A dispersive estimate for the linear wave equation with an electromagnetic potential

0.15 MB·English

by  Davide Catania

A DISPERSIVE ESTIMATE FOR THE LINEAR WAVE EQUATION WITH
AN ELECTROMAGNETIC POTENTIAL
DAVIDECATANIA
Abstract. WeconsiderradialsolutionstotheCauchyproblemforthelinearwaveequation
withasmallshort–rangeelectromagneticpotential(the“squareversion”ofthemasslessDirac
equationwithapotential)andzeroinitialdata. Weprovetwoaprioriestimatesthatimply,
6
0 in particular, adispersive estimate.
0
2

n
a 1. Introduction
J

2
In this paper, we investigate the dispersive properties of the linear wave equation with an

]
P electromagnetic potential, that is
A (1.1) (cid:3) u = F (t,x) [0, [ R3,
A
. ∈ ∞ ×
h where
t
a
(1.2) x= (x ,x ,x ), r = x,
m 1 2 3
| |
(1.3) (cid:3) = (cid:3) A ,
[ A t,r
− ·∇
1 (1.4) (cid:3) = ∂2 ∆ = ∂2 (∂2 +∂2 +∂2 ),
t − t − x1 x2 x3
v
∂
0 (1.5) = t .
3 ∇t,r ∂r
0 (cid:18) (cid:19)
1 The fact that the potential A = A(t,x) is electromagnetic means that A iR iR, where
∈ ×
0 i is the imaginary unit. This will play a crucial role in the development of the proof, since
6 electromagnetic potential are gauge invariant (see what follows).
0
We restrict ourselves to radial solutions u = u(t,r), with F = F(t,r) and
/
h
at (1.6) A= A(t,r) = A0(t,r) , A0,A1 iR.
m (cid:18)A1(t,r)(cid:19) ∈
: Weassumefurtherthatthepotentialdecreasessufficientlyrapidlywhenr approachesinfinity;
v
more precisely, we suppose that
i
X
r (1.7) 2−jh2−jiεA||ϕjA||L∞r 6δA
a j∈Z
X
(thatis,Aisashort–rangepotential),whereε > 0,δ isasufficientlysmallpositiveconstant
A A
independent of r (see Section 2) and the sequence (ϕj)j∈Z is a Paley–Littlewood partition of
unity,whichmeansthatϕ (r) = ϕ(2jr)andϕ :R+ R+ (R+ isthesetofallnon–negative
j
−→
real numbers) is a function so that
Keywords and phrases. Waveequation,linear,perturbed,potential,electromagnetic,short–range,Diracequa-
tion, massless, a priori estimate, dispersive estimate, decayestimate.
AMS Subject Classification: 35A08, 35L05, 35L15, 58J37, 58J45.
1
2 DAVIDECATANIA
a) suppϕ= r R : 2−1 6 r 6 2 ;
b) ϕ(r) > 0 { fo∈r 2−1 < r < 2; }
c) ϕ(2jr)= 1 for each r R+.
j∈Z ∈
In other words, ϕ (r) = 1 for all r R+ and
P j∈Z j ∈
(1.8) P suppϕ = r R : 2−j−1 6 r 6 2−j+1 .
j
{ ∈ }
It is well–known that there exists a unique global solution to the Cauchy problem
(cid:3) u= F (t,x) [0, [ R3,
A
(1.9) ∈ ∞ ×
(u(0,x) = ∂tu(0,x) = 0 x R3;
∈
in particular, this fact holds for the smaller class of radial solutions, that is for the problem
(cid:3) u= F (t,r) [0, [ R+,
A
(1.10) ∈ ∞ ×
(u(0,r) =∂tu(0,r) = 0 r R+.
∈
Let introduce the change of coordinates
. t r
(1.11) τ± = ±
2
.
and the standard notation s = √1+s2; our main result can be expressed as follows.
h i
Theorem 1.1. Let u be a radial solution to (1.9), i.e. a solution to (1.10), where A= A(t,r)
is an electromagnetic potential satisfying (1.7) for some δ > 0 and ε > 0. Then, for
A A
every ε > 0, there exist two positive constants δ and C (depending on ε) such that for each
δ ]0,δ], one has
A
∈
(1.12) τ+u L∞ 6 C τ+r2 r εF L∞.
|| || t,r || h i || t,r
Let introduce the differential operators
.
(1.13) ± = ∂t ∂r.
∇ ±
The proof of the previous a priori estimate follows easily from the following one.
Lemma 1.1. Under the same conditions of Theorem 1.1, for every ε > 0, there exist two
positive constants δ and C (depending on ε) such that for each δ ]0,δ], one has
A
∈
(1.14) τ+r −u L∞ 6 C τ+r2 r εF L∞.
|| ∇ || t,r || h i || t,r
An immediate consequence of Theorem 1.1 is the following dispersive estimate.
Corollary 1.1. Under the same conditions of Theorem 1.1, for every ε > 0, there exist two
positive constants δ and C (depending on ε) such that for each δ ]0,δ], one has
A
∈
C
(1.15) u(t,x) 6 τ+r2 r εF L∞
| | t || h i || t,r
for every t > 0.
The idea to prove the lemma is the following. First of all, the potential term in (1.10) can
be thought as part of the forcing term, that is (cid:3) u= F can be viewed as
A
.
(1.16) (cid:3)u= F = F +A u.
1 t,r
·∇
Then we can rewrite this equation in terms of τ± and ± (see Section 2), obtaining
∇
(1.17) + −v = G,
∇ ∇
A DISPERSIVE ESTIMATE FOR THE LINEAR WAVE EQUATION 3
where
. .
(1.18) v(t,r) = ru(t,r) and G(t,r) = rF (t,r).
1
Thislastequation canbeeasily integratedtoobtain arelatively simpleexplicitrepresentation
of ( −v)(τ+,τ−) in terms of G.
∇
Another fundamental step consists in taking advantage of the gauge invariance property of
the electromagnetic potential A, which means that, set
. A0 A1
(1.19) A± = ± ,
2
we can assume, with no loss of generality, that A 0 (see [2], p. 34). This implies that
+
≡
(1.20) A t,ru= A− −u+A+ +u = A− −u
·∇ ∇ ∇ ∇
and hence
(1.21) F1 = F +A− −u,
∇
thus
1
(1.22) G = rF +rA− −u= rF +A− −v+ A−v.
∇ ∇ r
Obviously, one has
(1.23) 2−j 2−j εA ϕjA− L∞ 6 δA.
h i || || r
j∈Z
X
These simplifications, combined with technical Lemma 2.1 and the estimate of Lemma 2.2,
allow us to easily obtain Lemma 1.1 and Theorem 1.1.
The dispersive properties of evolution equations are important for their physical meaning
and, consequently, they have been deeply studied, though the problem in its generality is still
open. ThedispersiveestimateobtainedinCorollary1.1providesthenaturaldecayrate,thatis
thesamerateonehasforthenon–perturbedwaveequation(see[11,13]),i.e. at−(n−1)/2 decay
intime,wherenisthespacedimension(inourcase,n= 3). Thegeneralizationtothecaseofa
potential–likeperturbationhasbeenconsideredwidely(see[1,3,4,5,7,10,14,16,17,18,19]),
also for the Schr¨odinger equation (see [8, 9, 12, 15]). Recently, D’Ancona and Fanelli have
considered in [6] the case
∂2u(t,x)+Hu = 0, (t,x) R R3,
(1.24) t ∈ ×
(u(0,x) = 0, ∂tu(0,x) = g(x),
where
.
(1.25) H = ( +iA(x))2 +B(x),
− ∇
(1.26) A :R3 R3, B :R3 R.
−→ −→
Under suitable condition on A, A and B, in particular
∇
3
C C
(1.27) A(x) 6 0 , ∂ A (x) + B(x) 6 0 ,
| | r r (1+ lgr )β | j j | | | r2(1+ lgr )β
h i | | j=1 | |
X
4 DAVIDECATANIA
with C > 0 sufficiently small, β > 1 and r = x, they have obtained the dispersive estimate
0
| |
C
(1.28) u(t,x) 6 22j r w1/2ϕ (√H)g ,
| | t ||h i β j ||L2
j>0
X
.
where w = r(1+ logr )β and (ϕ ) is a non–homogeneous Paley–Littlewood partition of
β j j>0
| |
unity on R3.
In this paper, restricting ourselves to radial solutions, we are able to obtain the result
in Corollary 1.1, which is optimal from the point of view of the estimate decay rate t−1 and
improveessentially theassumptionsonthepotential, assumingweakercondition (1.7)instead
of (1.27).
2. A priori estimates
First of all, we reformulate our problem taking advantage of the radiality of the solution u to
(1.10). Indeed, since ∆S2u(t,r) = 0 and v = ru, we have
2 1
(2.1) (cid:3)u(t,r) = (∂t2−∆x)u= ∂t2−∂r2− r∂r − r2∆S2 u(t,r)
(cid:18) (cid:19)
1 1
(2.2) = ∂2v(t,r) ∂2v(t,r)
r t − r r
1 1
(2.3) = + −v(t,r) = − +v(t,r).
r∇ ∇ r∇ ∇
Recalling (1.18) and (1.21), we get that the equation in (1.10) is equivalent to
(2.4) + −v = G.
∇ ∇
Letusnoticethatthesupportofu(t,r)iscontainedinthedomain (t,r) R2 :r > 0, t > r ,
{ ∈ }
therefore we have
(2.5) suppv(τ+,τ−) (τ+,τ−) R2 : τ− > 0, τ+ > τ− .
⊆ { ∈ }
From this fact, we get
τ+ τ+
(2.6) −v(τ+,τ−) = −v(τ−,τ−)+ G(s,τ−)ds = G(s,τ−)ds.
∇ ∇
Zτ− Zτ−
Let us observe that, for each s [τ−,τ+], we have
∈
(2.7) s6 τ+, s τ− 6 τ+ τ− = r,
− −
hence
τ+ τ+ s s τ− ε G(s,τ−)
G(s,τ−)ds 6 h − i | | ds
(cid:12)Zτ− (cid:12) Zτ− hsihs−τ−iε
(cid:12) (cid:12) τ+
(cid:12)(cid:12) (cid:12)(cid:12) 6 τ+ r εG L∞ s −1 s τ− −εds
|| h i || t,r h i h − i
Zτ−
for every ε > 0. Applying lemma 2.1 (see the end of this section), we conclude
(2.8) τ+ −v(τ+,τ−) 6 Cr τ+ r εG L∞;
|∇ | || h i || t,r
A DISPERSIVE ESTIMATE FOR THE LINEAR WAVE EQUATION 5
recalling that G satisfies (1.22), we obtain
τ+ −v(τ+,τ−) 6 Cr τ+ r εA− −v L∞ + τ+ r εr−1A−v L∞
|∇ | || h i ∇ || t,r || h i || t,r
(2.9)
(cid:16)
+ τ+ r εrF L∞ .
|| h i || t,r
(cid:17)
Now, if we choose for the moment ε 6 ε , we have
A
(2.10) r r εϕj(r)A−(t,r) 6 C2−j 2−j εA ϕjA− L∞
h i | | h i || || r
(here and in the following, we assume that C = C(ε) > 0 could change time by time), thus
r τ+ r εA− −v L∞ 6 C τ+ −v L∞ 2−j 2−j εA ϕjA− L∞
|| h i ∇ || t,r || ∇ || t,r h i || || r
(2.11) j∈Z
X
6 CδA τ+ −v L∞,
|| ∇ || t,r
wherewehaveusedthefactthat(ϕj)j∈Z isaPaley–Littlewoodpartitionofunityandproperty
(1.23).
Moreover, v(τ ,τ )= 0 because of (2.5), whence
+ +
τ+
(2.12) v(τ+,τ−)= −v(τ+,s)ds
− ∇
Zτ−
and consequently
τ+
(2.13) v(τ+,τ−) 6 −v(τ+,s) ds 6r −v L∞.
| | |∇ | ||∇ || t,r
Zτ−
Thus we have
(2.14) r εϕj(r)A−(t,r)v(τ+,τ−) 6 C2−j 2−j εA ϕjA− L∞ −v L∞,
h i | | h i || || r ||∇ || t,r
which implies
(2.15) r τ+ r εr−1A−v L∞ 6 CδA τ+ −v L∞.
|| h i || t,r || ∇ || t,r
Using (2.11) and (2.15) in (2.9), we deduce
(2.16) τ+ −v L∞ 6 C τ+r2 r εF L∞,
|| ∇ || t,r || h i || t,r
provided δ is sufficiently small. For instance, one can take δ such that 3C2δ 6 1.
A A A
From the definition of v, we have
(2.17) r −u = −v+u
∇ ∇
and hence
(2.18) τ+r −u 6 τ+ −v + τ+u,
| ∇ | | ∇ | | |
so
τ+r −u L∞ 6 τ+ −v L∞ + τ+u L∞
|| ∇ || t,r || ∇ || t,r || || t,r
6 C τ+r2 r εF L∞ + τ+r2 r εF1 L∞ ,
|| h i || t,r || h i || t,r
(cid:16) (cid:17)
6 DAVIDECATANIA
where we have used (2.16) and the inequality in Lemma 2.2. But
r2 r ε F1 6 r2 r ε A− −u +r2 r ε F
h i | | h i | |·|∇ | h i | |
(2.19) 6 r r εAϕj A− r −u +r2 r ε F
 h i | | |∇ | h i | |
j∈Z
X
6 CδAr −u +r2 rε F ,
|∇ | h i | |
thus
(2.20) τ+r2 r εF1 L∞ 6 CδA τ+r −u L∞ + τ+r2 r εF L∞
|| h i || t,r || ∇ || t,r || h i || t,r
and consequently
(2.21) τ+r −u L∞ 6 C τ+r2 r εF L∞
|| ∇ || t,r || h i || t,r
provided δ > 0 small enough, that is Lemma 1.1.
A
Now we use the fact that, because of (2.17), we have
(2.22) τ+u 6 τ+r −u + τ+ −v ;
| | | ∇ | | ∇ |
combining this estimate with (1.14) and (2.16), we finally conclude
(2.23) τ+u L∞ 6 C τ+r2 r εF L∞,
|| || t,r || h i || t,r
and also Theorem 1.1 is proven.
Lemma 2.1. For each ε> 0, there exists a positive constant C = C(ε) such that
τ+ Cr
s −1 s τ− −εds 6
h i h − i τ
Zτ− +
Proof. We distinguish two cases.
Case 1: τ+ > 2τ−. Let us notice that since r = τ+ τ− > τ+/2, in this case it is sufficient
−
to prove that
τ+
(2.24) s −1 s τ− −εds 6 C0(ε).
h i h − i
Zτ−
We observe that s τ− > s/2 provided s > 2τ−, so
−
τ+ τ−+1 τ++1
s −1 s τ− −εds 6 s −1ds+2ε s−(1+ε)ds
h i h − i h i
Zτ− Zτ− Zτ−+1
∞
1
6 +2ε s−(1+ε)ds
hτ−i Z1
6 1+C (ε).
1
Case 2: τ+ < 2τ−. We use the estimates s −1 < 2/τ+ and s τ− −ε 6 1 to get
h i h − i
τ+ 2 2r
(2.25) s −1 s τ− −εds 6 (τ+ τ−) = .
h i h − i τ − τ
Zτ− + +
This concludes the proof. (cid:4)
In the case A 0 (non–perturbed equation), we have the following version of the estimate
≡
in Theorem 1.1. It consists in a slight modification of estimate (1.8) shown in [7], p. 2269.
A DISPERSIVE ESTIMATE FOR THE LINEAR WAVE EQUATION 7
Lemma 2.2. Let u be the solution to
(cid:3)u= F (t,r) [0, [ R+,
(2.26) ∈ ∞ ×
(u(0,r) =∂tu(0,r) = 0 r R+.
∈
Then, for every ε > 0, there exists C > 0 such that
(2.27) τ+u L∞ 6 C τ+r2 r εF L∞.
|| || t,r || h i || t,r
Proof. Let notice that u is the solution to (1.10) with A 0. Then, from (2.9), we have
≡
(2.28) τ+ −v(τ+,τ−) 6 C τ+r2 r εF L∞,
|∇ | || h i || t,r
where v = ru. Using (2.13), we deduce
(2.29) τ+ u = τ+ v r−1 6 τ+ −v L∞
| | | | || ∇ || t,r
and hence the claim. (cid:4)
References
[1] M. Beals, Optimal L∞ decay for solutions to the wave equation with potential, Comm. Partial Diff.
Equations,19 (1994), 1319–1369.
[2] N.N. Bogoliubov and D.V. Shirkov,Quantumfields,TheBenjamin/CummingsPublishingCompany
Inc.,1983.
[3] N. Burq, F. Planchon, S. Stalker and S. Tahvildar–Zadeh,Strichartzestimatesforthewaveand
Schr¨odingerequationswith theinverse–squarepotential, J. Funct.Anal., 203 (2003), 519–549.
[4] S. Cuccagna, On the wave equation with a potential, Comm. Partial Diff. Equations, 25 (2000), 1549–
1565.
[5] S. Cuccagna and P.P. Schirmer, On the wave equation with a magnetic potential, Comm. Pure and
AppliedMath., 54 (2001), 135–152.
[6] P. D’Ancona and L. Fanelli, Decay estimates for the wave and Dirac equations with a magnetic
potential, Preprint.
[7] V. Georgiev, C. Heiming and H. Kubo, Supercritical semilinear wave equation with non–negative
potential, Comm. Partial Diff. Equations,26 (2001), 2267–2303.
[8] V. Georgiev, A. Stefanov and M. Tarulli, Smoothing–Strichartz estimates for the Schr¨odinger
equation with small magnetic potential, Preprint.
[9] V. Georgievand M. Tarulli,ScaleinvariantenergysmoothingestimatesfortheSchr¨odingerequation
with small magnetic potential, Preprint.
[10] V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential, Comm. Partial
Diff.Equations, 28 (2003), 1325–1369.
[11] J. GinibreandG. Velo,GeneralizedStrichartzinequalitiesforthewaveequation,J.Funct.Anal.,133
(1995), 50–68.
[12] J.–L.Journ´e,A.SofferandC.Sogge,DecayestimatesforSchr¨odingeroperators,Comm.PureAppl.
Math, 44(1991), 573–604.
[13] M. Keel and T. Tao, Endpoint Strichartzestimates, Amer.J. Math., 120 (1998), 955–980.
[14] F. Planchon, S. Stalker and S. Tahvildar–Zadeh,Dispersiveestimates for thewaveequationwith
inverse–squarepotential, Discrete ContinuousDynamicSystem,9 (2003), 427–442.
[15] I. Rodnianski and W. Schlag,Timedecayfor solutionsofSchr¨odingerequationswithrough andtime
dependentpotentials, Invent.Math., 155 (2004), 451–513.
[16] N. Visciglia, Existence globale pour une classe d’´equations d’ondes perturb´ees, C. R. Math. Acad. Sci.
Paris, 338 (2004), no. 1, 27–30.
[17] N. Visciglia, Semilinear wave equation with time dependent potential, Math. Methods Appl. Sci., 27
(2004), no. 18, 2153–2170.
[18] K. Yajima, The Wk,p–continuity of wave operators for Schr¨odinger operators, J. Math. Soc. Japan, 47
(1995), 551–581.
8 DAVIDECATANIA
[19] K.Yajima,Lp–boundednessofwaveoperatorsfortwo–dimensionalSchr¨odingeroperators,Comm.Math.
Phys.,208 (1999), 125–152.
Davide Catania, Dipartimento di Matematica, Universita` di Pisa, Largo B. Pontecorvo 5,
56127 Pisa, Italy; e–mail: [email protected]