Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

Подождите немного. Документ загружается.

358 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

1. Introduction

Nonlinear acoustics plays a role in several physical contexts. Our work is especially

motivated by high-intensity focused ultrasound (HIFU) being used in technical

and medical applications ranging from lithotripsy or thermotherapy to ultrasound

cleaning or welding and sonochemistry, see [1], [12], [18], [19], and the references

therein.

The Westervelt equation is given by

−

∼

+Δp

∼

Δ(p

∼

)=−



∼

(1.1)

with β

=1+B/(2A), where p

∼

denotes the acoustic pressure ﬂuctuations, c is

the speed of sound, b the diﬀusivity of the sound, 

the mass density, and B/A

the parameter of nonlinearity. For a detailed derivation of the PDE we refer to

[14], [18], [21], [33].

Throughout this paper we will assume that the domain Ω ⊂ R

, d ∈{1, 2, 3},

on which we consider the PDEs is open and bounded with C

smooth boundary Γ.

The Westervelt equation can be equivalently rewritten as:

(1 − 2ku)u

−c

Δu − bΔ(u

)=2k(u

)

, (1.2)

where k = β

/(c

) This is a quasilinear strongly damped wave equation with

potential degeneracy.

Quasilinear PDE’s have attracted considerable attention in the literature with

a large arsenal of mathematical tools developed for their treatment. Particularly

well studied, with optimal results available, are parabolic equations – see [3, 4, 29]

and references therein. In the case of hyperbolic like models, the intrinsic low

regularity and oscillatory dynamics puts additional demands on regularity of the

data as well as necessitates introduction of some sort of dissipation that may

include interior, boundary or partial interior damping – see [32, 34, 23, 5] and

references therein.

The distinctive feature of our work is that the model considered corresponds

to quasilinear internally damped wave equation with potential degeneracy and

nonhomogeneous boundary forcing. This calls for a careful setup of the state space

where the latter should provide certain topological invariance for the dynamics.

The above is achieved through a long chain of estimates that rely critically on

recent developments in regularity theory of structurally damped wave equations.

The lack of compactness of the resolvent operator is one of the sources of diﬃculties

to be contended with.

In fact, global well-posedness and decay rates for equation (1.2) which is

homogeneous on the boundary were obtained in [16].

Our main interest in this present work is global (in time) well-posedness the-

ory of Westervelt equation equipped with nonhomogeneous boundary data.Itis

known, that the presence of nonhomogeneous boundary conditions leads to rather

Well-posedness for the Westervelt Equation 359

subtle analysis, due to the fact that such inputs are modeled by “unbounded op-

erators” which are not deﬁned on the basic state space. (see [28]). This is clearly

seen when inspecting “boundary variation of parameters formula” which has been

used recently in the context of studying boundary control problems for parabolic

(analytic) semigroups (see [6, 22, 28], [2], [8]). Our methods include derivation of

such formula for damped wave equation and their use in the context of study-

ing regularity and long time stability of solutions driven by nonhomogeneity on

the boundary. It is our belief that the obtained linear results should also be of

independent interest in the context of boundary value problems associated with

“overdamping”. Analyticity and exponential decay rates valid for the semigroup

corresponding to strongly damped wave equation play a crucial role in the analysis.

1.1. Main results

The main goal of this paper is to provide results on (1) local existence, (2)global

existence and (3) exponential decay rates for the energy of solutions for the West-

ervelt equation with Dirichlet

u = g on ∂Ω (1.3)

boundary conditions and given initial data (u(0),u

(0)) = (u

In order to formulate our results we introduce the following energy functions:

u,0

(t)=



(t)|

+ |∇u(t)|



,t≥ 0

u,1

(t)=



(t)|

+ |∇u

(t)|

+ |Δu(t)|



,t>0

where |u|≡|u|

(Ω)

.Fort =0,

u,1

(0) ≡



|(1 − 2ku

)

−1

Δu

+ bΔu

+2ku

+ |∇u

(t)|

+ |Δu



In [16] Westervelt equation with homogeneous Dirichlet boundary conditions

was considered. The main goal of this paper is the treatment of nonhomogeneous

boundary conditions, with particular emphasis paid to regularity required to be

satisﬁed by the boundary data which obey the following compatibility conditions:

g(t =0) = u

∂Ω

(t =0) = u

∂Ω

. (1.4)

We deﬁne:

X ≡ C(0,T; H

3/2

(∂Ω)) ∩ C

(0,T; H

1/2

(∂Ω)) ∩ C

(0,T; H

−1/2

(∂Ω))

∩ H

(0,T; H

1/2

(∂Ω)) ∩ H

(0,T; H

−3/2

(∂Ω)). (1.5)

Our ﬁrst result pertains to local existence and uniqueness of solution:

Theorem 1.1. Let T>0 be arbitrary. There exist ρ

, ˜ρ

> 0 such that if

u,0

(0) + E

u,1

(0) ≤ ρ

, and g ∈ X, g

≤ ˜ρ

360 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

with the compatibility conditions (1.4), then there exists a unique solution (u, u

)

solving the Westervelt equation (1.2) (in a weak H

−1

(Ω) sense) and such that

u ∈ C(0,T; H

(Ω)) ∩ C

(0,T; H

(Ω)) ∩ C

(0,T; L

(Ω)) ∩ H

(0,T; H

(Ω)).

Our next theorem deals with global well-posedness.

Theorem 1.2. Let (u

,g) be such that E

u,0

(0) + E

u,1

(0) + |g|

< ∞. For any

M>0 there exist ρ>0, ˜ρ>0, such that solutions corresponding to initial and

boundary data with

u,1

(0) ≤ ρ (1.6)

sup

t≥0

l=0

g(t)

3/2−l

(∂Ω))

l=0



3−l

−3/2+2l

(∂Ω))

≤ ˜ρ

exist for all t>0 and satisfy E

u,1

(t)+E

u,0

(t) ≤ M for all t>0.

Finally, energy decays for the total energy are presented below:

Theorem 1.3. With Cauchy data (u

,g) given in Theorem 1.2 and such that

for all t>0 there exists ω

> 0

l=0

g(t)

3/2−l

(∂Ω)

g(t)

1/2

(∂Ω)

≤ C

−ω

, (1.7)

there exists a constant ω>0 such that

u,1

(t)+E

u,0

(t) ≤ C

−ωt

Remark 1.4. In one space dimension, one can also consider the case without damp-

ing, b = 0, which is relevant in certain applications, see, e.g., [15].

In this situation we cannot expect global existence but we still get the a local

result for small initial and boundary data, see Theorem 2 in [17].

Like in [16], a ﬁrst step of the proof is to prove energy estimates for a lin-

ear, nonautonomous and nonhomogeneous abstract wave equation related to the

nonlinear equation (1.2). The obtained results depend on the analyticity and maxi-

mal regularity of abstract damped wave equation with nonhomogeneous boundary

data. We shall take advantage of abstract version of variation of parameter formula

modeling nonhomogeneous boundary data associated with the strong damping. In

the case of purely parabolic problems, such variation of parameters formula was

critically used in [22, 2]. One of our technical goals is to extend this semigroup ap-

proach to models involving strong damping, which result should be of independent

interest.

Well-posedness for the Westervelt Equation 361

2. Strongly damped abstract wave equation

In what follows we consider a positive selfadjoint operator A : D(A) ⊂H→H,

where H is a suitable Hilbert space. We shall introduce the following notation

|u|≡|u|

, (u, v) ≡ (u, v)

; L

(Z) ≡ L

(0,T; Z) ,C(Z) ≡ C(0,T; Z) .

We shall impose

Assumption 2.1. The following continuous embeddings hold

D(A

1/2

) ⊂ L

(Ω) , with |w|≤C

1/2

w|,

D(A

1/2

) ⊂ L

(Ω) , with |w|

(Ω)

≤ C

1/2

w|,

D(A) ⊂ C(Ω) , with |w|

∞

(Ω)

≤ C

|Aw|.

In our considerations we take H = L

(Ω) where Ω is a bounded smooth

domain in R

, n =1, 2, 3. In view of the above, the Assumption 2.1 is automatically

satisﬁed with A = −Δ deﬁned on the domain D(A)=H

(Ω) ∩ H

(Ω). We will

be considering ﬁrst the following non-homogeneous and nonautonomous abstract

strongly damped wave equation:

α(t)u

+ c

Au + bAu

= f(t) (2.1)

with initial conditions u(0) = u

∈D(A

1/2

),u

(0) = u

∈H. Here α(t)isa

positive multiplier on H = L

(Ω), i.e.,

Assumption 2.2. There exist positive constants 0 <α, α<∞ such that ∀t ∈ [0,T]

(α(t)u, u) ≥ α(t)|u|

≥ α

|u|

(2.2)

(α(t)u, v)

(Ω)

≤ α(t)|u||v|≤α

|u||v|. (2.3)

The above conditions amount to say that α(t) ∈ M(H)and|α(t)|

M(H)

≤

α(t), where the multipliers space M(H) [30] is equipped with the norm

|α|

M(H)

=sup

|u|

|αu|

We are interested in studying regularity properties of solutions u, u

due to

the forcing f and initial conditions u

, u

The analysis will be conducted for t ∈ [0,T]whereT is either ﬁnite or inﬁnite

time horizon.

Remark 2.1. In reference to the original equation (1.2), the operator A = −Δ

where Δ is equipped with zero Dirichlet data. Moreover, α(t)=1− 2ku(t)and

f(t)=2ku

To take into account inhomogeneous boundary conditions, we will decompose

u = u

+¯g with u

having to satisfy homogeneous Cauchy (boundary and initial)

conditions and ¯g will denote an appropriate extension of the Cauchy data, i.e.,

¯g = g on ∂Ω , (2.4)

¯g(t =0)=u

, ¯g

(t =0) = u

. (2.5)

362 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

2.1. Regularity properties of damped wave equation

As in [16] we shall begin with a study of the semigroup generated by nondegenerate

operator

A =



0 I

−

A−



with the domain D(A)={(w, v) ∈D(A

1/2

) ×D(A

1/2

),c

w + bv ∈D(A)} where

α(t)=α>0.

In what follows we shall recall regularity properties of the generator e

∈

L(H)where

H ≡D(A

1/2

) ×H.

It was shown in [9] that e

is an analytic, strongly continuous semigroup de-

ﬁned on H. Moreover, the following characterization of the domain A is known [9]

D(A

)=D(A

1/2

) ×D(A

),θ≤ 1/2

and D(A

)]



= D(A

1/2

) ×[D(A

)]



,θ≤ 1/2 (2.6)

and the following analytic estimate is available:

L(H)

≤ Ce

−ωt

−θ

,t>0 (2.7)

where the constant ω is positive and depends on c, b, α [9].

In addition, the following regularity holds:



∞

1/2

dt ≤ C|x|

LF (·) ≡



A(·−s)

F (s)ds ∈L(L

(H))



∞

|LF (t)|

dt ≤ C



∞

|F (t)|



A(t−s)



f(s)



≤ C



−2ω(t−s)

−1/2

f(s)|

ds, t > 0. (2.8)

Remark 2.2. Theregularitylistedin(2.8)iswellknownandhasbeenoftenused[9]

on ﬁnite time horizon. It is important for the results of this paper the fact that these

estimates are uniform in time. Not surprisingly, this results from the fact that the

semigroup e

is exponentially decaying. Indeed, the ﬁrst estimate, on the strength

of (2.6) amounts to showing that for x ∈ H with Z(t) ≡ e

x =(z(t),z

(t)) one has

1/2

z|∈L

(0, ∞), |A

1/2

|∈L

(0, ∞).

But the ﬁrst relation follows just from exponential stability of the semigroup,

while the second relation follows from standard energy estimates available for the

damped wave equation.

Similarly, the second regularity statement (on (0, ∞)) follows from a standard

Fourier’s transform argument after accounting for the fact that the semigroup is

analytic and the spectrum of the generator A is in the left complex plane.

Well-posedness for the Westervelt Equation 363

The last estimate in (2.8) follows from standard by now Lyapunov function

argument [16].

Throughout the rest of this section we will consider time and space dependent

coeﬃcient α, and impose Assumptions 2.1, 2.2.

Theorem 2.3 (Theorem 2.1 of [16]). Consider (2.1) with

1. α ∈ L

∞

(M(H)) ∩ C

(H).

2. f ∈ L

(H) ∩ H

([D(A

1/2

)]



)

3. u

∈D(A), u

∈D(A

1/2

),and

α(0)

(f(0) − c

−bAu

) ∈H.

Then the following energy estimates hold

(t)+(b − ˆ −)



1/2

(s)|

≤ E

(0) +

4ˆ



−1/2

ds + C



|α

C(H)



(s)|

ds . (2.9)

(t)+(b − ˆ − )



1/2

(s)|

≤ E

(0) +

4ˆ



−1/2

ds + C



|α

C(H)



(s)|

ds , (2.10)

|Au(t)|

≤

|Au



|f|



, (2.11)

with C



and

(t) ≡



(α(t)u

(t),u

(t)) + c

1/2

u(t)|



These, by Gronwall’s inequality and |w

(s)|

≤

(s),forw = u and w = u

imply

u ∈ C(D(A)),u

∈ C(D(A

1/2

)),u

∈ C(H) ∩ L

(D(A

1/2

)). (2.12)

Note that on the strength of Assumption 2.2 we have

(t) ∼ E

u,0

(t) ≡



(t)|

+ |A

1/2

u(t)|



Moreover, we will repeatedly make use of the simple estimate

|w|

(Ω)

≤|w|

3/2

(Ω)

|w|

1/2

≤ C

3/2

1/2

3/2

|w|

1/2

≤ C

3/2

1/2

, (2.13)

following from Assumption 2.1, using H¨older’s inequality and Young’s inequality.

364 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

2.2. Decay rates for the homogeneous equation

We shall address next the issue of decay of the energy for the abstract wave

equation (2.1). We will consider decay rates in both lower level energy E

and

higher level energy

E(t) ≡ E

(t)+|Au(t)|

By Assumption 2.2 we have E

(t) ∼ E

u,0

(t) E(t) ∼ E

u,1

(t) .

For comparison, we quote the following result Theorem 2.3 from [16] for the

autonomous case.

Theorem 2.4. Let f =0,andα

≡ 0. Then there exist ω, ω

> 0 such that

• E

(t) ≤ e

1−ωt

(0) with ω = ω(α, b, c

)

•E(t) ≤ Ce

−ω

E(0) with ω

=min{

,ω} if

= ω and ω

<ωif

= ω.

2.2.1. Energy estimates for the variable coeﬃcient model. Back in the model with

time and space dependent α, we are dealing with

α(t, ·)u

+ c

Au + bAu

= f

where α

≤ α(t, x) ≤ α

, (t, x) ∈ R × Ωandα

∈ C(R; H). Our goal is to

estimate the lower and the higher level energy in terms of f and α.

Lower level energy

Here we can make use of the following result Proposition 2 in [16].

Proposition 2.5.

(T )+γ



(s) ds +



1/2

ds ≤ (C

+ C

(0)





−1/2

+(C

|α

C(H)

+ C

|α

C(H)

) |u



ds (2.14)

for some constants γ

,...,C

> 0 depending only on the coeﬃcients c, b and

the constants α

, C

Higher level energy

Here we quote Proposition 3 in [16].

Proposition 2.6.

E(T )+



|Au(t)|

+ |A

1/2

+ |A

1/2

dt (2.15)

≤ C

E(0) +



−1/2

+ C

|f|

+ C

|α

C(H)

(t)|

for some constants γ

,...,C

> 0 depending only on the coeﬃcients c, b and

the constants α

, C

Remark 2.7. The estimate in Proposition 2.6 will allow us to use the so-called

“barrier method” ([5, 32, 23, 34] for establishing global existence.

Well-posedness for the Westervelt Equation 365

2.3. Extension of nonhomogeneous boundary data to the interior

The purpose of this section is to provide appropriate extensions into the interior

of boundary-initial (Cauchy) data. This procedure allows to homogenize nonlinear

equation with a given boundary data. More speciﬁcally we will be considering

classical harmonic (Dirichlet) extensions that would lead to appropriately deﬁned

“parabolic” extensions. We begin by deﬁning the harmonic extension operator –

the so-called Dirichlet map deﬁned by

v = D

g, iﬀ



Δv =0, in Ω

v = g on ∂Ω

(2.16)

which for all s ∈ R is a bounded mapping

: H

(∂Ω) → H

s+1/2

(Ω) ∀s ∈ R. (2.17)

Remark 2.8. It is possible to use diﬀerent extensions of boundary data, for instance

via solutions of wave equation with nonhomogeneous boundary data. In fact, this

method, along with “hidden” regularity of wave equations [27], was successfully

used in [17] for the study of local existence of solutions. However, since we are

interested in global behavior, static harmonic extensions are more appropriate.

2.3.1. Weak solutions with nonhomogeneous boundary data. In this section we in-

troduce a “variation of parameter formula” representing weak solution to the non-

homogeneous boundary value problem driven by non-degenerate (α =1)damped

wave equation.

This is done with the help of harmonic extension D

. The above construct

leads naturally to a deﬁnition of “parabolic extension” – which is solution to

damped wave equation with non-homogeneous boundary data homogenized via

harmonic extension. The corresponding results, while used for the study of non-

linear problem, should be also of independent interest in linear theory pertinent

to boundary control, or more generally, theory of damped wave equation with

nonhomogeneous boundary data. They provide a counterpart of boundary varia-

tion of parameter formula used in the past in the context of heat equation with

nonhomogeneous boundary data [22, 2, 8, 28].

We consider

−c

Δw − bΔw

= f (t)in(0,T) × Ω (2.18)

w = g on ∂Ω

w(0) = w

(0) = w

in Ω.

Recalling the deﬁnition of D

we rewrite (2.18) as follows:

− c

Δ(I − D

γ)w − bΔ(I −D

γ)w

= f (t)in(0,T) × Ω (2.19)

w = g on ∂Ω (2.20)

where γ denotes the trace operator, i.e., γu = u|

366 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

Noting that for smooth and compatible Cauchy data g, w

, w −D

γw ∈

D(A), so that −Δ(w − D

g)=A(w − D

g), we rewrite (2.19) as the abstract

second-order ODE equation:

+ c

Aw + bAw

= c

g + bAD

+ f(t) (2.21)

where in splitting the brackets, we admit representation of the equation in the

dual space to D(A). This procedure is standard by now [28, 8] and references

therein. What is less standard, however, is rewriting of this second-order equation

as a ﬁrst-order system. While such procedure can be typically accomplished on

[D(A

∗

)]



, (see the references cited above) this is not the case for the damped wave

equation considered on H with Dirichlet boundary data. The reason for diﬃculty

is the structure of the domain D(A) along with the lack of global smoothing (hence

lack of compactness). We also recall that this diﬃculty can be circumvented when

considering the dynamics on L

(Ω) × L

(Ω), where A still generates an analytic

semigroup, though not a contraction. In fact, [7] provides a variation of constant

formula for boundary control acting on damped wave equation with the state space

(Ω) ×L

(Ω). However, the construction in [7], based on the analysis in [6], does

not apply to our choice of the state space given by H. In fact this remains true even

if we were to penalize the state space by negatives powers of A. The problem is

purely algebraic (and the same phenomenon was noticed later in [24, 25, 26] where

structurally damped problems with Dirichlet boundary actions were studied in the

context of optimization and Riccati theory.

To cope with this issue we need to modify the argument given in [7] and we

proceed as follows: Denote Y ≡



w − D



With the above notation and after noting that



−D

bAD



= A



−D



∈ [D(A

∗

)]



we can rewrite (2.21), in the variable Y =(w −D

g, w

), as a ﬁrst-order abstract

ODE well-deﬁned on [D(A

∗

)]



(t)=AY +



−D

bAD

+ f



or equivalently

(t)=AY (t)+A



−D

(t)





f(t)



the equation holding on [D(A

∗

)]



. Consequently, variation of parameters (applied

in dual spaces to the domain of A

∗

) yields

Y (t)=e

Y (0) +



A(t−s)



0 −D

(s)



ds +



A(t−s)



f(s)



ds .

(2.22)

Well-posedness for the Westervelt Equation 367

The above can be rewritten, with the explicit use of the structure of A and denoting

W (t) ≡ (w(t),w

(t)) as

W (t)=



(t)



+ e

Y (0) −



A(t−s)



(s)





A(t−s)



f(s)



ds.

The above leads to the following variation of parameter formula, which describes

weak (“ﬁnite energy”) solutions to the boundary model:

Lemma 2.9. Let g ∈ C(H

1/2

(Γ)), g

∈ C(H

−1/2+

(Γ)), γw

= g(0), w

∈ H

(Ω),

∈ L

(Ω), f ∈ L

−1

(Ω)).

The following representation holds pointwisely (in t) with values in H

(Ω) ×

(Ω):

W (t)=



g(t)



+ e



− D

g(0)



(2.23)

−



A(t−s)



(s)



ds +



A(t−s)



f(s)



ds.

The solution W ∈ C(H

(Ω) × L

(Ω)).

We denote by C(Z) the space of continuous functions on (0, ∞)withvalues

in Z. Similarly, L

(Z) mean norms over R

in time.

Remark 2.10. Variation of parameter formula given in (2.23) is a counterpart of

a well-known by know “parabolic boundary control formula” which has been used

extensively in both linear and non-linear boundary control theory [6, 28, 2, 8, 22]

and more speciﬁcally in the context of damped wave equation [7, 24]. This formula

provides a direct representation of the solution as depending on the boundary

input g. Speciﬁc feature of this formula is the presence of unbounded operator A

in the integrand involving boundary input g. Analyticity of the semigroup with

controlled singularity at zero allows to show that the H norm of this integrand is

locally integrable, as long as the range of the operator D

belongs to some positive

Sobolev space. But this is the case with harmonic extension where R(D

) ⊂

1/2

(Ω). The proof, written below, provides the details for this argument.

Proof. The formula (2.23) follows directly from (2.22). The original meaning of

all quantities is on the dual space [D(A

∗

)]



, see [28]. We only need to justify the

validity of the formula taking values in the state space H

(Ω) × L

(Ω). For this,

we refer to regularity of the damped wave equation that is inherited from the

analyticity of the semigroup.

The most critical term is the third term. Here we notice that by the (2.17)

∈ C(H



(Ω)) = C(D(A



), and by (2.6)





∈ C(D(A



)). Hence,

A(t−s)



(s)



≤ C

|t −s|

1−



−ω(t−s)

(s)|

−1/2+

(∂Ω)