Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism
aa r X i v : . [ m a t h . A P ] O c t GLOBAL WELL-POSEDNESS IN THE CRITICAL BESOV SPACESFOR THE INCOMPRESSIBLE OLDROYD-B MODEL WITHOUTDAMPING MECHANISM
QIONGLEI CHEN AND XIAONAN HAO
Abstract.
We prove the global well-posedness in the critical Besov spaces for the in-compressible Oldroyd-B model without damping mechanism on the stress tensor in R d for the small initial data. Our proof is based on the observation that the behaviors ofGreen’s matrix to the system of (cid:0) u, ( − ∆) − P ∇ · τ (cid:1) as well as the effects of τ changefrom the low frequencies to the high frequencies and the construction of the appropriateenergies in different frequencies. Introduction
We are concerned with the incompressible Oldroyd-B model of the non-Newtonian fluidin R + × R d (1.1) u t + u · ∇ u − ν ∆ u + ∇ p = µ ∇ · τ,τ t + u · ∇ τ + aτ + Q ( τ, ∇ u ) = µ D ( u ) , ∇ · u = 0 ,u (0 , x ) = u ( x ) , τ (0 , x ) = τ ( x ) . Here u ( t, x ) stands for the velocity fluid and τ ( t, x ) is the non-Newtonian part of stresstensor ( τ is a d × d symmetric matrix and [ ∇ · τ ] i = P j ∂ j τ i,j ). The pressure p is a scalarand coefficients ν, a, µ , µ are assumed to be non-negative constants. The bilinear term Q has the following form Q ( τ, ∇ u ) = τ W ( u ) − W ( u ) τ − b (cid:0) D ( u ) τ + τ D ( u ) (cid:1) , where b ∈ [ − , D ( u ) = (cid:0) ∇ u + ( ∇ u ) ⊤ (cid:1) , W ( u ) = (cid:0) ∇ u − ( ∇ u ) ⊤ (cid:1) are the defor-mation tensor and the vorticity tensor, respectively. If a = 0, we call the system (1.1)the Oldroyd-B model without damping mechanism which we investigate in this paper.The Oldroyd-B model is a typical prototypical model for viscoelastic fluids, which de-scribes the motion of some viscoelastic flows. For more detailed physical background andderivations about this model, one refers to [2, 7, 23].The well-posedness of the system (1.1) had been studied extensively. In the the case of a >
0, Guillop´e and Saut [19] proved that the strong solutions are local well-posed in theSobolev space H s . They [20] also showed that these solutions are global if the couplingparameter and the initial data are small enough. Their results were extended to the Date : October 16, 2018.2000
Mathematics Subject Classification.
Key words and phrases. incompressible Oldroyd-B model, critical spaces, global solution. L p framework by Fernandez-Cara, Guill´en and Ortega [15]. Chemin and Masmoudi [7]initiated the study of the global existence and uniqueness in the critical Besov spaces, andtheir results were improved later by Zi, Fang and Zhang [16] to the case of the non-smallcoupling parameter. For more results on the well-posedness and the blow-up criterion,one refers to [7, 8, 16, 17, 22] and references therein.Now let us say a few words about the so-called critical spaces, for the incompressibleNavier-Stokes equations(INS) u t + u · ∇ u − ν ∆ u + ∇ p = 0 , ∇ · u = 0 ,u (0 , x ) = u ( x ) , if ( u ( t, x ) , p ( t, x )) is a solution of (INS), then for λ > (cid:0) u λ ( t, x ) , p λ ( t, x ) (cid:1) , (cid:0) λu ( λ t, λx ) , λ p ( λ t, λx ) (cid:1) , is also a solution of (INS). Moreover, the functional space X is called critical to the system(INS) if the corresponding norm is invariant under the scaling (1.2). It is Obvious that˙ H d − is a critical space. Fujita and Kato [18] proved the wellposedness of (INS) in ˙ H d − ,see also [3, 4, 5, 6] and references therein for the other critical spaces. Although thesystem (1.1) does not have any scaling invariance, one may find that if the coupling term ∇ · τ as well as the damping term τ is neglected, and ( u, τ ) is a solution of (1.1), then for λ > (cid:0) u λ ( t, x ) , τ λ ( t, x ) , p λ ( t, x ) (cid:1) , (cid:0) λu ( λ t, λx ) , τ ( λ t, λx ) , λ p ( λ t, λx ) (cid:1) is also a solution of (1.1). This leads us to define the following critical spaces of the system(1.1) as in [7, 16], Definition 1.1.
A functional space is called critical to (1.1) if the associated norm isinvariant under the transformation ( u, τ ) → ( u λ , τ λ ) (up to a constant independent of λ ). It is obvious that ˙ H d × ˙ H d − and ˙ B dp p,q × ˙ B dp − p,q are critical spaces. And the reason whyone has to consider the well-posedness in such spaces has been fully explained in [7].As for the researches of other special cases of the system (1.1), we sketch some resultshere. If b = 0, the existence of the global weak solution had been proved by Lions andMasmoudi [21]. If µ = 0 and the equation of τ contains viscous term − ∆ τ , the globalexistence of small smooth solutions had been proved by Elgindi, Rousset and Liu [13, 14],and the similar result with general data had also been showed for more special models in[13]. When a = 0, Zhu Yi [24] constructed a global smooth solution in 3 D very recently.More precisely, Theorem 1.1. [24]
Let ν, µ , µ > and a = 0 . Suppose that ∇ · u = 0 , ( τ ) i,j = ( τ ) j,i and initial data Λ − u , Λ − τ ∈ H ( R ) . Then there exists a small constant ǫ such thatsystem (1.1) admits a unique global classical solution provided that k Λ − u k H + k Λ − τ k H ≤ ǫ, where Λ − = ( − ∆) . ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 3
It is noted that the required regularity of Theorem 1.1 is far from the regularity pre-scribed by the scaling (1.3), which inspires us to consider the well-posedness of (1.1)( a = 0) in critical Besov spaces just like Chemin and Masmoudi had done in [7, 16] for a > Theorem 1.2.
Let ν, µ , µ > and a = 0 . There exists a small constant ε such that if τ ∈ ˙ B d − , ∩ ˙ B d , , u ∈ ˙ B d − , with k u k ˙ B d − , + k τ k ˙ B d − , ∩ ˙ B d , ≤ ε, then the system (1.1) has a unique global solution ( u, τ ) such that u ∈ C ( R + ; ˙ B d − , ) ∩ L ( R + ; ˙ B d +12 , ); τ ∈ C ( R + ; ˙ B d − , ∩ ˙ B d , ) , P ∇ · τ ∈ L ( R + ; ˙ B d − , + ˙ B d , ) . Here P is the projection operator, and ˙ B s , is Besov space. One refers to Section 2 for itsdefinition.Remark . When d = 3, noting that H ֒ → ˙ B , (see Proposition 2.4), Theorem 1.2allows us to involve a class of functions of the initial data that Theorem 1.1 does notcontain. For example, χ p ( D ) | x | − σ , < σ ≤ / , where χ p ( D ) f , F − ( χ B (0 ,p ) ˆ f ) with the radial function χ B (0 ,p ) ∈ S ( R ) supported in theball B = { ξ ∈ R d , | ξ | ≤ p } . It is not difficult to check that χ p ( D ) | x | − σ ∈ ˙ B , ( R ); χ p ( D ) | x | − σ / ∈ H ( R ) . For the detailed proof, one please refers to Proposition 2.6.In [24], the author observed that ( u, P ∇ · τ ) satisfies some kind of the damped waveequations and has enough decay regardless of a = 0, then he proved Theorem 1.1 byconstructing two special time-weighted energies. However, the approach used in [24]seems not to work for the critical spaces. On the other hand, the method used in [7] forthe critical Besov spaces relies heavily on the damping effect ( a > a = 0. We find out that onepart of the stress tensor τ has damping effect while the stress tensor itself does not have.More precisely, motivated by the work of Danchin [10] on the compressible Navier-Stokesequations, we study the following mixed linear system(1.4) ( u t − ν ∆ u − µ Λ(Λ − P ∇ · τ ) = P E, (Λ − P ∇ · τ ) t + µ Λ u = Λ − P ∇ · F, where P is the projection operator,Λ = ( − ∆) and Λ − = ( − ∆) − . WELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM
Let G ( x, t ) be the Green matrix of system (1.4), we derive(1.5) ˆ G ( ξ, t ) = λ + e λ + t − λ − e λ − t λ + − λ − µ | ξ | e λ + t − e λ − t λ + − λ − − µ | ξ | e λ + t − e λ − t λ + − λ − λ + e λ − t − λ − e λ + t λ + − λ − ! , where λ + = − ν | ξ | + p ν | ξ | − µ µ | ξ | ,λ − = − ν | ξ | − p ν | ξ | − µ µ | ξ | . By analyzing the behaviors of b G ( x, t ) in different frequencies, we discover that u as well asthe low frequencies of Λ − P ∇· τ has a parabolic smoothing effect, and the high frequenciesof Λ − P ∇ · τ have a damping effect. Specifically, we set the energy in the low frequencies E lr ( t ) , sup t k u k l ˙ B d − , +sup t k Λ − P ∇· τ k l ˙ B d − , + Z t k u k l ˙ B d , d t ′ + Z t k Λ − P ∇· τ k l ˙ B d , d t ′ , (1.6)and the energy in the high frequencies E hr ( t ) , sup t k u k h ˙ B d − , + sup t k P ∇ · τ k h ˙ B d − , + Z t k u k h ˙ B d , d t ′ + Z t k P ∇ · τ k h ˙ B d − , d t ′ , (1.7)where k . k l and k . k h denotes the low and high part of corresponding norm (see Subsection2.3). Unfortunately, there seems no damping effect on another part of the tensor τ , andthe following estimates do not hold k τ k l ˙ B d − , ≤ C k Λ − P ∇· τ k l ˙ B d − , , k τ k l ˙ B d − , ≤ C k P ∇ · τ k h ˙ B d − , . Thus it is difficult to deal with some parts of the nonlinear terms (cid:0) e.g. Q ( τ, ∇ u ) (cid:1) by theoriginal energies E lr ( t ) , E hr ( t ). Naturally, we want to get more estimates containing term τ instead of P ∇ · τ , and this motivates us to construct the energies E l ( t ) , E h ( t ) as thefollowing,(1.8) E l ( t ) , sup t k u k l ˙ B d − , + sup t k τ k l ˙ B d − , + Z t k u k l ˙ B d , d t ′ + Z t k Λ − P ∇ · τ k l ˙ B d , d t ′ , and(1.9) E h ( t ) , sup t k u k h ˙ B d − , + sup t k τ k h ˙ B d , + Z t k u k h ˙ B d , d t ′ + Z t k P ∇ · τ k h ˙ B d − , d t ′ . Then the above-mentioned energies supply L ∞ T estimate of k τ k ˙ B d − , (cid:0) k τ k ˙ B d , (cid:1) in the low(high) frequencies. For more details, please see Section 3. Let us emphasize that if wehandle the nonlinear terms of τ involved, we shall appeal to the way used in [24], i.e., P ∇ · τ = P ( u · ∇ P ∇ · τ ) + some terms containing ∇ u. Notations.
Throughout this paper, C stands for the constant and changes from line toline. We use b u and F ( u ) to denote the Fourier transform of u . ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 5 Littlewood-Paley theory and Besov spaces
Littlewood-Paley decomposition.
Now we introduce the Littlewood-Paley de-composition, which relies on the dyadic partition of unity, and we can refer to [1] for moredetails. Let us choose two radial functions ϕ, χ ∈ S ( R d ) supported in C = { ξ ∈ R d , ≤| ξ | ≤ } and B = { ξ ∈ R d , | ξ | ≤ } respectively such that X j ∈ Z ϕ (2 − j ξ ) = 1 if ξ = 0 . Denote h ( x ) = F − (cid:0) ϕ ( ξ ) (cid:1) , we define the dyadic blocks as follows˙∆ j u = ϕ (2 − j D ) u = 2 jd Z R d h (2 j y ) u ( x − y )d y, ˙ S j u = χ (2 − j D ) u. Definition 2.1.
We denote by S ′ h the space of temperate distributions u such that lim j →−∞ ˙ S j u = 0 in S ′ . Remark . If a temperate distribution u is such that its Fourier transform F u is locallyintegrable near 0, then u belongs to S ′ h .Then the homogeneous Littlewood-paley decomposition is defined as(2.1) u = X j ∈ Z ˙∆ j u, for u ∈ S ′ h . With our choice of ϕ and χ , it is easy to verify that(2.2) ˙∆ j ˙∆ k u = 0 if | j − k | ≥ , and ˙∆ j ( ˙ S k − u ˙∆ k u ) = 0 if | j − k | ≥ . Next, let us introduce a useful lemma which will be repeatedly used throughout thispaper.
Lemma 2.2. [1]
Let ≤ p ≤ q ≤ + ∞ . Then for any γ ∈ ( N ∪ { } ) d , there exists aconstant C independent of f, j such that supp ˆ f ⊆ {| ξ | ≤ A j } ⇒ k ∂ γ f k L q ≤ C j | γ | + jd ( p − q ) k f k L p , supp ˆ f ⊆ { A j ≤ | ξ | ≤ A j } ⇒ k f k L p ≤ C − j | γ | sup | β | = | γ | k ∂ β f k L p . Homogeneous Besov space.Definition 2.2.
Let u ∈ S ′ ( R d ) , s ∈ R , and ≤ p, r ≤ ∞ , we set k u k ˙ B sp,r , k{ js k ˙∆ j u k L p } j k l r . We then define the space ˙ B sp,r , { u ∈ S ′ h , k u k ˙ B sp,r < ∞} . Remark . The definition of the space ˙ B sp,r does not depend on the choice of the couple( ϕ, χ ) defining the Littlewood-Paley decomposition.Let us now state some classical properties of the homogeneous Besov spaces. WELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM
Proposition 2.4.
For all s, s , s ∈ R , ≤ p, p , p , r, r , r ≤ + ∞ , we have the followingproperties:(i) If p ≤ p , r ≤ r , then ˙ B sp ,r ֒ → ˙ B s − dp + dp p ,r .(ii) If s = s and θ ∈ (0 , , then k u k ˙ B θs − θ ) s p,r ≤ k u k θ ˙ B s p,r k u k − θ ˙ B s p,r . (iii) ˙ H s ≈ ˙ B s , and C | s | +1 k u k ˙ B s , ≤ k u k ˙ H s ≤ C | s | +1 k u k ˙ B s , . (iv) If s > , then ˙ B s , ∩ L ∞ (especially ˙ B d , ) is an algebra. Proposition 2.5.
Let s > , u ∈ L ∞ ∩ ˙ B s , and v ∈ L ∞ ∩ ˙ B s , . Then uv ∈ L ∞ ∩ ˙ B s , and k uv k ˙ B s , . k u k L ∞ k v k ˙ B s , + k v k L ∞ k u k ˙ B s , . Let s , s ≤ d such that s + s > , u ∈ ˙ B s , and v ∈ ˙ B s , . Then uv ∈ ˙ B s + s − d , and k uv k ˙ B s s − d , . k u k ˙ B s , k v k ˙ B s , . Let u ∈ ˙ B d, ∞ and v ∈ ˙ B d, . Then uv ∈ ˙ B d, ∞ k uv k ˙ B d, ∞ . k u k ˙ B d, ∞ k v k ˙ B d, . We can refer to [1] for the proof of these propositions.
Proposition 2.6.
Let σ ∈ (1 , ] , then we get χ p ( D ) | x | − σ ∈ ˙ B , ( R ); χ p ( D ) | x | − σ / ∈ H ( R ) . Here χ p ( D ) f = F − ( χ B (0 ,p ) ˆ f ) .Proof. Noticing that F ( | x | − σ ) = C | ξ | σ − ∈ L loc ( R )and Remark 2.1, we get | x | − σ ∈ S ′ h . By direct computations we have˙∆ j | x | − σ = 2 j Z R h (cid:0) j ( x − y ) (cid:1) | y | − σ d y = 2 jσ Z R h (2 j x − j y ) | j y | − σ d2 j y = 2 jσ e h (2 j x ) , where e h ( x ) = R R h ( x − y ) | y | − σ d y . Since ( F h )( ξ ) = ϕ ( ξ ) ∈ D ( R / { } ), we obtain F e h ∈D ( R ). This implies k e h k L < + ∞ . From above analysis, we easily get k ˙∆ j | x | − σ k L = 2 j ( σ − ) k e h k L . Combining with the definition of the Besov space and supp χ p ⊂ B (0 , p ), we derive that k χ p ( D ) | x | − σ k ˙ B , . X j ≤ N p j ( σ − k e h k L < ∞ . ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 7
However, it is obvious that k χ p ( D ) | x | − σ k L = k χ p ( ξ ) | ξ | σ − k L = ∞ . (cid:3) Hybrid Besov spaces.
Let us now introduce the hybrid Besov spaces we will workin this paper.
Definition 2.3. [11]
Let s, t ∈ R . We set k u k ˙ B s,t = X j ≤ js k ˙∆ j u k L + X j> jt k ˙∆ j u k L . We then define the space ˙ B s,t , { u ∈ S ′ h , k u k ˙ B s,t < ∞} . We will often use the definition: k u k l ˙ B sp, , X j ≤ js k ˙∆ j u k p ; k u k h ˙ B sp, , X j> js k ˙∆ j u k p . We also define the norm of the space L rT ( ˙ B s,t ) k u k L rT ( ˙ B s,t ) = (cid:16) Z T k u k r ˙ B s,t d t (cid:17) r , with the usual change if r = ∞ . Furthermore, the norm of the time-space Besov space e L rT ( ˙ B s,t ) is defined by k u k e L rT ( ˙ B s,t ) = X j ≤ js k ˙∆ j u k L rT L + X j> js k ˙∆ j u k L rT L . Remark . (i) By the Minkowski inequality, we easily find that e L T ( ˙ B s,t ) = L T ( ˙ B s,t )and e L rT ( ˙ B s,t ) ⊆ L rT ( ˙ B s,t ) for r > Proposition 2.8. [11] (i) We have ˙ B s,s = ˙ B s , .(ii) If s ≤ t then ˙ B s,t = ˙ B s , ∩ ˙ B t , . Otherwise, ˙ B s,t = ˙ B s , + ˙ B t , .(iii) If s ≤ s and t ≥ t , then ˙ B s ,t ֒ → ˙ B s ,t . For the estimates of the product of two temperate distributions u and v in hybrid Besovspaces, we refer to the Appendix in section 5.2.4. Smoothing properties for the linear heat equation.Proposition 2.9. [1]
Let s ∈ R , ( p, r ) ∈ [1 , + ∞ ] , u ∈ ˙ B s − p,r and f ∈ e L T ( ˙ B s − p,r ) . Let u solve (2.3) ( ∂ t u − µ ∆ u = f,u (0 , x ) = u . WELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM
Then u ∈ e L ∞ T ( ˙ B s − p,r ) ∩ e L T ( ˙ B s +1 p,r ) and the following estimate holds: k u k e L ∞ T ( ˙ B s − p,r ) + µ k u k e L T ( ˙ B s +1 p,r ) ≤ k u k ˙ B s − p,r + C k f k e L T ( ˙ B s − p,r ) . If in addition r < + ∞ , then u ∈ C b ([0 , T ); ˙ B s − p,r ) . The following two Propositions are used for the proof of the uniqueness.
Proposition 2.10. [1]
Let ( p, r ) ∈ [1 , + ∞ ] and s ∈ (cid:0) − min( dp , dp ′ ); 1+ dp (cid:1) . Let u be a vectorfield such that ∇ u belongs to L (0 , T ; ˙ B dp p,r ∩ L ∞ ) . Suppose that f ∈ ˙ B sp,r , F ∈ L (0 , T ; ˙ B sp,r ) and that f ∈ L ∞ (0 , T ; ˙ B sp,r ) ∩ C ([0 , T ]; S ′ ) solves ( ∂ t f + u · ∇ f = F,f (0 , x ) = f . Let V ( t ) , R t k∇ u k ˙ B dpp,r ∩ L ∞ d t ′ . There exists a constant C depending only on s, p and d ,and such that the following inequality holds true for t ∈ [0 , T ](2.4) k f k e L ∞ t ( ˙ B sp,r ) ≤ e CV ( t ) (cid:16) k f k ˙ B sp,r + Z t e − CV ( t ) k F ( t ′ ) k ˙ B sp,r d t ′ (cid:17) . If r < + ∞ then f belongs to C ([0 , T ]; ˙ B sp,r ) . Proposition 2.11. [9, 12]
Let s ∈ R . Then for any ≤ p, r ≤ + ∞ and < ǫ ≤ , wehave k f k e L rT ( ˙ B sp, ) ≤ C k f k e L rT ( ˙ B sp, ∞ ) ǫ log (cid:16) e + k f k e L rT ( ˙ B s − ǫp, ∞ ) + k f k e L rT ( ˙ B s + ǫp, ∞ ) k f k e L rT ( ˙ B sp, ∞ ) (cid:17) . Priori estimate
Recalling (1.8) and (1.9), we denote(3.1) E ( t ) , E l ( t ) + E h ( t ) . We have the following priori estimate.
Proposition 3.1.
Let d ≥ and a = 0 . Assume that the system (1.1) has a solution ( τ, u ) on [0 , T ) . Then, there exist two positive constants C , C independent of T suchthat (3.2) E ( t ) ≤ C E + C E ( t ) , where E = k u k ˙ B d − , + k τ k ˙ B d − , d . Before the proof of the priori estimate, let us introduce two important Lemmas.First, we show the following rough energy estimate in frequency spaces. One refers to(1.6) and (1.7) for the definitions of E hr ( t ) and E lr ( t ). ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 9
Lemma 3.2.
Let d ≥ , a = 0 and ( u, τ ) be the solution of system (1.1) on [0 , T ) . Thefollowing estimate holds E lr ( t ) + E hr ( t ) ≤ C E + C Z t X j ≤ j ( d − ˜ E j / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ + C Z t X j> j ( d − (cid:16) ˜ F j / ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) + ˜ R j / k ˙∆ j u k L (cid:17) d t ′ , where C , C independent of T . Here we shall omit the definitions of ˜ E j , ˜ F j , ˜ R j for their specific forms do not affect theproof of Lemma 3.2. One can refer to the proof of Proposition 3.1 for their definitions. Proof.
Let us first fix a constant j ∈ Z , which will be chosen in Step 2. Throughoutthis part we shall suppose that j ≤ j as low frequencies and j > j as high frequencies.Moreover, without loss of generality, we assume that k ˙∆ j u k L , k ˙∆ j Λ − P ∇ · τ k L , k ˙∆ j P ∇ · τ k L = 0 . Step 1: Estimate of E lr ( t ) . Set e C = 2 j . Thanks to Lemma 2.2, for any function f , there exist two constants e C , e C such that(3.3) e C j k ˙∆ j f k L p ≤ k Λ ˙∆ j f k L p ≤ e C j k ˙∆ j f k L p . Applying the operator ˙∆ j to the system (1.4), we get(3.4) ( ( ˙∆ j u ) t + ν Λ ˙∆ j u − µ Λ ˙∆ j Λ − P ∇ · τ = ˙∆ j P E, ( ˙∆ j Λ − P ∇ · τ ) t + µ Λ ˙∆ j u = ˙∆ j Λ − P ∇ · F, with E = − u · ∇ u ; F = − u · ∇ τ − Q ( τ, ∇ u ) . For the sake of simplicity, we first suppose that
E, F = 0. Taking the L scalar productof the first equation of (3.4) with ˙∆ j u , of the second with ˙∆ j Λ − P ∇ · τ , we obtain thefollowing two identities:12 ddt k ˙∆ j u k L + ν k Λ ˙∆ j u k L − µ (Λ ˙∆ j (Λ − P ∇ · τ ) , ˙∆ j u ) = 0 , (3.5) 12 ddt k ˙∆ j (Λ − P ∇ · τ ) k L + µ j u, ˙∆ j Λ − P ∇ · τ ) = 0 . (3.6)To obtain an identity involving ( ˙∆ j P ∇ · τ, ˙∆ j u ), we take the scalar product of the firstequation of (3.4) with ˙∆ j P ∇ · τ , apply Λ to the second equation and take the L scalarproduct with ˙∆ j u , then sum up both equalities, which produces ddt ( ˙∆ j P ∇ · τ, ˙∆ j u ) + ν ( ˙∆ j Λ u, ˙∆ j P ∇ · τ ) − µ k ˙∆ j P ∇ · τ k L + µ k Λ ˙∆ j u k L = 0 . (3.7) Calculating µ (3.5) + µ (3.6) − K (3.7), we obtain12 ddt (cid:16) µ k ˙∆ j u k L + µ k ˙∆ j Λ − P ∇ · τ k L − K ( ˙∆ j P ∇ · τ, ˙∆ j u ) (cid:17) + (cid:0) µ ν − µ K (cid:1) k Λ ˙∆ j u k L + µ K k ˙∆ j P ∇ · τ k L − νK ( ˙∆ j Λ u, ˙∆ j P ∇ · τ )= 0 . (3.8)Using H¨older’s inequality, Young’s inequality and (3.3), we find that for all ǫ , ǫ > K | ( ˙∆ j P ∇ · τ, ˙∆ j u ) | ≤ ǫ k ˙∆ j u k + ǫ e C e C K k ˙∆ j Λ − P ∇ · τ k L ,νK | ( ˙∆ j Λ u, ˙∆ j P ∇ · τ ) | ≤ ν e C e C K ǫ k ˙∆ j Λ u k L + ǫ νK k ˙∆ j P ∇ · τ k L . Inserting the above two inequalities into (3.8) with ǫ = µ , ǫ = µ ν , and combining (3.3),we derive12 ddt (cid:18) µ k ˙∆ j u k L + (cid:16) µ − e C e C K µ (cid:17) k ˙∆ j Λ − P ∇ · τ k L (cid:19) + (cid:0) µ ν − µ K − ν e C e C K µ (cid:1) e C j k ˙∆ j u k L + µ e C K j k ˙∆ j Λ − P ∇ · τ k L ≤ . (3.9)Choosing a constant K such that 0 < K < min (cid:8) √ µ µ e C e C , µ µ νµ µ + ν e C e C (cid:9) , we ensure that thecoefficients of k ˙∆ j Λ − P ∇ · τ k L and k ˙∆ j u k L in (3.9) are positive. Thus in the generalcase ( E, F = 0) that containing the nonlinear terms, we obtain12 ddt (cid:16) k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L (cid:17) + 2 j (cid:16) k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L (cid:17) ≤ C ˜ E j , (3.10)for the constant C >
0. Dividing by k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L , (3.10) can be writtenas ddt (cid:16) k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L (cid:17) + 2 j (cid:16) k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L (cid:17) ≤ C ˜ E j / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L ) . (3.11)Multiplying both sides of (3.11) by 2 j ( d − , summing up by j ≤ j (actually we can choose j = 0, see Step 2), and then performing the time integration over [0 , t ], we conclude that E lr ( t ) ≤ C ( k u k l ˙ B d − , + k τ k l ˙ B d − , )+ C Z t X j ≤ j j ( d − ˜ E j / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . (3.12) Step 2: Estimate of E hr ( t ) . ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 11
By the similar argument as the way in the low frequencies, we first consider the linearterms of (3.4) as well. From the second equation of (3.4), we can easily obtain thefollowing inequality:(3.13) 12 ddt k ˙∆ j P ∇ · τ k L + µ j Λ u, ˙∆ j P ∇ · τ ) = 0 . Calculating ν (3.13) − µ (3.7) + K (3.5), we derive12 ddt (cid:16) K k ˙∆ j u k L + ν k ˙∆ j P ∇ · τ k L − µ ( ˙∆ j P ∇ · τ, ˙∆ j u ) (cid:17) + (cid:0) νK − µ (cid:1) k Λ ˙∆ j u k L + µ µ k ˙∆ j P ∇ · τ k L − µ K ( ˙∆ j P ∇ · τ, ˙∆ j u ) = 0 . (3.14)It is easy to check that for all ǫ , ǫ >
0, we have µ | ( ˙∆ j P ∇ · τ, ˙∆ j u ) | ≤ ǫ k ˙∆ j u k + µ ǫ k ˙∆ j P ∇ · τ k L ,µ K | ( ˙∆ j P ∇ · τ, ˙∆ j u ) | ≤ ǫ µ K k ˙∆ j u k L + µ K ǫ k ˙∆ j P ∇ · τ k L . (3.15)Noticing that j ≥ j + 1, the following inequality shows that the smoothing effect ofsystem (1.4) on u will lose,(3.16) k Λ ˙∆ j u k L ≥ e C j k ˙∆ j u k L ≥ e C e C k ˙∆ j u k L . Combining (3.14) ∼ (3.16), and choosing K = ǫ = µ ν ; ǫ = 2 K µ . we finally get 12 ddt (cid:16) µ ν k ˙∆ j u k L + ν k ˙∆ j P ∇ · τ k L (cid:17) + (cid:0) µ e C e C − µ µ ν (cid:1) k ˙∆ j u k L + µ µ k ˙∆ j P ∇ · τ k L = 0 . (3.17)We then define j = [log √ µ µ ν e C ] + 1 to ensure that the coefficient of k ˙∆ j u k in (3.17) ispositive. In light of the Remark 2.3, without loss of generality, we can define j = 0 hereand later. In the general case where the nonlinear terms E, F = 0, we get12 ddt (cid:16) k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L (cid:17) + ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) ≤ C ˜ F j . It is easy to see that the above estimate can be rewritten as ddt (cid:16) k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L (cid:17) + ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) ≤ C ˜ F j / ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) . (3.18) Multiplying both sides of (3.18) by 2 j ( d − , then summing up by j > j , we deduce afterperforming the time integration over [0 , t ], thatsup t k u k h ˙ B d − , + Z t + sup t k P ∇ · τ k h ˙ B d − , + Z t k u k h ˙ B d − , d t ′ + k P ∇ · τ k h ˙ B d − , d t ′ . k u k h ˙ B d − , + k τ k h ˙ B d , + Z t X j> j ( d − ˜ F j / ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L )d t ′ . (3.19)Next we show the smoothing effect on u . In the same way as the derivation of (3.5),we get ddt k ˙∆ j u k L + 2 j k ˙∆ j u k L ≤ C k ˙∆ j P ∇ · τ k L + C ˜ R j / k ˙∆ j u k L . Using the above inequality we deducesup t k u k h ˙ B d − , + Z t k u k h ˙ B d , d t ′ ≤ k u k h ˙ B d − , + C Z t k P ∇ · τ k h ˙ B d − , d t ′ + C Z t X j> j ( d − ˜ R j / k ˙∆ j u k L d t ′ . (3.20)Calculating (3.19) + η (3.20), and if we choose η small enough such that η C ≤ , thenthe term η C R t k P ∇ · τ k h ˙ B d − , d t ′ can be absorbed by the left side. Therefore, we obtain E hr ( t ) ≤ C (cid:0) k u k h ˙ B d − , + k τ k h ˙ B d , (cid:1) + C Z t X j> j ( d − (cid:16) ˜ F j / ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) + ˜ R j / k ˙∆ j u k L (cid:17) d t ′ . (3.21)Combining (3.12) and (3.21), we complete the proof of Lemma 3.2. (cid:3) Although we have derived the above rough energy estimate, it seems that some partsof the nonlinear terms could not be controlled by E hr ( t ) and E lr ( t ) effectively. Therefore,we have to introduce a more accurate estimate as follows. Lemma 3.3.
Let d ≥ , a = 0 and ( u, τ ) be the solution of system (1.1) on [0 , T ) . Thefollowing estimate holds E l ( t ) + E h ( t ) ≤ C E + C Z t X j ≤ j ( d − (cid:16) ˜ E j / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L ) + ˜ Q j / ( k ˙∆ j u k L + k ˙∆ j τ k L ) (cid:17) d t ′ + C Z t X j> j ( d − (cid:16) ˜ F j / ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) + ˜ R j / k ˙∆ j u k L + 2 j ˜ M j / k ˙∆ j τ k L (cid:17) d t ′ , where C , C independent of T . As pointed out in Lemma 3.2, here we also omit the definitions of ˜ E j , ˜ F j , ˜ Q j , ˜ R j , ˜ M j .Please refer to the proof of Proposition 3.1 for their definitions. ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 13
Proof.
Without loss of generality, we assume k ˙∆ j u k L , k ˙∆ j Λ − P ∇ · τ k L , k ˙∆ j P ∇ · τ k L , k ˙∆ j τ k L = 0 . Step 1: Estimate of E l ( t ) . In this step we will supply the estimate of sup t k τ k l ˙ B d − , to E lr ( t ).First, let us notice some cancellations on linear terms.( ˙∆ j ∇ p, ˙∆ j u ) = 0 , (3.22) ( ˙∆ j ∇ · τ, ˙∆ j u ) + ( ˙∆ j D ( u ) , ˙∆ j τ ) = 0 . (3.23)A standard energy computation of system (1.1) yields12 ddt k ˙∆ j u k L + ν k Λ ˙∆ j u k L = µ ( ˙∆ j ∇ · τ, ˙∆ j u ) − ( ˙∆ j ( u · ∇ u ) , ˙∆ j u ) . (3.24) 12 ddt k ˙∆ j τ k L = µ ( ˙∆ j D ( u ) , ˙∆ j τ ) − ( ˙∆ j ( u · ∇ τ ) , ˙∆ j τ ) − ( ˙∆ j Q ( τ, ∇ u ) , ˙∆ j τ )(3.25)Calculating µ (3.24) + µ (3.25), and thanks to (3.23), we finally get12 ddt (cid:16) k ˙∆ j u k L + k ˙∆ j τ k L (cid:17) ≤ C ˜ Q j / ( k ˙∆ j u k L + k ˙∆ j τ k L ) . By a straightforward computation in the low frequencies, the above inequality can beestimated as sup t k u k l ˙ B d − , + sup t k τ k l ˙ B d − , ≤ C ( k u k l ˙ B d − , + k τ k l ˙ B d − , )+ C Z t X j ≤ j ( d − ˜ Q j / ( k ˙∆ j u k L + k ˙∆ j τ k L )d t ′ . (3.26)Combining (3.26) and (3.12), we obtain E l ( t ) ≤ C ( k u k l ˙ B d − , + k τ k l ˙ B d − , d ) + C Z t X j ≤ j ( d − × (cid:16) ˜ E j / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L ) + ˜ Q j / ( k ˙∆ j u k L + k ˙∆ j τ k L ) (cid:17) d t ′ . (3.27) Step 2: Estimate of E h ( t ) . In this step we will supply the estimate of sup t k τ k h ˙ B d , to E hr ( t ).It follows from the equality (3.25) that ddt (cid:0) j k ˙∆ j τ k (cid:1) ≤ C j k ˙∆ j u k + C j ˜ M j / k ˙∆ j τ k L . The standard computation yieldssup t k τ k h ˙ B d , ≤ k τ k h ˙ B d , + C Z t k u k h ˙ B d , d t ′ + C Z t X j> j d ˜ M j / k ˙∆ j τ k L d t ′ . (3.28) Calculating (3.21) + η (3.28), and if we choose η small enough such that η C ≤ , thenthe term η C R t k u k h ˙ B d , d t ′ can be absorbed by the left side, we eventually obtain E h ( t ) ≤ C ( k u k h ˙ B d − , + k τ k h ˙ B d , ) + C Z t X j ≤ j ( d − (cid:16) ˜ F j / ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) + ˜ R j / k ˙∆ j u k L + 2 j ˜ M j / k ˙∆ j τ k L (cid:17) d t ′ . (3.29)Combining (3.27) and (3.29), we complete the proof of Lemma 3.3. (cid:3) Now let us turn to the proof of Proposition 3.1.
Proof.
We will apply Lemma 3.3 to prove (3.2). Let us give the definitions of the nonlinearterms in Lemma 3.3, the term ˜ E j = ˜ E j + ˜ E j + ˜ E j with˜ E j = − µ j P ( u · ∇ u ) , ˙∆ j u ) − µ ( ˙∆ j Λ − P ∇ · Q ( τ, ∇ u ) , ˙∆ j Λ − P ∇ · τ )+ K ( ˙∆ j P ∇ · Q ( τ, ∇ u ) , ˙∆ j u ) , ˜ E j = − µ ( ˙∆ j Λ − P ∇ · ( u · ∇ τ ) , ˙∆ j Λ − P ∇ · τ ) , ˜ E j = K ( ˙∆ j P ( u · ∇ u ) , ˙∆ j P ∇ · τ ) + K ( ˙∆ j P ∇ · ( u · ∇ τ ) , ˙∆ j u ) , and the term ˜ F j = ˜ F j + ˜ F j + ˜ F j with˜ F j = − ν ( ˙∆ j P ∇ · Q ( τ, ∇ u ) , ˙∆ j P ∇ · τ ) + µ j P ∇ · Q ( τ, ∇ u ) , ˙∆ j u ) − µ ν ( ˙∆ j P ( u · ∇ u ) , ˙∆ j u ) , ˜ F j = − ν ( ˙∆ j P ∇ · ( u · ∇ τ ) , ˙∆ j P ∇ · τ ) , ˜ F j = µ j P ( u · ∇ u ) , ˙∆ j P ∇ · τ ) + µ j P ∇ · ( u · ∇ τ ) , ˙∆ j u ) , and ˜ Q j = − µ ( ˙∆ j ( u · ∇ u ) , ˙∆ j u ) − µ ( ˙∆ j ( u · ∇ τ ) , ˙∆ j τ ) − µ ( ˙∆ j Q ( τ, ∇ u ) , ˙∆ j τ ) , ˜ R j = − ( ˙∆ j P ( u · ∇ u ) , ˙∆ j u ) , ˜ M j = − ( ˙∆ j ( u · ∇ τ ) , ˙∆ j τ ) − ( ˙∆ j Q ( τ, ∇ u ) , ˙∆ j τ ) , for the constant K is chosen in (3.9) in Lemma 3.2. Step 1: Estimate for ˜ E j , ˜ F j , ˜ Q j , ˜ M j , ˜ R j . First we consider the terms of ˜ E j , ˜ F j , ˜ Q j , ˜ M j , ˜ R j excluding Q ( τ, ∇ u ). Noting that ∇ · u = 0, we have(3.30) ( ˙∆ j P ( u · ∇ u ) , ˙∆ j u ) = ( ˙∆ j ( u · ∇ u ) , ˙∆ j u ) . According to Proposition 5.1 and Proposition 5.3 in Appendix, we have(3.31) | ( ˙∆ j ( u · ∇ u ) , ˙∆ j u ) | . c j − j ( d − k u k ˙ B d , k u k ˙ B d − , k ˙∆ j u k L , | ( ˙∆ j ( u · ∇ τ ) , ˙∆ j τ ) | . c j − jψ d − , d ( j ) k u k ˙ B d , k τ k ˙ B d − , d k ˙∆ j τ k L , ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 15 with P j ∈ Z c j ≤
1. Next we consider the terms of ˜ E j , ˜ F j , ˜ Q j , ˜ R j , ˜ M j including Q ( τ, ∇ u ).Applying Lemma 2.2 and H¨older’s inequality, for j ≤
0, we have(3.32) (cid:12)(cid:12) − µ ( ˙∆ j Λ − P ∇ · Q ( τ, ∇ u ) , ˙∆ j Λ − P ∇ · τ ) + K ( ˙∆ j P ∇ · Q ( τ, ∇ u ) , ˙∆ j u ) (cid:12)(cid:12) . (1 + 2 j ) k Q ( τ, ∇ u ) k L ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L ) . k Q ( τ, ∇ u ) k L ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L ) , (cid:12)(cid:12) − µ ( ˙∆ j Q ( τ, ∇ u ) , ˙∆ j τ ) (cid:12)(cid:12) . k Q ( τ, ∇ u ) k L k ˙∆ j τ k L , and j ≥
0, we have(3.33) (cid:12)(cid:12) − ν ( ˙∆ j P ∇ · Q ( τ, ∇ u ) , ˙∆ j P ∇ · τ ) + µ j P ∇ · Q ( τ, ∇ u ) , ˙∆ j u ) (cid:12)(cid:12) . j k Q ( τ, ∇ u ) k L ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L ) , (cid:12)(cid:12) − ( ˙∆ j Q ( τ, ∇ u ) , ˙∆ j τ ) (cid:12)(cid:12) . k Q ( τ, ∇ u ) k L ( k ˙∆ j τ k L + k ˙∆ j P ∇ · τ k L ) . Combining (3.30) ∼ (3.33), we finally obtain Z t X j ≤ j ( d − (cid:16) ˜ E j / ( k ˙∆ j u k + k ˙∆ j Λ − P ∇ · τ k L )+ ˜ Q j / ( k ˙∆ j u k L + k ˙∆ j τ k L (cid:17) d t ′ + Z t X j ≤ j ( d − (cid:16) ˜ F j / ( k ˙∆ j u k L + k ˙∆ j P ∇· τ k L )+ ˜ R j / k ˙∆ j u k L +2 j ˜ M j / k ˙∆ j τ k L (cid:17) d t ′ . (cid:0) sup t k τ k ˙ B d − , d + sup t k u k ˙ B d − , (cid:1) Z t k u k ˙ B d , d t ′ + Z t k Q ( τ, ∇ u ) k ˙ B d − , d d t ′ . (cid:0) sup t k τ k ˙ B d − , d + sup t k u k ˙ B d − , (cid:1) Z t k u k ˙ B d , d t ′ , (3.34)where in the second inequality we use Remark 5.2. Step 2: Estimate for ˜ E j , ˜ E j . Next we turn to the term ˜ E j . Our strategy is to apply Proposition 5.4 and divide ˜ E j into three parts. We have ˜ E j = ˜ E , j + ˜ E , j + ˜ E , j , where˜ E , j = − µ ( ˙∆ j Λ − P ( u · ∇ P ∇ · τ ) , ˙∆ j Λ − P ∇ · τ ) , ˜ E , j = − µ ( ˙∆ j Λ − P ( ∇ u · ∇ τ ) , ˙∆ j Λ − P ∇ · τ ) , ˜ E , j = µ ( ˙∆ j Λ − P ( ∇ u · ∇ ∆ − ∇ · ∇ · τ ) , ˙∆ j Λ − P ∇ · τ ) . Let us consider the most difficult term ˜ E , j . We claim that(3.35) k u ⊗ P ∇ · τ k l ˙ B d − , . k u k ˙ B d − , k P ∇ · τ k B d , d − + k u k ˙ B d , k P ∇ · τ k B d − , d − . Then by H¨older’s inequality, ∇ · u = 0 and (3.35), we obtain Z t X j ≤ j ( d − | ˜ E , j | / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . Z t k u ⊗ P ∇ · τ k l ˙ B d − , d t ′ . sup t k τ k ˙ B d − , d Z t k u k ˙ B d , d t ′ + sup t k u k ˙ B d − , Z t k P ∇ · τ k B d , d − d t ′ . (3.36)The proof of (3.35) is based on the Bony’s product decomposition u ⊗ P ∇ · τ = ˙ T u P ∇ · τ + ˙ T P ∇· τ u + ˙ R ( u, P ∇ · τ ) . where the definitions of ˙ T and ˙ R can be referred to Appendix. Then by Proposition 5.1,we obtain k T u P ∇ · τ + T P ∇· τ u k l ˙ B d − , . k T u P ∇ · τ + T P ∇· τ u k ˙ B d − , d − . k u k ˙ B d − , k P ∇ · τ k B d , d − , k R ( P ∇ · τ, u ) k l ˙ B d − , . k R ( P ∇ · τ, u ) k ˙ B d − , d . k u k ˙ B d , k P ∇ · τ k B d − , d − , which imply (3.35). Let us return to the terms ˜ E , j , ˜ E , j . Noticing that the incompressiblecondition on u , we have the following equalities:[ ∇ u · ∇ τ ] i , X j,k ∂ j u k ∂ k τ i,j = X j,k ∂ k ( ∂ j u k τ i,j ) , [ ∇ u · ∇ ∆ − ∇ · ∇ · τ ] i , X k ∂ i u k ∂ k ∆ − ∇ · ∇ · τ = X k ∂ k ( ∂ i u k ∇ ∆ − ∇ · ∇ · τ ) . (3.37)Combining (3.37) and Remark 5.2, we have Z t X j ≤ j ( d − (cid:12)(cid:12) ˜ E , j + ˜ E , j (cid:12)(cid:12) / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . Z t k∇ u · ∇ τ k l ˙ B d − , + k∇ u · ∇ ∆ − ∇ · ∇ · τ k l ˙ B d − , d t ′ . Z t k∇ u ⊗ τ k ˙ B d − , d + k∇ u ⊗ ∆ − ∇ · ∇ · τ k ˙ B d − , d d t ′ . sup t k τ k ˙ B d − , d Z t k u k ˙ B d , d t ′ . (3.38)Therefore, it follows from (3.36) and (3.38) that(3.39) Z t X j ≤ j ( d − | ˜ E j | / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . E ( t ) . ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 17
Now we consider ˜ E j . Applying Proposition 5.4, we have ˜ E j = ˜ E , j + ˜ E , j + ˜ E , j , where˜ E , j = K (cid:0) ( ˙∆ j P ( u · ∇ u ) , ˙∆ j P ∇ · τ ) + ( ˙∆ j P ( u · ∇ P ∇ · τ ) , ˙∆ j u ) (cid:1) , ˜ E , j = K ( ˙∆ j P ( ∇ u · ∇ τ ) , ˙∆ j u ) , ˜ E , j = − K ( ˙∆ j P ( ∇ u · ∇ ∆ − ∇ · ∇ · τ ) , ˙∆ j u ) . For the term ˜ E , j , noticing that j ≤ | ˜ E , j | . c j k u k ˙ B d , (cid:16) − j ( d − k u k ˙ B d − , k ˙∆ j P ∇ · τ k L + 2 − j ( d − k P ∇ · τ k ˙ B d − , d − k ˙∆ j u k L (cid:17) . Thanks to k ˙∆ j P ∇ · τ k L . j k ˙∆ j Λ − P ∇ · τ k L , and j ≤
0, we have Z t X j ≤ j ( d − | ˜ E , j | / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . (cid:0) sup t k u k ˙ B d − , + sup t k τ k ˙ B d − , d (cid:1) Z t k u k ˙ B d , d t ′ . (3.40)Dealing with the terms ˜ E , j , ˜ E , j in the same way as used in the proof of ˜ E , j , ˜ E , j , wehave Z t X j ≤ j ( d − (cid:12)(cid:12) ˜ E , j + ˜ E , j (cid:12)(cid:12) / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . sup t k τ k ˙ B d − , d Z t k u k ˙ B d , d t ′ . (3.41)Then combining (3.40) and (3.41), we deduce that(3.42) Z t X j ≤ j ( d − | ˜ E j | / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . E ( t ) . Step 3: Estimate for ˜ F j , ˜ F j . Using Proposition 5.4, we divide ˜ F j into three terms˜ F , j = − ν ( ˙∆ j P ( u · ∇ P ∇ · τ ) , ˙∆ j P ∇ · τ ) , ˜ F , j = − ν ( ˙∆ j P ( ∇ u · ∇ τ ) , ˙∆ j P ∇ · τ ) , ˜ F , j = ν ( ˙∆ j P ( ∇ u · ∇ ∆ − ∇ · ∇ · τ ) , ˙∆ j P ∇ · τ ) . By Proposition 5.3, we obtain | ( ˙∆ j ( u · ∇ P ∇ · τ ) , ˙∆ j P ∇ · τ ) | . c j − j ( d − k u k ˙ B d , k P ∇ · τ k ˙ B d − , d − k ˙∆ j P ∇ · τ k L , which implies Z t X j> j ( d − | ˜ F , j | / ( k ˙∆ j u k L + k ˙∆ j P ∇ · τ k L )d t ′ . sup t k τ k ˙ B d − , d Z t k u k ˙ B d , d t ′ . (3.43)Then we consider ˜ F , j and ˜ F , j . Thanks to Remark 5.2 and (3.37), we obtain Z t X j> j ( d − (cid:12)(cid:12) ˜ F , j + ˜ F , j (cid:12)(cid:12) / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . Z t k∇ u ⊗ τ k ˙ B d − , d + k∇ u ⊗ ∆ − ∇ · ∇ · τ k ˙ B d − , d d t ′ . sup t k τ k ˙ B d − , d Z t k u k ˙ B d , d t ′ . (3.44)Combining (3.43) and (3.44), we obtain(3.45) Z t X j ≤ j ( d − | ˜ F j | / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . E ( t ) . For the last term ˜ F j , we also have ˜ F j = ˜ F , j + ˜ F , j + ˜ F , j , where˜ F , j = µ (cid:0) ( ˙∆ j ( u · ∇ u ) , ˙∆ j P ∇ · τ ) + ( ˙∆ j ( u · ∇ P ∇ · τ ) , ˙∆ j u ) (cid:1) , ˜ F , j = µ j P ( ∇ u · ∇ τ ) , ˙∆ j u ) , ˜ F , j = − µ j P ( ∇ u · ∇ ∆ − ∇ · ∇ · τ ) , ˙∆ j u ) . Let us first consider ˜ F , j . By Proposition 5.3, we obtain (cid:12)(cid:12) ( ˙∆ j ( u · ∇ u ) , ˙∆ j P ∇ · τ ) + ( ˙∆ j ( u · ∇ P ∇ · τ ) , ˙∆ j u ) (cid:12)(cid:12) . c j k u k ˙ B d , (cid:0) − j ( d − k u k ˙ B d − , k P ∇ · τ k L + 2 − j ( d − k P ∇ · τ k ˙ B d − , d − k ˙∆ j u k L (cid:1) . Then, we have Z t X j> j ( d − | ˜ F , j | / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . (cid:0) sup t k u k ˙ B d − , + sup t k τ k ˙ B d − , d (cid:1) Z t k u k ˙ B d , d t ′ . (3.46)Dealing with the terms ˜ F , j , ˜ F , j in the same way as used in the proof of ˜ F , j , ˜ F , j , wehave Z t X j> j ( d − (cid:12)(cid:12) ˜ F , j + ˜ F , j (cid:12)(cid:12) / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . sup t k τ k ˙ B d − , d Z t k u k ˙ B d , d t ′ . (3.47) ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 19
Combining (3.46), (3.47), we gather Z t X j> j ( d − | ˜ F j | / ( k ˙∆ j u k L + k ˙∆ j Λ − P ∇ · τ k L )d t ′ . E ( t ) . (3.48)According to (3.34), (3.39), (3.42), (3.45) and (3.48), we complete the proof of Proposition3.1. (cid:3) The global existence and the uniqueness
This section is devoted to the proof of Theorem 1.2.4.1.
Approximate solutions and the uniform estimates.
The construction of ap-proximate solutions is based on the following local existence theorem.
Proposition 4.1.
Let ( u , τ ) be an initial data in H s with s strictly greater than d . Thena unique strictly positive time T exists so that a unique solution ( u, τ ) exists such that u ∈ C ([0 , T ); H s ) ∩ L loc (0 , T ; H s +1 ); τ ∈ C ([0 , T ); H s ) . Moreover, the solution ( u, τ ) can be continued beyond T if sup T k τ k ˙ B d − , d + Z T k u k ˙ B d , d t ′ < ∞ . Proof.
The proof is very similar to the Theorem 1.1 in [7], here we omit it since we onlyhave made a slight modification of its proof. (cid:3)
Set C n , { ξ ∈ R d | n − ≤ | ξ | ≤ n } . We build the approximate solution ( u n , τ n ) solvingthe system(4.1) u nt + u n · ∇ u n − ν ∆ u n − ∇ p = µ ∇ · τ n τ nt + u n · ∇ τ n + Q ( τ n , ∇ u n ) = µ D ( u n ) , ∇ · u n = 0 ,u n (0 , x ) = J n u ; τ n (0 , x ) = J n τ , where F ( J n U )( ξ ) = I C n ( ξ ) F U ( ξ ) with I C n the smooth cut-off functions supported in C n .Using the direct computations, we gather that(4.2) lim n →∞ k J n u − u k ˙ B d − , = 0; lim n →∞ k J n τ − τ k ˙ B d − , d = 0 . Indeed, it is easy to see that J n u , J n τ ∈ H s for all s >
0. Therefore, applying Proposition4.1, we can obtain that there exists a maximal existence time T n > s > u n , τ n ), u n ∈ C ([0 , T n ); H s ) ∩ L loc (0 , T n ; H s +1 ); τ n ∈ C ([0 , T n ); H s ) . Using the definition of the Besov space, it is easy to check that u n ∈ C ([0 , T n ); ˙ B d − , ) ∩ L loc (0 , T n ; ˙ B d +12 , ); τ n ∈ C ([0 , T n ); ˙ B d − , d ) , P ∇ · τ n ∈ L loc (0 , T n ; ˙ B d , d − ) . Let us define T ∗ n = sup { t ∈ [0 , T n ) | E n ( t ) ≤ e C E } . Firstly, we claim that T ∗ n = T n . Using the continuity argument, it suffices to show that for all n ∈ N , E n ( t ) ≤ e C E . In fact, set e C = 4 C , and choose E small enough such that E ≤ C C , then combineProposition 3.1, we obtain E n ( t ) ≤ (cid:0) C + 16 C C E (cid:1) E ≤ e C E . In conclusion, we construct a sequence of approximate solution ( u n , τ n ) on [0 , T n ) satis-fying(4.3) E n ( T n ) ≤ C E , for any n ∈ N . Thanks to (4.3), we can easily obtain that T n = ∞ by a direct applicationof Proposition 4.1. To sum up, we get that for all t >
0, we have(4.4) E n ( t ) ≤ C E . The existence.
In this part we will use a standard compact argument to show that,up to an extraction, the sequence (( u n , τ n )) n ∈ N converges in D ′ ( R + × R d ) to a solution( u, τ ) of (1.1) which has the desired regularity properties.It is convenient to split ( u n , τ n ) into linear part and discrepant part. More precisely,we denote by ( u nL , τ nL ) the solution to ∂ t u nL − ν ∆ u nL − µ ∇ · τ nL − ∇ p = 0 ,∂ t τ nL − µ D ( u nL ) = 0 , ∇ · u nL = 0 ,u nL (0 , x ) = J n u ; τ nL (0 , x ) = J n τ . By Proposition 3.1, we can easily get E nL ( t ) . E (0), where E nL ( t ) is the energy of ( u nL , τ nL )in form of (3.1). Also, we denote ( u L , τ L ) the solution to ∂ t u L − ν ∆ u L − µ ∇ · τ L − ∇ p = 0 ,∂ t τ L − µ D ( u L ) = 0 , ∇ · u L = 0 ,u L (0) = u ; τ L (0) = τ . It is easy to check that(4.5) u nL −→ u L in C ( R + ; ˙ B d − , ) ∩ L ( R + ; ˙ B d +12 , ) ,τ nL −→ τ L in C ( R + ; ˙ B d − , d ); P ∇ · τ nL −→ P ∇ · τ L in L ( R + ; ˙ B d , d − ) . ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 21
Denote ( u n , τ n ) , ( u n − u nL , τ n − τ nL ), thanks to (4.4), we also get u n ∈ C ( R + ; ˙ B d − , ) ∩ L ( R + ; ˙ B d +12 , ) ,τ n ∈ C ( R + ; ˙ B d − , d ); P ∇ · τ n ∈ L ( R + ; ˙ B d , d − ) . Lemma 4.2. (( u n , τ n )) n ∈ N is uniformly bounded in C loc ( R + ; ˙ B d − , ) × C loc ( R + ; ˙ B d − , ) .Proof. Recall that ∂ t τ n = µ D ( u n ) − u n · ∇ τ n − Q ( τ n , ∇ u n ) , which combining with (4.4) and Proposition 5.1, we gather that k ∂ t τ n k L T ( ˙ B d − , ) . k u n k L T ( ˙ B d , ) + sup T k τ n k ˙ B d , k u n k L T ( ˙ B d , ) < ∞ . Then we get τ n ∈ C loc ( R + ; ˙ B d − , ). On the other hand, ∂ t u n = µ P ∇ · τ n + ν ∆ u n + P u n · ∇ u n . Proposition 5.1 and (4.4) imply that k ∂ t u n k L T ( ˙ B d − , ) . T sup T k τ n k ˙ B d − , d + (sup T k u n k ˙ B d − , + 1) k u n k L T ( ˙ B d , ) < ∞ , which means that u n ∈ C loc ( R + ; ˙ B d − , ). (cid:3) Let us choose a sequence ( φ p ) p ∈ N of smooth cut-off functions supported in the ball B (0 , p + 1) of R d and equal to 1 in a neighborhood of B (0 , p ). Lemma 4.2 ensures that( φ p u n , φ p τ n ) is uniformly equicontinuous in C ([0 , p ]; ˙ B d − , × ˙ B d − , ) . We know that when s >
0, ˙ B s , ( K ) ∼ = B s , ( K ) for all compact set K and ˙ B s , ֒ → B s , when s ≤
0. Then we can get( φ p u n , φ p τ n ) is uniformly equicontinuous in C ([0 , p ]; B d − , × B d − , ) , (4.6) ( φ p u n , φ p τ n ) is uniformly bounded in C ([0 , p ]; B d − , × B d , ) . (4.7)Moreover, we have the facts B d − , ( K ) ֒ → ֒ → B d − , ( K ) and B d , ( K ) ֒ → ֒ → B d − , ( K ). Then,by Ascoli’s Theorem and Cantor’s diagonal process, we get a distribution ( u, τ ) such that(4.8) ( φ p u n , φ p τ n ) −→ ( φ p u, φ p τ ) in C ([0 , p ]; B d − , × B d − , ) . Denote u , u + u L , τ , τ + τ L , we easily have( u n , τ n ) −→ ( u, τ ) in S ′ ( R d × R + ) , With the help of (4.5), (4.7) and (4.8), following the argument as in [10], it is routine toverify that ( u, τ ) satisfies the system (1.1) in the distribution sense. Moreover, we caninfer from (4.4) that(4.9) u ∈ L ∞ ( R + ; ˙ B d − , ) ∩ L ( R + ; ˙ B d +12 , ); τ ∈ L ∞ ( R + ; ˙ B d − , d ) , P ∇ · τ ∈ L ( R + ; ˙ B d , d − ) . At last, we have to show the properties of continuity with respect to time. The conti-nuity of u is straightforward. Indeed from the equation of u , we have L loc ( R + ; ˙ B d − , )which imply u ∈ C ( R + ; ˙ B d − , ) . As for τ , by the same argument as in [10], we get τ ∈ C ( R + ; ˙ B d − , d ).4.3. The uniqueness.
Assume ( u , τ ) and ( u , τ ) are two solutions of (1.1) with thesame initial data. Denote δu = u − u , δτ = τ − τ , then ( δu, δτ ) satisfies(4.10) ( δu ) t − ν ∆ δu = µ ∇ · δτ − u · ∇ δu − δu · ∇ u , ( δτ ) t + u · ∇ δτ = µ D ( δu ) − δu · ∇ τ − Q ( τ , ∇ δu ) − Q ( δτ, ∇ u ) , ∇ · δu = 0 ,δu (0 , x ) = 0; δτ (0 , x ) = 0 . We shall work in the following functional spaces H T , (cid:0) L ∞ (0 , T ; ˙ B − d, ∞ ) ∩ e L (0 , T ; ˙ B d, ∞ ) (cid:1) d × L ∞ (0 , T ; ˙ B d, ∞ ) . We first state that ( δu, δτ ) ∈ H T . From the second equation of (4.10) and Proposition2.4, Proposition 2.5, we have k ( δτ ) t k L T ( ˙ B d, ∞ ) . k ( δτ ) t k L T ( ˙ B d − , ) . (cid:16) k u k L T ( ˙ B d , ) + k u k L T ( ˙ B d , ) (cid:17)(cid:16) k τ k L ∞ T ( ˙ B d , ) + k τ k L ∞ T ( ˙ B d , ) (cid:17) , which means that δτ ∈ C ([0 , T ]; ˙ B d, ∞ ). A similar discussion to δτ entails that δu ∈ C ([0 , T ]; ˙ B − d, ∞ ).Now, let us turn to the proof of estimates for δτ . Applying Proposition 2.10 to thesecond equation of (4.10), we have k δτ k e L ∞ t ( ˙ B d, ∞ ) . e C R t k∇ u k ˙ B d, d t ′ Z t e − C R t ′ k∇ u k ˙ B d, d s k F k ˙ B d, ∞ d t ′ . e C R t k∇ u k ˙ B d, d t ′ Z t k F k ˙ B d, ∞ d t ′ , where F = µ D ( δu ) − δu · ∇ τ − Q ( τ, ∇ δu ) − Q ( δτ, ∇ u ) . By Proposition 2.4 and Propo-sition 2.5, we get k F k L t ( ˙ B d, ∞ ) . k u k L t ( ˙ B d , ) k δτ k L ∞ t ( ˙ B d, ∞ ) + (cid:0) k τ k L ∞ t ( ˙ B d , ) + k τ k L ∞ t ( ˙ B d , ) (cid:1) k δu k L t ( ˙ B d, ) . We then finally obtain k δτ k L ∞ t ( ˙ B d, ∞ ) . e C R t k∇ u k ˙ B d , d t ′ (cid:16) k u k L t ( ˙ B d , ) k δτ k L ∞ t ( ˙ B d, ∞ ) + (cid:0) k τ k L ∞ t ( ˙ B d , ) + k τ k L ∞ t ( ˙ B d , ) (cid:1) k δu k L t ( ˙ B d, ) (cid:17) . (4.11) ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 23
From Proposition 2.11, we infer k δu k L t ( ˙ B d, ) . k δu k e L t ( ˙ B d, ∞ ) log (cid:18) e + k δu k e L t ( ˙ B d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) k δu k e L t ( ˙ B d, ∞ ) (cid:19) . Taking T small enough such that k u k L t ( ˙ B d , ) sufficiently small, then inserting the aboveinequality into (4.11), we deduce k δτ k L ∞ t ( ˙ B d, ∞ ) . C T k δu k e L t ( ˙ B d, ∞ ) log (cid:18) e + k δu k e L t ( ˙ B d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) k δu k e L t ( ˙ B d, ∞ ) (cid:19) , (4.12)where C T , exp (cid:0) C k u k L T ( ˙ B d , ) (cid:1)(cid:0) k τ k L ∞ T ( ˙ B d , ) + k τ k L ∞ T ( ˙ B d , ) (cid:1) < ∞ .For the estimate of δu . By Proposition 2.9, we get k δu k L ∞ t ( ˙ B − d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) . k E k e L t ( ˙ B − d, ∞ ) , where E = µ ∇ · δτ − u · ∇ δu − δu · ∇ u . It follows from (3.37), Remark 2.7 andProposition 2.5 that k E k e L t ( ˙ B − d, ∞ ) . k δτ k L t ( ˙ B d, ∞ ) + k u ⊗ δu k e L t ( ˙ B d, ∞ ) + k δu ⊗ u k e L t ( ˙ B d, ∞ ) . k δτ k L t ( ˙ B d, ∞ ) + (cid:0) k u k e L t ( ˙ B d, ) + k u k e L t ( ˙ B d, ) (cid:1) k δu k e L t ( ˙ B d, ∞ ) . Then in light of interpolation Theorem, we infer k δu k L ∞ t ( ˙ B − d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) . ( k u k e L t ( ˙ B d, ) + k u k e L t ( ˙ B d, ) )( k δu k L ∞ t ( ˙ B − d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) ) + k δτ k L t ( ˙ B d, ∞ ) . (4.13)Recall that u t − ν ∆ u = µ P ∇ · τ − P ( u · ∇ u ) . It is easy to verify that µ P ∇ · τ − P ( u · ∇ u ) ∈ L T ( ˙ B d, ) for the finite time T . Then fromProposition 2.9, we infer that u i ∈ e L ∞ T ( ˙ B d, ) ∩ e L T ( ˙ B d, ) ( i = 1 , u i ∈ e L T ( ˙ B d, )by interpolation theorem. Taking T small enough such that k u i k e L t ˙ B d, sufficiently small,then combining (4.12), (4.13), we have k δu k L ∞ t ( ˙ B − d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) . C T Z t k δu k e L t ′ ( ˙ B d, ∞ ) log (cid:18) e + k δu k e L t ′ ( ˙ B d, ∞ ) + k δu k e L t ′ ( ˙ B d, ∞ ) k δu k e L t ′ ( ˙ B d, ∞ ) (cid:19) d t ′ . (4.14)Denote X ( t ) , k δu k L ∞ t ( ˙ B − d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) , V ( t ) , k δu k e L t ( ˙ B d, ∞ ) + k δu k e L t ( ˙ B d, ∞ ) , we claimthat V ( T ) < ∞ . In fact k δu k e L T ( ˙ B d, ∞ ) + k δu k e L T ( ˙ B d, ∞ ) . X i =1 Z T ( k u i k ˙ B d, + k u i k ˙ B d, )d t ′ . X i =1 ( T k u i k L ∞ T ˙ B d − , + k u i k L T ˙ B d , ) . We can rewrite (4.14) as follows X ( t ) . C T Z t X ( t ′ ) log (cid:18) e + V ( T ) X ( t ′ ) (cid:19) . (4.15)As Z d rr log( e + V ( T ) r ) = + ∞ , Osgood lemma implies X ≡ , T ], whence also δτ ≡
0. Then a continuity argumentensures that ( u , τ ) = ( u , τ ) on R + .5. Appendix
This section is devoted to the estimates of the convection terms which were used inSection 3.First, let us give some definitions in paradifferential calculus in homogeneous spaces.We designate the homogeneous paraproduct of v by u as˙ T u v , X q ˙ S q − ˙∆ q v. and the homogeneous remainder of u and v as˙ R ( u, v ) , X q ˙∆ q u ˙ e ∆ q v, and ˙ e ∆ q = ˙∆ q − + ˙∆ q + ˙∆ q +1 . Formally, we have the following homogeneous Bony decomposition: uv = ˙ T u v + ˙ T v u + ˙ R ( u, v ) . The properties of continuity of homogeneous paraproduct and remainder on homogeneoushybrid Besov spaces are described as follows.
Proposition 5.1.
For all s , s , t , t such that s ≤ d and s ≤ d , the following estimateholds k T u v k ˙ B s t − d ,s t − d . k u k ˙ B s ,s k v k ˙ B t ,t . If min( s + t , s + t ) > , then k R ( u, v ) k ˙ B s t − d ,s t − d . k u k ˙ B s ,s k v k ˙ B t ,t . If u ∈ L ∞ , k T u v k ˙ B t ,t . k u k L ∞ k v k ˙ B t ,t , and, if min( t , t ) > , then k R ( u, v ) k ˙ B t ,t . k u k L ∞ k v k ˙ B t ,t . Remark . When d ≥
2, we have k uv k ˙ B d − , d . k u k ˙ B d , k v k ˙ B d − , d . ELL-POSEDNESS OF THE OLDROYD-B MODEL WITHOUT DAMPING MECHANISM 25
Proposition 5.3.
Let u be a vector with ∇ · u = 0 . Suppose that − − d < s , t , s , t ≤ d . The following two estimates hold (cid:12)(cid:12) ( ˙∆ j ( u · ∇ v ) , ˙∆ j v ) (cid:12)(cid:12) . c j − jψ s ,s ( j ) k u k ˙ B d , k v k ˙ B s ,s k ˙∆ j v k L , (cid:12)(cid:12) ( ˙∆ j ( u · ∇ v ) , ˙∆ j w )+( ˙∆ j ( u · ∇ w ) , ˙∆ j v ) (cid:12)(cid:12) . c j k u k ˙ B d , (2 − jψ s ,s ( j ) k v k ˙ B s ,s k ˙∆ j w k L + 2 − jψ t ,t ( j ) k w k ˙ B t ,t k ˙∆ j v k L ) . where the function ψ α,β ( j ) define as ψ α,β ( j ) = α if j ≤ , ψ α,β ( j ) = β , if j > , and P j ∈ Z c j ≤ . One can refer to [11] for the proof of above two Propositions. Here we only have madea slight modification since the incompressible condition on u .Next, we introduce a useful Proposition to deal with [ P div , u · ∇ ] type commutators. Proposition 5.4.
For any smooth tensor [ τ i,j ] d × d and d dimensional vector u , it alwaysholds that P ∇ · ( u · ∇ τ ) = P ( u · ∇ P ∇ · τ ) + P ( ∇ u · ∇ τ ) − P ( ∇ u · ∇ ∆ − ∇ · ∇ · τ ) , where the ith component of ∇ u · ∇ τ is [ ∇ u · ∇ τ ] i = X j ∂ j u · ∇ τ i,j , and also [ ∇ u · ∇ ∆ − ∇ · ∇ · τ ] i = ∂ i u · ∇ ∆ − ∇ · ∇ · τ. For more detailed derivations, one can refer to the proof of the three dimensions in [24].
Acknowledgements.
Q. Chen and X. Hao were supported by the National NaturalScience Foundation of China (No.11671045).
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Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
E-mail address : chen [email protected] The Graduate School of China Academy of Engineering Physics, Beijing 100088, China
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