Existence, regularity and weak-strong uniqueness for the three-dimensional Peterlin viscoelastic model
EExistence, regularity and weak-strong uniqueness for thethree-dimensional Peterlin viscoelastic model
Aaron Brunk Yong Lu M´aria Luk´aˇcov´a-Medvid’ov´aFebruary 5, 2021 ∗ Institute of Mathematics, Johannes Gutenberg-University MainzStaudingerweg 9, 55128 Mainz, [email protected], [email protected] † Department of Mathematics, Nanjing University22 Hankou Road, Goulou District, 210093 Nanjing, [email protected]
Abstract
In this paper we analyze the three-dimensional Peterlin viscoelastic model. By means of amixed Galerkin and semigroup approach we prove the existence of a weak solution. Furthercombining parabolic regularity with the relative energy method we derive a conditional weak-strong uniqueness result.
Contents a r X i v : . [ m a t h . A P ] F e b Parabolic Regularity and Conditional Energy Equality 17 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Conditional energy equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 We study a model for complex viscoelastic fluids where the Peterlin approximation is applied inorder to represent time evolution of the elastic conformation tensor, [33]. Dilute theory for complexfluids proposes that polymeric molecules can be represented by dumbbells that are suspended in aNewtonian solvent. A dumbbell is characterized by two beads connected by a spring. In general,a nonlinear spring force can be expressed as F ( R ) = γ ( | R | ) R , where R is the vector connectingthe two beads. The Peterlin approximation replaces the length of a spring by an averaged lengthof springs leading to F ( R ) ≈ γ ( (cid:104)| R | (cid:105) ) R . We note by passing that the well-known Oldroyd-Bmodel is based on the linear Hookean law, i.e. F ( R ) = H · R , where H is the spring constant. Instandard models for viscoelastic fluids diffusive terms are typically omitted yielding a hyperbolicequation for the time evolution of the elastic stress tensor. However, if the center-of-mass diffusionof polymer dumbbells is taken into account, a diffusive term arises in the evolution equation of theelastic stress tensor leading to a parabolic model, see e.g. [4] and the references therein.The aim of this paper is to extend and generalize our previous analytical results [25, 26] for thetwo-dimensional Peterlin viscoelastic model to three spaces dimensions. Besides showing global intime existence of weak solutions we also prove the weak-strong uniqueness principle by means ofthe relative energy method. The Peterlin viscoelastic model reads ∂ u ∂t + ( u · ∇ ) u = div (cid:0) η D u (cid:1) − ∇ p + div (cid:0) tr( C ) C (cid:1) ,∂ C ∂t + ( u · ∇ ) C = ( ∇ u ) C + C ( ∇ u ) (cid:62) + Φ(tr( C )) I − χ (tr( C )) C + ε ∆ C , (1.1)div (cid:0) u (cid:1) = 0 , T = tr( C ) C , where Φ := tr( C ) + a and χ := tr( C ) + a | tr( C ) | for a given a ≥
0. System (1.1) is consideredon Ω × (0 , T ), where Ω ⊂ R is a C ,β domain, β ∈ (0 , u , C ) | t =0 = ( u , C ) , u | ∂ Ω = , ∂ n C | ∂ Ω = . (1.2)The Peterlin model (1.1) of the incompressible Navier-Stokes equation, that are coupled in a non-linear way to a time evolution of the conformation tensor. The functions Φ , χ represent generalizedrelaxation terms.Complex viscoelastic fluids find their applications in everyday life. They are used to model poly-mers, blood or even in food industry. Mathematical literature dealing with analysis of viscoelasticfluid models is very broad. Existence and uniqueness of strong L p solutions for large times andsmall data or local-wellposedness for large data is studied in [17] for generalized Oldroyd-B orPeterlin models. A nice overview of available results on local existence of strong solutions can befound, e.g., in [15, 18] and the references therein. Global existence of weak solutions is a moredelicate problem. In [23] global existence of two- and three-dimensional cor-rotational Oldroyd-Bmodel has been proven. However, in the evolution equation for the elastic stress tensor ∇ u isreplaced by ( ∇ u − ∇ u (cid:62) ). In [13] local and global wellposedness in critical Besov spaces wasanalyzed. Non-blow up criteria for Oldroyd-B models were presented in [22]. Diffusive variationsof the Oldroyd-B model has been studied by Constantin and Kiegel [14] and Barrett and Boyavalin [1]. Global existence of weak solutions has been proven for two-dimensional viscoelastic fluids.In [30] the FENE-P model, that is based on the Peterlin approximation (P) and a finitely extensiblenonlinear elastic (FENE) spring potential has been studied and global existence of weak solutionshas been shown in three space dimensions. Existence of weak solutions for diffusive macro-micromodel based on the FENE or Hookean spring was studied by Barrett and S¨uli in [4, 5], seealso [7, 8, 24] for further developments for compressible viscoelastic fluids. We conclude thisintroductory part by referring to a recent work of Bathory, Buliˇcek and M´alek [10], where globalin time existence of weak solution have been proven for a special rate-type fluids in three spacedimensions.The diffusive Peterlin model has been studied from analytical an numerical point of view in ourrecent works [25, 27, 28, 19]. However due to missing a priori estimates for the conformation tensorall of these results are restricted to two-dimensional viscoelastic flows. In the present paper wecombine techniques from [2] and [26] in order to the extend global existence result to three spacedimensions.Moreover we analyze the properties and regularity of the conformation tensor. It turns out that thephysically relevant positive-(semi) definiteness, see [20], is crucial to obtain the existence result inthree space dimension. Furthermore, we prove a weak-strong uniqueness principle which is rarelyseen in these context except from [3]. In contrast to the Oldroyd-B model which is studied in [3]our model is more complex and needs a special treatment.The structure of this paper is as follows. In Section 2 we introduce suitable notation and recallsome necessary analytical tools. In Section 3 we state the concept of weak solution which we useand formulate the main results. Section 4 deals with the existence proof and the energy inequality,while Section 5 focuses on the parabolic regularity and the energy equality. Finally, in Section 63e apply the previous results in a relative energy method to obtain the conditional weak-stronguniqueness principle. Section 7 illustrates an possible application of the relative energy method inthe context of convergence of numerical schemes. In this section we introduce the notation and theoretical framework for the upcoming analysis ofthe Peterlin viscoelastic model (1.1). Let d ∈ { , } denote the space dimension. We denote byΩ T := Ω × (0 , T ) and Ω t := Ω × (0 , t ) the full space-time cylinder and the intermediate space-timecylinder, respectively. The norm of the Lebesgue space L p (Ω) is denoted by (cid:107)·(cid:107) p and the norm ofthe Bochner space L p (0 , T ; L q (Ω)) by (cid:107)·(cid:107) L p ( L q ) . Further, we set L (Ω) := C ∞ , div (Ω) (cid:107)·(cid:107) and L (Ω) d × dSP D := { D ∈ L (Ω) d × d | v T Dv > , ∀ v (cid:54) = 0 ∈ L ∞ (Ω) d } . Here C ∞ , div (Ω) stands for the set of smooth divergence free functions that are compactly supportedin Ω.We use the standard notation for the Sobolev spaces and introduce the notation V := H , div (Ω) d , H := L (Ω) d . Dual spaces of H (Ω) , W ,p (Ω) , V are denoted by H − (Ω) , W − ,p ∗ (Ω) , V ∗ , respectively. The defor-mation gradient is the symmetric part of the velocity gradient given byD u = 12 ( ∇ u + ∇ u (cid:62) ) . Proposition 2.1 ( L p -Matrix norm [31]) . For a matrix valued function D ∈ R d × d and p ≥ wehave (cid:107) tr( D ) (cid:107) pp := (cid:90) Ω (cid:32) d (cid:88) i =1 D ii (cid:33) p d x ≤ d p − (cid:90) Ω d (cid:88) i,j =1 | D ij | p d x =: d p − (cid:107) D (cid:107) pp . For symmetric positive(-semi) definite matrices both norms are equivalent, i.e. (cid:107) D (cid:107) pp ≤ (cid:107) tr( D ) (cid:107) pp ≤ d p − (cid:107) D (cid:107) pp . The norm (cid:107) tr( D ) (cid:107) p is the so-called trace norm. We denote by C : D = (cid:80) di =1 C ii D ii the Frobeniusinner product. Definition 2.2.
A symmetric matrix function C ( t ) ∈ R d × d can be decomposed as follows C ( t ) = Q ( t ) Λ ( t ) Q (cid:62) ( t )4or all t ∈ [0 , T ). We define the matrix logarithm log( C ( t )) for a symmetric positive definite (SPD)matrix function C ( t ) by ln C ( t ) = Q ( t ) ln( Λ )( t ) Q (cid:62) ( t ) . Furthermore for C ∈ C ([0 , T )) the following Jacobi formula holdsd C d t : C − = tr (cid:18) C − d C d t (cid:19) = dd t tr(ln C ) . (2.1) Lemma 2.3 ([2]) . Let D ∈ H (Ω) m × m ∩ C (Ω) m × m , m ∈ N , be a symmetric matrix function, whichis uniformly positive definite on Ω and satisfies homogeneous Neumann boundary conditions, then (cid:90) Ω ∆ D : D − d x = − (cid:90) Ω ∇ D : ∇ D − d x ≥ m (cid:90) Ω |∇ tr(log D ) | d x. (2.2) Lemma 2.4 (Gronwall) . Let f ∈ L ( t , T ) be non-negative and g, φ continuous functions on [ t , T ] .If φ satisfies φ ( t ) ≤ g ( t ) + (cid:90) tt f ( s ) φ ( s )d s, for all t ∈ [ t , T ] then φ ( t ) ≤ g ( t ) + (cid:90) tt f ( s ) g ( s ) exp (cid:18)(cid:90) ts f ( τ )d τ (cid:19) d s, for all t ∈ [ t , T ] . If g is moreover non-decreasing, then φ ( t ) ≤ g ( t ) exp (cid:18)(cid:90) tt f ( τ )d τ (cid:19) , for all t ∈ [ t , T ] . We proceed by recalling some regularity results for parabolic Neumann problems. We start byintroducing fractional-order Sobolev spaces. Let Ω be the whole space R d or a bounded Lipschitzdomain in R d . For any k ∈ N , β ∈ (0 ,
1) and s ∈ [1 , ∞ ), we define W k + β,s (Ω) := (cid:8) v ∈ W k,s (Ω) : (cid:107) v (cid:107) W k + β,s (Ω) < ∞ (cid:9) , where (cid:107) v (cid:107) W k + β,s (Ω) := (cid:107) v (cid:107) W k,s (Ω) + (cid:88) | α | = k (cid:18)(cid:90) Ω (cid:90) Ω | ∂ α v ( x ) − ∂ α v ( y ) | s | x − y | d + βs d x d y (cid:19) s . Consider the parabolic initial-boundary value problem: ∂ t ρ − ε ∆ ρ = h in Ω T ; ρ (0 , · ) = ρ in Ω; ∂ n ρ = 0 in (0 , T ) × ∂ Ω . (2.3)Here ε > ρ and h are known functions, and ρ is the unknown solution. In what follows theregularity result will be useful, see for example Section 7.6.1 in [32].5 emma 2.5. Let < β < , < p, q < ∞ , Ω ⊂ R d be a bounded C ,β domain with β ∈ (0 , , ρ ∈ W − p ,q n , h ∈ L p (0 , T ; L q (Ω)) , where W − p ,q n is the completion of the linear space { v ∈ C ∞ (Ω) : ∂ n v | ∂ Ω = 0 } with respect to thenorm of W − p ,q (Ω) . Then there exists a unique function ρ satisfying ρ ∈ L p (0 , T ; W ,q (Ω)) ∩ C ([0 , T ]; W − p ,q (Ω)) , ∂ t ρ ∈ L p (0 , T ; L q (Ω)) solving (2.3) in Ω T . In addition, ρ satisfies the Neumann boundary condition in (2.3) in the senseof the normal trace, which is well defined since ∆ ρ ∈ L p (0 , T ; L q (Ω)) . Moreover, ε − p (cid:107) ρ (cid:107) L ∞ ( W − p ,q ) + (cid:107) ∂ t ρ (cid:107) L p ( L q ) + ε (cid:107) ρ (cid:107) L p ( W ,q ) ≤ C ( p, q, Ω) (cid:2) ε − p (cid:107) ρ (cid:107) W − p ,q (Ω) + (cid:107) h (cid:107) L p ( L q ) (cid:3) . Lemma 2.6.
Let
X, Y be Banach spaces and assume X is reflexive and is continuously embeddedin Y , then L ∞ (0 , T ; X ) ∩ C w ([0 , T ]; Y ) = C w ([0 , T ]; X ) , see [8]. The aim of this section is to present the main results of the paper: the global existence of weaksolutions in three space dimensions and the weak-strong uniqueness principle. We start by definingthe weak solutions to the Peterlin viscoelastic system (1.1).
Definition 3.1.
Given the initial data ( u , C ) ∈ [ H × L (Ω) × SP D ]. The couple ( u , C ) iscalled a weak solution of (1.1) if u ∈ C w ([0 , T ]; L (Ω)) ∩ L (0 , T ; V ) ∩ C ([0 , T ]; L q (Ω)) ∩ W , (0 , T ; V ∗ ) , C ∈ C w ([0 , T ]; L (Ω)) ∩ L (0 , T ; H (Ω)) ∩ L (Ω T ) ∩ C ([0 , T ]; L q (Ω)) ∩ W , (0 , T ; H − (Ω)) , χ (tr( C )) C ∈ L (Ω T ) , Φ(tr( C )) ∈ L (Ω T ) , for any 1 ≤ q < For any t ∈ (0 , T ] and any test function v ∈ C ∞ ([0 , T ]; C ∞ c (Ω; R )) , (cid:90) t (cid:90) Ω u · ∂ v ∂t d x d τ − (cid:90) t (cid:90) Ω ( u · ∇ ) u · v d x d τ − (cid:90) t (cid:90) Ω η D u : D v d x d τ = (cid:90) t (cid:90) Ω tr( C ) C : ∇ v d x d τ + (cid:90) Ω u ( t ) · v ( t ) d x − (cid:90) Ω u · v (0) d x. (3.1) • For any t ∈ (0 , T ] and any test function D ∈ C ∞ ([0 , T ] × Ω; R × )) , (cid:90) t (cid:90) Ω C : ∂ D ∂t d x d τ − (cid:90) t (cid:90) Ω ( u · ∇ ) C : D d x d τ − (cid:90) t (cid:90) Ω ε ∇ C :: ∇ D d x d τ = (cid:90) t (cid:90) Ω χ (tr( C )) C : D d x d τ − (cid:90) t (cid:90) Ω Φ(tr( C )) tr D d x d τ + (cid:90) Ω C ( t ) : D ( t ) d x − (cid:90) Ω C : D (0) d x. (3.2) Theorem 3.2 (Existence of weak dissipative solutions) . For given initial data ( u , C ) ∈ [ H × L SP D (Ω) × ] and a final time T > there exists a global in time weak solution of thePeterlin viscoelastic system (1.1) in the sense of Definition 3.1. Moreover, it satisfies fora.e. t ∈ (0 , T ) the energy inequality (cid:18)(cid:90) Ω | u ( t ) | + 14 | tr( C ( t )) | d x (cid:19) + (cid:90) Ω t η | D u | + ε |∇ tr( C ) | + 12 | tr( C ) | + a | tr( C ) | d x d τ ≤ (cid:90) Ω t | tr( C ) | + a C ) d x d τ + (cid:18)(cid:90) Ω | u (0) | + 14 | tr( C (0)) | d x (cid:19) . (3.3) If a = 0 the conformation tensor C is symmetric positive semi-definite a.e. in Ω × [0 , T ) .If a > and the initial datum C satisfies additionally tr(log C ) ∈ L (Ω) then C is sym-metric positive definite a.e. in Ω × [0 , T ) and enjoys the additional regularity tr(log C ) ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) , tr( C − ) , tr( C − )tr( C ) ∈ L (0 , T ; L (Ω)) . Furthermore, for a.a. t ∈ (0 , T ) additionally the free energy inequality holds, i.e (cid:18)(cid:90) Ω | u ( t ) | + 14 | tr( C ( t )) | −
12 tr(log C ( t )) d x (cid:19) (3.4)+ (cid:90) Ω t η | D u | + ε |∇ tr( C ) | + ε |∇ tr (ln C ) | + 12 χ (tr( C ))tr( T + T − − I ) d x d τ (cid:18)(cid:90) Ω | u (0) | + 14 | tr( C (0)) | −
12 tr(log C (0)) d x (cid:19) . Remark 3.3.
Note that in (3.4) one can also write the energy inequality with ε (cid:12)(cid:12) C − / ∇ CC − / (cid:12)(cid:12) instead of ε |∇ tr (ln C ) | , see [29]. In this case energy equality holds for a smooth solution. Remark 3.4.
The positive definiteness condition a > u we obtain by parabolicregularity a solution ( u m , C m ), where C m is the classical solution of (1 . corresponding to finite-dimensional velocity u m . By the energy method and parabolic regularity we obtain approximationindependent bounds and pass to the limit in the equations for u m and C m and in the energyinequality. Furthermore, we prove the positive definiteness for a > Theorem 3.5 (Conditional energy equality) . Moreover, if the initial datum C ∈ W , n (Ω) ,then C satisfies the following higher-order estimates C ∈ L (0 , T ; W , (Ω)) + L ˜ s (0 , T ; W , ˜ r (Ω)) , ∂ C ∂t ∈ L (Ω T ) + L ˜ s (0 , T ; L ˜ r (Ω)) , (3.5) where (˜ s, ˜ r ) satisfies s + 3˜ r = 4 , < ˜ s < , < ˜ r < . (3.6) If the weak solution ( u , C ) obtained in Theorem 3.2 satisfies C ∈ L s (0 , T ; L r (Ω)) with s + 3 r ≤ , < s < ∞ , < r < ∞ , (3.7) then for almost all τ ∈ (0 , T ] there holds (cid:90) Ω | C ( τ ) | d x + (cid:90) τ (cid:90) Ω ε |∇ C | d x d t + (cid:90) τ (cid:90) Ω | tr( C ) | | C | + a | tr( C ) || C | d x d t = (cid:90) Ω | C | d x + (cid:90) τ (cid:90) Ω | tr( C ) | + a tr( C ) d x d t + (cid:90) τ (cid:90) Ω (cid:2) ( ∇ u ) C + C ( ∇ u ) (cid:62) (cid:3) : C d x d t. (3.8)We give some remarks on the additional integrability assumption.8 emark 3.6. • Following the well known Serrin type blow-up criterion on the Leray-Hopf weak solutions tothe three dimensional incompressible Navier-Stokes equations, see [35, 36, 11, 21], it can beshown that if the weak solution ( u , C ) satisfies u , C ∈ L s (0 , T ; L r (Ω)) with ( s, r ) satisfying(3.7), then the weak solution ( u , C ) is regular. • We only assume C ∈ L s (0 , T ; L r (Ω)), while the Serrin type criterion requires both u and C are in L s (0 , T ; L r (Ω)) with s, r satisfying (3.7). • A similar result can be obtained by assuming u ∈ L (Ω T ). • In the special case of two space dimensions the first result holds with ˜ s = ˜ r = . Furthermore,the second result holds without further integrability assumptions.We continue by introducing the relative energy E as the first-order Taylor expansion of the energy.Let u , U and C , H be two velocity vectors and conformation tensors, respectively. We introducethe relative energy E ( u , C | U , H ) := E kin + E el + E frob (3.9)where E kin ( u | U ) = 12 (cid:90) Ω | u − U | d x, E el ( C | H ) = 14 (cid:90) Ω | tr( C − H ) | d x, E frob ( C | H ) = 12 (cid:90) Ω | C − H | d x. Since the elastic relative energy is not definite, i.e. E el ( C | H ) = 0 (cid:59) C = H , we penalize therelative energy by (cid:107) C − H (cid:107) , i.e. the relative Frobenius energy.Due to the convexity the following properties of E hold1) E ( u , C | U , H )( t ) = 0 ⇐⇒ u ( t ) = U ( t ) , C ( t ) = H ( t ) a.e. in Ω , (3.10)2) E ( u , C | U , H ) ≥ E ( u , C | U , H ) ≥ (cid:107) u − U (cid:107) , (cid:107) tr( C − H ) (cid:107) , (cid:107) C − H (cid:107) . For further study it is convenient to set E ( u , C | U , H ) := E kin + E el , E ( C | H ) := E frob . (3.11)We note by passing that the relative energy resulting from the energy inequality (3.3) is moreconvenient for the forthcoming investigations than those derived from the free energy, cf. (3.4). Theorem 3.7 (Relative energy inequality) . Let ( u , C ) be a global weak solution of thePeterlin viscoelastic system (1.1) starting from the initial data ( u , C ) . Assume that C ∈ s (0 , T ; L r (Ω)) with ( s, r ) satisfying (3.7) . Let ( U , H ) be a more regular weak solution of(1.1) satisfying additionally U ∈ L (0 , T † ; L (Ω)) ∩ L (0 , T † ; H (Ω)) , H ∈ L ∞ (0 , T † ; L (Ω)) ∩ L (0 , T † ; H (Ω)) , for some T † ≤ T . Let ( U , H ) start from the initial data ( U , H ) . Then the relative energygiven by (3.9) satisfies the inequality E ( t ) + b D ≤ E (0) + (cid:90) t g ( τ ) E ( τ ) d τ, (3.12) for almost all t ∈ (0 , T † ) . Here g ∈ L (0 , T † ) , and D is given by (6.27) and b > . Let us comment about the above result.
Remark 3.8.
We note that Theorem 3.7 can be also proven by assuming u ∈ L (0 , T ; L (Ω)).The same result holds true with a relative energy (3.9) where E frob is replaced by λ E frob , for aarbitrary λ >
0. Thus, the impact of the penalty contribution can be made small.
Theorem 3.9 (Weak-strong uniqueness) . Let the assumptions of Theorem 3.7 hold and letthe initial data coincide, i.e. u = U , C = H . Then any weak solution ( u , C ) in the senseof Definition 3.1 coincides with the more regular weak solution ( U , H ) almost everywhere in Ω × (0 , T † ) . Remark 3.10. • The above result implies the (local) uniqueness in the class of more regular solutions. • In two space dimensions by reviewing Theorem 3.5, and the corresponding remark 3.8 andquick inspection of the arguments used in the proof of Theorem 3.7, we can conclude theuniqueness of weak solutions for the initial data C ∈ W , n (Ω). In order to analyse the Peterlin viscoelastic system (1.1) we consider several energy type estimatesas follows. By formally taking the inner product of (1 . with u and (1 . with tr( C ) I / t (cid:18)(cid:90) Ω | u | + 14 | tr( C ) | d x (cid:19) (4.1)10 (cid:90) Ω η | D u | + ε |∇ tr( C ) | + 12 (cid:0) | tr( C ) | + a | tr( C ) | (cid:1) − (cid:0) | tr( C ) | + a tr( C ) (cid:1) d x ≤ . Estimating the last integral of (4.1) by the H¨older inequality we find after applying the GronwallLemma 2.4 u ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) , (4.2)tr( C ) ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) ∩ L (Ω T ) . Since for a smooth solution the matrix C is positive definite we find also that C ∈ L (Ω T ), due tothe norm equivalence in Proposition 2.1. Now we can take the Frobenius inner product of (1 . with C / t (cid:18)(cid:90) Ω | C | d x (cid:19) + ε (cid:90) Ω |∇ C | + 12 (cid:0) | tr( C ) C | + a | tr( C ) || C | (cid:1) d x ≤ (cid:90) Ω tr( C ) + a tr( C ) d x + 2 (cid:90) Ω ( ∇ uC ) : C d x. (4.3)The first integral of (4.3) can be treated as in (4.1). The second integral of (4.3) can be boundedas follows 2 (cid:90) Ω ( ∇ uC ) : C d x ≤ (cid:107)∇ u (cid:107) (cid:107) C (cid:107) ≤ (cid:107)∇ u (cid:107) + (cid:107) C (cid:107) . (4.4)Using again the Gronwall Lemma 2.4 on (4.3) with (4.4) yields the regularity C ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) . (4.5)The free energy inequality (3.4) can be formally derived by taking the inner product of (1 . with u and (1 . with tr( C ) I / − C − / t (cid:16) (cid:90) Ω | u | + 14 | tr( C ) | −
12 tr(log C ) d x (cid:17) (4.6)+ (cid:90) Ω η | D u | + ε |∇ tr( C ) | − ε ∇ C : ∇ C − d x + 12 (cid:90) Ω | tr( C ) | + a | tr( C ) | − d | tr( C ) | + ad tr( C ) d x − (cid:90) Ω d | tr( C ) | + ad | tr( C ) | − tr( C )tr( C − ) − a tr( C − ) d x ≤ . To proceed, we first expand the diffusion term involving the inverse matrix by (cid:90) Ω ∇ C : ∇ (cid:0) C − (cid:1) d x = − d (cid:88) i =1 (cid:90) Ω ∂ C ∂x i : C − ∂ C ∂x i C − d x − d (cid:88) i =1 (cid:90) Ω tr (cid:18) C − / ∂ C ∂x i C − / C − / ∂ C ∂x i C − / (cid:19) d x = − d (cid:88) i =1 (cid:13)(cid:13)(cid:13)(cid:13) C − / ∂ C ∂x i C − / (cid:13)(cid:13)(cid:13)(cid:13) := − (cid:13)(cid:13) C − / ∇ CC − / (cid:13)(cid:13) ≤ − d (cid:107)∇ tr(log C ) (cid:107) . Here we have used the cyclic property of the trace, symmetry of C , the existence of a square root C / and Lemma 2.3. Rewriting (4.6) yieldsdd t (cid:18)(cid:90) Ω | u | + 14 | tr( C ) | −
12 tr(log C ) d x (cid:19) (4.7)+ (cid:90) Ω η | D u | + ε |∇ tr( C ) | + ε |∇ tr (ln C ) | d x + 12 (cid:90) Ω (cid:0) tr( C ) + a tr( C ) (cid:1) tr( T + T − − I ) d x ≤ . Since C is symmetric positive definite we obtain from (4.7) the additional informationtr(log C ) ∈ L ∞ (0 , T ; L (Ω)) , ∇ tr(log C ) , C − / ∇ CC − / ∈ L (Ω T ) ,a tr( C − ) , tr( C − )tr( C ) ∈ L (Ω T ) . The goal of this section is to derive an approximation scheme based on the Galerkin method forthe velocity u , see [26]. Let v j , j = 1 , . . . , ∞ be smooth basis functions of V = span { v j } ∞ j =1 . Herethe v j are divergence-free and subjected to the homogeneous Dirichlet boundary conditions. Thenwe define the m -th Galerkin approximation of u by u m ( x, t ) = m (cid:88) j =1 g jm ( t ) v j ( x ) , u m (0) = u m . (4.8)Furthermore, C m ( u m ) denotes the solution of the parabolic problem (1 . for C m . Due to standardtheory for ordinary differential equations there exists a finite-dimensional approximation of thevelocity u m . Uniform bounds imply the existence up to time T for all m . Further, the parabolicregularity, see [34], and the bounds on the velocity u m show that there is a conformation stresstensor C m ∈ C ((0 , T ]; C (Ω)). Since C m is positive definite for every m , see [26], we obtain thefollowing regularity result by integrating the Galerkin approximations of (4.1) and (4.3) in time u m ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) , (4.9)tr( C m ) ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) ∩ L (Ω T ) , C m ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) ∩ L (Ω T ) . Using (4.9) yields χ (tr( C m )) C m ∈ L (Ω T ) , Φ(tr( C m )) ∈ L (Ω T ) . (4.10)12 .2 Compact embeddings In order to derive suitable integrability of the time derivative we rewrite the Navier-Stokes (3.1)as operator equation ∂ u m ∂t = −A u m + B u m + H C m , (4.11)with the operators A , B , H defined by A : V → V ∗ (cid:104)A u , v (cid:105) := (cid:90) Ω η D u : D v , v ∈ V, B : V → V ∗ (cid:104)B u , v (cid:105) := − (cid:90) Ω ( u · ∇ ) u · v , v ∈ V, H : H (Ω) → V ∗ (cid:104)H C , v (cid:105) := − (cid:90) Ω tr ( C ) C : ∇ v , v ∈ V. By using Sobolev embedding, the following estimate holds (cid:90) T (cid:13)(cid:13)(cid:13)(cid:13) ∂ u m ∂t (cid:13)(cid:13)(cid:13)(cid:13) pV ∗ d t ≤ c (cid:90) T (cid:107) D u m (cid:107) p + (cid:107) tr( C m ) C m (cid:107) p + (cid:107) u m (cid:107) p (cid:107)∇ u m (cid:107) p d t. (4.12)Using the regularity result (4.9) we find that ∂ u m ∂t ∈ L (0 , T ; V ∗ ), i.e. taking p = in (4.12).Next we consider the evolution equation for the conformation tensor (1 . which can be rewrittenas an operator equation of the form ∂ C m ∂t + ε ∆ C m = F m (4.13)with F m := − ( u m · ∇ ) C m + ( ∇ u m ) C m + C m ( ∇ u m ) T − χ (tr ( C m )) C m + Φ(tr ( C m )) I . We calculate (cid:90) T (cid:107) F m (cid:107) pH − d t ≤ (cid:90) T (cid:107) u m (cid:107) p (cid:107)∇ C m (cid:107) p + (cid:107) C (cid:107) p (cid:107)∇ u m (cid:107) p + (cid:107) χ (tr ( C m )) C m (cid:107) p / + (cid:107) Φ(tr ( C m )) (cid:107) p d t. (4.14)Standard calculations using (4.9), (4.10) show that F m ∈ L (0 , T ; H − (Ω)), i.e. p = in (4.14).This implies by bootstrapping that ∂ C m ∂t ∈ L (0 , T ; H − (Ω)) , C m ∈ L (0 , T ; H (Ω)) . Using these estimates and the Aubin-Lions Lemma we have for suitable subsequences the followingconvergence results u m (cid:42) (cid:63) u ∈ L ∞ (0 , T ; L (Ω)) , C m (cid:42) (cid:63) C ∈ L ∞ (0 , T ; L (Ω)) , (4.15)13 m (cid:42) u ∈ L (0 , T ; V ) ∩ L (Ω T ) , C m (cid:42) C ∈ L (0 , T ; H (Ω)) ∩ L (Ω T ) , u m → u ∈ L (0 , T ; L p (Ω)) for p < , C m → C ∈ L (0 , T ; L p (Ω)) for p < , u m (cid:42) u a.e. in Ω × (0 , T ) , C m (cid:42) C a.e. in Ω × (0 , T ) ,∂ u m ∂t (cid:42) ∂ u ∂t ∈ L (0 , T ; H − (Ω)) , ∂ C m ∂t (cid:42) ∂ C ∂t ∈ L (0 , T ; H − (Ω)) . Furthermore, considering the Galerkin approximation of the energy inequality (4.7) we obtain (cid:13)(cid:13) tr( C m ) − C m ) (cid:13)(cid:13) L ∞ ( L ) + (cid:107)∇ tr(log C m ) (cid:107) L ( L ) + (cid:13)(cid:13) C − / m ∇ C m C − / m (cid:13)(cid:13) L ( L ) ≤ c, (4.16) a (cid:13)(cid:13) tr( C − m ) (cid:13)(cid:13) L ( L ) + (cid:13)(cid:13) tr( C m )tr( C − m ) (cid:13)(cid:13) L ( L ) ≤ c, tr(log C m ) (cid:42) tr(log C m ) ∈ L (0 , T ; H (Ω)) . In this section we will pass to the limit in the Galerkin approximation of (3.1) and (3.2) as m → ∞ . Here we will focus on the limiting process in the main nonlinearities of (3.1) and (3.2).Let ϕ ∈ L ∞ (0 , T ) be a time dependent test function. We start with the Galerkin approximationof the Navier-Stokes equations (3.1) and consider the elastic stress tensor term P ,m := (cid:90) T (cid:90) Ω (tr( C m ) C m − tr( C ) C ) : ∇ v ϕ d x d t = (cid:90) T (cid:90) Ω (tr( C m − C )) C m : ∇ v ϕ + tr( C )( C m − C ) : ∇ v ϕ d x d t ≤ (cid:90) T ( (cid:107) C m − C (cid:107) (cid:107) tr( C ) (cid:107) + (cid:107) tr( C m − C ) (cid:107) (cid:107) C m (cid:107) ) (cid:107)∇ v (cid:107) (cid:107) ϕ (cid:107) ∞ d t ≤ c (cid:16) (cid:107) C m − C (cid:107) L ( L ) (cid:107) tr( C ) (cid:107) L ( L ) + (cid:107) tr( C m − C ) (cid:107) L ( L ) (cid:107) C m (cid:107) L ( L ) (cid:17) . Since tr( C m ) , C m are strongly convergent to tr( C ) , C in L (0 , T ; L (Ω)), cf. (4.15), we get P ,m → m → ∞ .We now turn to the conformation tensor equation (3 .
2) and first consider P ,m := (cid:90) T (cid:90) Ω χ (tr( C m )) C m : D ϕ ( t ) d x d t. (4.17)The integrand of P ,m is bounded in L r (Ω T ) for r = + , which yields r = >
1. Therefore bythe Vitali Lemma (4.17) converges to its limit P since the integrand is continuous and convergenta.e. in Ω T , cf. (4.15).Finally, we consider the upper convective derivative P ,m := (cid:90) T (cid:90) Ω (cid:104) ( ∇ u m ) C m − ( ∇ u ) C + C m ( ∇ u m ) T − C ( ∇ u ) T (cid:105) : D ϕ d x d t (4.18)14 (cid:90) T (cid:90) Ω (cid:104) ( ∇ u m − ∇ u ) C m + ∇ u ( C m − C ) + C m ( ∇ u m − ∇ u ) T (4.19)+ ( C m − C )( ∇ u ) T (cid:105) : D ϕ d x d t. Thanks to the strong convergence of C m in L (0 , T ; L (Ω)) and the weak convergence of ∇ u m in L (Ω T ), cf. (4.15), P ,m → m → ∞ . The limit process in other terms is standard, cf. [26]. In this section we will consider the limiting process in the energy inequalities (3.3), (3.4). Fur-thermore, for the limit in the free energy inequality (3.4) we need to prove that the limitingconformation tensor C is positive definite a.e. in Ω T .First we consider the limit in the discrete version of (4.1), c.f. (3.3). We observe that due to theconvergence given by (4.15) we can apply the same arguments as in [12] to derive (cid:18)(cid:90) Ω | u ( t ) | + 14 | tr( C ( t )) | d x (cid:19) + (cid:90) Ω t η | D u | + ε |∇ tr( C ) | + 12 (cid:0) | tr( C ) | + a | tr( C ) | (cid:1) d x d τ ≤ (cid:90) Ω t | tr( C ) | + a tr( C ) d x d τ + (cid:18)(cid:90) Ω | u (0) | + 14 | tr( C (0)) | d x (cid:19) . (4.20)Here we have used the strong convergence of C m , cf. (4.15), to pass to the limit in the first integralon the right hand side of (4.20).In what follows we want to prove a similar limit for the Galerkin approximation of (3.4). Here wefollow the ideas in [1, 3, 6, 9]. In order to identify the limit correctly we first need to prove thepositive definiteness of the limit C , since all approximations C m are positive definite by construc-tion. Repeating the same calculations yielding to (4.7) for the Galerkin approximations we deducethat (cid:90) Ω T tr( C − m ) d x d t ≤ c ( a ) , (4.21)where the constant c ( a ) depends inversely on a , i.e. it blows up for a →
0. Estimate (4.21) impliesby using the positive definiteness of C m the following estimates on C − m (cid:90) Ω T (cid:12)(cid:12) C − m (cid:12)(cid:12) d x d t, (cid:90) Ω T tr( C − m ) d x d t ≤ c ( a ) . (4.22)With these result at hand we can prove the following useful lemma by contradiction. Lemma 4.1.
Let a > and C be the limit of the sequence of positive definite solutions C m of (4.13). Then the limit C is positive definite a.e. in Ω T . If a = 0 we can conclude positivesemi-definiteness of the limit solution C a.e. in Ω T . roof. Assume the existence of a set D ⊂ Ω T having non-zero measure such that C ( x, t ) is notpositive definite for ( x, t ) ∈ D . By construction C is the limit of positive definite sequence C m which yields that C is positive semi-definite, due to the strong convergence of C m in L (Ω T ), cf.(4.15).This implies that C has at least one zero eigenvalue in D . Thus, there exists a vector function v ∈ L ∞ (Ω T ) d such that | v | = 1 in D and v = in Ω T \ D , such that v T Cv = 0 a.e. in Ω T . Weestimate the measure of D by | D | = (cid:90) D | v | d x d t = (cid:90) Ω T | v | d x d t = (cid:90) Ω T (cid:12)(cid:12) C − / m C / m v (cid:12)(cid:12) d x d t ≤ (cid:18)(cid:90) Ω T (cid:12)(cid:12) C − / m (cid:12)(cid:12) d x d t (cid:19) (cid:18)(cid:90) Ω T (cid:12)(cid:12) C / m v (cid:12)(cid:12) d x d t (cid:19) ≤ (cid:18)(cid:90) Ω T (cid:12)(cid:12) C − m (cid:12)(cid:12) d x d t (cid:19) (cid:18)(cid:90) Ω T (cid:12)(cid:12) v T C m v (cid:12)(cid:12) d x d t (cid:19) ≤ c ( a ) (cid:18)(cid:90) Ω T (cid:12)(cid:12) v T C m v (cid:12)(cid:12) d x d t (cid:19) . (4.23)It is easy to see that if a > c ( a ) is bounded and the right side of the inequality (4.23)converges as m → ∞ since C m converges strongly to C in L (Ω T ), cf. (4.15). However, byassumption v T Cv = 0 a.e. in Ω T . Consequently, | D | = 0, which is a contradiction and impliesthat C is positive definite a.e. in Ω T .In the case a > C − m ) = h ◦ C m , tr( C m )tr( C − m ) = h ◦ C m , tr(log C m ) = h ◦ C m , (4.24)where h i : (0 , ∞ ) → R , i = 1 , . . . , C m , C > T andconverge for a.e. ( x, t ) in Ω × (0 , T ), cf. (4.15), it is easy to see that h ◦ C m −→ h ◦ C , h ◦ C m −→ h ◦ C , h ◦ C m −→ h ◦ C a.e. in Ω T (4.25)see, e.g., [16, Exercise 2.37]. Applying (4.25) for (4 . we conclude that tr(log C m ) = tr(log C )a.e. in Ω T , i.e. tr(log C m ) (cid:42) tr(log C ) ∈ L (0 , T ; H (Ω)) . (4.26)We proceed with the terms in (4 . . Since C m is positive definite a.e. in Ω T we conclude byvirtue of C m C − m = I that C − m > × (0 , T ). Consequently, we obtain tr ( C − m ) > T . Application of the Fatou Lemma yields (cid:90) Ω t tr( C − ) d x d τ = (cid:90) Ω t tr( C − m ) d x d τ ≤ lim inf m →∞ (cid:90) Ω t tr( C − m ) d x d τ. (4.27)16e have used here tr( C − m ) = tr( C − ) a.e. in Ω T . Similarly, we derive for a.a. t ∈ (0 , T ) (cid:90) Ω t tr( C − )tr( C ) d x d τ = (cid:90) Ω t tr( C − m )tr( C m ) d x d τ ≤ lim inf m →∞ (cid:90) Ω t tr( C − m )tr( C m ) d x d τ. (4.28)Having obtained the convergences (4.15), (4.26), (4.27), (4.28) we can pass to the limit in the freeenergy inequality and obtain for a.a. t ∈ (0 , T ) (cid:18)(cid:90) Ω | u ( t ) | + 14 | tr( C ( t )) | −
12 tr(log C ( t )) d x (cid:19) (4.29)+ (cid:90) Ω t η | D u | + ε |∇ tr( C ) | + ε |∇ tr (ln C ) | + 12 (cid:0) tr( C ) + a tr( C ) (cid:1) tr( T + T − − I ) d x d τ ≤ (cid:18)(cid:90) Ω | u (0) | + 14 | tr( C (0)) | −
12 tr(log C (0)) d x (cid:19) . This section is devoted the proof of Theorem 3.5. We shall show that the weak solution ( u , C ) ob-tained in Theorem 3.2 satisfies the energy equality (3.8) under additional integrability assumption C ∈ L s (0 , T ; L r (Ω)) with ( s, r ) satisfying (3.7). We rewrite equation (1.1) for each component C i,j , ≤ i, j ≤ , as ∂ C i,j ∂t − ε ∆ C i,j = h ,i,j + h ,i,j , in Ω T ,∂ n C i,j = 0 on (0 , T ) × ∂ Ω , C i,j (0 , · ) = ( C ) i,j in Ω , (5.1)where h := − ( u · ∇ ) C , h := ( ∇ u ) C + C ( ∇ u ) (cid:62) + Φ(tr( C )) I − χ (tr( C )) C . (5.2)Using the integrability of a weak solution in Definition 3.1, and applying H¨older’s inequality gives h ∈ L (0 , T ; L (Ω)) ∩ L (0 , T ; L (Ω)) , h ∈ L (Ω T ) . (5.3)By interpolation, we derive h ∈ L ˜ s (0 , T ; L ˜ r (Ω)) , s + 3˜ r = 4 , ≤ ˜ s ≤ , ≤ ˜ r ≤ . (5.4)17n particular, choosing ˜ s = ˜ r gives h ∈ L (0 , T ; L (Ω)). Hence, for each 1 ≤ i, j ≤
3, the sourceterm in (5.1) satisfies h ,i,j + h ,i,j ∈ L (0 , T ; L (Ω)). Recall that the initial datum C ∈ W , n (Ω) ⊂ W , n (Ω) . Thus, applying Lemma 2.5 implies that the unique solution C i,j to (5.1) satisfies C ∈ L (0 , T ; W , (Ω)) , ∂ t C ∈ L (0 , T ; L (Ω)) . (5.5)On the other hand, we can decompose this unique solution C i,j = C (1) i,j + C (2) i,j where C (1) i,j and C (2) i,j solves ∂ C (1) i,j ∂t − ε ∆ C (1) i,j = h ,i,j , in Ω T ,∂ n C (1) i,j = 0 on (0 , T ) × ∂ Ω , C (1) i,j (0 , x ) = 0 in Ω (5.6)and ∂ C (2) i,j ∂t − ε ∆ C (2) i,j = h ,i,j , in Ω T ,∂ n C (2) i,j = 0 on (0 , T ) × ∂ Ω , C (2) i,j (0 , · ) = ( C ) i,j in Ω , (5.7)respectively. Using (5.3), (5.4) and applying Lemma 2.5 to (5.6), (5.7) gives C (1) i,j ∈ L ˜ s (0 , T ; W , ˜ r (Ω)) , ∂ C (1) i,j ∂t ∈ L ˜ s (0 , T ; L ˜ r (Ω)) , (5.8)and C (2) i,j ∈ L (0 , T ; W , (Ω)) , ∂ C (2) i,j ∂t ∈ L (Ω T ) , (5.9)where (˜ s, ˜ r ) satisfies (5.4) except the board-line cases ˜ s = 1 , ˜ r = and ˜ s = 2 , ˜ r = 1. Consequently,we have completed the proof of the first part of Theorem 3.5. We proceed with the proof of the the energy equality (3.8) under additional integrability assump-tion C ∈ L s (0 , T ; L r (Ω)) for some ( s, r ) satisfying (3.7) i.e.2 s + 3 r ≤ , < s < ∞ , < r < ∞ . (5.10)18ecall the main result of Section 5.1: C = C (1) + C (2) with C (1) ∈ L ˜ s (0 , T ; W , ˜ r (Ω)) , ∂ C (1) ∂t ∈ L ˜ s (0 , T ; L ˜ r (Ω)) , (5.11)and C (2) ∈ L (0 , T ; W , (Ω)) , ∂ C (2) ∂t ∈ L (Ω T ) , (5.12)for all (˜ s, ˜ r ) satisfying 2˜ s + 3˜ r = 4 , ≤ ˜ s ≤ , ≤ ˜ r ≤ . It is easy to see that the Lebesgue conjugate numbers (˜ s (cid:48) , ˜ r (cid:48) ) satisfy2˜ s (cid:48) + 3˜ r (cid:48) = 1 , ≤ ˜ s (cid:48) ≤ ∞ , ≤ ˜ r (cid:48) ≤ ∞ , which is a subcase of (5.10). Thus, C ∈ L (Ω T ) ∩ L s (0 , T ; L r (Ω)) ⊂ (cid:0) L (Ω T ) + L ˜ s (0 , T ; L ˜ r (Ω)) (cid:1) (cid:48) . Together with the energy estimates for u and C in (3.1), one can verify by a density argumentthat C can be chosen as a test function in (3.2). This gives for each t ∈ (0 , T ] that (cid:90) t (cid:90) Ω C : ∂ C ∂t d x d τ − (cid:90) t (cid:90) Ω ( u · ∇ ) C : C d x d τ − (cid:90) t (cid:90) Ω ε |∇ C | d x d τ = (cid:90) t (cid:90) Ω χ (tr( C )) C : C d x d τ − (cid:90) t (cid:90) Ω Φ(tr( C ))tr C d x d τ + (cid:90) Ω | C ( t ) | d x − (cid:90) Ω | C | d x. (5.13)We next claim that for a.a. t ∈ (0 , T ) (cid:90) t (cid:90) Ω C : ∂ C ∂t d x d τ = 12 (cid:90) Ω | C ( t ) | d x − (cid:90) Ω | C | d x, (cid:90) t (cid:90) Ω ( u · ∇ ) C : C d x d τ = 0 . (5.14)To prove (5.14), we employ the classical Friedrichs mollification in spatial variable. Let φ δ is astandard Friedrichs mollifier in R d with δ ∈ (0 ,
1) small. Define C δ ( t, x ) := φ δ ∗ C ( t, x ). Foreach t ∈ [0 , T ], here we treat C ( t, · ) as a function that is extended in W , (Ω ) where Ω := { x ∈ R d ; dist ( x, ∂ Ω) < } . It is easy to verify that for each 0 < δ < C δ ( t, · ) ∈ C ∞ c (Ω ) withΩ := { x ∈ R d ; dist ( x, ∂ Ω) < } that (cid:107) ∂ αx C δ ( t, · ) (cid:107) L ∞ ( R d ) ≤ c ( α, δ ) (cid:107) C ( t, · ) (cid:107) L (Ω) , for all α ∈ N .
19e start by showing the second equality in (5.14). We have (cid:90) t (cid:90) Ω ( u · ∇ ) C : C d x d τ = (cid:90) t (cid:90) Ω ( u · ∇ ) C : ( C − C δ ) d x d τ + (cid:90) t (cid:90) Ω ( u · ∇ )( C − C δ ) : C δ d x d τ + (cid:90) t (cid:90) Ω ( u · ∇ ) C δ : C δ d x d τ. (5.15)We first consider the last term in (5.15). Since C δ ( t, · ) ∈ C ∞ c ( R d ) and u ( t, · ) ∈ W , (Ω) , div ( u ) = 0for a.a. t ∈ (0 , T ), we obtain (cid:90) Ω ( u · ∇ ) C δ : C δ d x = (cid:90) Ω ( u · ∇ ) 12 | C δ | d x = − (cid:90) Ω div ( u ) 12 | C δ | d x = 0 . (5.16)We then consider the second term in (5.15) (cid:90) t (cid:90) Ω ( u · ∇ )( C − C δ ) : C δ d x d τ = (cid:90) t (cid:90) Ω (cid:0) ( u · ∇ ) C − (( u · ∇ ) C ) δ (cid:1) : C δ d x d τ + (cid:90) t (cid:90) Ω (cid:0) (( u · ∇ ) C ) δ − ( u · ∇ ) C δ (cid:1) : C δ d x d τ. (5.17)In (5.4) we have shown that( u · ∇ ) C ∈ L ˜ s (0 , T ; L ˜ r (Ω)) for all (˜ s, ˜ r ) such that 2˜ s + 3˜ r = 4 , ≤ ˜ s ≤ , ≤ ˜ r ≤ . (5.18)Then for each (˜ s, ˜ r ) satisfying (5.18), for a.e. t ∈ (0 , T ), there holds (cid:107) ( u · ∇ ) C ( t, · ) − (( u · ∇ ) C ) δ ( t, · ) (cid:107) ˜ r → , as δ → , (cid:107) ( u · ∇ ) C ( t, · ) − (( u · ∇ ) C ) δ ( t, · ) (cid:107) ˜ r ≤ (cid:107) ( u · ∇ ) C ( t, · ) (cid:107) ˜ r ∈ L ˜ s (0 , T ) . Lebesgue’s dominated convergence theorem implies that (cid:107) ( u · ∇ ) C − (( u · ∇ ) C ) δ (cid:107) L ˜ s ( L ˜ r ) → , as δ → . (5.19)We then obtain (cid:90) t (cid:90) Ω (cid:0) ( u · ∇ ) C − (( u · ∇ ) C ) δ (cid:1) : C δ d x d τ ≤ (cid:107) ( u · ∇ ) C − (( u · ∇ ) C ) δ (cid:107) L s (cid:48) ( L r (cid:48) ) (cid:107) C δ (cid:107) L s ( L r ) ≤ c (cid:107) ( u · ∇ ) C − (( u · ∇ ) C ) δ (cid:107) L ˜ s ( L ˜ r ) (cid:107) C (cid:107) L s ( L r ) , (5.20)which converges to 0 as δ →
0. Here we have used the fact that there exists (˜ s, ˜ r ) satisfying (5.18)such that s (cid:48) ≤ ˜ s, r (cid:48) ≤ ˜ r. To show the convergence of the second term in (5.17), we introduce the Friedrichs’ commutatorlemma, see , e.g., Lemma 6.7 in [32]. 20 emma 5.1.
Suppose that d ≥ . Let < q, β < ∞ and q + β ≤ . Let ≤ α ≤ ∞ and α + p ≤ .Suppose that ρ ∈ L α (0 , T ; L βloc ( R d )) , u ∈ L p (0 , T ; W ,qloc ( R d )) . Then (cid:107) ( u · ∇ ρ ) δ − ( u · ∇ ρ δ ) (cid:107) L s ( L rloc ( R d )) → as δ → with s = 1 α + 1 p , r = 1 q + 1 β . Moreover, u · ∇ ρ = div ( ρu ) − ρ div ( u ) . Since u ∈ L (0 , T ; W , (Ω)) and C ∈ L (Ω T ), Lemma 5.1 yields (cid:107) (( u · ∇ ) C ) δ − ( u · ∇ ) C δ (cid:107) L ( L ) → δ → . Together with the uniform boundedness (cid:107) C δ (cid:107) L ( L ) = (cid:107)(cid:107) C δ (cid:107) (cid:107) L (0 ,T ) ≤ (cid:107)(cid:107) C (cid:107) (cid:107) L (0 ,T ) ≤ (cid:107) C (cid:107) L ( L ) ≤ C, we deduce (cid:90) t (cid:90) Ω (cid:0) (( u · ∇ ) C ) δ − ( u · ∇ ) C δ (cid:1) : C δ d x d τ → . (5.21)For a.e. t ∈ (0 , T ) there holds (cid:107) C ( t, · ) − C δ ( t, · ) (cid:107) r → δ → , (cid:107) C ( t, · ) − C δ ( t, · ) (cid:107) r ≤ (cid:107) C ( t, · ) (cid:107) r ∈ L s (0 , T ) . (5.22)Lebesgue’s dominated convergence theorem implies that (cid:107) C − C δ (cid:107) L s ( L r ) → δ → . (5.23)Together with the estimates ( u · ∇ ) C ∈ L s (cid:48) (0 , T ; L r (cid:48) (Ω), applying H¨older’s inequality implies (cid:90) t (cid:90) Ω ( u · ∇ ) C : ( C − C δ ) d x d τ → δ → . (5.24)By (5.15), (5.16), (5.17), (5.20), (5.21) and (5.24), we obtain (5.14) .Now we consider the first equality in (5.14). Similarly as (5.19) and (5.23) we can show (cid:107) C − C δ (cid:107) L ( L ) → , (cid:13)(cid:13)(cid:13)(cid:13) ∂ C ∂t − ∂ C δ ∂t (cid:13)(cid:13)(cid:13)(cid:13) L ( L )+ L s (cid:48) ( L r (cid:48) ) → . (5.25)21his together with (5.23) implies (cid:90) t (cid:90) Ω C : ∂ C ∂t d x d τ = lim δ → (cid:90) t (cid:90) Ω C δ : ∂ C δ ∂t d x d τ. (5.26)Standard density argument with respect to the time variable implies ∂ | C δ | ∂t = 2 C δ : ∂ C δ ∂t , (5.27)where ∂ | C δ | ∂t stands for the weak time derivative of | C δ | . Define F δ ( t ) := 12 (cid:90) Ω | C δ ( t ) | d x. We have the following lemma for F δ : Lemma 5.2.
The function F δ ∈ L ∞ (0 , T ) and F δ admits a weak derivative F (cid:48) δ ∈ L (0 , T ) . More-over F (cid:48) δ ( t ) = 12 (cid:90) Ω ∂ | C δ | ∂t d x = (cid:90) Ω C δ : ∂ C δ ∂t d x. Proof of Lemma 5.2.
Let φ n ( x ) ∈ C ∞ c (Ω) such that φ n = 1 on (cid:26) x ∈ Ω; dist( x, ∂ Ω) > n (cid:27) . Clearly φ n → L r (Ω) , for all 1 ≤ r < ∞ . Given ψ ∈ C ∞ c (0 , T ) we compute (cid:90) T F δ ( t ) ψ (cid:48) ( t ) d t = 12 (cid:90) T (cid:90) Ω | C δ ( t ) | d x ψ (cid:48) ( t ) d t = 12 (cid:90) T (cid:90) Ω | C δ ( t ) | ψ (cid:48) ( t ) φ n ( x ) d x d t + 12 (cid:90) T (cid:90) Ω | C δ ( t ) | ψ (cid:48) ( t )(1 − φ n ( x )) d x d t. (5.28)Direct calculation gives12 (cid:90) T (cid:90) Ω | C δ ( t ) | | ψ (cid:48) ( t ) || (1 − φ n ( x )) | d x d t ≤ (cid:107) C δ (cid:107) L ( L ) (cid:107) ψ (cid:48) (cid:107) L ∞ (0 ,T ) (cid:107) − φ n (cid:107) → , as n → ∞ . (5.29)22y (5.27) we derive12 (cid:90) T (cid:90) Ω | C δ ( t ) | ψ (cid:48) ( t ) φ n ( x ) d x d t = − (cid:90) T (cid:90) Ω ∂ | C δ | ∂t ψ ( t ) φ n ( x ) d x d t = − (cid:90) T (cid:90) Ω C δ : ∂ C δ ∂t ψ ( t ) φ n ( x ) d x d t = − (cid:90) T (cid:90) Ω C δ : ∂ C δ ∂t ψ ( t ) d x d t − (cid:90) T (cid:90) Ω C δ : ∂ C δ ∂t ψ ( t )( φ n ( x ) −
1) d x d t. (5.30)Since C ∈ L ∞ (0 , T ; L (Ω)) and ∂ C ∂t ∈ L (Ω T ) + L (0 , T ; L (Ω)), we obtain C δ ∈ L ∞ (0 , T ; C ∞ c ( R d )) , ∂ C δ ∂t ∈ L (0 , T ; C ∞ c ( R d )) . Thus, (cid:90) T (cid:90) Ω | C δ | (cid:12)(cid:12)(cid:12)(cid:12) ∂ C δ ∂t (cid:12)(cid:12)(cid:12)(cid:12) | ψ ( t ) || ( φ n ( x ) − | d x d t ≤ (cid:107) C δ (cid:107) L ∞ (Ω T ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ C δ ∂t (cid:13)(cid:13)(cid:13)(cid:13) L ( L ) (cid:107) ψ (cid:107) L ∞ (0 ,T ) (cid:107) φ n − (cid:107) → , as n → ∞ . (5.31)Applying (5.28)–(5.31) we obtain (cid:90) T F δ ( t ) ψ (cid:48) ( t ) d t = − (cid:90) T (cid:18)(cid:90) Ω C δ : ∂ C δ ∂t d x (cid:19) ψ ( t ) d t. (5.32)Together with the fact C δ ∈ L ∞ (0 , T ; C ∞ c ( R d )) and ∂ C δ ∂t ∈ L (0 , T ; C ∞ c ( R d )), we finally deduce F (cid:48) δ ( t ) = (cid:90) Ω C δ : ∂ C δ ∂t d x ∈ L (0 , T ) . (5.33)Lemma 5.2 implies that F δ ∈ W , (0 , T ). The Sobolev embedding implies that F δ ∈ C , ([0 , T ])is H¨older continuous. Consequently, it is rather easy to verify by density argument that for each t ∈ [0 , T ] there holds (cid:90) t F (cid:48) δ ( τ ) d τ = F δ ( t ) − F δ (0) . (5.34)Since C ∈ C w ([0 , T ]; L (Ω)) then for a.e. t ∈ (0 , T ), we have F δ (0) = 12 (cid:90) Ω | C ,δ | d x → (cid:90) Ω | C | d x, F δ ( t ) = 12 (cid:90) Ω | C δ ( t ) | d x → (cid:90) Ω | C ( t ) | d x. (5.35)Hence, by (5.26), (5.27), Lemma 5.2, (5.34) and (5.34), we finally deduce (5.14) . We thus finishedthe proof of (5.14). As a consequence, together with (5.13), we obtain the equality (3.8) andcomplete the proof of Theorem 3.5. 23 Relative Energy
The goal of this section is to expand the relative energy (3.9) by using the corresponding energyinequalities and inserting the weak and more regular solutions as suitable test functions. We recallthat (3.3) can be written in the following abstract form: E ( u , C )(0) + (cid:90) t D ( u , C ) d τ ≤ E ( u , C )(0) + (cid:90) t F ( C ) d τ, (6.1) D ( u , C ) = (cid:90) Ω η | D u | + ε |∇ tr( C − H ) | + 12 χ (tr( C ))tr( C ) d x, (6.2) F ( C ) = (cid:90) Ω
12 Φ(tr( C ))tr( C ) d x. (6.3)To this end we expand the relative energy (3.9) as follows E ( t ) = E ( u , C )( t ) + E ( U , H )( t ) − R ( u , C | U , H )( t ) , E ( t ) − E (0) = − (cid:90) t D ( u , C ) − F ( C ) + D ( U , H ) − F ( H ) d τ − (cid:90) t ∂∂t R ( u , C | U , H ) d τ. Here we used the energy inequality/equality, respectively and the reminder is defined by R ( u , C | U , H ) = (cid:90) Ω u · U + 12 tr( C )tr( H ) − | C − H | d x. Next, we will expand the time derivative of the reminder − (cid:90) t ∂∂t R ( u , C | U , H ) d τ = I (cid:122) (cid:125)(cid:124) (cid:123) − (cid:90) Ω t U · ∂∂t u + u · ∂∂t U d x d τ (6.4) I (cid:122) (cid:125)(cid:124) (cid:123) − (cid:90) Ω t tr (cid:18) ∂∂t C (cid:19) tr( H ) + tr( C )tr (cid:18) ∂∂t H (cid:19) d x d τ (6.5)+ 12 (cid:90) Ω t ∂∂t | C − H | d x d τ (cid:124) (cid:123)(cid:122) (cid:125) I . (6.6)First we focus on the terms I and I . The term I will be treated separately. In what follows wewill apply the H¨older, the Young and interpolation inequalities repeatedly.For the term I we insert v = U in weak formulation (3.1) for ( u, C ) and v = u into a weakformulation for ( U , C ). Note that by density and the assumptions for this theorem, these are valid24est functions, and we obtain I = (cid:90) Ω t ( u · ∇ ) u · U + ( U · ∇ ) U · u + 2 η D u : D U + tr( C ) C : ∇ U + tr( H ) H : ∇ u d x d τ. (6.7)We star with the viscosity term by considering I and the velocity gradient terms from D ( u , C ), D ( U , H ) P := (cid:90) Ω t − η | D u | − η | D U | + 2 η D u : D U d x d τ = − (cid:90) Ω t η | D u − D U | d x d τ. (6.8)Next we deal with the convective terms. Applying integration by parts and adding the zero − (( u − U ) · ∇ ) U · U in the third line below yields P := (cid:90) Ω t ( u · ∇ ) u · U + ( U · ∇ ) U · u d x d τ = (cid:90) Ω t ( u · ∇ ) u · U − ( U · ∇ ) u · U d x d τ = (cid:90) Ω t (( u − U ) · ∇ ) u · U − (( u − U ) · ∇ ) U · U d x d τ = (cid:90) Ω t (( u − U ) · ∇ )( u − U ) · U d x d τ ≤ c (cid:90) t (cid:107) u − U (cid:107) (cid:107)∇ u − ∇ U (cid:107) (cid:107) U (cid:107) d τ ≤ δ (cid:90) t (cid:107) D u − D U (cid:107) + c ( δ ) (cid:90) t (cid:107) U (cid:107) E ( u , C | U , H )( τ ) d τ. (6.9)Note that we have used the regularity U ∈ L (0 , T ; L (Ω)). Here and hereafter δ, c ( δ ) > (cid:90) Ω t tr( C ) C : ∇ U + tr( H ) H : ∇ u d x d τ (6.10)will be treated together with the trace part of the relative energy later.For the term I we insert D = C in weak formulation (3.2) for ( u, C ) and D = H into a weakformulation for ( U , C ). Note that by density and the assumptions for this theorem, these are validtest functions, and we find I = (cid:90) Ω t u · ∇ tr( C )tr( H ) + 12 U · ∇ tr( H )tr( C ) − tr( C ∇ u )tr( H ) + tr( H ∇ U )tr( C )+ 12 χ (tr( C ))tr( C )tr( H ) + 12 χ (tr( H ))tr( C )tr( H ) −
12 Φ(tr( C ))tr( H ) −
12 Φ(tr( H ))tr( C ) + ε ∇ tr( C ) · ∇ tr( H ) d x d τ. (6.11)25irst we consider the last term of I and the tensor gradient term from D ( u , C ) , D ( U , H ) to obtain P := − ε (cid:90) Ω t |∇ tr( C ) | + |∇ tr( H ) | − ∇ tr( C ) · ∇ tr( H ) d x d τ = − ε (cid:90) Ω t |∇ tr( C − H ) | d x d τ. (6.12)We continue with the convective terms P := 12 (cid:90) Ω t u · ∇ tr( C )tr( H ) + U · ∇ tr( H )tr( C ) d x d τ = 12 (cid:90) Ω t ( U − u ) · ∇ tr( H )tr( C ) d x d τ = 12 (cid:90) Ω t ( U − u ) · ∇ tr( H )tr( C − H ) d x d τ = − (cid:90) Ω t ( U − u ) · ∇ tr( C − H )tr( H ) d x d τ ≤ δ (cid:107)∇ tr( C − H ) (cid:107) L ( L ) + δ (cid:107)∇ u − ∇ U (cid:107) L ( L ) + c ( δ ) (cid:90) t (cid:107) tr( H ) (cid:107) E ( u , C | U , H )( τ ) d τ. (6.13)Here we have used the regularity tr( H ) ∈ L (0 , T ; L (Ω)). Next, the relaxation terms and thecorresponding terms from D ( u , C ) , D ( U , H ) yield P := 12 (cid:90) Ω t − χ (tr( C ))tr( C ) − χ (tr( H ))tr( H ) + χ (tr( C ))tr( C )tr( H ) + χ (tr( H ))tr( C )tr( H ) d x d τ = 12 (cid:90) Ω t − χ (tr( C ))tr( C − H ) − ( χ (tr( C )) − χ (tr( H )))tr( H )tr( C − H ) d x d τ = 12 (cid:90) Ω t − χ (tr( C ))tr( C − H ) − (cid:90) Ω t χ (cid:48) ( ζ )tr( H )tr( C − H ) d x d τ (6.14) ≤ − (cid:90) Ω t χ (tr( C ))tr( C − H ) d x d τ + δ (cid:107)∇ tr( C − H ) (cid:107) L ( L ) (6.15)+ c ( δ ) (cid:90) t (cid:107) tr( H ) (cid:107) (cid:107) χ (cid:48) ( ζ ) (cid:107) E ( u , C | U , H )( τ ) d τ ≤ − (cid:90) Ω t χ (tr( C ))tr( C − H ) d x d τ + δ (cid:107)∇ tr( C − H ) (cid:107) L ( L ) (6.16)+ c ( a, δ ) (cid:107) tr( H ) (cid:107) L ∞ ( L ) (cid:90) t (cid:107) tr( C + H ) (cid:107) E ( u , C | U , H )( τ ) d τ. Here ζ denotes the convex combination of tr( C ) , tr( H ) from the mean-value theorem. In the lastintegral we have used the regularity tr( H ) ∈ L ∞ (0 , T ; L (Ω)).We focus on the second relaxation terms and the corresponding terms from F ( C ) , F ( H ) and find P := 12 (cid:90) Ω t Φ(tr( C ))tr( C ) + Φ(tr( H ))tr( H ) − Φ(tr( C ))tr( H ) − Φ(tr( H ))tr( C ) d x d τ = 12 (cid:90) Ω t (Φ(tr( C )) − Φ(tr( H )))tr( C − H ) d x d τ = 12 (cid:90) Ω t tr( C − H ) d x d τ ≤ c (cid:90) t E ( u , C | U , H )( τ ) d τ. (6.17)26inally we estimate the term corresponding to the upper convected derivative of I , cf. (6.11) andthe coupling terms to the Navier-Stokes equations from I , cf. (6.10) P := (cid:90) Ω t tr( C ) C : ∇ U + tr( H ) H : ∇ u − tr ( C ∇ u ) tr( H ) − tr ( H ∇ U ) tr( C ) d x d τ = (cid:90) Ω t tr( C − H ) C : ∇ u − tr( C − H ) H : ∇ U d x d τ = (cid:90) Ω t tr( C − H ) C : ( ∇ u − ∇ U ) + tr( C − H )( C − H ) : ∇ U d x d τ ≤ δ (cid:107) D u − D U (cid:107) L ( L ) + δ (cid:107)∇ tr( C − H ) (cid:107) L ( L ) + δ (cid:107)∇ ( C − H ) (cid:107) L ( L ) + c ( δ ) (cid:90) t ( (cid:107) C (cid:107) sr + (cid:107)∇ U (cid:107) ) E ( u , C | U , H )( τ ) d τ. (6.18)The regularity ∇ U ∈ L (0 , T ; L (Ω)) and C ∈ L s (0 , T ; L r (Ω)) have been used. To obtain theabove estimate we have applied (cid:90) t (cid:107)∇ u − ∇ U (cid:107) (cid:107) tr( C − H ) (cid:107) p (cid:107) C (cid:107) q d τ ≤ δ (cid:107) D u − D U (cid:107) L ( L ) + c ( δ ) (cid:90) t (cid:107) tr( C − H ) (cid:107) p (cid:107) C (cid:107) q d τ for p + q = , p <
6. Next we interpolate (cid:107) tr( C − H ) (cid:107) p between L and L with a given θ ∈ (0 , L into H and applying the Young inequality yield c ( δ ) (cid:90) t (cid:107) tr( C − H ) (cid:107) p (cid:107) C (cid:107) q d τ ≤ δ (cid:107)∇ tr( C − H ) (cid:107) L ( L ) + c ( δ ) (cid:90) t (cid:107) tr( C − H ) (cid:107) (cid:107) C (cid:107) θ − θ d τ. Setting r = − θ yields (cid:107) C (cid:107) θ − θ = (cid:107) C (cid:107) rr − r . But this is exactly the case when s + r = 1 for s ∈ (0 , ∞ ) , r ∈ (3 , ∞ ).Note that P is the only term in the trace-part where we have to estimate C − H in the Frobeniusnorm instead of the trace norm.Summing up the estimates (6.8)-(6.9) and (6.12)-(6.18) while using the notation of (3.11) yields E ( u , C | U , H )( t ) + (1 − δ ) (cid:90) t D ( u , C | U , H )( τ ) d τ ≤ E ( u , C | U , H )(0) (6.19)+ c (cid:90) t g ( τ ) E ( u , C | U , H )( τ ) d τ + δ (cid:107)∇ ( C − H ) (cid:107) L ( L ) , D ( u , C | U , H ) = η (cid:107) D u − D U (cid:107) + ε (cid:107)∇ tr ( C − H ) (cid:107) + 12 (cid:13)(cid:13)(cid:13)(cid:112) χ (tr( C ))tr ( C − H ) (cid:13)(cid:13)(cid:13) ,g ( τ ) = 1 + (cid:107) C (cid:107) sr + (cid:107)∇ U (cid:107) + (cid:107) tr( H ) (cid:107) L ∞ ( L ) (cid:107) tr( C + H ) (cid:107) + (cid:107) tr( H ) (cid:107) + (cid:107) U (cid:107) . .2 Frobenius Energy In this section we deal with the term I arising from the expansion of the relative energy. It shouldbe noted that the structure of the estimates are quite similar to the trace part of the equations.Since the weak solutions for C is regular enough direct calculations yield I = (cid:90) Ω t dd t | C − H | d x d τ = (cid:90) Ω t tr (cid:18) ∂∂t ( C − H )( C − H ) (cid:19) d x d τ =2 (cid:90) Ω t tr (cid:18) ∂∂t C · ( C − H ) (cid:19) + tr (cid:18) ∂∂t H · ( H − C ) (cid:19) d x d τ = (cid:90) Ω t − ( u · ∇ ) C : ( C − H ) + [ ∇ uC + C ∇ u (cid:62) ] : ( C − H ) − χ (tr( C )) C : ( C − H ) + Φ(tr( C ))tr( C − H ) − ε ∇ C : ∇ ( C − H ) d x d τ + (cid:90) Ω t − ( U · ∇ ) H : ( H − C ) + [ ∇ UH + H ∇ U (cid:62) ] : ( H − C ) − χ (tr( H )) H : ( H − C ) + Φ(tr( H ))tr( H − C ) − ε ∇ H : ∇ ( H − C ) d x d τ. (6.20)In the above computations we have employed Theorem 3.5 and inserted D = H as test function inthe weak formulation (3.2) for C and D = C as test function in the weak formulation (3.2) for H .We treat the above terms pairwise and start with the diffusive terms Q := − (cid:90) Ω t ε ∇ C : ∇ ( C − H ) + ε ∇ H : ∇ ( H − C ) d x d τ = − ε (cid:90) Ω t |∇ ( C − H ) | d x d τ. (6.21)The convective terms can be estimated as Q := (cid:90) Ω t − ( u · ∇ ) C : ( C − H ) − ( u · ∇ ) H : ( H − C ) d x d τ = − (cid:90) Ω t [( u · ∇ ) C : ( C − H ) − ( U · ∇ ) H ] : ( C − H ) d x d τ = − (cid:90) Ω t [( u · ∇ )( C − H ) : ( C − H ) − (( U − u ) · ∇ ) H ] : ( C − H ) d x d τ = − (cid:90) Ω t (( U − u ) · ∇ )( C − H ) : H d x d τ ≤ δ (cid:107)∇ ( C − H ) (cid:107) L ( L ) + δ (cid:107) D u − D U (cid:107) L ( L ) + c ( δ ) (cid:90) t (cid:107) H (cid:107) E ( u , C | U , H )( τ ) d τ. (6.22)Here we have applied the regularity H ∈ L (0 , T ; L (Ω)).28he terms arising from the upper convected derivative can be bounded in the following way Q := (cid:90) Ω t [ ∇ uC + C ∇ u (cid:62) ] : ( C − H ) + [ ∇ UH + H ∇ U (cid:62) ] : ( H − C ) d x d τ = (cid:90) Ω t [ ∇ uC + C ∇ u (cid:62) − ∇ UH − H ∇ U (cid:62) ] : ( C − H ) d x d τ = (cid:90) Ω t tr( ∇ uC · ( C − H ) − ∇ UH · ( C − H )) d x d τ = (cid:90) Ω t tr( ∇ ( u − U ) C ( C − H ) + ∇ U ( C − H ) ) d x d τ (6.23) ≤ δ (cid:107) D u − D U (cid:107) L ( L ) + δ (cid:107)∇ ( C − H ) (cid:107) L ( L ) + c ( δ ) (cid:90) t ( (cid:107)∇ U (cid:107) + (cid:107) C (cid:107) sr ) E ( C | H )( τ ) d τ. This estimate can be obtained following the same idea as in (6.18). Again we need the regularity ∇ U ∈ L (0 , T ; L (Ω)) and C ∈ L s (0 , T ; L r (Ω)). Next we focus on the relaxation terms. We startwith Q := (cid:90) Ω t − χ (tr( C )) C : ( C − H ) − χ (tr( H )) H : ( H − C ) d x d τ = (cid:90) Ω t − χ (tr( C )) | C − H | − ( χ (tr( C )) − χ (tr( H ))) H : ( C − H ) d x d τ = (cid:90) Ω t − χ (tr( C )) | C − H | d x d τ + (cid:90) Ω t χ (cid:48) ( ζ ) H : ( C − H )tr( C − H ) d x d τ ≤ − (cid:90) Ω t χ (tr( C )) | C − H | d x d τ + c ( δ ) (cid:90) t ( (cid:107) χ (cid:48) ( ζ ) (cid:107) (cid:107) H (cid:107) ) E ( C | H )( τ ) d τ ≤ − (cid:90) Ω t χ (tr( C )) | C − H | d x d τ + δ (cid:107)∇ ( C − H ) (cid:107) L ( L ) + c ( δ, a ) (cid:90) t (cid:107) tr( C + H ) (cid:107) (cid:107) H (cid:107) E ( C | H )( τ ) d τ ≤ − (cid:90) Ω t χ (tr( C )) | C − H | d x d τ + δ (cid:107)∇ ( C − H ) (cid:107) L ( L ) + c ( δ, a ) (cid:107) H (cid:107) L ∞ ( L ) (cid:90) t (cid:107) tr( C + H ) (cid:107) E ( C | H )( τ ) d τ. (6.24)Here ζ denoted the a convex combination of tr( C ) , tr( H ) from the mean-value theorem. Again theregularity H ∈ L ∞ (0 , T ; L (Ω)) have been used.The second relaxation term is controlled by Q := (cid:90) Ω t Φ(tr( C ))tr( C − H ) + Φ(tr( H ))tr( H − C ) d x d τ (cid:90) Ω t (Φ(tr( C )) − Φ(tr( H )))tr( C − H ) d x d τ = (cid:90) Ω t tr( C − H ) d x d τ ≤ c (cid:90) t E ( C | H )( τ ) d τ. (6.25)Summing up the estimates (6.21)-(6.25) yields E ( C | H )( t ) + (1 − δ ) (cid:90) t D ( C | H )( τ ) d τ ≤ E ( C | H )(0) + c (cid:90) t E ( u , C | U , H )( τ ) g ( τ ) d τ + 2 δ (cid:107) D u − D U (cid:107) , D ( C | H ) = (cid:13)(cid:13)(cid:13)(cid:112) χ (tr( C ))( C − H ) (cid:13)(cid:13)(cid:13) + ε (cid:107)∇ ( C − H ) (cid:107) ,g ( τ ) = 1 + (cid:107) tr( C + H ) (cid:107) (cid:107) H (cid:107) L ∞ ( L ) + (cid:107)∇ U (cid:107) + (cid:107) C (cid:107) sr . (6.26) In order to obtain the desired result we have to combine estimates (6.19) and (6.26) to obtain thefull relative energy inequality E ( u , C | U , H )( t ) + (1 − δ ) (cid:90) t D ( u , C | U , H )( τ ) d τ ≤ E ( u , C | U , H )(0) + c (cid:90) t g ( τ ) E ( u , C | U , H )( τ ) d τ, D ( u , C | U , H ) = η (cid:107) D u − D U (cid:107) + ε (cid:107)∇ tr ( C − H ) (cid:107) + 12 (cid:13)(cid:13)(cid:13)(cid:112) χ (tr( C ))tr ( C − H ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:112) χ (tr( C )) C − H (cid:13)(cid:13)(cid:13) + ε (cid:107)∇ C − ∇ H (cid:107) ,g ( τ ) ≤ (cid:107)∇ U (cid:107) + (cid:107) H (cid:107) L ∞ ( L ) (cid:107) tr( C + H ) (cid:107) + (cid:107) U (cid:107) + (cid:107) C (cid:107) sr . (6.27)Since D is by construction non-negative, b is positive for δ small enough and g ∈ L (0 , T † ) we canapply the Gronwall Lemma 2.4 and obtain E ( u , C | U , H )( t ) ≤ E ( u , C | U , H )(0) exp (cid:18) c (cid:90) t g ( τ ) d τ (cid:19) , (6.28)which concludes to proof of Theorem 3.7. Observe that for u = U , C = H it holds that E ( u , C | U , H )(0) = 0, cf. (3.10). We obtain Theorem 3.9. In this section we illustrate an application of the relative energy inequality (6.27) in numericalanalysis. The relative energy is an appropriate distance to measure the convergence of numericalschemes. In this paper we apply the Lagrange-Galerkin finite element method, that is based on30iecewise linear approximations per element. We work here with tetrahedral meshes. The materialderivative is approximated by means of characteristics, [27, 28]. Specifically, our computationaldomain Ω = [0 , , is triangulated uniformly with a mesh size h . The main ingredients of themethod in [27, 28] are the following • P elements for u , p, C . • Brezzi-Pitk¨aranta stabilization for the pressure p . • Fixed-point iteration to solve a coupled nonlinear system for ( u , p, C ).We recall that in three space dimensions the relative energy is given by E ( u , C | U , H ) := (cid:90) Ω | u − U | + 14 | tr( C − H ) | + 12 | C − H | d x. In order to measure the convergence order of the Lagrange-Galerkin method we fix the numericalsolution with the highest resolution to be the reference solution u ref , C ref . Now the error can becomputed as E h ( u h , C h | u ref , C ref ) . In order to measure the rate of convergence we compute the experimental error of convergence(EOC) by EOC h = log (cid:18) E h ( u h , C h | u ref , C ref ) E h/ ( u h/ , C h/ | u ref , C ref ) (cid:19) . Remark 7.1.
In our situation the relative energy satisfies the properties of a norm. However, ingeneral the relative energy does not need to satisfy symmetry and the triangle inequality. Henceone has to be careful when computing the EOC.
Experiment:
We choose the following initial data u (0) = u (0) = u (0) = sin(2 πx ) sin(2 πy ) sin(2 πz ) , C = 1 √ I . The model parameters are set to η = 2 , ε = 1 , a = 0 with final time T = 1.Our extensive numerical experiments for d = 2 , a = 0. Furthermore, at least experimentally the free energy decreases in time.31igure 1: Experiment: Time evolution of the relative energy on a hierarchy of meshes(left) andthe convergence order(right). M is the number of points per side, i.e. it relates to h .The first observation is that experimentally the scheme is converging. As time evolves the con-vergence order of approximately 2 is achieved as expected for squared norms, see also our recentworks on the error analysis of the Lagrange-Galerkin method using more standard tools [27, 28]. In this work we have proven the existence of global weak solutions for the Peterlin viscoelasticsystem (1.1) in three space dimensions, see Theorem 3.2. Our proof is based on a combination ofthe Galerkin method for the incompressible Navier-Stokes equations and a semigroup approach forthe time evolution of the conformation tensor. This approach allows to deduce the positive-(semi)definiteness of the tensor C which is necessary from the physical point of view. In fact, in twospace dimension it is possible to prove existence of weak solutions without showing positive-(semi)definiteness. On the other hand in three space dimensions this property is a crucial ingredient ofthe existence proof of the conformation tensor.Moreover, we provided a regularity result, see Theorem 3.5, applying methods from the semigrouptheory. Using the latter result allows us to apply the relative energy method, see Theorem 3.7which ultimately provides the weak-strong uniqueness principle, see Theorem 3.9. The relativeenergy method has wide applications. We illustrate its use as a suitable metric in the experimentalconvergence study of the Lagrange-Galerkin method. In future, it will be interesting to analysetheoretically the convergence of Lagrange-Galerkin method using the relative energy.32 cknowledgement This research of A.B. and M.L. was supported by the German Science Foundation (DFG) un-der the Collaborative Research Center TRR 146 Multiscale Simulation Methods for Soft Matters(Project C3). The research of the Y.L. has been supported by the Recruitment Program of GlobalExperts of China. We would like to thank P. Tolksdorf, M. Bachmayr, B. She and H. Egger forfruitful discussions on the topic.
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