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ANALYSIS OF A VISCOSITY MODEL FOR CONCENTRATED POLYMERS MIROSLAVBULÍCˇEK,PIOTRGWIAZDA,ENDRESU¨LI,ANDAGNIESZKAS´WIERCZEWSKA-GWIAZDA Abstract. Thepaperisconcernedwithaclassofmathematicalmodelsforpolymericfluids,whichinvolvesthecou- plingoftheNavier–Stokesequationsforaviscous,incompressible,constant-densityfluidwithaparabolic-hyperbolic integro-differentialequationdescribingtheevolutionofthepolymerdistributionfunctioninthesolvent,andapar- abolic integro-differential equation for the evolution of the monomer density function inthe solvent. The viscosity 6 coefficient,appearinginthebalanceoflinearmomentumequationintheNavier–Stokessystem,includesdependence 1 ontheshear-rateaswellasontheweight-averagedpolymerchainlength. Thesystemofpartialdifferentialequations 0 underconsiderationcapturestheimpactofpolymerizationanddepolymerizationeffectsontheviscosityofthefluid. 2 Weprovetheexistenceofglobal-in-time,large-dataweaksolutionsunderfairlygeneralhypotheses. n a J 7 1. Viscosity models for polymers – formulation of the problem ] Contemporary approaches to the modelling of polymeric fluids have exploited multi-scale descriptions in an P essential way. Mathematical models have thus been built by coupling systems of partial differential equations A describingthemotionofthesolventwithequationsthattracktheevolutionofthemicroscopicbehaviourofthesolute . inthesolution. Inthispaperwefocusonthemodellingandtheanalysisoftwophenomena: wewishtoexplorehow h t the rheologicalproperties of the fluid are affected by the presence of, possibly very long, chains of macromolecules; a and, second, we aim to investigate the possible mechanisms for polymerization and depolymerization effects that m can also heavily depend on the properties of the flow. The solvent is considered to be an incompressible fluid and [ weassumethatthepossiblechangesofthedensityarenegligiblerelativetootherphenomena. Thus,theunderlying 2 system of equations under consideration consists of the balance of linear momentum and the incompressibility v constraint, i.e., 6 ∂ (̺v)+div (̺v⊗v)−div TTT=̺f, 6 t x x (1) 7 div v =0. x 5 0 These equations are assumed to be satisfied in a time-space cylinder Q := (0,T)×Ω, with Ω ⊂ Rd, d ∈ {2,3}, . being the physical flow domain. Here, v : Q → Rd denotes the velocity of the solvent, ̺ is the constant density, 1 0 f : Q → Rd represents the density of volume forces andTTT : Q → Rd×d is the Cauchy stress. In order to close the 5 system (1), one also needs to relate the Cauchy stress to other quantities describing the qualitative behaviour of 1 the material. Despite their importance, in the present paper we shall, for the sakeof simplicity, neglect all thermal : v effects and will focus instead on mechanical properties of the fluid in the isothermal setting. In particular, we i are interested in models that link the Cauchy stress to the shear rate and to the contribution of the microscopic X polymer molecules, in order to investigate how the level of interaction between polymer chains, their lengths, and r a their configuration influence the Cauchy stress. That the presence of macromolecules in a solvent dramatically changes the properties of the flow was already observed in [13], and has been, ever since, the focus of mathematical models. In general terms, one can find two different approaches to the problem: the empirical (rheological) one, where one typically fits the observed data in the modelandintroducesempiricalformulaeforthe viscosity,ν,basedontheratioofthe magnitudeofthe Cauchy stressandthemagnitudeoftheshearrate;andasecondone,basedonmodellingthemicrostructureofthepolymer under consideration using tools from statistical mechanics. The second approachhas been particularly common in the case of dilute polymers, i.e., when the concentration of polymer molecules in the solvent is very low, so that 2000 Mathematics Subject Classification. 35Q35, 76D03,35M13,35Q92. Key words and phrases. Non-Newtonianfluid,polymers,monomers,weaksolution,large-dataexistence. M. Bul´ıˇcek thanks the project MORE (ERC-CZ project LL1202 financed by the Ministry of Education, Youth and Sports, Czech Republic) and the Neuron Fund for Support of Science. M. Bul´ıˇcek is a member of the Neˇcas Center for Mathematical Modeling. A. S´wierczewska-Gwiazda acknowledges the support of the project IdP2011/00066. The research was partially supported by the Warsaw Center of Mathematics and Computer Science. P. Gwiazda is a coordinator of the International Ph.D. Projects Programme ofthe Foundation forPolishScience operatedwithintheInnovative Economy Operational Programme2007–2013 (Ph.D.Programme: MathematicalMethodsinNaturalSciences). E.Su¨liacknowledgesthefinancialsupportoftheprojectMORE(ERC-CZprojectLL1202 financed by the Ministry of Education, Youth and Sports, Czech Republic) and the Oxford Centre for Nonlinear Partial Differential Equations (OxPDE). 1 $LaTeX: 2016/1/8 $ 2 M.BULÍCˇEK,P.GWIAZDA,E.SU¨LI,ANDA.S´WIERCZEWSKA-GWIAZDA the interactions of the macromolecules can be neglected. In such models the presence of macromolecules in the solvent is typically modelled by additively supplementing the Cauchy stress tensor with an elastic stress tensor. This thenleadsto aclosedsystemofpartialdifferentialequations; foraderivationofthe modelandfurtherrelated mathematical results and references we refer the reader to Barrett & Su¨li [2, 3, 4, 5, 6] and Bul´ıˇcek, Ma´lek & Su¨li [9]; see also [18] and [1]. On the other hand, in the case of concentrated polymers one has to take into account the interactions of the polymer molecules (see, for example, [12]). Here we shall be concerned with models in which the only important property of the microstructure that influences the macroscopic flow equations is the polymer chain length, in the sense that the viscosity coefficient appearing in the balance of linear momentum equation is considered to be a functionofthe polymerchainlength,resultinginafamilyofviscosity models. Thereisanevidentcontrastbetween mathematical studies of models that ignore the dependence of the viscosity coefficient on the lengths of polymer chains, and the vast set of experimental data indicating the influence of the polymer chain length on the viscosity, see[7,17,13,19,20]amongothers. Inparticular,noneoftheapproachesmentionedaboveconsideredtheimpactof polymerizationand depolymerizationeffects onthe flow properties. The goalof this paper is therefore to explore a class ofviscosity models that incorporatethe influence of polymerizationanddepolymerizationonthe macroscopic flowvariables. Specifically,weprovidearigorousproofoftheexistenceofaglobalintime, large-dataweaksolution to the class of models under consideration. 1.1. Viscosity models: the relationship between the viscosity, the shear rate and the polymer chain length. It was experimentally observed more than seventy years ago already (cf. [13], for example,) that the properties of the solvent heavily, and nonlinearly, depend on both the number of polymer chains in the solvent and the polymer chain lengths. Flory had studied the relationship between the viscosity of the solvent and the number of polymer chains in the solvent for linear polyesters at constant temperature, pressure and shear-rate. Guided by experimental evidence he proposed that for linear polyesters the logarithm of the viscosity should be a linear function of the square-root of the so-called weight-averaged chain length. Moreover, it was also observed that this relationshipis independent ofthe type ofdistributionfunction for the species inthe polymer. To be more concrete, if we denote by ψ(r) the distribution function of polymer chains of length r (meaning that the polymer chainconsistsof r monomers)anddenoting the minimal lengthofa polymerby r , the totalweightof the polymer 0 can be considered to be proportional to ∞ rψ(r)dr. Zr0 This is indeed the case if one considers r ≫ 1, but must be corrected for lower values of r , see [13]. Thus, the 0 0 weight-averagedchain length can be defined as ∞r2ψ(r)dr ψ˜:= r0 . ∞ rψ(r)dr Rr0 It was empirically shown in [13] that R lnν ∼ ψ˜, q in the case when one considers linear polyesters; see [14] for a more detailed account. These experiments indicate that, generally,one does not need to considerspecific values of the distribution function but that its averagein the above sense will suffice from the point of view of accurate macroscopic modelling. This was also conjectured to be the case in many other situations, and quite recently also in the case of amines at high pressures, see [20]. In that paper, the authors showed that the logarithm of the viscosity is not necessarily related to the square-root of the averagedchainlengthbutitcanbeamoregeneralfunction,thoughstilldependentonlyontheaveragedproperties and not on the fine details of ψ(r). Further, it was common belief (based on laboratory experiments at nonconstant shear rates) that many such phenomena canbe explainedby using a shear-dependentviscosityinsteadof a polymer-length-dependentviscosity. However,this is notthe case forconcentratedpolymers. Forexample, in[17], the authorsstudiedthe simultaneous influence of the shear-rateandthe polymer-lengthonthe viscosityfor polydisperse polymer melts. It wasobserved that,uptoacertaincriticalvalueoftheshearrate,thefluidbehavesasaNewtonianfluidwithviscositydepending only on the chain length, while for large shear-rates significant shear-thinning was observed. Moreover,it was also shown that, asymptotically, the viscosity depends only on the shear-rate and the influence of the chain length in large shear-rateregions can be neglected. In addition, it also transpired that the weight-averagedchain length was not an appropriate quantity in the cases considered, and the authors suggested instead the following general form $LaTeX: 2016/1/8 $ ANALYSIS OF A VISCOSITY MODEL FOR CONCENTRATED POLYMERS 3 of weighting: ∞ ω(r)rψ(r)dr (2) ψ˜:= r0 , ∞ R r0 rψ(r)dr where the function ω represents the most significantRlengths of the polymer chains. Surprisingly, during the last decade, it was observed that there exist materials (e.g. polystyrene-decalin, polyethylen) that exhibit the opposite phenomenon: shear-thickening,see[16]. Forallthese reasons,weconsideramuchmoregeneralformofthe Cauchy stress: TTT:=SSS−qIII, where q is the pressure and SSS represents the viscous part of the Cauchy stress; we further assume that SSS is of the form (3) SSS(ψ˜,DDD v):=ν(ψ˜,|DDD v|)DDD v, x x x whereDDD v is the symmetric velocity gradientand ψ˜is a generalweight-averagedchain length function of the form x (2), which one can simplify (without loss of generality) to the form ∞ ψ˜:= γ(r)ψ(r)dr. Zr0 We shall not specify the particular form of ν, but will instead admit a general class of viscosities so as to enable the consideration of both shear-thinning and shear-thickening fluids. 1.2. Polymerization models. In the above system the microscopic parameter that is taken into account is the polymer chain length, and we describe the process of elongation of these chains (by binding free monomers to polymer chains) as well as the process of breaking longer chains into shorter ones. We shall view this microscopic structure as an evolution of two populations: the population of polymer chains and the population of monomers. This kind of description is akin to the model for prion dynamics considered in [15]; see also [10, 11]. As the model formulated in the current paper can be seen as an extension of the prion proliferation model in those papers to the case with spatial effects (transport by a solvent and spatial diffusion) we shall discuss it in more detail. The authors of [15, 10, 11] study how the healthy prion protein and infectious prion protein populations interact in an infected organism. The infectious prionsareabnormalpathogenicconformationsofthe normalones. The proposed modelconsidersthe infectious prionproteinsto be a polymericformofnormalprionproteins. Polymersofinfected prions can split into shorter chains. Such a splitting transforms one infectious polymer into two shorter infectious polymers, which can then attach again to normal prion proteins. However, when the length of a part of a split polymerfallsbelowacertaincriticalvalue,itimmediatelydegradesintoanormalprionprotein,namelyamonomer. By ψ we denote the distribution function of polymer chains of length r >r that satisfy the following equation: 0 ∞ (4) ∂ ψ(t,r)+τφ(t)∂ ψ(t,r)=−β(r)ψ(t,r)+2 β(r˜)κ(r,r˜)ψ(t,r˜)dr˜. t r Zr Thetermτφ(t)∂ ψ(t,r)representsthegaininlengthofpolymerchainsduetopolymerizationwithrateτ >0,β(r) r is the fragmentation rate, namely the length-dependent likelihood of splitting of polymers to monomers, κ(r,r˜) is the probability that a polymer chain will split into two shorter polymer chains of length r and r˜−r, respectively, the term −β(r)ψ(t,r) is the lossof polymer chains subjectto the splitting rate β(r), and the lastterm is the count of all the polymer chains of length r resulting from the splitting of polymer chains of length greater than r. Theevolutionofmonomersisdescribedbymeansofthefunctionφ(t),whichistheconcentrationoffreemonomers at time t, satisfying the equation d r0 ∞ ∞ (5) φ(t)=2 r β(r˜)κ(r,r˜)ψ(t,r˜)dr˜dr−φ(t) τψ(t,r)dr. dt Z0 Zr Zr0 The first term on the right-hand side represents the monomers gained when a polymer chain splits with at least one polymer chain shorter than the minimum length r , while the second term is the loss of monomers as they are 0 polymerized. 1.3. The complete model. We conclude this introductory section with the precise statement of the complete system of equations under consideration. For a given Lipschitz domain Ω ⊂ Rd, d ∈ {2,3}, and a given final time T >0, we consider the balance of linear momentum and the incompressibility constraint on Q:=(0,T)×Ω in the form (after scaling by the constant density ̺) ∂ v(t,x)+div (v(t,x)⊗v(t,x))+∇ q(t,x)−div SSS(ψ˜(t,x),DDD v(t,x))=f, t x x x x (6) div v(t,x)=0, x $LaTeX: 2016/1/8 $ 4 M.BULÍCˇEK,P.GWIAZDA,E.SU¨LI,ANDA.S´WIERCZEWSKA-GWIAZDA where v : Q → Rd denotes the velocity of the fluid (solvent), q : Q → R is the pressure, and f : Q → Rd is the density of the external body forces. The viscous part of the Cauchy stressSSS:Q→Rd×d is given by the formula (7) SSS(ψ˜(t,x),DDD v(t,x)):=ν(ψ˜(t,x),|DDD v(t,x)|)DDD v(t,x). x x x Here, DDD v denotes the symmetric velocity gradient, i.e., DDD v := 1(∇ v +(∇ v)T), and the viscosity coefficient x x 2 x x ν :R ×R →R is allowedto depend onthe shear-rate|DDD v| andonthe averagedpolymer distributionfunction + + + x ψ˜:Q→R , defined by + (8) ψ˜(t,x):= γ(r)ψ(t,x,r)dr, R Z 0 where R := (r ,∞) and γ : R → R is a continuous nonnegative function, representing the weight function 0 0 0 + associated with the averaging of polymer chain lengths. The distribution function ψ :Q×R →R , which in the 0 + absence of fluid motion satisfies (4), is now assumed to satisfy the following equation: ∂ ψ(t,x,r)+v(t,x)·∇ ψ(t,x,r)+τ(r)φ(t,x)∂ ψ(t,x,r)−A(r)∆ ψ(t,x,r) t x r x (9) ∞ =−β(r,v,DDD v)ψ(t,x,r)+2 β(r˜,v,DDD v)κ(r,r˜)ψ(t,x,r˜)dr˜, x x Zr inQ×R . Therearethereforetwoadditionalterms comparedwith (4): the convective/transporttermv·∇ ψ due 0 x tothemotionofthesolvent,andthediffusiontermA∆ ψassociatedwiththeBrownianforceactingonthepolymer x molecules immersed in the solvent. The parameter r ∈ [0,∞) is a fixed minimal length of a polymer molecule, 0 A(r) :R →R denotes the rate of diffusion for a particular value of r and is assumed here to be a nonincreasing 0 + function of r, τ : [r ,∞) → R is the polymerization rate, β : R ×Rd ×Rd×d → R is the fragmentation rate 0 + 0 + of polymer chains, which, unlike (4), will also be allowed to depend on the macroscopic quantities in the model, namelyonthe fluidvelocityandthe shearrate,and,finally,κ(r,r˜)denotestheprobabilitythatapolymermolecule of length r˜will split into two polymer molecules of lengths r and r˜−r, respectively. The function φ appearing in (9) is the concentration of free monomers satisfying, in the absence of fluid motion, the identity (5), and which, in the presence of a moving solvent, is assumed to satisfy the following equation: ∂ φ(t,x)+v(t,x)·∇ φ(t,x)−A ∆ φ(t,x) t x 0 x (10) ∞ r0 ∞ =−φ(t,x) ∂ (rτ(r))ψ(t,x,r)dr +2 r β(r˜,v,DDD v)κ(r,r˜)ψ(t,x,r˜)dr˜dr r x Zr0 Z0 Zr0 in the space time cylinder Q. Here, the rate of diffusion A (causedby Browniannoise)is assumedto be a positive 0 constant and the additional transport term v·∇ φ is due to the flow of the solvent. x We shall further assume that the velocity satisfies a Navier slip boundary condition, i.e., v·n=0 on ∂Ω, (11) (SSSn)τ =−α∗v on ∂Ω, where α∗ ≥ 0 is the domain-wall friction coefficient, n is the unit outward normal vector to ∂Ω, and for any v we have denoted by vτ :=v−(v·n)n the projection of v on the tangent hyperplane to the boundary. The function φ is assumed to satisfy a homogeneous Neumann boundary condition with respect to the x variable, i.e., (12) ∇xφ·n=0 on ∂Ω, and for ψ we also prescribe a homogeneous Neumann boundary condition with respect to the x variable, i.e., (13) ∇xψ·n=0 on ∂Ω. We shall further assume that ψ vanishes at infinity with respect to r, i.e., (14) lim ψ(t,x,r)=0. r→∞ Finally, to complete the statement of the problem, we prescribe the following set of initial conditions: for the velocity field we assume that (15) v(0,x)=v (x) in Ω, div v =0 in Ω, v ·n=0 on ∂Ω, 0 x 0 0 and for φ and ψ we assume that (16) φ(0,x)=φ (x) in Ω, φ ≥0, 0 0 (17) ψ(0,x,r)=ψ (x,r) in Ω×(r ,∞), ψ ≥0. 0 0 0 $LaTeX: 2016/1/8 $ ANALYSIS OF A VISCOSITY MODEL FOR CONCENTRATED POLYMERS 5 1.4. Assumptions on the parameters. In what follows K will signify a universal positive constant. We adopt the following assumptions on the data: (A1) The diffusion rate A:R →R is a continuous nonincreasing strictly positive function defined on R such 0 + 0 that lim A(r)=0. r→∞ (A2) The polymerization rate τ is a nondecreasing bounded globally Lipschitz continuous function such that τ(r )=0, τ′(r )>0, and there exists a positive constant K such that 0 0 τ(r) (18) K−1r ≤τ(r)+rτ′(r) and τ(r)+τ′(r)+rτ′(r)+ ≤K. 0 r (A3) The fragmentation rate β :R ×Rd×Rd×d →R is a smooth bounded function, which is increasing with 0 + respect to its first variable, and satisfies, for all (r,v,DDD)∈R ×Rd×Rd×d, 0 (19) 0<β(r,v,DDD)≤K. Moreover,we require that the function η, defined by ∂ β(r,u,DDD) r (20) η(r):= sup , u∈Rd,DDD∈Rd×d β(r,u,DDD) is measurable and nonnegative, and ∞ (21) 0≤(1+r)η(r) ≤K, η(r)dr ≤K. Zr0 f (A4) The kernel function κ is, for simplicity, defined as 1 if r˜>r and 0<r <r˜, 0 (22) κ(r,r˜):= r˜  0 otherwise.  It therefore follows that, for all α>0,  ∞ r˜α−1 (23) rα−1κ(r,r˜)dr = for r˜>r . 0 α Z0 (A5) There exists a positive real number θ >0 such that, for all r∈[r ,∞), one has 0 (24) γ(r)≤K(1+r)θ. (A6) We assume that ν : R ×R → R is a continuous function such that, for some p > 2d and for all + + + d+2 ξ,ξ˜ ∈Rd×d fulfilling ξ 6=ξ˜, one has (cf. (3)) |SSS(·,ξ)|≤K(1+|ξ|)p−1, (25) SSS(·,ξ)·ξ ≥K−1|ξ|p−K, (SSS(·,ξ)−SSS(·,ξ˜))·(ξ−ξ˜)>0. 1.5. The main result. Beforeintroducing the definitionof weak solution to problem(6)–(17) we shallsummarize our notational conventions. We denote by T ∈ (0,∞) the length of the time interval and by Ω ⊂ Rd, d ∈ {2,3}, a bounded domain in Rd with C1,1-boundary ∂Ω; we then write Ω ∈ C1,1. We also set Q := (0,T)× Ω and Γ:=(0,T)×∂Ω. For p ∈ [1,∞] we define the Lebesgue space Lp(Ω) and the Sobolev spaces W1,p(Ω) in the usual way, and we denote the trace on ∂Ω of a Sobolev function u by tru. If X is a Banach space, then Xd := X ×···×X and we write X∗ for the dual space of X; Lp(0,T;X) denotes the Bochner space of X-valued Lp functions defined on (0,T). For (scalar, vector- or tensor-valued) functions f and g and · signifying the product of real numbers, scalar product of vectors, or scalar product of tensors, as the case may be, we shall write (f,g):= f(x)·g(x)dx if f ·g ∈L1(Ω), ZΩ (f,g) := f(S)·g(S)dS if f ·g ∈L1(∂Ω), ∂Ω Z∂Ω hg,fi:=hg,fiX∗,X if f ∈X and g ∈X∗. We alsorequire the space C (0,T;Lp(Ω)) consisting of all u∈L∞(0,T;Lp(Ω)) suchthat (u(t),ϕ)∈C([0,T])for weak all ϕ∈C(Ω). $LaTeX: 2016/1/8 $ 6 M.BULÍCˇEK,P.GWIAZDA,E.SU¨LI,ANDA.S´WIERCZEWSKA-GWIAZDA We introduce certain subspaces (and their dual spaces) of the space of vector-valued Sobolev functions from W1,p(Ω)d,whichhavezeronormalcomponentontheboundary. First,wedefine,intheusualway,foranyp∈[1,∞), Lp :={v ∈D(Ω)d; div v =0}k·kp, n,div x where by D(Ω) we mean the set of all infinitely smooth functions with a compact support in the set Ω. Then, by V and V we denote div V :={v ∈Wd+2,2(Ω)d; v·n=0 on ∂Ω}, Vdiv :=V ∩L2n,div. Note that V ⊂ W1,∞(Ω)d and therefore we can finally introduce the following spaces for any p ∈ [1,∞), and p′ :=p/(p−1): W1,p :=Vk·k1,p, W−1,p′ := W1,p ∗, n n n W1,p :=V k·k1,p, W−1,p′ :(cid:0)= W(cid:1)1,p ∗. n,div div n,div n,div (cid:16) (cid:17) Moreover,for any α>1 we introduce the space Lp(R ):={ϕ∈Lp(R ): rαϕp(r)dr <∞}; analogously,we let α 0 0 R 0 Lp(Ω×R ):={ϕ∈Lp(Ω×R ): rαϕp(x,r)drdx<∞}, R :=(r ,∞). α 0 0 Ω R0 0 0R Theorem 1.1. Let the assumptioRnsR(A1)–(A6) be satisfied. Then, for any Ω ∈ C1,1, T ∈ (0,∞), any v , f, and 0 any nonnegative ψ , φ satisfying 0 0 (26) v0 ∈L2n,div, f ∈Lp′(0,T;Wn−1,p′), ψ0 ∈L1(Ω;L1θ∗(R0))∪L2(Ω;L23(R0)), φ0 ∈L∞(Ω), 1 with some θ∗ >θ and θ∗ ≥1, there exists a quadruple (q,v,ψ,φ) and q∗ >1 such that 1 1 (27) q ∈Lq∗(Q), (28) v ∈Cweak(0,T;L2n,div)∩Lp(0,T;Wn1,,pdiv), (29) ψ ∈L∞(0,T;L23(Ω×R0))∩L2(0,T;L2loc(R0;W1,2(Ω)))∩L1(0,T;L1(Ω;L1θ∗(R0))), 1 (30) φ∈L∞((0,T)×Ω)∩L2(0,T;W1,2(Ω)), with ∂ v ∈ Lq∗(0,T;W−1,q∗), ∂ ψ ∈ Lq∗(0,T;(W1,2(R ×Ω)∩W1,(q∗)′(R ×Ω))∗) and ∂ φ ∈L2(0,T;W−1,2(Ω)), t n,div t 0 0 t which attains the initial conditions in the following sense: (31) lim kv(t)−v k2+kφ(t)−φ k2+kψ(t)−ψ k2 =0, 0 2 0 2 0 2 t→0+ and solves, for almost all t ∈ (0,T), the equations (6)–(13) in the following sense: for all w ∈ W1,1 such that n ∇ w ∈L∞(Ω)d×d, x (32) h∂tv,wi+(SSS,∇xw)−(v⊗v,∇xw)+α∗(v,w)∂Ω =hf,wi+(q,divxw); for all ϕ∈W1,∞(Ω;D(R)), h∂ ψ,ϕi−(vψ,∇ ϕ)−(∂ (τϕ),φψ)+(A(r)∇ ψ,∇ φ) t x r x x (33) ∞ =−(βψ,ϕ)+2 βκψdr˜,ϕ ; (cid:18)Zr (cid:19) and for all z ∈W1,2(Ω), h∂ φ,zi−(vφ,∇ z)+A (∇ φ,∇ z) t x 0 x x ∞ r0 ∞ (34) =− φ ∂ (rτ)ψdr,z +2 r βκψdr˜dr,z . r (cid:18) Zr0 (cid:19) (cid:18)Z0 Zr0 (cid:19) A key difficulty in the mathematical analysis of the problem under consideration is that the partial differential equation (9), governing the evolution of the distribution of polymer chains, is a hyperbolic equation with respect to the variable r with a nonlocal term, which is nonlinearly coupled to the evolution equations (6) and (10). $LaTeX: 2016/1/8 $ ANALYSIS OF A VISCOSITY MODEL FOR CONCENTRATED POLYMERS 7 2. Analytical framework An essential part of the existence proof in the fluid part of the problem relies on Lipschitz approximations of Bochner functions taking values in Sobolev spaces. We recall from [8] the following lemma, which collects the properties of the approximationin a simplified setting (omitting the generality of Orlicz spaces used in [8]). Lemma 2.1. LetΩ⊂Rd, d∈{2,3},be abounded open set, let T >0 bethelength of thetimeinterval and suppose that 1 + 1 =1, with p∈(1,∞). For any function HHH and arbitrary sequences {un}∞ and {HHHn}∞ , we consider p p′ n=1 n=1 an :=|HHHn|+|HHH| and bn :=|DDD un|, x such that, for a certain C∗ >1, |an|p′ +|bn|pdxdt+ess.sup kun(t)k2 ≤C∗, t∈(0,T) 2 (35) ZQ un →0 a.e. in Q:=(0,T)×Ω. ¯ In addition, let {GGGn}∞ and {fn}∞ be such that GGGn is symmetric and n=1 n=1 (36) GGGn →0 strongly in L1(Q)d×d, ¯ (37) fn →0 strongly in L1(Q)d, ¯ and suppose that the following identity holds in D′(Q)d: (38) ∂ un+div (HHHn−HHH+GGGn)=fn. t x Then, there exists a β >0 such that, for arbitrary Qh ⊂⊂Q and for arbitrary λ∗ ∈(pp−11,∞) and arbitrary k ∈N, there exists a sequence of {λn}∞ , a sequence of open sets {En}∞ , En ⊂Q, and a sequence {un,k}∞ bounded k n=1 k n=1 k n=1 in L∞(0,T;W1,∞(Ω)d), such that, for any 1≤s<∞, loc loc (39) λnk ∈ λ∗,p−ppk−−11(λ∗)pk , ∀n∈N, (cid:20) (cid:21) (40) un,k →0 strongly in Ls(Q )d, h ¯ (41) kDDDx(un,k)kL∞(Qh) ≤C(h,Ω)λnk, (42) un,k =un in Q \En, h k C∗ (43) limsup|Q ∩En|≤C(h,Ω) . h k (λ∗)p n→∞ Moreover, 1 1 (44) limsup (|HHHn|+|HHH|)|DDD (un,k)|dxdt≤C(h,C∗) + , x (λ∗)p−1 kβ n→∞ ZQh∩Ekn (cid:18) (cid:19) and the following bound holds for all g ∈D(Q ): h T 1 1 β (45) −liminf h∂ un,un,kgidt≤C(g,h,C∗) + . n→∞ t (λ∗)p−1 k Z0 (cid:18) (cid:19) 3. Uniform a priori estimates Inthissection,weshallformallyderivetheuniformestimatesthatwillplayacrucialroleintheproofofexistence of weak solutions to the problem. All of the statements below are valid for sufficiently smooth solutions. First, we recall a minimum principle. However, we will prove it for the following slightly modified problem (we shall only indicate the dependence of functions on the variable r, except in cases when it is necessary to emphasize the dependence on the other independent variables as well): ∞ (46) ∂ ψ+v·∇ ψ−A∆ ψ =−βψ−τφ∂ ψ +2 β(r˜,·)κ(r,r˜)ψ (·,r˜)dr˜, t x x r + + Zr ∞ r0 ∞ (47) ∂ φ+v·∇ φ−A ∆ φ=−φ ∂ (rτ)ψ dr+2 r β(r˜,·)κ(r,r˜)ψ (·,r˜)dr˜dr, t x 0 x r + + Zr0 Z0 Zr0 where ψ :=max(ψ,0). + Lemma 3.1 (Minimum principle for ψ and φ). Let ψ and φ be nonnegative; then, the functions ψ and φ that 0 0 solve the coupled system (46) and (47) are also nonnegative. $LaTeX: 2016/1/8 $ 8 M.BULÍCˇEK,P.GWIAZDA,E.SU¨LI,ANDA.S´WIERCZEWSKA-GWIAZDA It directly follows from Lemma 3.1 that we can replace ψ by ψ and then (46), (47) reduce to (9), (10). + Proof. We test the equation (46) by ψ , where ψ = min(ψ,0) and integrate with respect to x and r. Using the − − Neumann boundary condition on ψ we get, after integration by parts, that 1 d ∞ kψ (t)k2+ A(·)|∇ψ |2+β(r,·)|ψ |2 drdx 2 dt − 2 − − (48) ZΩZr0 1 ∞ (cid:0) (cid:1) ∞ ∞ = (div v)|ψ |2+τφψ ∂ ψ drdx+2 ψ β(r˜,·)κ(r,r˜)ψ (·,r˜)dr˜drdx, x − − r + − + 2 ZΩZr0 ZΩZr0 Zr where we have used the fact that v·n = 0 on ∂Ω. Then, the first term on the right-hand side is identically zero since div v = 0. The same is true for the second term on the right-hand side since ψ ∂ ψ = 0. Finally, since β x − r + is nonnegative, we see that the last term is nonpositive. Consequently, we get d kψ (t)k2 ≤0, dt − 2 and since we have assumed that ψ ≥ 0, we deduce that ψ ≥ 0 almost everywhere. Using a similar procedure, we 0 obtain an identical result also for φ. (cid:3) Our next estimate is a maximum principle for φ. Lemma3.2(Maximumprincipleforφ). ThereexistsaconstantC,dependingonlyonK,suchthat,ifφ ∈L∞(Ω), 0 then (49) ess.sup kφ(t)k ≤max(K2,kφ k ). t∈(0,T) ∞ 0 ∞ Proof. We begin the proof with a pointwise bound on the right-hand side of (10). Using the nonnegativity of ψ and φ (Lemma 3.1) and the assumptions on τ and β and the definition of κ, we observe that ∞ r0 ∞ −φ ∂ (rτ)ψdr+2 r β(r˜,·)κ(r,r˜)ψ(r˜)dr˜dr r Zr0 Z0 Zr0 ∞ r0 ∞ ψ(r˜) ≤−φ (τ +r∂ τ)ψdr+K r dr˜dr r r˜ Zr0 Z0 Zr0 ∞ r0 ∞ ψ(r˜) ≤−K−1r φ ψdr+K r dr˜dr 0 r Zr0 Z0 Zr0 0 ∞ Kr ∞ =−K−1r φ ψdr+ 0 ψdr 0 2 Zr0 Zr0 ∞ ≤−K−1r ψ(r)dr φ−K2 0 (cid:18)Zr0 (cid:19) ∞ (cid:0) (cid:1) ≤−K−1r ψ(r)dr (φ−M), 0 (cid:18)Zr0 (cid:19) where M is defined as M :=max(K2,kφ k ). Hence, by multiplying (47) with (φ−M) and integrating over Ω, 0 ∞ + we get (using integration by parts, the Neumann data and the fact that the velocity is a solenoidal function) that d k(φ−M) k2 ≤0. dt + 2 Consequently, we immediately arrive at (49). (cid:3) The next result concerns conservation of mass. Lemma 3.3 (Conservationof mass). Let the pair of functions (ψ,φ) be a solution to (9), (10); then, the following identity holds: d ∞ (50) φ(t,x)dx+ r ψ(t,x,r)dxdr =0. dt (cid:20)ZΩ Zr0 ZΩ (cid:21) Consequently, for a.a. t∈(0,T), ∞ ∞ (51) φ(t,x)dx+ r ψ(t,x,r)dxdr = φ (x)dx+ r ψ (x,r)dxdr =:E . 0 0 0 ZΩ Zr0 ZΩ ZΩ Zr0 ZΩ $LaTeX: 2016/1/8 $ ANALYSIS OF A VISCOSITY MODEL FOR CONCENTRATED POLYMERS 9 Proof. First, we integrate (9) over Ω. Since div v =0, the transport term is equal to div (vψ) and then both the x x transport and diffusion terms vanish thanks to v·n = 0 on ∂Ω and the zero Neumann boundary condition for ψ. Hence, we obtain ∂ ψ(t,x,r)dx+ φ(t,x)τ∂ ψ(t,x,r)dx t r (52) ZΩ ZΩ ∞ =− β(r,·)ψ(t,x,r)dx+2 κ(r,r˜)β(r˜,·)ψ(t,x,r˜)dr˜dx. ZΩ ZΩZr In the next step we multiply (52) by r and integrate over the interval (r ,∞). Consequently, 0 d ∞ ∞ r ψ(t,x,r)dxdr+ rτφ(t,x)∂ ψ(t,x,r)dxdr r dt (53) Zr0 ZΩ Zr0 ZΩ ∞ ∞ ∞ =− rβ(r,·)ψ(t,x,r)dxdr+2 r κ(r,r˜)β(r˜,·)ψ(t,x,r˜)dr˜dxdr. Zr0 ZΩ Zr0 ZΩ Zr Finally, assuming that ψ vanishes sufficiently quickly at infinity1, we canintegrate by parts with respect to r (note that since τ(r )=0 the second boundary term also vanishes); this yields the identity 0 d ∞ ∞ r ψ(t,x,r)dxdr− ∂ (rτ)φ(t,x)ψ(t,x,r)dxdr r dt (54) Zr0 ZΩ Zr0 ZΩ ∞ ∞ ∞ =− rβ(r,·)ψ(t,x,r)dxdr+2 r κ(r,r˜)β(r˜,·)ψ(t,x,r˜)dr˜dxdr. Zr0 ZΩ Zr0 ZΩ Zr Next we integrate (10) overΩ. Similarly as above,after integrationby parts,the transportandthe diffusion terms vanish and we get d ∞ φ(t,x)dx=− φ(t,x)∂ (rτ)ψ(t,x,r)drdx r dt (55) ZΩ ZΩZr0 r0 ∞ +2 r β(r˜,·)κ(r,r˜)ψ(t,x,r˜)dr˜drdx. ZΩZ0 Zr0 Moreover,using the fact that κ(r,r˜)=0 for r˜<r , we can rewrite the last term to obtain the identity 0 d ∞ φ(t,x)dx=− φ(t,x)∂ (rτ)ψ(t,x,r)drdx r dt (56) ZΩ ZΩZr0 r0 ∞ +2 r β(r˜,·)κ(r,r˜)ψ(t,x,r˜)dr˜drdx. ZΩZ0 Zr Thus, by summing (54) and (56) and using Fubini’s theorem, we have that d ∞ φ(t,x)dx+ rψ(t,x,r)drdx dt (57) (cid:18)ZΩ ZΩZr0 (cid:19) ∞ ∞ ∞ =− rβ(r,·)ψ(t,x,r)drdx+2 r κ(r,r˜)β(r˜,·)ψ(t,x,r˜)dr˜drdx. ZΩZr0 ZΩZ0 Zr Finally, we evaluate the last term. Changing the order of integrationand using the fact that κ(r,r˜)=0 for r˜≤r , 0 we deduce that ∞ ∞ 2 r κ(r,r˜)β(r˜,·)ψ(t,x,r˜)dr˜drdx ZΩZ0 Zr ∞ ∞ =2 χ rκ(r,r˜)β(r˜,·)ψ(t,x,r˜)dr˜drdx r˜≥r ZΩZ0 Z0 ∞ r˜ =2 rκ(r,r˜)β(r˜,·)ψ(t,x,r˜)drdr˜dx ZΩZ0 Z0 ∞ r˜ ∞ (23) =2 rκ(r,r˜)dr β(r˜,·)ψ(t,x,r˜)dr˜dx = r˜β(r˜,·)ψ(t,x,r˜)dr˜dx. ZΩZr0 Z0 ! ZΩZr0 Consequently, we see that the right-hand side of (57) is identically zero, and we deduce (50). (cid:3) We shall now develop further bounds on the function φ. 1Attheleveloftheseformalcomputations,itisassumedthatψ vanishessufficientlyrapidlyatinfinity;theargumentwillbemade rigorousinSection4byfixingthefunctionspaceinwhichthetripe(v,ψ,φ),understoodasaweaksolutiontotheproblem,issought. $LaTeX: 2016/1/8 $ 10 M.BULÍCˇEK,P.GWIAZDA,E.SU¨LI,ANDA.S´WIERCZEWSKA-GWIAZDA Lemma 3.4 (Parabolic regularity of φ). Let ψ , φ be nonnegative, φ ∈L∞(Ω) and E <∞ (cf. (51)); then, 0 0 0 0 (58) |∇φ|2dxdt≤C(kφ k ,E ,K,T). 0 ∞ 0 ZQ Proof. Let us rewrite equation (10) as follows: (59) ∂ φ+v·∇ φ−A ∆φ=F, t x 0 where F is the right-handside of (10). First, we deduce a pointwise bound on F. Using the minimum principle for ψ in conjunction with the properties of β and τ, we get ∞ r0 ∞ F(t,x)=−φ(t,x) ∂ (rτ)ψ(t,x,r)dr+2 r β(r˜,·)κ(r,r˜)ψ(t,x,r˜)dr˜dr r Zr0 Z0 Zr0 r0 ∞ r0 ∞ ≤2 r β(r˜,·)κ(r,r˜)ψ(t,x,r˜)dr˜dr ≤2K r κ(r,r˜)ψ(t,x,r˜)dr˜dr (60) Z0 Zr0 Z0 Zr0 ∞ r0 ∞ =2K rκ(r,r˜)dr ψ(t,x,r˜)dr˜≤K r ψ(t,x,r˜)dr˜ 0 Zr0 (cid:18)Z0 (cid:19) Zr0 ∞ ≤K rψ(t,x,r)dr. Zr0 Thus, uponmultiplying (59)by φ, integratingoverΩ, usingthe nonnegativityofφ, partialintegration(notingthat all boundary terms again vanish) together with (51) and Lemma 3.2 give: 1 d ∞ kφ(t)k2+A k∇φ(t)k2 ≤K φ(t,x) rψ(t,x,r)drdx 2 dt 2 0 2 ZΩ Zr0 ∞ ≤Kkφ(t)k rψ(t,x,r)drdx ∞ ZΩZr0 (49),(51) ≤ KE max(K2,kφ k ). 0 0 ∞ Hence(58)directlyfollows. Wenoteinpassingthatwehavenotexplicitlyindicatedthedependenceoftheconstant C(kψ k ,E ,K,T) appearing in (58) on A and |Ω|. (cid:3) 0 ∞ 0 0 The next step is to establish a bound on high-order moments of the function ψ. Lemma 3.5 (High-ordermomentsofψ). Let ψ be nonnegative and suppose that it satisfies (9); then, for all α≥0, ∞ ∞ (61) ess.sup rαψ(t,x,r)drdx≤C(kφ k ,α,K,T) rαψ (x,r)drdx. t∈(0,T) 0 ∞ 0 ZΩZr0 ZΩZr0 Proof. We first integrate (9) over Ω to get ∞ ∂ ψ(t,x,r)dx=− β(r,·)ψ(t,x,r)dx+2 β(r˜,·)κ(r,r˜)ψ(t,x,r˜)dr˜dx t ZΩ ZΩ ZΩZr − τ(r)φ(t,x)∂ ψ(t,x,r)dx. r ZΩ We then multiply the result by rα, where α≥0, and integrate over (r ,∞) to deduce that 0 d ∞ ∞ rαψ(t,x,r)drdx=− rαβ(r,·)ψ(t,x,r)drdx dt ZΩZr0 ZΩZr0 ∞ ∞ ∞ +2 rα β(r˜,·)κ(r,r˜)ψ(t,x,r˜)dr˜drdx− τ(r)rαφ(t,x)∂ ψ(t,x,r)drdx. r ZΩZr0 Zr ZΩZr0 $LaTeX: 2016/1/8 $