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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



•< 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



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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.



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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



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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 + x§) 1/2
u = (v 2 + v\ + v§) 1/2
C = xjv^ + X£V2 + X3V3



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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\ + v§) 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



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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) - — + cos a u - — - cos a 0(t,r) - —

3 1 3 r 3u



3 F
+ (0/u - u/r) sina — =

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.



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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 £

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

I© | < 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.



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Let the constant A(T) satisfy

|h| < 2tt / | F — 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



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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



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.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 <
->■ 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 + — sin a

r



2

-i- u -2

+ — 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)



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.2(t) < .*(*„> + *»3 A l/3 « 3M2/3 A l/3
li) r(0 < t, r(t ) > i in »hich from (5.1) and parts a,b
At)« 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



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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



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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).



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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.



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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.



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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)«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 • 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



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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

V«E = - 4tt fdv , V«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 € 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



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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] .



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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

Online LibraryStephen WollmanExistence and uniqueness theory of the Vlasov equation → online text (page 1 of 2)