Non-uniform dependence on initial data for the Whitham equation

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by  Mathias Nikolai Arnesen

NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF
WHITHAM TYPE
MATHIASNIKOLAIARNESEN
Abstract. WeconsidertheCauchyproblem
∂tu+u∂xu+L(∂xu)=0,
6
u(0,x)=u0(x)
1
0 onthetorusandonthereallineforaclassofFouriermultiplieroperators L,andprovethat
2 thesolutionmapu07→u(t)isnotuniformlycontinuous inHs(T)orHs(R)fors> 32. Under
certainassumptions,theresultalsoholdfors>0. Theclassofequationsconsideredincludes
p
inparticulartheWhithamequationandfractionalKorteweg-deVriesequationsandweshow
e
that,ingeneral,theflowmapcannotbeuniformlycontinuousifthedispersionofLisweaker
S
thanthatoftheKdVoperator. Theresultisprovedbyconstructingtwosequencesofsolutions

6 convergingtothesamelimitattheinitialtime,whilethedistanceatalatertimeisbounded
2 belowbyapositiveconstant.

]
P
A 1. Introduction
.
h
We consider the Cauchy problem
t
a
∂ u+u∂ u+L(∂ u)=0, (1.1a)
m t x x
u(x,0)=u (x), x R or x T, t R. (1.1b)
[ 0 ∈ ∈ ∈
on the torus T and on the real line R. The operator L is a Fourier multiplier operator with
3
symbol m(ξ), meaning that
v
0 Lf(ξ)=m(ξ)f(ξ). (1.2)
5
2 A concrete example is the Whitham equation where m(ξ) = tanh(ξ). The Whitham equation
c b ξ
0 was introduced by Whitham in 1967 as a better alternativeqto the Korteweg–de Vries (KdV)
0
equationformodellingshallowwaterwaves[20],andfeaturestheexactlineardispersionrelation
.
2 for travelling gravity water waves (see [14] for a rigorous justification of (1.1a) as a model for
0
shallow water waves and [16] for a derivation of it from the Euler equations via exponential
6
scaling).
1
: The recent papers [6] and [8] concern local well-posedness for the Whitham equation and
v
related nonlinear and nonlocal dispersive equations with nonlinearities of low regularity. These
i
X results are, when comparable, in line with the earlier investigations [1] [19]. For problems with
r homogeneoussymbolsandsmoothnonlinearities,localwell-posednesshasbeenloweredtos 3
a in [15] using dispersive properties, with the lower bound for s depending on the strength of≤th2e
dispersion. The paper at hand concerns further regularity of the flow map, or rather the lack
thereof. We prove that in the periodic case, the flow map is not uniformly continuous on any
bounded set of Hs(T) for s> 3 for any symbol m that is even and locally bounded, and on the
2
reallinetheflowmapisnotuniformlycontinuousonanyboundedsetofHs(R)fors> 3 ifmdoes
2
Date:September 27,2016.
The author gratefully acknowledges the support of the project Nonlinear water waves (Grant No. 231668)
fromtheResearchCouncilofNorway.
1
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 2
notgrow”too”quickly. Theresultsarealsoextendedto0<s 3 undercertainconditions. The
≤ 2
paperismotivatedbyaseriesofsimilarresultsforothermodelequations(e.g.,fortheCamassa–
Holm (CH) equation [9] and the Benjamin–Ono (BO) equation [13]), and indeed for the Euler
equations themselves [10], as well as recent investigations into non-local dispersive equations
of Whitham type with very general, and in particular also inhomogeneous, symbols m ([7],[3],
[8]) and recent work connected to well-/ill-posedness for the Whitham equation specifically. In
particular, we mention the recent positive verifications of two conjectures of Whitham, namely
thatforcertaininitialdatathesolutionexhibitswave-breakinginfinitetime[11]andtheexistence
of a highest cusped wave [5].
Our results are contained in Theorems 1.1 and 1.2 below, for the periodic case and the real
line case, respectively.
Theorem 1.1 (Non-uniform continuity on T). Assume that m L∞(R) is even, there exists
∈ loc
N >0 such that m(ξ) is continuous for ξ >N and that
| |
m(ξ) . ξ p
| | | |
for some p>0 when ξ 1. Then:
| |≫
(i) If s > 3, the flow map u u(t) for the Cauchy problem (1.1a)-(1.1b) on the torus is
not uni2formly continuous0fr7→om any bounded set in Hs(T) to C([0,T);Hs(T)).
(ii) Let 0<s 3. When the flow map exists on Hs(T), (i) remains true.
≤ 2
Theorem 1.2 (Non-uniform continuity on R). Assume that m is even, m L∞(R) and there
∈ loc
exists N >0, 0 γ <2 and a constant C >0 such that
≤
m(ξ+y) m(ξ) C y ξ γ−1 (1.3)
| − |≤ | || |
for all ξ > N and y sufficiently small. In particular, this means that m(ξ) is continuous for
| | | |
ξ >N and that m(ξ) . ξ γ for large ξ . Then:
| | | | | | | |
(i) If s> 3, the flow map u u(t) for the Cauchy problem (1.1a)-(1.1b) on the line is not
uniform2ly continuous fro0m7→any bounded set in Hs(R) to C([0,T);Hs(R)).
(ii) Let 1 < r < 2 and 0 < s < r. If the Cauchy problem (1.1a)-(1.1b) is locally well-posed
2 2
in Hs(R) in the sense of Theorem 2.1 and m satisfies the lower bound
m(ξ) & ξ r
| | | |
for ξ 1 in addition to (1.3), then (i) is true also for 0<s< r.
| |≫ 2
Remark 1.3. Localwell-posednessof (1.1a)isingeneralknownonlyfors> 3 (cf. Theorem2.1
2
below), hence we have to assume the existence of the flow map in part (ii) of Theorem 1.1 and
1.2. For some specific choices of m well-posedness results for s 3 are known (see for instance
≤ 2
[15]).
Remark1.4. Theadditionalcondition m(ξ) & ξ r for ξ 1andtheboundsonsinTheorem
1.2 (ii) come from using conservation la|ws for| (1.|1a|) to|bo|u≫nd the Hσ(R) norm of the solution
in terms of the norm of the initial data for σ > s (cf. the end of Section 4). These conditions
can be improved upon in cases where more conservation laws are known, as done in [13] for the
BO equation in order to cover all s>0.
The assumptions of Theorems 1.1 and 1.2 cover, for example, the Whitham equation, and
the fractional Korteweg–de Vries (fKdV) equation where m(ξ) = ξ α for any α 0 in the
periodic case and 0 α < 2 on R. One would expect the strength|of|the dispersion≥to be the
≤
essentialpropertydecidingtheregularityoftheflowmap,withstrongerdispersiongivinggreater
regularity. Theorem 1.2 shows that this is the case, for while the restriction γ < 2 in Theorem
1.2 appears in the prooffrom our construction of a specific approximate solutionto (1.1a), it is,
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 3
in fact, optimal. When γ = 2 our assumptions includes the KdV equation for which the flow
mapis knowntobe locallyLipschitzinHs(R)fors> 3 [12],meaningthatTheorem1.2is not
−4
true in this case. For 0 γ < 2 it was proved in [18] that for m(ξ) such that p(ξ) = ξm(ξ) is
≤
differentiable and satisfying p′(ξ) ξ γ for 0 γ <2, which implies that the the assumptions
of Theorem 1.2 are satisfied,|the fl|o≤w|m|ap cann≤ot be C2 in Hs(R) for any s R. Our findings
∈
areinagreementwith,andimprovesupon,theseresuls. Intheperiodcase(1.1a)has,inasense,
no dispersive effect as it is invariant under the transformation
u(x,t) v(x,t)=u(x tω,t)+ω.
7→ −
Having no restriction on p in Theorem 1.1 is therefore perfectly in line with the notion that the
strength of the dispersion is the decisive factor for the regularity of the flow map. To avoidthis
situation, one often considers initial data having zero mean f(x)dx = f(0). In this case the
T
flow map of the KdV equation is known to be Lipschitz continuous in Hs(T) for s 0 [4] and
thus Theorem1.1 fails for p=2. In fact, it fails for the KdVRin Hs(T) for sb> 1 [12≥]. We will,
−2
however,not consider this case here and make no assumption on the mean of the initial data in
the periodic case.
We will prove Theorems 1.1 and 1.2 using a method based on [13], where nonuniform de-
pendence on initial data was established for the BO equation on R, describing the effect of
a low-frequency perturbation on a high-frequency wave. In [17] the proof of [13] for the BO
equation on the line is adapted to the simpler periodic case for the fKdV equation. For the
periodic case,the argumentsareeasilyextendedto operatorswithmoregeneralsymbolsm, and
the proof we present for Theorem 1.1 is a straightforward extension of that of [17] and [13].
Non-uniformcontinuityforfractionalKdVequationsonthe line hasnotbeenprovedforgeneral
order 0 α < 2, but as the symbol m(ξ) = ξ α is homogeneous the equation enjoys scaling
≤ | |
properties similar to those of the BO equation and the procedure of [13] should therefore be
applicable without too muchdifficulty. The Whitham equationhowever,orindeed any equation
with inhomogeneous symbol m, does not share these properties, and different argumentation is
therefore required (see Section 4 and in particular Proposition 4.4).
Our paper is structured as follows: Section 2 is devoted to some preliminary results on local
well-posedness, existence time and energy estimates that are crucial ingredients in the proofs of
Theorems1.1and1.2. InSections 3and4 weproveTheorem1.1andTheorem1.2,respectively.
2. Preliminaries
In this section we state results on the existence of solutions to equation (1.1a) with initial
data (1.1b) andestimates onthe existence time andHs-normofthe solutions. All the resultsin
this section hold equally on T and on R, and we will denote by Hs either Hs(T) or Hs(R). The
main result is the following:
Theorem 2.1 ([8]). Assumethat m is even, m L∞(R) and that m(ξ). ξ p for some p when
∈ loc | | |
ξ >1. Then, for s> 3 andu Hs thereisamaximal T >0dependingonly on u , anda
| | 2 0 ∈ k 0kHs
unique solution u to (1.1a)-(1.1b) in the class C([0,T);Hs). The solution depends continuously
on the initial data, i.e. the map u u(t) is continuous from Hs to C([0,T);Hs).
0
7→
Moreover,wehavethefollowinglowerboundfortheexistencetimeT andrelationshipbetween
the Hs norm of the solution u at time t and the Hs norm of the initial data:
Lemma 2.2. Let s > 3. If u is the solution to (1.1a) with initial data u Hs described in
2 0 ∈
Theorem 2.1, then there exists a constant c , depending only on s, such that
s
u
0 Hs
u(t) Hs k k . (2.1)
k k ≤ 1 tc u
s 0 Hs
− k k
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 4
In particular, the maximal existence time T in Theorem 2.1 satisfies
1
T .
≥ c u
s 0 Hs
k k
Remark 2.3. Lemma 2.2 is a typical result for equations of the form (1.1a) and can be proved
by standardarguments,butwe givea proofherefor the sakeofcompleteness. We proveLemma
2.2 on the line. How to extend the proof to the periodic case should be clear. The proof follows
the proof of Proposition1 in [9], an equivalent result for the CH equation, but is in fact simpler
due to the the operator L being skew-symmetric and linear.
In order to prove Lemma 2.2, we introduce the operators Λs defined by
Λsf(ξ)=(1+ξ2)s/2f(ξ), s R.
∈
Note that Λsf L2(R) = f Hs(R).
k k k k d b
Proof. The proof relies on the following differential inequality for the solution u that we will
establish:
1 d
u(t) 2 c u(t) 3 . (2.2)
2dtk kHs(R) ≤ sk kHs(R)
Solving (1.1a) for ∂ u, we get
t
∂ u= u∂ u L(∂ u).
t x x
− −
In order to make all the terms be in Hs(R), we mollify, which we write as
J f =j f.
ε ε
∗
Thus we consider the equation
∂ J u= J (u∂ u) L(∂ J u), (2.3)
t ε ε x x ε
− −
where writing L(∂ J u) in the last term is justified as follows: Firstly, writing L(u) as a con-
x ε
volution F−1(m(ξ)) u, associativity and commutativity of convolution gives that J and L
ε
∗
commutes,
J L(∂ u)=j (F−1(m(ξ)) ∂ u)=F−1(m(ξ)) (j ∂ u)
ε x ε x ε x
∗ ∗ ∗ ∗
=L(J ∂ u).
ε x
Secondly, it can easily be shown that J ∂ u=∂ J u using integration by parts.
ε x x ε
Applying the operator Λs to both sides of (2.3), then multiplying the resulting equation by
Λs(J u) and integrating it for x R gives
ε
∈
1 d
J u(t) 2 = Λs(J (u∂ u))Λs(J u)dx Λs(L(∂ J u))Λs(J u)dx. (2.4)
2dtk ε kHs(R) − R ε x ε − R x ε ε
Z Z
First we consider the last term on the right hand side:
Λs(L(∂ J u))Λs(J u)dx= (1+ξ2)s/2J u(ξ)(1+ξ2)s/2m(ξ)∂\J u(ξ)dξ
x ε ε ε x ε
R R
Z Z
= (1+ξ2)smd(ξ)iξJ u( ξ)J u(ξ)dξ
ε ε
R −
Z
=0,
d d
where the lastinequality follows fromm being even. For the first term on the righthand side of
(2.4), we know from the proof of Proposition 1 in [9] that
Λs(J (u∂ u))Λs(J u)dx c ∂ u u 2
| R ε x ε |≤ sk x k∞k kHs(R)
Z
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 5
(the proof relies on commutator estimates for the operators Λs). Thus we have that
1 d
2dtkJεu(t)k2Hs(R) ≤csk∂xukL∞kuk2Hs(R).
Integrating from 0 to t on both sides, we get
1 1 t
J u(t) 2 J u(0) 2 c ∂ u(τ) u(τ) 2 dt,
2k ε kHs(R)− 2k ε kHs(R) ≤ s k x k∞k kHs(R)
Z0
and letting ε 0, we have that
→
1 1 t
2ku(t)k2Hs(R)− 2ku(0)k2Hs(R) ≤cs k∂xu(τ)kL∞ku(τ)k2Hs(R)dτ.
Z0
From this we deduce that
1 d
2dtku(t)k2Hs(R) ≤csk∂xu(t)kL∞ku(t)k2Hs(R). (2.5)
Since s > 3, the Sobolev embedding Hs−1(R) ֒ L∞(R) holds and we thus get (2.2). Now let
2 →
y(t)= u(t) 2 . Then (2.2) implies
k kHs(R)
1 dy
y−3/2 c .
s
2 dt ≤
Integrating from 0 to t gives
1 1
c t,
s
y(0) − y(t) ≤
and we obtain (2.1). From (2.1) wpe immedpiately get that ku(t)kHs(R) is finite when t <
1 , and thus we get the lower bound on the maximal existence time T. (cid:3)
csku0kHs(R)
We also have energy estimates for arbitrary Sobolev norm. The statements below are rather
rough, as we are not interested in optimizing the constants for which the inequalities are true.
Corollary 2.4. Let s > 3. Given u Hs, let u be the corresponding solution. Then, for any
2 0 ∈
T <(c u )−1 and all t [0,T ], one has
0 s 0 Hs 0
k k ∈
u(t) .exp(Ct u ) u , (2.6)
Hr 0 Hs 0 Hr
k k k k k k
for all r >0, for some constant C depending only on r and distance between (c u )−1 and
s 0 Hs
k k
T .
0
Proof. Note that in the arguments establishing (2.5) in the proof of Lemma 2.2, it was nowhere
used that s in the order of the Sobolev norm was the same s as in the statement of the Lemma,
so (2.5) holds for any r >0 in place of s. Thus
d
u(t) Hr cr ∂xu(t) L∞ u(t) Hr
dtk k ≤ k k k k
for any r >0. From Gr¨onwall’s inequality, Sobolev embeddings and (2.1) we then get that
t
u(t) Hr exp(cr ∂xu(τ) L∞dτ) u0 Hr
k k ≤ k k k k
Z0
t
C exp(c u(τ) dτ) u
1 r Hs 0 Hr
≤ k k k k
Z0
t
C exp(c C u dτ) u
1 r 2 0 Hs 0 Hr
≤ k k k k
Z0
=C exp(c C t u ) u
1 r 2 0 Hs 0 Hr
k k k k
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 6
for all r >0 andt [0,T ], where C is anembedding constantand C >0 depends only onthe
0 1 2
∈
difference (c u )−1 T .
s 0 Hs 0
k k −
(cid:3)
3. The periodic case
This section is devoted to proving lack of uniform continuity for the flow map of equation
(1.1a) on T. That is, we will prove Theorem 1.1. This will be done in two steps. First, we
construct two sequences of approximate solutions in Hs(T) that converge to the same limit at
time 0, while remaining bounded apart at any later time. Then we show that the approximate
solutions are sufficiently close to real solutions, thereby establishing lack of uniform continuity.
The proof is based on [13] and [17].
Theapproximatesolutionsconsistofalow-frequencytermandahigh-frequencytermandare
constructed as follows: For ω R and n N, we set
∈ ∈
uω(x,t)=ωn−1+n−scos( nm(n)t+nx ωt).
n − −
By direct calculation, one can show that for n N and α R,
∈ ∈
sin(nx α) Hσ(T) nσ, (3.1)
k − k ≃
and similarly for cosine as well. Thus, for ω bounded, we have
kuωn(·,t)kHs(T) ≃1, for all t∈R, n∈N.
In particular, uω Hs(T) for all n N and all t R and the Hs(T) norm is bounded above
uniformly in n nN∈. ∈ ∈
∈
The next lemma measures how far away the functions uω are from solving equation (1.1a) in
n
the spaces Hσ(T):
Lemma 3.1. Set
E =∂ uω+uω∂ uω+L(∂ uω), (3.2)
t n n x n x n
the error of uω as an approximate solution to (1.1a). Then, for σ R, the error E satisfies
n ∈
E Hσ(T) .n−2s+1+σ.
k k
Proof. By straightforwardcalculations, we find
∂ uω(x,t)=n−s(nm(n)+ω)sin( nm(n)t+nx ωt),
t n − −
∂ uω(x,t)= n−s+1sin( nm(n)t+nx ωt),
x n − − −
L(∂ uω(x,t))= n−s+1m(n)sin( nm(n)t+nx ωt).
x n − − −
Inserting uω(x,t) into (1.1a) and using the above equalities, we get the following expression for
n
the error:
E =∂ uω+uω∂ uω+L(∂ uω)
t n n x n x n
= n−2s+1sin( nm(n)t+nx ωt)cos( nm(n)t+nx ωt)
− − − − −
1
= n−2s+1sin[2( nm(n)t+nx ωt)].
−2 − −
The statement now follows from (3.1). (cid:3)
Lemma 3.2. For n 1 and t 0,
≫ ≥
ku1n(·,t)−u−n1(·,t)kHs(T) &sin(t).
Moreover
ku1n(·,0)−u−n1(·,0)kHs(T) →0, as n→∞.
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 7
Proof. Using the basic trigonometric identity cos(α β) = cos(α)cos(β) sin(α)sin(β) with
± ∓
α= nm(n)t+nx and β =t, and (3.1) we get
−
ku1n(·,t)−u−n1(·,t)kHs(T)
= 2n−1+n−s[cos( nm(n)t+n t) cos( nm(n)t+n +t)] Hs(T)
k − ·− − − · k
= 2n−1+2n−ssin( nm(n)t+n)sin(t) Hs(T)
k − · k
2
&2n−s|sin(t)|ksin(−nm(n)t+n·)kHs(T)− n
1
sin(t) .
≃| |− n
This proves the first statement. Setting t = 0 in the calculations above, it is plain to see that
the second statement also holds. (cid:3)
Now we show that the approximate solutions uω are sufficiently close to real solutions vω of
n n
(1.1a) for n 1.
≫
Lemma 3.3. Let vω(x,t) be the Hs(T) solution to the Cauchy problem
n
∂ vω +vω∂ vω +L(∂ vω)=0,
t n n x n x n
vω(x,0)=ωn−1+n−scos(nx).
n
That is, vω is a solution to equation (1.1a) with initial data given by uω evaluated at time t=0.
n n
Then the following holds:
(i) If s> 3, there exists T >0 independent of n such that for any k >s,
2 0
kuωn(t)−vnω(t)kHs(T) .n(1−s)(1−ks), 0≤t≤T0, n≫1.
(ii) If 0<s 3, there exists 0 T .ns−σ, where σ > 3 can be arbitrarily close to 3, such
≤ 2 ≤ n 2 2
that
kuωn(t)−vnω(t)kHs(T) .n−(1/2)(1−s/k)
for any k> 3 and 0 t T .
2 ≤ ≤ n
Proof. We prove (i) first; that is, we assume s > 3 so that the Cauchy problem is locally well-
2
pexoissetdenicneHofsv(Tω). HAss(kTu)ωnukpHtso(Ts)o≃me1tfiomreaTll n ∈1tNh,atTchaenorbeemco2n.1sidaenrdedLienmdempaen2d.2engtuoafrnan.tLeeestttinhge
n ∈ ≃
T be strictly smaller than the T given by Lemma 2.2, for instance T = 1T, we have that
0 0 2
kvnω(t)kHs(T) .kuωnkHs(T)
for all 0 t T .
0
≤ ≤
Set w =uω vω. Straightforward calculations, using the expression (3.2) for E and that vω
n − n n
is an exact solution to (1.1a), show that w solves the initial value problem
∂ w =E+w∂ w ∂ (wuω) L(∂ w) (3.3)
t x − x n − x
w(,0)=0.
·
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 8
Multiplying by w on both sides of (3.3), we see that
1 d
w(t) 2 = wEdx
2dtk kL2(T) T
Z
+ w2∂ wdx
x
T
Z
w∂ (wuω)dx
− T x n
Z
wL(∂ w)dx.
x
− T
Z
Using Parseval’s identity and that m(ξ) is even, we see that the last integralvanishes. The first
term on the right-hand side is easily estimated by Ho¨lder’s inequality:
wEdx E L2(T) w L2(T).
(cid:12)ZT (cid:12)≤k k k k
(cid:12) (cid:12)
The second term is easily seen t(cid:12)o vanish b(cid:12)y writing w2∂ w =∂ (w3). For the third term we use
(cid:12) (cid:12) x x
integration by parts and Ho¨lder’s inequality:
1 1
(cid:12)ZTw∂x(wuωn)dx(cid:12)= 2(cid:12)ZTuωn∂x(w2)dx(cid:12)= 2(cid:12)ZTw2∂xuωndx(cid:12).k∂xuωnkL∞kwk2L2(T).
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Combin(cid:12)ing these estima(cid:12)tes we(cid:12) get the followin(cid:12)g ine(cid:12)quality: (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
1 d
2dtkw(t)k2L2(T) .kEkL2(T)kwkL2(T)+k∂xuωnkL∞kwk2L2(T).
From the definition of uωn(x,t) it follows that k∂xuωn(t)kL∞ . n−s+1, and using Lemma 3.1 we
then conclude that
1 d
2dtkw(t)k2L2(T) .n−2s+1kwkL2(T)+n−s+1kwk2L2(T),
which implies that
d
dtkw(t)kL2(T) .n−s+1kwkL2(T)+n−2s+1. (3.4)
Recalling that w(,0)=0, we conclude that
·
kuωn(t)−vnω(t)kL2(T) .n−2s+1 (3.5)
for all 0 t T .
0
≤ ≤
As vω is a solution to (1.1a), Corollary 2.4 implies that for k >s,
n
kvnω(t)kHk(T) .nk−s,
for t [0,T ]. We thereby get the ”rough” estimate
0
∈
kuωn(t)−vnω(t)kHk(T) ≤kuωn(t)kHk(T)+kvnω(t)kHk(T)
.kuωn(t)kHk(T)+kuωn(0)kHk(T)
.n−s+k (3.6)
for k >s and 0 t T . Interpolating between (3.5) and (3.6) for k >s, we get
0
≤ ≤
kuωn(t)−vnω(t)kHs(T) ≤kuωn(t)−vnω(t)k1L−2(sT/)kkuωn(t)−vnω(t)ksH/kk(T)
.n(1−s)(1−ks).
This proves part (i).
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 9
Now we turn to the case where 0 < s ≤ 32. As kuωnkHσ(T) . nσ−s, Theorem 2.1 and Lemma
2.2 imply that vnω ∈Hσ(T) exists and satisfies kvnωkHσ(T) .nσ−s for 0 ≤t ≤ T ≃ ns−σ for any
σ > 3. Taking k > 3, we get that the estimate (3.6) holds for 0 t T ns−σ. Moreover,
2 2 ≤ ≤ ≃
(3.4) still holds and using Gr¨onwall’s inequality we conclude that
kuωn(t)−vnω(t)kL2(T) .n−2s+1T (3.7)
for 0 T .ns−σ. Interpolating between (3.7) and (3.6) for k > 3, we get that
≤ 2
kuωn(t)−vnω(t)kHs(T) .n−(1/2)(1−s/k)
for 0 t T ns−σ. (cid:3)
≤ ≤ ≃
We are now able to prove Theorem 1.1:
Proof of Theorem 1.1. Let vω(x,t) be the Hs(T) solution to the Cauchy problem
n
∂ vω +vω∂ vω +L(∂ vω)=0,
t n n x n x n
vω(x,0)=ωn−1+n−scos(nx).
n
Assume first that s> 3. By Lemma 3.3 we have that
2
kvn1(t)−vn−1(t)kHs(T)
≥ku1n(t)−u−n1(t)kHs(T)−ku1n(t)−vn1(t)kHs(T)−ku−n1(t)−vn−1(t)kHs(T)
&ku1n(t)−u−n1(t)kHs(T)−n(1−s)(1−ks).
As (1 s)(1 s)<0 for s>1 and k >s, Lemma 3.2 then implies that
− − k
kvn1(t)−vn−1(t)kHs(T) &|sin(t)|
for all 0 t T and n 1. Moreover,
0
≤ ≤ ≫
kvn1(0)−vn−1(0)kHs(T) →0
as n . This proves part (i).
→∞
When 0<s 3 the above arguments do not lead to a contradictionas the times t for which
≤ 2
theyholdgotozero;weneedthesolutionstogoapartmuchsooner. Asnotedin[17],theessential
observation is that if u(x,t) solves (1.1a) with initial data u , then v(x,t) = u(x ωt,t)+ω
0
−
solves (1.1a) with initial data u +ω, as is easily verified. The arguments in [17] can be applied
0
directly from this point, but we repeat them here or the sake of completeness.
Let v0 be a solution to the Cauchy problem above for ω = 0, and define v˜ω(x,t) := v0(x
n n n −
ωt,t)+ω. We pick t [n−1+ε,ns−σ] for some ε>0 sufficiently small and set
n
∈
π π
ω =(nt )−1 , ω = (nt )−1 .
1 n 2 n
2 − 2
At time t=0, we get
kv˜nω1(·,0)−v˜nω2(·,0)kHs(T) ≃|ω1−ω2|.n−ε →0
NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR EQUATIONS OF WHITHAM TYPE 10
as n . At t=t we can use Lemma 3.3 (ii):
n
→∞
kv˜nω1(·,tn)−v˜nω2(·,tn)kHs(T) =kvn0(·−ω1tn,tn)+ω1−vn0(·−ω2tn,tn)−ω2kHs(T)
&kvn0(·−ω1tn,tn)−u0n(·−ω1tn,tn)kHs(T)
+kvn0(·−ω2tn,tn)−u0n(·−ω2tn,tn)kHs(T)
+ku0n(·−ω1tn,tn)−u0n(·−ω2tn,tn)kHs(T)
ω ω
1 2
−| − |
1+n−(1/2)(1−s/k) n−ε,
≃ −
where we calculated
ku0n(x−ω1tn,tn)−u0n(x−ω2tn,tn)kHs(T)
= n−s(cos( nm(n)tn+nx π/2) cos( nm(n)tn+nx+π/2)) Hs(T)
k − − − − k
= 2n−ssin( nm(n)tn+nx) Hs(T)
k − k
1.
≃
Taking n , this concludes the proof of part (ii). (cid:3)
→∞
4. Non-uniform continuity on the real line
In this section we prove the lack of uniform continuity for the flow map of the Whitham
equation (1.1a) on R. That is, we will prove Theorem 1.2. As in the periodic case (cf. Section
3), Theorem1.2 will be provenby constructing two sequences of approximatesolution in Hs(R)
that convergeto the same limit at time 0, while remaining bounded apartat any later time and
showing that the approximate solutions are sufficiently close to real solutions. The idea for the
proof is from [13].
In the sequel, δ will always denote a number 1<δ <2 that we may choose freely and λ will
be apositiveparameter. Forconvenienceofnotationwewilldenotef (x):=f( x )forfunctions
λ λδ
f :R R and λ>0. The following lemma will be useful in the sequel.
→
Lemma 4.1 ([13]). Let ϕ S(R), 1<δ <2 and α R. Then for any s 0 we have that
∈ ∈ ≥
1
λl→im∞λ−δ/2−skϕλcos(λ·+α)kHs(R) = √2kϕkL2(R).
The statement holds true also if cos is replaced by sin.
Lemma 4.1 can be found as Lemma 2.3 in [13] and a proof is given there.
Weconstructatwo-parameterfamilyofapproximatesolutionsuω,λ =uω,λ(t,x),following[9].
Each function uω,λ consists of two parts;
uω,λ =u +uh.
l
The high frequency part uh is given by
uh =uh,ω,λ(t,x)=λ−δ/2−sϕ (x)cos( λm(λ)t+λx ωt),
λ
− −
where ϕ is a C∞ function such that
1, if x <1,
ϕ(x)= | |
(0, if x 2.
| |≥