A sharp condition for the well-posedness of the linear KdV-type equation

0.23 MB·English

by  Timur Akhunov

A SHARP CONDITION FOR THE WELL-POSEDNESS OF THE LINEAR
KDV-TYPE EQUATION
TIMURAKHUNOV
2
1
0
2 Abstract. AninitialvalueproblemforaverygenerallinearequationofKdV-typeisconsidered.
Assuming non-degeneracy of the third derivative coefficient this problem is shown to be well-
t
c posedunderacertainsimplecondition,whichisanadaptationofMizohata-typeconditionfrom
O the Schro¨dinger equation to the context of KdV. When this condition is violated ill-posedness
is shown by an explicit construction. These results justify formal heuristics associated with
9 dispersiveproblemsandhaveapplications tonon-linearproblemsofKdV-type.
1

]
P 1. Introduction
A
This paper is concerned with the study of the equation
.
h
at (1) ∂tu+Lu=f for (t,x)∈(0,T]×R , where L= 3 a (t,x)∂i
m (u(0,x)=u0(x) i=0 i x
X
[
where ai are real-valued functions.

2 This is the most generallinear form of the KdV, one of the most studied dispersive equations, and
v used as an important model in understanding behavior of linear and non-linear waves. Such an
8
equationwithnon-constantdispersivecoefficienta describesnonisotropicdispersionanditsstudy
5 3
is of use for the quasi-linear analogues of (1).
6
1
. Another motivation, for the study of the well-posedness of (1) is understanding the relative
9
strength of dispersive and non-dispersive effects present in the equation. In particular, from the
0
2 geometrical optics expansion for the equation, c.f. the classical book of Whitham [13], the disper-
1 sive coefficient a guides the propagationof the wave packets, while the term a ∂2 can lead to the
3 2 x
: growth of the amplitudes of the wave packets of (1). In light of these heuristics, it is natural to
v
i expect that well-posedness requires non-degeneracy of a3, which prevents the collapse of the wave
X
packets, namely 0 < ε a 1 for some ε, and a condition on a to ensure dispersion domi-
r nates anti-diffusioneffec≤ts.|C3r|a≤ig-εGoodman[4]provedwell-posedness2inthe Sobolevspaces Hs for
a
a a 0 under the non-degeneracy of coefficient a and ill-posedness for some degenerate cases
2 1 3
≡ ≡
of a . In a follow up paper, Craig-Kappeler-Strauss[3] provedwell-posedness with non-degenerate
3
dispersion and a 0, as well as extensions to the quasi-linear analogues. These results were
2
− ≥
1+
extended in [1] to allow for the ”anti-diffusion” in a2, as long as x 2 a2 C, under some addi-
h i | |≤
tional assumptions on other coefficients, and to systems of equations.
In the current paper, the condition on the diffusion coefficient a is extended to a sharp one for
2
thewell-posednessinHs,wherewell-posednessmeansexistence ofC0 Hs distributionalsolutions
[0,T]
2010 Mathematics Subject Classification. Primary: 35Q53.
Key words and phrases. KdV, linear, dispersive, partial differential equations, energy method, Mizohata
condition.
1
2 TIMURAKHUNOV
of (1),thatareunique anddepend continuously on data inthe C0 Hs topology. Namelyacondi-
[0,T]
tiononthediffusioncoefficienta alongtheflowisobtained,thatseparateswell-posednessfromill-
2
posedness (in the sense of violating continuous dependence) of (1) with non-degeneratedispersion.
t
ThisisqualitativelysimilartothenecessityofaMizohatocondition sup b(x+sω)ωds <
| x,t|ω|=1 0 ℑ · |
for the well-posedness Schr¨odinger equation ∂ u+i u+b(x) u=0 in [10], see also [5], [6], [9]
∞ t △ ∇ R
andreferencesthereinformorerefinedresultsonthevariablecoefficientSchr¨odingerequation. The
well-posedness is proved by the ”gauged energy method” and the condition on the gauge captures
the a condition. Ill-posedness is proved by an explicit geometricaloptics construction in the time
2
independent coefficient case that has consequence for general coefficients.
While preparing this paper for publication, I have learned of a preprint by Ambrose-Wright [2]
that treats an analogue of (1) in the periodic case. Their argument for the well-posedness is also
based on the ”gaugedenergy method”, however in the case of R the smoothness of the coefficients
does notimply integrabilitythat is oftenneeded. Additionally, this paper also provesthat (1) pos-
sesses a local smoothing effect, which is not present in the periodic case. The ill-posedness result
in [2] is done by a spectral method, using a change of variables that works in periodic case, but is
unbounded on the real line.
The methods used in both positive and negative arguments of this paper can be extended to
nonlinear problems, which will be a subject of future work.
The rest of the paper is organized as follows. In the section 2 the main results of the paper are
stated. Well-posedness is proved in the section 3, and ill-posedness in section 4.
Some results of this paper were obtained during my Ph.D. studies at the University of Chicago,
under the supervision of Carlos Kenig. I would like to thank Carlos Kenig and Cristian Rios for
helpful discussions.
2. Main results.
The following functional space notation is used. Let
N = f(x) CN(R): ∂if L∞ for all 0 i N , = N, and
BHxs = {f S∈′ : f =x ξ∈sfˆ(ξ) < ≤, w≤here}xB= ∩n1B+ x2.
Hs L2
{ ∈ k k kh i k ∞} h i | |
Finally, when dealing with mixed norm spaces it is convenient to denote
p
L∞ X X for the spaces X as above.
[0,T] x ≡ T
The following assumptions are made for the coefficients of (1)
(A1): Dispersivecoefficienta (t,x)isnon-degenerate. Thatis,thereareconstantsΛ λ>0,
3
≥
such that
λ a (t,x) Λ
3
≤| |≤
uniformly for (x,t) R [0,T].
∈ ×
(A2): Regularity of the coefficients. For all N 0.
≥
a C0 N+3 C1 1.
• 3 ∈ [0,T]Bx ∩ [0,T]Bx
a C0 N+2 C1 0.
• 2 ∈ [0,T]Bx ∩ [0,T]Bx
a C0 N+1
• 1 ∈ [0,T]Bx
a C0 N.
• 0 ∈ [0,T]Bx
(A3): Weak diffusion. x a2(y,t)dy C1 L∞.
0 |a3(y,t)| ∈ [0,T] x
Note, that by (A1) and (A2), a has a constant sign.
R 3
A SHARP CONDITION FOR THE WELL-POSEDNESS OF THE LINEAR KDV-TYPE EQUATION 3
For N 0 define
≥
1 3 3 x a (y,t)
CN =ka3kL∞T +ka3kL∞T + i=0kaikBTN+i +i=2k∂taikL∞T +kZ0 |a23(y,t)|dykL∞T
X X
x a (y,t)
+ ∂t 2 dy L∞
k a (y,t) k T
Z0 | 3 |
For the well-posedness arguments, positive constants will depend on C for some N and will not
N
be made explicit.
Theorem1. Supposethecoefficientsof (1)satisfy(A1)-(A3). Then forall s R,(1)is well-posed
∈
in Hs. That is for any (u ,f) Hs L1 Hs there exists a unique u C0 Hs satisfying (1) in
0 ∈ × [0,T] ∈ [0,T]
the sense of distributions. In addition, there exists C =C(s)
T
(2) sup u(t) Hs CeCT( u0 Hs + f(t) Hsdt)
k k ≤ k k k k
0≤t≤T Z0
Moreover, for any δ > 1, the solution additionally satisfies u L2 Hs+1 and there is a
2 ∈ [0,T] hxi−2δdx
C˜ =C˜(s,δ)
T
(3) khxi−δ∂xukL2[0,T]Hxs ≤C˜(1+√T)eC˜T(ku0kHs +Z0 kf(t)kHsdt)
Estimate (2) implies continuous dependence for (1), while estimate (3) is a manifestation of a
local smoothing effect of (1).
Remark 2. If in addition, f C0 Hs−3, then for s > 31 the unique solution from the Theorem
∈ [0,T] 2
1 is classical by the Sobolev embedding.
Remark 3. If the coefficients of (1), in addition, satisfy (A1) - (A3) on [ T,0], then the trans-
−
formation of the equation by t t changes the sign of all a , while again preserving all of the
i
→ −
assumptions. Therefore, Theorem 1 extends to [ T,0].
−
Moreover, the transformation x x in (1) changes the sign of a for odd i, but preserves the
i
→ −
assumptions (A1) - (A3). Without of loss of generality a >0 will be assumed.
3
Ill-posedness result complements the Theorem 1 and is proved by a different method.
Theorem 4. Suppose the coefficients of (1) satisfy (A1), (A2) and
(A3N): sup x a2(y,0)dy =
x>0 0 |a3(y,0)| ∞
Then for all T >0 anRd s∈R (1) is ill-posed in C[00,T]Hs forward in time. More precisely, there is
no continuous function C(t,t ) for 0 t t T, such that
0 0
≤ ≤ ≤
(4) sup u(t) C(t,t ) u(t )
Hs 0 0 Hs
k k ≤ k k
t0≤t≤T
whenever u solves (1) on [0,T] with f 0. Equivalently (2) fails on any [0,T].
≡
Remark 5. The transformation x x shows that (A3N) is equivalent to
→−
0 a (y,0)
sup 2 dy = .
a (y,0) ∞
x<0Zx | 3 |
However,theequivalencebreaksdownifabsolutevaluesareremovedfroma in(A3). Thusa >0
3 3
can be assumed without loss of generality, as long as (A3N) is replaced with
(A3N’): a >0. Furthermore,
3
x a (y,0) 0 a (y,0)
sup 2 dy = or sup 2 dy =
x>0Z0 a3(y,0) ∞ x<0Zx a3(y,0) ∞
4 TIMURAKHUNOV
Remark 6. By reversing the time t t as in the Remark 3, Theorem 4 shows that
→−
x a (y,0)
sup 2 dy =
a (y,0) −∞
x>0Z0 | 3 |
leads to ill-posedness on [ T,0]. Thus the condition x a2(y,0)dy L∞ is crucial for the well-
− 0 |a3(y,0)| ∈
posedness and the condition (A3) for the Theorem 1 is sharp for well-posedness on [ T,T].
R −
3. Well-posedness
The mainingredient in the proof of the Theorem1 is statedas the following Proposition,which
is an a priori L2 estimate for a slightly more general version of (1), that comes from commuting
derivatives.
∂ u+L u=f for (t,x) (0,T] R
t A
(5) ∈ × , where L =L+A (t,x,∂ )
A 0 x
(u(0,x)=u0(x)
with L from (1). The following assumptions are made on A C0 S0, the Pseudo-Differential
0 ∈ [0,T]
operator of standard symbol class of order 0 (Cf. Chapter VI of [11]):
(A4): TheS0 semi-normsofA areboundedfort [0,T]andtheirsizedependsonconstants
0
∈
C from (A1)–(A3).
N
Proposition 7. Suppose that the coefficients a of (1) satisfy (A1)–(A3) and A satisfies (A4).
i 0
Then there exists a constant C and for any δ > 1 there is a constant C˜, such that for any u
2 ∈
C1 L2 C0 H3, the triple (u,u ,f) with u and f defined by (5) satisfies
[0,T] ∩ [0,T] 0 0
T
(6) sup u(t) CeCT( u + f(t) dt)
L2 0 L2 L2
k k ≤ k k k k
0≤t≤T Z0
T
(7) x −δ∂ u C˜(1+√T)eC˜T( u + f(t) dt)
kh i x kL2[0,T]×x ≤ k 0kL2 Z0 k kL2
Remark 8. If A 0, then N =0 in (A2) can be chosen for the Proposition 7.
0
≡
The proofof the Proposition7 is done by a changeof variables(gauge)followedby the application
of the energy estimates. The proof is broken into several preliminary results.
A gauge is a smooth invertible function, which for the purposes of the argument needs to have
3 bounded derivatives:
Definition 9. A function φ C0 3 C1 0 is called a gauge, if
∈ [0,T]Bx∩ [0,T]B
φ(x,t)>0 with 1 L∞ .
• φ ∈ [0,T]×R
• kφ1kL∞[0,T]×R +kφkBT3 +k∂tφkL∞T ≤C(C0,δ) with CN from (A1)–(A3).
Suppose that φ(x,t) is a gauge. Define
v =φ−1u
Definition 9 implies that v C1 L2 C0 H3 if and only if u C1 L2 C0 H3 and
∈ [0,T] ∩ [0,T] ∈ [0,T] ∩ [0,T]
substitution of v into (5) gives:
∂ v+L v =φ−1f
t φ
(8)
(v(x,0)=φ−1u0
where
L =a ∂3+ a +φ−13a ∂ φ ∂2+ a +φ−1(2a ∂ φ+3a ∂2φ) ∂
φ 3 x 2 3 x x 1 2 x 3 x x
+ a(cid:0)0+φ−1(∂tφ+a1(cid:1)∂xφ+(cid:0)a2∂x2φ+a3∂x3φ) I+φ−1A0((cid:1)φ )
(cid:0) (cid:1)
A SHARP CONDITION FOR THE WELL-POSEDNESS OF THE LINEAR KDV-TYPE EQUATION 5
Remark 10. From the definition of the gauge,
1
u v and x −δ∂ u x −δ∂iv
k kL2 ≈k kL2 kh i x kL2[0,T]×x ≈ kh i x kL2[0,T]×x
i=0
X
with comparability constants dependent only on the constant in the Definition 9. Therefore, to
prove Proposition 7 it suffices to prove (6) and (7) for v satisfying (8).
The energy method involves multiplying (8) by v to estimate ∂ v 2 by v 2 :
tk kL2 k kL2
∂ v 2 = 2Re(L v,v)+(f,φv)
t φ
| | −
Z
The following Lemma summarizes the energy estimates for L or L :
φ
Lemma 11. Consider an operator L=a ∂3+a ∂2+a ∂ +a , where a –a satisfy (A2). Then
3 x 2 x 1 x 0 3 0
for v C0 H3
∈ [0,T]
3
Re(Lv,v)=( a + ∂ a ∂ v,∂ v)+(b v,v)
2 x 3 x x 0
− 2
(cid:20) (cid:21)
for b =a 1(∂ a ∂2a +∂3a ), where (u,v) is an L2 pairing.
0 0− 2 x 1− x 2 x 3 x
Proof of Lemma 11. The computation is immediate by computing the adjoint L∗ of L using the
Calculus of PDO. Alternatively, as L is a differential operator, the same computation can be also
done by a repeated integration by parts. Indeed, the operator ∂k is skew-adjoint for odd k, which
x
impliesthatprincipalpartsofoddordertermsareeliminatedbyintegrationbyparts. Forexample
(a ∂ v,v)= (v,a ∂ v) (∂ a v,v)= (a ∂ v,v) (∂ a v,v)
1 x 1 x x 1 1 x x 1
− − − −
An identical computation shows
1 1
Re(a ∂2v,∂ v)= (∂ a ∂ v,∂ v) and Re(∂2a ∂ v,v)= (∂3a v,v)
3 x x −2 x 3 x x x 3 x −2 x 3
Using these identities and more integration by parts establishes
3 1
Re(a ∂3v,v)= (∂ a ∂ v,∂ v) (∂3a v,v)
3 x 2 x 3 x x − 2 x 3
A similar analysis for Re(a ∂2v,v) completes the proof. (cid:3)
2 x
Applying Lemma 11 to L , shows that the only term of order higher than 0 is
φ
[2a + 6a3∂xφ 3∂ a ]∂ v,∂ v . Thus, if this term were negative, an a priori estimate would be
2 φ − x 3 x x
(cid:16)obtained for v. This motivates t(cid:17)he choice of a gauge φ that should satisfy
2a +φ−16a ∂ φ 3∂ a 0
2 3 x x 3
− ≤
A choice of equality in this equation can be made and this choice is enough for the estimate (6),
but by exploiting the inequality the local smoothing estimate (7) is proved. The exact choice of a
gauge is summarized in the following Lemma
Lemma 12. For δ > 1, let φ(x,t) be a solution of the ODE
2
6a∂ φ= 3∂ a c x −2δ 2a φ
x x δ 2
− h i −
(φ(t,0)=(cid:16)1 (cid:17)
where c = 0 or 1. Then φ is a gauge in the sense of the Definition 9, and is independent of δ if
δ
c =0.
δ
6 TIMURAKHUNOV
Proof. The ODE for φ is solved explicitly as
φ(x,t)= a(x,t)e−R0x 3aa23((yy,,tt))dye−R0x 6a3(cyδ,td)hyyi2δ
sa(t,0)
By (A3) e−R0x 3aa23((yy,,tt))dy ≈1. (A1) implies aa((xt,,0t)) ≈1.
Finally, as δ > 1, q
2
e−R0x 6a3(cyδ,td)hyyi2δ = 1, if cδ =0
(C(δ), if cδ =1
A computation for ∂ φ and ∂jφ for j =1, 2 and 3 and using (A1)–(A3) finishes the proof. (cid:3)
t x
Proof of Proposition 7. By the Remark 10 it suffices to prove the Propositionfor v satisfying (8).
Applying the Lemma 11 for L implies that
φ
6a ∂ φ
∂ v 2dx=( 2a + 3 x 3∂ a ∂ v,∂ v)+(˜b v,v)
t | | 2 φ − x 3 x x 0
Z (cid:20) (cid:21)
2Re(A (φv),φv)+(f,φv)
0
−
where ˜b is obtained from the Lemma 11 applied to L . With φ chosen from the Lemma 12, this
0 φ
implies
∂ v 2dx c (x −2δv,v)+(˜b v,v) 2Re(A (φv),φv)+(f,φv)
t δ 0 0
| | ≤− h i −
Z
By (A4), A :L2 L2 is bounded. Moreover,by the Definition 9 and(A2), φ L∞ and˜b L∞.
0 0
→ ∈ ∈
Hence
∂ v 2 C( v 2dx+ v f ) x −δ∂ v 2
t | | ≤ | | k kL2k kL2 −kh i x kL2
Z Z
For c =0 an application of Grownwall Lemma implies (6) for v.
δ
Whereas moving ∂ v term to the left hand side for c =1 and integrating in time gives
x δ
T T
x −δ∂ v 2dt C ( v 2dx+ v f )dt+ v 2 v 2
kh i x k ≤ | | k kL2k kL2 k 0kL2 −k kL2
Z0 Z0 Z
T
(C(1+T) 1) sup v(t) 2 + v 2 +( f(t) dt)2
≤ − k kL2 k 0kL2 k kL2
0≤t≤T Z0
Using (6) completes the proof of (7). (cid:3)
Proposition7 can be strengthened to an Hs estimate.
Proposition 13. Let L be as in (1), whose coefficients a satisfy (A1)–(A3). Then for any s R
i
∈
there exist constants C(s) and C˜(s,δ) for any δ > 1, such that for any u C1 Hs C0 Hs+3
2 ∈ [0,T] ∩ [0,T]
the following estimates hold
T
(9) sup u(t) CeCT( u(0) + ∂ u+Lu dt)
Hs Hs t Hs
k k x ≤ k k x k k x
0≤t≤T Z0
T
sup u(t) CeCT( u(T) + ∂ u+L∗u dt)
Hs Hs t Hs
k k x ≤ k k x k− k x
0≤t≤T Z0
A SHARP CONDITION FOR THE WELL-POSEDNESS OF THE LINEAR KDV-TYPE EQUATION 7
where L∗ is the adjoint of L. Moreover
T
x −δ∂ u C˜(1+√T)eC˜T( u + f(t) dt)
kh i x kL2[0,T]Hxs ≤ k 0kHs Z0 k kHs
Corollary 14. By the Theorem 23.1.2 on page 387 in [8], the proof of the Theorem 1 reduces to
the Proportion 13.
The Proposition 13 is reduced to the Proposition 7. Observe, that
f =∂ u+Lu if and only if Jsf =∂ Jsu+LJsu+[JsL]J−sJsu
t t
where Js is a Pseudo Differential Operator with symbol ξ s. Therefore to prove (9) it suffices to
h i
show that the Proposition 7 applies to the operator L˜ =L+[JsL]J−s.
Lemma 15. Let L˜ =L+[JsL]J−s with L from (1) that satisfies (A1) and (A2). Then
2
L˜ =a ∂3+a + (a +a˜ )∂i +A (t,x,∂ )
3 x 0 i i x s x
(10) i=1
X
s(s 1)
with a˜ =s∂ a and a˜ =s∂ a + − ∂2a
2 x 3 1 x 2 2 x 3
where A S0, whose semi-norms depend on the coefficient bounds (A2) for N =N(s) and hence
s
∈
satisfies (A4).
Furthermore, the coefficients a˜ for i=1, 2 satisfy (A2)–(A3).
i
Proof. FromthefirsttermintheCalculusofPDO[JsL]J−s S2. Afurtherexpansionof[Js,a ∂3]
∈ 3 x
gives:
i−|α|
σ([Js,a ∂3])= ∂α ξ s∂α(a (iξ)3) mod Ss
3 x α! ξh i x 3
1≤X|α|≤2
s(s 1)
=s∂ a (iξ)2 ξ s+ − ∂2a iξ ξ s mod Ss
x 3 h i 2 x 3 h i
where the substitution ξ2 = ξ 2 1 was used and the terms of order s were absorbed into the
h i −
remainder. Performing a similar computation for the remaining terms in [JsL] and composition
with J−s verifies (10).
It is immediate from (10) that a˜ satisfies (A2). To verify (A3) observe that
i
x a˜ (y,t) a (x,t)
2 dy =ssign(a )log 3 C1 L∞
a (y,t) 3 a (0,t) ∈ [0,T] x
Z0 | 3 | 3
by (A1) and (A2). (cid:3)
Remark 16. A simple computation shows that the adjoint L∗ of the operator L from (1) is
L∗ = a ∂3+(a 3∂ a )∂2+(a +2∂ a 3∂2a )∂
− 3 x 2− x 3 x 1 x 2− x 3 x
+(a ∂ a +∂2a ∂3a )
0− x 1 x 2− x 3
whereas a substitution t T t transforms (1) to
→ −
∂ u(T t)+Lu(T t)=f(T t)
t
− − − −
(u(T t) t=0=u(T)
− |
Both L∗ and L(T t) satisfy (A1)–(A3).
−
Corollary 17. Lemma 15, Remark 16 and the Proposition 7 imply the Proposition 13.
This completes the proof of Theorem 1 by the Corollary 14.
8 TIMURAKHUNOV
4. Ill-posedness
Ill-posednessisprovedbyjustifyingtheformalgeometricalopticsargumentforaspecialchoiceof
initialdata. Thecaseoftimeindependentdispersivecoefficientiscommonlystudiedintheliterature
(Cf. [5] and [6] for the variable dispersion Schr¨odinger equation). The same simplification is done
in this work and the construction is flexible enough to be useful in the generality of (1). Thus (1)
is replaced by
∂ u+L u=f for (t,x) (0,T] R
t 0
(11) ∈ ×
(u(0,x)=u0(x)
for L = 3 a (x,0)∂i + 1 a (t,x)∂i with coefficient a from L. The coefficients a and a
0 i=2 i x i=0 i x i 1 0
are insignificant for the argument and no change is done for them. To simplify notation, denote
P P
a (x)=a (x,0) for i=2 and 3.
i i
The geometrical optics argument involves making an ansatz for the solution of (1) of the form
u = eiSw or (11) and converting them into a system of simpler equations. Indeed, a substitution
of such ansatz into (11) gives
f =eiS(i∂ S ia ∂ S3)+eiS ∂ w 3a ∂ S2∂ w [3a ∂ S∂2S+a ∂ S2]w
t − 3 x t − 3 x x − 3 x x 2 x
+eiS a3(i∂x3S·w+3i∂x2S∂xw(cid:8)+3i∂xS∂x2w+∂x3w)+a2(i∂x2S·w+2i∂xS∂xw(cid:9))
+eiS(cid:8)a2∂x2w+a1(i∂xS·w+∂xw)+a0w (cid:9)
Formall(cid:8)y assuming that ∂ S ξ and w (cid:9) 1 for ξ 1, leads to an eikonal equation for S to
x
≈ ≈ ≫
eliminate ξ3 terms, a transport equation for w to eliminate ξ2 terms and the remaining terms left
in f. More precisely,an ansatz ofthe form u=eiSw solves(11) for appropriateu andf provided
0
∂ S a (x)(∂ S)3 =0
t 3 x
(12) −
x −1
(S(ξ,0,x)=ξ a 3(y)dy
0 3
∂ w 3a (x)(R∂ S)2∂ w [3a (x)∂ S∂2S+a (x)(∂ S)2]w =0
(13) t − 3 x x − 3 x x 2 x
(w(ξ,0,x)=w0(x)
f =eiSa i∂3S w+3i∂2S∂ w+3i∂ S∂2w+∂3w
(14) 3 x · x x x x x
+eiS a2(i∂x2S·w+2i∂xS(cid:2)∂xw+∂x2w)+a1(i∂xS·w+∂xw)+a0w(cid:3)
(cid:2) (cid:3)
The first two equations are solved by the method of characteristics, Cf. [7], and the third is com-
pletelydeterminedbyS andw. However,the methodofcharacteristics,ingeneral,givesonlylocal
solutions,anditwasimportantto makeachoice forS(ξ,0,x)in (12)to ensureS has asolutionon
[0, ) R.
∞ ×
When dispersion coefficient is constant, which after rescaling is equivalent to a (x,t) 1, (12)
3
≡
is solvedwith a simple use of the dispersive relationby setting S =ξ3t+ξx. Using an analogue of
(A3N)inthiscase,ill-posednesswasprovedin[1],byanalogywiththeSchr¨odingerequationaspre-
sentedinlecturenotesbyCarlosKenigon”The Cauchy Problem for the Quasi-linear Schr¨odinger Equation”.
This analysis is expanded below to account for non-constant dispersion.
The main ingredient in the proof of the Theorem 4 is the following Theorem, whose proof is
based on the geometrical optics ansatz above.
Theorem 18. Suppose the coefficients of (11) satisfy (A1), (A2) and (A3N) as given in the
Theorem 4.
A SHARP CONDITION FOR THE WELL-POSEDNESS OF THE LINEAR KDV-TYPE EQUATION 9
Let n N. Then there exists a T >0 and u=u (ξ,t,x) CNC1 CN+3(R), such that
∈ n n ∈ ξ [0,T] 0
T
u (ξ, n,x) 2n and u(ξ,0,x) =1
k n ξ2 kL2x ≥ k kL2x
sup [∂ u +L u ](ξ,t,x) C ξ
t n 0 n L2 n
(15) 0≤t≤Tnk k x ≤
ξ2
sup u (ξ,t,x) C ξi for i=0,...,3.
n Hi n
k k x ≤
0≤t≤Tn
ξ2
Moreover, adding a Pseudo-differential operator A S0, that satisfies (A4), toL does not change
0 0
∈
the conclusion above.
Theorem 18 is proved after a number of preparatory Lemmas in the subsection. After that the
Theorem 4 is proved in the section 4.2.
x −1
4.1. Analysis of (12)–(14). Note, that the choice of S(ξ,0,x)=ξ a 3(y)dy is well-defined by
0 3
(A1), but will not be bounded as x . This choice makes ∂ S(x,0,ξ) = ξ3 constant for all x
→ ∞ t R
from the equation (12). If S(x,t) solves (12), then Ξ(t)=∂ S(x(t),t), ω(t)=∂ S(x(t),t) and x(t)
x t
satisfy
H(ω,x,Ξ)=ω a (x)Ξ3 =0
3
−
To achieve dH = 0 a following system of ”characteristic” ODEs is selected, which is later used to
dt
construct S:
ddxt = ∂∂HΞ =−3a3(x)Ξ2 ddΞt =−∂∂Hx =∂xa3(x)Ξ3
(16) (x(0)=x0 (Ξ(0)=ξ0 =ξa3−31(x0)
dω = ∂H =0
dt − ∂t
(ω(0)=a3(x0)ξ03 =ξ3
The last two equations in (16) are easy to solve explicitly: ω(t)=ξ3 and Ξ(t)=ξa−13(x(t)). This
3
simplifiesthesystemabovetoasingleODE,whereanindexξ isaddedtox(t)=x (t)toemphasize
ξ
the ξ dependence
dxξ = 3ξ2a31(x )
(17) dt − 3 ξ
(xξ(0)=x0
1 1
ξ = 0 for the construction, so x is unambiguous. As the bounds on a3(x) and ∂ [a3(x)] are
6 0 3 x 3
uniform in x by (A1) and (A2), Picard iteration gives a unique global solution for each x , Cf.
0
Chapter 1 of [12].
Remark 19. Observe, that re-scaling time t t for ξ = 0 implies by uniqueness of solutions to
→ ξ2 6
(17), that
t
x ( )=x (t)
ξ ξ2 1
where x (t) satisfies (17) for ξ = 1. Using this rescaling, the index ξ = 0 can be suppressed to
1
6
x(t)=x (t)
1
The properties of this ODE are summarized as follows, where as in the Remark 5, a > 0 is
3
assumed.
Lemma 20. There exists a global in time flow x x(t) that depends smoothly on data with
0
→
∂x(t) a (x(t))
(18) = 3 3
∂x0 s a3(x0)
10 TIMURAKHUNOV
As (17) is autonomous in time, for each (x,T) there exists a unique x , such that x(t) satisfies
0
(17) and x (T)=x.
1
Moreover, the flow is geodesic on R, that is for every x , x there exists a unique T, such that x(t)
0
satisfies (17) for all t and x(T)=x and
(19) x(T) x T
0
− ≈−
with proportionality constants dependent on the (A1) bounds.
Proof. The global existence follows from Picard iteration as observed before the Remark 19. Re-
versing the time direction t T t in (17) shows the unique dependence of x on (x,T) from the
0
→ −
equation.
By the Implicit Function Theorem for a fixed x=x(x (x,t),t):
0
∂x dx (x,t) dx
0
+ =0
∂x dt dt
0
which gives (18) using (17).
Finally,integrating(17)intime,using(A1)andcontinuityofx (t)provesthegeodesic property.
1
(cid:3)
Remark 21. UsingRemark19,Lemma20holdsfortheunrescaledflowx (t)=x (ξ2t)withobvious
ξ 1
changes. Thatisgiven(ξ,x,t)thereisauniquex ,suchthatthesolutionof (17)satisfiesx (t)=x,
0 ξ
(18) holds and the flow map in (17) is geodesic with (19) replaced by x (T,x ) x Tξ2.
ξ 0 0
− ≈−
This leads to the following construction, which reconstructs S from the characteristicequations
(16)
Lemma 22. Define S(ξ,x,t) = S(ξ,x ,0)+ tΞ(s)dxξ(s) +ω(s)ds, where x is defined as in
0 0 dt 0
the Remark 21 and x (s), Ξ(s) and ω(s) solve (16). Then S is well-defined on R R, satisfies
ξ
R ×
∂S(x,t)=Ξ(t), ∂S(x,t)=ω and solves (12) for all (x,t).
∂x ∂t
Proof. Using (16) and the choice of data S(ξ,x,0), S(ξ,x,t) can be simplified to
S(ξ,x,t)=ξ x0a−31(y)dy 2ξ3t
3 −
Z0
Differentiating this formula using ∂x0 and ∂x0 using the Lemma 20 and Remark 21 completes
∂t ∂x
the proof. (cid:3)
Next (13) is analyzed. Let x(t)=x (t) andΞ(t) be as in (16). With this notation(13) becomes
ξ
∂ w(ξ,t,x(t)) 3a (x(t))Ξ2(t)∂ w(t,x(t))
t 3 x
−
=[3a (x(t))Ξ(t)∂ Ξ(t)+a (x(t))Ξ2(t)]w(t,x(t))
 3 x 2
w(ξ,0,x0)=w0(x0)
which isan ODE in t as dxξ = 3a (x(t))Ξ2(t). Solving this ODE for w(t,x(t)) gives
dt − 3
(20) w(ξ,t,x)= s3 aa33((xx0))e31Rxx0 aa32((yy))dyw0(x0)
where the identity ∂ Ξ(t)= −∂xa3(x)Ξ(t), derived from Ξ(t)=ξa−13(x(t)), was used and where x
x 3a3(x) 3 0
is defined as in the Remark 21.