The starting point of

this analysis is the linear theorem [3, Theorem I, p. 189] for

linear symmetric hyperbolic systems. This theorem is used as

-22-

in [ 3] to prove local existence and uniqueness for a general sys-

tem with operator coefficients of the type mentioned above.

Solutions are obtained as continuous mappings of t into classes of

Sobolev spaces on R . The proof given is in fact a modification

of that in [ 3] . Having generalized Kato's work we can then give

a proof for local existence and uniqueness for the Vlasov-Maxwell

system. The proof we give is for an initial distribution with

compact support. For this class of data the Vlasov-Maxwell sys-

tem can be put into a form so that the general theorem applies.

As far as we know the existence and uniqueness theorem pre-

sented in this paper is the first such result obtained for the

fully non-linear three-dimensional Vlasov-Maxwell system. Pre-

viously some results have been obtained for the system in lower

dimensions and with various additional modifications or restric-

tions, [1], [ 2] , [ 4] .

-23-

II. Function spaces and notation

We are adopting as much as possible the notation of [3], Sore chanaes

in notation, however, are made.

Rj^ - m dimensional Euclidean space

P - A real or complex Hilbert space which in the present context

will be identified with ^ for some value of m.

m

For x 6 IV , |x| = y^Xi 2 ) 1 / 2

u(R m ) - a real P valued functions defined on ^

L^R^P) - the linear space of P valued functions on R^ such that

u|lo,m= ( 1 |u| 2 dx)l/ 2 <

H s (R m ,p) - the space of P value functions on ^ for which the

distributions derivatives of order

1 2

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