Regularity and stability of the semigroup associated with some interacting elastic systems I: A degenerate damping case
aa r X i v : . [ m a t h . A P ] M a r REGULARITY AND STABILITY OF THE SEMIGROUP ASSOCIATED WITH SOMEINTERACTING ELASTIC SYSTEMS I: A DEGENERATE DAMPING CASE
KA¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOUA
BSTRACT . In this paper, we examine regularity and stability issues for two damped abstractelastic systems. The damping involves the average velocity and a fractional power θ , with θ in [ − , , of the principal operator. The matrix operator defining the damping mechanism for thecoupled system is degenerate. First, we prove that for θ in (1 / , , the underlying semigroupis not analytic, but is differentiable for θ in (0 , ; this is in sharp contrast with known resultsfor a single similarly damped elastic system, where the semigroup is analytic for θ in [1 / , ;this shows that the degeneracy dominates the dynamics of the interacting systems, preventinganalyticity in that range. Next, we show that for θ in (0 , / , the semigroup is of certainGevrey classes. Finally, we show that the semigroup decays exponentially for θ in [0 , , andpolynomially for θ in [ − , . To prove our results, we use the frequency domain method,which relies on resolvent estimates. Optimality of our resolvent estimates is also established.Several examples of application are provided. C ONTENTS
1. Problem formulation and statement of main results 12. Some technical Lemmas 43. Proof of Theorem 1.1 54. Proof of Theorem 1.2 115. Proof of Theorem 1.3 196. Examples of application 22References 231. P
ROBLEM FORMULATION AND STATEMENT OF MAIN RESULTS
Let H be a Hilbert space with inner product ( ., . ) and norm | . | . Let A be a positive unboundedself-adjoint operator, with domain D ( A ) dense in the Hilbert space H .Set V = D ( A ) . We assume that V ֒ → H ֒ → V ′ , each injection being dense and compact, where V ′ denotes the topological dual of V .Let a, b , and γ be positive constants. Let θ ∈ [ − , , and consider the evolution system(1.1) y tt + aAy + γA θ ( y t + z t ) = 0 in (0 , ∞ ) z tt + bAz + γA θ ( y t + z t ) = 0 in (0 , ∞ ) y (0) = y ∈ V, y t (0) = y ∈ H, z (0) = z ∈ V, z t (0) = z ∈ H. Mathematics Subject Classification.
Key words and phrases.
Regularity, stability, semigroup, interacting elastic systems.
Introduce the Hilbert space H = V × H × V × H , over the field C of complex numbers, equippedwith the norm || Z || = a | A u | + | v | + b | A w | + | z | , ∀ Z = ( u, v, w, z ) ∈ H . Throughout this note, we shall assume:(1.2) ∃ a > | u | ≤ a | A u | , ∀ u ∈ V. Introduce the operator(1.3) A θ = I − aA − γA θ − γA θ I − γA θ − bA − γA θ with domain D ( A θ ) = n ( u, v, w, z ) ∈ V ; aAu + γA θ ( v + z ) ∈ H, and bAw + γA θ ( v + z ) ∈ H o . One easily checks that for every Z = ( u, v, w, z ) ∈ D ( A ) ,(1.4) ℜ ( A θ Z, Z ) = − γ | A θ ( v + z ) | ≤ . so that the operator A θ is dissipative. Further, the operator A θ is densely defined, so A θ isclosable on H . Therefore, the Lumer-Phillips Theorem shows that the operator A generates astrongly continuous semigroup of contractions ( S θ ( t )) t ≥ on the Hilbert space H . One also checksthat i R ⊂ ρ ( A θ ) where ρ ( A θ ) denotes the resolvent set of A θ . This shows that the semigroup ( S θ ( t )) t ≥ is stronglystable on the Hilbert space H , thanks to the strong stability criterion of e.g. Arendt and Batty[1].This work was inspired by those of Chen and Russel [6, 7], where they considered thefollowing abstract elastic system ddt (cid:18) uv (cid:19) = (cid:18) I − A − B α (cid:19) (cid:18) uv (cid:19) on D ( A ) × H . The operator A and B α are positive operator on the Hilbert space H satisfying(1.2), and are equivalent in a certain sense. They proved some results of regularity of suchsystem. Similar results was proved by considering thermoelastic system, see for example [10]where the authors present a complete regularity and stability analysis of the abstract system y tt = − Ay + δA α z in (0 , ∞ ) z t = − δA α y t − κA β in (0 , ∞ ) y (0) = y , y t (0) = y , z (0) = z . The operator A is as above, δ = 0 , κ > are real numbers, and α, β ∈ [0 , .Our purpose in this work is twofold: • For θ ∈ (0 , , what type of regularity should we expect for the semigroup ( S θ ( t )) t ≥ ? Inother words, is the semigroup analytic for certain values of θ or not? If the semigroupis not analytic, is it of a certain Gevrey class? The reader should keep in mind that, inaccordance with regularity results for a single elastic system damped as above, regularity EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 3 is expected only for θ ∈ (0 , , e.g. [6, 7, 20]. In particular, responding to two conjecturesof G. Chen and Russell [5], S.P. Chen and Triggiani [6, 7] proved for a single elasticsystem with a more general damping operator, which is equivalent to a fractional power α of the principal operator, that the semigroup is analytic for α in [1 / , , and of Gevreyclass s > / α for α in (0 , / . • For θ ∈ [ − , , what type of stability should we expect for the semigroup ( S θ ( t )) t ≥ ?Thus, the main objective of our work is to analyze how the interplay of the dynamics of thecoupled system affects the regularity or stability of the underlying semigroup.Our findings are summarized in the following results: Theorem 1.1.
Assume that the constants a and b are distinct.(i) For every θ ∈ (1 / , , the semigroup ( S θ ( t )) t ≥ is not analytic. In particular, for every θ ∈ (1 / , , and every r in (2(1 − θ ) , , we have : lim sup | λ |→∞ | λ | r (cid:13)(cid:13) ( iλI − A θ ) − (cid:13)(cid:13) L ( H ) = ∞ . (1.5) (ii) For every θ ∈ (0 , , the semigroup ( S θ ( t )) t ≥ is differentiable, namely, there exists a positiveconstant C such that we have the resolvent estimate: log( | λ | ) (cid:13)(cid:13) ( iλI − A θ ) − (cid:13)(cid:13) L ( H ) ≤ C, ∀ λ ∈ R with | λ | ≥ λ . (1.6) for some large enough λ > . The differentiability of the semigroup is valid for all t > K ,where [18, p. 57] : K := lim sup | λ |→∞ h log( | λ | ) (cid:13)(cid:13) ( iλI − A θ ) − (cid:13)(cid:13) L ( H ) i . Theorem 1.2.
Assume that the constants a and b are distinct. For every θ ∈ (0 , / , thesemigroup ( S θ ( t )) t ≥ is of Gevrey class δ for every δ > /s , with s = 2 θ if θ ∈ (0 , / , and s = 3 θ/ (1 + 2 θ ) if θ ∈ (1 / , / . More precisely, there exists a positive constant C such thatwe have the resolvent estimate: | λ | s (cid:13)(cid:13) ( iλI − A θ ) − (cid:13)(cid:13) L ( H ) ≤ C, ∀ λ ∈ R . (1.7) In particular, the resolvent estimate (1.7) is optimal, in the sense that, for every θ ∈ (0 , / andevery r ∈ (2 θ, , we have: lim sup | λ |→∞ | λ | r (cid:13)(cid:13) ( iλI − A θ ) − (cid:13)(cid:13) L ( H ) = ∞ . (1.8) Theorem 1.3.
Assume that the constants a and b are distinct. (1) For every θ ∈ [0 , , the semigroup ( S θ ( t )) t ≥ is exponentially stable. More precisely, thereexist positive constants K and ξ such that: || S θ ( t ) || L ( H ) ≤ K exp( − ξt ) , ∀ t ≥ . (1.9)(2) For every θ ∈ [ − , , the semigroup ( S θ ( t )) t ≥ is not exponentially stable. However, thesemigroup is polynomially stable, so that there exists a positive constant K with: || S θ ( t ) Z || L ( H ) ≤ K || Z || D ( A θ ) (1 + t ) − θ , ∀ Z ∈ D ( A θ ) , ∀ t ≥ . (1.10) KA¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU
Furthermore, the polynomial decay rate is optimal. More precisely, for every θ in [ − , and every r in [0 , − θ ) , we have : lim sup | λ |→∞ | λ | − r (cid:13)(cid:13) ( iλI − A θ ) − (cid:13)(cid:13) L ( H ) = ∞ . Remark 1.1.
One readily checks that if a = b and we set q = y − z , where the pair ( y, z ) isthe solution of System (1.1) , then q satisfies the system (1.11) q tt + aAq = 0 in (0 , ∞ ) q (0) = y − z ∈ V, q t (0) = y − z ∈ H. System (1.11) is a conservative system; no regularity should then be expected of System (1.1) ,since it can be decoupled into q = y − z and p = y + z , where the p -system is damped as (1.1) ,and as such, enjoys the regularity demonstrated in [6, 7, 20] , the q -system being conservativeenjoys no regularity whatsoever. If we further assume y = z or y = z , then the energy ofsystem (1.11) is nonzero for all times; so no stability is to be expected of System (1.1) . Remark 1.2.
Concerning the resolvent estimate leading to the differentiability of the semigroupassociated with System (1.1) , our proof shows that we can have the stronger resolvent estimate: ∀ r ≥ , ∃ λ = λ ( r ) > | λ | )) r (cid:13)(cid:13) ( iλI − A θ ) − (cid:13)(cid:13) L ( H ) ≤ C, ∀ λ ∈ R with | λ | ≥ λ . (1.12) Remark 1.3.
It is quite surprising that analyticity fails for the coupled elastic system for θ in (1 / , . In that same range, the semigroup is differentiable though, except for θ = 1 . We remindthe reader that for a single similarly damped elastic system, analyticity holds for θ in [1 / , , [6, 7] . However, one should keep in mind that in the present situation, we are dealing with adegenerate system, as the matrix defining the damping is degenerate; it then appears that for θ in (1 / , , the degeneracy dominates the dynamics of the system, and precludes analyticity ofthe underlying semigroup. We think that the situation would be different in the non-degeneratecase, which we plan to analyze in a subsequent work. For θ in (0 , / , we obtain the sameregularity result as for a single similarly damped elastic system. For θ in (1 / , / , we obtaina weaker regularity result, and we think that in this range too, the degeneracy is at work, butits action is milder than in the range (1 / , .As for our polynomial decay estimate, we point out that for θ in [ − , , no exponential decayis to be expected, as the damping plays the role of a compact perturbation of an otherwiseconservative system; in this case a result of Gibson [8] or Triggiani [25] ensures that exponentialdecay fails. Anyway the optimality of our polynomial decay estimate also precludes the exponentialdecay of the semigroup. We also point out that we obtain the same polynomial decay rate asfor a single similarly damped elastic system.
2. S
OME TECHNICAL L EMMAS
Lemma 2.1. ( [11, 19] ) Let A be the generator of a bounded C semigroup ( S ( t )) t ≥ on aHilbert space H . Then ( S ( t )) t ≥ is exponentially stable if and only if: i) i R ⊂ ρ ( A ) , and ii) sup {|| ( ib − A ) − || ; b ∈ R } < ∞ , where ρ ( A ) denotes the resolvent of A . Lemma 2.2. ( [4] ) Let A be the generator of a bounded C semigroup ( S ( t )) t ≥ on a Hilbertspace H such that i R ⊂ ρ ( A ) , where ρ ( A ) denotes the resolvent of A . Then ( S ( t )) t ≥ is EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 5 polynomially stable, viz. , there are positive constants M and α that are independent of theinitial data such that || S ( t ) Z || H ≤ M || Z || D ( A ) (1 + t ) α , ∀ t ≥ , Z ∈ D ( A ) . if and only if ∃ C > || ( ib − A ) − || L ( H ) ≤ C | b | α , ∀ b ∈ R with | b | ≥ . General and weaker versions of Lemma 2.2 may be found, respectively, in [21] and [2, 26, 3, 15].
Lemma 2.3. ( [18, Chap. 2, p. 57] ) Let T = ( T ( t )) t ≥ be a strongly continuous semigroup ona Hilbert space X , with || T ( t ) ≤ M e ωt . Let A denote the infinitesimal generator of A . If forsome µ ≥ ω : lim sup | λ |→∞ (cid:2) log( | λ | ) || ( µ + iλI − A ) − || L ( X ) (cid:3) = C < ∞ . Then T = ( T ( t )) t ≥ is differentiable for t > C . Lemma 2.4. ( [17, Chap. 1, p. 5] ) Let T = ( T ( t )) t ≥ be a strongly continuous semigroup ofcontractions on a Hilbert space X , with infinitesimal generator A . Suppose that i R ⊂ ρ ( A ) , where ρ ( A ) denotes the resolvent of A .The semigroup T = ( T ( t )) t ≥ is analytic if and only if lim sup | λ |→∞ | λ ||| ( iλI − A ) − || L ( X ) < ∞ . Lemma 2.5. ( [20] ) Let T = ( T ( t )) t ≥ be a strongly continuous and bounded semigroup on aHilbert space X . Suppose that the infinitesimal generator A of the semigroup T satisfies thefollowing estimate, for some < α < : lim sup | λ |→∞ | λ | α || ( iλI − A ) − || L ( X ) < ∞ . (2.1) Then T = ( T ( t )) t ≥ is of Gevrey class δ for t > , for every δ > α .
3. P
ROOF OF T HEOREM
Part 1: The semigroup is not analytic for θ ∈ (1 / , θ ∈ (1 / , θ ∈ (1 / , . Here, we proceed as in [14, 22]. Let θ ∈ (1 / , . We are going to show that there exist a sequence of positive real numbers ( λ n ) n ≥ ,and for each n , an element Z n ∈ D ( A ) such that for every r ∈ (2(1 − θ ) , , one has: lim n →∞ λ n = ∞ , || Z n || = 1 , lim n →∞ λ − rn || ( iλ n − A θ ) Z n || = 0 . (3.1)Indeed, if we have sequences λ n and Z n satisfying (3.1), then we set V n = λ − rn ( iλ n − A θ ) Z n , U n = V n || V n || . (3.2) KA¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU
Therefore, || U n || = 1 and lim n →∞ λ rn || ( iλ n − A θ ) − U n || = lim n →∞ || V n || = ∞ , (3.3)which would establish the claimed result, thereby completing the proof of Theorem 1.1. Thus, itremains to prove the existence of such sequences.We shall borrow some ideas from [22]. For each n ≥ , we introduce the eigenfunction e n , with | e n | = 1 , Ae n = ω n e n , and where ( ω n ) is an increasing sequence of positive real numbers with lim n →∞ ω n = ∞ . We seek Z n in the form Z n = ( a n e n , iλ n a n e n , c n e n , iλ n c n e n ) , with λ n and thecomplex numbers a n and c n chosen such that Z n fulfills the desired conditions.For this purpose, set α = ( a + b ) / , β = ( a − b ) / , and for every n ≥ , set: λ n = √ αω n . (3.4)With that choice, we readily check that:(3.5) ( iλ n − A θ ) Z n = (cid:2) ( aω n − λ n ) a n + iλ n γω θn ( a n + c n ) (cid:3) e n − λ n + bω n ) c n + iλ n γω θn ( a n + c n )) e n = (cid:20) ( β + iα γω θ − n ) a n + iα γω θ − n c n (cid:21) ω n e n (cid:20) ( − β + iα γω θ − n ) c n + iα γω θ − n a n (cid:21) ω n e n , by (3.4) . If for each n ≥ , we set a n = − iα γω θ − n c n β + iα γω θ − n , and ω n c n = α n + iβ n , (3.6)where α n and β n are real numbers to be specified later on. It then follows from (3.5):(3.7) ( iλ n − A ) Z n = (cid:20)(cid:18) − β + iα γω θ − n + αγ ω θ − n β + iα γω θ − n (cid:19) c n (cid:21) ω n e n = − β β + iα γω θ − n ω n c n e n EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 7
Therefore, we have (keeping in mind that θ > / ): lim n →∞ λ − rn || ( iλ n − A ) Z n || = lim n →∞ β β + αγ ω θ − n | c n | λ − rn ω n | e n | = lim n →∞ α − r β ω − rn β + αγ ω θ − n ( α n + β n ) (3.8) = lim n →∞ α − r − γ − β ω − r − θn ( α n + β n ) = 0 , as r > − θ ) , andprovided that the sequences ( α n ) and ( β n ) are built in such a way that they converge to somesuitable real numbers, and || Z n || = 1 , for n large enough.One checks that || Z n || = aω n | a n | + λ n | a n | + bω n | c n | + λ n | c n | = ( a + α ) ω n | a n | + ( b + α ) ω n | c n | )= (cid:20) ( a + α ) αγ ω θ − n β + αγ ω θ − n + ( b + α ) (cid:21) ω n | c n | (3.9) = (cid:20) ( a + α ) (cid:18) − β β + αγ ω θ − n (cid:19) + ( b + α ) (cid:21) ω n | c n | = (cid:20) a + b ) − ( a + α ) β β + αγ ω θ − n (cid:21) ( α n + β n ) , as α = a + b. Now, we shall build sequences ( α n ) and ( β n ) . For each n ≥ , set α n = ( α + r n ) , β n = ( β + r n ) , where the real numbers α and β are chosen with(3.10) a + b )( α + β ) = 1 , and r n is to be conveniently chosen in the sequel, such that r n → as n ր ∞ .Notice that with those choices, we now have, setting, ζ n = ( a + α ) β / α [( b + α ) β + 4 α γ ω θ − n ] :(3.11) || Z n || = h a + b ) − ( a + α ) β β + αγ ω θ − n i ( α n + β n )= h a + b ) − ( a + α ) β β + αγ ω θ − n i ( α + β + 2( α + β ) r n + 2 r n )= 1 − ( a + α ) β β + αγ ω θ − n ( α + β )+ h a + b ) − ( a + α ) β β + αγ ω θ − n i (2( α + β ) r n + 2 r n )= 1 − ( a + α ) β α ( β + αγ ω θ − n ) + h α − ( a + α ) β β + αγ ω θ − n i (2( α + β ) r n + 2 r n )= 1 − ( a + α ) β α ( β + αγ ω θ − n ) + ( b + α ) β +4 α γ ω θ − n β + αγ ω θ − n (2( α + β ) r n + 2 r n ) , as a + b )( α + β ) = 1 , and α − a = b + α .Thus, || Z n || = 1 if and only if the quadratic equation(3.12) r n + 2( α + β ) r n − ζ n = 0 , has at least one real root.Now, the discriminant of the quadratic equation is positive, and its roots are given by: r ± n = − ( α + β ) ± p ( α + β ) + 2 ζ n . KA¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU
Notice that ζ n → as n ր ∞ . Accordingly, as r n must converge to zero, for α + β > , wechoose r n = r + n , and for α + β < , we choose r n = r − n . When α + β = 0 , any of the tworoots will do. This completes the proof of the claimed lack of analyticity. Part 2: Differentiability for θ ∈ (0 , θ ∈ (0 , θ ∈ (0 , . From now on, we shall use the following nota-tions || u || τ = | A τ u | ∀ τ ∈ [ − , , ∀ u ∈ V. Let r > be an arbitrary real number. Let λ = λ ( r ) > be so large that for every λ ∈ R with | λ | ≥ λ :(3.13) log( | λ | ) | λ | r ≤ . Thanks to Lemma 2.3, (note that we may choose µ = 0 in our case), it is enough to prove thatthere exists a positive constant C such that: sup (cid:8) log( | λ | ) || ( iλI − A θ ) − || L ( H ) ; λ ∈ R with | λ | ≥ λ (cid:9) ≤ C . (3.14)The constant C may vary from line to line and depends on the parameters of the system, butnot on the frequency variable λ .To prove (3.14), we will show that there exists C > such that for every U ∈ H , one has: log( | λ | ) || ( iλI − A θ ) − U || ≤ C || U || , ∀ λ ∈ R , | λ | ≥ λ . (3.15)Thus, let λ ∈ R with | λ | ≥ λ , U = ( f, g, h, ℓ ) ∈ H , and let Z = ( u, v, w, z ) ∈ D ( A θ ) such that ( iλ − A θ ) Z = U. (3.16)Multiply both sides of (3.16) by Z , then take the real part of the inner product in H to derive: γ | A θ ( v + z ) | = Re ( U, Z ) ≤ || U |||| Z || . (3.17)Equation (3.16) may be rewritten as:(3.18) iλu − v = f,iλv + aAu + γA θ ( v + z ) = g,iλw − z = h,iλz + bAw + γA θ ( v + z ) = ℓ. Thus, (3.15) will be established as soon as we prove the following estimate: log( | λ | ) || Z || ≤ C || U || , ∀ λ ∈ R with | λ | ≥ λ . (3.19)First, we will estimate | λ ||| v + z || − . Then using an interpolation inequality, get an estimate for log( | λ | ) | v + z | . Afterwards, we shall estimate − | λ | )) ℜ ( v, z ) ; the latter two estimates thenyield estimates for both | v | and | z | . Finally the latter estimates will be used to estimate || u || and || w || , thereby completing the proof of (3.19).Estimate | λ ||| v + z || − .It follows of (3.18): iλ ( v + z ) = − aAu − bAw − γA θ ( v + z ) + g + ℓ. Therefore | λ ||| v + z || − ≤ C ( || u || + || w || + | A θ − ( v + z ) | + || U || ) . EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 9
Now, we have the following estimate(3.20) | A θ − ( v + z ) | ≤ C || v + z || − θ θ − | A θ ( v + z ) | θ θ ≤ C || ( v + z ) || − θ θ − ( || U |||| Z || ) θ θ , thanks to (3.17) . Consequently(3.21) | λ ||| v + z || − ≤ C ( || Z || + | λ | − − θ θ || λ ( v + z ) || − θ θ − ( || U |||| Z || ) θ θ + || U || ) . Using Young inequality, we then derive(3.22) | λ ||| v + z || − ≤ C ( || Z || + | λ | − − θ θ ( || U |||| Z || ) + || U || ) . Reporting (3.22) in (3.20), we find(3.23) | A θ − ( v + z ) | ≤ C | λ | − − θ θ ( || Z || + | λ | − − θ θ ( || U |||| Z || ) + || U || ) − θ θ ( || U |||| Z || ) θ θ . Estimate log( | λ | ) | v + z | . To this end, we rely on an interpolation inequality, (keeping in mind(3.13) and calling upon (3.22)):(3.24) log( | λ | ) | v + z | ≤ C log( | λ | ) || v + z || θ θ − || v + z || θ θ ≤ C log( | λ | ) | λ | θ θ || λ ( v + z ) || θ θ − || v + z || θ θ ≤ C log( | λ | ) | λ | θ θ (cid:16) || Z || + | λ | − − θ θ ( || U |||| Z || ) + || U || (cid:17) θ θ ( || U |||| Z || ) θ ≤ C (cid:16) || Z || θ +12 θ +2 || U || θ +2 + || U || || Z || + || U || θ +12 θ +2 || Z || θ +2 (cid:17) . Estimate − | λ | )) ℜ ( v, z ) . Taking the inner product of both sides of the second equationin (3.18) and bz , we derive(3.25) b ℜ ( v, z ) = ℜ iλ (cid:16) − ab ( A u, A z ) − γ ( A θ ( v + z ) , bA θ z ) + b ( g, z ) (cid:17) . Proceeding similarly, but now using the fourth equation in (3.18), and av , we find(3.26) − a ℜ ( v, z ) = ℜ iλ (cid:16) ab ( A w, A v ) + γ ( A θ ( v + z ) , aA θ v ) − a ( ℓ, v ) (cid:17) . Combining (3.25) and (3.26), then multiplying both sides by (log( | λ | )) , we derive, (keeping inmind β = a − b = 0 ):(3.27) − β (log( | λ | )) ℜ ( v, z ) = (log( | λ | )) ℜ abiλ (cid:16) ( A w, A v ) − ( A u, A z ) (cid:17) +(log( | λ | )) ℜ iλ (cid:16) γ ( A θ ( v + z ) , aA θ v − bA θ z ) + b ( g, z ) − a ( ℓ, v ) (cid:17) . Using the first and third equations of (3.18), it follows(3.28) (cid:16) ( A w, A v ) − ( A u, A z ) (cid:17) = (cid:16) ( A w, iλA u − A f ) − ( A u, iλA w − A h ) (cid:17) . Now, notice that(3.29) ℜ abiλ (cid:16) ( A w, iλA u ) − ( A u, iλA w ) (cid:17) = ℜ abiλ (cid:16) ( − iλ )( A w, A u ) + iλ ( A u, A w ) (cid:17) = ℜ ab (cid:16) − ( A w, A u ) + ( A u, A w ) (cid:17) = 0 as the complex number inside the parentheses is purely imaginary.Consequently, (3.27) reduces to(3.30) − | λ | )) ℜ ( v, z ) = (log( | λ | )) ℜ abiβλ (cid:16) − ( A w, A f ) + ( A u, A h ) (cid:17) +(log( | λ | )) ℜ iβλ (cid:16) γ ( A θ ( v + z ) , aA θ v − bA θ z ) + b ( g, z ) − a ( ℓ, v ) (cid:17) . from which, we derive without any particular difficulty(3.31) − | λ | )) ℜ ( v, z ) ≤ C | λ | )) | λ | ( || U |||| Z || + || v + z || θ ( || v || θ + || z || θ )) . We have the interpolation inequality, (the same holds for || z || θ ): || v || θ ≤ C | v | − θ || v || θ ≤ C || Z || − θ || iλu − f || θ ≤ C ( | λ | θ || Z || + || Z || − θ || U || θ ) . Reporting that in (3.31), and using (3.17), we find, (keeping (3.13) in mind):(3.32) − | λ | )) ℜ ( v, z ) ≤ C | λ | )) | λ | − θ (cid:16) || U |||| Z || + || Z || || U || + || Z || − θ || U || θ (cid:17) ≤ C (cid:16) || U |||| Z || + || Z || || U || + || Z || − θ || U || θ (cid:17) . Squaring (3.24) and combining obtained estimate with (3.32), we derive(3.33) (log( | λ | )) ( | v | + | z | ) ≤ C (cid:16) || U |||| Z || + || Z || || U || + || Z || − θ || U || θ (cid:17) + C (cid:16) || Z || θ +1 θ +1 || U || θ +1 + || U || θ +1 θ +1 || Z || θ +1 (cid:17) It remains to estimate log( | λ | ) | A u | and log( | λ | ) | A w | . Since the process is the same for bothestimates, we just prove one of them.Estimate (log( | λ | )) | A u | . The first step consists in estimating λ − (log( | λ | )) | A v | , then usingthe first equation in (3.18) to derive the desired estimate.For this purpose, we note that the second equation in (3.18) may be recast as: aAv = λ v − aAf − iλγA θ ( v + z ) + iλg. Taking the inner product of both sides of that equation and λ − (log( | λ | )) v , we obtain(3.34) aλ − (log( | λ | )) || A v | = log( | λ | )) | v | − a (log( | λ | )) λ ℜ ( A f, A v )+ (log( | λ | )) λ ℜ h − iγ ( A θ − ( v + z ) , A v ) + i ( g, v ) i . We shall now estimate the last three terms in (3.34). Applying the Cauchy-Schwarz inequality,we obtain the estimate (cid:12)(cid:12)(cid:12)(cid:12) − a (log( | λ | )) λ ℜ ( A f, A v ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ a (log( | λ | )) λ | A f || A v |≤ a λ − (log( | λ | )) || A v | + C | A f | , using (3.13) ≤ a λ − (log( | λ | )) || A v | + C || U || EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 11
Similarly, one shows, invoking (3.23) (cid:12)(cid:12)(cid:12)(cid:12) γ (log( | λ | )) λ ℜ ( iA θ − ( v + z ) , A v ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ a λ − (log( | λ | )) || A v | + C (log( | λ | )) | A θ − ( v + z ) | ≤ a λ − (log( | λ | )) || A v | + C (log( | λ | )) | λ | − − θ θ ( || Z || θ || U || θ θ + || U |||| Z || + || U || θ || Z || θ θ ) ≤ a λ − (log( | λ | )) || A v | + C ( || Z || θ || U || θ θ + || U |||| Z || + || U || θ || Z || θ θ ) , thanks to (3.13) . and, calling upon (3.13) once more: (cid:12)(cid:12)(cid:12)(cid:12) (log( | λ | )) λ ℜ i ( g, v ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | g || v | ≤ C || U |||| Z || . Gathering those estimates, and reporting the resulting estimate in (3.34), we find aλ − (log( | λ | )) || A v | ≤ C ( || U || + || Z || θ || U || θ θ + || U |||| Z || + || U || θ || Z || θ θ ) . (3.35)Using the first equation in (3.18), we easily derive a (log( | λ | )) || A u | ≤ C ( || U || + || Z || θ || U || θ θ + || U |||| Z || + || U || θ || Z || θ θ ) . (3.36)In a similar way, one proves b (log( | λ | )) || A w | ≤ C ( || U || + || Z || θ || U || θ θ + || U |||| Z || + || U || θ || Z || θ θ ) . (3.37)Combining (3.33), (3.36) and (3.37), one easily derives(3.38) (log( | λ | )) || Z || ≤ C (cid:16) || U |||| Z || + || Z || || U || + || Z || − θ || U || θ (cid:17) + C (cid:16) || Z || θ +1 θ +1 || U || θ +1 + || U || θ +1 θ +1 || Z || θ +1 (cid:17) + C ( || U || + || Z || θ || U || θ θ + || U |||| Z || + || U || θ || Z || θ θ ) . Finally, applying Young inequality in (3.38), one gets the claimed estimate log( | λ | ) || Z || ≤ C || U || , thereby completing the proof of Theorem 1.1 ⊔⊓
4. P
ROOF OF T HEOREM || u || τ = | A τ u | ∀ τ ∈ [ − , , ∀ u ∈ V. This proof is divided into two parts. In the first part, we shall prove the claimed Gevreyregularity, then in the second part, we will prove the stated optimality of the resolvent.
Part 1: Gevrey regularity.
Thanks to Lemma 2.5, it will be enough to prove that there existsa positive constant C such that: sup (cid:8) | λ | s || ( iλI − A θ ) − || L ( H ) ; λ ∈ R , with | λ | > (cid:9) ≤ C , (4.1) where the constant s is the one stated in the theorem; the proof of the theorem will show howwe got the stated values for s . As in the last section, the constant C may vary from line toline, and C depends on the parameters of the system, but not on the frequency variable λ .To prove (4.1), we will show that there exists C > such that for every U ∈ H , one has: | λ | s || ( iλI − A θ ) − U || ≤ C || U || , ∀ λ ∈ R , with | λ | > , (4.2)with the exponent s as stated.Thus, let λ ∈ R with | λ | > , U = ( f, g, h, ℓ ) ∈ H , and let Z = ( u, v, w, z ) ∈ D ( A θ ) such that ( iλ − A θ ) Z = U. (4.3)Multiply both sides of (4.3) by Z , then take the real part of the inner product in H to derivethe dissipativity estimate: γ | A θ ( v + z ) | = Re ( U, Z ) ≤ || U |||| Z || . (4.4)Equation (4.3) may be rewritten as:(4.5) iλu − v = f,iλv + aAu + γA θ ( v + z ) = g,iλw − z = h,iλz + bAw + γA θ ( v + z ) = ℓ. Thus, (4.2) will be established if we prove the following estimate: | λ | s || Z || ≤ C || U || , ∀ λ ∈ R with | λ | > . (4.6)To prove (4.6), first, we establish the following estimates for constituents of the velocity componentsdefined below:(4.7) | λ || v | + | λ | | A v | ≤ || U || , | λ || z | + | λ | | A z | ≤ || U || . (4.8) | λ | s | v + z | ≤ ε || Z || + C ε || U || , ∀ ε > and(4.9) | λ | s |ℜ ( v , z ) | ≤ ε || Z || + C ε || U || , ∀ ε > . From those estimates, we shall derive(4.10) | λ | s ( | v | + | z | ) ≤ ε || Z || + C ε || U || , ∀ ε > , then using the latter estimate we shall establish(4.11) | λ | s ( | A u | + | A w | ) ≤ ε || Z || + C ε || U || , ∀ ε > . Finally, gathering the last two estimates, one easily derives (4.6), thereby completing the proofof the claimed Gevrey regularity.The attentive reader may fairly wonder why in this section we have to decompose the velocitycomponents, instead of proceeding as in the previous section, viz , without breaking them up.We could do that, but we would only get, for every θ in (0 , / : | λ | θ θ +1 || Z || ≤ C || U || , (4.12)which yields a weaker resolvent estimate than any of the claimed values for s .We now turn to establishing the estimates stated above. We shall proceed in three steps. Inthe first step, (4.8) will be established, in the second step, we will prove (4.9), then combine(4.7)-(4.9) to derive (4.10). In the third step, we shall use (4.10) to prove (4.11), and finally EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 13 derive the desired estimate (4.6).
Step 1.
In this step, we are going to show that for every ε > , there exists a positive constant C ε , independent of λ such that: | λ | s ( | v + z | ) ≤ ε || Z || + C ε || U || , ∀ ε > , ∀ λ ∈ R with | λ | > , (4.13)where v and z are constituents of the velocity components v and z respectively, and are givenbelow.To prove (4.10), the main idea, adapted from the work of Liu and Renardy [16] dealing withthe semigroup analyticity of thermoelastic plates, is to use an appropriate decomposition of eachvelocity component. Therefore, set v = v + v , and z = z + z with(4.14) iλv + Av = g, iλv = − aAu − γA θ ( v + z ) + Av and(4.15) iλz + Az = ℓ, iλz = − bAw − γA θ ( v + z ) + Az One easily proves (4.7) just by taking the appropriate inner products, then estimate real andimaginary parts separately. Next, we shall estimate | λ ||| v + z || − . For this purpose, notice that(4.14) and (4.15) yield iλ ( v + z ) = − aAu − γA θ ( v + z ) + Av − bAw + Az . It then follows, (keeping in mind that θ ≤ / ):(4.16) | λ ||| v + z || − ≤ C ( || u || + || w || + | A θ − ( v + z ) | + || v || + || z || ) ≤ C ( || u || + || w || + | v | + | z | + || v || + || z || ) ≤ C ( || Z || + | λ | − || U || ) , thanks to (4.7) . Now, we have the interpolation inequality | v + z | ≤ C || v + z || θ θ − || v + z || θ θ . From which, one derives, making use of the dissipativity estimate (4.4):(4.17) | λ | s | v + z | ≤ C | λ | s − θ θ || λ ( v + z ) || θ θ − || v + z − v − z || θ θ ≤ C | λ | s − θ θ ( || Z || + | λ | − || U || ) θ θ ( || v + z || θ + || v || θ + || z || θ ) θ ≤ C | λ | s − θ θ ( || Z || + | λ | − || U || ) θ θ ( || U || || Z || + || v || θ + || z || θ ) θ . We also have the interpolation inequality || v || θ ≤ C | v | − θ || v || θ so that invoking (4.7), we obtain(4.18) || v || θ ≤ C | λ | θ − − θ || U || ≤ C | λ | θ − || U || . A similar estimate holds for || z || θ . Therefore (4.17) becomes:(4.19) | λ | s | v + z | ≤ C | λ | s − θ θ ( || Z || + | λ | − || U || ) θ θ ( || U || || Z || + | λ | θ − || U || ) θ ≤ C | λ | s − θ θ ( || Z || θ +12+2 θ || U || θ + | λ | − θ θ || U || θ +12+2 θ || Z || θ )+ C | λ | s − θ θ ( | λ | θ − θ || U || θ || Z || θ θ + | λ | − θ || U || ) . Now, we are going to replace Z with | λ | s Z , then find the best value possible for s ; this iswhere the stated values for s in the theorem are obtained. Doing that in (4.19) leads to(4.20) | λ | s | v + z | ≤ C | λ | s − θ θ − (2 θ +1) s θ ||| λ | s Z || θ +12+2 θ || U || θ + C | λ | − ( θ + s )2+2 θ + s − θ θ || U || θ +12+2 θ ||| λ | s Z || θ + C (cid:18) | λ | s − θ (1+ s )1+ θ + θ − θ || U || θ ||| λ | s Z || θ θ + | λ | − θ + s − θ θ || U || (cid:19) . At this stage, we draw the reader’s attention to the fact that we need all exponents of | λ | to beless than or equal to zero; in the last two terms, this is true for any s in [0 , . However, onereadily checks that for the first term, we need s ≤ θ , and for the second term, s ≤ θ/ (2 θ + 1) .Now, we have θ ≤ θ/ (2 θ + 1) , for θ in (0 , / , and for θ in (1 / , / , the inequality isreversed. Hence the values of s stated in the theorem. In the next step, we let s be genericagain, and find intervals for s , which we then compare to those found here.Since all exponents of | λ | to be less than or equal to zero in (4.20), it readily follows(4.21) | λ | s | v + z | ≤ C (cid:16) ||| λ | s Z || θ +12+2 θ || U || θ + || U || θ +12+2 θ ||| λ | s Z || θ (cid:17) + C (cid:16) || U || θ ||| λ | s Z || θ θ + || U || (cid:17) . Applying Young inequality, one then derives (4.8).
Step 2.
We shall prove (4.9) in this step. To this end, we recall the equations satisfied by v and z : iλv = − aAu − γA θ ( v + z ) + Av , iλz = − bAw − γA θ ( v + z ) + Az Taking the inner product of the v -equation and bz , and the inner product of the z -equationand − av , then taking real parts, we find respectively:(4.22) b ℜ ( v , z ) = ℜ iλ (cid:16) − ab ( A u, A z ) − γ ( A θ ( v + z ) , bA θ z ) + b ( A v , A z ) (cid:17) and(4.23) − a ℜ ( v , z ) = ℜ iλ (cid:16) ab ( A w, A v ) + γ ( A θ ( v + z ) , aA θ v ) − a ( A z , A v ) (cid:17) . Combining (4.22) and (4.23), then multiplying both sides by | λ | s , we derive, (keeping in mind β = a − b = 0 ):(4.24) − β | λ | s ℜ ( v , z ) = | λ | s ℜ abiλ (cid:16) ( A w, A v ) − ( A u, A z ) (cid:17) + | λ | s ℜ iλ (cid:16) γ ( A θ ( v + z ) , aA θ v − bA θ z ) + b ( A v , A z ) − a ( A z , A v ) (cid:17) . We note that(4.25) (cid:16) ( A w, A v ) − ( A u, A z ) (cid:17) = (cid:16) ( A w, A v − A v ) − ( A u, A z − A z ) (cid:17) . EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 15
Using the first and third equations of (4.5), it follows(4.26) (cid:16) ( A w, A v ) − ( A u, A z ) (cid:17) = (cid:16) ( A w, iλA u − A f ) − ( A u, iλA w − A h ) (cid:17) . Now, notice that(4.27) ℜ abiλ (cid:16) ( A w, iλA u ) − ( A u, iλA w ) (cid:17) = ℜ abiλ (cid:16) ( − iλ )( A w, A u ) + iλ ( A u, A w ) (cid:17) = ℜ ab (cid:16) − ( A w, A u ) + ( A u, A w ) (cid:17) = 0 as the complex number inside the parentheses is purely imaginary.Consequently, (4.24) reduces to(4.28) − β | λ | s ℜ ( v , z ) = | λ | s ℜ abiλ (cid:16) ( A w, − A f − A v ) + ( A u, A h + A z ) (cid:17) + | λ | s ℜ iλ (cid:16) γ ( A θ ( v + z ) , aA θ v − bA θ z ) + b ( A v , A z ) − a ( A z , A v ) (cid:17) , from which, we derive(4.29) | λ | s | ℜ ( v , z ) | ≤ C | λ | s − ( || U |||| Z || + || v + z || θ ( || v || θ + || z || θ ))+ C | λ | s − (cid:16) |ℜ i ( A v , A z ) | + |ℜ i ( A z , A v ) | (cid:17) . We have the interpolation inequality, (the same holds for || z || θ ):(4.30) || v || θ ≤ C | v | − θ || v || θ ≤ C ( | v | + | v | ) − θ ( || v || + || v || ) θ ≤ C ( || Z || + | λ | − || U || ) − θ ( | λ ||| Z || + || U || ) θ ≤ C ( | λ | θ || Z || + || Z || − θ || U || θ + | λ | θ − || U || − θ || Z || θ + | λ | θ − || U || ) . Also, given that Av = g − iλv , and z = z − z , we readily check, using (4.7):(4.31) (cid:12)(cid:12)(cid:12) ℜ i ( A v , A z ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) ℜ i ( g − iλv , z − z ) (cid:12)(cid:12) ≤ ( | g || z | + | g || z | + | λ || v || z | + | λ || v || z | ) ≤ C ( || U |||| Z || + | λ | − || U || ) . Similarly, one shows(4.32) (cid:12)(cid:12)(cid:12)(cid:12) ℜ i ( A z , A v ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( || U |||| Z || + | λ | − || U || ) . Reporting (4.30), (4.31) as well as (4.32) in (4.29), and using (4.4), we find(4.33) | λ | s |ℜ ( v , z ) | ≤ C | λ | s − (cid:16) || U |||| Z || + | λ | − ||| U || + ( || U |||| Z || ) ( || v || θ + || z || θ ) (cid:17) ≤ C | λ | s − (cid:16) || U |||| Z || + | λ | − ||| U || + | λ | θ || Z || || U || + || U || θ || Z || − θ (cid:17) + C | λ | s − (cid:16) || Z || θ || U || − θ + | λ | θ − || U || || Z || (cid:17) . Proceeding as in Step 1, we replace || Z || in (4.33) with ||| λ | s Z || , thereby getting(4.34) | λ | s |ℜ ( v , z ) | ≤ C (cid:16) | λ | s − || U ||||| λ | s Z || + | λ | s − ||| U || + | λ | θ − s ||| λ | s Z || || U || (cid:17) + C (cid:16) | λ | s − − (3 − θ ) s || U || θ ||| λ | s Z || − θ + | λ | s − − (1+2 θ ) s ||| λ | s Z || θ || U || − θ (cid:17) + C | λ | s + θ − || U || ||| λ | s Z || . One readily checks that, not only all exponents of | λ | are less than or equal to zero, but alsoeach maximal value of s is larger than any of the values of s obtained in Step 1. Therefore,we have(4.35) | λ | s |ℜ ( v , z ) | ≤ C (cid:16) || U ||||| λ | s Z || + ||| U || + ||| λ | s Z || || U || (cid:17) + C (cid:16) || U || θ ||| λ | s Z || − θ + ||| λ | s Z || θ || U || − θ (cid:17) + C || U || ||| λ | s Z || . Squaring (4.21) and combining obtained estimate with (4.35), we derive(4.36) | λ | s ( | v | + | z | ) ≤ C (cid:16) ||| λ | s Z || θ +11+ θ || U || θ + || U || θ +11+ θ ||| λ | s Z || θ (cid:17) + C (cid:16) || U || θ ||| λ | s Z || θ θ + || U || (cid:17) + C (cid:16) || U ||||| λ | s Z || + ||| U || + ||| λ | s Z || || U || (cid:17) + C (cid:16) || U || θ ||| λ | s Z || − θ + ||| λ | s Z || θ || U || − θ (cid:17) + C || U || ||| λ | s Z || . The application of Young inequality, then yields(4.37) | λ | s ( | v | + | z | ) ≤ ε || Z || + C ε || U || , ∀ ε > . One readily checks from (4.7):(4.38) | λ | s ( | v | + | z | ) ≤ C || U || . The combination of (4.37) and (4.38) then leads to the desired estimate (4.10).It remains to estimate | λ | s | A u | and | λ | s | A w | . Since the process is the same for both estimates,we just prove one of them. That will be the object of the next and final step. Step 3.
In this step, we shall estimate | λ | s | A u | . First, we hall estimate λ s − | A v | , then usethe first equation in (4.5) to derive the desired estimate.For this purpose, we note that the second equation in (4.5) may be recast as: aAv = λ v − aAf − iλγA θ ( v + z ) + iλg. Taking the inner product of both sides of that equation and λ s − v , we obtain(4.39) aλ s − | A v | = λ s | v | − aλ s − ℜ ( A f, A v )+ λ s − ℜ h − iγ ( A θ − ( v + z ) , A v ) + i ( g, v ) i . We shall now estimate the last three terms in (4.39). Applying the Cauchy-Schwarz inequality,we obtain the estimate, (keeping in mind s ≤ ): (cid:12)(cid:12)(cid:12) − aλ s − ℜ ( A f, A v ) (cid:12)(cid:12)(cid:12) ≤ aλ s − | A f || A v |≤ a λ s − | A v | + C | A f | ≤ a λ s − | A v | + C || U || . EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 17
Similarly, one shows, (as θ ≤ ): (cid:12)(cid:12)(cid:12) γλ s − ℜ ( iA θ − ( v + z ) , A v ) (cid:12)(cid:12)(cid:12) ≤ a λ s − | A v | + C λ s | A θ − ( v + z ) | ≤ a λ s − | A v | + C λ s ( | v | + | z | ) . Finally, using the Cauchy-Schwarz inequality once more, we find, (as s ≤ ) (cid:12)(cid:12) λ s − ℜ i ( g, v ) (cid:12)(cid:12) ≤ C | λ | s | g || v | ≤ C || U || λ | s | v | ≤ C ( || U || + λ s | v | ) . Gathering those estimates, and reporting the resulting estimate in (4.39), we get(4.40) aλ s − || A v | ≤ C ( || U || + λ s ( | v | + | z | )) . Using the first equation in (4.5), we readily derive aλ s || A u | ≤ C ( || U || + λ s ( | v | + | z | )) . (4.41)In a similar way, one proves bλ s || A w | ≤ C ( || U || + λ s ( | v | + | z | )) . (4.42)Combining (4.41), (4.42), one finds(4.43) λ s || Z || ≤ C ( || U || + λ s ( | v | + | z | )) Finally, invoking (4.10), one gets the claimed estimate | λ | s || Z || ≤ C || U || , thereby completing the proof of (4.6) Part 2: Resolvent estimate optimality.
Let θ ∈ (0 , / . We are going to show that there exista sequence of positive real numbers ( λ n ) n ≥ , and for each n , an element Z n ∈ D ( A ) such thatfor every r ∈ (2 θ, , one has: lim n →∞ λ n = ∞ , || Z n || = 1 , lim n →∞ λ − rn || ( iλ n − A θ ) Z n || = 0 . (4.44)Indeed, if we have sequences λ n and Z n satisfying (4.44), then we set V n = λ − rn ( iλ n − A θ ) Z n , U n = V n || V n || . (4.45)Therefore, || U n || = 1 and lim n →∞ λ rn || ( iλ n − A θ ) − U n || = lim n →∞ || V n || = ∞ , (4.46)which would establish the claimed result, thereby completing the proof of Theorem 1.2. Thus, itremains to prove the existence of such sequences.For each n ≥ , let e n be the eigenfunction of the operator A , and ω n be its correspondingeigenvalue as in the proof of Theorem 1.1. As in that proof, we seek Z n in the form Z n = ( a n e n , iλ n a n e n , c n e n , iλ n c n e n ) , with λ n and the complex numbers a n and c n chosen suchthat Z n fulfills the desired conditions.We recall that α = ( a + b ) / , β = ( a − b ) / , but now, for every n ≥ , we set: λ n = √ aω n . (4.47) With that choice, we readily check that:(4.48) ( iλ n − A θ ) Z n = (cid:2) ( aω n − λ n ) a n + iλ n γω θn ( a n + c n ) (cid:3) e n − λ n + bω n ) c n + iλ n γω θn ( a n + c n )) e n = (cid:20) ia γω θ + n a n + ia γω θ + n c n (cid:21) e n (cid:20) ( − β + ia γω θ − n ) c n + ia γω θ − n a n (cid:21) ω n e n , by (4.47) . If for each n ≥ , we now set a n = − (1 + 2 iβa − γ − ω − θn ) c n . (4.49)It then follows from (4.48):(4.50) ( iλ n − A ) Z n = βω n c n e n Therefore, we have: lim n →∞ λ − rn || ( iλ n − A ) Z n || = lim n →∞ β | c n | λ − rn ω n | e n | = 4 β a − r lim n →∞ ω θ − rn ω − θn | c n | (4.51) = 0 , for r > θ, andprovided that the sequence ( ω − θn | c n | ) converges to some nonzero real number, and || Z n || = 1 .One checks that || Z n || = aω n | a n | + λ n | a n | + bω n | c n | + λ n | c n | = 2 aω n | a n | + ( b + a ) ω n | c n | )= 2 a (1 + 4 β a − γ − ω − θn ) ω n | c n | + ( b + a ) ω n | c n | ) (4.52) = (cid:16) (3 a + b ) ω θ − n + 8 β γ − (cid:17) ω − θn | c n | , so that we might just choose c n = ω θ − n r(cid:16) (3 a + b ) ω θ − n + 8 β γ − (cid:17) (4.53)to get || Z n || = 1 as desired.This completes the proof of Theorem 1.2. ⊔⊓ EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 19
5. P
ROOF OF T HEOREM
Part 1: Decay estimates.
The notations being as the proof of Theorem 1.2, and given that wehave i R ⊂ ρ ( A θ ) , (5.1)it remains to prove the following two resolvent estimates: ∀ θ ∈ [0 ,
1] : sup (cid:8) || ( iλI − A θ ) − || L ( H ) ; λ ∈ R (cid:9) ≤ C (5.2)and ∀ θ ∈ [ − ,
0) : sup n | λ | θ || ( iλI − A θ ) − || L ( H ) ; λ ∈ R , with | λ | > o ≤ C , (5.3)where in each estimate, and in the sequel, the constant C depends on the parameters of thesystem, but never on the frequency variable λ .We note that (5.1) and (5.2) yield the desired exponential decay estimate thanks to [11, Theorem3] or [19, Corollary 4], while (5.1) and (5.3) yield the claimed polynomial decay estimate thanksto [4, Theorem 2.4]. It remains to prove (5.2) and (5.3). The proof of those two estimatesare similar, but the proof of (5.3) is technically more involved; so we shall prove only (5.3).However, we will be careful in our estimates, so that the proof of (5.2) can easily follow alongour proof.To prove (5.3), we will show that for every θ in [ − , , there exists C > such that forevery U ∈ H , one has: | ( iλI − A θ ) − U || ≤ C | λ | − θ || U || , ∀ λ ∈ R , with | λ | > . (5.4)Thus, let θ in [ − , , and let λ ∈ R with | λ | > . Also let U = ( f, g, h, ℓ ) ∈ H , and Z = ( u, v, w, z ) ∈ D ( A θ ) such that ( iλ − A θ ) Z = U. (5.5)Thus, (5.4) will be established if we prove the following estimate: || Z || ≤ C | λ | − θ || U || , ∀ λ ∈ R with | λ | > . (5.6)Multiply both sides of (5.5) by Z , then take the real part of the inner product in H to derivethe dissipativity estimate: γ | A θ ( v + z ) | = ℜ ( U, Z ) ≤ || U |||| Z || . (5.7)Equation (5.5) may be rewritten as:(5.8) iλu − v = f,iλv + aAu + γA θ ( v + z ) = g,iλw − z = h,iλz + bAw + γA θ ( v + z ) = ℓ. Using the first and third equations in (5.8), it follows from the dissipativity estimate λ | A θ ( u + w ) | ≤ C ( || U |||| Z || + || U || ) . (5.9)To prove (5.6), we note that using first and third equations in (5.8), one derives the two equations(5.10) ( aAu = λ u − γA θ ( v + z ) + g + iλf,bAw = λ w − γA θ ( v + z ) + ℓ + iλh. Taking the inner product of the first equation in (5.10) and u , and doing the same for the secondequation and w , we find(5.11) a || u || = λ | u | − γ ( A θ − ( v + z ) , A u ) + ( g + iλf, u ) and(5.12) b || w || = λ | w | − γ ( A θ − ( v + z ) , A w ) + ( ℓ + iλh, w ) . The application of the Cauchy-Schwarz inequality readily yields, (keeping in mind that θ − ≤ θ ):(5.13) (cid:12)(cid:12)(cid:12) − γ ( A θ − ( v + z ) , A u ) (cid:12)(cid:12)(cid:12) ≤ C | A θ ( v + z ) || A u | ≤ C || U || || Z || , by (5.7) , (5.14) | ( g, u ) | ≤ | g || u | ≤ C || U |||| Z || and(5.15) | ( iλf, u ) | ≤ | f || λ || u | ≤ C || U || + λ | u | . Reporting those estimates in (5.11), we find(5.16) a || u || ≤ λ | u | + C ( || U || || Z || + || U |||| Z || + || U || ) . Similarly, one shows(5.17) b || w || ≤ λ | w | + C ( || U || || Z || + || U |||| Z || + || U || ) . At this stage, we want to draw the reader’s attention to the fact that in the proof of (5.13), wehave relaxed the range of θ , so that our estimate is valid for every θ in [ − , .To complete our proof of the resolvent estimate, it remains to estimate λ ( | u | + | w | ) . To thisend, first, we shall estimate λ | u + w | , then λ |ℜ ( u, w ) | , and combine those two estimates toget the desired estimate.Estimate of λ | u + w | . We have the interpolation inequality, (keeping in mind that θ < ): | u + w | ≤ C | A θ ( u + w ) | − θ | A ( u + w ) | − θ − θ , from which we derive the following inequality(5.18) λ | u + w | ≤ C λ | A θ ( u + w ) | − θ | A ( u + w ) | − θ − θ ≤ C | λ | − θ − θ ( || U || || Z || + || U || ) − θ || Z || − θ − θ , thanks to (5.9) ≤ C | λ | − θ − θ ( || U || − θ || Z || − θ − θ + || U || − θ || Z || − θ − θ ) . Estimate of λ ℜ ( u, w ) . For this purpose, first, take the inner product of the first equation of(5.10) and bw , next, do the same for the other equation and − au , and take real parts to get ab ℜ ( A u, A w ) = bλ ℜ ( u, w ) − γ ℜ ( A θ − ( v + z ) , bA w ) + ℜ ( g + iλf, bw ) , and − ab ℜ ( A w, A u ) = − aλ ℜ ( w, u ) + γ ℜ ( A θ − ( v + z ) , aA u ) − ℜ ( ℓ + iλh, au ) . EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 21
Adding the two equations, we find (recalling β = a − b ):(5.19) βλ ℜ ( u, w ) = γ ℜ ( A θ − ( v + z ) , aA u − bA w ) + ℜ ( g + iλf, bw ) − ℜ ( ℓ + iλh, au ) . Thanks to the first and third equations in (5.8), one readily checks ( iλf, bw ) = ( f, − biλw ) = ( f, − bz − bh ) , and ( iλh, − au ) = ( h, aiλu ) = ( h, av + af ) . Consequently βλ ℜ ( u, w ) = γ ℜ ( A θ − ( v + z ) , aA u − bA w )+ ℜ (( g, bw ) − ( ℓ, au ) − ( f, b ( z + h )) + ( h, a ( v + f ))) . The application of the Cauchy-Schwarz inequality then leads to | γ ℜ ( A θ − ( v + z ) , aA u − bA w ) | ≤ C | A θ ( v + z ) | ( | A u | + | A w | ) ≤ C || U || || Z || , by (5.7)and | ( g, bw ) − ( ℓ, au ) − ( f, b ( z + h )) + ( h, a ( v + f )) | ≤ C ( || U |||| Z || + || U || ) . Hence(5.20) λ |ℜ ( u, w ) | ≤ C ( || U || || Z || + || U |||| Z || + || U || ) . The combination of (5.18) and (5.20) then yields(5.21) λ ( | u | + | w | ) ≤ C | λ | − θ − θ ( || U || − θ || Z || − θ − θ + || U || − θ || Z || − θ − θ )+ C ( || U || || Z || + || U |||| Z || + || U || ) . Using the first and third equations in (5.8), we then derive(5.22) | v | + | z | ≤ C | λ | − θ − θ ( || U || − θ || Z || − θ − θ + || U || − θ || Z || − θ − θ )+ C ( || U || || Z || + || U |||| Z || + || U || ) . Gathering (5.16), (5.17), (5.21) and (5.22), we obtain(5.23) || Z || ≤ C | λ | − θ − θ ( || U || − θ || Z || − θ − θ + || U || − θ || Z || − θ − θ )+ C ( || U || || Z || + || U |||| Z || + || U || ) . Finally, applying Young inequality, and keeping in mind that | λ | > , we get || Z || ≤ C | λ | − θ || U || , hence the claimed resolvent estimate. The claimed semigroup decay estimate then follows from[4, Theorem 2.4]. Part 2: Optimality of the polynomial decay estimate.
Let θ ∈ [ − , . Let the sequences ( λ n ) and ( Z n ) be given as in the second part of Theorem 1.2. Then we already have for each n : lim n →∞ λ n = ∞ , and || Z n || = 1 . Let r ≥ . We shall now prove lim n →∞ λ rn || ( iλ n − A θ ) Z n || = 0 , provided r < − θ .Proceeding as in the proof of Theorem 1.2, we get lim n →∞ λ rn || ( iλ n − A θ ) Z n || = lim n →∞ β | c n | λ rn ω n | e n | = 4 β a r lim n →∞ ω θ + rn ω − θn | c n | (5.24) = 0 , for r < − θ, since the sequence ( ω − θn | c n | ) converges to a positive real number, thanks to (4.53).Consequently, setting for each n: V n = λ rn ( iλ n − A θ ) Z n , U n = V n || V n || , (5.25)it then follows, || U n || = 1 and lim n →∞ λ − rn || ( iλ n − A θ ) − U n || = lim n →∞ || V n || = ∞ . (5.26)This proves the claimed estimate, and the proof of Theorem 1.3 is complete. ⊔⊓
6. E
XAMPLES OF APPLICATION
Let Ω be a bounded domain in R N with smooth boundary Γ . Typical examples of applicationincludes, but are not limited to(1) Interacting membranes.
We consider the following system y tt − a ∆ y + γ ( − ∆) µ y t + γ ( − ∆) µ z t = 0 in Ω × (0 , ∞ ) ,z tt − b ∆ z + γ ( − ∆) µ y t + γ ( − ∆) µ z t = 0 in Ω × (0 , ∞ ) ,y = 0 , z = 0 on Γ × (0 , ∞ ) , with initial conditions y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , z ( x,
0) = z ( x ) , z t ( x,
0) = z ( x ) , where a, b and γ are positive constants. We take H = L (Ω) , A = − ∆ with D ( A ) = H (Ω) ∩ H (Ω) . Then A is a densely defined, positive unbounded operator on the Hilbertspace H satisfying (1.2) by Poincar´e inequality, and D ( A ) is dense in the Hilbert space H . Moreover, V := D ( A ) = H (Ω) and the injections V ֒ → H ֒ → V ′ = H − (Ω) aredense and compact.Thus, the above satisfies all results in Theorems 1.1, 1.2, and 1.3.(2) Interacting plates.
Consider the system of coupled plate equations given by y tt + a ∆ y + γ ∆ µ y t + γ ∆ µ z t = 0 in Ω × (0 , ∞ ) ,z tt + b ∆ z + γ ∆ µ y t + γ ∆ µ z t = 0 in Ω × (0 , ∞ ) ,y = 0 , ∂y∂ν = 0 , z = 0 , ∂z∂ν = 0 on Γ × (0 , ∞ ) , with initial conditions y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , z ( x,
0) = z ( x ) , z t ( x,
0) = z ( x ) , where a, b and γ are positive constants. Let the operator A defined in the Hilbert space L (Ω) by: A = ∆ with D ( A ) = H (Ω) ∩ H (Ω) . EGULARITY ... SEMIGROUP ... INTERACTING ELASTIC SYSTEMS 23
It can be proved that that A is positive, densely defined operator satisfying (1.2). The domainof A is V := H (Ω) , and moreover, the injections H (Ω) ֒ → L (Ω) ֒ → H − (Ω) are dense andcompact.Hence, the system of coupled plate equations satisfies all results in Theorems 1.1, 1.2, and 1.3.R EFERENCES [1] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer.Math. Soc., (1988), 837-852.[2] A. B´atkai, K.-J. Engel, J. Pr¨uss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., (2006), 1425-1440.[3] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol.Equ., (2008), 765-780.[4] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., (2010), 455-478.[5] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart.Appl. Math. (1982), 433-454.[6] S.P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems,Pacific J. Math., (1989), 15-55.[7] S. P. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation:the case < α < / , Proc. Am. Math. Soc. (1990), 401-415.[8] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback.SIAM J. Control and Opt., (1980), 311-316.[9] Z.J Han and Z. Liu, Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigtdamping. ESAIM Control Optim. Calc. Var., (2019), Paper No. 51, 14 pp.[10] J. Hao, Z. Liu and J. Yong, Regularity analysis for an abstract system of coupled hyperbolic and parabolicequations. J. Differential Equations, (2015), 4763-4798.[11] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,Ann. Differential Equations, (1985), 43-56.[12] F. L. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Controland Optimization., (1988), 714-724[13] M. A. Jorge Silva, T. F. Ma and J. E. Mu˜noz Rivera, J. Mindlin-Timoshenko systems with Kelvin-Voigt:analyticity and optimal decay rates. J. Math. Anal. Appl., (2014), 164-179.[14] V. Keyantuo, L. Tebou and M. Warma, A Gevrey class semigroup for a Thermoelastic plate model with afractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete Contin. Dyn. Syst. A, (2020), 2875-2889.[15] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,Z. Angew. Math. Phys., (2005), 630-644.[16] Z. Liu and M. Renardy, A note on the equations of thermoelastic plate, Appl. Math. Lett., (1995), 1-6.[17] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC, 1999.[18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied MathematicalSciences, 44, Springer-Verlag, New York, 1983.[19] J. Pr¨uss, On the spectrum of C -semigroups, Trans. Amer. Math. Soc., (1984), 847-857.[20] S. W. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Controllability, Thesis(Ph.D.), University of Minnesota. 1989, 182 pp.[21] J. Rozendaal, D. Seifert and R. Stahn, Optimal rates of decay for operator semigroups on Hilbert spaces,Adv. Math., (2019), 359-388.[22] L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations, DCDS B., (2010), 1601-1620.[23] L. Tebou, Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plateequation with perturbed boundary conditions, C. R. Math. Acad. Sci. Paris., (2013), 539-544.[24] L. Tebou, Regularity and stability for a plate model involving fractional rotational forces and damping.Submitted.[25] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc.Amer. Math. Soc., 105 (1989), 375-383. [26] K. F. Vu and Yu. I. Lyubich, A spectral criterion for almost periodicity for one-parameter semigroups.(Russian) Teor. Funktsi ¨A Funktsional. Anal. i Prilozhen. No. (1987), 36-41; translation in J. Soviet Math. (1990), no. 6, 644-647UR A NALYSIS AND C ONTROL OF
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