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Global solutions for dynamical systems modeling motion of charged particles
Nonlintwr
Analysis,
Theory,
Mehods
Pergamon
Lb Applications, Vol. 30, No. 7, pp. 46634614. 1997 Proc. 2nd World Congress of Nonlinear Analysrs Q 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
PII: so362.546X(97)00234-4
GLOBAL SOLUTIONS MODELING MOTION
FOR DYNAMICAL OF CHARGED
SYSTEMS PARTICLES
CHAOCHENGHUANG* ‘Department
of M&uxnstics
Dayton.
OH 45435.
nod USA,
Statistics,
Iknail:
Wright
State
University
chuang0math.wright.eddu
1. INTRODUCTION
Consider
the following
non-local
dynamical
system in R” :
(14 (W (1.3) with
the initial
and boundary
conditions
cp (z, t) (ln IzI)-’
is bounded
v(z,t)-0
as IzI-+oo
2C1(2,o)=f,
1Lt(?O)=tio((x),
as 1x1 ---) 00 for n. = 2 forn23
(1.4) (1.5)
where X, p are two constants, 1L-l is the inverse of the mapping T,!J(., t) for f&d t, J (.) is the Jacobian, and xa, is the characteristic function of a bounded domain fro. The system (l.l)-(1.5) models motion of a cloud of electrically charged particles under their self-induced electric field [2]. In this model, $ is the particle trajectory, ‘p is the electric potential generated by the particle charges, P is the particle density, and PO is the initial density defined for all 5 in R”. The initial condition in (1.5) describe the physical situation that initially the particle is accumulated in a bounded region. The problem (l.l)-( 1.5) can be formulated as a non-local integredifferential equation (see $2). By using methods in [8], one may be able to study existence of weak solutions 4 in the Sobolev spaces W’pP. In this paper, we are interested in studying existence of a solution, in certain classical sense, for system (l.l)-( 1.5) for all t > 0. We call such a solution a global solution. Existence of uniqueness of solutions for the system for a short time have been investigated in [2]. However, there are examples [2] that show that a solution of the system in general will blow up at finite time. In the case X = 0, n = 3, it is shown in [2] that a global solution exists if the initial data are nearly radially symmetric. We shall show that this result will remain true for any constants X and p. In the special case that /I = 0, we shall establish global existence for any initial data.
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We finally point out that the techniques used to study more general system
developed
to handle
system (l.l)-(
1.5) can also be
p$+$ ati= A(hx,t) ?cl(x, 0)= 2,g (x,0)=lclo (4
0.6) (1.7)
where A is a non-local operator. Many evolution problems, such as vortex dynamics for inviscid fluids and some free boundary problems can be formulated as system (1.6) and (1.7). The paper is organized as follows. After introducing some preliminaries in Section 2, we present some global existence results for system (l.l)-(1.5) in Section 3. Section 4 is devoted to the special case where p = 0. Finally in Section 5, we shall provide with several applications of system (1.6) and (1.7) to vortex dynamics and free boundary problems. 2. PRELIMINARIES
AND
We begin with reformulating system (l.l)-(1.5) formula, it follows from the boundary condition as
LOCAL
EXISTENCE
m ’ t o an integrodifferential equation. By Green’s (1.4) that the solution of (1.2) can be represented
p(x,t)=-jG(x-t)Pdz Rn where G is the Green’s
where
function
for the Laplace
l/cc is the area of the unit sphere.
in R”. Hence
operator
Set fk=ti(Qo,t),
the particle
region at time t. It then follows that system (l.l)-(1.5)
p (x,t) = xn,fiW)J qqx,0) = 2, y Notice that when P = Po($-‘),I(@-‘)
(lq
(2.2)
= Ilo (x).
is H”ld o er continuous,
= -du
becomes
(2.3)
one can verify that, for x E Q,
- P(x,t)I, Jv, tip (64d.z
(2.4)
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where the symbol P, means the principal values of the followed integrals and I is the identity matrix. This formula will be frequently used later on. For convenience, we shall omit the notation Pv. All singular integrals in the paper should be understood as their principal values. For any function f (x, t) is defined in Gt for t < T, where Gt C_ R” depends on t, we shall use the notation f E C,m+a (Gc) to specify that f (., t) E Cmfa (Gt) for any fixed t. Denote by If @N,+, and Ilf (411m+a 7respectively, the Holder semi-norm and Holder norm in x defined by
If WI m+o,Gt Ilf (al m+o,Gt
=
@f(x,t) sup =#vEGt, IPI=m
- @f(y,t) Ix-YT
’
sup (@f (3, t) ( + If W,,+, . - sEGt, ICll
In the case Gt = Rs x {t} with Rs C R”, we denote by ~~o’(1 (fIc x (0, T)) the set of all functions f (x, t) such that f (-, t) E C”“+‘I (no) for any fixed t and f (x, .) E C” ((0, T)) for any fixed x. For convenience, we introduce the following notations
DEFINITION 2.1. We call 1c,(x,t) a C”+O: solution of (2.1)-(2.3) in de x [O,T) if $(x,t) , holds Dtlclht), @ti(x,t) E C 1+cr (no) , $(x, t)-’ (x, t) E C1+a (fit) , for t < T, and (2.1)-(2.3) for 5 E 00. Throughout the paper, we assume that PO E C” (R”) ( $0 (,, t) E C’-+O (Icl,) , PO L 0, and that there exists a function
0s (x) E Cl+= (R”)
such that
iI20 = (190 (x) < O}, 000 (x) # 0 for x E 8%. Existence and uniqueness in [2] PROPOSITION 2.1. The existence result general existence result PROPOSITION 2.2. any t 2 0,
of a C’+cr solution
for system (2.1)-(2.3)
for a short time was established
System (2.1)-(2.3) admits a unique solution for t < T for some T > 0. will remain true for more genera1 system. We conclude this section by a for system (1.6) and (1.7). Suppose that the operator B ($) , mapping from C1+a to C”, satisfies, for
IP ($1(t)lI* 5 k ( JIG-1(t)lll,a,nt + Ilr/,(t)Ill+~,*J 1 where k(r) is a continuous function. Then the system (2.1), (2.3) with P = xii0 (q-l) I3 (+) admits at least one solution for t < T, for some T > 0. Moreover, V2v (., t) , defined by (2.4), is bounded in R”, V*cp (., t) is of CQ in a,, and the following estimates hold
I CIPIO(1 + bnwm) + IPI, d@w)O lIv2~(t~l11,x~ I c(lPl, + lPlo4) (1+ b(wm))l) ? llV2~ (t)lll+Q%
(2.5)
(2.6)
4666
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where c is a universal
constant,
d(!&)
Congress
of Nonlinear
is the diameter,
Analysts
and (2.7)
where B (z, t) = f& ($J- ’ (z, t)) . Proof The estimates (2.5) and (2.6) can be derived by slightly modifying the proof of Lemma 4.1 & 4.2 in [2]. It remains to show existence. For any M, T > 0 to be chosen later on, we define a set K (M, T) of vector value functions in nQ x [O, T) as follows:
K (M,T) = t@bt t) E .@ : lIti Mll~+a,n~I M 1111 by .&no I M,
II, (x,0)= x7tit (x,0) = &I (xl , IW - II I 1/q. Next we define, for p # 0, a mapping
A from K (M, T) to a functional
A(ti)(x,t)=x+ ( 1- ‘iAt” > ‘$,, (x)- ] q
space by
iBp ($)(x,T)eXT/*drds,
0
(2.8)
0
where
In the case p = 0, A ($) is defined as the limit of (2.8) as /J + 0. Since (V?l, - II < I/2, $J-’ (., t) exists and maps Rt onto Re. The mapping thus is well defined. By apply the estimates (2.5) and (2.6), we deduce that, for T < 1, 1 - .c-wr IkW)lll+a,na
2
c+c
(
x
p
)
+ ]
T
0
I
c+c (
1 - ,-WP p x ’ +c]
where c is a constant, i (M) is a continuously similar manner, we can derive
1 /BP (qb) (T)II~+~,~~
ex”f‘d7ds
0
T]k(M)e’T/pdrds 0
increasing
5 c+i(M)T2,
0
function
of M, depending
on k (M) . In a
IIA (+1(x, .I II&5 i WI T’-” and IVA (11) - 11 5 k(M)
2’.
We now choose M such that h(M) 2 2c, 2’ = c/2i (M) . Then A ($J) E K (M, T) . Hence, the mapping A maps K (M, T) into itself. For any @, 4 E K (M, T) . Set Rt = $J (no, t) , fit = 4 (G, t) and define
p(t)= I1L(t)- 4 (t)lo,no’ vp (t)= IvlL0) - 04 w IO&
Second World Congress of Nonlinear Analysts By changing
variables
in the expressions
4667
for BP ($) and BP (4) , we obtain
dz
where J (.) is the Jacobian.
Since V$-’
and 04-l
We write,
are bounded,
for E > 0 to be determined
later,
we have
We now choose E = p (7) . It follows ~B~(~C,(~,T))--B~(~C,(Z,~))I Consequently,
Ic(v~(~)+~(~)lln~(~)l).
we derive (2.9)
l~(~(~,T))-A(W(z.T))l~cj~j(~~(r)+p(T)lln~(T)~)e~~‘~dTdS 0
0
Define a sequence 11, (x, t) by
lclo = 2, &+I
(z, 4 = A (TM (~4 .
Since 11, E K(M,T) , this sequence {&,} is precompact Hence we can select an subsequence, still denote it as {&,} such that q& + This implies
7) in C$“’
that 1c,E K (M, T) (by checking
under the C$7’7-norm for any 7 < o. , and a function 1(, E C.$‘r17 (no x [0, T))
- norm.
from the definitions)
and by (2.9),
l&+1 (5,t) - A ($1b, t)I = IA ($4 b, t) - A ($4(z, t)l 2 CTo:y$T (I wn (t) - w w lo,*,) -+ 02y~T(lb 0) - II (4 IO,RoIln IA (t) - 1c,@I10,Ro I) . Let n + co, we fmd that T/Jis a hxed point for A, i.e., A ($J) = 1/1.The proof of existence is complete.
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3. A GENERAL
GLOBAL
EXISTENCE
THEOREM
In this section, we shall establish global existence for system (l.l)-(1.5) with initial data that are nearly radially symmetric. We consider first the situation when the initial data are radially symmetric, i.e.,
f&3 = Bl,PO(z) =s (14), $0(4 =40(14)5, 40(0)=0. We now look for solutions
T/Jof system (l.l)-(1.5)
in the forms of
P(2,t)=B(12l,t),.~(~,t)=9(l~l,t)r System (l.l)-(1.3)
ti(~d~=4w)j$
then reduces to
a%j alj ^ PT+xat=-g(&r,t),t) g+---=n-lag r -P (T, t)
(1.1’) (1.2’)
4-1 n-1 ( t) rn-’>
P(r, t) = xhl(4-1)J(4-1)(T, It follows that, by integrating
where x1 is the characteristic
the initial
(1.3’)
(1.2’) and changes of variables,
?P(r,t)
a+ --m-11 zP “-‘P(p,t)dp=--+ -= ar J 0
with
(3.1)
function
J
x~P)~~(P)P”-~
dp,
0
of the unit interval
[0, 11. We thus formally
arrive at
condition 4 CT, 0) = r, 2
(T-,0) = &J (7.).
LEMMA 3.1. Suppose that p 1 0, X2 + p2 # 0, and &(r) admits a unique global solution 4 (T, t) for r > 0. Furthermore, (a) if p # 0, X > 0, then the solution satisfies o<&(f,t),
,liSn,rj(r,t)
=m,
(3.3) >_ 0. Then system (3.2) and (3.3) we have the following estimates:
,Iiir&(T,t)=O,
r<$(r,t)
forr,t>O;
(3.4) (3.5)
(b)ifp#O,X
^
0 < q!+ (T, t) , $((T, t) N cl
( t + e-At/P)
r, for t N co,
(3.6)
Second World
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of Nonlinear
Analysts
where cl is a constant depending only on A, p and the initial Proof. Since the right-hand side of (3.2) is Lipschitz in follows from the standard ODE theory that for r > 0, there t < T for some T > 0, and that the solution can be extended hand, by integration, we have
4669
data. 4 for 12, > 0 for any fixed T- > 0, it exists a ur$que solution 4 (r, t) for as long as $J (T, t) > 0. On the other
(3.7) where r h(r)
=
/xd~)ib(~W~
dp.
0
Hence & (r, t) > 0, 4 (T, t) > 0, as long as the solution exists up to t. Consequently, the solution can be extended for all t > 0. Consider now the case A > 0. Since 4 (r, t) is increasing in t, the limit 4 (T, 00) = &II 121(T, t) exists. It follows by taking
t ---) 00 in (3.7) that
Hence, (3.4) holds if we can show & (r, co) = 0. To estimate (3.2). Then, for n > 2, we have
&
p 8 (4t)’ 27 +A (IL), = -gg Hence, by integrating
we multiply
by & on both sides of
(2+).
this equation,
(3.8)
=
^ 2 2h (r) @cl + (n _ 1)/@-2
(>
Dividing (3.8) by e2xt’fi and taking t --+ 00, we obtain and h (7) 5 li$lo n-l?, we deduce from (3.8) that
2h(r) +
(n
-
1)p-2
& (r, co) = 0. Since $ (r, s) 2 4 (T, 0) = T
[l
+ ?i~e2AdP&]
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The inequality (3.5) thus follows. Next, we consider the case X1< 0. Notice that (3.9) keeps unchanged for X < 0. It remains to establish the lower bounds for +. From (3.8), since & > 0, we have
(tit^>2ewP-> (&>2, This establishes the assertion (3.6) for X <*O. The case X = 0 can be argued in a similar manner. We observe that in general, a solution ?I) (T, t) of (3.2) and (3.3) may not be a solution of (l.l’)(1.3’) if & (r,t) = 0 f or some T, t. On the other hand, one can verify that the solution 4 (T, t) of (3.2) and (3.3) can generate a C1+a solution of (3.1’)-(3.3’) (hence a Cl+” solution of (l.l)-(1.5)) via (3.1) if and only if & (r,t) > 0, In the case X = 0, it is shown in [2] that in general the situation 4, (T, t) = 0 may happen even if initially $0 (r) > 0. However, we can show that if Jo 2 0, & (r) = br, and & (r) = p for some constants b,p 2 0, then & (r, t) > 0 for any 0 < T 5 1, t 2 0. More general conditions can also be derived by using the analysis in a similar manner to the proof of Lemma 3.1. Using Lemma 3.1 and techniques in [2], we can now show the following global existence result. THEOREM 3.1. Suppose that PO (11 =
PO
(14) + ~6 (~1, $0 (~1 =
40
(14) i
+ &lb),
dist (&I, &I
I E,
where & and $0 satisfy the conditions in Lemma 3.1, 4 and $1 have compact supports. Suppose also the corresponding solution to (3.2) and (3.3) exists globally. Then, for small E, system (l.l)(1.5) admits a unique global C1+a (no) solution. 4. GLOBALSOLUTIONSWHENp=O
In this section, we consider Jacobian J (@) satisfies
dJ= dt It follows that 5(+)
the case p = 0. For simplicity,
the
-J (Icl)true (V2P (N) = J (@Ip ($) = PO(z).
= 1 f tPo and consequently
p= Hence (2.1)-(2.3)
we take X = 1. From (l.l)-(1.3),
POW) ml (q-l) + 1.
(4.1)
reduces to
(4.2) LEMMA
4.1. Suppose that 1c,is a C1+a solution Cl Iz - yl l’m)
where ,B (t) = exp (-ct)
IIlc,(~,t)--(Y,t)lIczl~-Yl
, cl, c2 and c are constants.
of (5.1). Then p(t)
9
(4.3)
Second World
Proof.
Congress
Since P (z, t) in (4.1) is bounded,
of Nonlinear
467 1
Analysts
we have (see [S])
J
fors,yEat.
iit
(4.3) thus follows from Gronwall’s inequality. LEMMA 4.2. For any T > 0, let & = a/3(T),
where P(t) is defined in the Lemma
4.1. Then
llV@0) II&,mt 5 4”) 9 where c(T) is a constant depending only on the initial Proof. By Lemma 4.1, we know that
IW)lti Since nt = (0 (2, t) = 00 (+-l
data and T.
5 c(T).
(2, t)) < 0} , it follows that VB satisfies dV0 ~+(U.v)V8=-(Vu)TV8,
where u=C()
JGP((z,t) Ix - 4
dz, Vzl=cs
fit
It follows that, by integrating
Jn,
V zKP(z,t)
dz + PI (see (2.4)).
12--I
along $J, VB (11, t) = V&I (z) - j (VU)~ VO ($, s) ds. 0
By replacing cy in (2.5) with &, we find that [VZL[0,~~ Ic(l is defined as in (2.7) with Q = /3~. Hence
IV4-o,n,
+lnl~~d(f%)
+cd(%)l),
where
~c+cj(l+ln/6:d(R~)+cd~~~)~),v~(~)~ods.
6,’
(4.4)
0
By [4, Lemma
Suppose
4.11, we have
that we can show
J(WT VQ(s)(pT 5 c(lVQ(s)lo + IVeb)lp~) (1 +lnlG,TdW + I. Then
by using a generalized
Gronwall’inequality
(see [5]), it follows from (4.4) and (4.5) that
IV0 (s)lo + IV0 (41p~ I c(T).
(4.6)
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This would complete the proof. It remains to establish (4.6). Set
We write
(VU)~
V0 as
(VtL)Tve = cclJ y--$”
P (z, t) VB (2, t) dr + P (2, t) V6’ (x, t)
fit
= co (+--L))T
J b---In fit Jn,lx-4
+ce
=
(VO (x, t) - W(z,
t)) P (2, t) dr + P(z,
(“(x-~~~T~~(~,t)(~(z,t)-~(x,t))dz
t) VB (x, t)
+ce~(x,t)
AcIx-4 J(“(x-z~)TV+,t)dz
Jl+&+&+&
(4.7) By [I, Appendix], the terms Jl, Jz and 53 in (4.7) can be bounded by the left-hand side of (4.6). To estimate J4, we notice that the ith component of (0 (x - .z))~ Ix - ~1~” VB (z, t) can be written as, for n > 2, ui (x - z) . VB (2, t) = Di l~~z;n lx - zy Since A 1x - zI-~+~ DiV =
= =
(z,
t) =
= 0, we obtain 1 (3 - tin-2
VD; lx-‘,I,-2
4( j#i
’ VB
.ve(z,t)= (
Div Ix-zln-2 l
-w)
~W’(z,t) -Dte(z,t)e;) -c Dj~j,x~~l,,~2 Die(~,t) ) j#l (
DjDiix-li”-2)D,B(r.t)00,
-A,x-‘Z(,-2G
1 (x - t(n-2
(DjDjlx_11,-2)D.B(r,t)]
. (Dj0 (t, t) < - Die (~9 t) E’j) 1
where c is the ith unit base vector. Noticing that D#(z,t) t$ - D;B(z,t) C’j is a divergence and tangent to V8, the normal vector to EQ, it follows from the divergence theorem that
J
_ :>:;,,")V@(Z,~)
=C
J (VDj1x-iln-2) j#i fir
free
.(Dje(~,t)e’i-DiB(z,t)~j)=O.
Rt Therefore, 54 = 0. The proof is complete. By applying Lemma 4.2 and modifying the methods used in [5], we can establish global existence. THEOREM 4.1. There exists a unique global C1+O solution for system (5.1). Proof. To reduce the length of the paper, we only outline the proof. By Lemma 4.1 & 4.2, we can derive 1$(x, t)l < c(T) for t 5 T. It then follows that IQ (t)l, 5 c(T) _ Hence the solution can be extend to t 5 T, for any T > 0.
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5. APPLICATIONS In this section, we present two examples to demonstrate that some problems in fluid dynamics some free boundary problems can be treated as non-local dynamical systems (1.6) and (1.7). EXAMPLE 1. Consider dynamics of vortex patches for inviscid Euler systems [7]:
w=vxu, with initial
condition
(vortex
V.u=Q,
patch) who)
By Biot-Savart
and
law, we can represent
=wo(4x&y
the velocity
‘~1in terms of the vorticity
u(z,t)=J K(s-z)w(z,t)dz, K(z)=4
i3
w as
Jf, :y)
R”
The vorticity
system then reduces to
(?J, (~4- 4w(z,t) dz, ti(~0) =z, J8~ w(z) = ~~(~-'(i,t),t)Wg(Vil(z,t))xn.. dll, dt=
where $ is the fluid particle trajectory. This system is investigated solution blows up at t = 2” if and only if
in [4]. It is shown that a Cl+=
/T(IwMo,n,+ P(t)lo,n,)dt= 00. 0
EXAMPLE 2. Consider in type-11 superconductors:
the free boundary
wt = V
problem
(]w] VH)
modeling
evolution
v, = -sign(w)
(54
= 0 on Ft = asZ,, g
on rt,
where n, is the moving domain initiated at L!c, rz is the outward of the moving boundary rt, and the bracket [.] denotes the jump function for the equation (5.1). Then H&t)
=
J nr
regions
in Rt,
AH - H = -w in Rt, AH - H = 0 in R2\f&,
WI = [g]
of mixed-state
K(r-z)w(z,t)
dz.
normal, V, is the normal velocity across rt. Let K(z) be the Green
4614
Second
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Defme J!Jby
dt = -szgn (w) VH (Q, t) , + (z,O) = z. We find that w (2, t) = (J($)-’ formulated as
~0) (q!~-l (z, t)) . It f o 11ows that the free boundary
problem
can be
z =-sign(w) /WC (Ic, (5,t)- 2)(J(?/J-l wo) (11-l (2, t,)dz, ?(, (5,O) =I. Global
existence
of solutions
for this system is established
in [5].
REFERENCES [l] [2] [3] [4] (51 [S] [7] [S]
A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152, 1928 (1993). A. Friedman and C. Huang, Averaged motion of charged particles under their self-induced electric field, Indiana Univ. Math. J. 43, 1167-1225 (1994). D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 3nd edition, SpringerVerlag, New York, 1985. C. Huang, On boundary regularity of vortex patches for 3D incompressible Euler systems, to appear. C. Huang and T. Svobodny, Evolution of Mixed-State Regions in Type-11 Superconductors, to appear. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39, S187-220 (1986). P. G. Saffman, Vortex dynamics, Cambridge, 1992. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.
Analysis,
Theory,
Mehods
Pergamon
Lb Applications, Vol. 30, No. 7, pp. 46634614. 1997 Proc. 2nd World Congress of Nonlinear Analysrs Q 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
PII: so362.546X(97)00234-4
GLOBAL SOLUTIONS MODELING MOTION
FOR DYNAMICAL OF CHARGED
SYSTEMS PARTICLES
CHAOCHENGHUANG* ‘Department
of M&uxnstics
Dayton.
OH 45435.
nod USA,
Statistics,
Iknail:
Wright
State
University
chuang0math.wright.eddu
1. INTRODUCTION
Consider
the following
non-local
dynamical
system in R” :
(14 (W (1.3) with
the initial
and boundary
conditions
cp (z, t) (ln IzI)-’
is bounded
v(z,t)-0
as IzI-+oo
2C1(2,o)=f,
1Lt(?O)=tio((x),
as 1x1 ---) 00 for n. = 2 forn23
(1.4) (1.5)
where X, p are two constants, 1L-l is the inverse of the mapping T,!J(., t) for f&d t, J (.) is the Jacobian, and xa, is the characteristic function of a bounded domain fro. The system (l.l)-(1.5) models motion of a cloud of electrically charged particles under their self-induced electric field [2]. In this model, $ is the particle trajectory, ‘p is the electric potential generated by the particle charges, P is the particle density, and PO is the initial density defined for all 5 in R”. The initial condition in (1.5) describe the physical situation that initially the particle is accumulated in a bounded region. The problem (l.l)-( 1.5) can be formulated as a non-local integredifferential equation (see $2). By using methods in [8], one may be able to study existence of weak solutions 4 in the Sobolev spaces W’pP. In this paper, we are interested in studying existence of a solution, in certain classical sense, for system (l.l)-( 1.5) for all t > 0. We call such a solution a global solution. Existence of uniqueness of solutions for the system for a short time have been investigated in [2]. However, there are examples [2] that show that a solution of the system in general will blow up at finite time. In the case X = 0, n = 3, it is shown in [2] that a global solution exists if the initial data are nearly radially symmetric. We shall show that this result will remain true for any constants X and p. In the special case that /I = 0, we shall establish global existence for any initial data.
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We finally point out that the techniques used to study more general system
developed
to handle
system (l.l)-(
1.5) can also be
p$+$ ati= A(hx,t) ?cl(x, 0)= 2,g (x,0)=lclo (4
0.6) (1.7)
where A is a non-local operator. Many evolution problems, such as vortex dynamics for inviscid fluids and some free boundary problems can be formulated as system (1.6) and (1.7). The paper is organized as follows. After introducing some preliminaries in Section 2, we present some global existence results for system (l.l)-(1.5) in Section 3. Section 4 is devoted to the special case where p = 0. Finally in Section 5, we shall provide with several applications of system (1.6) and (1.7) to vortex dynamics and free boundary problems. 2. PRELIMINARIES
AND
We begin with reformulating system (l.l)-(1.5) formula, it follows from the boundary condition as
LOCAL
EXISTENCE
m ’ t o an integrodifferential equation. By Green’s (1.4) that the solution of (1.2) can be represented
p(x,t)=-jG(x-t)Pdz Rn where G is the Green’s
where
function
for the Laplace
l/cc is the area of the unit sphere.
in R”. Hence
operator
Set fk=ti(Qo,t),
the particle
region at time t. It then follows that system (l.l)-(1.5)
p (x,t) = xn,fiW)J qqx,0) = 2, y Notice that when P = Po($-‘),I(@-‘)
(lq
(2.2)
= Ilo (x).
is H”ld o er continuous,
= -du
becomes
(2.3)
one can verify that, for x E Q,
- P(x,t)I, Jv, tip (64d.z
(2.4)
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where the symbol P, means the principal values of the followed integrals and I is the identity matrix. This formula will be frequently used later on. For convenience, we shall omit the notation Pv. All singular integrals in the paper should be understood as their principal values. For any function f (x, t) is defined in Gt for t < T, where Gt C_ R” depends on t, we shall use the notation f E C,m+a (Gc) to specify that f (., t) E Cmfa (Gt) for any fixed t. Denote by If @N,+, and Ilf (411m+a 7respectively, the Holder semi-norm and Holder norm in x defined by
If WI m+o,Gt Ilf (al m+o,Gt
=
@f(x,t) sup =#vEGt, IPI=m
- @f(y,t) Ix-YT
’
sup (@f (3, t) ( + If W,,+, . - sEGt, ICll
In the case Gt = Rs x {t} with Rs C R”, we denote by ~~o’(1 (fIc x (0, T)) the set of all functions f (x, t) such that f (-, t) E C”“+‘I (no) for any fixed t and f (x, .) E C” ((0, T)) for any fixed x. For convenience, we introduce the following notations
DEFINITION 2.1. We call 1c,(x,t) a C”+O: solution of (2.1)-(2.3) in de x [O,T) if $(x,t) , holds Dtlclht), @ti(x,t) E C 1+cr (no) , $(x, t)-’ (x, t) E C1+a (fit) , for t < T, and (2.1)-(2.3) for 5 E 00. Throughout the paper, we assume that PO E C” (R”) ( $0 (,, t) E C’-+O (Icl,) , PO L 0, and that there exists a function
0s (x) E Cl+= (R”)
such that
iI20 = (190 (x) < O}, 000 (x) # 0 for x E 8%. Existence and uniqueness in [2] PROPOSITION 2.1. The existence result general existence result PROPOSITION 2.2. any t 2 0,
of a C’+cr solution
for system (2.1)-(2.3)
for a short time was established
System (2.1)-(2.3) admits a unique solution for t < T for some T > 0. will remain true for more genera1 system. We conclude this section by a for system (1.6) and (1.7). Suppose that the operator B ($) , mapping from C1+a to C”, satisfies, for
IP ($1(t)lI* 5 k ( JIG-1(t)lll,a,nt + Ilr/,(t)Ill+~,*J 1 where k(r) is a continuous function. Then the system (2.1), (2.3) with P = xii0 (q-l) I3 (+) admits at least one solution for t < T, for some T > 0. Moreover, V2v (., t) , defined by (2.4), is bounded in R”, V*cp (., t) is of CQ in a,, and the following estimates hold
I CIPIO(1 + bnwm) + IPI, d@w)O lIv2~(t~l11,x~ I c(lPl, + lPlo4) (1+ b(wm))l) ? llV2~ (t)lll+Q%
(2.5)
(2.6)
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where c is a universal
constant,
d(!&)
Congress
of Nonlinear
is the diameter,
Analysts
and (2.7)
where B (z, t) = f& ($J- ’ (z, t)) . Proof The estimates (2.5) and (2.6) can be derived by slightly modifying the proof of Lemma 4.1 & 4.2 in [2]. It remains to show existence. For any M, T > 0 to be chosen later on, we define a set K (M, T) of vector value functions in nQ x [O, T) as follows:
K (M,T) = t@bt t) E .@ : lIti Mll~+a,n~I M 1111 by .&no I M,
II, (x,0)= x7tit (x,0) = &I (xl , IW - II I 1/q. Next we define, for p # 0, a mapping
A from K (M, T) to a functional
A(ti)(x,t)=x+ ( 1- ‘iAt” > ‘$,, (x)- ] q
space by
iBp ($)(x,T)eXT/*drds,
0
(2.8)
0
where
In the case p = 0, A ($) is defined as the limit of (2.8) as /J + 0. Since (V?l, - II < I/2, $J-’ (., t) exists and maps Rt onto Re. The mapping thus is well defined. By apply the estimates (2.5) and (2.6), we deduce that, for T < 1, 1 - .c-wr IkW)lll+a,na
2
c+c
(
x
p
)
+ ]
T
0
I
c+c (
1 - ,-WP p x ’ +c]
where c is a constant, i (M) is a continuously similar manner, we can derive
1 /BP (qb) (T)II~+~,~~
ex”f‘d7ds
0
T]k(M)e’T/pdrds 0
increasing
5 c+i(M)T2,
0
function
of M, depending
on k (M) . In a
IIA (+1(x, .I II&5 i WI T’-” and IVA (11) - 11 5 k(M)
2’.
We now choose M such that h(M) 2 2c, 2’ = c/2i (M) . Then A ($J) E K (M, T) . Hence, the mapping A maps K (M, T) into itself. For any @, 4 E K (M, T) . Set Rt = $J (no, t) , fit = 4 (G, t) and define
p(t)= I1L(t)- 4 (t)lo,no’ vp (t)= IvlL0) - 04 w IO&
Second World Congress of Nonlinear Analysts By changing
variables
in the expressions
4667
for BP ($) and BP (4) , we obtain
dz
where J (.) is the Jacobian.
Since V$-’
and 04-l
We write,
are bounded,
for E > 0 to be determined
later,
we have
We now choose E = p (7) . It follows ~B~(~C,(~,T))--B~(~C,(Z,~))I Consequently,
Ic(v~(~)+~(~)lln~(~)l).
we derive (2.9)
l~(~(~,T))-A(W(z.T))l~cj~j(~~(r)+p(T)lln~(T)~)e~~‘~dTdS 0
0
Define a sequence 11, (x, t) by
lclo = 2, &+I
(z, 4 = A (TM (~4 .
Since 11, E K(M,T) , this sequence {&,} is precompact Hence we can select an subsequence, still denote it as {&,} such that q& + This implies
7) in C$“’
that 1c,E K (M, T) (by checking
under the C$7’7-norm for any 7 < o. , and a function 1(, E C.$‘r17 (no x [0, T))
- norm.
from the definitions)
and by (2.9),
l&+1 (5,t) - A ($1b, t)I = IA ($4 b, t) - A ($4(z, t)l 2 CTo:y$T (I wn (t) - w w lo,*,) -+ 02y~T(lb 0) - II (4 IO,RoIln IA (t) - 1c,@I10,Ro I) . Let n + co, we fmd that T/Jis a hxed point for A, i.e., A ($J) = 1/1.The proof of existence is complete.
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3. A GENERAL
GLOBAL
EXISTENCE
THEOREM
In this section, we shall establish global existence for system (l.l)-(1.5) with initial data that are nearly radially symmetric. We consider first the situation when the initial data are radially symmetric, i.e.,
f&3 = Bl,PO(z) =s (14), $0(4 =40(14)5, 40(0)=0. We now look for solutions
T/Jof system (l.l)-(1.5)
in the forms of
P(2,t)=B(12l,t),.~(~,t)=9(l~l,t)r System (l.l)-(1.3)
ti(~d~=4w)j$
then reduces to
a%j alj ^ PT+xat=-g(&r,t),t) g+---=n-lag r -P (T, t)
(1.1’) (1.2’)
4-1 n-1 ( t) rn-’>
P(r, t) = xhl(4-1)J(4-1)(T, It follows that, by integrating
where x1 is the characteristic
the initial
(1.3’)
(1.2’) and changes of variables,
?P(r,t)
a+ --m-11 zP “-‘P(p,t)dp=--+ -= ar J 0
with
(3.1)
function
J
x~P)~~(P)P”-~
dp,
0
of the unit interval
[0, 11. We thus formally
arrive at
condition 4 CT, 0) = r, 2
(T-,0) = &J (7.).
LEMMA 3.1. Suppose that p 1 0, X2 + p2 # 0, and &(r) admits a unique global solution 4 (T, t) for r > 0. Furthermore, (a) if p # 0, X > 0, then the solution satisfies o<&(f,t),
,liSn,rj(r,t)
=m,
(3.3) >_ 0. Then system (3.2) and (3.3) we have the following estimates:
,Iiir&(T,t)=O,
r<$(r,t)
forr,t>O;
(3.4) (3.5)
(b)ifp#O,X
^
0 < q!+ (T, t) , $((T, t) N cl
( t + e-At/P)
r, for t N co,
(3.6)
Second World
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Analysts
where cl is a constant depending only on A, p and the initial Proof. Since the right-hand side of (3.2) is Lipschitz in follows from the standard ODE theory that for r > 0, there t < T for some T > 0, and that the solution can be extended hand, by integration, we have
4669
data. 4 for 12, > 0 for any fixed T- > 0, it exists a ur$que solution 4 (r, t) for as long as $J (T, t) > 0. On the other
(3.7) where r h(r)
=
/xd~)ib(~W~
dp.
0
Hence & (r, t) > 0, 4 (T, t) > 0, as long as the solution exists up to t. Consequently, the solution can be extended for all t > 0. Consider now the case A > 0. Since 4 (r, t) is increasing in t, the limit 4 (T, 00) = &II 121(T, t) exists. It follows by taking
t ---) 00 in (3.7) that
Hence, (3.4) holds if we can show & (r, co) = 0. To estimate (3.2). Then, for n > 2, we have
&
p 8 (4t)’ 27 +A (IL), = -gg Hence, by integrating
we multiply
by & on both sides of
(2+).
this equation,
(3.8)
=
^ 2 2h (r) @cl + (n _ 1)/@-2
(>
Dividing (3.8) by e2xt’fi and taking t --+ 00, we obtain and h (7) 5 li$lo n-l?, we deduce from (3.8) that
2h(r) +
(n
-
1)p-2
& (r, co) = 0. Since $ (r, s) 2 4 (T, 0) = T
[l
+ ?i~e2AdP&]
Second World Congress of Nonlinear Analysts
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The inequality (3.5) thus follows. Next, we consider the case X1< 0. Notice that (3.9) keeps unchanged for X < 0. It remains to establish the lower bounds for +. From (3.8), since & > 0, we have
(tit^>2ewP-> (&>2, This establishes the assertion (3.6) for X <*O. The case X = 0 can be argued in a similar manner. We observe that in general, a solution ?I) (T, t) of (3.2) and (3.3) may not be a solution of (l.l’)(1.3’) if & (r,t) = 0 f or some T, t. On the other hand, one can verify that the solution 4 (T, t) of (3.2) and (3.3) can generate a C1+a solution of (3.1’)-(3.3’) (hence a Cl+” solution of (l.l)-(1.5)) via (3.1) if and only if & (r,t) > 0, In the case X = 0, it is shown in [2] that in general the situation 4, (T, t) = 0 may happen even if initially $0 (r) > 0. However, we can show that if Jo 2 0, & (r) = br, and & (r) = p for some constants b,p 2 0, then & (r, t) > 0 for any 0 < T 5 1, t 2 0. More general conditions can also be derived by using the analysis in a similar manner to the proof of Lemma 3.1. Using Lemma 3.1 and techniques in [2], we can now show the following global existence result. THEOREM 3.1. Suppose that PO (11 =
PO
(14) + ~6 (~1, $0 (~1 =
40
(14) i
+ &lb),
dist (&I, &I
I E,
where & and $0 satisfy the conditions in Lemma 3.1, 4 and $1 have compact supports. Suppose also the corresponding solution to (3.2) and (3.3) exists globally. Then, for small E, system (l.l)(1.5) admits a unique global C1+a (no) solution. 4. GLOBALSOLUTIONSWHENp=O
In this section, we consider Jacobian J (@) satisfies
dJ= dt It follows that 5(+)
the case p = 0. For simplicity,
the
-J (Icl)true (V2P (N) = J (@Ip ($) = PO(z).
= 1 f tPo and consequently
p= Hence (2.1)-(2.3)
we take X = 1. From (l.l)-(1.3),
POW) ml (q-l) + 1.
(4.1)
reduces to
(4.2) LEMMA
4.1. Suppose that 1c,is a C1+a solution Cl Iz - yl l’m)
where ,B (t) = exp (-ct)
IIlc,(~,t)--(Y,t)lIczl~-Yl
, cl, c2 and c are constants.
of (5.1). Then p(t)
9
(4.3)
Second World
Proof.
Congress
Since P (z, t) in (4.1) is bounded,
of Nonlinear
467 1
Analysts
we have (see [S])
J
fors,yEat.
iit
(4.3) thus follows from Gronwall’s inequality. LEMMA 4.2. For any T > 0, let & = a/3(T),
where P(t) is defined in the Lemma
4.1. Then
llV@0) II&,mt 5 4”) 9 where c(T) is a constant depending only on the initial Proof. By Lemma 4.1, we know that
IW)lti Since nt = (0 (2, t) = 00 (+-l
data and T.
5 c(T).
(2, t)) < 0} , it follows that VB satisfies dV0 ~+(U.v)V8=-(Vu)TV8,
where u=C()
JGP((z,t) Ix - 4
dz, Vzl=cs
fit
It follows that, by integrating
Jn,
V zKP(z,t)
dz + PI (see (2.4)).
12--I
along $J, VB (11, t) = V&I (z) - j (VU)~ VO ($, s) ds. 0
By replacing cy in (2.5) with &, we find that [VZL[0,~~ Ic(l is defined as in (2.7) with Q = /3~. Hence
IV4-o,n,
+lnl~~d(f%)
+cd(%)l),
where
~c+cj(l+ln/6:d(R~)+cd~~~)~),v~(~)~ods.
6,’
(4.4)
0
By [4, Lemma
Suppose
4.11, we have
that we can show
J(WT VQ(s)(pT 5 c(lVQ(s)lo + IVeb)lp~) (1 +lnlG,TdW + I. Then
by using a generalized
Gronwall’inequality
(see [5]), it follows from (4.4) and (4.5) that
IV0 (s)lo + IV0 (41p~ I c(T).
(4.6)
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Second Wodd
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This would complete the proof. It remains to establish (4.6). Set
We write
(VU)~
V0 as
(VtL)Tve = cclJ y--$”
P (z, t) VB (2, t) dr + P (2, t) V6’ (x, t)
fit
= co (+--L))T
J b---In fit Jn,lx-4
+ce
=
(VO (x, t) - W(z,
t)) P (2, t) dr + P(z,
(“(x-~~~T~~(~,t)(~(z,t)-~(x,t))dz
t) VB (x, t)
+ce~(x,t)
AcIx-4 J(“(x-z~)TV+,t)dz
Jl+&+&+&
(4.7) By [I, Appendix], the terms Jl, Jz and 53 in (4.7) can be bounded by the left-hand side of (4.6). To estimate J4, we notice that the ith component of (0 (x - .z))~ Ix - ~1~” VB (z, t) can be written as, for n > 2, ui (x - z) . VB (2, t) = Di l~~z;n lx - zy Since A 1x - zI-~+~ DiV =
= =
(z,
t) =
= 0, we obtain 1 (3 - tin-2
VD; lx-‘,I,-2
4( j#i
’ VB
.ve(z,t)= (
Div Ix-zln-2 l
-w)
~W’(z,t) -Dte(z,t)e;) -c Dj~j,x~~l,,~2 Die(~,t) ) j#l (
DjDiix-li”-2)D,B(r.t)00,
-A,x-‘Z(,-2G
1 (x - t(n-2
(DjDjlx_11,-2)D.B(r,t)]
. (Dj0 (t, t) < - Die (~9 t) E’j) 1
where c is the ith unit base vector. Noticing that D#(z,t) t$ - D;B(z,t) C’j is a divergence and tangent to V8, the normal vector to EQ, it follows from the divergence theorem that
J
_ :>:;,,")V@(Z,~)
=C
J (VDj1x-iln-2) j#i fir
free
.(Dje(~,t)e’i-DiB(z,t)~j)=O.
Rt Therefore, 54 = 0. The proof is complete. By applying Lemma 4.2 and modifying the methods used in [5], we can establish global existence. THEOREM 4.1. There exists a unique global C1+O solution for system (5.1). Proof. To reduce the length of the paper, we only outline the proof. By Lemma 4.1 & 4.2, we can derive 1$(x, t)l < c(T) for t 5 T. It then follows that IQ (t)l, 5 c(T) _ Hence the solution can be extend to t 5 T, for any T > 0.
Second World
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5. APPLICATIONS In this section, we present two examples to demonstrate that some problems in fluid dynamics some free boundary problems can be treated as non-local dynamical systems (1.6) and (1.7). EXAMPLE 1. Consider dynamics of vortex patches for inviscid Euler systems [7]:
w=vxu, with initial
condition
(vortex
V.u=Q,
patch) who)
By Biot-Savart
and
law, we can represent
=wo(4x&y
the velocity
‘~1in terms of the vorticity
u(z,t)=J K(s-z)w(z,t)dz, K(z)=4
i3
w as
Jf, :y)
R”
The vorticity
system then reduces to
(?J, (~4- 4w(z,t) dz, ti(~0) =z, J8~ w(z) = ~~(~-'(i,t),t)Wg(Vil(z,t))xn.. dll, dt=
where $ is the fluid particle trajectory. This system is investigated solution blows up at t = 2” if and only if
in [4]. It is shown that a Cl+=
/T(IwMo,n,+ P(t)lo,n,)dt= 00. 0
EXAMPLE 2. Consider in type-11 superconductors:
the free boundary
wt = V
problem
(]w] VH)
modeling
evolution
v, = -sign(w)
(54
= 0 on Ft = asZ,, g
on rt,
where n, is the moving domain initiated at L!c, rz is the outward of the moving boundary rt, and the bracket [.] denotes the jump function for the equation (5.1). Then H&t)
=
J nr
regions
in Rt,
AH - H = -w in Rt, AH - H = 0 in R2\f&,
WI = [g]
of mixed-state
K(r-z)w(z,t)
dz.
normal, V, is the normal velocity across rt. Let K(z) be the Green
4614
Second
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Defme J!Jby
dt = -szgn (w) VH (Q, t) , + (z,O) = z. We find that w (2, t) = (J($)-’ formulated as
~0) (q!~-l (z, t)) . It f o 11ows that the free boundary
problem
can be
z =-sign(w) /WC (Ic, (5,t)- 2)(J(?/J-l wo) (11-l (2, t,)dz, ?(, (5,O) =I. Global
existence
of solutions
for this system is established
in [5].
REFERENCES [l] [2] [3] [4] (51 [S] [7] [S]
A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152, 1928 (1993). A. Friedman and C. Huang, Averaged motion of charged particles under their self-induced electric field, Indiana Univ. Math. J. 43, 1167-1225 (1994). D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 3nd edition, SpringerVerlag, New York, 1985. C. Huang, On boundary regularity of vortex patches for 3D incompressible Euler systems, to appear. C. Huang and T. Svobodny, Evolution of Mixed-State Regions in Type-11 Superconductors, to appear. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39, S187-220 (1986). P. G. Saffman, Vortex dynamics, Cambridge, 1992. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.