Pointwise decay for solutions of the 2D Neumann exterior problem for the wave equation. II

Pointwise Decay for Solutions of the 2 D Neumann exterior problem

for the wave equation - II.

PAOLO SECCHI(*)

ABSTRACT- We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition. We are interested to the asymptotic be- havior for large times for the solution, and in particular to the dependence on the norms of the initial data in the estimate for the pointwise decay rate. In the paper we improve an estimate of this kind, proved by the author in a pre- vious paper, by a cut-off technique which use in particular a new estimate of the local energy decay of the free space solution.

1. Introduction.

LetVbe an exterior domain inR²; the boundary¯Vis a smooth, con- vex and compact hypersurface. Given rD0 , we denote V_r4VOB_r, whereB_r4 ]xR²NNxN Er(. Below,r₀D0 is a fixed constant such that V^c%B_r0 (V^c is the complement of V). We set Q4[ 0 ,Q)3V,S4 4[ 0 ,Q)3¯V.

We study the decay property of solutions to the mixed problem for

(*) Indirizzo dell’A.: Dipartimento di Matematica - via Valotti 9, 25133 Bre- scia, Italy.

the wave equation with Neumann boundary condition (¯²_tt2D)u40

¯_nu40 u( 0 , x)4f(x) ,

¯_tu( 0 , x)4g(x)

in Q, on S,

in V. (1)

In the previous paper [6] we showed the decay rate ( 11 1t)^{21 /2}log²(e1t), slightly slower than the optimal rate ( 11t)^{21 /2} of the free space solution. The aim of the present paper is to improve the de- pendence on the data, in a form suitable for applications. As in [6], our proof is a combination by a cut-off argument of the estimate of the local energy decay following from the analysis of Kleinman and Vainberg [1], Morawetz [2], Vainberg [7] and decay estimates for the free space sol- ution. Differently from [6], we use a new estimate of the local decay of the free space solution. In order to get a decay rate of local energy in the presence of an obstacle, some assumption on its shape should be taken, in order to exclude the existence of closed ray solutions. In fact, for the Dirichlet problem, Ralston [5] has shown that if there is a closed ray sol- ution, there is no rate of decay. For the Dirichlet problem the obstacle should benon-trapping, see [3]; for the Neumann problem, the analysis of Kleinman and Vainberg [1], Morawetz [2], Vainberg [7] gives the de- cay rate for convex bodies. This is the reason why in this paper we take the boundary convex. The result of this paper will be applied to the study of the Euler compressible flow in a forecoming paper.

Let us introduce some notation. For a multi-index a4(a1,a2) we set ¯^a4¯₁^a1¯₂^a2, NaN 4a₁1a₂, where ¯₁4¯/¯x₁,¯₂4¯/¯x₂. Let W^m,p(V) be the usual Sobolev space of orderm,m41 , 2 ,Rand order of integrability pF1 , and let VQV_Wm,p denote its norm. If p42 we set W^{m, 2}(V)4H^m(V) with normVQV_Hm. The norm ofL²(V) is denoted byVQ QV, the norm ofL^p(V), 1GpG Q, byNQN^p. For simplicity we use the ab- breviated notation W^m,p,H^m,L^p. Let us define the weighted Sobolev space

WA^m,p

4WA^m,p(V)»4 ]fL^p: VfV_WAm,pE Q(

where

VfV_WAm,p»4

g

^!

h

ClearlyVfV_Wm,pGVfV_WAm,p. Ifp42 we setWA^{m, 2}

4HA^mwith normVQV_HAm. If m40 we set WA^{0 ,p}

4LA^p with norm VQV_LAp. Similarly we introduce the spaces W^m,p(R²), H^m(R²), L^p(R²), WAm,p(R²), HAm(R²) and LAp(R²) with norms denoted by VQV_Wm,p(R²), VQV_Hm(R²), VQV_Lp(R²), VQV_WAm,p(R²), VQ QV_HAm(R²) and VQV_LAp(R²) respectively. We will also use the same symbol for spaces of vector valued functions.

THEOREM 1.1. Suppose u is a solution of the exterior problem (1.1).

Assume the initial data satisfy fWA^{3 , 1}OHA³,gWA^{2 , 1}OHA². Then there exists a constant CD0 such that, for every tF0 ,

(2) N¯_tu(t,Q)NQ1 N˜u(t,Q)NQGC( 11t)^{21 /2}log²(e1t)3 3

[

HA3

1 /2(VfV_WA3 , 11VfV_HA3)^{1 /2}1VgV

HA2

1 /2(VgV_WA2 , 11VgV_HA2)^{1 /2}

]

2. Proof of Theorem 1.1.

Let us take functions fA,gA:R²KR such that fA

4f, gA 4g on V, and such that fAWA^{3 , 1}(R²)OHA³(R²), gAWA^{2 , 1}(R²)OHA²(R²),

VfAV_HA3(R²)1VgAV_HA2(R²)GC(VfV_HA31VgV_HA2), VfAV_WA3 , 1(R²)1VgAV_WA2 , 1(R²)GC(VfV_WA3 , 11VgV_WA2 , 1) . (3)

For this, observe that it’s enough to take extensions over the bounded set V^c with the regularity fAH³(R²), gAH²(R²), since the required behavior at infinity is already furnished by f,g.

Let u₁ be the solution of the Cauchy problem (¯²_tt2D)u140

u₁( 0 , x)4fA(x),

¯_tu₁( 0 , x)4gA(x)

in [ 0 ,Q)3R²,

in R². (4)

From [4], Theorem 2.1, we have

N¯_tu₁(t)N^LQ(R2)1 N˜u₁(t)N^LQ(R2)GC( 11t)^{21 /2}V(˜fA,gA)V_W2 , 1(R²). (5)

ChoosingrDr₀andx(x)C₀^Q(R²) so thatx(x)41 ifNxNGrand40 if NxN Fr11 , we put

u₂4u2( 12x)u₁, G4 2u₁Dx22˜u₁Q˜x

The function u₂ is the solution of the initial boundary value problem (¯²_tt2D)u₂4G

¯_nu₂40

u₂( 0 , x)4xf(x),

¯_tu₂( 0 , x)4xg(x)

in Q, on S,

in V. (6)

Observe that suppG(t,Q)’]xNrG NxN Gr11( for all tF0 , and suppxf’Vr11, suppxg’Vr11. From (17) in the Appendix with w4u2, w₀4xf, w₁4xg, we obtain

(7) N¯_tu₂(t)NL^Q(Vr12)1N˜u₂(t)NL^Q(Vr12)GC_r(11t)²¹(V˜(xf)V_H21VxgV_H2)

1Cr

0 t

( 11t2s)²¹VG(s)V_H2ds.

We estimate VG(s)V_H2. First of all we observe that VG(s)V_H2GC_rVu₁(s)V_H3(B_r11).

We apply to u4u1 the local decay estimate given in Corollary 3.1 and use (3). This yields

VG(s)V_H2GC_rM₁( 11s)²¹ (8)

where

M₁4VfV

HA3

1 /2(VfV_WA3 , 11VfV_HA3)^{1 /2}1VgV

HA2

1 /2(VgV_WA2 , 11VgV_HA2)^{1 /2}. We obtain from (7), (8) and (25) in the Appendix that

(9) N¯_tu₂(t)N^LQ(Vr12)1N˜u₂(t)N^LQ(Vr12)GC_r(11t)²¹(V˜(xf)V_H21VxgV_H2)

1C_rM1

0 t

( 11t2s)²¹( 11s)²¹ds

GC_rM₁( 11t)²¹log (e1t) (tF0 ,

sinceVfV_HmGCVfV_HAm. Choosingc(x)C₀^Q(R²) so thatc(x)41 ifNxN F Fr12 and 40 if NxN Gr11 , we observe that

cxf40 , cxg40 , cG40 . Let us define

H4 2u₂Dc22˜u₂Q˜c. The function cu2 solves the Cauchy problem

(¯²_tt2D)(cu₂)4H cu2( 0 , x)40 ,

¯_t(cu₂)( 0 , x)40

in [ 0 ,Q)3R²,

in R². (10)

From (5) and the Duhamel’s principle we get

N¯_t(cu₂)(t)NL^Q(R²)1 N˜(cu₂)(t)NL^Q(R²)

GC

0 t

( 11t2s)^{21 /2}VH(s,Q)V_W2 , 1(R²)ds

GC_r

0 t

( 11t2s)^{21 /2}Vu₂(s)V_H3(Vr12)ds.

(11)

On the other hand, applying (18) in the Appendix to the solutionu2of (6) yields

Vu₂(t)V_H3(V_r12)GC_r( 11t)²¹(VxfV_H31VxgV_H2) 1C_r

0 t

( 11t2s)²¹VG(s)V_H2ds

GC_rM₁( 11t)²¹1C_rM₁

0 t

( 11t2s)²¹( 11s)²¹ds GC_rM1( 11t)²¹log (e1t) .

(12)

Then from (11), (12) and (27) in the Appendix one has

N¯_t(cu₂)(t)N^LQ(R2)1 N˜(cu₂)(t)N^LQ(R2) GC_rM₁

0 t

( 11t2s)^{21 /2}( 11s)²¹log (e1s)ds

GC_rM₁( 11t)^{21 /2}log²( 11t) . (13)

Moreover, from (12) and a Sobolev imbedding we have

Nu₂(t)N^LQ(Vr12)GC_rM₁( 11t)²¹log (e1t) . (14)

Since u4( 12x)u₁1u₂, we have

N¯_tu(t)NQ1 N˜u(t)NQ

G N( 12x)¯_tu₁(t)NQ1 N˜( ( 12x)u₁(t) )NQ1 N¯_tu₂(t)NQ1 N˜u₂(t)NQ

G N¯_tu₁(t)N^LQ(R2)1 N˜u₁(t)N^LQ(R2)1CNu₁(t)N^LQ(Br110B_r)

1N¯_t(cu₂(t) )N^LQ(R2)1 N˜(cu₂(t) )N^LQ(R2)

1N¯_tu₂(t)N^LQ(Vr12)1 N˜u₂(t)N^LQ(Vr12)1CNu₂(t)N^LQ(Vr12).

From (3), (5), Corollary 3.1 and a Sobolev imbedding, (9), (13), (14) and Lemma 3.3 we finally obtain

N¯_tu(t)NQ1 N˜u(t)NQ

GC( 11t)^{21 /2}V(˜fA,gA)V_W2 , 1(R²)1C_rM₁( 11t)²¹ 1C_rM₁( 11t)^{21 /2}log²( 11t)

1C_rM₁( 11t)²¹log (e1t)

GC_rM1( 11t)^{21 /2}log²(e1t) (tF0 . (15)

3. Appendix.

Let us consider the initial boundary value problem (¯²_tt2D)w4G

¯_nw40 w( 0 ,x)4w₀,

¯_tw( 0 ,x)4w₁

in Q, on S,

in V. (16)

From [6] we have

LEMMA 3.1. Let(w₀,w₁)and G(t,Q)have compact support for each tD0. Assume (˜w₀,w₁)H²,G(t,Q)H² for each tD0. Then the sol- ution w of (16) satisfies the estimate

N¯_tw(t)N^LQ(VR)1 N˜w(t)N^LQ(VR)GC_R( 11t)²¹(V˜w₀V_H21Vw₁V_H2) 1C_R

0 t

( 11t2s)²¹VG(s,Q)V_H2ds. (17)

If (w0,w1)H³3H², G(t,Q)H² for each tD0 , then Vw(t)V_H3(V_R)GCR( 11t)²¹(Vw0V_H31Vw1V_H2)

1C_R

0 t

( 11t2s)²¹VG(s,Q)V_H2ds. (18)

The above inequalities hold for every RDr₀and tF0 ;C_Rdepends on R, the support of the data and the geometry of ¯V.

Let us consider the initial value problem (¯²_tt2D)u40

u( 0 , x)4f(x),

¯_tu( 0 , x)4g(x)

in [ 0 ,Q)3R²,

in R². (19)

In the following lemma all norms, if not explicitly indicated other- wise, are evaluated over R².

LEMMA 3.2. The solution u of the Cauchy problem(19)satisfies the estimate

Vu(t)V_H1(B_R)GC_R( 11t)²¹

[

HA1

1 /2(VfV_WA1 , 11VfV_HA1)^{1 /2}1VgV

LA2

1 /2(VgV_LA11VgV_LA2)^{1 /2}

]

PROOF. For the sake of brevity here we writeVQVinstead ofVQV_L2(R²). First we assumeg40. Let us denote byu×(t,j) the Fourier transfom of u w.r.t. the x-variables:

u

×(t,j)4

R²

u(t,x)e^2ijQxdx.

Moreover,f×denotes the Fourier transform off. From the Cauchy prob- lem it follows that u×(t,j)4cos (tNjN) f×(j). Then

u(t,x)4 1 ( 2p)²

R²

cos (tNjN) f×(j)e^ijQxdj.

Let 0GtG1. By the Plancherel theorem

Vu(t)V4( 2p)²¹Vcos (tNjN) f×VG( 2p)²¹Vf×V4VfVGVfV_LA2. (20)

Let tF1. By integrating by parts we get u(t,x)4 2 1

( 2p)²t

R²

sin (tNjN)

{

NjN

f×(j)1f×(j)

NjN 1ijQx NjN

f×(j)

}

We decompose the integral as the sum of integrals overB_a4 ]NjN Ea( and]NjN Da(respectively, whereawill be chosen below. Accordinglyu is written in the formu4u₁1u₂. Then, passing to polar coordinates, we show that

(21) Nu₁(t,x)NG 1 (2p)²t

S¹

0

a

m

^V

V

L^Q(B_a)1(11rR)Vf×V_LQ(B_a)

n

G CR

t a( 11a)V( 11 NxN) fV_L1

for every NxNER, since

V

¯NjN f×

V

L^QGVxfV_L1, Vf×V_LQGVfV_L1.

Setaxb4( 11 NxN²)^{1 /2} and denote byxa(r) the characteristic function of ]rFa(. Again by the Plancherel theorem and by V ^¯

¯NjN

f×VGCVxfV, we obtain

Vu2(t)V_L2(B_R)GCRVaxb²¹u2(t,x)V GC_R

t

V

R²

x_a(NjN) sin (tNjN)

{

}

^V

1CR

t

V

axbQ

R²

x_a(NjN) sin (tNjN) j

NjN f×(j)e^ijQxdj

V

GC_R

t

V

{

NjN

f×(j)1f×(j) NjN

} ^V

1C_R

t

V

f×(j)

V

G C_R

t

g ^V

f×(j)

V

g

h

h

a (VxfV1VfV) . (22)

We choosea4VfV

LA1 21 /2

VfV

LA2

1 /2. Adding (20)-(22) after the substitution ofa gives

Vu(t)V_L2(BR)GC_R( 11t)²¹VfV

LA1 /22

(VfV_LA11VfV_LA2)^{1 /2}. (23)

Then, we take the spatial derivatives of (19) and apply the estimate just shown, obtaining the estimate of˜u in terms of ˜f. Finally, we assume f40. Since the proof is similar to the first case we omit the de- tails. r

COROLLARY 3.1. The solution u of the Cauchy problem (19) satis- fies the estimate

Vu(t)V_H3(B_R)GC_R(11t)²¹

[

HA3

1/2(VfV_WA3,11VfV_HA3)^1/21VgV

HA2

1/2(VgV_WA2,11VgV_HA2)^1/2

]

LEMMA 3.3. If fLA¹OLA², then VfV_L1GCVfV

LA1 1 /2VfV

LA2 1 /2. If fWA^{m, 1}OHA^m, then

VfV_Wm, 1GCVfV

WAm, 1 1 /2 VfV

HAm 1 /2.

PROOF. Applying the Hölder inequality gives VfV_L1(R²)4

R²

[(11NxN)NfN]^1/2[(11NxN)NfN]^1/2(11NxN)²¹dx

G

u

R²

(11NxN)NfNdx

v

u

R²

(11NxN)²NfN²dx

v

u

R²

(11NxN)²⁴dx

v

GCVfV

LA1 1/2VfV

LA2 1/2. (24)

This inequality immediately yields the second thesis. r

We finally report some elementary estimates used above. For the proof see [6].

LEMMA 3.4. There exists a constant CD0 such that for all tF0

0 t

( 11t2s)²¹( 11s)²¹dsGC( 11t)²¹log ( 11t), (25)

0 t

( 11t2s)²¹( 11s)^{21 /2}dsGC( 11t)^{21 /2}log ( 11t).

(26)

0 t

( 11t2s)^{21 /2}( 11s)²¹log (e1s)dsGC( 11t)^{21 /2}log²( 11t).

(27)

R E F E R E N C E S

[1] R. KLEINMAN - B. VAINBERG, Full-low frequency asymptotic expansion for second-order elliptic equations in two dimensions, Math. Methods Appl. Sci., 17 (1994), pp. 989-1004.

[2] C. S. MORAWETZ, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), pp. 229-264.

[3] C. S. MORAWETZ, Notes on time decay and scattering for some hyperbolic problems, Reg. Conf. Series Appl. Math., SIAM 1975.

[4] R. RACKE,Lectures on Nonlinear Evolution Equations: Initial Value Prob- lems, Vieweg Verlag, 1992.

[5] J. V. RALSTON,Solutions of the wave equation with localized energy, Comm.

Pure Appl. Math.,22 (1969), pp. 807-824.

[6] P. SECCHI,Pointwise decay for solutions of the 2D Neumann exterior prob- lem for the wave equation, preprint.

[7] B. VAINBERG,On the short wave asymptotic behaviour of solutions of station- ary problems and the asymptotic behaviour as tK Qof solutions of non-sta- tionary problems, Russian Math. Surveys, 30 (1975), pp. 1-58.

Manoscritto pervenuto in redazione il 15 aprile 2002.