proceedings

André Unterberger

From pseudodifferential analysis to modular form theory

Jean Ginibre

Giorgio Velo

JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
JEAN GINIBRE
GIORGIO VELO
Long range scattering and modified wave operators
for Hartree equations
Journées Équations aux dérivées partielles (1999), p. 1-9
<http://www.numdam.org/item?id=JEDP_1999____A17_0>
© Journées Équations aux dérivées partielles, 1999, tous droits réservés.
L’accès aux archives de la revue « Journées Équations aux dérivées partielles » (http://www.
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Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Journees Equations aux derivees partielles
Saint-Jean-de-Monts, 31 mai-4 juin 1999
GDR 1151 (CNRS)
Long range scattering and modified wave operators
for Hartree equations
Jean Ginibre Giorgio Velo
Abstract
We study the theory of scattering for the Hartree equation with long range
potentials. We prove the existence of modified wave operators with no size
restriction on the data and we determine the asymptotic behaviour in time of
solutions in the range of the wave operators.
In this lecture, we report on some recent work [5] [6] on the theory of scattering
for the Hartree equation
i9tU + _A ^ = (V ^ H2
)?/ (1)
^
where u is a complex valued function defined in space time JR714'
1
, A is the Laplacian
in j0y\ V is a real valued even function defined in 1R71 and ^ denotes the convolution
in space. The stationary version of (1) has been introduced in the late twenties as
a simplified model for complex atoms. An improved version taking into account the
Pauli principle, namely the Hartree-Fock equation, has been introduced in the early
thirties and is still used currently in the theory of nuclear collisions. If V in (1) is
replaced by a delta function, the equation becomes the cubic nonlinear Schrodinger
equation so that (1) can also be regarded as a regularized version of the latter.
We restrict our attention here to potentials V of the form
V(x) = X\x\-7
 (2)
for some 7 > 0, although the results presented below extend to time dependent V
of the type
V{x) = X t^ \x\-^ (3)
for suitable ^ with 0 < IJL < n.
The Cauchy problem for the equation (1) is known to be globally well posed in H1
under assumptions on V which in the special case (2) reduce to 0 < 7 < Min(4,n),
This talk was given while one of the authors (G. V.) was visiting the Institut des Hautes Etudes Scientifiques, Bures￾sur-Yvette, France. He is very grateful to this institution for the kind hospitality extended to him in a stimulating
environment.
XVII-l
and in addition 7 < 2 for negative A [1] [4]. It is then a natural question to study
the asymptotic behaviour in time of the global solutions, and that problem can
be addressed through the theory of scattering. One of the basic problems in that
theory is that of classifying the previous asymptotic behaviours by relating them to
a set of model functions V == {v = v{u+)} parametrized by some data u^. and with
suitably chosen and preferably simple asymptotic behaviour in time. For each v in
V one looks for a solution u of (1) such that u(t) behaves as v(t) when t —)- +00 in a
suitable sense. If v(t) can be parametrized by its value v(0) at t = 0, one can then
define the wave operator for positive time as the map Q+ : v{0) —^ u{0). A similar
question can be considered for t —^ -oo. We restrict our attention to positive time.
In the same way as in linear scattering theory, two different situations arise. If
V decays sufficiently fast at infinity, the previous construction can be performed by
taking for V the set of solutions of the free Schrodinger equation, namely
v(t} = U(t) u+ = exp (i ( A) ^ . (4)
\ ^ /
This is the short range case, corresponding to 7 > 1 in (2). Existence of the wave
operators for (1) in that case is proved in [11].
If V decays too slowly at infinity, namely for 0 < 7 < 1, the previous (ordinary)
wave operators are known not to exist. In fact, one can prove the following result [9].
Proposition 1. Let u e C(2R,L2
) be a solution of (1 ) with V given by (2 ) and
0 < 7 < 1, and let u^. 6 L2
 be such that
lim || u(t) - U(t} u. Us = 0
t-^oo ' ' "
Then u^. = 0 and u = 0.
This is the long range case. In that case it is known in linear scattering theory
that the asymptotic model functions v have to be modified by a suitable phase,
thereby leading to the construction of modified wave operators. There is a vast
literature on that subject for which we refer to [2] [10]. Here we want to perform the
same construction for the equation (1), namely to construct modified wave operators
for that equation.
The introduction of a phase in the model functions can be done in several ways.
Let cj) be a real function of space time. One can replace v in (4) by
^i (*) = U{t) exp [-z<^, -zV)] u^ . (5)
This is the choice made in most of the literature on linear scattering.
A different choice, better suited for our purpose, has been proposed in [12] [13].
It uses the following structure of the group U(t). In fact U(t) can be represented as
U(t) = M(t) D(t) F M(t) (6)
where M(t) is the operator of multiplication by the function
M(t)=exp[^x2
/2t] , (7)'
XVII-2
D(t) is the dilation operator defined by
(D(t)f)(x) = (zt)-"/2
 f(x/t) (8)
and F is the Fourier transform. The fact that M(t) tends strongly to 1 (for instance
in -L2
) when t —> 00 suggests to replace v^ by
v^t) ^M(t) D(t) Fexp[-i4>(t,-iV)]u+
= M(t) D(t) exp[-^(t, x)]w+ (9)
= expp<^, ;r/()]M(<) D(t) w+
where w+ = Fu+.
In order to construct solutions u of (1) that are asymptotic to v^, it is useful to
represent those solutions as
u(t) = M(t} D(t) expH y(t, x)}w(t) (10)
and try to ensure that w(t) ->• w+ and that y(t) behaves as (f>{t) as t ->• 00. The
parametrization (10) has been introduced in [7] [8] and used there to prove the
existence of global solutions with small initial data for the equation (1) in long
range situations.
The evolution equation for (w, y) is obtained by substituting (10) into (1). Using
the fact that
D(t)-1
 (V* H2
) D(t) = r^ po(w,w) (11)
where
50(^1,^2) = A Re |;r|~7
 * (wiWa) (12)
we obtain
{i9t + (2t2
)-1
 A - r7
 go{w, w)} exp(-^)w = 0 (13)
or equivalently
^i9t + (2t2
)~l
 A - z (2^)~1
 (2V(^ • V + (Ay)) I w
+{^-(2t2
)"1
 IV^-r^po^wdw^O . (14)
However, the parametrization (10) is redundant and we have only one evolution
equation for two unknown functions. In the same way as in gauge theory, we arbi￾trarily add a second equation by taking the last bracket in (14) to be zero, thereby
obtaining the following auxiliary system
9tW = i (2t2
)"1
 Aw + (2t2
)'1
 (2Vy? • Vw + (Ay))w)
(15)
9tV=(2t2
)~l
 IV^+r^w.w) .
The construction of modified wave operators for the equation (1) will be per￾formed by first constructing wave operators for the auxiliary system (15) and then
recovering u from (10).
XVII-3
The first step consists in showing that the auxiliary system yields a well defined
dynamics for large time. In fact, the Cauchy problem for (15) is well posed for
large time in suitable spaces, with w(t) == 0(1) and ^p(t) = O^1
"
7
) for large t
(see Proposition 2 below). That asymptotic behaviour for large time is obviously
compatible with (15).
The next step in the construction of the wave operators for (15) consists in
choosing an asymptotic comparison dynamics. This is obtained by dropping all the
terms that are integrable at infinity in time in the RHS of (15). For orientation
we first consider the simple case where 1/2 < 7 < 1. In that case we obtain the
asymptotic system
QfWo = 0
(16)
9tVo =t 7 go{wo,wo)
which is immediately solved by
Wo(t) = W+
(17)
yo{t) = (1 ~ 7)-1
 (^-7
- 1) 5o(w^w+)
with initial condition (^o(l)
 = 0. One sees easily that (17) provides an adequate
description of the asymptotic behaviour of solutions of (15). In fact, it follows from
the first equation of (15) that QfW = O^"1
"
7
) so that w{t) has a limit w+ as t —> oo
and that
w(t) - w+ = 0 (r7
) . (18)
Taking the difference of the second equations of (15) and (16), one obtains
9t{y - <A)) = (2<2
)"1
 |V^[2
 + r7
 go{w - w^ w + w^) (19)
the RHS of which is 0(t~27) and therefore integrable at infinity for 7 > 1/2. There￾fore (p — yo has a limit ^4. as t —^ oo and
^t)-vo(t)-^=0(t1
-
2
^) (20)
(see Proposition 3 below).
The main step in the construction of the wave operator for the system (15) now
consists in obtaining solutions of that system satisfying (18) (20) for given (w+, ^4-).
The main difficulty consists in the fact that we want to solve the Cauchy problem
for the system (15) with infinite initial time, with initial data for y which blow up
at that time. That difficulty shows up already when solving the Cauchy problem
for the system (15) with finite initial time to with the appropriate initial condition
at to (see Proposition 2). In that situation, the solution turns out to be defined
only in an interval [T, oo) where T = 0(^~7
) when to —> oo. In order to circumvent
that difficulty, we proceed as follows. For given (w+,^+) we choose to large and we
construct the solution (w^.^o) of (15) with initial condition (w+.^o^o) + ^+) at
to. Estimating the difference {wto — w-^-, ^o — (po ~ '0+)?
 w^ show that the solution
can be extended to a fixed interval [T, oo) independent of to. We can then take the
XVII-4
limit of (wto, Vto) as to -> oo, thereby obtaining the required solution (w, y) of (15)
(see Proposition 4).
The extension of the previous construction to the general case 0 < 7 < 1 proceeds
as follows. Let p > 0 be an integer. We expand w and y as
^ = S w^ + 9p+i =
 w? + 9p+i
0<m<p
^ = £ ^m + ^p+1 = 0p + ^p-H
0<m<p
with the understanding that as t -> oo
w^)=o(r^) , g^)=o(r^)
(21)
^(^) = 0 (^-(m+ih) ^ ^^(^) = ^ (^-(p+ih) .
(22)
Substituting (21) into (15) and identifying the various powers of (~7
 yields the
system of equations
9f w^i = (2t2
)-1
 ^ (2V^-V+(A^.))w^
0<J<:m
(23)
(9,^^i=(2<2
)
 x ^ V^-.V^-.+r7
 ^ 5o(w^w^i_,)
0<j<m o<j<m4-l
for 0 < m + 1 < p.
The system (23) is triangular and can be solved by successive integrations over
time, with initial conditions
wo(oo) = w+ , w^(oo) =0 for m > 0
(^(1)=0 for0<m< p ,
(24)
thereby reproducing the asymptotic behaviour (23) at least for 0 < m < p and
{p + 1)7 < 1. One can then show that for (p + 2)7 > 1, one can construct solutions
(w, y) of (15) satisfying the asymptotic behaviour
w{t) - Wp(t) = 0 (r^)7
)
y(t) - ^(t) - ^ = 0 (t1
-^^)
(25)
(see Proposition 4).
The previous constructions enable us to define the preliminary wave operator for
the system (15) as the map
Qo : (w+^+) -^ (w,y) (26)
where (w, y) is the solution of (15) with asymptotic behaviour (25). Combining (26)
with (10) yields a map
(w+,^+) —> (w,(^) —> u = M Dexp{-iy)w
XVIL5
which is however unsatisfactory as a wave operator for (p for two reasons. Firstly
it depends on too many variables, namely on (^+,^4-), instead of only w^. = Fu^.
Secondly it has no chance of being injective since the same u can be obtained from
different pairs (w,y?). We now briefly discuss that problem. Two solutions (w,(^)
and (w'.y?') of the system (15) are said to be gauge equivalent if they give rise to
the same n, namely ifexp(—zy?)w = expf^—zy/)^. .Similarly, two pairs of asymptotic
variables (w+,^4.) and (w^,^) are said to be gauge equivalent if exp(—w-^)w^. =
exp(—%^)w^. One can then show that gauge equivalent solutions of the system (15)
have gauge equivalent asymptotic variables, and conversely that gauge equivalent
pairs of asymptotic variables have gauge equivalent images under Qo- One then
defines the wave operator for u as the map
0:^4- —> (Fz^, 0) —> (w, y) —>u= M D exp(~zy0w (27)
and that map is injective.
We now substantiate the previous heuristic discussion with mathematical state￾ments. We first define the function spaces where to solve the system (15). We look
for (w, (p) such that
(w, ^-^) e (C n L°°) (J, ^ © Y1
) (28)
for some interval I = [T.oo), where k and t are positive integers, H1
^ is the usual
Sobolev space, and
Y^ = L°° n H^ n H^2
^
1
 n H^2
where ro = oo for n even, r-o = 2n for n odd, and H^ are the homogeneous Sobolev
spaces. We shall use the notion | • |^ and | • \i for the norms in Hk
 and V^, ambiguity
being eliminated by the context. Pairs of integers (A:, t) will be said to be admissible
if
k ^ £ , t > n/2
i + 2 + 7 < Min(n/2 + 2A-, n + k)
k> n/2 if £+2+^=n+k
k > 2 if 7=1 and n is even .
Admissible pairs exist only for n > 3, and our results hold only in that case. If (k, £)
is admissible, so is also {k+j,£+j) for any positive integer j.. For n=3,0<7^1 ,
the pair (2,2) is admissible.
For simplicity, we state the results by giving the estimates in the form appropriate
to the case where 7~1
 is not an integer. If 7~1
 is an integer, additional logarithms
occur in the estimates. In all the subsequent results, we assume that n >_ 3 and
0<7<1 .
We first state the results on the Cauchy problem for the system (15) with finite
initial time.
XVIL6
Proposition 2. Let (k,£ ) be an admissible pair and let (wo,^o) € ^ ® Y^ Then
there exists To = To(wo,(po) such that for all to ^ To, the system (15) with initial
data w(to) = Wo, y{to) = ^-7^o has a unique solution (w, y) such that
(w, r-1
^) e (c n L°°) ([r, oo), ^ ® y^) (29)
for some T <, to. One can take T = C T^ ^~7
 and the solution satisfies the estimate
|^)|, <C(Max(Mo))1
"
7
 . (30)
Note that neither T nor the estimate (30) are uniform in to. The main ingredient
in the proof of Proposition 2, as well as in those of the subsequent ones, consists
of energy estimates. In the subsequent results, that method, of proof will produce
various losses of derivatives.
The next result is the existence of asymptotic variables (w+, ^4.) for the solutions
of the system (15) obtained through Proposition 2.
Proposition 3. Let (k,£) be an admissible pair. Let p be an integer such that
(p + 2)7 > 1. Let (w, y) be a solution of the system (15 ) such that
(w, r-1
^) e (C n L°°) ([r, oo), j^+Max(p+i,2) ^ y^P) (31)
for some T > 1. Then
( 1 ) The following limit exists
lim w(t) = w+ (32)
t—>00 ' '
strongly in H^ and weakly in H^^.
(2) Let Wp and (j)p be defined by (21) in terms of the solutions of the system (23)
with initial condition (24). Then the following limit exists
}^v(t) - (f)p{t) = ^ (33)
strongly in Y^~1
. Furthermore the following estimates hold
\w(t) - Wp(t)\,_,^ C t-^^
(34)
W}-W-^\^<Ct1
-^^ .
The main technical result is the existence of solutions (w, y) of the system (15) with
prescribed asymptotic variables (w+,^4-).
Proposition 4. Let {k,£) be an admissible pair. Let p be an integer such that
(p+ 2)7 > 1. Let W+ € ^+Max(p+l,2) ^ ^fi^ ^ ^ ^ ^ ^ ^ ^ ^ ^
Proposition 3. Let ^ e Y^1
. Then there exists T = T(w+,^) and there exists a
unique solution (w,y) of the system (15) such that
(w, ^-^) 6 (C n L°°) ([T, oo), Hk
 © Y^ (35) = (29)
XVII-7
and such that the estimates (34) hold.
With Proposition 4 available, one can define the auxiliary wave operator Qo by
(26) and the wave operator Q by (27). In order to describe the asymptotic behaviour
of solutions u of (1) in the range of Q, we need additional notation. We define the
operator
J(t) = x + zW = M(t) D(t) z'V D(tY M{tY (36)
so that in view of (10)
J^ u{t) = M(t) Z5(^)(^V)m
exp(-^(<))w(<) .
One sees easily that if (w,y?) satisfy (35), then exp(~z^)w e C([T, oo), ^A>), so
that ^ obtained through (10) from such a (w,y) belongs to the space ^([T.oo))
defined by
X
k
(I)={^<J(t)>k
ueC(I.L 2
)] (37)
where < • >=< 1 + | • |2
 >1
/
2
. In particular it follows from Proposition 4 that the
wave operator Q maps F^+Max(p+i,2) ^ ^^ ^ ^ suitable A: and T.
We finally collect the properties of solutions u of (1) in the range of Q that follow
from the previous results.
Proposition 5. Let k be the first member of an admissible pair. Let p be an integer
such that (p+ 2)7 > 1. Let u^ € F^+Max(p+i,2) j^
( 1 ) There exists T = T{u^) and there exists a unique solution u of the equation
( 1 ) defined in [T, oo) which can be represented through (10 ) in terms of a solution
(w,y) of the system (15 ) obtained in Proposition 4 with w+ = Fu^., ^ = 0. The
solution u belongs to Xk
([T,oc)).
(2) The map Q : zz+ -^ u is injective.
(3) u satisfies the estimate
\\< J(t} >k
 {exp(icf>^x/t))u(t) - M(t) D(t) F u^} ^ < C t1
-^2
^ . (38)
(4) Let r satisfy 0 < 6(r} = n/2 - n/r < Min(A:,n/2), 6(r) < n/2 if k = n/2.
Then u satisfies the estimate
|| u(t) - exp (-z^ x/t)) M(t) D(t) F u+ \\r < C t1
-^2
^-6
^ (39)
where | ' \\r denotes the norm in 27 (IT?72).
Remark. The asymptotic estimates (34) (38) (39) hold in the form stated here for
simplicity only under the condition (p+ 1)7 < 1. They have to be suitably modified
if that condition does not hold (see [6]).
References
[1] T. Cazenave, An introduction to nonlinear Schrodinger equations, Text. Met.
Mat. 26, Inst. Mat., Rio de Janeiro (1993).
XVII-8
[2] J. Derezinski, C. Gerard, Scattering Theory of Classical and Quantum N￾Particle Systems, Springer, Berlin, 1997.
[3] J. Ginibre, T. Ozawa, Long range scattering for nonlinear Schrodinger and
Hartree equations in space dimension n > 2, Commun. Math. Phys. 151
(1993), 619-645.
[4] J. Ginibre, G. Velo, On a class of nonlinear Schrodinger equations with non￾local interaction, Math. Z. 170 (1980), 109-136.
[5] J. Ginibre, G. Velo, Long range scattering and modified wave operators for
some Hartree type equations I, Rev. Math. Phys., to appear.
[6] J. Ginibre, G. Velo, Long range scattering and modified wave operators for
some Hartree type equations II, Preprint, Orsay 1999.
[7] N. Hayashi, P. I. Naumkin, Scattering theory and large time asymptotics of
solutions to Hartree type equations with a long range potential, Preprint, 1997.
[8] N. Hayashi, P. I. Naumkin, Remarks on scattering theory and large time
asymptotics of solutions to Hartree type equations with a long range potential,
SUT J. of Math. 34 (1998), 13-24.
[9] N. Hayashi, Y. Tsutsumi, Scattering theory for Hartree type equations, Ann.
IHP (Phys. Theor.) 46 (1987), 187-213.
[10] L. Hormander, The Analysis of Linear Partial Differential Operators^ Vol. IV,
Springer, Berlin, 1985.
[11] H. Nawa, T. Ozawa, Nonlinear scattering with nonlocal interaction, Commun.
Math. Phys. 146 (1992), 259-275.
[12] T. Ozawa, Long range scattering for nonlinear Schrodinger equations in one
space dimension, Commun. Math. Phys. 139 (1991), 479-493.
[13] D. R. Yafaev, Wave operators for the Schrodinger equation, Theor. Mat. Phys.
45 (1980), 992-998.
LABORATOIRE DE PHYSIQUE THEORIQUE (UNITE MIXTE DE RECHERCHE 8627
- CNRS), UNIVERSITE DE PARIS XI, BATIMENT 210, 91405 ORSAY CEDEX,
FRANCE
DIPARTIMENTO DI FISICA, UNIVERSITA DI BOLOGNA AND INFN, VIA IRNE￾RIO ?46, 40126 BOLOGNA, ITALY
ginibreQqcd.th.u-psud.fr
velo@bo.infn.it
XVII-9

Long range scattering and modified wave operators for Hartree equations

Daniel Tataru

The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation

András Vasy

Propagation of singularities in many-body scattering in the presence of bound states

Mikhail V. Novitskii

Two lectures on spectral invariants for the Schrödinger operator

Dragoş Iftimie

JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
DRAGO ¸S IFTIMIE
Évolution de tourbillon à support compact
Journées Équations aux dérivées partielles (1999), p. 1-8
<http://www.numdam.org/item?id=JEDP_1999____A4_0>
© Journées Équations aux dérivées partielles, 1999, tous droits réservés.
L’accès aux archives de la revue « Journées Équations aux dérivées partielles » (http://www.
math.sciences.univ-nantes.fr/edpa/) implique l’accord avec les conditions générales d’utili￾sation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression sys￾tématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi￾chier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Journées Equations aux dérivées partielles
Saint-Jean-de-Monts, 31 mai-4 juin 1999
GDR 1151 (CNRS)
/
Evolution de tourbillon à support compact
Drago§ Iftimie
Résumé
On considère l'équation d'Euler incompressible dans le plan. Dans le cas
où le tourbillon est positif et à support compact on montre que le support
du tourbillon croît au plus comme 0[(Hog^)]1
/
4
, améliorant la borne 0(t1
/
3
)
obtenue par C. Marchioro. Dans le cas où le tourbillon change de signe, on
donne un exemple de tourbillon initial tel que la croissance du diamètre du
support du tourbillon est exactement 0(t). Enfin, dans le cas du demi-plan
et du tourbillon initial positif et à support compact, on montre que le centre
de masse se déplace parallèlement à l'axe avec une vitesse minoré par une
constante positive; de plus, la distance d'un point du support du tourbillon à
l'axe est au plus en 0[(Hogt)1
/
3
].
Tous les résultats qui suivent ont été obtenus en collaboration avec Thomas C.
Sideris. Le théorème 3 est aussi le résultat d'une collaboration avec Nicolas Depauw.
Un fluide parfait incompressible évoluant dans R2
 obéit aux équations d'Euler
suivantes:
f
 9tV+v ' Vî; = -VP,
div v(t, •) = 0,
^\t=0=VQ￾Ici v : R+ x R2
 -^ R2
 est le champ de vitesse du fluide et P : R+ x R2
 —^ R est la
pression. Une quantité importante pour l'étude de ces équations est le tourbillon
UJ = 9^ — 02^1.
En prenant le rotationnel de l'équation de v on trouve
9t u + v ' Vu = 0, o;(0, x) = uj^x} (1)
ce qui signifie que le tourbillon est transporté par le flot
Q^ =z;(^), ^(0,^) =rc.
La condition d'incompressibilité permet de retrouver r à partir de uj par l'intermédiaire
de la loi de Biot-Savart
. . . I f (x-y ^
v[t.x ) = ^-iS^-v^^^
IV-l
OÙ X^ == (—^2^l)-
L'existence et runicité globale des solutions classiques a été montré pour la
première fois par Wolibner [12]. Le cadre des solutions faibles a été abordé par
Yudovich [13] (o;o £ L1
 n L°°), DiPerna et Majda [4] (o;o € L1
 n Z^), Delort [3]
(nappes de tourbillon) et d'autres auteurs.
Malgré cette théorie d'existence et d'unicité globale assez complète, peu de choses
sont connues sur le comportement à l'infini des solutions. Plusieurs lois de conserva￾tion sont à notre disposition pour étudier ce comportement: conservation des normes
If de uj, de la masse J* uj{t,x) dx, du centre de masse f xuj{t,x} dx et du moment
d'inertie f Irrpù^t.rr) dx.
Comme le tourbillon est transporté par le flot de v, on voit tout de suite que si
le tourbillon est à support compact au moment initial il reste à support compact
pour tout temps. De plus, le fait que v soit borné en espace et temps montre que la
taille du support du tourbillon est au plus en 0(t). Dans la suite, on se propose de
voir comment et dans quelles conditions cette borne triviale peut être améliorée.
Il y a très peu de solutions explicites des équations d'Euler. Les solution sta￾tionnaires sont les premiers exemples. Ensuite, on peut aisément vérifier que si
le tourbillon initial est la fonction caractéristique de l'intérieur d'une ellipse, son
évolution est donné par une rotation. Enfin, le couple de Batchelor est un tourbillon
antisymétrique par rapport à un axe qui se déplace parallèlement à cette axe à vi￾tesse constante (voir [2], page 534). Ainsi, aucun exemple explicite ne montre une
croissance du diamètre du support du tourbillon et le seul exemple qui montre une
croissance de la distance à l'origine fait intervenir un tourbillon qui change de signe.
À défaut de solutions explicites, on peut se rabattre sur des solutions approchées
qui consistent à prendre le tourbillon comme une somme de masses de Dirac avec
certains poids. Il n'y a pas assez de régularité pour définir une solution du système
d'Euler, l'équation est obtenue dans ce cas en ignorant les termes qui ne sont pas
définis. Marchioro et Pulvirenti [10] montrent néanmoins que si le tourbillon initial
est approché au sens des distributions d'une manière particulière mais pas trop
restrictive, alors ceci reste vrai pour tout temps. Malheureusement, cette convergence
n'est pas uniforme en temps, donc rien ne peut être déduit sur le comportement à
l'infini des vraies solutions. Cependant, on peut voir que pour ce système approché le
support du tourbillon reste borné si le tourbillon est positif. De plus et toujours pour
ce système approché, il existe des tourbillons changeant de signe avec une croissance
du diamètre du support qui est exactement en 0(t).
Venons-en aux résultats disponibles dans la littérature. Marchioro [8] montre
que dans le cas d'un tourbillon positif appartenant à L1
 DL00
, la distance à l'origine
du support du tourbillon est au plus en 0{t1
^
3
). Lopes Filho et Nussenzveig Lopes
[7] étendent ce résultat au cas des tourbillons L1
 n V et Hounie, Lopes Filho et
Nussenzveig Lopes [5] obtiennent des résultats partiels dans le cas des nappes de
tourbillon. Enfin, dans le cas d'un domaine extérieur à un compact et du tourbillon
positif, Marchioro [9] montre que la croissance du support du tourbillon est au plus
en O(^) pour tout a > 1/2.
Les trois théorèmes qui suivent sont les résultats nouveaux présentés dans ce
travail.
IV-2
Théorème 1 Soituj{t^x} la solution des équations d^Euler avec un tourbillon initial
positif et à support compact UJQ € L^R2
) n L°°(R2
). Il existe une constante Co
telle que, pour tout temps t >_ 0, le support de uj{t, •) est contenu dans la boule
|.r|<Co[(l+t)log(2+t)]1
/
4
.
Théorème 2 On considère un tourbillon initial qui est obtenu en prenant une fonc￾tion positive appartenant à L1
 n L°° à support dans le premier quadrant et en la
prolongeant par antisymétrie par rapport aux axes de coordonnées. Il existe une
constante Co telle que le diamètre du support du tourbillon à l'instant t soit minoré
par C^t.
Théorème 3 Considérons les équations'd^Euler dans le demi-plan x^ > 0 avec
des conditions au bord de glissement. Supposons que le tourbillon initial est positif
et à support compact, UJQ 6 L^R2
) n L°°(R2
). Alors le centre de masse P(t} =
fxu}(t^x) dx se déplace parallèlement à Vaxe avec une vitesse minorée par une
constante positive. De plus, la distance d^un point du support du tourbillon à l'axe est
au plus en 0[(t\ogt)1
^
3
] „ En d'autres termes, il existe une constante Co > 0 telle que
?2 = est., Pi(t) > C^t pour t suffisamment grand et x^ < Co [(1 +1) log(2 + t)]1/3
pour tout x € support, •).
Remarque 1. L'exemple donnée dans le théorème 2 est inspiré d'une part par l'ana￾logue discret qui montre une croissance en Ct et d'autre part par un exemple similaire
de Bahouri et Chemin [1] qui est utilisé pour des problèmes de régularité minimale
du flot.
Remarque 2. Dans le cadre du théorème 1, Serfati [11] montre indépendamment et
simultanément la borne [t\ogo- " o log(^)]1
/
4
. Gamblin [6] donne une autre preuve
du théorème 1 en utilisant des estimations sur les moments d'ordre supérieur.
Remarque 3. Les théorèmes 1 et 3 restent vrais pour des solutions faibles uj e Z^nZA
p > 2 car les constantes dans les preuves ne dépendent que des normes L1
 et 27'.
Remarquons aussi que les bornes obtenues sont valables en fait pour toute ligne de
flot.
Les preuves détaillées des théorèmes 1 et 2 se trouvent dans [6]. D'autre part,
la preuve du théorème 3 reprend, dans un cas plus simple, les idées principales des
preuves des théorèmes 1 et 2. Il n'y a donc pas lieu de répéter ces preuves ici. On
va se contenter de montrer le théorème 3 dans la section suivante.
1. Preuve du théorème 3.
Dans la suite, (7, Ci,... désignent des constantes qui dépendent de UJQ et qui
peuvent changer d'une inégalité à l'autre. L'ensemble D désigne le demi-plan x^ > 0.
Remarquons d'emblée que les conservations du centre de masse et du moment
d'inertie du tourbillon ne sont plus valables dans D. On ne pourra utiliser que la
conservation des normes LP.
Le lemme suivant nous sera fort utile dans la suite.
Lemme 1 Soit a ç. (0,2), S C R2
 et h : S —^ R+ une fonction appartenant à
L
l
(S)nL P
{S),p> ^. Alors
r k(y \ 2-a-2/p a / i^y ) i ^ r\\h\\ 2-2/p
 ll^ll 2
-
2
^
A \x - y^ ll'1
"^) \\1^\\LP(S) '
IV-3
Preuve du lemme. Soit k un nombre positif quelconque. On peut estimer par l'inégalité
de Hôlder
f^-1
 ! ^+
1° j S IT-—ni A^a
SD{\x-y\>k} Sn{\x-y\<k]
WL^S) ,„„ II 1
——fc———+N^)||—— ^
< " "- ^ +
|a-|"ll£?-T(|.c|<Â:)
IHI-^) , ^HLII L2-a-2/p
——^——+^l|"lkP(5) 'ça K
i
Le choix À; = (ll^lk^ll/illjy1
^))
 2 2/p
 termine la démonstration.
Revenons à la preuve du théorème 3. On suppose, pour simplifier, que fy <jj[x} dx =
1. On commence par montrer l'affirmation sur le centre de masse.
Il n'est pas difficile de vérifier que si l'on prolonge uj par antisymétrie par rapport
à l'axe x-i = 0, le tourbillon résultant obéit aux équations d'Euler posées dans R2
.
La loi de Biot-Savart dans 3R2
 donne ainsi la loi de Biot-Savart dans x^ > 0:
.(.}- 1 / (^-v^ ^-y^\ ,., ^-^JAv^w-v^w)^^
où y = (î/i, -^2) désigne le complexe conjugué de y. Un calcul très simple montre
alors que
".M = ^/"f-^+f^U)*- 27r JD\ \x-y\2
 \x-y\2
} "" " v M/
^) = !/• fa-y.)^ (^
v
 JD \x - ynx - y\2
 ' ' v /
Soit
P(t) = / xu(t,x)dx,
JD
le centre de masse du tourbillon. On a par (1) et après une intégration par parties
P'{t} = / x0tu{t, x) dx = - l xv(x} • Vo;(.r) dx = [ v{x}^{x} dx. (3)
JD JD JD
La loi de Biot-Savart (1)-(2) implique maintenant
w = i/f^^w^^^
P,{t) =0 .
où on a utilisé l'antisymétrie des expressions ^^^(x^Çy} et ^~^o
•
^2j
/
|a;(3;)a;(j/)
par rapport au changement de variables (x,y) <—> (y,x). On obtient tout de suite
une nouvelle loi de conservation
/ x^ujÇx) dx = est.
JD
IV-4
Montrons maintenant qu'il existe une constante C > 0 telle que P[ > C. Par
abus de notation, on désigne par uj la prolongée du tourbillon par antisymétrie par
rapport à l'axe x^ = 0. On a déjà remarqué que le nouveau tourbillon vérifie les
équations d'Euler dans R2
. Il est facile de voir que l'énergie de la vitesse dans tout
l'espace est conservée et que cette quantité est égale à
EQ = —— // \og\x-y\u;{x)u;(y)dxdy
^ J J R2><]R2
=
 à/A.^^^^^^^^
=
 ^/L10^^^)^)^^-
Remarquons que le noyau obtenu ci-dessus est positif, contrairement au noyau qui
correspond à un tourbillon positif dans R2
. Une application de l'inégalité de Hôlder
donne
Ea <
-
c
^" /L (^çr h (1 + S)]'^M^^
 l/t
.
avec q > 1 à choisir convenablement. On utilise maintenant l'inégalité log(l +1) <
CT^—, 1 - 1/ç <, a < 1, avec t = ,^p ce qui implique 1 +1 = Ijr^. On obtient
ainsi ainsi
ff fl^I!!y-lLg(l+-4
^-)1^^)&^
J J zA ^2+2/2; L v \x-y\2
^
 v / vy/
^
 c ff. ^ . y^^ - y^^ dx dy
^ ^/L^-^^'1
"^"^^^
= c//^]^=^iÀJWy}dxdy
= 2C x^{x)( —————uj{y}dy\dx
j D 'J D ^ ~ y\ '
où on a choisi a = 3/2 — 1/ç ce qui est loisible si q < 2. Le lemme permet donc de
dire que
EQ ^ CiP^Y-^P^ = CT^O)1
^?!')1
-
1
^,
ce qui implique bien que P[ est minoré par une constante. Remarquons par ailleurs
que le fait que v soit borné en espace et temps et la relation (3) implique que P[ est
majoré par une autre constante.
On passe maintenant à la deuxième partie. On va montrer qu'il existe C\ > \/3
telle que |z'2(.r)| ^ C\x^2
 pour tout x tel que 2:2
 >
. Ci{t\ogt)1
^ et t assez grand.
IV-5
Ceci impliquera qu'une particule fluide ne peut pas s'échapper de la zone 3-2 <
Ci(tlogé)1
/
3
.
Comme |a- - y\2
 > max(.rj,a-2?/2) on peut estimer à l'aide de (2) et du lemme
|^)| ^ 2
7r
! / ^V^^dy +2 / .^-^.^dy
 \Jy,<^/2 \x - y^x -y\2
 • TT 4>^/2 |.r - y^x - y\2 vy/ y
<
: — -,—
2—u(y)dy+C \ -———-u)(y) dy ^ Jy^i2 \x-y\ 4>^/2 \x - y\ ""
^ ^ f ï-^—i^)^+c'flHk°° [ ^(y)dy
1/2
X2 Jyî<X2/2 ^2 - V2\ \ Jy2>X2/2 'V2>X2/2
1/2 c t / r
^ ~2 y2^(y}dy+C[\\uj\\L^ \ ^{y)dy
-2 '' D \ Jy2>Xî/2
La preuve du théorème sera terminée une fois la proposition suivante démontré:
Proposition 1 Pour tout k > 0 il existe une constante Co telle que
f ^y)dy<,c
^,
Jyî>X2/2 X2
pour tout Xî > Co [(1 +1) log(2 + *)]1/3
.
Preuve de la proposition. Soit
/r(t)= ! 1^(x2——^}u^x)dx,
JJDD \ ^ /
où A = A(r) 6 (0,1) est à choisir convenablement et
e
8
n(s)=
l+e8
On voit facilement que
fr(t)^v(0) t U{t,x}dx.
Jx2>r
Donc, pour montrer la proposition il suffit d'estimer
W <. %
pour tout r > Co[(l + ^)log(2 + ^)]1
/
3
. Pour cela, on va déduire une inégalité
différentielle vérifiée par fr.
L'équation de uj ainsi que la loi de Biot-Savart (2) impliquent
/;(t) = /\(^^)^(^)^
= -t r^(x2^-r
-)v(x)•^(x)dx
=
 TrS^i^V^^^
2
 H ^/^2-î-\(^l-yi).3'22/2 / . / ., ,
=
 ~T î 1 ^ \—\——J1———T^i——=^^[x)iLi{y)dxdy. ^^rJ J p2 V \r / \x - y^x - y\2
 ' •• '
IV-6
En utilisant le changement de variables (x, y) ^—> (y, x) on obtient finalement
fiH.\ _
 1 [ [ ^/'^-I^ ^/^2-^1 (.Cl - yï)x2V2 , . , ., . , .
fr( ) ~^rJ J ^ rl [~^~)~11 (-^>J ^-y^x-y^^^^- (4)
Le théorème des accroissements finis implique
^^'(^/v
I
)
=
^•''^
où ç est situé entre x2^
1
- et ^r
. Il est très facile de vérifier que |7/'(5)| < rj(s) pour
tout s et que la fonction rf est positive croissante. On peut donc conclure que
^M^I^^H^)).
En insérant cette relation dans (4) on obtient
m
 ^ ^/AJ<^)+
<^)](.&-^)-^
=
 ^//nA^)]xf^^wy)dxd^^
L(x,y)
Pour x-î < r/ 2 on majore L(x,y) <: e~^ et pour 3-2 > r/ 2 on majore
T s \ ^ ( X 2 ~
 r\yî ^ ^ ^î- r
L^y}^^-^-}^-^{-^-^. \
En utilisant aussi la conservation de la masse et de J^ x^(x) dx on trouve la majo￾ration suivante:
W < ^-^ + ^frW￾Le lemme de Gronwall donne maintenant
fr(t) ^ fr(0)e^ + re^~^.
On a bien évidemment que fr{0) < Ce~^ pour r as ez grand. On obtient donc
fr(t) <: Ce^~^(l+r).
Si l'on suppose que t < ^- on trouve
fr{t)<Ce-^(l+r}.
Le choix A == [4{k + l)logr]~\ qui conduit à r > ^(Hog^)1
/
3
, conclut la preuve
de la proposition.
IV-7
Références
[I] H. Bahouri et J.-Y. Chemin. Equations de transport relatives à des champs de
vecteurs non-lipschitziens et mécanique des fluides. Arch. Rational Mech. Anal.,
127, n° 2, 1994, pp. 159-181.
[2] G. K. Batchelor. An introduction to fluid dynamics. Cambridge University
Press, 1967.
[3] J.-M. Delort. Existence de nappes de tourbillon en dimension deux. J. Amer.
Math. Soc., 4, n° 3, 1991, pp. 553-586.
[4] R. J. DiPerna et A. J. Majda. Concentrations in regularizations for 2-D incom￾pressible flow. Comm. Pure Appl. Math., 40, n° 3, 1987, pp. 301-345.
[5] J. Hounie, M. C. Lopes Filho, et H. J. Nussenzveig Lopes. Bounds on thé dis￾persion of vorticity in 2D incompressible, inviscid flows with a priori unbounded
velocity. To appear in SIAM J. Math. Anal.
[6] D. Iftimie, T.C. Sideris et P. Gamblin. On thé évolution ofcompactiy supported
planar vorticity. To appear in Comm. Part. Diff. Eqns.
[7] M. C. Lopes Filho et H. J. Nussenzveig Lopes. An extension of Marchioro ^s
bound on thé growth of a vortex patch to flows with Lp
 vorticity. SIAM J. Math.
Anal., 29, n° 3, 1998, pp. 596-599.
[8] C. Marchioro. Bounds on thé growth of thé support of a vortex patch. Comm.
Math. Phys., 164, n° 3, 1994, pp. 507-524.
[9] C. Marchioro. On thé growth of thé vorticity support for an incompressible non￾vis cous fluid in a two'dimensional exterior domain. Math. Methods Appl. Sci.,
19, n° 1, 1996, pp. 53-62.
[10] C. Marchioro et M. Pulvirenti. Vortices and localization in Euler flows. Comm.
Math. Phys., 154, n° 1, 1993, pp. 49-61.
[II] P. Serfati. Borne en temps des caractéristiques de l'équation d'Euler 2D à
tourbillon positif et localisation pour le modèle point-vortex. Manuscrit.
[12] W. Wolibner. Un théorème sur l'existence du mouvement plan d^un fluide
parfaite homogène, incompressible, pendant un temps infiniment long. Math.
Z., 37, 1933, pp. 698-726.
[13] V. I. Yudovich. Non-stationary flows of an idéal incompressible fluid. Z. Vycisl.
Mat. i Mat. Fiz., 3, 1963, pp. 1032-1066.
IRMAR, UNIVERSITÉ DE RENNES 1, CAMPUS DE BEAULIEU, 35042 RENNES
CEDEX (FRANCE)
iftimieOmaths.univ-rennesl.fr
IV-8

Évolution de tourbillon à support compact

Vladimir Kozlov

Vladimir Maz'ya

JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
VLADIMIR KOZLOV
VLADIMIR MAZ’YA
Boundary singularities of solutions to quasilinear elliptic equations
Journées Équations aux dérivées partielles (1999), p. 1-9
<http://www.numdam.org/item?id=JEDP_1999____A7_0>
© Journées Équations aux dérivées partielles, 1999, tous droits réservés.
L’accès aux archives de la revue « Journées Équations aux dérivées partielles » (http://www.
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Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Journees Equations aux derivees partielles
Saint-Jean-de-Monts, 31 mai-4 juin 1999
GDR 1151 (CNRS)
Boundary singularities of solutions to quasilinear
elliptic equations
Vladimir Kozlov Vladimir Maz'ya
Abstract
Asymptotic formulae for solutions to boundary value problems for linear
and quasilinear elliptic equations and systems near a boundary point are dis￾cussed. The boundary is not necessarily smooth. The main ingredient of
the proof is a spectral splitting and reduction of the original problem to a
finite-dimensional dynamical system. The linear version of the corresponding
abstract asymptotic theory is presented in our new book "Differential Equa￾tions with Operator Coefficients", Springer, 1999
In the article [W], S. Warschawski obtained an asymptotic formula for conformal
mappings of curvilinear strips under rather weak restrictions to their boundaries.
His proof is based on methods of geometric function theory. Here we state a corollary
of Warschawskfs result, which was the starting point of the present work.
Let Q be the domain [z = x + iy : y > f(x)} in the complex plane, where /
is Lipschitz and /(O) ==0 . By w we denote a conformal mapping of Q onto the
half-plane {w = u + iv : u > 0}, w(0) = 0. Suppose that f'{x} —> 0 as x approaches
0 except possibly for a set of measure zero and that
Ai/w+i.n-^n^oo.
Jo P
Then the real part of w admits the asymptotic representation as z —> 0:
^(z)~c(y-/(.r))exp{-^yl
(/(p,)+/(-p))c
^}, (1)
where c is a constant. This relation can be interpreted as an asymptotic formula
for a solution to the Laplace equation in a neighbourhood of the origin with zero
Dirichlet data on the curve [z : y = f(x)}.
In the present lecture we state asymptotic representations similar to (1) for
solutions to the Dirichlet problem for arbitrary order linear and even quasilinear
elliptic equations and systems in the multi-dimensional Lipschitz domain
Q={^=(^^)e^:^>/(^)}.
VII-l
Let /(O) = 0 and
flIV/ll^-^oo, (2)
u 0 r
with B^ denoting the ^-dimensional ball with radius p centred at the origin.
The representations just mentioned are examples of application of a recently
developed asymptotic theory of nonlinear abstract higher order ordinary differen￾tial equations. ^From 1990 we worked on an asymptotic theory of arbitrary order
linear differential equations with unbounded operator coefficients in a Hilbert or
Banach space. The impetus was given by our previous and simultaneous studies of
singularities of solutions to elliptic boundary value problems (see [KMR]). The first
real publication summarizing our linear abstract theory has just appeared as the
Springer book [KM1] (previously we wrote seven preprints concerning this topic).
In Introduction to the book we mentioned that reach possibilities for generaliza￾tions to nonlinear operator equations were left aside. Some of these possibilities are
discussed in what follows.
We study solutions to the equation
A(Di)u(t) =Y^Dr3
^f,(t:u(t)^.^Dt---q
u{t)) (3)
.7=0 •
on a semiaxis t > to^ where A{Dt) is an ordinary differential operator of order £ with
unbounded operator coefficients in a Hilbert space and Ao, • . • ,A/g are nonlinear
operators. We write the right-hand side in this form in order to cover more general
nonlinearities: the operators A/j are not assumed to be differentiable. Our conditions
on A(Df) are dictated by an analogy with linear elliptic operators written in the
variational form and are the same as in [KM1].
Our main concern is with the asymptotic behaviour of solutions as t —^ +00.
We show that for a certain class of equations (3) the question of asymptotics can be
reduced to that for a finite dimensional dynamical system perturbed by a "weak7
'
non-local nonlinear operator. This is a far-reaching generalization of a similar result
for linear differential equations with operator coefficients obtained in [KM1] (see
Ch. 13). However, since the right-hand side is now written in the generic divergence
form, our present result is new also for the linear case and, apparently, even for
linear ordinary differential equations with scalar coefficients.
In order to avoid technicalities we give a very approximate description of a general
result concerning a solution u{t) to (3). Let u be subject to a growth condition of
the type
l^-<?(M+i) ^ ^W ^or ^S6
 P0 u ^^ ^ (4)
where Wi
~
q
 is a Sobolev space and the majorant M(t) behaves like exp{—kot) in
a certain rough sense. Estimates of this type are usually obtained in applications
by using specific features of the equation: one relies upon monotonicity properties
of differential operators, the maximum modulus principle and more refined tricks.
Dealing with an abstract theory we take (4) for granted.
VII-2
Under some natural assumptions on the nonlinearity we obtain the representation
K,
col(u(t),...,D!-q
-\{t))=^h,(t)co$U,(t),...,Di-<l
-
l
U,{t))+v(t). (5)
5=1
The vector functions Us are solutions to A{Dt)U(t) = 0 of the form exp(zV)P(t)
with \y being eigenvalues of the pencil A($ on the line SA = fco and with P being
polynomials with constant vector coefficients. The vector h = (/ii,... , /^) satisfies
the finite dimensional perturbed dynamical system
Dth(t)=N(t^h(t))+K[h}(t). (6)
For the components of N we have the representation
N.[/^) = EW^; E MW^),... ^ ^(^~^(<))IA9
"^)
J=0 5=1 5=1
where 14 are solutions of the adjoint equation ^t*(£)()V(<) = 0 which are connected
with Us by a biortogonality condition. Here, (•|-) is the scalar product. By K we
denote a non-local nonlinear operator.
The vector function v can be regarded as a remainder term. We give estimates
which show that K and v are weak in a certain sense.
Our main technical tool is a comparison principle which shows that solutions to
a certain ordinary differential equation majorize solutions of equation (3) (compare
with [KM1], Sect.5, and [KM2]). It is this principle that we use to obtain estimates
for the vector function v and the operator K.
System (6) is the corner stone of our asymptotic theory. On one hand, it can be
applied to construct solutions of (3) with the vector h(t) asymptotically close to a
solution of the dynamical system
Da(t)=N{t:x(t)). (7)
On the other hand, one can try to show that solutions to (3) subject to the growth
restriction (4) have the asymptotic representation (5), where the vector h satisfies
(7).
Consider the differential equation in the domain B^ n R^:
C2m(9Mx) = ^ (-^(A^, {Q^U{X} }|,|^)), (8)
|Q|<m
where 9^ = V. By ^2m(0 we mean a homogeneous polynomial of degree 2m with
real constant coefficients.
We assume that (-1)^2^(0 is positive for ^ e W1
 \ 0. Without loss of gen￾erality we suppose that the coefficient of ^m(^) in 9^ is equal to (-I)771
. By
Na = ^Q^. {^}|/3|<m) we denote functions subject to the Caratheodory condi￾tions, that is these functions are measurable in x for all {Yg}^^ and continuous
in {yo}\Q\<m for almost all x.
VII-3
The solution u, which is understood in a variational sense, belongs to the local
Sobolev space W^m
(B^W^\ 0), p e (0,oo), and is subject to zero Dirichlet
conditions on (B^ n 9R^) \ 0.
Under some restrictions on the operators and solution we obtain the asymptotic
formula
u(x) - X(r)^ ,
where the function X satisfies the ordinary differential equation
9rX(r) + r71
-
1
 t E N^ X(r){ 9^ }^)9^E(x)d0 = 0 , (9)
^-1
 ,
where 5^~1
 is the upper unit hemisphere and x = rO, r = \x\. By E we denote the
solution of
C2m(9Mx) = 0 (10)
in R^, positive homogeneous of degree m—n and subject to the Dirichlet conditions
on the hyperplane Xn = 0:
9
3
^E=0 for 0<j<m-2 , and 9^1
E=.8{x'},
where 8 is the Dirac 5-function. Up to the change of variables ( = logr~1
 this
equation is a special case of the dynamical system (7).
Let us discuss the special case when equation (8) is linear, that is we consider
the Dirichlet problem
C2mWu= ^ {-9^{N^x}9^u} inR^nBi, (11)
\a\,\/3\<m
9^u(x\ 0) = 0 for k = 0,1,... , m - 1 and x ' e (R71"
1
 n Bi) \ 0. (12)
Let Na0 satisfy
f E lllr^"IQ^
IA^^||ioc(^)P-l
^<oo. (13)
1/0
 \a\^\<m
where Qp = [x € Q : p > |.r| > y0/e} . Now equation (9) is written as
9rX(r) + r71
-
1
 t V N^{x)9^x^9^E(x)d0X{r) = 0
j Qn—\
t/5+ \a\,W<m
and therefore any solution of (11), (12) subject to
\u{x) | ^ C^r-71
-
1
-^ with ^ > 0 (14)
admits one of the following asymptotic representations:
VII-4
Either
or
u(x) ~ cE(x) exp { f ^ A^(0<9f2?(0 ^d(i\
1
 ^KI^KI |Q|,|/3|^
 J
u(x) ~ c^ exp { - ! ^ A./^)^ <^(0^} .
1
 •^Kl^la^m J
We formulate two special cases which seem to be of independent interest.
Consider the equation
n
" S <9^ (aij(x)Q^u) =0 in R^ n Bi
2,J=1
complemented by the boundary condition
H(:r',0)=0 forrr'eR^nBi.
Let
/*! "
/ ^ ll%•(•)-^lli-(^,)P~l
^<^)
 •
^o , ._i / ElM
zj=l
Then any solution u subject to
\u(x)\ ^Cp1
-^ (15)
admits the asymptotic representation
u(x)^cx^xp{^w f i>.n(o-m(^i-")rfd.
1 7r
 J\x\<\a,\<\ ^i
 )
Another example is provided by the Schrodinger equation
(iV + ^(x^u - P(x)u = 0 inR^nBi
complemented by zero Dirichlet condition on the hyperplane x^ = 0. Here m is a
magnetic vector potential and P is an electric potential. The condition (13) becomes
rsup^lm^l+^IP^D^Oc.
Jolo ee^ip ^p P
Then all solutions subject to (15) admit the asymptotic representation
, , f r(n/2) r /-, , .. . n(m(0,a \ ^2
 )
„(,) ~ €,„ exp { - ±^ j^ [PW - H2
 - -^^.)^^} .
Now we come back to the Lipschitz domain Q and consider equation (8) in B^nQ.
Let u be a variational solution of (8) in the local Sobolev space ^^{B^ n Q \ 0}
and subject to zero Dirichlet conditions on (B^ n 9Q) \ 0.
VII-5
The results obtained are new even for solutions to the linear equation (10).
Therefore we begin by describing the asymptotic behaviour of solutions to (10). We
show that any solution to equation (10) in B^ D 0 satisfying the zero Dirichlet con￾ditions on (B^n9^l)\0 and the estimate (14) is subject to the following asymptotic
alternatives:
Either
u(x) - c E(x^ Xn - /Or')) exp { f f^Q^E^^ Q}dA
U\x\<\^\<\ 'n
 J
or
u(x) - c Or, - /(^rexp {- t f(^E^^)d^} , (16)
I J\x\<^\<! ) '
where c = const.
In the special case of the polyharmonic operator C^m == (-A)'71 the above alter￾natives become:
Either
(xn-f^}^ ( r(n/2) r .. . d^ }
u(x) ~ c—————— exp {m , / f ((•}——>
^ \ ^ !w^ ^\n
}
or
^)^(..-swr^{-mW /^w)§].
If m = 1 and n = 2 the last formula coincides with the Warschawski asymptotic
formula (1).
We give two examples of application of our asymptotic theory to quasilinear
partial differential equations. Consider the equation
^m{Qx)u = IV^)^)/^-^ in B? n Q.
where k is a non-negative integer such that m — n < k < m. Now, we deal with
an arbitrary solution u € W^^ n Q \ 0) satisfying zero Dirichlet conditions on
(B^ n (90) \ 0. We replace (2) by the more stringent condition
|log|a:'|| |V/(:r')| < C for small [4 (17)
Then one of the following alternatives is valid:
Either u satisfies (16) or
u{x)^h(\x\)E(x^Xn-f(xf
))^
where h is a solution to the equation
h\r) + ^hir) - ^{r}^^'^-^ = 0
r- r
with
/(r)= f f^Q^E^e'W
J5"-2
VII-6
and
a= [ \{^kE)(0)$M)/k^
n
~
m)0
m
de .
^-1
The second example deals with an arbitrary solution u to the equation
Ai7 = la;!"4
!^!2
^
in B^ n ^ which is continuous in B^ n Q \ C? and vanishes on (B^ n <9Q) \ 0. Using
our asymptotic scheme we show that under condition (17)
u[x}^^h(\x${xn-f{x1
}}
where e is a constant unit vector and h is a solution of the equation
h\r} - ^(r) - ^(r) = 0 .
r- r
Here -'-^u^
and
a = { 6^d6 .
Jsr1
In conclusion we add that the same ideas led us in [KM3] to an asymptotic
description of all solutions to the Neumann problem for the two-dimensional Riccati
equation:
^u+a{x)(^^u)2
+2(3(x)9^u9^u+^x)(9^u)2
=0 (18)
in the sector K§ = [x = (rri,^) € R2
 : 0 < r < 6,0. € (0,y?)}, where (r,0) are the
polar coordinates of x, and y € (0, 27r],
9eu\e=o = 9eu\e=^ = 0 for r < 5.
Here a, /3 and 7 are measurable functions. We suppose that for almost all x € Ks
and for all (^2) € K2
^ (^i2
 + ^22
) < ^)^ + 2/?(.r)66 + 7(^22
 < A (^2
 + ^) (19)
with positive constants A and A. We assume that > belongs to the Sobolev space
W^(Ks\0).
There exist two possibilities: either u is unbounded and then
u(x) =Q(r)+c,+o(l!. (20)
where
Q(r}=
[^ds / r f a(x)x2
,+20{x)x,X2+-f{x)x^ , ^
/ —I / / ———————r~T"i———————dx^dx^ / ^
Jr S \J ]^K.\K_ |.E|4
Jr S\ j J^KAK. \X 4
IT > a ^J JX^KS\K,
VII-7
-1
or u is bounded and has the same asymptotics
u(x) = Co + cir^ cos(7T0/y) + o(r^) (21)
as in the case of the Neumann problem for ^\u = 0. Here c^ Co and ci are real
constants.
Clearly,
A log log - < Q(r) < A log log - for r < 6/e . (22)
If the coefficients a, (3 and 7 are constant we write the asymptotic expansion for
unbounded solutions:
«M~dlogl,gr-+c.+f:pt<'og"
>
^.'•g
),
^—^ flop-r-1
(logr-i)^^
k=l
where
ri= ( a + ^ g sin2
^ a-7 sin2y\
^ 2 0^ 4 (^ y
^a + 7 . sin2
^ a — 7 sin2y^-1
~T~ • ° ~^ + ~T~ ~p~) ?
and Pk{r, 0) are polynomials of degree <^ k m r whose coefficients are smooth func￾tions of 6 € [0, y]. If u is bounded it admits the asymptotic representation
m:r) ~- {x) Co + ^r^/^_i(logr^ (23)
A;=l
where Co = const and pk are polynomials in the first argument with smooth coeffi￾cients on [0, y?].
These results and their proofs can be extended to the case when 0 is the center
of the disk Kg = [x : r < 6}. One should only put y = TT in (21) and (23). In other
words we also describe the asymptotic behaviour of solutions to equation (18) which
are either bounded at 0 or have an isolated singularity there.
It is worth noting that equation (18) and the Neumann conditions as well as as￾sumption (19) about the coefficients a, (3 and 7 are preserved under conformalmap￾pings. Therefore (20) and ( 21) along with asymptotics ofconformal mappings (see
[W]) imply asymptotic representations of solutions at infinity and near boundary
singularities other than corners, for example, cusps.
The requirement (19) can not be removed. In fact, the function U{x) = logr~1
 4-
cos 0 is a solution of the homogeneous Neumann problem for the equation
^—^[WI^O
1 + sin2
 0
in the upper half-plane. Bv (22) the singularity of this solution is stronger than that
in (21).
VII-8
References
[KMl] Kozlov, V., Maz'ya, V.: Differential Equations with Operator Coefficients
(with Applications to Boundary Value Problems for Partial Differential Equa￾tions), Monographs in Mathematics, Springer-Verlag, 1999.
[KM2] Kozlov, V., Maz'ya, V.: Comparison principles for nonlinear operator differ￾ential equations in Banach spaces, Differential Operators and Spectral Theory
(M.Sh. Birman's 70th Anniversary Collection), Amer. Math. Soc. TransL, Ser.
2, 189 (1999).
[KM3] Kozlov, V., Maz'ya, V.: Angle singularities of solutions to the Neumann
problem for the two-dimensional Riccati's equation, Asymptotic Analysis 19
(1999), 57-79.
[KMR] Kozlov, V., Maz'ya, V. and Rossmann J.: Elliptic boundary value problems
in domains with point singularities, Mathematical Surveys and Monographs 52
(1997), Amer. Math. Soc.
[W] Warschawski, S.E.: On con formal mapping of infinite strips, Trans. Amer.
Math. Soc. 52 (1942), 280-335.
DEPARTMENT OF MATHEMATICS, LINKOPING UNIVERSITY, S-581 83 LINKOPING,
SWEDEN
vlmaz@clapton.mai.liu.se
VII-9

Boundary singularities of solutions to quasilinear elliptic equations

Wei-Min Wang

Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in $\mathbf {Z}^d$

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From pseudodifferential analysis to modular form theory

Long range scattering and modified wave operators for Hartree equations

The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation

Propagation of singularities in many-body scattering in the presence of bound states

Two lectures on spectral invariants for the Schrödinger operator

Évolution de tourbillon à support compact

Boundary singularities of solutions to quasilinear elliptic equations

Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in $\mathbf {Z}^d$

Injectivité de la transformation de Radon sur les grassmanniennes

Monoïdes libres dans les groupes hyperboliques

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