Uniqueness and Stability Results for Non-Linear Johnson-Segalman Viscoelasticity and Related Models

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.2111 DYNAMICAL SYSTEMS SERIES B Volume 19, Number 7, September 2014 pp. 2111–2132

UNIQUENESS AND STABILITY RESULTS FOR NON-LINEAR JOHNSON-SEGALMAN VISCOELASTICITY AND RELATED MODELS

Franca Franchi, Barbara Lazzari and Roberta Nibbi Department of Mathematics, University of Bologna Piazza di Porta S. Donato, 5 40126 Bologna, Italy

To Professor Mauro Fabrizio on his 70th birthday, in appreciation of his great ability of leading with generous friendship and an evergreen enthusiasm for research through the years.

Abstract. In this paper we have proved exponential asymptotic stability for the corotational incompressible diffusive Johnson-Segalman viscolelastic model and a simple decay result for the corotational incompressible hyperbolic Maxwell model. Moreover we have established continuous dependence and uniqueness results for the non-zero equilibrium solution. In the compressible case, we have proved a Höldercontinuous dependence theorem upon the initial data and body force for both models, whence follows a result of continuous dependence on the initial data and, therefore, uniqueness. For the Johnson-Segalman model we have also dealt with the case of negative elastic viscosities, corresponding to retardation effects. A comparison with other type of viscoelasticity, showing short memory elastic effects, is given.

1. Introduction. The flow behavior of non-Newtonian fluids has been a topic of recent extensive interest, due to their large number of physical, engineering and physiological applications; a vast amount of papers can be found in literature, which include analytical, numerical and experimental studies. This area of research has become increasingly relevant, owing to the advent and then the growth of the syn- thetic polymer industries and for the great range of non-Newtonian materials (blood, paints, food mixing, . . . ), playing an essential role in our everyday life. Viscoelastic response is often used as a probe in polymer rheology, as it is sensitive to the materials chemistry and microstructure. Comparative studies between various models are qualitatively carried on in the linear context, even if only a few polymers are linearly viscoelastic (see e.g. [25]). The Johnson-Segalman viscoelastic model (J-S model), firstly proposed to allow non-affine deformations [17], exhibits the ability to provide a useful description for diluted suspensions of a high-molecular-weight polymer into a largely viscous solvent, which may be generally considered a liquid. So, most recent literature is concerned only with incompressibile J-S fluids, within the single fluid approach, and often in presence of a magnetic field.

2010 Mathematics Subject Classification. Primary: 76A05, 35B30; Secondary: 76A10. Key words and phrases. Johnson-Segalman viscolestic fluids, Maxwell viscoelastic fluids, continuous dependence, stability, relaxation/retardation effects.

2111 2112 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI

An effective method for regularizing, which is natural for viscoelastic fluids, is just to add a viscosity term in the momentum equation; in this way, the polymer is not frozen in the fluid, but it quickly relaxes to its non-stretching configuration. Mean- while, this experimental property creates an interesting link with electromagnetic theories, according to the recent Christov’s revisitation [4] of a frame indifferent (continuum) formulation of electromagnetism. In view of this proposal, the diffusive J-S model allows for recovering a generalized resistive magnetohydrodynamics (MHD) theory, which circumvents the difficulty associated with the frozen-in magnetic field, and breaks the constraint of ideal MHD, as advocated by experimental observations. For a dilute polymer solution, described through the J-S model, the total (viscous) stress tensor Π is given by the sum of a Newtonian solvent part Πs and an elastic part Πe, due to the presence of the polymer solutions, which satisfy the following constitutive equations: 2µ Π = 2µ D + ζ − s (tr D) , s s s 3 I (1) DΠe 2µe τ + Πe = 2µeD + ζe − (tr D)I , Dt a 3 where D is the symmetric part of the velocity gradient L = ∇v, τ is a positive constant relaxation time and µs and µe are the constant solvent and elastic shear viscosities, while ζs and ζe are the constant solvent and elastic bulk viscosities. These parameters are supposed to meet the thermodynamic restrictions µs ≥ 0, µ = µe + µs ≥ 0, ζs ≥ 0, ζ = ζe + ζs ≥ 0 and, for a sufficiently diluted polymeric solution, it is experimentally shown that µs > µe and also µ > µe . DΠe The notation Dt a stands for the objective convected time derivative, which can be chosen from the one-parametric family of the Gordon-Schowalter derivatives (G-S derivative) [8], defined as follows DΠe = Π˙ e + ΠeW − WΠe + a(DΠe + ΠeD) , −1 ≤ a ≤ 1, (2) Dt a where the spin tensor W is the antisymmetric part of L, a is the dimensionless slip parameter, accounting for the slip between the molecular network and the continuum medium, and the superposed dot denotes the material time derivative. For a = ±1,2 leads to the upper/lower convected derivatives, whereas for a = 0 gives the corototational or Jaumann derivative. In the non-objective case the corotational (or upper/lower convected) derivative reduces to the material one and becomes a partial derivative in the linear limit. The non-linear invariant J-S model provides a non instaneous response character, which is quite embodied in the constitutive equation of rate type for the elastic stress tensor, with a single relaxation time and elastic viscosities. For constant rheological parameters, the J-S viscoelastic model is of Oldroyd B type; further- more, as already noted, the introduction of the regularizing Newtonian component is empirically motivated and allows for viscoelastic hysteresis, in analogy with other electromagnetic and thermal theories with lagging (see e.g. [5, 29]). The J-S model ranges from the classical Newtonian and Maxwell models, according as Πe or Πs be null; in turn, the simplified Maxwell model furnishes a fairly good description of the rheological behavior of purely polymer solutions or melts. Moreover, also for deformations occurring at time scales either longer or shorter than the relaxation UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2113 time, the viscoelastic fluid behaves either like a simple Newtonian fluid or diplays an elastic behavior. Indeed, a polymer liquid/fluid has a spectrum of relaxation times, so the relaxation time present in1 2 must be interpreted as the longest relaxation time exhibited by the real polymeric situation. Indeed, the J-S model, for constant parameters, has the additional advantage of being described through a constitutive equation of Jeffreys type, with a relaxation time τΠ = τ, a retardation time τD and a dimensionless retardation parameter r, given by the ratio between τD and τΠ. In linear/linearized (about the rest null state) viscoelasticity, for low relaxation/retardation times, this creates a special link with other dual-phase-lagging (DPL) thermal and electromagnetic theories, accounting for short elastic memory effects. So, besides the relaxation effects, another important mathematical and physical aspect, which has not yet received particular attention, is the retardation effect of the deformation rate tensor on the elastic stress tensor (corresponding to negative elastic viscosities or, equivalently, to suppose the dimensionless retardation parameter r > 1) on the stability analysis. It is empirically proven that, strongly negative viscosities would lead to instabilities and, consequently, to ill- posed problems. Meanwhile, one needs to realize that viscoelastic instabilities are of fundamental experimental importance in understanding the physics of complex fluids and also of practical relevance to material processing and fluid characteriza- tions (see e.g. [23] for a comparison between experimental data and scission model prediction). The incompressible corotational J-S model is very versatile and strong, and provides an almost complete mathematical and also numerical formulation, which is easily incorporated into boundary layer theories for engineering applications; further, also in presence of a magnetic field, it has been used to describe the non instantaneous response of blood, under physiological conditions, and the peristaltic motions in a flux tube, like an endoscope (see e.g.[13, 14]). In this connection, it is noteworhy to emphasize that the peristaltic pumps occur in many practical applications involving biomechanical systems, such as in dialysis and heart-lung-machines, but also in nuclear industry, and have been the object of many recent papers (see e.g. [15]). An interesting result about structural stability is given in [24]; the wave propagation properties have been recently investigated in [20,21] and, from a Lagrangian point of view, in [11]. In the framework of non-Newtonian fluids another interesting tool is the ability to control viscoelastic stresses in ascertaining characterizing properties of the materials; the controllability of the J-S model is investigated in [31] and in [26] for the Maxwell fluid. The interaction between (microscopic) elastic properties and the (macroscopic) fluid motion, in analogy with other theories with micro/macro scales effects accounting for lagging phenomena, gives not only a variety of interesting rheological phenomena, but also a powerful challenge in analysis and numerical simulation. Moreover, as in recent years there has been a remarkable interest in the employ- ment of nano particles to improve the mechanical, electrical, thermal and also flux tubes properties of polymers, the J-S model may furnish a proper description of nano fluids. We therefore believe that well-posedness results may be useful not only in connection with the longtime behavior of micro-macro scales models, but also for the stability of numerical routines. 2114 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI

The compressional effects of the viscoelasticity are relatively unaddressed in the recent literature, even if also in polymeric processing operations, pressure and flow rate may be large and these effects become so important in influencing the resulting phenomena (see e.g. [10] and references therein). From theories of meta continuum mechanics, it is well known that viscoelastic behavior is manifested by transverse shear oscillations: the experimental observation of torsional oscillations in white dwarfs points out the possibility of an elastic behavior joined to the viscous one in an astrophysical setting, like the interstellar medium. This suggests that stellar matter fits fairly well the characteristics of a non Newtonian compressible viscoelastic fluid, where the elastic/viscous properties join together; in this framework, the compressibility effects cannot be avoided. In this paper we are primarily concentrated on some analytical issues concerning the well-posedness of the non linear corotational diffusive J-S model, in combina- tion with the viscous-elastic splitting technique and the decomposition tool into the deviatoric and spherical components; a comparative study with the non linear hyperbolic Maxwell model, as a special limiting case, is also carried out. In the development of mathematical computations, the introduction of the non linear objective G-S convected time derivative raises a few troublesome problems, above all for the Maxwell model, but its presence is necessary as the elastic stress tensor is carried on by the fluid flows, which rotate, deformate and move in the flow itself. By following the same guidelines, through energy arguments, but sometimes with different mathematical strategies strictly related to the different rheological aspects, in order to overcome the difficulties due to a few challenging non linear terms, we investigate continuous dependence and uniqueness properties for both models and comment on the results. Besides the increasingly wide-spread use of the diffusive J-S and hyperbolic Maxwell models to describe phenomena occurring in polymeric fluids, we state in advance that our mathematical findings can be viewed as pertaining a viscoelastic liquid/fluid, that is not necessarily a polymer solution. The paper is organized as follows. In Section2 we set up a few preliminaries concerning notations, techniques, model equations and rheological aspects. In Section 3 we address the corotational J-S and Maxwell models, within the incompressibility constraint; by employing energy arguments, a detailed account of the non linear stability analysis, for null and non null basic solutions, is given. On the one hand, with regard to the relaxation effects, the diffusive J-S model relies on the regularizing presence of the solvent stress tensor which “helps” the stability and allows for recovering the exponential stability of the null solution, against the simple continuous dependence result obtained for the hyperbolic Maxwell model. On the other hand, for negative elastic shear viscosities, the solvent dissipative term dominates the elastic dissipative one, only for the retardation times 1 < r < 3/2; whenever this condition is violated, we get instabilities and consequently ill-posed problems. For the non null solution, we prove continuous dependence upon the initial data and then uniqueness results for both models. The solvent dissipative term still shows its predominant role also on the non linear rotational terms, due to the spin tensor, so for the J-S model the stability result easily follows. On the contrary, for the Maxwell model, instability problems and change of type may be strictly related to the introduction of an invariant convected time derivative of G-S type, like the corotational one; so, besides a stronger boundedness requirement, we have to manage a quite tricky tool in order to control the rotational terms, working with a generalized energy functional and writing down a new, but generally approximated, UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2115 equation for the vorticity vector, referred to as a new essential variable. Finally the last section is devoted to the compressible flows case: for the J-S model we establish a Höldercontinuous dependence theorem upon the initial data and body forces, which can be extended to comprising also the objective time derivative and the non strong retardation effects; for the non objective hyperbolic model, following the same procedure, by an additional boundedness requirement, we arrive at a similar Hölderstability result which, however, cannot be reasonably generalized to comprise the invariant time derivative. Our study is based on the eulerian formulation and we use the single fluid approach in dealing with the dilute polymer solution; the elastic polymer stress tensor is supposed symmetric (see e.g. [1]) as follows from almost all kinetic theories for polymeric liquids, the possible asymmetric contribution being negligibly small.

2. Some rheological aspects and mathematical formulation. As is usual in dealing with these rate type models, we may decompose the symmetric elastic stress tensor into the deviatoric (traceless) and the spherical (diagonal) parts, according to the well-known formula [1] 1 Π = hΠ i + σ , tr hΠ i = 0 , σ = tr Π . e e eI e e 3 e 1 Taking into account that the spin tensor W is traceless and D = hDi + 3 ∇·vI, equation1 2 can be decomposed as follows h ˙ i τ hΠei − W hΠei + hΠei W + hΠei = 2µe hDi 2 − τa[hhDi hΠ i + hΠ i hDii + 2σ hDi + ∇·v hΠ i] , e e e 3 e 2 τσ˙ + σ = ζ ∇·v − τa [ tr (hΠ i hDi) + 3σ ∇·v] . e e e 3 e e As we can see, for the upper/lower convected models, the two equations are related via the last terms, whereas for the corotational model we have just two separated equations for hΠei and σe respectively. Further the spherical part σe plays a role similar to a pressure. Therefore we define the new pressure variable pe = −σe whose evolution equation is 2 τp˙ + p = −ζ ∇ · v + τa [ tr (hΠ i hDi) − 3p ∇·v] . e e e 3 e e The advantage of this decomposition is that we now work with a tensorial equation for (five components) hΠei and a scalar equation for the elastic pressure pe. For the J-S model, the usefulness of the decomposition of the elastic stress tensor Πe is noteworthy, both for mathematical analysis and for numerical simulations (see e.g. [2]). The equations governing the general objective compressible J-S model are the following: ρ˙ + ρ∇ · v = 0 , 1 ρv˙ = −∇(p + p ) + ∇ · hΠ i + µ ∆v + ζ + µ ∇(∇ · v) + ρb , e e s s 3 s (3) h ˙ i τ hΠei − W hΠei + hΠei W + hΠei = 2µe hDi 2 − τa[hhDi hΠ i + hΠ i hDii + 2σ hDi + ∇·v hΠ i] , e e e 3 e 2116 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI 2 τp˙ + p = −ζ ∇ · v + τa [ tr (hΠ i hDi) − 3p ∇·v] , e e e 3 e e where ρ, p, v and b are the density, the pressure, the velocity and the external body force, respectively. For purely polymeric melts we take µs = ζs = 0 and the simplified system3 governs the classical objective compressible Maxwell model. As well known, this model has the desirable feature that it is hyperbolic, but it doesn’t include the dissipative terms due to the presence of the solvent stress. It is noteworthy that the general (compressible) objective J-S model can be described also by a constitutive equation of Jeffreys type (see e.g.[1]) DΠ DD 3ζ − 2µ D(∇·v) τΠ + Π = 2µ D + τD + ∇·v + τD I , Dt a Dt a 3 Dt a (4) where τΠ = τ is the relaxation parameter and τD is the retardation parameter. Moreover, in linear/linearized (about the rest null state) viscoelasticity, the partial time derivative replaces the non linear Gordon-Schowalter derivative and4 becomes 3ζ − 2µ τ Π + Π = 2µ [D + τ D ] + [∇·v + τ ∇·v ] Π t D t 3 D t I which, for small τΠ and τD, is equivalent to the following integral equation with fading memory t Z τ t−s D τ τΠΠ(t) =2µ τDD(t) + 1 − e Π D(s) ds −∞ τΠ t 3ζ − 2µ Z τ t−s D τ + τD∇·v(t) + 1 − e Π ∇·v(s) ds I . 3 −∞ τΠ The correlation between integral constitutive equations with rapidly fading memory and short memory effects described by Taylor expansions, can be found in other physical theories displaying memory effects: we think of generalized electromagnetic theories, accounting for a retarded/advanced behavior of the electric field on the current density vector (see e.g. [5]), or of non Fourier dual-phase-lag heat conduction theories (see e.g. [29, 30] and reference therein). By introducing the dimensionless retardation parameter r = τD , the model is often characterized by r and by the τΠ positive total, shear and bulk, viscosities µ and ζ , so that µs = rµ and µe = (1−r)µ, ζs = rζ and ζe = (1 − r)ζ. In most recent literature (see e.g. [16] and references therein), it is assumed (1 − r) > 0. The special limiting cases r = 0 and r = 1 correspond to the Maxwell and Newtonian models, respectively. Among the various Gordon-Schowalter derivatives, we choose to work with the corotational derivative. The motivation in twofold. Firstly, in the case of the null slip parameter (a = 0), the previous procedure allows for recovering two distinct constitutive equations of rate type, so that hΠei and pe can be computed separately. Further, when a = 0, besides the advection term, the only nonlinear effects of the objective Gordon-Schowalter derivatives are of rotational type, due to the presence of the spin tensor W and, from the identity (hΠei W − W hΠei) · hΠei = 0 (the dot denoting the scalar product in Lin(R3)), we always obtain the relation DΠe · hΠei = Π˙ e · hΠei . Dt a=0 Hence the corotational derivative preserves the invariant form of the entropy pro- duction (see e.g. [22]). UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2117

Throughout this paper, we consider evolutive problems in the space time domain Ω × [0, T ], where Ω is a bounded region in R3 with boundary Σ suﬃciently regular to allow applications of the divergence theorem. Moreover we adopt the usual notations k · k and (·, ·)2 to denote the norm and the inner product in L2(Ω), while the index t denotes the partial time derivative.

3. Decay results and uniqueness for the incompressible model. In this section we conﬁne our attention to the corotational incompressible J-S model in the space time domain Ω × [0, T ]. Under this hypothesis, system3 reduces to the form ∇ · v = 0 ,

v˙ = −∇(p + pe) + ∇ · hΠei + µs∆v + b , h ˙ i (5) τ hΠei − W hΠei + hΠei W + hΠei = 2µe hDi ,

τp˙e + pe = 0 , where, without any loss of generality, we put ρ = 1. In the sequel, we deal with classical solutions χi = (v, p, hΠei , pe) to5, namely 1 1 p is of class C with respect to the space variables; hΠei and pe are of class C with respect to both the space-time variables, whereas v is of class C2 with respect to the space variables, but of class C1 with respect to the time. We begin by supposing that the incompressible corotational ﬂuid has been un- dergoing a mechanically isolated canister ﬂow, that is to say, the classical no slip 0 0 condition v = 0 holds on the boundary Σ and (v, b )2 = 0, where b is the part of the body force which is not derivable from a potential.

3.1. Polymeric solution with positive solvent viscosity. The results of this subsection are obtained under the additional assumption µe > 0.

3.1.1. Null equilibrium solution. Consider now the equilibrium solution given by v = 0, Πe = 0, pe = 0, and introduce a positive definite energy measure, as follows τ E(t) = µ kv(t)k2 + k hΠ i (t)k2 + kp (t)k2 (6) e 2 e e for any perturbed solution (v, hΠei , pe), with assigned initial data (v0, hΠei0, pe0). We can establish the following asymptotic stability result. Theorem 3.1 (Exponential stability). For any canister flow of the corotational incompressible J-S fluid which is mechanically isolated, the energy E decays exponentially in time, when µs > 0.

Proof. Observe that the momentum equation5 2 may be recast in the so called Gromeka Lamb form [4] 1 v + ∇v2 + curl v × v = −∇(p + p ) + ∇ · hΠ i + µ ∆v + b (7) t 2 e e s and, by6, an evolution equation for E is easily found as d E(t) =2µ (v(t), v (t)) + τ (hΠ i (t), hΠ i (t)) + (p (t), p (t)) dt e t 2 e e t 2 e et 2 Z (8) 2 = − µe |v(t)| + 2p(t) + 2pe(t) v(t) · n ds Σ 2118 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI Z τ 2 2 − |∇ hΠei (t)| + |pe(t)| v(t) · n ds 2 Σ Z T + 2µe hΠei (t) + µsL (t) v(t) · n ds Σ 2 2 2 − k hΠei (t)k − 2µeµsk∇v(t)k − kpe(t)k , where, together with well-know identities, we have used the equality

(hΠei W − W hΠei , hΠei)2 = (hΠei hΠei , W)2 = 0 which holds in view of the symmetry of the tensor hΠei hΠei. Hence8, in view of the vanishing boundary conditions for v, easily yields d µ 2 E(t) ≤ − min s , E(t) , (9) dt λ τ where λ > 0 is the Poincar´econstant. If µs > 0, inequality9 assures the exponential decay of E. Hence the null velocity is exponentially asymptotically stable, as in the classical Newtonian context. As an immediate consequence, we can state the following uniqueness result. Corollary 1 (Uniqueness). Within the class of mechanically isolated canister ﬂows, the null state v = 0, Πe = 0, pe = 0 is the unique equilibrium solution, up to an arbitrary pressure term p depending on the time only.

Proof. If E(0) = 0 and b = 0, we have v = 0, Πe = 0, pe = 0, then5 2 yields ∇p = 0, whence the pressure p depends on the time only. Remark 1 (Growth result for the backwards problem). We can also consider the analogous canister ﬂow problem, but backward in time. In view of the formal time reversal t → −t, and carrying on the same strategy, we arrive at the energy evolution8, with a diﬀerent sign on the left hand side; hence, in view of homogeneous boundary conditions on v, we recover the growth result for the energy E.

Remark 2 (Purely polymeric solution melt: µs = 0). If we consider the classical corotatonial Maxwell model, we have µs = 0 and consequently from8 we derive d E(t) ≤ 0 . dt Whence we have a (non exponential) decay of E, which is suﬃcient for the uniqueness result stated in Corollary1. As we expect, the diﬀusive character of the J-S type viscoelasticy is stabilizing, which allows for recovering exponential stability, whereas for the hyperbolic Maxwell model we obtain only a continuous dependence result on the initial data. 3.1.2. Non zero equilibrium solution. For the rest of this subsection, we deal with non zero equilibrium solutions. We want to investigate the initial-boundary value problem Pi in Ω × [0, T [ described by5, together with the initial conditions

v(x, 0) = v0(x) , hΠei (x, 0) = hΠei0(x) , pe(x, 0) = pe0(x) , x ∈ Ω (10) and the boundary condition

v(x, t) = vΣ(x, t) , (x, t) ∈ Σ × [0, T [ , (11) where vΣ is a known function. UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2119

∗ Let χi and χi + χi be two classical solution to Pi, for the same initial and ∗ boundary data and body force. Then χi solves ∇ · v∗ = 0 , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ vt + (v · ∇)v + [(v + v ) · ∇]v = −∇(p + pe) + ∇ · hΠei + µs∆v , ∗ ∗ ∗ ∗ ∗ ∗ ∗ τ {hΠeit + [(v + v ) · ∇] hΠei + (v · ∇) hΠei − W hΠei + hΠei (W + W ) (12) ∗ ∗ ∗ ∗ ∗ + hΠei W − W (hΠei + hΠei)} + hΠei = 2µeD , ∗ ∗ ∗ ∗ ∗ τ [pet + ∇pe · (v + v) + ∇pe · v ] + pe = 0 , together with homogeneous initial and boundary conditions. At this point, we state in advance a remark: the terms which are difficult to ∗ ∗ ∗ ∗ estimate are hΠei W and W (hΠei + hΠei), related to the spin tensor W which, in turn, is related to the vorticity vector curl v∗. Indeed, the difficulty may be easily overcome for the J-S model, because of its diffusive character. On the contrary, for the hyperbolic Maxwell model (µs = 0) this in not possible anymore. The idea is to deal with the vorticity vector as a new variable, and hence it is necessary to introduce a generalized energy which involves the square of the L2-norm of curl v∗. To this aim, we need also to isolate an additional equation for the vorticity. We start with the definition of a generalized energy ∗ ∗ 2 τ ∗ 2 ∗ 2 ∗ 2 EM (t) = µekv (t)k + k hΠei (t)k + kpe(t)k + kcurl v (t)k 2 (13) = E∗(t) + kcurl v∗(t)k2 .

We can obtain the new equation for the vorticity, by taking the curl of5 2 and the curl div of5 3, and then by combining the resulting equations. In view of well-known identities, and neglecting lower order terms, owing to the smallness of the relaxation time, we ﬁnd

curl vt + curl (curl v × v) = −µcurl curl curl v + curl b . (14)

Equation 14 would be an exact vorticity equation if curl div hΠei = 0, which happens in some physical examples, for special ﬂows (see e.g. [3, 18]). Working with equation 14, of necessity, we add, besides 10 and 11, both an initial condition and a boundary condition on curl v. To begin, we lay down some preliminary a priori estimates. The ﬁrst Lemma is strictly related to equation 14, and its application is essential to prove continuous dependence and uniqueness results for the Maxwell model. Lemma 3.2. Let (v, curl v) and (v+v∗, curl v+curl v∗) be two classical solutions to the equation 14, satsfying the same initial and boundary conditions, for the same body force and suppose that, for some constant A < ∞, sup {|v|, |S|} ≤ A, (15) Ω×[0,T [ where S is the symmetric part of ∇[curl v], then we have the following vorticity inequality: d kcurl v∗(t)k2 ≤ A A kcurl v∗(t)k2 + 2kv∗(t)k2 , (16) dt 1 for a computable positive constant A1 depending on A. Proof. It is a simple matter to show that (v∗, curl v∗) obeys the perturbation equation in Ω × [0, T [ ∗ ∗ ∗ ∗ ∗ curl vt + curl [curl (v + v ) × v + curl v × v] = −µ curl curl curl v (17) 2120 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI together with homogeneous initial and boundary data. In the sequel we make use of the following identities curl curl curl v∗ · curl v∗ = ∇ · (curl curl v∗ × curl v∗) + (curl curl v∗)2 , curl (v∗ × v) · curl v∗ = −(curl v∗∇)v · curl v∗ + (v · ∇)curl v∗ · curl v∗ = −∇ · [(v · curl v∗)curl v∗ − v(curl v∗)2] + curl curl v∗ × curl v∗ · v , (18) curl [curl (v + v∗) × v∗] · curl v∗ = −∇ · {[v∗ · curl (v + v∗)]curl v∗} + 2v∗ · S curl v∗ . Performing the scalar product in L2 of 17 by curl v∗, taking in account 18 and applying the divergence theorem, we get 1 d kcurl v∗k2 = − µkcurl curl v∗k2 2 dt ∗ ∗ ∗ ∗ − (curl curl v × curl v , v)2 − (2v , S curl v )2 Z (19) − [µcurl curl v∗ × curl v∗ − (v · curl v∗)curl v∗] · nds Σ Z − {[(curl v∗)2v − [v∗ · curl (v + v∗)]curl v∗]}· nds . Σ Using the arithmetic-geometric mean (a-g mean) inequality and the requirement 15, we also derive ξ 1 |(curl curl v∗ × curl v∗, v) | ≤A kcurl curl v∗k2 + kcurl v∗k2 2 2 2ξ ∗ ∗ ∗ 2 ∗ 2 |(2v , S curl v )2| ≤A kv k + kcurl v k , 2µ for a suitable positive parameter ξ, to be chosen. Hence, by selecting ξ = A , in wiew of the vanishing boundary conditions, 19 yields the desired L2 vorticity A+4µ inequality 16, for A1 = 2µ . Lemma 3.3. Suppose that

sup | hΠei | ≤ A < ∞ , (20) Ω×[0,T [ then the following relations hold ∗ ∗ ∗ ∗ 2 ∗ 2 (hΠei (t)W (t) − W (t) hΠei (t), hΠei (t))2 ≤ A k hΠei (t)k + kcurl v (t)k , ∗ ∗ ∗ (hΠei (t)(W(t) + W (t)), hΠe(t)i)2 = 0 . (21) ∗ ∗ ∗ Proof. Let `1, `2 and `3 be the principal elastic stresses of the perturbation tensor ∗ ∗ hΠei, so that hΠei admits the spectral decomposition 3 ∗ X ∗ ∗ ∗ hΠei = ì ei ⊗ ei (22) i=1 ∗ ∗ in the usual indicial notation, ei being the principal direction corresponding to ì . In view of the definition of the inner product in Lin(R3), on account of 22, we have ∗ ∗ ∗ ∗ ∗ ∗ ∗ (hΠei W − W hΠei) · hΠei = tr (hΠei W hΠei − W hΠei hΠei) 3 3 X ∗ ∗ ∗ ∗ ∗ X ∗ ∗ ∗ ∗ = ì (hΠei W − W hΠei)ei · ei = 2 ì W ei · hΠei ei . i=1 i=1 UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2121

∗ ∗ 1 ∗ ∗ Next, as W ei = 2 curl v × ei , we get " 3 # ∗ ∗ ∗ X ∗2 ∗ 2 (hΠei W − W hΠei) · hΠei ≤ sup | hΠei | `i + |curl v | . (x,t)∈Ω×[0,T [ i=1

Hence 211 follows straightaway. As for 212, we have: 3 ∗ ∗ ∗ X ∗2 ∗ ∗ ∗ hΠei (W + W ) · hΠei = `i (W + W )ei · ei = 0 , i=1 ∗ ∗ 1 ∗ ∗ because (W + W )ei = 2 curl (v + v ) × ei .

2 ∗ Finally, we can isolate the L estimate for the elastic pressure perturbation pe Lemma 3.4. Suppose that

sup |∇pe| ≤ A < ∞ , (23) Ω×[0,T [ then we get d τ 2 τ k(p∗(t)k2 ≤ A βkv∗(t)k2 + − τk(p∗(t)k2 , (24) dt e β Aτ e for β > 0, at our disposal. Proof. Using the a-g mean inequality and taking account of 23, it follows τ 2 β τ(p∗, ∇p · v∗) ≤ A kp∗k2 + kv∗k2 . (25) e e 2 2β e 2

2 ∗ Performing the scalar product in L of 124 by pe, and using standard energy arguments, together with 25 and homogeneous boundary conditions, the above pressure estimate follows, directly.

In what follows, we need to restrict the classical solutions to a class of suitable bounded functions. If we denote by A a positive computable constant, we consider the (basic) solution χi in the following constraint set Mi

sup {| hΠei |, |v|, |∇ hΠei |, |∇pe|} ≤ A. (26) (x,t)∈Ω×[0,T [ We are able to prove continuous dependence theorems on initial data and hence uniqueness results for both viscoelastic models. To this aim, we carry on our com- ∗ ∗ parative study simultaneously, by using the energy functionals E and EM for the J-S and Maxwell models, respectively. ∗ To begin we observe that, taking into account 16, the evolution law for EM satisﬁes the following inequality

d ∗ d ∗ d ∗ 2 EM (t) = E (t) + kcurl v (t)k dt dt dt (27) d ≤ E∗(t) + A A kcurl v∗(t)k2 + 2kv∗(t)k2 . dt 1 So, we primarily address our attention to E∗. Upon its differentiation, in view of 12, by managing suitable identities and then applying the divergence theorem, in a 2122 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI straightforward matter we arrive at its evolution law d E∗(t) =2µ (v∗(t), v∗(t)) + τ(hΠ∗i (t), hΠ∗i (t)) + τ(p∗(t), p∗ (t)) dt e t 2 e e t 2 e et 2 Z ∗ ∗ 2 ∗ = − µe 2(v(t) · v (t)) + |v (t)| + 2(p(t) + pe(t)) v (t) · n ds Σ Z h ∗ ∗ τ ∗ 2 ∗ 2 ∗ i + 2µe hΠei (t)v (t) − |hΠei (t)| + |pe(t)| (v(t) + v (t)) · n ds Σ 2 ∗ 2 ∗ 2 ∗ ∗ − k hΠei (t)k − kpe(t)k + (µev(t), curl v (t) × v (t))2 ∗ ∗ ∗ ∗ − τ((v (t) · ∇) hΠei (t) − hΠei (t)W (t) + W (t) hΠei (t), hΠei (t))2 ∗ ∗ ∗ ∗ ∗ − (hΠei (t)(W(t) + W (t)), hΠei (t))2 − τ(pe(t), ∇pe(t) · v (t))2 Z ∗T ∗ ∗ 2 + 2µsµe L (t)v (t) · n ds − k∇v (t)k . Σ (28) ∗ ∗ Note that the troublesome elastic dissipative term 2µe hΠei · D is cancelled. Equation 28 simplifies for the Maxwell model, as we let µs = 0. On the other hand, only for the J-S model, we may use the following identity Z ∗ ∗ µe ∗ 2 ∗ ∗ (µev, curl v × v )2 = − |v | v · n ds + (µe(v · ∇)v , v)2 , (29) 2 Σ together with the known decomposed form of the solvent diffusive term ∗ 2 ∗ 2 ∗ 2 2µsµek∇v k = 2µsµekD k + 2µsµekW k . (30) We can now formalize the continuous dependence theorem on the initial data for the J-S model.

∗ Theorem 3.5 (J-S model: µs > 0). Let χi and χi + χi be the basic and the perturbed solutions to Pi satisfying the same boundary conditions and body forces, but different initial data. If the basic solution χi satisfies the constraint requirement 26, the perturbation energy E∗ meets the Gronwall type inequality d E∗(t) ≤ M E∗(t) , (31) dt JS for a positive computable parameter MJS. Proof. Consider now the evolution law 28, and take into account the replacement 29 and the identity 30. In view of the homogeneous boundary conditions, the boundary integrals vanish. We can evaluate the various terms by employing the a-g mean and Cauchy Schwartz (C-S) inequalities within the constraint set. In particular, we present the following estimates which are crucial for the underling stability discussion: 2 ∗ ∗ A ∗ 2 1 ∗ 2 |(µe(v · ∇)v , v)2| ≤ µe α1kv k + k∇v k , 2 2α1 ∗ ∗ ∗ 2 ∗ 2 1 ∗ 2 τ |(hΠei W − W hΠei , hΠei)2| ≤ τ A α2k hΠei k + kW k , (32) α2 2 ∗ ∗ τ ∗ 2 α3 ∗ 2 τ |((v · ∇) hΠei , hΠei)2| ≤ A k hΠei k + kv k , 2α3 2 for positive parameters α1, α2 and α3 at our disposal. Finally, we substitute the estimates 32 together with 25 into the modified 28. Upon collecting terms we easily UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2123 obtain the following energy inequality d ∗ τ ∗ 2 µe ∗ 2 E (t) ≤ − 2µsµe kW (t)k + − 2µsµeλk k∇v (t)k dt α2 2α1 2 A Aβ τAα3 ∗ 2 + µe α1 + + kv (t)k 2 2µe 2µe (33) τ 2 Aτ 2 ∗ 2 + − + + 2A α2 k hΠei (t)k 2 τ α3 τ 2 2Aτ + − + kp∗(t)k2 , 2 τ β e where λk is the positive Korn constant. Selecting 1 τ α1 = , α2 = , 2µsλk 2µsµe we achieve the Gronwall type energy inequality 31, where A β + τα3 τ Aτ 2τ MJS = A max + , + , , 4µsλk 2µe α3 µsµe β for any choice of α3 and β. As for the null solution case, it is an easy matter to state, and then prove, the following uniqueness result. ∗ ∗ ∗ ∗ ∗ Corollary 2 (Uniqueness). If χi = (v , p , pe, hΠei) is a classical solution to 12, subject to condition 26, satisfying homogeneous boundary conditions for v∗ and with ∗ ∗ ∗ null initial data, then (v , pe, hΠei) vanishes identically for all t ∈ [0, T [, within an arbitrary time dependent pressure p∗. It is worthwhile noting that, from inequality 33, we might still provide an exponential stability result for the non null basic solution, within the constraint set 26, even if with a new constant K strongly related, not only to the rheological parameters, but also to the geometry of the domain Ω. Finally, starting from the inequality 27 and taking into account 28 with µs = 0, we can recover the same stability results for the Maxwell model, under a further stronger condition on the basic solution. In fact, if we follow step by step the previous strategy, making use of the Lemmas 3.3, 3.4 and, applying the following estimate A |(µ v, curl v∗ × v∗) | ≤ µ kvk2 + kcurl vk2 , e 2 2 e we are able to establish the following continuous dependence result on the initial data for the Maxwell model. ∗ Theorem 3.6 (Maxwell model: µs = 0). Let χi and χi + χi be the basic and the perturbed solutions to Pi satisfying the same boundary conditions and body forces, but different initial data. If the basic solution χi satisfies the new constraint requirement sup {| hΠei |, |v|, |∇ hΠei |, |∇pe|, |S|} ≤ A, (x,t)∈Ω×[0,T [ ∗ where S is defined in Lemma 3.2. The perturbation energy EM meets the Gronwall type inequality d E∗ (t) ≤ M E∗ (t) , (34) dt M M M 2124 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI with 2 5 + 2τ + µe τ + 2 A + 4µe(1 + τ) + 2µe MM = A max , , 2τ, . 2µe 2 4µe Theorem 3.6 easily yields a uniqueness result which generalizes the analogous one obtained in [9], for a non-objective incompressible Maxwell model.

3.2. Polymeric solution with negative viscosity. We observe that the conditions for compatibility with thermodynamics require µs ≥ 0 and µe + µs ≥ 0. This allows us to consider models with µe < 0 as long as |µe| < µs. Constitutive equations that provide for such a choice are used to describe retardation phenomena in polymeric liquids. The hypothesis µe < 0, or, equivalently, the dimensionless parameter r > 1, suggests τD > τΠ in4; this requirement is in disagreement with the causality principle. However, as in analogous Jeffreys type heat conduction theories, the retardation behavior is empirically possible (see e.g. [30]). In this subsection we investigate the stability of the null equilibrium solution when µe < 0. To this aim we introduce the new positive definite energy measure τ E (t) = |µ |kv(t)k2 + k hΠ i (t)k2 + kp (t)k2 (35) 1 e 2 e e for any perturbed solution (v, hΠei , pe) with assigned initial data (v0, hΠe0i , pe0). We can still establish the following exponential decay result. Theorem 3.7 (Exponential stability with r > 1). For any canister retarded flow of a corotational incompressible J-S viscoelastic fluid, which is mechanically isolated, the energy E1 decays exponentially in time, whenever 3|µe|−µs < 0 or, equivalently, 3 r < 2 . Proof. We follow the same energy technique as in Theorem 3.1 to obtain d E (t) = −k hΠ i (t)k2 − 2|µ |µ k∇v(t)k2 − kp (t)k2 − 4|µ |(∇v(t), hΠ i (t)) . dt 1 e e s e e e 2 (36) Then, let us employ the a-g-mean inequality, in the form 2 µs + |µe| 2 4µsµe 2 2|µe|(∇v(t), hΠei (t))2 ≤ k hΠei (t)k + k∇v(t)k , 4µs µs + |µe| to derive

d |µe| − µs 2 3|µe| − µs 2 2 E1(t) ≤ k hΠei (t)k + 2 |µe|µsk∇v(t)k − kpe(t)k . (37) dt 2µs |µe| + µs

Whenever µs > 3|µe|, by applying the Poincar´einequality it results d E (t) ≤ −cE (t) , dt 1 1 where 2 µs − |µe| µs − 3|µe| c = min , , 2 λµs . τ τµs µs + |µe|

In a similar way it is also possible to extend the validity of Theorem 3.5 to the µe negative case, provided that µs > 3|µe|. UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2125

4. Continuous dependence and uniqueness in compressible barotropic J-S viscoelastic fluids. In this section we extend our non linear stability analysis to the compressible J-S model and prove a Höldercontinuous dependence theorem, which yields a uniqueness result consequently. The simplified case of a purely polymeric solution (µs = ζs = 0) is then investigated and the results are compared and discussed. From now on, we assume that ρ ≥ ρ¯ > 0 and, as we are interested in barotropic motions, we adopt the following state equation for the pressure p:

p = φ(ρ) , (38)

φ ∈ C2([¯ρ, +∞[) being a known function, with φ0 > 0 being the squared sound wave speed. In order to simplify the computations, we work with the material time derivative in place of the previous co-rotational one, so that system3 reduces ρ˙ + ρ∇ · v = 0 , 0 1 ρv˙ = −φ (ρ)∇ρ − ∇pe + ∇ · hΠei + µs∆v + ζs + µs ∇(∇ · v) + ρb , 3 (39) ˙ τhΠei+ hΠei = 2µe hDi ,

τp˙e + pe = −ζe∇ · v .

To complete the mathematical description, we must specify appropriate boundary and initial conditions. The nature of the boundary conditions is strictly connected to the presence of the Newtonian solvent stress tensor Πs; hence, besides the classical adherence condition for v on Σ, it is necessary to assign also ρ, hΠei and pe on the portion of Σ, where the ﬂuid enters.The boundary conditions are the following

v(x, t) = vΣ(x, t) ,

ρ(x, t) = ρΣ(x, t) , hΠei(x, t) = hΠeiΣ(x, t) , if νn(x, t) < 0 , (40)

pe(x, t) = peΣ(x, t) , if νn(x, t) < 0 , where (x, t) ∈ Σ × [0, T ], n is the unit outward normal to Σ, and νn = v · n, vΣ, ρΣ, hΠeiΣ, peΣ denote prescribed functions of the indicated arguments. The initial conditions are

v(x, t) = v0(x) , ρ(x, 0) = ρ0(x) , hΠei(x, 0) = hΠei0(x) , pe(x, 0) = pe0(x) , (41) where v0, ρ0, hΠei0, pe0 represent known smooth functions. We denote the initial boundary value problem 39–41 by Pc and, by a classical 1 solution χc = (v, ρ, hΠei , pe) to Pc, we mean that ρ, hΠei and pe are of class C with respect to the space and the time variables, while v is of class C2 with respect to the space variables and of class C1 with respect to the time. We now show that a classical solution to Pc depends Höldercontinuosly on the initial data and on the body force in the interval [0,T ], for some T < ∞. Let ∗ ∗ ∗ ∗ ∗ χc = (v, ρ, hΠei , pe) and χc + χc = (v + v , ρ + ρ , hΠei + hΠei , pe + pe) be two classical solution to Pc in Ω × [0, T [, for the same boundary data, as given in 41, ∗ ∗ but for different initial data χc0, χc0 + χc0 and different body forces b and b . ∗ ∗ If we denote the difference variables as χc , it is easy to show that χc satisfies the 2126 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI following system of equations ∗ ∗ ∗ ∗ ∗ ∗ ∗ ρt + (v + v ) · ∇ρ + v · ∇ρ + ρ∇·v + ρ ∇·(v + v ) = 0 , ∗ ∗ ∗ ∗ ∗ ρ [vt + [(v + v) · ∇]v + (v · ∇)v − b ] ∗ ∗ ∗ ∗ ∗ + ρ [vt + vt + [(v + v ) · ∇](v + v ) − (b + b )] + [φ0(ρ + ρ∗) − φ0(ρ)]∇(ρ + ρ∗) + φ0(ρ)∇ρ∗ − ∇·hΠ∗i + ∇p∗ e e (42) 1 = +µ ∆v∗ + ζ + µ ∇(∇ · v∗) , s s 3 s ∗ ∗ ∗ ∗ ∗ ∗ τ [hΠeit + [(v + v ) · ∇) hΠei + (v · ∇) hΠei] + hΠei = 2µe hD i , ∗ ∗ ∗ ∗ ∗ ∗ τ(pet + (v + v ) · ∇pe + v · ∇pe + pe + ζe∇·v = 0 , ∗ ∗ ∗ ∗ ∗ together with prescribed initial condition χc0 = (v0, ρ0, Π0, pe0) and homogeneous boundary conditions on Σ × [0, T ]. For later reference, we define the energy measure E∗ as follows Z 2 ∗ ∗ p ∗ E (t) = e (x, t) dx = µe ρ(t)v (t) Ω (43) 2 1 p 0 −1 ∗ ∗ 2 2µeτ ∗ 2 + φ (ρ(t))ρ (t)ρ (t) + τk hΠe(t)i k + kpe(t)k . 2 ζe We observe that E∗(0) provides a measure of the initial data. To obtain continuous dependence results, it is necessary to restrict the classical solutions to a class of suitable bounded functions. If we denote by A a positive computable constant, we consider the (basic) solution χc and the (perturbed) solution ∗ χc + χc in the following constraint set M 0 0 0 φ (ρ) sup |v|, |vt|, |∇φ (ρ)| , |ρt|, |∇ρ|, φ (ρ)|∇ρ|, |∇ hΠei |, |∇pe|, ≤ A, Ω×[0,T [ ρ t ∗ ∗ ∗ ∗ sup {|v + v |, |vt + vt |, |∇(v + v )|, |∇(ρ + ρ )|} ≤ A. Ω×[0,T [ (44) We need to assume also a bound for the body force, namely sup |b + b∗| ≤ B. (45) Ω×[0,T [ For the J-S model, the Hölderstability result can be formalized as follows.

∗ Theorem 4.1 (Höldercontinuous dependence for J-S model). If χc and χc + χc are two solutions to Pc of the class M, for the same boundary conditions, but for different initial data and body forces and ∗ 1 ∗ 2 max{E (0),T kρ 2 b k } ≤ , (46) [0,T ] for some T ∈]0, T [ and ∈]0, 1[, then there exist two positive constants N and δ, δ ∈]0, 1[, independent on , such that E∗(t) ≤ N(T )δ , t ∈ [0,T ] . (47) Inequality 47 shows the Höldercontinuity on the initial data and the body force. Proof. Among the various identities that are to be employed, we present the following one which is crucial for the proof, but also strictly related to the J-S model ρ(v∗ ·∇)v·v∗ = ∇·(ρ(v∗ · v)v∗)−ρ(v∗·v)∇·v∗ −ρ(v∗ ·∇)v∗·v−(v∗·v)∇ρ·v∗ . (48) UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2127

The terms which are now tricky to estimate are the ∇v∗ and ∇·v∗ terms; for the J-S model, in analogy with the compressible Newtonian case, these terms can be controlled through the solvent viscous (dissipative) terms. We take the scalar product in L2(Ω) of the four equations 42 by respectively 0 −1 ∗ ∗ ∗ 2µe ∗ 2µeφ (ρ)ρ ρ , 2µev , hΠ i and p , and then add the resulting equations; in e ζe e such a way troublesome terms, like the elastic ‘dissipative’ terms simplify. Use of classical identities, together with the divergence theorem, and ﬁnally upon collecting terms, yield straightaway Z d ∗ ∗ ∗ ∗ ∗ ∗ 2 E (t) + u (t) · n ds + c (t) − 2µe(t)(ρ(t)b (t), v (t))2 = −k hΠe(t)i k dt Σ (49) 1 ∗ 2 ∗ 2 1 ∗ 2 − 2µe kpe(t)k + µsk∇v (t)k + ζs + µs k∇·v (t)k , ζe 3 where the ﬂux term u∗ is now given as follows ∗ ∗ ∗ ∗ ∗ ∗ ∗ u (t) =e (t)(v(t) + v (t)) − 2µe [hΠe(t)i v (t) + pe(t)v (t)] 0 ∗ ∗ 00 ∗ ∗ ∗ + 2µe [φ (ρ(t))ρ (t)v (t) + φ (ˆρ(t))(ρ(t) + ρ (t))ρ (t)v (t)] 3ζ + µ + 2µ ρ(t)(v∗(t)· v(t))v∗(t) − µ L∗T (t) + s s ∇·v∗(t) v∗(t) e s 3 I 3ζ + µ =u∗ (t) + 2µ ρ(t)(v∗(t)· v(t)) − µ L∗T (t) − s s ∇·v∗(t) v∗(t) M e I s 3 I (50) and Z ∗ ∗ ∗ p 0 −1 ∗ 2 c (t) = − e (t)∇ · (v(t) + v (t)) dx − µek (φ (ρ(t))ρ (t))tρ (t)k Ω p ∗ 2 ∗ 2 ∗ 0 −1 − µek ρt(t)v (t)k − µe |ρ (t)| (v(t) + v (t)), ∇φ (ρ(t))ρ (t) 2 ∗ ∗ + τ (v (t) · ∇) hΠe(t)i , hΠe(t)i 2 2µeτ ∗ ∗ + (σe (t)∇σe(t), v (t))2 ζe ∗ 0 −1 ∗ ∗ + 2µe ρ (t) φ (ρ(t))ρ (t)∇ρ(t) + vt(t) + vt (t) , v (t) 2 (51) ∗ ∗ ∗ ∗ + 2µe (ρ (t) [((v(t) + v (t)) · ∇)(v(t) + v (t))] , v (t))2 ∗ 00 ∗ ∗ ∗ + 2µe (ρ (t)[φ (ρ(t)∇(ρ(t) + ρ (t)) − (b(t) + b (t))] , v (t))2 ∗ ∗ ∗ ∗ − 2µe[(ρ(t)∇·v (t)v (t), v)2(t) + (ρ(t)(v (t) ·∇)v (t), v(t))2] ∗ ∗ − 2µe((∇ρ(t) · v (t))v (t), v(t))2 ∗ ∗ ∗ ∗ ∗ =cM (t) − 2µe[(ρ(t)∇·v (t)v (t), v)2(t) + (ρ(t)(v (t) ·∇)v (t), v(t))2] ∗ ∗ − 2µe((∇ρ(t) · v (t))v (t), v(t))2 . Employing the a-g mean and C-S inequalities, together with the constraint set M, on the last µe-terms in 51, we get ∗ ∗ 1 ∗ 2 1 ∗ 2 2µe(ρ∇·v v , v)2 ≤ µeA k∇·v k + α1kρ 2 v k , α1 ∗ ∗ 1 ∗ 2 1 ∗ 2 2µe(ρ(v ·∇)v , v)2 ≤ µeA k∇v k + α2kρ 2 v k , (52) α2 2 ∗ ∗ A 1 ∗ 2 2µ ((∇ρ · v )v , v) ) ≤ 2µ kρ 2 v k , e 2 e ρ¯ 2128 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI

∗ for positive α1 and α2 at our disposal. Concerning the other cM -terms, an iterated use of the C-S inequality, together with the constraint set M, easily provides the following estimate ∗ ∗ cM (t) ≤ ME (t) (53) for a suitable, positive, computable constant M. Including the estimates 52 and 53 into 49, on account of homogeneous boundary conditions and, ﬁnally, upon re-gathering terms, we arrive at the energy inequality straightaway

d ∗ ∗ ∗ ∗ 2 2µe ∗ 2 E (t) ≤ 2µe(t)(ρ(t)b (t), v (t))2 − k hΠe(t)i k − kpe(t)k dt ζe ∗ A 1 ∗ 2 + ME (t) + µ A α + α + kρ 2 v k e 1 2 ρ¯ A ∗ 2 A 1 ∗ 2 + 2µe − µs k∇v (t)k + − ζs − µs k∇·v (t)k . 2α2 2α1 3 (54) 3A A Next, select α1 = and α2 = , so that the last two terms are simpliﬁed 2(3ζs+µs) 2µs and 54 reduces to the form d E∗(t) ≤ 2µ (t)(ρ(t)b∗(t), v∗(t)) + M 0E∗(t) . (55) dt e 2 In this manner, when b∗ = 0, inequality 54 demonstrates continuous dependence on the initial data for a classical solution to Pc of class M. In order to recover the H¨olderstability, we use a standard procedure by introducing the classical Bernoulli transformation Ψ(e t) = e−αtΨ∗(t) for an arbitrary function Ψ∗, α being a positive constant, at our future disposal. Then, multiplication of 55 by e−2αt yields the new energy equation

d 0 −αt ∗ Ee(t) ≤ (M − 2α)Ee(t) − 2µe(t)(e ρ(t)b (t), v(t))2 . (56) dt e Owing to the a-g mean inequality and 46, we have Z t Z t −αy ∗ 1 2 µe 2µe (e ρ(y)b (y), ve(y))2 dy ≤ 2µe kρ 2 (y)ve(y))k dy + (57) 0 0 2αT for each t ∈ [0,T ], with T ∈ [0, T [. Upon integration over [0, t], inequality 56, in view of 46 and 57, yields the basic energy inequality for t ∈ [0,T ] µ Z t e−2αtE∗(t) ≤ 1 + e + (M 00 − 2α) e−2αyE∗(y) dy . (58) 2αT 0 Applying Gronwall’s lemma, we obtain µ 00 e−2αtE∗(t) ≤ 1 + e e(M −2α)T . (59) 2αT (δ−1) Setting α = 2T log , for some δ ∈]0, 1[, we arrive at the H¨oldertype estimate 47, with ( 1 M 00T 00 log 1 + 2αT e if M ≥ − T N(T ) = 1 00 log 1 + 2αT if M < − T UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2129

By Theorem 4.1 the H¨oldercontinuity property holds on compact intervals [0,T ] ⊆ [0, T [. As a corollary, we can formulate and prove a uniqueness theorem as follows:

∗ Corollary 3 (Uniqueness). Let χc and χc + χc be two classical solutions to Pc of class M for the same body force and satisfying the same initial and boundary ∗ conditions. Then the difference variables χc vanish identically on [0, T [. Proof. Since b∗ = 0 and E∗(0) = 0, we have = 0 in 46. Inequality 59 yields ∗ ∗ E (t) ≤ 0, hence χc (t) = 0 on [0, T [. These well-posedness results can be easily extended to the invariant formulation of the compressible J-S model, not only for relaxation effects (0 < r < 1), but also in presence of retardation effects (r > 1). Thus, for completeness, two remarks are in order. Remark 3 (Corotational compressible J-S model, with 0 < r < 1 ). When we refer to the corotational compressible J-S model, the presence of the rotational terms, due to the spin tensor W, leads to the necessity of using the inequality ∗ ∗ ∗ 2 ∗ 2 1 ∗ 2 τ |(hΠei W − W hΠei , hΠei)2| ≤ τ A α4k hΠei k + kW k , α4 where kW∗k2 may be still controlled by the solvent dissipative terms, through the τ formula 30, by choosing α4 = , as for the incompressible case. Following the 2µsµe ∗ 2 same procedure, we arrive at a modified 54, where the new coefficient of k hΠei k is 2 given by τ − 2 + τA and is consequently inserted into the E∗-term. Concerning 2 τ µeµs ∗ 2 A the k∇v k term, its coefficient is 2µe − λkµs ; hence, through the further 2α2 A choice α2 = , we still achieve the Höldercontinuity property. 2µsλk Remark 4 (Compressible J-S model with r > 1 ). It is interesting to investigate also the retardation effects on the stability analysis, corresponding to negative elastic viscosities. After the re-definition of the energy 43 with |µe| and |ζe| replacing µe and ζe, and following the same procedure, among the variety of previous energy terms, we point out the introduction of the elastic “dissipative” terms ∗ ∗ ∗ ∗ −4|µe| [(hD i , hΠei)2 − (pe∇, v )2] into equation 49. By employing suitable a-g mean inequalities, like in Theorem 3.7, we still get the Hölderstability property whenever 3|µe| − µs < 0 or, equivalently, 3 r < 2 . In conclusion, for the J-S model, invariant or not, with r > 0, only the retardation effects, with strongly negative elastic viscosities, can lead to instabilities and, hence, yield ill-posed problems.

4.1. Compressible polymeric melts. In order to complete our comparative study between the diffusive J-S model and the hyperbolic Maxwell model, we consider the purely compressible polymeric solution case, corresponding to µs = ζs = 0. The previous method works likewise well for the non objective Maxwell model: of necessity, the energy argument must require something more, because of the absence of the solvent viscous dissipative terms. Consequently, for the Maxwell model, the identity 48 cannot be employed. The difficulty is overcome through an additional 2130 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI boundedness constraint on the velocity gradient of the basic state, which allows us to avoid 48. Hence 49 reduces to the form d Z E∗(t)+ u∗ (t) · n ds + c∗ (t) + 2µ (ρ(t)(v∗(t) · ∇)v(t), v∗(t)) dt M M e 2 Σ (60) ∗ ∗ ∗ 2 2µe ∗ 2 − 2µe(ρ(t)b (t), v (t))2 = −k hΠe(t)i k − kpe(t)k . ζe As we can see, the only difference, with respect to the previous procedure, is related to the necessity of evaluating the new term, as follows: ∗ ∗ 1 ∗ 2 1 ∗ 2 |2µe(ρ(v · ∇)v, v )2| ≤ sup |∇v| kρ 2 v k ≤ Akρ 2 v k , Ω×[0,T [ where, without misunderstanding, we assume the additional constraint sup |∇v| ≤ A. (61) Ω×[0,T [ ∗ This evaluation, combined with the previous cM -inequality 53, easily provides the new estimate ∗ ∗ ∗ ∗ cM (t) + 2µe(ρ(v · ∇)v, v )2 ≤ MM E (t) , (62) which, in turn, inserted into 60, yields the basic energy inequality 55, with the 0 new parameter MM , in place of M . Hence, we still recover the Hölderstability and related well-posed results, by employing the same methods as before, for the J-S model. Hölderstability can be stated for solutions in the constraint set MM , defined by 44 together with 61, as follows.

Theorem 4.2 (Höldercontinuous dependence for Maxwell model). If χc and ∗ χc +χc are two solutions to Pc of the class MM , for the same boundary conditions, but for different initial data and body forces, and ∗ 1 ∗ 2 max{E (0),T kρ 2 b k } ≤ , (63) [0,T ] for some T ∈]0, T [ and ∈]0, 1[, then there exist two positive constants NM and δ, δ ∈]0, 1[, independent on , such that ∗ δ E (t) ≤ NM (T ) , t ∈ [0,T ] . (64) Inequality 64 shows the Höldercontinuity on the initial data and the body force. In carrying on our comparative analytical study, we realize that the only challenging terms are those due to the invariance requirement. As already remarked, for the hyperbolic, incompressible/compressible, Maxwell model, these terms are exclusively related to the spin tensor terms or, more generally, to the velocity gradient terms, present in the definition of the invariant corotational, or G-S convected time derivative. Hence, apart from the null basic solution case, where the difficulties have been overcome thanks to the particular choice of objective corotational derivative, in the stability proof for the non null solution, we managed a quite tricky tool to be able to control the above terms, working with a generalized energy functional, involving also the L2-norm of the vorticity vector, and writing down a new, but generally approximated, equation for the vorticity, as a new essential variable. Thus, we state in advance that, in our opinion, the same techniques, or different energy arguments, cannot be reasonably extended to include also compressional flows, without requiring strong restrictions on the model itself. In spite of this, our remark is not surprising, but simply realistic. Meanwhile, we stress that, in contrast UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2131 to the J-S model, the forwards Maxwell model exhibits a single positive relaxation time and, in view of the thermodynamic restrictions, it is only the introduction of an objective time derivative that is responsible for instability mechanisms and consequent change of type [18], just leading to ill-posed problems. In fact, by investigating the wave propagation properties of the model, in the framework of singular surfaces of the first order, we may easily show that the exis- tence of transverse acceleration (visco)elastic shear waves, is related to the validity of two conditions (see e.g.[7]), just depending on the elastic stress tensor and due to the jumps of the velocity gradient terms into the G-S convected time derivative. The hyperbolicity is lost whenever one or both conditions are not satisfied; in this case we find imaginary wave speeds and hence the Maxwell model displays instability phenomena, as observed empirically.

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