Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
Подождите немного. Документ загружается.
20
Karen
A.
Ames
where
u
is
a
smooth function such that
v(0)
=
1,
u(p1)
=
0,
and
d(p)
<
0.
We note that the constraints required to stabilize prob-
lem (4.18), regarded
as
a
modeling perturbation of (4.19), are much
stronger than the usual
C2
constraint that stabilizes other ill posed
problems against various sources of error. We do not go into details
about these constraints but refer the reader to Bennett's comprehen-
sive account of such nonstandard problems.
B.
An end problem
for
a one dimensional nonlinear heat equation
In this example, we wish to relate the solution of
"="[p(l$12)g]
at
ax
o<x<xo,
t>tl
to that of the ill posed problem
aw
a2w
----
-0
o<x<xo,
t>tl
at
ax2
(4.21)
(4.22)
Here again the data were assumed to be "close" in the appropri-
ate measure and
p(luZl2)
satisfied certain order conditions. Bennett
obtained
a
Holder continuous dependence inequality on the perturba-
tion parameter
c
for solutions belonging to the appropriate constraint
classes.
The papers
of
Ames (1982a,b) might also be interpreted
as
contin-
uous dependence on modeling investigations. Ames compared solu-
tions of Cauchy problems for certain classes of first and second order
operator equations with solutions of associated perturbed problems.
Neither the original problem nor the perturbed problem was required
to be well posed. Holder stability inequalities relating solutions of
the perturbed and unperturbed problems were obtained by Ames
using logarithmic convexity arguments. The class of problems and
Improperly Posed Problems for Nonlinear PDEs
21
type of derivable results can be exemplified with
a
comparison of the
solution of
'Ilt
+
Mu
=
F(t,u)
t
E
[O,T)
(4.23)
40)
=
fl
with the solution
of
the perturbed problem
Wt
+
Mw
+
rM2w
=
F(t,w)
t
E
[O,T)
(4.24)
40)
=
f2.
Here
r
is
a
small positive parameter,
M
is
a
symmetric, time indepen-
dent linear operator (bounded or unbounded) defined
on
a
Hilbert
space, and the nonlinear term
F
satisfies
a
uniform Lipschitz condi-
tion in its second argument. Ames showed that if
u
and
w
belong
to the appropriate constraint sets and if the Cauchy data are close
(in
a
suitably defined sense), then the difference
u
-
w
in
a
certain
measure is
of
order
~fi(~)
for
0
5
t
<
T
where
p(t)
>
0.
We emphasize that the studies attempting to address the ques-
tion of continuous dependence
on
modeling must be regarded only
as
mathematical illustrations. Unfortunately, we cannot interpret the
results
of
these investigations
as
answers to the modeling questions
that arise in actual physical problems.
In
fact, the question
of
sta-
bilizing improperly posed problems for partial differential equations
against errors in modeling might well be the major unresolved issue
in
the field of ill posed problems.
Bibliography
Adelson,
L.
(1973)
Singular perturbations
of
improperly
posed
prob-
lems.
SIAM
J.
Math. Anal.,
4, 344-366.
Adelson,
L.
(1974)
Singular perturbation
of
an improperly posed
Cauchy problem,
SIAM
J.
Math. Anal.,
5,417-424.
Agmon,
S.
(1966)
Unicite' et Convexite' duns les Probltmes Diffe'rentiels,
University
of
Montreal Press, Montreal.
22
Karen
A.
Ames
Ames,
K.
A.
(1982a)
On the comparison
of
solutions
of
related prop-
erly and improperly posed Cauchy problems
for
first order op-
emtor equations,
SIAM
J.
Math. Anal.,
13, 594-606.
Ames,
K.
A.
(1982b)
Comparison results
for
related properly and
improperly posed Cauchy problems
for
second order operator
equations,
J.
Differential Equations,
44, 383-399.
Ames,
K.
A., Levine,
H.
A. and Payne, L. E.
(1987)
Improved con-
tinuous dependence results
for
a
class
of
evolutionary equations,
Inverse and Ill-Posed Problems (ed.
H.
W.
Engl and
C.
W.
Groetsch) Academic Press,
443-450.
Ames,
K.
A., Payne, L.
E.
and Straughan,
B.
(1989)
On the possi-
bility
of
global solutions
for
variant models
of
viscous flow on
unbounded domains,
Int.
J.
Engng. Sci.,
27, 755-766.
Bell,
J.
B.
(1981a)
The nonchamcteristic Cauchyproblem
for
a
class
of
equations wtih time dependence I, Problems
in
one space
dimension,
SUM
J.
Math. Anal.,
12, 759-777.
Bell,
J.
B.
(1981b)
The nonchamcteristic Cauchy
pmbkrn
for
a
class
of
equations with time dependence
11,
Multidimensional
problems,
SIAM
J.
Math. Anal.,
12, 778-797.
Bennett, A. D.
(1986)
Continuous dependence on modeling in the
Cauchy pmblem
for
second order nonlinear partial diferential
equations,
Ph.D. Thesis, Cornell University.
Cannon,
J.
R. and Knightly,
G.
H.
(1970)
Some continuous
de-
pendence theorems
for
viscous
fluid
motions,
SIAM
J.
Appl.
Math.,
18,
627-640.
Carasso, A. and Stone, A.
P.
(1975)
Improperly Posed Boundary
Value Problems,
Pitman Research Notes in Mathematics,
Vol.
1,
Pitman Press, London.
Crooke,
P.
S.
and Payne, L.
E.
(1984)
Continuous dependence on
geometry
for
the backward heat equation,
Math. Methods in
the Applied Sciences,
6, 433-448.
Improperly Posed Problems for Nonlinear PDEs
23
Engl,
H.
W.
and Groetsch,
C.
W.,
ed.
(1987)
Inverse and Ill-Posed
Problems,
Notes and Reports in Mathematics in Science and
Engineering, Vol.
4,
Academic Press, Inc.
Ewing,
R.
E.
(1975)
The approximation
of
certain pambolic equa-
tions backwad
in
time
by
Sobolev equations,
SIAM
J.
Math.
Anal.,
6,283-294.
Fujita,
H.
(1966)
On
the blowing up
of
solutions to the Cauchy
problem
for
ut
=
Au
+
u’+~,
J.
Fac. Sci. Univ. Tokyo,
13,
109- 124.
Fujita,
H.
(1970)
On some nonexistence and nonuniqueness the-
orems
for
nonlinear pambolic equations,
Proc. Symp. Pure
Math.,
18, 105-113.
Galdi, G.
P.
and Straughan,
B.
(1988)
Stability
of
solutions to
the Navier-Stokes equations backward
in
time,
Arch. Ratio-
nal Mech. And.,
101, 107-114.
Glassey, R.
T.
(1973)
Blow
up
theorems
for
nonlinear wave equa-
tions,
Math.
Z.,
132, 183-203.
Glassey, R.
T.
(1981)
Finite time blow up
for
solutions
of
nonlinear
wave equations,
Math.
Z.,
177, 323-340.
Hills,
R.
N.
and Knops, R.
J.
(1974)
Concavity and the evolutionary
properties
of
a class
of
geneml materials,
Proc.
Roy.
SOC.
Edinburgh,
72, 239-243.
John,
F.
(1955)
A
note on “improper” problems in partial differen-
tial equations,
Comm. Pure Appl. Math.,
8, 494-495.
John,
F.
(1960)
Continuous dependence on data
for
solutions
of
partial differential equations with a prescribed bound,
Comm.
Pure. Appl. Math.,
13, 551-585.
John,
F.
(1979)
Blow up
of
solutions
of
nonlinear wave equations
in three dimensions,
Man. Math.
28, 235-268.
24
Karen
A.
Ames
Kaplan,
S.
(1963)
On the gmwth
of
solutions
of
quasilinear pambolic
equations,
Comm.
Pure Appl. Math.,
16, 305-330.
Knops,
R.
J.
ed
(1973)
Symposium on Non-Well-Posed Problems
and Logarithmic Convexity,
Springer Lecture
Notes,
#316.
Knops, R.
J.,
Levine,
H.
A. and Payne, L.
E.
(1974)
Nonexistence,
instability and gmwth theorems
for
solutions
of
a class
of
ab-
stmct nonlinear equations with applications to nonlinear elas-
todynamics,
Arch. Rational Mech. And.,
55, 52-72.
Knops, R.
J.
and Payne,
L.
E.
(1968)
On the stability
of
solutions
of
the Navier-Stokes equations backward in time,
Arch. Rational
Me&. And.,
29,331-335.
for
the equations
of
classical elastodynamics,
Proc.
Phil.
SOC.,
66, 481-491.
Knops, R.
J.
and Payne,
L.
E.
(1979)
Uniqueness and continu-
ous
dependence
of
the null solution in the Cauchy problem
for
a nonlinear elliptic system.,
Mathematical Research-Inverse
and Improperly Posed Problems in Differential Equations (ed.
G.
Anger), Vol.
1,
151-160.
Knops. R.
J.
and Payne,
L.
E.
(1983)
Some uniqueness and con-
tinuous dependence theorems
for
non-linear elastodynamics in
exterior domains,
Appl. Anal.,
15, 33-51.
Knops,
R.
J.
and Payne,
L.
E.
(1988)
Improved estimates
for
contin-
uous
data dependence in linear elastodynamics,
Proc.
Camb.
Knops,
R.
J.
and Payne,
L.
E.
(1969)
Continuous data dependence
Camb.
Phil.
SOC.,
103, 535-559.
Knops,
R.
J.
and Straughan,
B.
(1976)
Non-existence
of
global
so-
lutions to non-linear Cauchy problems arising in mechanics,
Trends in Applications
of
Pure Mathematics to Mechanics, Pit-
man, London.
Lattks,
R.
and Lions,
J.
L.
(1969)
The
Method
of
Quasireuersibil-
ity, Applications to Partial Differential Equations,
American
Elsevier, New York.
Improperly Posed Problems for Nonlinear PDEs
25
Lavrentiev,
M.
M. (1956)
On the Cauchy pmblem
for
the Luphce
equation,
Izvest. Akad. Nauk SSR, Ser. Math., 120, 819-842.
Lavrentiev, M. M. (1967)
Some Improperly Posed Problems in Math-
ematical Physics,
Springer-Verlag, New York.
Lavrentiev, M. M., Romanov,
V.
G.
and Sisatskii,
S.
P.
(1986)
Ill-
Posed Problems
of
Mathematical Physics and Analysis, Tmns-
lations
of
Mathematical Monographs,
64,
American Mathemat-
ical Society, Providence.
Lees,
M.
and Protter, M.
H.
(1961)
Unique continuation
for
pambolic
differential equations and differential inequalities,
Duke Math.
J.,
28, 369-382.
Levine,
H.
A. (1970a)
Logarithmic convexity and the Cauchy prob-
lem
for
some abstmct second order differential inequalities,
J.
Differential Equations,
8,
34-55.
Levine,
H.
A. (1970b)
Logarithmic convexity, first order differential
inequalities and some applications,
Trans. Amer. Math. SOC.,
152, 299-319.
Levine,
H.
A.
(1973)
Some nonexistence and instability theorems
for
solutions
of
formally pambolic equations
of
the form Put
=
-Au
+
F(u),
Arch. Rational Mech. Anal.,
51,
371-386.
Levine,
H.
A. (1974a)
Instability and nonexistence
of
global
solutions
to nonlinear wave equations
of
the form Putt
=
-Au
+
F(u),
Trans. Amer. Math. SOC., 192, 1-21.
Levine,
H.
A. (1974a)
Some additional remarks on the nonexistence
of
global
solutions to nonlinear wave equations,
SIAM
J.
Math.
Anal.
5,
138-146.
Levine,
H.
A. (1974~)
A
note on a nonexistence theorem
for
non-
linear wave equations,
SIAM
J.
Math. Anal.,
5,
644-648.
Levine,
H.
A. (1990)
The
role
of
critical exponents in blow up the-
orems,
SIAM Review, 32, 262-288.
26
Karen
A.
Ames
Levine,
H.
A. and Payne,
L.
E.
(1974a) Nonezistence theorems
for
the heat equation with nonlinear boundary conditions and
for
the
porous
medium equation backward
in
time,
J.
Differential
Equations, 16, 319-334.
Levine,
H.
A. and Payne,
L.
E.
(1974b) Some nonezistence theorems
for initial boundary value problems with nonlinear boundary
constmints, Proc. Amer. Math. SOC., 46, 277-284.
Levine,
H.
A. and Payne, L.
E.
(1976) Nonezistence for global weak
solutions
for
classes of nonlinear wave and pambolic equations,
J.
Math. Anal. Appl.,
55,
329-334.
Lewy,
H.
(1957) An example
of
a
smooth linear partial differential
equation without solution, Ann.
of
Math., 66,
155-158.
Miller,
K.
(1973a) Non-unique continuation
for
certain
ODE’S
in
Hilbert
space
and
for uniformly parabolic elliptic equations
in
self-adjoint divergence form, Symp. on Non-well-posed Prob-
lems
and
Logarithmic Convexity, Springer Lecture Notes
#
316, 85-101.
Miller,
K.
(1973b) Stabilized quasi-reversibilite and other nearly-
best-possible methods for non-well-posed pmblems, Symp. on
Non-well-posed Problems and Logarithmic Convexity, Springer
Lecture
Notes
#316, 161-176.
Miller,
K.
(1974) Nonunique continuation for uniformly pambolic
and
elliptic equations in self-adjoint divergence form with Hdder
continuous coeficients, Arch. Rational Mech. Anal., 54, 105-
117.
Morozov, V. A. (1984) Methods
for
Solving Incorrectly Posed Prob-
lems, Springer-Verlag, Berlin-Heidelberg-New York.
Murray, A. (1972) Uniqueness
and
continuous dependence for the
equations of elastodynamics without strain energy function, Arch.
Rat. Mech. Anal., 47, 195-204.
Improperly Posed Problems for Nonlinear PDEs
27
Murray, A. C. and Protter,
M.
H.
(1973)
The asymptotic behav-
ior
of
solutions
of
second order systems
of
partial differential
equation,
J.
Differential Equations,
13,
57-80.
Payne,
L.
E.
(1975)
Improperly Posed Problems in Partial Difler-
ential Equations,
Regional Conference Series in Applied Math-
ematics, Vol. 22, SIAM.
Payne,
L.
E.
(1985)
Improved stability estimates
for
classes
of
ill-
posed Cauchy problems,
Applicable Analysis, 19, 63-74.
Payne,
L.
E.
(1987a)
On stabilizing ill posed problems against errors
in geometry and modeling,
Inverse and 111 Posed Problems (ed.
H.
W.
Engl and C.
W.
Groetsch), Academic Press, 443-450.
Payne,
L.
E.
(1987b)
On geometric and modeling perturbations in
partial diflerential equations,
L.M.S. Symposium on Non-Classical
Continuum Mech. Proc.
ed.
by R.
J.
Knops and A. A. Lacey,
Cambridge Univ. Press, 108-128.
Payne,
L.
E.
(1989)
Continuous dependence on geometry with appli-
cations
in
continuum mechanics,
Continuum Mechanics and
its
Applications, ed.
G.
A.C. Graham and
S.
K.
Malitz, 877-890.
Payne, L.
E.
and Sather,
D.
(1967)
On singular perturbations
of
non-well-posed problems,
Annali
di
Mat. Pura ed. Appl., 75,
21 9-230.
Payne, L.
E.
and Straughan,
B.
(1990a)
Emr
estimates
for
the
tempernture
of
a
piece
of
cold ice, given data on
only
part
of
the boundary,
Nonlinear Anal., Theory, Meths. Applicns., 14,
443-452.
Payne, L.
E.
and Straughan,
B.
(1990b)
Effects
of
emrs in the
initial-time geometry on the solution
of
the heat equation in
an exterior domain,
Q.
J.
Mech. Appl. Math., 43, 75-86.
Payne,
L.
E.
and Straughan,
B.
(1990~)
Improperly posed and non-
standard problems
for
parabolic partial differential equations,
28
Karen
A.
Ames
Ian Sneddon
70th
Anniversary Volume,
(G.
Eason
&
R.
W.
Ogden, eds.), Ellis-Horwood Pub. Co.,
273-299.
Payne,
L.
E.and Straughan, B. Continuous dependence on geometry
for
the backward heat equation
in
an ezterior domain, Proc.
Lond. Math. SOC. (to appear).
Persens,
J.
(1986)
On stabilizing
ill
posed
problems in partial dif-
ferential equations under perturbations
of
the geometry
of
the
domain, Ph.D. Thesis, Cornell University.
Plis, A.
(1960)
Nonuniqueness
in
Cauchy
’s
problem
for
differential
equations
of
elliptic type,
J.
Math. Mech.,
9, 557-562.
Plis, A.
(1963)
On nonuniqueness in the Cauchy problem
for
an
elliptic second order diferential equation, Bull. Acad. Polon.
Sci.,
11, 95-100.
Pucci, C.
(1955)
Sui problem; di Cauchy non “ben posti, Rend. Ac-
cad.
Nu.
Lincei,
18, 473-477.
Sattinger, D.
(1968)
On
global
solutions
of
nonlinear hyperbolic
equations, Arch. Rational Mech. Anal.,
30,
148-172.
Serrin,
J.
(1963)
The initial value problem
for
the Navier-Stokes
equations in Nonlinear Problems, Univ. Wisconsin Press, Madi-
son.
Showalter,
R.
E.
(1974)
The final value problem
for
evolution equa-
tions,
J.
Math. Anal. Appl.,
47, 563-572.
Showalter, R.
E.
(1975)
Quasi-reuersibility
of
first and second
or-
der parabolic evolution equations, Pitman Research Notes in
Mathematics, Vol.
1,
76-84.
Song,
J.
C.
(1988)
Some stability crtieria
in
fluid
and
solid mechan-
ics, Ph.D. Thesis, Cornell University.
Straughan, B.
(1975a)
Uniqueness and stability
for
the conduction-
diffusion solution to the Bousssinesq equations backward in time,
Proc. Roy.
SOC.
A,
347,435-446.
Improperly Posed Problems for Nonlinear PDEs
29
Straughan,
B.
(1975b)
firther global nonexistence theorems
for
ab-
stract nonlinear wave equations,
Proc. Amer. Math. SOC.,
48,
381-390.
Straughan,
B.
(1976)
Global nonexistence
of
solutions to Ludyzhen-
skaya
's
variants
of
the Navier-Stokes equations backward in
time,
Proc. R. SOC. Edinburgh,
75A, 165-170.
Straughan,
B.
(1977)
Uniqueness and continuous dependence the-
orems
for
the conduction-diffusion solution to the Boussinesq
equations on an exterior domain,
J.
Math. Anal. Appl.,
57,
203-233.
Straughan,
B.
(1982)
Instability, Nonexistence
and
Weighted Energy
Methods in
Fluid
Dynamics and Related Theories,
Pitman Re-
search Notes in Mathematics Vol.
74,
Pitman Press,
London.
Straughan,
B.
(1983)
Backward uniqueness and unique continuation
for
solutions to the Navier-Stokes equations on an exterior do-
main,
J.
Math. pures
et
appl.,
62, 49-62.
Tikhonov, A. N.
(1963)
On the solution
of
ill-posed problems and
the method
of
regularisation,
Dokl.
Akad.
Nauk
SSSR,
151,
501-504.
Tikhonov, A. N.
and
Arsenin, V.
Y.
(1977)
Solution
of
Ill-Posed
Problems,
Wiley Press.
Tsutsumi,
M.
(1971)
Some nonlinear evolution equations
for
second
order,
Proc.
Japan. Acad.,
47, 450-455.
Tsutsumi,
M.
(1972)
Existence and non-existence
of
global solutions
for
nonlinear parabolic equations,
Publ.
Research Inst. Math.
Sci., Kyoto
Univ.,
8, 211-229.