Long time Solutions for a Burgers-Hilbert Equation via a Modified Energy Method

0.11 MB·English

by  John K. Hunter

.
LONG TIME SOLUTIONS FOR A BURGERS-HILBERT
EQUATION VIA A MODIFIED ENERGY METHOD
JOHNK.HUNTER,MIHAELAIFRIM,DANIELTATARU,ANDTAKKWONGWONG
3
1
Abstract. Weconsideraninitialvalueproblemforaquadraticallynonlinear
0
inviscidBurgers-Hilbertequation that models themotion of vorticitydiscon-
2
tinuities. We use a modified energy method to prove the existence of small,

n smoothsolutionsovercubicallynonlineartime-scales.
a
J

9

] 1. Introduction
P
A We consider the following initial value problem for an inviscid Burgers-Hilbert
equation for u(t,x):
.
h
u +uu =H[u],
t t x
a (1)
m u(0,x)=u0(x),
[ whereHisthespatialHilberttransformonR,andtheinitialdatau issufficiently
0
small
1
v ku0(x)kH2(R) ≤ǫ≪1.
7
Here, Hk(R) denotes the standard Sobolev space of functions with k weak L2-
4
9 derivatives.
1 Equation (1) was proposed in [1] as a model equation for nonlinear waves with
1. constant frequency. It also provides a formal asymptotic approximation for the
0 small-amplitude motion of a planar vorticity discontinuity located at y = u(t,x)
3 [1, 5]. Moreover,even though the equation is quadratically nonlinear, this approx-
1
imation is valid over cubically nonlinear time-scales.
:
v Equation (1) is nondispersive, and solutions of the linearized equation oscillate
i in time but do not exhibit any dispersive decay. Nevertheless, in comparison with
X
the inviscid Burgers equation, small smooth solutions of (1) have an enhanced,
r
a cubically nonlinear lifespan. This enhanced lifespan was observed numerically in
[1] and proved in [2] by use of a change of the independent variable, which was
suggested by a normal form transformation of the equation.
In this paper we give a different, and simpler, proof of the enhanced lifespan
of smooth solutions of (1) from the one in [2]. Our main result is the following
theorem:
Theorem 1. a) Let k ≥ 2. Suppose that the initial data u ∈ Hk(R) for the
0
equation (1) satisfies
ku0kH2(R) ≤ǫ≪1.
Date:January10,2013.
ThefirstauthorwaspartiallysupportedbytheNSFundergrantnumberDMS-0072343.
The third author was partially supported by the NSF grant DMS-0801261 as well as by the
SimonsFoundation.
1
2 HUNTER,IFRIM,TATARU,WONG
Then there exists a solution u ∈ C Iǫ;Hk(R) ∩C1 Iǫ;Hk−1(R) of (1) defined
on the time-interval Iǫ = −α/ǫ2,α(cid:0)/ǫ2 , wher(cid:1)e α >(cid:0)0 is a univers(cid:1)al constant, so
that (cid:2) (cid:3)
kukL∞(Iǫ;Hs(R)) .ku(0)kHs(R), 0≤s≤k.
b) Consider u ,u two solutions of (1) as in part (a). Then
1 2
ku1−u2kL∞(Iǫ;L2(R)) .ku1(0)−u2(0)kL2(R).
The quadratically nonlinear terms in (1) are non-resonant, and a standard ap-
proach for problems of this type is to use a normal form transformation (see for
instance[7])toremovethem. However,thisisaquasi-linearproblem,sothenormal
formtransformationisunbounded,andthismethodcannotbeapplieddirectly. Our
alternative approachis to use the normal form transformation in order to derive a
modified energy functional for the original problem (1) which evolves according to
a cubic law.
ThesamemodifiedenergymethodalsoappliestothelinearizationoftheBurgers-
Hilbert equation, and allows us to control differences of solutions on a cubic time-
scale.
We remarkthatwithsomeadditionalworkonecanreplacethe H2 normforthe
data inpart(a) ofthe abovetheoremwith Hs0 for any s0 >3/2. Similarly, the L2
norm in part (b) can be replaced by Hs for a smaller range 0≤s≤s −1.
0
2. The Normal Form Transformation
Straightforwardenergy estimates applied to (1) yield
d
dtk∂xkuk2L2 .kuxkL∞kuk2Hk
for k ≥ 2, which gives a lifespan of smooth solutions of the order ǫ−1, as in the
standard local existence theory for quasi-linear hyperbolic PDEs [3, 4].
The quadratically nonlinear terms in (1) can be removed by a normal form
transformation [1, 2]
(2) v =u+H[Hu·Hu ].
x
The transformed equation has the form
(3) v +Q(u)=H[v]
t
where Q(u) is cubic in u but involves two spatial derivatives. Straightforward
energy estimates for (3) yield
d
k∂kvk2 .ku k2 kuk2 .
dt x L2 x L∞ Hk+1
However, since the above normal form transformation is not invertible, the energy
estimatesforvdonotclose. Thus,thev-equationiscubicallynonlinear,butthereis
a loss of derivatives. On the other hand, the u-equationis quadratically nonlinear,
but there is no loss of derivatives.
The reason for this disparity in energy estimates is that the Hk-norms of u and
v are not comparable. From (2), we have
(4) k∂kvk2 =k∂kuk2 +2h∂ku,∂kH[Hu·Hu ]i+k∂kH[Hu·Hu ]k2 .
x L2 x L2 x x x x x L2
BURGERS-HILBERTEQUATION 3
The second term on the right-hand side of (4) is comparable to the Hk-norm of u
because
1
h∂ku,∂kH[Hu·Hu ]i= k+ hHu ,(∂kHu)2i+l.o.t.
x x x (cid:18) 2(cid:19) x x
where l.o.t. stands for lower order terms. The third term on the right-hand side of
(4) is not comparable to the Hk-norm, but it is quartic and therefore irrelevant to
the cubically nonlinear energy estimates.
Thisdiscussionsuggeststhatwedefineamodifiedenergyfunctionalbydropping
the higher-derivative quartic term from the right-hand side of (4) to get
1
(5) E (u)= k∂kuk2 +h∂ku,∂kH[Hu·Hu ]i.
k 2 x L2 x x x
As we will show,this modified energy is equivalent to the standard Hk-energy and
satisfies cubically nonlinear estimates without a loss of derivatives.
3. The Modified Energy Method
In this section, we use the modified energy functional introduced in (5) to prove
Theorem1. We first show thatE energy is equivalent to the standardHk energy.
k
Lemma 2. Let E be defined by (5). Then
k
1
Ek(u)= 2k∂xkuk2L2(1+O(kHuxkL∞)).
Proof. Denote
T f :=H[Hu·Hf ].
u x
This operator is essentially skew-adjoint. Moreover,we have
hf,T fi=−hHf,Hu·Hf i
u x
1
= 2ZRHux|Hf|2dx=O(kHuxkL∞)kfk2L2.
We write
∂kH[Hu·Hu ]=T ∂ku+l.o.t.
x x u x
wherel.o.t.standsfortermsoftheformH[∂jHu ·∂k−1−jHu ]with0≤j ≤k−1.
x x x x
For the leading term we use the previous estimate for T , and for the lower order
u
terms we use interpolation,
k∂xjHux·∂xk−1−jHuxkL2 .kHuxkL∞k∂xk−1uxkL2
Thus, we obtain
h∂xku,∂xkH[Hu·Hux]i=O(kHuxkL∞)k∂xkukL2
(cid:3)
The energy estimate is discussed in the following lemma:
Lemma 3. For the energy (5) the estimate below holds: for any δ >0,
d
E (u)≤C ku k2 kuk2
dt k k,δ x H21+δ Hk
where C is a constant depending on k and δ only.
k,δ
4 HUNTER,IFRIM,TATARU,WONG
Proof. A direct computation yields
d
− E (u)= ∂k(uu )∂kH[Hu·Hu ] dx
dt k ZR x x x x
+ ∂ku ∂kH[H[uu ]·Hu ]+∂kH[Hu·H[uu ] ] dx.
ZR x x x x x x x
(cid:8) (cid:9)
Note that all cubic terms cancel. This is because the energy was determined using
the normalform transformation,which at the level of the energy amounts to elim-
inating the cubic terms. Using the antisymmetryof H and integratingby parts we
rewrite the above expression in the form
d
− E (u)= −∂kH(uu )∂k[Hu·Hu ]+∂kHu ∂k[Hu·H(uu )]dx.
dt k ZR x x x x x x x x
We observethatineachterminthe expansionofthe aboveintegrandhas the form
H∂αuH∂βuH(∂γu ∂δu)
where
α+β+γ+δ =2k+2
We observe that as long as
1≤α,β,γ,δ ≤k, α+β ≥3,δ+γ ≥3
such terms can be estimated directly in terms of the right hand side of the energy
relation in the Lemma. We will call such terms “good”.
We return to the expression above, and observe that a cancellation occurs if
the second ∂k operator in each of the two expressions applies to the second factor.
x
Using a full binomial expansion, we are left with the expressions
I = −∂kH(uu )·∂αHu·∂k−αHu +∂kHu ·∂αHu·∂k−αH(uu )dx
α ZR x x x x x x x x x x
where 1≤α≤k. Distributing the remaining derivatives, we get good terms if any
derivatives fall on the u factor (an additional integration by parts is needed to see
this for the second term). Thus we have
I = −H(u∂ku )·∂αHu·∂k−αHu +∂kHu ·∂αHu·H(u∂k−αu )dx+good.
α ZR x x x x x x x x x x
If u is commuted out the two expressions above cancel. Hence
I = −[H,u]∂ku ·∂αHu·∂k−αHu +∂kHu ·∂αHu·[H,u]∂k−αu dx+good.
α x x x x x x x x x x
ZR
The commutator [H,u]f vanishes unless the frequency of u is at least as large
as the frequency of f. Hence we can always move derivatives from f onto u. Then
for the first term in I we can use directly the commutator estimate below:
α
(6) k[H,u]∂xkuxkL2 .kuxkL∞k∂xkukL2.
This is a consequence of the commutator estimates result obtained by Dawson,
McGahagan, and Ponce in [6]. We recall it below:
Lemma 4. Let H denote the Hilbert transform. Then for any p∈(1,∞) and any
l,m∈Z+ there exists c=c(p,l,m)>0 such that
k∂xl[H,a]∂xmfkLp ≤ck∂xl+makL∞kfkLp.
BURGERS-HILBERTEQUATION 5
For the second in I we need to integrate once by parts since the first factor
α
has too many derivatives. If α < k this is done directly. If α = k then we move
a derivative on the third factor. We are left with two terms which are nontrivial,
namely
J := |∂kHu|2[H,u ]u dx
k Z x x x
and
Jk−1 := |∂xkHu|2[H,u]uxxdx.
Z
For these we need the pointwise bound
(7) k[H,ux]uxkL∞ +k[H,u]uxxkL∞ .kuxk2H21+δ.
Indeed,bothtermsontheleftcanbeestimatedinH21+δ usingstandardLittlewood-
Paley decompositions. (cid:3)
By a Gronwall type argument, the two lemmas above lead to an ǫ−2 lifespan,
proving part (a) of Theorem 1.
Next,inordertoprovepart(b)ofTheorem1,weconsiderthelinearizedBurgers-
Hilbert equation:
(8) w +wu +uw =Hw.
t x x
Beginningwiththenormalformtransformation(2)fortheBurgers-Hilbertequa-
tion, we obtain the normal form transformation for the linearization (8):
q :=w+|∂ |(Hw·Hu).
x
where |∂ |=H ◦ ∂ . Inspired by this we can define a modified linearized energy
x x
(9) E (w)= w2+2|∂ |w·Hu·Hwdx
lin x
Z
This is equivalent to the L2 norm of w,
Elin(w)=(1+O(kHuxkL∞)kwk2L2.
This is proved using integration by parts in order to write
E (w)= w2−|∂ |u·(Hw)2 dx.
lin x
Z
We want to prove that E satisfies good cubic bounds; a Gronwall type argu-
lin
ment implies part (b) of Theorem 1.
Lemma 5. For the energy (9) the estimate below holds: for any δ >0,
d
E (w)≤C ku k2 kwk2
dt lin k,δ x H21+δ L2
where C is a constant depending on k and δ only.
k,δ
Proof. A straightforwardcomputation leads to
d
E (w)= 2Hw·H[wu ]·Hu +2Hw·H[uw ]·Hu
lin x x x x
dt Z
+|Hw|2·|∂ |(uu )dx
x x
= 2Hw·H[wu ]·Hu +2Hw·[H,u]∂ w·Hu
x x x x
Z
+|Hw|2∂ ([H,u]∂ u) dx.
x x
6 HUNTER,IFRIM,TATARU,WONG
The first term is estimated directly and for the second we use the commutator
bound (6). The third term can be controlled because we can rewrite
∂ ([H,u]∂ u)=[H,u ]∂ u+[H,u]∂2u,
x x x x x
where the right hand side can be estimated by the pointwise bound (7). (cid:3)
References
[1] J.BielloandJ.K.Hunter,NonlinearHamiltonianwaveswithconstantfrequencyandsurface
waves on vorticitydiscontinuities,Comm.PureAppl.Math.63,2009, 303-336.
[2] J.K.HunterandM.Ifrim,Enhancedlifespanofsmooth solutionsofaBurgers-Hilbertequa-
tion,SIAMJ.Math.Anal.,44,2012,2039-2052.
[3] T.Kato,TheCauchyproblemforquasi-linearsymmetrichyperbolicsystems,Arch.Rational
Mech.Anal.58,1975,181-205.
[4] A. J. Majda, Compressible fluid flow and systems of conservation laws in several space
variable,Springer-Verlag,Volume53,1984.
[5] J. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incom-
pressible fluids,PhysicaD7,1983, 305-323.
[6] G.Ponce, F.LinaresandD.Pilod,Well-Posedness for a higher-order Benjamin-Ono equa-
tion.Proc.AMS.136,2008, 2081-2090.
[7] J.Shatah,NormalformsandquadraticnonlinearKlein-Gordonnonlinearities,Comm.Pure
Appl.Math.38,1985,685-696.
Departmentof Mathematics,University ofCaliforniaatDavis
E-mail address: [email protected]
Departmentof Mathematics,McMaster University
E-mail address: [email protected]
Departmentof Mathematics,University ofCaliforniaatBerkeley
E-mail address: [email protected]
Departmentof Mathematics,University ofPennsylvania
E-mail address: [email protected]