Taylor M.E. Partial Differential Equations III: Nonlinear Equations
Подождите немного. Документ загружается.
408 15. Nonlinear Parabolic Equations
[CH] J. Cannon and C. D. Hill, Existence, uniqueness, stability, and monotone de-
pendence in a Stefan problem for the heat equation, J. Math. Mech. 17(1967),
1–20.
[CZ] H.-D. Cao and X.-P. Zhu, A complete proof of the Poincar´e and geometrization
conjectures – application of the Hamilton-Perelman theory of the Ricci flow,
Asia J. Math. 10 (2006), 169–492.
[Car] G. Carpenter, A geometrical approach to singular perturbation problems, with
application to nerve impulse equations, J. Diff. Equ. 23(1977), 335–367.
[Cher] S. S. Chern (ed.), Seminar on Nonlinear Partial Differential Equations,MSRI
Publ. #2, Springer, New York, 1984.
[CHMM] A. Chorin, T. Hughes, M. McCracken, and J. Marsden, Product formulas and
numerical algorithms, CPAM 31(1978), 206–256.
[Chow] B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom. 33(1991), 325–334.
[Con] C. Conley, On travelling wave solutions of nonlinear diffusion equations,
pp. 498–510 in Lecture Notes in Physics #38, Springer, New York, 1975.
[DeT] D. DeTurk, Deforming metrics in the direction of their Ricci tensors, J. Diff.
Geom. 18 (1983), 157–162.
[DiB] E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York, 1993.
[DF] E. DiBenedetto and A. Friedman, H¨older estimates for nonlinear degenerate
parabolic systems, J. Reine Angew. Math. 357(1985), 1–22.
[Don] S. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic
surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50(1985), 1–26.
[Dong] G. Dong, Nonlinear Partial Differential Equations of Second Order,Transl.
Math. Monog., AMS, Providence, R. I., 1991.
[EL] J. Eells and L. Lemaire, A report on harmonic maps, Bull. Lond. Math. Soc.
10(1978), 1–68.
[ES] J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Am.
J. Math. 86(1964), 109–160.
[Ev] W. Everitt (ed.), Spectral Theory and Differential Equations, LNM #448,
Springer, New York, 1974.
[FS] E. Fabes and D. Stroock, A new proof of Moser’s parabolic Harnack inequality
via the old ideas of Nash, Arch. Rat. Mech. Anal. 96(1986), 327–338.
[Fi] P. Fife, Asymptotic states of equations of reaction and diffusion, Bull. AMS
84(1978), 693–724.
[Frd] M. Freidlin, Functional Integration and Partial Differential Equations,
Princeton University Press, Princeton, N. J., 1985.
[Fr1] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall,
Englewood Cliffs, N.J., 1964.
[Fr2] A.
Friedman, Variational Principles and Free Boundary Problems, Wiley,
New York, 1982.
[Giu] E. Giusti (ed.), Harmonic Mappings and Minimal Immersions, LNM #1161,
Springer, New York, 1984.
[Grin] P. Grindrod, Patterns and Waves, the Theory and Applications of Reaction-
Diffusion Equations, Clarendon, Oxford, 1991.
[Ham] R. Hamilton, Harmonic Maps of Manifolds with Boundary, LNS #471, Springer,
New York, 1975.
[Ham2] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom.
17(1982), 255–307.
[Ham3] R. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71(1988).
References 409
[HW] R. Hardt and M. Wolf (eds.), Nonlinear Partial Differential Equations in Dif-
ferential Geometry, IAS/Park City Math. Ser., Vol. 2, AMS, Providence, R. I.,
1995.
[Hen] D. Henry, Geometric Theory of Semilinear Parabolic Equations, LNM #840,
Springer, New York, 1981.
[Hild] S. Hildebrandt, Harmonic mappings of Riemannian manifolds, pp. 1–117 in
E. Giusti (ed.), Harmonic Mappings and Minimal Immersions, LNM #1161,
Springer, New York, 1984.
[HRW1] S. Hildebrandt, H. Raul, and R. Widman, Dirichlet’s boundary value prob-
lem for harmonic mappings of Riemannian manifolds, Math. Zeit. 147(1976),
225–236.
[HRW2] S. Hildebrandt, H. Raul, and R. Widman, An existence theory for harmonic
mappings of Riemannian manifolds, Acta Math. 138(1977), 1–16.
[Ho] L. H¨ormander, Non-linear Hyperbolic Differential Equations, Lecture Notes,
Lund University, 1986–1987.
[Iv] A. Ivanov, Quasilinear degenerate and nonuniformly elliptic and parabolic equa-
tions of second order, Proc. Steklov Inst. Math. 160(1984), 1–287.
[J1] J. Jost, Lectures on harmonic maps, pp. 118–192 in E. Giusti (ed.), Harmonic
Mappings and Minimal Immersions, LNM #1161, Springer, New York, 1984.
[J2] J. Jost, Nonlinear Methods in Riemannian and Kahlerian Geometry,Birkh¨auser,
Boston, 1988.
[K] T. Kato, Quasi-linear equations of evolution, with applications to partial differ-
ential equations, pp. 25–70 in W. Everitt (ed.), Spectral Theory and Differential
Equations, LNM #448, Springer, New York, 1974.
[KP] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes
equations, CPAM 41(1988), 891–907.
[KSt] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities
and Their Applications, Academic, New York, 1980.
[KPP] A. Kolmogorov, I. Petrovskii, and N. Piskunov, A study of the equations of dif-
fusion with increase in the quantity of matter, and its applications to a biological
problem, Moscow Univ. Bull. Math. 1(1937), 1–26.
[Kru] S. Krushkov, A priori estimates for weak solutions of elliptic and parabolic
differential equations of second order, Dokl. Akad. Nauk. SSSR 150(1963),
748–751. Engl. transl. Sov. Math. 4(1963), 757–761.
[Kry] N. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order,
D.Reidel, Boston, 1987.
[KryS] N. Krylov and M. Safonov, A certain property of solutions of parabolic equa-
tions with measurable coefficients, Math. USSR Izv. 16(1981), 151–164.
[KMP] K. Kunisch, K. Murphy, and G. Peichl, Estimates on the conductivity in the one-
phase Stefan problem I: basic results, Boll. Un. Mat. Ital. B 9(1009), 77–103.
[LSU] O
. Ladyzhenskaya, B. Solonnikov, and N. Ural’tseva, Linear and Quasilinear
Equations of Parabolic Type, AMS Transl. 23, Providence, 1968.
[Leu] A. Leung, Systems of Nonlinear Partial Differential Equations, Kluwer, Boston,
1989.
[Lie] G. Lieberman, The first initial-boundary value problem for quasilinear second
order parabolic equations, Ann. Sc. Norm. Sup. Pisa 13(1986), 347–387.
[Mars] J. Marsden, On product formulas for nonlinear semigroups, J. Funct. Anal.
13(1973), 51–72.
[McK] H. McKean, Application of Brownian motion to the equation of Kolmogorov–
Petrovskii–Piskunov, CPAM 28(1975), 323–331.
410 15. Nonlinear Parabolic Equations
[Mei] A. Meirmanov, The Stefan Problem, W. deGruyter, New York, 1992.
[MT] J. Morgan and G. Tian, Ricci Flow and the Poincar’e Conjecture,ClayMath.
Monogr. #3, AMS, Providence, RI, 2007.
[Mos1] J. Moser, A new proof of DeGiorgi’s theorem concerning the regularity problem
for elliptic differential equations, CPAM 13(1960), 457–468.
[Mos2] J. Moser, A Harnack inequality for parabolic differential equations, CPAM
15(1964), 101–134.
[Mos3] J. Moser, On a pointwise estimate for parabolic differential equations, CPAM
24(1971), 727–740.
[Mos4] J. Moser, A rapidly convergent iteration method and nonlinear partial differen-
tial equations, I, Ann. Sc. Norm. Sup. Pisa 20(1966), 265–315.
[Mur] J. Murray, Mathematical Biology, Springer, New York, 1989.
[Na] J. Nash, Continuity of solutions of parabolic and elliptic equations, Am.J.Math.
80(1958), 931–954.
[NPS] W.-M. Ni, L. Peletier, and J. Serrin (eds.), Nonlinear Diffusion Equations and
Their Equilibrium States, MSRI Publ., Vols. 12–13, Springer, New York, 1988.
[Per1] G. Perelman, The entropy formula for the Ricci flow and its geometric applica-
tions, math.DG/0211159, 2002.
[Per2] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain
three-manifolds, math.DG/0307245, 2003.
[Per3] G. Perelman, Ricci flow with surgery on three-manifolds, math.DG/0303109,
2003.
[Po] J. Polking, Boundary value problems for parabolic systems of partial differential
equations, Proc. Symp. Pure Math. X(1967), 243–274.
[Rab] J. Rabinowitz, A graphical approach for finding travelling wave solutions to
reaction-diffusion equations, Senior thesis, Math. Dept., University of North
Carolina, 1994.
[Rau] J. Rauch, Global existence for the Fitzhugh–Nagumo Equations, Comm. PDE
1(1976), 609–621.
[RaSm] J. Rauch and J. Smoller, Qualitative theory of the Fitzhugh–Nagumo equations,
Adv. Math. 27(1978), 12–44.
[Rot] F. Rothe, Global Solutions of Reaction-Diffusion Equations, LNM #1072,
Springer, New York, 1984.
[Rub] L. Rubenstein, The Stefan Problem, Transl. Math. Monogr. #27, AMS, Provi-
dence, R. I., 1971.
[Sch] R. Schoen, Analytic aspects of the harmonic map problem, pp. 321–358 in
S. S. Chern (ed.), Seminar on Nonlinear Partial Differential Equations,MSRI
Publ. #2, Springer, New York, 1984.
[ScU1] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff.
Geom. 17(1982), 307–335.
[ScU2] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem
for harmonic maps, J. Diff. Geom. 18(1983), 253–268.
[Siu] Y. - T. S i u , Lectures on Hermitian–Einstein Metrics for Stable Bundles and
K¨ahler–Einstein Metrics,Birkh¨auser, Basel, 1987.
[Smo
] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer,
New York, 1983.
[Str] M. Struwe, Variational Methods, Springer, New York, 1990.
[Str2] M. Struwe, Geometric evolution problems, pp. 259–339 in R. Hardt and
M. Wolf (eds.), Nonlinear Partial Differential Equations in Differential Geom-
etry, IAS/Park City Math. Ser., Vol. 2, AMS, Providence, R. I., 1995.
References 411
[Tay] M. Taylor, Pseudodifferential Operators and Nonlinear PDE,Birkh¨auser,
Boston, 1991.
[Tro] A. Tromba, Teichmuller Theory in Riemannian Geometry, ETH Lectures in
Math., Birkh¨auser, Basel, 1992.
[Tso] K. Tso, Deforming a hypersurface by its Gauss–Kronecker curvature, CPAM
38(1985), 867–882.
[Wan] W.-P. Wang, Multiple impulse solutions to McKean’s caricature of the nerve
equation, CPAM 41(1988), 71–103; 997–1025.
[Ye] R. Ye, Global existence and convergence of Yamabe flow, J. Diff. Geom.
39(1994), 35–50.
16
Nonlinear Hyperbolic Equations
Introduction
Here we study nonlinear hyperbolic equations, with emphasis on quasi-linear
systems arising from continuum mechanics, describing such physical phenom-
ena as vibrating strings and membranes and the motion of a compressible fluid,
such as air.
Sections 1–3 establish the local solvability for various types of nonlinear hy-
perbolic systems, following closely the presentation in [Tay]. At the end of 1 we
give some examples of some equations for which smooth solutions break down in
finite time. In one case, there is a weak solution that persists, with a singularity.
This is explored more fully later in the chapter.
In 4 we prove the Cauchy–Kowalewsky theorem, in the nonlinear case, us-
ing the method of Garabedian [Gb2] to transform the problem to a quasi-linear,
symmetric hyperbolic system.
In 5 we derive the equations of ideal compressible fluid flow and discuss some
classical results of Bernoulli, Kelvin, and Helmholtz regarding the significance of
the vorticity of a fluid flow.
In 6 we begin the study of weak solutions to quasi-linear hyperbolic systems
of conservation law type, possessing singularities called shocks. Section 6 is de-
voted to scalar equations, for which there is a well-developed theory.
We then study k k systems of conservation laws, with k 2,in 7–10,
restricting attention to the case of one space variable. Section 7 is devoted to the
“Riemann problem,” in which piecewise-constant initial data are given. Section 8
discusses the role of “entropy” and of “Riemann invariants” for systems of con-
servation laws. These concepts are used in 9, where we establish a result of
R. DiPerna [DiP4] on the global existence of entropy-satisfying weak solutions
for a class of 2 2 systems, in one space variable.
The first nonlinear hyperbolic system we derived, in 1 of Chap. 2, was the
system for vibrating strings. We return to this in 10. Far from setting down a
definitive analysis, we make note of some further subtleties that arise in the study
M.E. Taylor, Partial Differential Equations III: Nonlinear Equations,
Applied Mathematical Sciences 117, DOI 10.1007/978-1-4419-7049-7
4,
c
Springer Science+Business Media, LLC 1996, 2011
413
414 16. Nonlinear Hyperbolic Equations
of vibrating strings, giving rise to problems that have by no means been overcome.
This starkly illustrates that in the study of nonlinear hyperbolic equations, a great
deal remains to be done.
1. Quasi-linear, symmetric hyperbolic systems
In this section we examine existence, uniqueness, and regularity for solutions to a
system of equations of the form
(1.1)
@u
@t
D L.t; x; u;D
x
/u C g.t; x; u/; u.0/ D f:
We derive a short-time existence theorem under the following assumptions. We
suppose that
(1.2) L.t; x; u;D
x
/v D
X
j
A
j
.t; x; u/@
j
v
and that each A
j
is a K K matrix, smooth in its arguments, and furthermore
symmetric:
(1.3) A
j
D A
j
:
We suppose g is smooth in its arguments, with values in R
K
I u D u.t; x/ takes
values in R
K
.Wethensay(1.1) is a symmetric hyperbolic system. For sim-
plicity, we suppose x 2 M D T
n
, though any compact manifold M can be
treated with minor modifications, as can the case M D R
n
. We will suppose
f 2 H
k
.M /; k > n=2 C 1.
Our strategy will be to obtain a solution to (1.1) as a limit of solutions u
"
to
(1.4)
@u
"
@t
D J
"
L
"
J
"
u
"
C g
"
; u
"
.0/ D f;
where
(1.5) L
"
v D
X
j
A
j
.t;x;J
"
u
"
/@
j
v
and
(1.6) g
"
D J
"
g.t; x; J
"
u
"
/:
In (1.4), f might also be replaced by J
"
f , though this is not crucial. Here, fJ
"
W
0<" 1g is a Friedrichs mollifier. For M D T
n
, we can define J
"
by a Fourier
series representation:
1. Quasi-linear, symmetric hyperbolic systems 415
(1.7)
J
"
v
O.`/ D '."`/Ov.`/; ` 2 Z
n
;
given ' 2 C
1
0
.R
n
/, real-valued, '.0/ D 1.Forany">0,(1.4) can be re-
garded as a system of ODEs for u
"
, for which we know there is a unique solution,
for t close to 0. Our task will be to show that the solution u
"
exists for t in an
interval independent of " 2 .0; 1 and has a limit as " & 0 solving (1.1).
To do this, we estimate the H
k
-norm of solutions to (1.4). We begin with
(1.8)
d
dt
kD
˛
u
"
.t/k
2
L
2
D 2.D
˛
J
"
L
"
J
"
u
"
;D
˛
u
"
/ C 2.D
˛
g
"
;D
˛
u
"
/:
Since J
"
commutes with D
˛
and is self-adjoint, we can write the first term on the
right as
(1.9) 2.L
"
D
˛
J
"
u
"
;D
˛
J
"
u
"
/ C 2.ŒD
˛
;L
"
J
"
u
"
;D
˛
J
"
u
"
/:
To estimate the first term in (1.9), note that, by the symmetry hypothesis (1.3),
(1.10) .L
"
C L
"
/v D
X
j
Œ@
j
A
j
.t;x;J
"
u
"
/v;
so we have
(1.11) 2.L
"
D
˛
J
"
u
"
;D
˛
J
"
u
"
/ C
kJ
"
u
"
.t/k
C
1
kD
˛
J
"
u
"
k
2
L
2
:
Next, consider
(1.12) ŒD
˛
;L
"
v D
X
j
D
˛
.A
j"
@
j
v/ A
j"
D
˛
.@
j
v/
;
where A
j"
D A
j
.t;x;J
"
u
"
/. By the Moser estimates from Chap.13, 3 (see
Proposition 3.7 there), we have
(1.13)
kŒD
˛
;L
"
vk
L
2
C
X
j
kA
j"
k
H
k
k@
j
vk
L
1
CkrA
j"
k
L
1
k@
j
vk
H
k1
;
provided j˛jk. We use this estimate with v D J
"
u
"
. We also use the estimate
(1.14) kA
j
.t;x;J
"
u
"
/k
H
k
C
k
kJ
"
u
"
k
L
1
1 CkJ
"
u
"
k
H
k
;
which follows from Proposition 3.9 of Chap. 13. This gives us control over the
terms in (1.9), hence of the first term on the right side of (1.8). Consequently, we
obtain an estimate of the form
(1.15)
d
dt
ku
"
.t/k
2
H
k
C
k
kJ
"
u
"
.t/k
C
1
1 CkJ
"
u
"
.t/k
2
H
k
:
416 16. Nonlinear Hyperbolic Equations
This puts us in a position to prove the following:
Lemma 1.1. Given f 2 H
k
;k>n=2C1, the solution to (1.4)existsfort in an
interval I D .A; B/, independent of ", and satisfies an estimate
(1.16) ku
"
.t/k
H
k
K.t/; t 2 I;
independent of " 2 .0; 1.
Proof. Using the Sobolev imbedding theorem, we can dominate the right side of
(1.15)byE
ku
"
.t/k
2
H
k
,soku
"
.t/k
2
H
k
D y.t/ satisfies the differential inequality
(1.17)
dy
dt
E.y/; y.0/ Dkf k
2
H
k
:
Gronwall’s inequality yields a function K.t/, finite on some interval Œ0; B/,giving
an upper bound for all y.t/ satisfying (1.17). Using time-reversibility of the class
of symmetric hyperbolic systems, we also get a bound K.t/ for y.t/ on an interval
.A; 0.ThisI D .A; B/ and K.t/ work for (1.16).
We are now prepared to establish the following existence result:
Theorem 1.2. Provided (1.1) is symmetric hyperbolic and f 2 H
k
.M /, with
k>n=2C 1, then there is a solution u, on an interval I about 0, with
(1.18) u 2 L
1
.I; H
k
.M // \ Lip.I; H
k1
.M //:
Proof. Take the I above and shrink it slightly. The bounded family
u
"
2 C.I;H
k
/ \ C
1
.I; H
k1
/
will have a weak limit point u satisfying (1.18). Furthermore, by Ascoli’s theorem,
there is a sequence
(1.19) u
"
! u in C.I;H
k1
.M //;
since the inclusion H
k
H
k1
is compact. Also, by interpolation inequalities,
fu
"
W 0<" 1g is bounded in C
.I; H
k
.M // for each 2 .0; 1/,sosince
the inclusion H
k
,! C
1
.M / is compact for small >0if k>n=2C 1,we
can arrange that
(1.20) u
"
! u in C.I;C
1
.M //:
1. Quasi-linear, symmetric hyperbolic systems 417
Consequently, with " D "
,
(1.21)
J
"
L.t; x; J
"
u
"
;D/J
"
u
"
C J
"
g.t; x; J
"
u
"
/
! L.t; x; u;D/u C g.t; x; u/ in C.I M/;
while clearly @u
"
=@t ! @u=@t weakly. Thus (1.1) follows in the limit from (1.4).
Let us also note that, with y.t/ Dku
"
.t/k
2
H
k
,wehave,by(1.15),
(1.22)
dy
dt
a.t/y C a.t/; a.t/ D C
k
kJ
"
u
"
.t/k
C
1
;
so
(1.23) y.t/ e
b.t/
y.0/ C
Z
t
0
a.s/e
b.s/
ds
;
with b.t/ D
R
t
0
a.s/ ds. It follows that we have fu
"
W 0<" 1g bounded
in C.I;H
k
/ \ Lip.I; H
k1
/, as long as k>n=2C 1, with convergence
(1.19)–(1.21). A careful study of the estimates shows that I can be taken to be
independent of k (provided k>
1
2
n C 1). In fact, a stronger result will be estab-
lished in Proposition 1.5.
There are questions of the uniqueness, stability, and rate of convergence of u
"
to u, which we can treat simultaneously. Thus, with " 2 Œ0; 1, we compare a
solution u to (1.1) with a solution u
"
to
(1.24)
@u
"
@t
D J
"
L.t; x; J
"
u
"
;D/J
"
u
"
C J
"
g.t; x; J
"
u
"
/; u
"
.0/ D h:
Set v D u u
"
, and subtract (1.24) from (1.1). Suppressing the variables .t; x/,
we have
(1.25)
@v
@t
D L.u;D/vCL.u;D/u
"
J
"
L.J
"
u
"
;D/J
"
u
"
Cg.u/ J
"
g.J
"
u
"
/:
Write
(1.26)
L.u;D/u
"
J
"
L.J
"
u
"
;D/J
"
u
"
D
L.u;D/L.u
"
;D/
u
"
C .1 J
"
/L.u
"
;D/u
"
C J
"
L.u
"
; D/.1 J
"
/u
"
C J
"
L.u
"
;D/ L.J
"
u
"
;D/
J
"
u
"
;
and
(1.27)
g.u/ J
"
g.J
"
u
"
/ D Œg.u/ g.u
"
/ C .1 J
"
/g.u
"
/
C J
"
Œg.u
"
/ g.J
"
u
"
/: