On the Jordan-Moore-Gibson-Thompson equation: well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time
aa r X i v : . [ m a t h . A P ] O c t ON THE JORDAN–MOORE–GIBSON–THOMPSON EQUATION:WELL-POSEDNESS WITH QUADRATIC GRADIENTNONLINEARITY AND SINGULAR LIMIT FOR VANISHINGRELAXATION TIME
BARBARA KALTENBACHER
Alpen-Adria-Universit¨at Klagenfurt, Institut f¨ur Mathematik,Universit¨atsstraße 6567, 9020 Klagenfurt, [email protected]
VANJA NIKOLI´C
Technical University of Munich, Department of Mathematics,Boltzmannstraße 3, 85748 Garching, [email protected]
In this paper, we consider the Jordan–Moore–Gibson–Thompson equation, a third orderin time wave equation describing the nonlinear propagation of sound that avoids theinfinite signal speed paradox of classical second order in time strongly damped modelsof nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We showwell-posedness in an acoustic velocity potential formulation with and without gradientnonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respec-tively. Moreover, we consider the limit as the parameter of the third order time derivativethat plays the role of a relaxation time tends to zero, which again leads to the classicalKuznetsov and Westervelt models. To this end, we establish appropriate energy estimatesfor the linearized equations and employ fixed-point arguments for well-posedness of thenonlinear equations. The theoretical results are illustrated by numerical experiments.
Keywords : nonlinear acoustics, energy estimates, singular limitAMS Subject Classification: 35L77, 35L72, 35L80,
1. Introduction
Nonlinear propagation of sound arises in numerous applications. We here especiallymention high-intensity ultrasound used in medical imaging and therapy, but alsofor industrial purposes, such as ultrasound cleaning or welding; see, e.g., Refs. 1,8, 23, and the given references therein. For the physical fundamentals of nonlinearacoustics, we refer to, e.g., Refs. 3, 9, 14, 24, 30, 31, 32, 41.Its physical and mathematical description involves the acoustic particle velocity ~v , the acoustic pressure p , as well as the mass density ̺ , which can be decomposedinto constant and fluctuating components ~v = ~v + ~v ∼ , p = p + p ∼ , ̺ = ̺ + ̺ ∼ , Barbara Kaltenbacher, Vanja Nikoli´c where in the applications mentioned above, the ambient flow vanishes; i.e, ~v = 0.Furthermore, we have the balances of mass, momentum and sometimes of energy,complemented with an equation of state that relates the mass density to the pres-sure. Combination of these balance and material laws yields wave-type partial dif-ferential equations that are often second order in space and time, but higher orderin time equations play an important role as well. It is one of these third order intime equations that we focus on in this paper.One of the most established models of nonlinear acoustics is Kuznetsov’sequation24 , p ∼ tt − c ∆ p ∼ − δ ∆ p ∼ t = (cid:18) ̺ c B A p ∼ + ̺ | ~v | (cid:19) tt , (1.1)where c is the speed of sound, δ is the diffusivity of sound δ = 1 ̺ (cid:18) µ V ζ V (cid:17) + κ̺ (cid:16) c V − c p (cid:19) , and the velocity is related to the pressure via some balance of forces, ̺ ~v t = −∇ p . (1.2)By ignoring local nonlinear effects modeled by the quadratic velocity term, we arriveat the Westervelt equation47 p ∼ tt − c ∆ p ∼ − δ ∆ p ∼ t = β a ̺ c p ∼ tt , (1.3)with β a = 1 + B/ (2 A ). In terms of the acoustic velocity potential ψ satisfying ~v = −∇ ψ and p = ̺ ψ t , these equations can be rewritten as ψ tt − c ∆ ψ − δ ∆ ψ t = (cid:18) c B A ( ψ t ) + |∇ ψ | (cid:19) t , (1.4)and ψ tt − c ∆ ψ − δ ∆ ψ t = (cid:18) β a c ( ψ t ) (cid:19) t , (1.5)respectively.As has been observed in, e.g., Ref. 16, the use of classical Fourier’s law q = − K ∇ ϑ where ϑ , q , and K denote the absolute temperature, heat flux vector, and thermalconductivity, respectively, leads to an infinite signal speed paradox, which appearsto be unnatural in wave propagation. Therefore in Ref. 16, several other constitutiverelations for the heat flux are considered within the derivation of nonlinear acousticwave equations. Among these is the Maxwell–Cattaneo law τ q t + q = − K ∇ ϑ, n the Jordan–Moore–Gibson–Thompson equation where τ is a positive constant accounting for relaxation (the relaxation time), whosecombination with the above mentioned balance equations and the equation of stateleads to the third order in time PDE model: τ ψ ttt + ψ tt − c ∆ ψ − b ∆ ψ t = (cid:18) c B A ( ψ t ) + |∇ ψ | (cid:19) t , (1.6)where b = δ + τ c . (1.7)This model is known in the literature as the Jordan–Moore–Gibson–Thompson(JMGT) equation20 and we refer to Ref. 16 for its derivation. If one neglects localnonlinear effects modeled by the quadratic velocity term in (1.6), one arrives at τ ψ ttt + ψ tt − c ∆ ψ − b ∆ ψ t = (cid:18) β a c ( ψ t ) (cid:19) t , (1.8)analogously to the reduction of the Kuznetsov to the Westervelt equation20. Wewill refer to equation (1.6) as the Kuznetsov-type and to (1.8) as the Westervelt-type JMGT equation. Obviously, equations (1.6) and (1.8) formally reduce to (1.4)and (1.5) upon setting τ = 0. The present work is, in part, devoted to the rigorousjustification of passing to the limit τ → τ ψ ttt + αψ tt − c ∆ ψ − b ∆ ψ t = f , (1.9)often called the Moore–Gibson–Thompson equation, is studied using semigrouptechniques; see also Refs. 6 and 40. As it turns out, the exponential stability of thetrajectories depends on the critical parameter given by γ := α − τ c b . (1.10)In the case of a constant coefficient α , exponential decay of the energy function E [ ψ ]( t ) = (cid:8) | ψ t ( t ) | + |∇ ψ ( t ) | + | ψ tt ( t ) | + |∇ ψ t ( t ) | + | ∆ ψ ( t ) | (cid:9) (1.11)requires γ to be strictly positive. The case γ < γ = 0marginally stable. An intuitive explanation for this phenomenon is the following:According to the linear wave part of the equation, we can trade αψ tt for c ∆ ψ , thusalso τ ψ ttt for τc α ∆ ψ t in order to relate (1.9) back to the linearization of (1.5) αψ tt − c ∆ ψ − bα γ ∆ ψ t = f , which is a strongly damped wave equation.We mention that the Moore–Gibson–Thompson equation (1.9) is also studiedin Ref. 29, where the problem of identifying γ ( t ) from boundary measurements isconsidered, and in Ref. 26 and 25, where the effect of additive convolution memoryterms acting on ∆ u and ∆ u t , and their combination, respectively, is investigated.For a reformulation of the nonlinear equation (1.8) in terms of the acoustic Barbara Kaltenbacher, Vanja Nikoli´c pressure p = ̺ ψ t , global in time well-posedness and exponential decay of theenergy E [ p ]( t ) is proven in Ref. 20 for small initial data ( p , p , p ) ∈ H (Ω) ∩ H (Ω) × H (Ω) × L (Ω), where Ω is a bounded C smooth domain.One of the key elements in the above cited works on the analysis of equations(1.8) and (1.9) is introduction of the auxiliary state z := ψ t + c b ψ . (1.12)Indeed, in this manner, the third order in time equation (1.9) is reduced to a linear(weakly) damped wave equation for z , τ z tt + γz t − b ∆ z − γ c b z + γ c b ψ = f , (1.13)where − γ c b z + γ c b ψ is a lower order term.This approach, first of all, illustrates the fact that γ should be non-negativeto guarantee a damping behaviour of the term γz t . Secondly, it displays the keydifference to the strongly damped second order equations (1.4) and (1.5). As pointedout in Subsection 6.2.1 of Ref. 33, equation (1.9) does not give rise to an analyticsemigroup; see also Remark 1.3 in Ref. 19. Consequently, the operator driving theevolution does not exhibit maximal parabolic regularity27 and the Implicit functiontheorem argument from, e.g., Ref. 36 cannot be transferred to the present setting.
2. Main results
This paper contributes to the analysis of the JMGT equation in two ways.Firstly, we prove well-posedness with a quadratic gradient nonlinearity arisingwhen taking into account local nonlinear effects; cf. the additional ( |∇ ψ | ) t termon the right hand side in (1.6) compared to (1.8). We base our approach on energyestimates for a reformulation of (1.6) in the form τ ψ ttt + (1 − kψ t ) ψ tt − c ∆ ψ − b ∆ ψ t = 2 ∇ ψ · ∇ ψ t , (2.1)where we use the abbreviation k = c B A . The sign of k will not matter in whatfollows, whereas we assume the coefficients b and c to be strictly positive. We relyon the formulation of the equations in terms of the acoustic velocity potential ψ and not the acoustic pressure20, since it allows to include more easily the quadraticvelocity term ( |∇ ψ | ) t on the right-hand side. We note that the energy estimatesrequired for this purpose differ from those provided in Ref. 20 for the equation (1.8)without the quadratic gradient nonlinearity.Secondly, we consider the limit τ → τ →
0, and analo-gously for the Westervelt-type equation. For this purpose, the energy estimates wewill derive are crucial. These estimates differ for the Kuznetsov-type and for theWestervelt-type version of the JMGT equation, which is why we treat these modelsin separate sections. n the Jordan–Moore–Gibson–Thompson equation The rest of the paper is organized as follows. In Section 3, we consider the lin-earized equation (1.9) with a fixed positive coefficient α that is possibly space andtime dependent, but bounded away from zero, and an inhomogeneity f , as well asa fixed positive τ . We prove well-posedness of this linearized model together withan energy estimate.Section 4 contains a well-posedness proof for the Westervelt version (1.8) of theequation by setting α = 1 − kψ t and f = 0. The proof is based on the equation (2.1),but with zero right-hand side, as the gradient nonlinearity is not present in (1.8).This fact allows to prove local in time well-posedness for small inital data, evenwithout any sign condition on the parameter γ = α − τc b . However, the energy esti-mates from Section 3 do not cover the gradient nonlinearity in the Kuznetsov-typeversion of the JMGT equation, so that higher order energy estimates are needed.We derive them in Section 5. These involve the auxiliary function z and requirepositivity of both α and γ , where the latter follows from positivity and bounded-ness away from zero of α for τ sufficiently small.Section 6 provides the corresponding well-posedness result for the equation (1.6)based on reformulation (2.1); i.e., setting α = 1 − kψ t and f = 2 ∇ ψ · ∇ ψ t . Startingfrom a sufficiently small positive value and letting τ tend to zero clearly preservesthe sign structure of the coefficients, in particular of γ and b , so that the energybounds from Sections 4 and 6 can be used for justifying the limiting process τ → Remark 2.1 (On medium parameters).
We require strict positivity of con-stants c , δ , and τ (hence, also of b = δ + τ c ) appearing in the equations for provingwell-posedness of initial-boundary value problems (1.6) and (1.8). These are, in-deed, very natural assumptions from a physical point of view; typical values ofthese parameters in different media can be found, for example, in Refs. 4 and 43.It is also known that, in order to establish global well-posedness for the limitingproblems (1.4) and (1.5) in space dimensions higher than one, strict positivity of δ is needed17 ,
18. The constant k does not need to have a particular sign in ouranalysis, but will typically be non-negative in applications. Theoretical preliminaries and assumptions
We set here the notation and collect some useful theoretical results that we oftenuse in the analysis. Throughout the paper, the spatial domain Ω ⊂ R d , where d ∈ { , , } , is assumed to be sufficiently smooth to admit integration by parts aswell as second order elliptic regularity.We consider the PDEs on a bounded space time cylinder Ω × (0 , T ) and imposehomogeneous Dirichlet boundary conditions on ∂ Ω for simplicity ψ = 0 on ∂ Ω × (0 , T ); (2.2) Barbara Kaltenbacher, Vanja Nikoli´c i.e., ( − ∆) : H (Ω) → H − (Ω) is the Laplace operator equipped with homogeneousDirichlet boundary conditions. We expect that Neumann and impedance (absorb-ing) boundary conditions can be treated analogously, but lead to modifications inthe energy estimates.The third order in time equations (1.6) and (1.8) are complemented with initialconditions ψ (0) = ψ , ψ t (0) = ψ , ψ tt (0) = ψ , (2.3)whereas for the limiting second order in time equations (1.4), (1.5) as τ →
0, theinitial condition on ψ tt naturally disappears, which will be seen also in the energyestimates.2.1.1. Notation
For simplicity of notation, we often omit the time interval and the spatial domainwhen writing norms, i.e., k · k L p L q denotes the norm on L p (0 , T ; L q (Ω)). We alsoabbreviate the L (Ω) inner product by ( · , · ) L and the L (Ω) norm (as well as theabsolute value) by | · | . We employ the notation L p (0 , T ; Z ), W s,p (0 , T ; Z ) for theBochner-Sobolev spaces of time dependent functions.More specifically, we use dedicated function spaces for the solutions of the con-sidered equations. We collect their notation here for future reference: X W := W , ∞ (0; T ; H (Ω) ∩ H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)) , ∩ H (0 , T ; L (Ω)) X K := L ∞ (0 , T ; H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)) ∩ W , ∞ (0; T ; H (Ω)) , ¯ X W := W , ∞ (0; T ; H (Ω) ∩ H (Ω)) ∩ H (0 , T ; H (Ω))¯ X K := L ∞ (0 , T ; H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)) ∩ H (0; T ; H (Ω)) , (2.4)with partly τ -dependent norms induced by the energy estimates to be derived k ψ k X W := k ψ k W , ∞ H + k ψ tt k L H + τ k ψ tt k L ∞ H + τ k ψ ttt k L L , k ψ k X K := k ψ k L ∞ H + k ψ t + c b ψ k L ∞ H + k ψ tt + c b ψ t k L H + τ k ψ tt + c b ψ t k L ∞ H , k ψ k X W := k ψ k W , ∞ H + k ψ tt k L H , k ψ k X K := k ψ tt + c b ψ t k L H + k ψ t + c b ψ k L ∞ H + k ψ k L ∞ H . (2.5)2.1.2. Helpful inequalities
Throughout the paper, we often employ the continuous embeddings H (Ω) ֒ → L (Ω), H (Ω) ֒ → L ∞ (Ω) k v k L (Ω) ≤ C Ω H ,L k v k H (Ω) , k v k L ∞ (Ω) ≤ C Ω H ,L ∞ k v k H (Ω) (2.6) n the Jordan–Moore–Gibson–Thompson equation as well as boundedness of the operator ( − ∆) − : L (Ω) → H (Ω) ∩ H (Ω), thePoincar´e-Friedrichs inequality, k v k H (Ω) ≤ C Ω( − ∆) − k − ∆ v k L (Ω) , k v k H (Ω) ≤ C Ω P F k∇ v k L (Ω) , (2.7)and the trace theorem k ν · ∇ v k H − / ( ∂ Ω) ≤ C Ω tr k v k H (Ω) , k v k H / ( ∂ Ω) ≤ C Ω tr k v k H (Ω) , (2.8)see, e.g., Lemma 4.3 in Ref. 34, where ν denotes the outward unit normal on theboundary of Ω.
3. Analysis of the linear damped wave equation (1.13)We now turn our attention to the linear equation (1.9) and study the followinginitial-boundary value problem: τ ψ ttt + α ( x, t ) ψ tt − c ∆ ψ − b ∆ ψ t = f in Ω × (0 , T ) ,ψ = 0 on ∂ Ω × (0 , T ) , ( ψ, ψ t , ψ tt ) = ( ψ , ψ , ψ ) in Ω × { } , (3.1)under the non-degeneracy assumption that for some α > α ( t ) ≥ α on Ω a.e. in Ω × (0 , T ) . (3.2)We assume that the coefficient α and the source term f have the following regularity: α ∈ L ∞ (0 , T ; L ∞ (Ω)) ∩ L ∞ (0 , T ; W , (Ω)) ,f ∈ H (0 , T ; L (Ω)) . (3.3)Moreover, ψ is assumed to satisfy the initial conditions (2.3) with( ψ , ψ , ψ ) ∈ X W := H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) × H (Ω) . (3.4)For an analysis of equation (1.9) with constant coefficient α , under the slightlyweaker assumptions( ψ , ψ , ψ ) ∈ ( H (Ω) ∩ H (Ω)) × H (Ω) × L (Ω)and f ∈ L (0 , T ; L (Ω)), we refer to Corollary 1.2 in Ref. 19. There it is shownthat ψ ∈ C (0 , T ; H (Ω) ∩ H (Ω)) ∩ C (0 , T ; H (Ω)) ∩ C (0 , T ; L (Ω)) by meansof semigroup techniques. The assumptions (3.3) and (3.4) naturally arise from theenergy estimates in the well-posedness proof below and lead to the stronger (ascompared to Ref. 19) regularity stated in (3.5) below. Theorem 3.1.
Let c , b , τ > , and let T > be a fixed time horizon. Let thenon-degeneracy assumption (3.2) and the regularity assumptions (3.3) and (3.4) hold. Then there exists a unique solution ψ of the problem (3.1) that satisfies ψ ∈ X W := W , ∞ (0; T ; H (Ω) ∩ H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) . (3.5) Barbara Kaltenbacher, Vanja Nikoli´c
Furthermore, the solution fullfils the estimate τ k ψ ttt k L L + τ k ψ tt k L ∞ H + k ψ tt k L H + k ψ t k L ∞ H ≤ C ( α, T, τ ) (cid:0) | ψ | H + | ψ | H + τ | ψ | H + k f k L ∞ L + k f t k L L (cid:1) . (3.6) The constant above is given by C ( α, T, τ ) = C (cid:0) T + k α k L ∞ L ∞ (cid:1) exp (cid:0) C (cid:0) τ k∇ α k L ∞ L + 1 + T (cid:1) T (cid:1) , where C , C > do not depend on τ, T , or α .If additionally we assume that k∇ α k L ∞ L is sufficiently small so that k∇ α k L ∞ L < αC Ω H ,L (3.7) holds, then (3.6) is valid with an upper bound that is independent of τ , i.e., C ( α, T, τ ) = C ( α, T ) = C (cid:0) T + k α k L ∞ L ∞ (cid:1) exp ( C (1 + T ) T ) . (3.8) Proof.
We carry out the proof by via Galerkin approximations in space, relyingon energy estimates; cf. Refs. 10, 44. Note that the initial data are meaningful sinceregularity (3.5) implies ψ ∈ C ([0 , T ]; H (Ω) ∩ H (Ω)) ,ψ t ∈ C w ([0 , T ]; H (Ω) ∩ H (Ω)) ,ψ tt ∈ C w ([0 , T ]; H (Ω)) , where C w denotes the space of weakly continuous functions; see Lemma 3.3 inRef. 45. Step 1: Discretization in space.
Let { w i } i ∈ N denote the eigenfunctions of theDirichlet-Laplacian operator − ∆. Then { w i } i ∈ N can be normalized to form an or-thogonal basis of H (Ω) ∩ H (Ω) and to be orthonormal with respect to the L (Ω)scalar product.Fix n ∈ N and denote V n = span { w , . . . , w n } . We seek an approximate solutionin the form of ψ n = n X i =1 ξ i ( t ) w i ( x ) , (3.9)where ξ i : (0 , T ) → R , i ∈ [1 , n ]. The initial data are chosen as ψ n ( x ) = n X i =1 ξ i, w i ( x ) , ψ n ( x ) = n X i =1 ξ i, w i ( x ) , ψ n ( x ) = n X i =1 ξ i, w i ( x ) , where the coefficients ξ i, , ξ i, , ξ i, ∈ R are given by ξ i, = ( ψ , w i ) L , ξ i, = ( ψ , w i ) L , ξ i, = ( ψ , w i ) L , n the Jordan–Moore–Gibson–Thompson equation for i ∈ [1 , n ]. In this way, we have by construction that k ψ n k H ≤ k ψ k H and ψ n −→ ψ in H ∩ H , k ψ n k H ≤ k ψ k H and ψ n −→ ψ in H ∩ H , k ψ n k H ≤ k ψ k H and ψ n −→ ψ in H ; (3.10)see Lemma 7.5 in Ref. 42. We then consider the following approximation of ourproblem ( τ ψ nttt + αψ ntt − c ∆ ψ n − b ∆ ψ nt , φ ) L = ( f, φ ) L , for every φ ∈ V n pointwise a.e. in (0 , T ) , ( ψ n (0) , ψ nt (0) , ψ ntt (0)) = ( ψ n , ψ n , ψ n ) . (3.11)We introduce matrices I n = [ I ij ], M n = [ M ij ], K n = [ K ij ], C n = [ C ij ], and vector F n = [ F i ], where I nij = ( w i , w j ) L = δ ij , M nij ( t ) = ( αw i , w j ) L ,K nij = − c (∆ w i , w j ) L , D nij = − b (∆ w i , w j ) L ,F ni = ( f, w i ) L (3.12)for i, j ∈ [1 , n ], where δ ij denotes the Kronecker delta. By setting ξ n = [ ξ . . . ξ n ] T , ξ n = [ ξ , . . . ξ n, ] T , ξ n = [ ξ , . . . ξ n,n ] T , and ξ n = [ ξ , . . . ξ n, ] T , problem (3.11)can be rewritten as ( τ I n ξ nttt + M n ξ ntt + D n ξ nt + K n ξ n = F n ( t ) , ( ξ n (0) , ξ nt (0) , ξ ntt (0)) = ( ξ n , ξ n , ξ n ) . (3.13)After additionally rewriting (3.13) as a first-order system, existence of an absolutelycontinuous solution [ ξ n , ξ nt , ξ ntt ] T on [0 , T n ] for some T n ≤ T follows from standardODE theory; see, for example, Chapter 1 in Ref. 44. To see that ξ n ∈ H (0 , T n ),we can employ a bootstrap argument, | ξ ttt | L (0 ,T n ) = τ | − M n ξ ntt − D n ξ nt − K n ξ n + F n | L (0 ,T n ) ≤ C ( k α k L ∞ L ∞ + k f k L L ) . (3.14)We, therefore, conclude that (3.11) has a solution ψ n ∈ H (0 , T n ; V n ). The upcomingenergy estimate will allow us to extend the existence interval to [0 , T ]. Step 2: Energy estimates.
Our next goal is to obtain a bound for ψ n that isuniform with respect to n . To this end, we test our problem (3.11) with a suitabletest function. First estimate.
Testing the first equation in (3.11) with φ = − ∆ ψ ntt ∈ V n and Barbara Kaltenbacher, Vanja Nikoli´c integrating over (0 , t ), where t ≤ T n , yields the energy identity τ |∇ ψ ntt ( t ) | + b | − ∆ ψ nt ( t ) | + k√ α ∇ ψ ntt k L t L = τ |∇ ψ ntt (0) | + b | − ∆ ψ nt (0) | − Z t ( ψ ntt ∇ α, ∇ ψ ntt ) L d s − c ( − ∆ ψ n , − ∆ ψ nt ) L (cid:12)(cid:12)(cid:12) t + c Z t ( − ∆ ψ nt , − ∆ ψ nt ) L d s + ( f, − ∆ ψ nt ) L (cid:12)(cid:12)(cid:12) t − Z t ( f t , − ∆ ψ nt ) L d s =: rhs ( t ) , (3.15)where we have skipped the argument ( s ) under the time integral for notationalsimplicity and used the abbreviation L t L for L (0 , t ; L (Ω)). To derive (3.15), wehave used the following three identities:( αψ ntt , − ∆ ψ ntt ) L = ( α ∇ ψ ntt , ∇ ψ ntt ) L + ( ψ ntt ∇ α, ∇ ψ ntt ) , and c Z t ( − ∆ ψ n , − ∆ ψ ntt ) L d s = c ( − ∆ ψ n , − ∆ ψ nt ) L (cid:12)(cid:12)(cid:12) t − c Z t ( − ∆ ψ nt , − ∆ ψ nt ) L d s, as well as Z t ( f, − ∆ ψ ntt ) L d s = ( f, − ∆ ψ nt ) L (cid:12)(cid:12)(cid:12) t − Z t ( f t , − ∆ ψ nt ) L d s. We note that f ∈ H (0 , T ; L (Ω)) ֒ → C ([0 , T ]; L (Ω)). We next estimate rhs ( t )from above. We introduce here a constant that depends on the initial data to sim-plify the notation: C ( ψ , ψ , ψ ; τ )= τ |∇ ψ ntt (0) | L + b | − ∆ ψ nt (0) | L + | f (0) | L | − ∆ ψ nt (0) | L + c | − ∆ ψ n (0) | L | − ∆ ψ nt (0) | L = τ |∇ ψ n (0) | L + b | − ∆ ψ n | L + | f (0) | L | − ∆ ψ n | L + c | − ∆ ψ n | L | − ∆ ψ n | L . By applying H¨older’s inequality, we get rhs ( t ) ≤ C ( ψ , ψ , ψ ; τ ) + k∇ α k L ∞ L k ψ ntt k L L k∇ ψ ntt k L t L + c | − ∆ ψ n ( t ) | L | − ∆ ψ nt ( t ) | L + c k − ∆ ψ nt k L t L + k f k L ∞ L | − ∆ ψ nt ( t ) | L + k f t k L t L k − ∆ ψ nt k L t L . We further estimate the right-hand side with the help of Young’s ε -inequality xy ≤ ε x + ε y , (3.16) n the Jordan–Moore–Gibson–Thompson equation and choosing ε = b or ε = 1, and the embedding results to obtain for a.e. t ∈ [0 , T n ], rhs ( t ) ≤ C ( ψ , ψ , ψ ; τ ) + C Ω H ,L k∇ α k L ∞ L k∇ ψ ntt k L t L + c b | − ∆ ψ n ( t ) | L + b | − ∆ ψ nt ( t ) | L + c k − ∆ ψ nt k L t L + b k f k L ∞ L + b | − ∆ ψ nt ( t ) | L + 12 k f t k L L + 12 k − ∆ ψ nt k L t L . (3.17)We can estimate the term k− ∆ ψ n ( t ) k L appearing on the right-hand side as follows k − ∆ ψ n k L ∞ t L ≤ √ t k − ∆ ψ nt k L t L + | − ∆ ψ | L . (3.18)Altogether, we get τ |∇ ψ ntt ( t ) | L + α k∇ ψ ntt k L t L + b | − ∆ ψ nt ( t ) | L ≤ C ( ψ , ψ , ψ ; τ ) + C Ω H ,L k∇ α k L ∞ L k∇ ψ ntt k L t L + c b T k − ∆ ψ nt k L t L + c b | − ∆ ψ | L + c k − ∆ ψ nt k L t L + b k f k L ∞ L + k f t k L L + k − ∆ ψ nt k L t L . (3.19)If the smallness assumption (3.7) holds, then the term containing k∇ α k L ∞ L canbe absorbed into the left-hand side. A priori bound for ψ n . Applying Gronwall’s inequality to (3.19), and taking thesupremum over t ∈ (0 , T n ) then yields τ k∇ ψ ntt k L ∞ L + k∇ ψ ntt k L L + k − ∆ ψ nt k L ∞ L ≤ ˜ C ( α, T n , τ ) (cid:0) | ψ n | H + | ψ n | H + τ | ψ n | H + k f k L ∞ L + k f t k L L (cid:1) . (3.20)By employing the upper bounds for the approximate initial data stated in (3.10)and the inequality T n ≤ T , we further have τ k∇ ψ ntt k L ∞ L + k∇ ψ ntt k L L + k − ∆ ψ nt k L ∞ L ≤ ˜ C ( α, T n , τ ) (cid:16) | ψ | H + | ψ | H + τ | ψ | H + k f k L ∞ L + k f t k L t L (cid:17) . (3.21)The constant above is given by˜ C ( α, T, τ ) = ˜ C exp (cid:16) ˜ C (cid:0) τ k∇ α k L ∞ L + 1 + T (cid:1) T (cid:17) , or, if assumption (3.7) holds, by˜ C ( α, T, τ ) = ˜ C ( T ) = ˜ C exp (cid:16) ˜ C (1 + T ) T (cid:17) , where ˜ C , ˜ C > n or τ . Since the right-hand side of (3.21) doesnot depend on T n , we can extend the existence interval to [0 , T ], i.e. T n = T . Second estimate.
By testing (3.11) with φ = τ ψ nttt ∈ V n and integrating over(0 , T ), we obtain τ k ψ nttt k L L ≤ k − αψ ntt + c ∆ ψ n + b ∆ ψ nt + f k L L k τ ψ nttt k L L , (3.22) Barbara Kaltenbacher, Vanja Nikoli´c from which we have τ k ψ nttt k L L ≤ k α k L ∞ L ∞ k ψ ntt k L L + c k − ∆ ψ n k L L + b k − ∆ ψ nt k L L + k f k L L . (3.23)The terms k− ∆ ψ n k L L , k− ∆ ψ nt k L L can be further estimated similarly to (3.18), k − ∆ ψ n k L L = Z T (cid:12)(cid:12)(cid:12)(cid:12) − ∆ ψ n + Z t − ∆ ψ nt ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) L dt ! / ≤ √ T | − ∆ ψ n | L (Ω) + Z T (cid:12)(cid:12)(cid:12)(cid:12)Z t − ∆ ψ nt ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) L dt ! / ≤ √ T | − ∆ ψ n | L (Ω) + Z T t dt ! / k − ∆ ψ nt k L ∞ L ≤ √ T | − ∆ ψ n | L (Ω) + q T k − ∆ ψ nt k L ∞ L , k − ∆ ψ nt k L L ≤ √ T k − ∆ ψ nt k L ∞ L . The term k ψ ntt k L L by means of the Poincar´e-Friedrichs inequality, so that by using(3.21) we obtain energy estimate (3.6) with ψ n in place of ψ . Step 3: Passing to the limit.
On account of estimate (3.21) and standard com-pactness results, together with the fact that the spatial and temporal domains(0 , T ) and Ω are bounded, we know that there exist a subsequence, denoted againby { ψ n } n ∈ N , and a function ψ such that ψ nttt − ⇀ ψ ttt weakly in L (0 , T ; L (Ω)) ,ψ ntt − ⇀ ψ tt weakly- ⋆ in L ∞ (0 , T ; H (Ω)) ,ψ nt − ⇀ ψ t weakly- ⋆ in L ∞ (0 , T ; H (Ω) ∩ H (Ω)) ,ψ n − ⇀ ψ weakly- ⋆ in L ∞ (0 , T ; H (Ω) ∩ H (Ω)) . (3.24)Our next task is to prove that ψ solves (3.1). We test (3.11) with η ∈ C ∞ c (0 , T ) andintegrate over time to obtain − Z T ( τ ψ nttt , w i ) L η ( t ) d t = − Z T ( αψ ntt − c ∆ ψ n − b ∆ ψ nt − f, w i ) L η ( t ) d t, (3.25)for all i ∈ [1 , n ]. Thanks to (3.24), letting n → ∞ in (3.25) leads to − Z T ( τ ψ ttt , w i ) L η ′ ( t ) d t = − Z T ( γψ tt − c ∆ ψ − b ∆ ψ t − f, w i ) L η ( t ) d t, n the Jordan–Moore–Gibson–Thompson equation for all i ∈ N and η ∈ C ∞ (0 , T ). By construction, ∪ n ∈ N V n is dense in L (Ω), so ψ solves the PDE in (3.1) in the L (0 , T ; L (Ω)) sense. Due to the embeddings ψ n ∈ W , ∞ (0 , T ; H (Ω) ∩ H (Ω)) ֒ → ֒ → C ([0 , T ]; H (Ω) ∩ H (Ω)) ,ψ nt ∈ W , ∞ (0 , T ; H (Ω)) ֒ → ֒ → C ([0 , T ]; H (Ω)) ,ψ ntt ∈ H (0 , T ; L (Ω)) ֒ → C ([0 , T ]; L (Ω)) , we know that ψ n (0) → ψ (0) in H (Ω) ∩ H (Ω) ,ψ nt (0) → ψ t (0) in H (Ω) ,ψ ntt (0) → ψ tt (0) in L (Ω) . Thanks to (3.10), we can then infer that ψ (0) = ψ , ψ t (0) = ψ , and ψ tt (0) = ψ .Altogether, we conclude that ψ is a solution of the initial-boundary value problem(3.1). Step 4: Energy inequality for ψ . We can take the limit inferior as n → ∞ of (3.21), (3.23), and via the weak and the weak- ⋆ lower semi-continuity of normsobtain the final estimate (3.6). Uniqueness of a solution follows by the linearity ofthe equation, together with the fact that the homogeneous equation only has thezero solution, by the above energy estimates.
4. Well-posedness of the nonlinear Westervelt-type wave equationfor τ > After having studied the linearized equation, we now proceed to the nonlinear model(1.8). For proving well-posedness of (1.8), we introduce the fixed-point operator T that maps φ to a solution ψ of τ ψ ttt + (1 − kφ t ) ψ tt − c ∆ ψ − b ∆ ψ t = 0 , (4.1)on some ball B X W ρ = { ψ ∈ X W : ψ (0) = ψ , ψ t (0) = ψ , ψ tt (0) = ψ , k ψ k X W := τ k ψ ttt k L L + τ k ψ tt k L ∞ H + k ψ tt k L H + k ψ k W , ∞ H ≤ ρ } (4.2)in the space X W , defined in (2.4). Note that the operator is well-defined on accountof Theorem 3.1.For establishing T as a self-mapping on B X W ρ , it is crucial to prove that α =1 − kφ t is in L ∞ (0 , T ; L ∞ (Ω)) ∩ L ∞ (0 , T ; W , (Ω)) and that the smallness condition(3.7) of Theorem 3.1 holds, provided φ ∈ B Xρ . Smallness of φ will also be requiredfor verifying the non-degeneracy condition α ( t ) ≥ α > τ . In particular, it holds for arbitrarily small τ and Barbara Kaltenbacher, Vanja Nikoli´c therefore allows for taking limits as τ → T , based on the fact that ˆ ψ = ψ − ψ = T ( φ ) − T ( φ ) solves τ ˆ ψ ttt + (1 − kφ t ) ˆ ψ tt − c ∆ ˆ ψ − b ∆ ˆ ψ t = k ˆ φ t ψ tt , (4.3)with homogeneous initial and boundary conditions (where ˆ φ = φ − φ ), would re-quire to prove that α = 1 − kφ t and f = k ˆ φ t ψ tt are in L ∞ (0 , T ; L ∞ (Ω)) ∩ L ∞ (0 , T ; W , (Ω)) and H (0 , T ; L (Ω)), respectively. This regularity, however,would only be possible in an O ( √ τ ) neighborhood because of the φ ttt term arisingin f t . Therefore, we do not prove contractivity, but base our existence proof onSchauder’s theorem, similarly to the approach in Ref. 21. Theorem 4.1.
Let c , b > , k ∈ R and let T > . Then there exist ρ > and ρ > such that for all ( ψ , ψ , ψ ) ∈ X W = H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) × H (Ω) satisfying k ψ k H (Ω) + k ψ k H (Ω) + τ k ψ k H (Ω) ≤ ρ , (4.4) there exists a solution ψ ∈ X W of τ ψ ttt + ψ tt − c ∆ ψ − b ∆ ψ t = (cid:0) k ( ψ t ) (cid:1) t in Ω × (0 , T ) ,ψ = 0 on ∂ Ω × (0 , T ) , ( ψ, ψ t , ψ tt ) = ( ψ , ψ , ψ ) in Ω × { } , (4.5)such that it holds τ k ψ ttt k L L + τ k ψ tt k L ∞ H + k ψ tt k L H + k ψ k W , ∞ H ≤ ρ . (4.6) Proof.
Our proof relies on Schauder’s fixed-point theorem applied to the operator T : B X W ρ ∋ φ ψ, where ψ solves (4.1). To obtain the self-mapping property of T , we have to verifythe condition (3.7) of Theorem 3.1 as well as α ( t ) ≥ α >
0. We thus estimate the L ∞ (0 , T ; L ∞ (Ω)) ∩ L ∞ (0 , T ; W , (Ω)) norm of α = 1 − kφ t . In view of the bounds k∇ α k L ∞ L = | k | k∇ φ t k L ∞ L ≤ | k | C Ω H ,W , k φ k W , ∞ H ≤ | k | C Ω H ,W , ρ, k α − k L ∞ L ∞ = | k | k φ t k L ∞ L ∞ ≤ | k | C Ω H ,L ∞ k φ k W , ∞ H ≤ | k | C Ω H ,L ∞ ρ, the smallness condition (3.7) and the non-degeneracy condition α ( t ) ≥ α > ρ < (cid:0) | k | max (cid:8) C Ω H ,L ∞ , C Ω H ,L C Ω H ,W , (cid:9)(cid:1) − . The self-mapping property follows from the estimate (3.6), with f = 0 and ρ ≤ (cid:0) C (cid:0) + T (cid:1) exp ( C (1 + T ) T ) (cid:1) − ρ . The set B X W ρ is a weak* compact and convex subset of the Banach space X W ,defined in (2.4). The weak* continuity of T can be established as follows: For any n the Jordan–Moore–Gibson–Thompson equation sequence ( φ n ) n ∈ N ⊆ B X W ρ that weakly* converges to φ ∈ B X W ρ in X W , we alsohave ( T ( φ n )) n ∈ N ⊆ B X W ρ . Thus, by compactness of the embedding X W → W , ∞ (0 , T ; L ∞ (Ω)), there ex-ists a subsequence ( φ n ℓ ) ℓ ∈ N such that 1 − kφ n ℓ t converges to 1 − kφ t strongly in L ∞ (0 , T ; L ∞ (Ω)) and T ( φ n ℓ ) converges weakly* in X W to some ψ ∈ B X W ρ , whichby definition of B X W ρ satisfies the initial and homogeneous Dirichlet boundary con-ditions. It is readily checked that ψ also solves the PDE (4.1), which, by uniquenessin Theorem 3.1, implies ψ = T ( φ ). A subsequence-subsequence argument yieldsweak* convergence in X W of the whole sequence ( T ( φ n )) n ∈ N to T ( φ ).We can therefore conclude existence of a fixed point of T in B X W ρ from the gen-eral version of Schauder’s fixed-point theorem in locally convex topological spaces;see Ref 11, which we here quote for the convenience of the readerLet L be a locally convex topological linear space and K a compactconvex set in L . Let M ( K ) be the family of all closed convex (non-empty) subsets of K . Then for any upper semicontinuous point-to-set transformation f from K into M ( K ), there exists a point x ∈ K such that x ∈ f ( x ).We use this theorem with the single valued point-to-set relation (i.e., mapping) f = T , the weak*topology on X W , and K = B X W ρ .
5. Higher energy estimates
Due to the appearance of k f t k L L on the right-hand side of estimate (3.6) inTheorem 3.1, we cannot rely only on this estimate for the Kuznetsov-type JMGTequation (1.6) since f t = 2 ∇ φ · ∇ φ tt + 2 |∇ φ t | . Existence of solutions can still be based on Theorem 3.1, (case k∇ α k L ∞ L <α/C Ω H ,L with a τ -independent bound on the energy) because f is still in theright space. However, k f t k L L = 2 k∇ ψ · ∇ ψ tt + |∇ ψ t | k L L can only be shown to be bounded by √ τ , so it might be large as τ →
0. This doesnot matter for proving existence according to Theorem 3.1, but excludes a fixed-point argument for proving well-posedness of the nonlinear equation (1.6) in thissetting.To be able to take limits as τ →
0, we thus need higher order energy estimates. Inparticular, we need to derive τ -independent bounds on k ψ k L ∞ H which will enableus to estimate f = 2 ∇ ψ · ∇ ψ t in the required norms. We replace estimate (3.6) by Barbara Kaltenbacher, Vanja Nikoli´c an estimate on the auxiliary function z and complement this with a higher order inspace estimate on ψ .In order to carry out these new error estimates, we now turn our attention tostudying the equation (1.9), restated again here for convenience: τ ψ ttt + α ( x, t ) ψ tt − c ∆ ψ − b ∆ ψ t = f in Ω × (0 , T ) , (1.9)together with its equivalent reformulation (1.13) using (1.12), (1.10), i.e., τ z tt + γz t − b ∆ z − γ c b z + γ c b ψ = f in Ω × (0 , T ) , (1.13)where we recall that the auxiliary state is given by z = ψ t + c b ψ. (1.12)We assume that for some α > γ > α ( t ) ≥ α, γ ( t ) = α ( t ) − τ c b ≥ γ on Ω for a.e. t ∈ (0 , T ) . (5.1) Theorem 5.1.
Let c , b , and let T > . Assume that • α ∈ W , (0 , T ; H (Ω)) ∩ L ∞ (0 , T ; W , (Ω) ∩ L (0 , T ; H (Ω) , • f ∈ H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) , • ( ψ , ψ , ψ ) ∈ X K := H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) × H (Ω) ,and that the non-degeneracy condition (5.1) holds. Then there exists ¯ τ > suchthat for τ ∈ (0 , ¯ τ ) and sufficiently small k∇ γ k L ∞ L , there exists a unique solution ( ψ, z ) of the problem τ z tt + ( α − τ c b ) z t − b ∆ z − γ c b z + γ c b ψ = f in Ω × (0 , T ) ,z = ψ t + c b ψ in Ω × (0 , T ) ,ψ = 0 on ∂ Ω × (0 , T ) , ( ψ, ψ t , ψ tt ) = ( ψ , ψ , ψ ) in Ω × { } , (5.2) that satisfies ( ψ, z ) ∈ L ∞ (0 , T ; H (Ω)) × ( L ∞ (0 , T ; H (Ω)) ∩ W , ∞ (0; T ; H (Ω))) ;in other words, ψ ∈ X K := L ∞ (0 , T ; H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)) ∩ W , ∞ (0; T ; H (Ω)) . (5.3) Furthermore, there exists a C ( γ, T ) > , which does not depend on τ , such that k ψ k L ∞ H + k ψ t + c b ψ k L ∞ H + k ψ tt + c b ψ t k L H + τ k ψ tt + c b ψ t k L ∞ H ≤ C ( γ, T ) (cid:0) | ψ | H + | ψ | H + τ | ψ | H + k f k L H (cid:1) . (5.4)Note that while f ∈ H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) is required, in the right-hand side of the energy estimate only k f k L H , but not k f k H L appears. n the Jordan–Moore–Gibson–Thompson equation Proof. Step 1: Existence of a solution.
Theorem 3.1 implies existence of asolution ( ψ, z ) of (5.2) with regularity as stated in (3.5) and z ∈ L ∞ (0; T ; H (Ω) ∩ H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) . Note that z t inherits the homogeneous Dirichlet boundary conditions from ψ . Thusit only remains to establish the higher order energy estimates. For this purpose, wereturn to the Galerkin approximation (3.11) and define z n = ψ n + c b ψ n . Step 2: A priori estimates.
As in Section 3, our goal is to obtain a bound for ψ n that is uniform with respect to n . To this end, we test the spatially discretizedversion of our problem (5.2) with two test functions. The first energy identity.
Problem (3.11) can be equivalently rewritten as (cid:16) τ z ntt + γz nt − b ∆ z n − c b γz n + γ c b ψ n , φ (cid:17) L = ( f, φ ) L , for every φ ∈ V n pointwise a.e. in (0 , T ) ,z n = ψ nt + c b ψ n , ( ψ n (0) , ψ nt (0) , ψ ntt (0)) = ( ψ n , ψ n , ψ n ) , (5.5)where V n is defined as in the proof of Theorem 3.1, as the span of the first n eigenfunctions of the Laplacian with homogeneous Dirichlet boundary conditions.Multiplying the first equation in (5.5) by − ∆ z nt ∈ V n and integrating over Ω and(0 , t ) yields the energy identity τ |∇ z nt ( t ) | + Z t |√ γ ∇ z nt | d s + b | − ∆ z n ( t ) | = τ |∇ z nt (0) | − Z t ( z nt ∇ γ, ∇ z nt ) L d s + b | − ∆ z n (0) | − c b Z t ( γ ∇ z n , ∇ z nt ) L d s − c b Z t ( z n ∇ γ, ∇ z nt ) L d s + Z t ( ∇ f − γ c b ∇ ψ n − c b ψ n ∇ γ, ∇ z nt ) L d s − Z t h ν · ∇ z nt , f i H − / ( ∂ Ω) ,H / ( ∂ Ω) d s =: rhs ( t ) , (5.6)where we have skipped the argument ( s ) under the time integral for notationalsimplicity. To derive (5.6), we have used the identity( γz nt , − ∆ z nt ) L = |√ γ ∇ z nt | + ( z nt ∇ γ, ∇ z nt ) L . Furthermore, we have made use of − Z t ( γ c b z n , − ∆ z nt ) L d s = − Z t ( γ c b ∇ z n , ∇ z nt ) L d s − Z t ( z n ∇ γ c b , ∇ z nt ) L d s Barbara Kaltenbacher, Vanja Nikoli´c as well as the identity Z t ( f − γ c b ψ n , − ∆ z nt ) L d s = Z t ( ∇ f − γ c b ∇ ψ n − c b ψ n ∇ γ, ∇ z nt ) L d s − Z t h ν · ∇ z nt , f i H − / ( ∂ Ω) ,H / ( ∂ Ω) d s. The second energy identity.
Our aim is to obtain a bound on ψ in the H (Ω)norm. To this, end we test (3.11) with φ = ( − ∆) ψ n ∈ V n (due to the fact that ψ n is a linear combination of eigenfunctions of − ∆) which yields the second energyidentity c Z t |∇ ( − ∆) ψ n | d s + b |∇ ( − ∆) ψ n ( s ) | (cid:12)(cid:12)(cid:12) t = − τ ( ∇ ψ ntt ( s ) , ∇ ( − ∆) ψ n ( s )) L (cid:12)(cid:12)(cid:12) t + τ | − ∆ ψ nt ( s ) | (cid:12)(cid:12)(cid:12) t + Z t ( − ∆[ αψ nt ] , − ∆ ψ nt ) L d s − ( α ( s ) ∇ ψ nt ( s ) + ψ t ( s ) ∇ α ( s ) , ∇ ( − ∆) ψ n ( s )) L (cid:12)(cid:12)(cid:12) t + Z t ( ψ nt ∇ α t + α t ∇ ψ nt , ∇ ( − ∆) ψ n ) L d s + Z t ( ∇ f, ∇ ( − ∆) ψ n ) L d s − Z t h ν · ∇ ( − ∆) ψ n , f i H − / ( ∂ Ω) ,H / ( ∂ Ω) d s =: rhs ( t ) , (5.7)Above, we have made use of( ψ nttt , ( − ∆) ψ n ) L = ( ∇ ψ nttt , ∇ ( − ∆) ψ n ) L = dd t h ( −∇ ψ ntt , ∇ ( − ∆) ψ n ) L − | − ∆ ψ nt | i , and the fact that − ∆ ψ nt = 0 on ∂ Ω. Morever, we rewrote the α term as follows Z t ( αψ ntt , ( − ∆) ψ n ) L d s i.b.p. in space = Z t ( ∇ [[ αψ nt ] t − α t ψ t ] , ∇ ( − ∆) ψ n ) L d s i.b.p. in time = ( ∇ [ α ( s ) ψ nt ( s )] , ∇ ( − ∆) ψ n ( s )) L (cid:12)(cid:12)(cid:12) t − Z t ( ∇ [ αψ nt ] , ∇ ( − ∆) ψ nt ) L d s − Z t ( ∇ [ α t ψ nt ] , ∇ ( − ∆) ψ n ) L d s i.b.p. in space = ( ∇ [ α ( s ) ψ nt ( s )] , ∇ ( − ∆) ψ n ( s )) L (cid:12)(cid:12)(cid:12) t − Z t ( − ∆[ αψ nt ] , − ∆ ψ nt ) L d s − Z t ( ∇ [ α t ψ nt ] , ∇ ( − ∆) ψ n ) L d s, where we used again that − ∆ ψ nt = 0 on ∂ Ω. Note that under the assumptions madeon α , we have that ( αψ nt )( t ) ∈ H (Ω) ∩ H (Ω) for almost every t ∈ (0 , T ), since n the Jordan–Moore–Gibson–Thompson equation ψ nt ( t ) ∈ H (Ω) ∩ H (Ω).The left-hand sides of our two energy identities (5.6) and (5.7) can be estimatedfrom below by τ |∇ z nt ( t ) | + Z t |√ γ ∇ z nt | d s + b | − ∆ z n ( t ) | ≥ (cid:16) τ k∇ z nt k L ∞ t L + γ k∇ z nt k L t L + b k − ∆ z n k L ∞ t L (cid:17) , (5.8)and by c Z t |∇ ( − ∆) ψ n | d s + b |∇ ( − ∆) ψ n ( t ) | ≥ (cid:16) c k∇ ( − ∆) ψ n k L t L + b k∇ ( − ∆) ψ n k L ∞ t L (cid:17) , (5.9)respectively.We will consider the weighted sum (5.6) plus λ > lhs ( t ) = (cid:16) τ k∇ z nt k L ∞ t L + γ k∇ z nt k L t L + b k − ∆ z n k L ∞ t L (cid:17) + λ (cid:16) c k∇ ( − ∆) ψ n k L t L + b k∇ ( − ∆) ψ n k L ∞ t L (cid:17) . (5.10)It then remains to estimate the right-hand sides, rhs ( t ) and λ rhs ( t ). Estimates of the right-hand sides.
For estimating the right-hand sides in (5.6)and (5.7), we can then use the norms of z , z t , and ψ appearing in the lower bounds(5.8) and (5.9). Furthermore, we can employ the continuous embeddings (2.6) as wellas boundedness of ( − ∆) − : L (Ω) → H (Ω) ∩ H (Ω) and the Poincar´e-Friedrichsinequality (2.7). Additionally, we employ the identities ψ t = z − c b ψ, ψ tt = z t − c b z + c b ψ. To simplify the notation, we introduce two constants depending on the initialdata, C ( ψ , ψ , ψ ; τ ) = τ |∇ z t (0) | + b | − ∆ z (0) | = τ |∇ ψ + c b ∇ ψ | + b | − ∆ ψ − c b ∆ ψ | , as well as C ( ψ , ψ , ψ ; τ ) = b |∇ ( − ∆) ψ | + τ ( ∇ ψ , ∇ ( − ∆) ψ ) L − τ |− ∆ ψ | + ( α (0) ∇ ψ + ψ ∇ α (0) , ∇ ( − ∆) ψ ) L . By applying H¨older’s inequality and the trace theorem, we get for the right-hand Barbara Kaltenbacher, Vanja Nikoli´c side in (5.6), rhs ( t ) ≤ C ( ψ , ψ , ψ ; τ ) + k∇ z nt k L t L k z nt k L t L k∇ γ k L ∞ L + k∇ z nt k L t L k∇ z n k L t L c b k γ k L ∞ L + k∇ z nt k L t L k z n k L t L ∞ c b k∇ γ k L ∞ L + k∇ z nt k L t L (cid:16) k∇ f k L L + c b k γ k L ∞ L k∇ ψ n k L t L ∞ + c b k∇ γ k L ∞ L k ψ n k L t L ∞ (cid:17) + k ν · ∇ z nt k L t H − / ( ∂ Ω) k f k L H / ( ∂ Ω) , a.e. in time. We further obtain, by making use of Young’s inequality (3.16) and theembedding results, rhs ( t ) ≤ C ( ψ , ψ , ψ ; τ ) + C Ω H ,L k∇ γ k L ∞ L k∇ z nt k L t L + γ k∇ z nt k L t L + γ (cid:16) C Ω( − ∆) − C Ω H ,W , k − ∆ z n k L t L c b k γ k L ∞ L + C Ω( − ∆) − C Ω H ,L ∞ k − ∆ z n k L t L c b k∇ γ k L ∞ L + k∇ f k L L + c b ( k γ k L ∞ L + C P F k∇ γ k L ∞ L ) C Ω( − ∆) − C Ω H ,L ∞ k∇ ( − ∆) ψ n k L t L + ( C Ω tr ) C Ω P F k f k L H (cid:17) , (5.11)since for the Galerkin discretization by eigenfunctions of the Laplacian, we have( − ∆) ψ n ∈ H (Ω) ∩ H (Ω) for smooth Ω. All terms on the right-hand side exceptfor g rhs := C ( ψ , ψ , ψ ; τ ) + γ (cid:0) (1 + ( C Ω tr ) C Ω P F ) k f k L H + C Ω( − ∆) − C Ω H ,W , k − ∆ z n k L t L c b k γ k L ∞ L + c b k γ k L ∞ L C Ω( − ∆) − C Ω H ,L ∞ k∇ ( − ∆) ψ n k L t L ) (5.12)can be absorbed into the left-hand side (5.10) by making k∇ γ k L ∞ L small. Theright-hand side in (5.7) can be estimated as follows rhs ( t ) ≤ C ( ψ , ψ , ψ ; τ ) + τ k∇ ( − ∆) ψ n k L ∞ t L k∇ z nt − c b ∇ z n + c b ∇ ψ n k L ∞ t L + τ k − ∆ z n + c b ∆ ψ n k L ∞ t L + k − ∆ z n + c b ∆ ψ n k L ∞ t L (cid:16) k − ∆ z n + c b ∆ ψ n k L ∞ t L k α k L L ∞ + 2 k∇ z n − c b ∇ ψ n k L ∞ t L k∇ α k L L + k z n − c b ψ n k L ∞ t L ∞ k − ∆ α k L L (cid:17) + k∇ ( − ∆) ψ n k L ∞ t L (cid:16) k∇ f k L L + k∇ z n − c b ∇ ψ n k L ∞ t L ( k α k L ∞ L + k α t k L L + k z n − c b ψ n k L ∞ t L ∞ ( k∇ α k L ∞ L + k∇ α t k L L ) + ( C Ω tr ) C Ω P F k f k L H (cid:17) , n the Jordan–Moore–Gibson–Thompson equation where we have estimated the boundary term by means of the trace theorem − Z t h ν · ∇ ( − ∆) ψ n , f i H − / ,H / d s ≤ ( C Ω tr ) k ( − ∆) ψ n k L ∞ t H k f k L H ≤ ( C Ω tr ) C Ω P F k∇ ( − ∆) ψ n k L ∞ t L k f k L H . We further have for λ > λ · rhs ( t ) ≤ λ C ( ψ , ψ , ψ ; τ ) + λ b k∇ ( − ∆) ψ n k L ∞ t L + λ τb τ k∇ z nt − c b ∇ z n + c b ∇ ψ n k L ∞ t L + λ τ k − ∆ z n + c b ∆ ψ n k L ∞ t L + λ k − ∆ z n + c b ∆ ψ n k L ∞ t L (cid:16) k α k L L ∞ + 2 C Ω( − ∆) − C Ω H ,W , k∇ α k L L + C Ω( − ∆) − C Ω H ,L ∞ k − ∆ α k L L (cid:17) + λ b k∇ ( − ∆) ψ n k L ∞ t L + λ b (cid:16) (cid:0) C Ω tr (cid:1) C Ω P F (cid:17) k f k L H + λ k − ∆ z n − c b ( − ∆) ψ n k L ∞ t L b (cid:16) C Ω( − ∆) − (cid:17) (cid:16) C Ω H ,W , ( k α k L ∞ L + k α t k L L ) + C Ω H ,L ∞ ( k∇ α k L ∞ L + k∇ α t k L L ) (cid:17) , (5.13)where by making τ and λ small, all terms except for those containing the initialdata and the inhomogeneity, λ · g rhs := λC ( ψ , ψ , ψ ; τ ) + λ b (cid:0) C Ω tr ) C Ω P F (cid:1) k f k L H (5.14)can be absorbed into the left-hand side given in (5.10).We now combine the energy estimates obtained from (5.6), and λ times (5.7)with a small constant λ > g rhs and λ g rhs remain on the right-hand side, cf. (5.10), (5.12), (5.14). Therewith, we end up withan inequality of the form η ( t ) ≤ C (cid:18)Z t η ( s ) d s + k ψ k H + k ψ k H + τ k ψ k H + k f k L H (cid:19) , (5.15)for η ( t ) = (cid:16) τ k∇ z nt k L ∞ (0 ,t ; L ) + γ k∇ z nt k L (0 ,t ; L ) + b k − ∆ z n k L ∞ (0 ,t ; L ) (cid:17) + λ (cid:16) c k∇ ( − ∆) ψ n k L (0 ,t ; L )) + b k∇ ( − ∆) ψ n k L ∞ (0 ,t ; L ) (cid:17) , to which we employ Gronwall’s lemma.To obtain a uniform bound on the full H (Ω) norm of ψ n , we combine the |∇ ( − ∆) ψ n | L term with | − ∆ z n | L and the fact that ψ n ( x, t ) = e − ( c /b ) t ψ ( x ) + Z t e − ( c /b )( t − s ) z n ( x, s ) Barbara Kaltenbacher, Vanja Nikoli´c for t ∈ (0 , T ). In this way, we have | ψ n ( t ) | H (Ω) ≤ C Ω( − ∆) − (cid:16) |∇ ( − ∆) ψ n ( t ) | L + | ( − ∆) ψ n ( t ) | L (cid:17) ≤ C Ω( − ∆) − (cid:16) |∇ ( − ∆) ψ n ( t ) | L + | e − ( c /b ) t ( − ∆) ψ | L + (cid:12)(cid:12)(cid:12)(cid:12)Z t e − ( c /b )( t − s ) ( − ∆) z ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12) L (cid:17) for a.e. t ∈ (0 , T ). Altogether, we get the estimate k ψ n k L ∞ H + k ψ nt + c b ψ n k L ∞ H + k ψ tt + c b ψ nt k L H + τ k ψ ntt + c b ψ nt k L ∞ H ≤ C ( γ, T ) (cid:0) | ψ | H + | ψ | H + τ | ψ | H + k f k L H (cid:1) , (5.16)with a constant C ( γ, T ) > τ , provided k∇ γ k L ∞ L is sufficientlysmall. Step 3: Passing to the limit.
On account of estimate (5.16) and the Banach-Alaoglu theorem, we know that there exists a subsequence, denoted again by { ψ n } n ∈ N , and a function ˜ ψ such that ψ n − ⇀ ˜ ψ weakly- ⋆ in L ∞ (0 , T ; H (Ω) ∩ H (Ω)) ,ψ nt − ⇀ ˜ ψ t weakly- ⋆ in L ∞ (0 , T ; H (Ω) ∩ H (Ω)) ,ψ nt − ⇀ ˜ ψ t weakly in L (0 , T ; H (Ω)) ,ψ ntt − ⇀ ˜ ψ tt weakly in L (0 , T ; H (Ω)) ,ψ ntt − ⇀ ˜ ψ tt weakly- ⋆ in L ∞ (0 , T ; H (Ω)) . By uniqueness of limits ˜ ψ has to coincide with the solution ψ according to Theorem3.1, which thus satisfies (5.4).
6. Well-posedness for the nonlinear Kuznetsov-type wave equationfor τ >
We next intend to employ the Banach fixed-point theorem to prove well-posednessfor equation (1.6). To this end, we introduce the operator T that maps φ to asolution ψ of τ ψ ttt + (1 − kφ t ) ψ tt − c ∆ ψ − b ∆ ψ t = 2 ∇ φ · ∇ φ t , on some ball B X K ρ = n ψ ∈ X K : ψ (0) = ψ , ψ t (0) = ψ , ψ tt (0) = ψ , and k ψ k X K := τ k z t k L ∞ H + k z t k L H + k z k L ∞ H + k ψ k L ∞ H ≤ ρ , for z = ψ t + c b ψ o (6.1) n the Jordan–Moore–Gibson–Thompson equation in the space X K , defined in (2.4). Thus, for establishing T as a self-mappingon B X K ρ , it is crucial to prove that α = 1 − kφ t and f = 2 ∇ φ · ∇ φ t are in W , (0 , T ; H (Ω)) ∩ L ∞ (0 , T ; W , (Ω) ∩ L (0 , T ; H (Ω) and L (0 , T ; H (Ω)), re-spectively, and that the derivatives of α are small when φ ∈ B X K ρ .Concerning non-degeneracy, we assume that τ ∈ (0 , ¯ τ ] with ¯ τ < bc so that γ ∗ := 1 − ¯ τ c b > . (6.2)Therefore, keeping k γ − γ ∗ k L ∞ L ∞ = k α − k L ∞ L ∞ = k k φ t k L ∞ L ∞ ≤ m (6.3)with m < γ ∗ allows to choose γ := 1 − ¯ τ c b − m > , α := 1 − m > τ , which will also be important for the considerations inSection 7. Thus we also need to verify that (6.3) follows from φ ∈ B X K ρ .To additionally obtain contractivity, based on the fact that the difference ˆ ψ = ψ − ψ = T ( φ ) − T ( φ ) solves τ ˆ ψ ttt + (1 − kφ t ) ˆ ψ tt − c ∆ ˆ ψ − b ∆ ˆ ψ t = k ˆ φ t ψ tt + 2 ∇ ˆ φ · ∇ φ t + 2 ∇ φ · ∇ ˆ φ t , (6.5)with homogeneous initial and boundary conditions (where ˆ φ = φ − φ ), we needto prove that α = 1 − kφ t and f = k ˆ φ t ψ tt + 2 ∇ ˆ φ · ∇ φ t + 2 ∇ φ · ∇ ˆ φ t are in W , (0 , T ; H (Ω)) ∩ L ∞ (0 , T ; W , (Ω) ∩ L (0 , T ; H (Ω) and L (0 , T ; H (Ω)), andthat the derivatives of α are small, provided φ , φ ∈ B X K ρ . Moreover, k f k L H needs to be estimated by a multiple of k ˆ φ k X K with a small factor. Theorem 6.1.
Let c , b , T > , k ∈ R . Then there exist ¯ τ , ρ > , ρ > such thatfor all ( ψ , ψ , ψ ) ∈ X K = H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) × H (Ω) satisfying k ψ k H (Ω) + k ψ k H (Ω) + τ k ψ k H (Ω) ≤ ρ , (6.6) and all τ ∈ (0 , ¯ τ ) , there exists a unique solution ψ ∈ X K of τ ψ ttt + ψ tt − c ∆ ψ − b ∆ ψ t = (cid:0) k ( ψ t ) + |∇ ψ | (cid:1) t in Ω × (0 , T ) ,ψ = 0 on ∂ Ω × (0 , T ) , ( ψ, ψ t , ψ tt ) = ( ψ , ψ , ψ ) in Ω × { } , (6.7) which satisfies the estimate τ k z t k L ∞ H + k z t k L H + k z k L ∞ H + k ψ k L ∞ H ≤ ρ . Barbara Kaltenbacher, Vanja Nikoli´c
Proof.
We first prove that T is a self-mapping on B X K ρ . For this purpose, weestimate α and f , assuming that φ ∈ B X K ρ and abbreviating w = φ t + c b φ : k∇ α t k L L = | k | k∇ w t − c b ∇ w + c b ∇ φ k L L ≤ | k | √ T (1 + c b √ T + c b ) ρ, k∇ α k L ∞ L = | k | k∇ w − c b ∇ φ k L ∞ L ≤ | k | C Ω H → W , (1 + c b ) ρ, k − ∆ α k L L = | k | k − ∆ w + c b ∆ φ k L L ≤ | k | √ T (1 + c b ) ρ, k γ − γ ∗ k L ∞ L ∞ = k α − k L ∞ L ∞ = | k | k w − c b φ k L ∞ L ∞ ≤ | k | C Ω H → L ∞ (1 + c b ) ρ. (6.8)Moreover, we find that k∇ f k L L = 2 (cid:13)(cid:13) ∇ φ ∇ φ t + ∇ φ t ∇ φ (cid:13)(cid:13) L L ≤ (cid:16) k∇ φ k L ∞ L k∇ w − c b ∇ φ k L L + k∇ w − c b ∇ φ k L L k∇ φ k L ∞ L ∞ (cid:17) ≤ C Ω H → W , C Ω H ,W , + C Ω H ,W , ∞ (cid:16) c b (cid:17) ρ =: C f ρ . Therefore, energy estimate (5.4) yields k ψ k X K ≤ C ( γ, T ) (cid:16) C f ρ + k ψ k H (Ω) + k ψ k H (Ω) + τ k ψ k H (Ω) (cid:17) ≤ ρ , provided that the initial data are small in the sense of (6.6) with ρ < C ( γ, T ) C f ,ρ ≤ min ( p − C ( γ, T ) C f ρ C ( γ, T ) C f , mkC Ω H → L ∞ (1 + c /b ) ) , (6.9)which implies that T maps B X K ρ into itself.For proving contractivity of T , we estimate α analogously to (6.8), and withabbreviations ˆ w := ˆ φ t + c b ˆ φ , w i := φ i t + c b φ i , z i := ψ i t + c b ψ i , i ∈ { , } . We have k∇ f k L L ≤ | k | (cid:16) k∇ ˆ w − c b ∇ ˆ φ k L ∞ L k z t − c b z + c b ψ k L L + k ˆ w − c b ˆ φ k L ∞ L ∞ k∇ z t − c b ∇ z + c b ∇ ψ k L L (cid:17) + 2 (cid:16) k∇ ˆ φ k L ∞ L k∇ w − c b ∇ φ k L L + k∇ ˆ w − c b ∇ ˆ φ k L L k∇ φ k L ∞ L ∞ (cid:17) + 2 (cid:16) k∇ φ k L ∞ L k∇ ˆ w − c b ∇ ˆ φ k L L + k∇ w − c b ∇ φ k L L k∇ ˆ φ k L ∞ L ∞ (cid:17) . n the Jordan–Moore–Gibson–Thompson equation From here it folllows that k∇ f k L L ≤ ( | k |k ψ k X K + 2 k φ k X K + 2 k φ k X K ) (cid:0) C Ω H → W , C Ω H → L + C Ω H → L ∞ (cid:1) × (cid:16) c b + c b (cid:17) k ˆ φ k X K ≤ ˆ C f ρ k ˆ φ k X K . By applying estimate (5.4) to equation (6.5) with homogeneous initial conditions,we obtain k ˆ ψ k X K ≤ p C ( γ, T ) ˆ C f ρ k ˆ φ k X K , which after possibly decreasing ρ yields contractivity.Since B X K ρ is closed, we can make use of Banach’s contraction principle toconclude existence and uniqueness of a solution. Remark 6.1 (On global well-posedness).
Note that C ( γ, T ) in (5.4) dependson the final time due to the use of Gronwall’s inequality. We do not expect thatglobal in time well-posedness can be proven in the nonlinear case due to the factthat we must deal with a quadratic nonlinearity and only have weak damping inthe equation for z .
7. Singular limit for vanishing relaxation time
We next focus on proving a limiting result for equations (1.6) and (1.8) as τ → b = δ + τ c and that the norms on the spaces X W and X K , defined in(2.4), depend on τ , whereas the radius ρ of the balls (4.2) and (6.1) is independentof τ .As already stated in the notational preliminaries (2.5), we denote by k ·k ¯ X W and k · k ¯ X K the respective τ -independent part of the norms defined in (4.2) and (6.1), k ψ k X W := k ψ tt k L H + k ψ k W , ∞ H , k ψ k X K := k ψ tt + c b ψ t k L H + k ψ t + c b ψ k L ∞ H + k ψ k L ∞ H . Moreover, we recall the spaces for the initial data X W := H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) × H (Ω) ,X K := H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) × H (Ω) , the only difference being in the regularity of ψ . Therewith, we can formulate alimiting result for (1.6) and (1.8). Theorem 7.1.
Let c , b , T > , and k ∈ R . Then there exist ¯ τ , ρ > suchthat for all ( ψ , ψ , ψ ) ∈ X W , the family ( ψ τ ) τ ∈ (0 , ¯ τ ) of solutions to (4.5) accordingto Theorem 4.1 converges weakly* in ¯ X W to a solution ¯ ψ ∈ ¯ X W of (1.5) withhomogeneous Dirichlet boundary conditions (2.2) and initial conditions ¯ ψ (0) = ψ , ¯ ψ t (0) = ψ . Barbara Kaltenbacher, Vanja Nikoli´c
The statement remains valid with the equations (4.5) , (1.5) , the spaces X W , ¯ X W and Theorem 4.1 replaced by the equations (6.7) , (1.4) , the spaces X K , ¯ X K and Theorem 6.1, respectively. Proof.
From the energy estimates in Theorems 4.1 (or 6.1), we have uniformboundedness of ( ψ τ ) τ ∈ (0 , ¯ τ ) in ¯ X W (or in ¯ X K ) and therefore existence of a weakly*¯ X W (or ¯ X K ) convergent sequence ( ψ ℓ ) ℓ ∈ N with τ ℓ ց
0. By compactness of embed-dings, this sequence also converges strongly in C (0 , T ; L (Ω)) ∩ C (0 , T ; W , (Ω)).Its limit ¯ ψ therefore lies in ¯ X W (or in ¯ X K ) and satisfies the initial conditions¯ ψ (0) = ψ , ¯ ψ t (0) = ψ .To prove that ¯ ψ also satisfies the respective PDEs, we test with arbitrary func-tions v ∈ C ∞ (0 , T ; C ∞ (Ω)) and invoke the Fundamental Lemma of Calculus ofVariations. To this end, we introduce an abbreviation for the nonlinear term in therespective equations, namely N ( ψ ) = ( β a c ( ψ t ) for (1.5), (1.8) c B A ( ψ t ) + |∇ ψ | for (1.4), (1.6),= k ( ψ t ) + σ |∇ ψ | with σ = ( . Now the τ -dependent and the limiting equation can be rewritten in both Westerveltand Kuznetsov cases as τ ψ ttt + ψ tt − c ∆ ψ − ( δ + τ c )∆ ψ t − ( N ( ψ )) t = 0 (7.1)and ψ tt − c ∆ ψ − δ ∆ ψ t − ( N ( ψ )) t = 0 , (7.2)respectively.Note that by the regularity inherent in the spaces X W , X K and ¯ X W , ¯ X K , cf.(2.4), ψ τ satisfies equation (7.1) in L (0; T ; L (Ω)). Inserting ¯ ψ into the left-handside of (7.2) yields an L (0; T ; L (Ω)) function. Therewith, we get, for ˆ ψ ℓ := ¯ ψ − ψ ℓ and any v ∈ C ∞ (0 , T ; C ∞ (Ω)) that Z T Z Ω (cid:16) ¯ ψ tt − c ∆ ¯ ψ − δ ∆ ¯ ψ t − N ( ¯ ψ ) t (cid:17) v d x d t = Z T Z Ω (cid:16) ˆ ψ ℓ tt − c ∆ ˆ ψ ℓ − δ ∆ ˆ ψ ℓ t − ( N ( ¯ ψ ) t − N ( ψ ℓ )) t − τ ℓ ψ ℓttt − τ ℓ c ∆ ψ ℓ t (cid:17) v d x d t = I − II − III . Above, we have thatI = Z T Z Ω (cid:16) ˆ ψ ℓ tt − c ∆ ˆ ψ ℓ − δ ∆ ˆ ψ ℓ t (cid:17) v d x d t → ℓ → ∞ n the Jordan–Moore–Gibson–Thompson equation due to the weak* convergence to zero of ˆ ψ ℓ in ¯ X W (or ¯ X K ). Moreover,II = Z T Z Ω (cid:16) ( N ( ¯ ψ ) − N ( ψ ℓ )) (cid:17) t v d x d t = − Z T Z Ω (cid:16) ( N ( ¯ ψ ) − N ( ψ ℓ )) (cid:17) v t d x d t = − Z T Z Ω (cid:16) k ( ¯ ψ t + ψ ℓt ) ˆ ψ ℓ t + σ ( ∇ ¯ ψ + ∇ ψ ℓ ) · ∇ ˆ ψ ℓ (cid:17) v t d x d t → ℓ → ∞ due to the boundedness of ( ψ ℓ ) ℓ ∈ N in ¯ X W (or ¯ X K ) by ρ , and the strong convergenceto zero of ˆ ψ ℓ in C (0 , T ; L (Ω)) ∩ C (0 , T ; W , (Ω)). Finally,III = τ ℓ Z T Z Ω (cid:16) ψ ℓttt + c ∆ ψ ℓ t (cid:17) v d x d t = τ ℓ Z T Z Ω (cid:16) ψ ℓtt + c ∆ ψ ℓ (cid:17) v t d x d t → ℓ → ∞ due to the boundedness of ( ψ ℓ ) ℓ ∈ N in ¯ X W (or ¯ X K ), and τ ℓ → ψ τ ) τ ∈ (0 , ¯ τ ) . Remark 7.1 (On compatibility conditions).
Note that, in contrast to Ref. 21,no compatibility condition on ψ is needed, since no continuity of the limit ¯ ψ tt withrespect to time arises and in the used energy estimates the ψ term vanishes as τ → Remark 7.2 (On strong convergence).
We could look directly at the equationsolved by the difference ˆ ψ = ψ τ − ¯ ψ of solutions to the JMGT and the Westerveltequation: ˆ ψ tt − c ∆ ˆ ψ − δ ∆ ˆ ψ t − k (( ψ τ + ¯ ψ ) ˆ ψ ) t = − τ ψ τttt − τ c ∆ ψ τt . (7.3)However, showing that ˆ ψ tends to zero as τ → τ ψ τttt : we only know this it is bounded by ρ in L (0 , T ; L (Ω)) according to estimate (4.6), but not that it tends to zero. Ananalogous argument can be made for the Kuznetsov-type JMGT equation. Comparison to the regularity results in the literature
We note that Theorem 7.1 also contains a regularity result on the solutions ¯ ψ ∈ ¯ X W and ¯ ψ ∈ ¯ X K of the Westervelt (1.5) and the Kuznetsov (1.4) equations withhomogeneous Dirichlet boundary conditions (2.2) and initial conditions ¯ ψ (0) = ψ ∈ H (Ω) ∩ H (Ω) or H (Ω), ¯ ψ t (0) = ψ ∈ H (Ω).By comparing this regularity with the regularity results on the Westervelt equa-tion from Refs. 17, 35 and with those for the Kuznetsov equation from Ref. 18, 36, Barbara Kaltenbacher, Vanja Nikoli´c
37, and noting that in Refs. 17, 35, 18, 36, u is the acoustic pressure, i.e., relatedto ψ by u = ̺ ψ t , we get • Westervelt equation: – Ref. 17: ( u , u ) ∈ ( H (Ω) ∩ H (Ω)) × H (Ω)and additionally (1 − ku ) − [ c ∆ u + b ∆ u + ku ] ∈ L (Ω) ⇒ u ∈ C (0 , T ; L (Ω)) ∩ H (0 , T ; H (Ω)) ∩ C (0 , T ; H (Ω)); – Ref. 35: ( u , u ) ∈ ( H (Ω) ∩ H (Ω)) × H (Ω) ⇒ u ∈ H (0 , T ; L (Ω)) ∩ H (0 , T ; H (Ω) ∩ H (Ω)); – here: ( ψ , ψ ) ∈ (cid:0) H (Ω) ∩ H (Ω) (cid:1) and additionally (1 − kψ ) − [ c ∆ ψ + b ∆ ψ ] ∈ H (Ω) ⇒ u ∈ H (0 , T ; H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)). • Kuznetsov equation: – Ref. 37: ( ψ , ψ ) ∈ H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) ⇒ ψ ∈ C (0 , T ; H (Ω) ∩ H (Ω)) ∩ H (0 , T ; H (Ω)); – Ref. 18: ( u , u ) ∈ ( H (Ω) ∩ H (Ω)) × H (Ω)and additionally (1 − ku ) − [ c ∆ u + b ∆ u + ku + 2 |∇ u | ] ∈ L (Ω) ⇒ u ∈ C (0 , T ; L (Ω)) ∩ H (0 , T ; H (Ω)) ∩ C (0 , T ; H (Ω)); – Ref. 36: ( u , u ) ∈ ( H (Ω) ∩ H (Ω)) × H (Ω) ⇒ u ∈ H / (0 , T ; H (Ω)) ∩ H (0 , T ; H (Ω)); – here: ( ψ , ψ ) ∈ H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω)and additionally (1 − kψ ) − [ c ∆ ψ + b ∆ ψ + ∇ ψ · ∇ ψ ] ∈ H (Ω) ⇒ u ∈ H (0 , T ; H (Ω)) ∩ W , ∞ (0 , T ; H (Ω)) ∩ L ∞ (0 , T ; H (Ω)).We point out that these works also contain results on global in time existenceand exponential decay of solutions, as well as, in case of Refs. 5, 37, on theCauchy problem, and, in case of Refs. 35, 36, in general, non-Hilbert L p (Ω) and W s,p (Ω) spaces. Moreover, we wish to point to Ref. 7, where local in time well-posedness of a class of quasilinear wave equations without strong damping wasshown, which also comprises the Westervelt equation with δ = 0. It yields theregularity u ∈ C (0 , T ; H (Ω)) ∩ C (0 , T ; H (Ω)) ∩ C (0 , T ; H (Ω)) for initial data( u , u ) ∈ H (Ω) ∩ H (Ω) × H (Ω) ∩ H (Ω) with ∆ u | ∂ Ω = 0. n the Jordan–Moore–Gibson–Thompson equation
8. Numerical results
As an illustration of our theoretical findings, we solve and compare numericallyequations (1.5) and (1.8) in a one-dimensional channel geometry. For the medium,we choose water with parameters c = 1500 m / s , δ = 6 · − m / s , ρ = 1000 kg / m , B/A = 5;cf. Chapter 5 in Ref. 22. Recall that b = δ + τ c ; the choice of the relaxationparameter τ is given below. Discretization in space is performed by employing B-splines as basis functions within the framework of Isogeometric Analysis (IGA);see Refs. 2, 15. For a detailed insight into the application of Isogeometric Analysisin nonlinear acoustics, we refer to Refs. 13, 38. We use quadratic basis functionswith the maximum C global regularity and have 251 degrees of freedom for thechannel length l = 0 . − .After discretizing in space, we end up with a semi-discrete matrix equation andproceed with a time-stepping scheme. For the Westervelt equation (1.8), we employthe standard Newmark relations39 for second-order equations: ψ n +1 = ψ n + ∆ t ˙ ψ n + (∆ t ) (cid:16) (1 − β )¨ ψ n + 2 β ¨ ψ n +1 (cid:17) , ˙ ψ n +1 = ˙ ψ n + ∆ t (cid:16) (1 − γ )¨ ψ n + γ ¨ ψ n +1 (cid:17) , (8.1)realized through a predictor-corrector scheme analogously to Algorithm 1 in Ref. 38.In (8.1), ∆ t denotes the time step size. The vectors ψ n , ˙ ψ n , and ¨ ψ n denote the dis-crete acoustic potential, its first time derivative, and its second time derivative,respectively, at the time step n .For the Jordan–Moore–Gibson–Thompson equation with Westervelt-type non-linearity (1.8), we use an extension of the Newmark relations to third-order modelssimilar to the one employed in Appendix B.2 of Ref. 12: ψ n +1 = ψ n + ∆ t ˙ ψ n + (∆ t )
2! ¨ ψ n + (∆ t ) (cid:16) (1 − β )... ψ n + 6 β ... ψ n +1 (cid:17) , ˙ ψ n +1 = ˙ ψ n + ∆ t ¨ ψ n + (∆ t ) (cid:16) (1 − γ )... ψ n + 2 γ ... ψ n +1 (cid:17) , ¨ ψ n +1 = ¨ ψ n + ∆ t (cid:16) (1 − η )... ψ n + η ... ψ n +1 (cid:17) . (8.2)The average acceleration scheme corresponds to taking the Newmark parame-ters ( β, γ ) = (1 / , /
2) in (8.1) for the Westervelt equation and ( β, γ, η ) =(1 / , / , /
2) in (8.2) for the Jordan–Moore–Gibson–Thompson equation (1.8),which is what we use in all the experiments.We set the initial conditions to( ψ , ψ , ψ ) = (cid:18) , A exp (cid:18) − ( x − . σ (cid:19) , (cid:19) , (8.3) Barbara Kaltenbacher, Vanja Nikoli´c with A = 8 · m / s and σ = 0 .
01, meaning that we normalize potential (whichis determined by ~v = −∇ ψ only up to a constant) such that it vanishes at t = 0,drive the system by an inital pressure (based on the idetity ρψ t = p ) concentratedat x = 0 .
1, and assume vanishing initial acceleration. Discretization in time isperformed with 800 time steps for the final time T = 45 µ s. The spatial and tem-poral refinement always remain the same for both equations and different values ofthe relaxation time τ . All the numerical results are obtained with the help of theGeoPDEs package in MATLAB46.Figure 1 displays on the left side snapshots of the pressure wave u = ̺ψ t ob-tained by employing equation (1.8) with the relaxation time set to τ = 0 . µ s. Weobserve the nonlinear steepening of the wave as it propagates. On the right, wesee how the pressure profile changes with decreasing relaxation time. The pressurewave for τ = 0 µ s is computed by solving the Westervelt equation. · − · − .
12 0 .
16 0 . x p r e ss u r e [ M P a ] τ = 0 . µ s t = 0 s t = 16 . µ s t = 33 . µ s t = 45 µ s 0 .
14 0 .
16 0 .
18 0 . x p r e ss u r e [ M P a ] t = 45 µ s τ = 1 µ s τ = 0 . µ s τ = 0 µ s Fig. 1. (left)
Snapshots of the pressure u = ̺ψ t for a fixed relaxation time τ = 0 . µ s (right) Pressure wave for different relaxation parameters τ at final time. To further illustrate the results from Section 7, we solve equation (1.8) withthe relaxation time varying over τ ∈ [10 − , µs and compute the difference to thesolution of the Westervelt equation (1.5). We plot the relative errors in the ¯ X W norm, defined in (2.5), and in the C ([0 , T ]; H (Ω)) norm:error ¯ X W ( τ ) = k ψ τ − ¯ ψ k ¯ X W k ¯ ψ k ¯ X W , error CH ( τ ) = k ψ τ − ¯ ψ k CH k ¯ ψ k CH ;see Figure 2. The numerical errors decrease with the parameter τ , in agreementwith the theoretical results of Theorem 7.1. Figure 2 even indicates a strongerresult, i.e., strong convergence in the ¯ X W norm. For τ = 10 − s, the errors amountto error CH ( τ ) ≈ . · − and error ¯ X W ( τ ) ≈ . · − . The error plots alsosuggest a lower rate of convergence with respect to τ in the ¯ X W norm. n the Jordan–Moore–Gibson–Thompson equation · − . . . . . . τ [s] e rr o r C H Error in C ([0 , T ]; H (Ω)) · − . . . . . . . τ [s] e rr o r ¯ X W Error in ¯ X W Fig. 2. Relative errors for varying relaxation time in (left) C ([0 , T ]; H (Ω)) and (right) ¯ X W . Acknowledgments
The second author acknowledges the funding provided by the Deutsche Forschungs-gemeinschaft under the grant number WO 671/11-1.
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