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The Fujita-Kato Theorem for some Oldroyd-B Models
Francesco de Anna, Marius Paicu
To cite this version:
The Fujita-Kato theorem for some Oldroyd-B models
July 1, 2020
Francesco De Anna
Institut f¨ur Mathematik, Universit¨at W¨urzburg, Germany e-mail: francesco.deanna@mathematik.uni-wuerzburg.de
Marius Paicu
Institut de Math´ematiques de Bordeaux, Universit´e de Bordeaux, France e-mail: Marius.Paicu@math.u-bordeaux.fr
article info abstract
Keywords: Oldroyd-B model, finite energy solutions, Lipschitz flow, criti-cal regularities.
MSC: 35Q35, 35B65, 76D05, 76N10.
In this paper, we investigate the Cauchy problem associated to a system of PDE’s of Oldroyd type. The considered model describes the evolution of certain viscoelastic fluids within a corotational framework. The non corotational setting is also addressed in dimension two.
We show that some widespread results concerning the incompressible Navier-Stokes equations can be extended to the considered systems.
In particular we show the existence and uniqueness of global-in-time classical solutions for large data in dimension two. This result is supported by suitable condition on the initial data to provide a global-in-time Lipschitz regularity for the flow, which allows to overcome specific challenging due to the non time decay of the main forcing terms.
Secondly, we address the global-in-time well posedness in dimension d ≥ 3. We prove the propagation of Lipschitz regularities for the flow. For this result, we just assume the initial data to be sufficiently small in a critical Lorentz space.
1. Introduction
The modeling and analysis of the hydrodynamics of viscoelastic fluids has attracted much attention over the last decades [2, 5, 10, 21, 22, 25]. Generally, the physical state of matter of a material can be determined by the degree of freedom for the movement of its constitutive molecules. Increasing this degree of freedom, the most of materials evolves from a solid state to a liquid phase and eventually to a gas form. Nevertheless, there exist in nature several materials that present characteristics in the between of an isotropic fluid and a crystallized solid. These materials are usually classified as viscoelastic fluids, since they generally behave as a viscous fluids as well as they share some properties with elastic materials. The most common ones, for instance, are related to memory effects. We refer the reader to [4, 17, 26–28] for an overview of the Physics behind the modeling of these complex fluids.
This article is devoted to the analysis of the following evolutionary system of PDE’s, describing the hydrodynamics of specific incompressible viscoelastic fluids:
∂tτ + u · ∇τ − ω τ + τ ω = µ D R+× Rd,
∂tu + u · ∇u − ν∆u + ∇p = div τ R+× Rd,
div u = 0 R+× Rd,
(u, τ)|t=0 = (u0, τ0) Rd.
(1)
Here µ ≥ 0, the constant ν > 0 stands for the viscosity of the fluids, while the state variables correspond to u = u(t, x), the velocity field of a particle x ∈ Rn at a time t ∈ R, and τ = (τ(t, x)i j)i, j=1,...,d, the conformation
tensor in Rd×d describing the internal elastic forces that the constitutive molecules exert on each other. To simplify our analysis, we assume our non-Newtonian fluid to occupy the entire whole space Rd, with dimension d ≥ 2. The evolutionary equation for the conformation tensor τ is then driven by the vorticity tensor ω = (ω(t, x))i, j=1,...d, which
stands for the skew-adjoint part of the deformation tensor ∇u: ω =
∇u − t∇u
2 .
The system can be seen as a simplified version of the more general Oldroyd-B model (cf. [5]): ∂tτ + u · ∇τ + a τ + Q(u, τ ) = µ2D R+× Rd,
∂tu + u · ∇u − ν∆u + ∇p = µ1div τ R+× Rd,
div u = 0 R+× Rd,
(u, τ)|t=0 = (u0, τ0) Rd.
(2)
The main parameters ν, a, µ1, µ2 are assumed to be non-negative and they are specific to the characteristic of the
considered material. In particular [22] ν, a and b correspond respectively to θ /Re, 1/We and 2(1 − θ )/(We Re), where Re is the Reynolds number, θ is the ratio between the so called relaxation and retardation times and We is the Weissenberg number (see also [11], equations (1.5)–(1.6)). The bilinear term Q(u, τ) reads as follows:
Q(u, τ) = τ ω − ωτ + b(D τ + τ D), (3)
where the so-called slip parameter b is a constant value between [−1, 1] and D = (∇u + ∇uT)/2 is the symmetric contribution of the deformation tensor ∇u. The corotational term τ ω − ωτ describes how molecules are twisted by the underlying flow, while the term depending on b describes how molecules are stretched and deformed by the flow itself.
For the sake of our analysis, in this paper we mainly impose the following restriction on the main parameters of the Oldroyd-B model:
µ1 = 1 µ2 = µ ≥ 0, b= 0,
from which one obtain our main system (1). The first condition is introduced just for the sake of a clear presentation, while the second and third conditions will play a major role in the analysis techniques we will perform in the forthcoming sections. The case of µ2 = 0 can roughly be interpreted as the case of the infinite Weissenberg number
limit. Furhtermore, following [5, 11], we determine that the limit model µ = 0 occurs when θ , the ratio between the so called relaxation and retardation times, is converging to 1. Indeed, when µ1= 1, we have µ2 = 2νλ1(1 − θ ) where
λ1 is the retardation time and θ = λλ21 with λ2 being the relaxation time. We finally note that some different Oldroyd
models with infinite Weissenberg number (and so µ2= 0) have been studied in [16]. We additionally assume that
a = 0, increasing the challenging of our model, since no damping effect is now assumed on the evolution of the conformation tensor τ. We claim that all our results hold also for the damped case given by a > 0, which is somehow much easier to treat.
We present an overview of some recent results concerning systems (1) and (2). In [13], the authors dealt with the existence and uniqueness of local strong solutions for system (2) in Sobolev space Hs(Ω), for a sufficiently-smooth
bounded domain Ω and a sufficiently large regularity s > 0. The same authors in [14] showed that these solutions are global if the initial data are sufficiently small as well it is small the coupling between the main terms of the constitutive equations.
Lions and Masmoudi [22] addressed the corotational case of system (2), given by b = 0 and showed the existence and uniqueness of global-in-time weak solutions in a bidimensional setting.
Lei, Liu and Zhou [20] proved existence and uniqueness of classical solutions near equilibrium of system (2) for small initial data, assuming the domain to be periodic or to be the whole space.
Bresh and Prange [3] analyze the low Weissenberg asymptotic limit of solutions for system (2) in a corotational setting b = 0. They focus on the specific formulation of the Oldroyd-B system in which the main parameter of (2) are explicitly defined in terms of the Weissenberg number We, which compares the viscoelastic relaxation time to a time scale relevant to the fluid flow. The authors study the weak convergence towards the Navier–Stokes system, as We→ 0. Furthermore, they take into account the presence of defect measures in the initial data and show that they do not perturb the Newtonian limit of the corotational system.
Chemin and Masmoudi [5] proved the existence and uniqueness of strong solutions of the Oldroyd-B model (2), within the framework of homogenous Besov spaces with critical index of regularity. The authors particularly showed local and global-in-time existence of solutions for large and small initial data, respectively, under the assumption of a smallness condition on the coupling parameters of system (2). In [30], Zi, Fang and Zhand extended the mentioned result, relaxing this smallness condition. The main results of this manuscript should be seen as suitable improvements of [5,30], within the setting of system (1). In particular, we relax the assumption of small initial data for global-in-time solution, considering small functions in weak Lebesgue spaces (cf. Theorem 1.6).
Elgindi and Rouss´et [9] addressed the well-posedness of a system related to (1). On the one hand they introduced a dissipative and regularizing mechanism on the evolution of the deformation tensor τ. On the other hand they considered the case of a null viscosity ν = 0, i.e. of an Euler type equation for the flow u. Neglecting the bilinear term (3), they proved the global existence of classical solution for large initial data in dimension two. Furthermore, for a general bilinear term (3), they showed the existence of global-in-time classical solution for small initial data. This initial data belong to a somehow-similar space as the one introduced in our Theorem 1.1.
In this article, we aim to show that three of the most widespread results about the Navier-Stokes equations can be extended to the Oldroyd-B model (1), under suitable condition on the initial data. More precisely:
• Existence of global-in-time classical solutions in dimension two for large initial data, • Uniqueness in dimension two of strong solutions,
• Existence and uniqueness of global-in-time strong solution in dimension d ≥ 3, with a Fujita-Kato [12] smallness condition for the initial data.
The first problem we address in this article is the existence and uniqueness of global-in-time solutions when dealing with large initial data in L2(R2). In the case of the classical Navier-Stokes system, existence of weak solutions `a la Leray was proven in [18] while the uniqueness was showed by Lions and Prodi in [19]. In our contest, specific difficulties arise when transposing these results to the Oldroyd-B system. This difficulties should be recognized within the intrinsic structure of the model (1):
• the equation of the conformation tensor τ is of hyperbolic type, sharing the majority of the behaviours of a standard transport equation,
• the fluid equation is driven by a forcing term which behaves as the gradient of the conformation tensor, complicating the behavior of the flow u for large value of time.
transposing these techniques to our system. For instance, when dealing with nonlinearities such as
−ωτ + τω ω = ∇u −
t∇u
2 ,
in the τ-equation, one should recognize the typical difficulty related to the product of two weakly convergent sequences, when passing to the limit of suitable approximate solutions. In order to overcome this issue, we assume some extra regularity on the initial tensor τ0 that will somehow allow to achieve suitable strong convergences of solutions. One
can evince how the hyperbolic behavior of the conformation equation counteracts against the propagation of this regularity. Typically, this can be overcome when dealing with sufficiently regular flow, such as velocity field u that are Lipschitz in space. This Lipschitz condition, however, is above the properties of Leray type solutions and this leads in considering an additional regularities also for the initial velocity field u0. We can hence summarize our first result in
the following statement:
Theorem 1.1. Consider system (1) within the bidimensional case d = 2, µ = 0, b = 0 and a ≥ 0. Let the initial data u0 be a free divergence vector field in L2(R2) ∩ ˙B2/p−1p,1 , and τ0 an element of L2(R2) ∩ ˙B2/pp,1, for an index
p∈ (2, ∞). When µ = 0, then the system (1) admits a global-in-time classical solution (u, τ) within the functional framework u ∈ L∞loc(R+; L2(R2)) ∩ L2loc(R+, ˙H1(R2)), τ ∈ L∞(R+, L2(R2)), u ∈ L∞loc(R+; ˙B 2 p−1 p,1 ) ∩ L 1 loc(R+, ˙B 2 p+1 p,1 ), τ ∈ L ∞ loc(R+, ˙B 2 p p,1).
The local Lebesgue conditions in time can be replaced with global ones when a > 0. This solution is unique if p∈ [1, 4]. In addition, the Besov regularities satisfy
k τ(t) k ˙ B 2 p p,1 ≤ e−atk τ0k ˙ B 2 p p,1 exp Cν−1Υ1,ν(t, u0, τ0) , k u(t) k ˙ B 2 p −1 p,1 + ν ˆ t 0 k u(s) k ˙ B 2 p +1 p,1 ds ≤ k u0k ˙ B 2 p −1 p,1 + Θa(t)k τ0k ˙ B 2 p p,1 × × exp Cν−1Υ1ν(t, u0, τ0) ν−1Υ2ν(t, u0, τ0) + 1 . where Θa(t) = t if a = 0 while Θa(t) = (1 − e−at)/a if a > 0. Furthermore Υ1ν(t, u0, τ0) and Υ
2
ν(t, u0, τ0) are
two smooth functions depending on the time T and on the norms of (u0, τ0) in L2(R2) ∩ ˙B−1∞,1× L2(R2) ∩ ˙B0∞,1 (cf.
Definition 1.3).
The result is extended to the case of µ > 0, for a local in time classical solution defined on [0, Tmax), whose lifespan
Tmax> 0 satisfies the lower bound
Tmax≥ sup n T > 0 such that Cµ ν2T 2Ψ 2,ν,µ(T, u0, τ0)e 2C νΨ2,ν,µ(T,u0,τ0) 1−e−aT a kτ0kB0˙ ∞,1+2C µ νΨ2,ν,µ(T,u0,τ0)T < 1o, for a fixed positive constant C and a suitable function Ψ2,ν,µ(T, u0, τ0) (cf. Definition 1.3 below).
One of the main novelty of the theorem is the arbitrariness of the parameter a ≥ 0. When a > 0 then a damping mechanism insures that any classical solution is uniformly bounded in time. However, the theorem provides also the existence and uniqueness of global-in-time classical solutions when the damping mechanism is neglected, i.e. a = 0. In this scenario the norms of the solution grow up exponentially in time, never blowing up.
When µ = µ2 > 0 in (2) and b > 0 in (3) are sufficiently small, we are further capable to provide a unique
Theorem 1.2. Consider system (1) within the bidimensional case d = 2, a > 0, µ > 0 and b > 0. There exists a constant c > 0 depending on the parameter a > 0 such that if
|b| + µ ≤ c 1 + exp n kτ0kL2(Rd)+ kτ0k ˙ B d p p,1∩ ˙B d p +1 p,1 + ku0kL2(Rd)+ ku0k ˙ B d p −1 p,1 ∩ ˙B d p p,1 o ,
i.e. if µ, |b| are sufficiently small in relation to the initial data and the parameter a, then there exists a global-in-time classical solution for problem (1).
As pointed out, the initial condition (u0, τ0) that belongs to ˙B2/p−1p,1 × ˙B 2/p
p,1 (cf. Section 2 for some details about
these functional spaces) is the precursor of the Lipschitz behavior of the fluid u. Nevertheless, the real regularity which unlock the Lipschitz condition for u is somehow hidden in the above statement. We overview some specific about that: when considering the case p = 2, that is (u0, τ0) ∈ ˙B02,1× ˙B12,1, we are dealing with a strict subcase of
the framework (u0, τ0) ∈ L2(R2) × ˙H1(R2), where ˙H1(R2) stands for the homogeneous Sobolev space. It is well
known that just considering the simplified case of a transport equation ∂tτ + u · ∇τ = 0,
the Sobolev regularity ˙H1(R2) is propagated by a Lipschitz flow with the following exponential growth:
k τ(t) kH˙1(R2) ≤ k τ0kH˙1(R2)exp ˆ t 0 k u kLip .
Coupling this inequality with the structure of system (1) would eventually lead to a bound for the Lipschitz regularity of the flow u of the following type:
d
dtk u kLip ≤ C(k u0kLip, k τ0kH˙1(R2)) exp
ˆ t
0
k u kLip
,
for which no global-in-time bound is automatically determined. Hence, we will first propagate the norm of the initial data within a largest functional framework than the one specifically stated in Theorem 1.1, namely we will propagate the following regularities:
u0 ∈ ˙B−1∞,1 and τ0 ∈ ˙B0∞,1,
in which ˙B2/p−1p,1 and ˙B2/pp,1 are embedded, respectively. We will show that this particular choice is essential to achieve a linear growth of the Lipschitz regularity:
k τ(t) kB˙0 ∞,1 ≤ k τ0kB˙0∞,1 1 + ˆ t 0 k u kLip ⇒ d dtk u kLip ≤ C(k u0kLip, k τ0kH˙1(R2)) 1 + ˆ t 0 k u kLip .
Definition 1.3. In Theorem 1.1 we have avoided to explicitly present the form of Υ1ν(T, u0, τ0) and Υ2ν(T, u0, τ0)
for the sake of a compact formulation. We report here their exact expression. We first introduce the functional Γa,ν(T ) :=
r
1 − e−2aT
2aν when a > 0, Γa,ν(T ) := ν
−1 2T 1 2 when a = 0 Φν ,µ ,a(T, u0, τ0) = ku0kL2(R2)+ ν−1k u0k2 L2(R2)+ kτ0kL2(R2)Γa,ν(T ) + ν−1kτ0k2 L2(R2)Γa,ν(T )2 2 if µ = 0, n 1 + Γa,ν(T )µ 1 2ku0k L2(R2)+ ν−1ku0k2 L2(R2)+ + µ−12 + Γa,ν(T )kτ0kL2(R2)+ ν−1kτ0k2 L2(R2)Γa,ν(T )2 o2 if µ > 0, Ψ1,ν, µ, a(T, u0, τ0) = C ν− 3 2Φ ν , µ , a(T, u0, τ0)2 + ν− 5 4Φ ν , µ , a(T, u0, τ0)k u0kL2(R2) , Ψ2, ν, µ, a(T, u0, τ0) = C n Γa,ν(T ) µku0k2L2(R2)+ k τ0k2L2(R2) 1 2 + ν−54Φ ν , µ , a(T, u0, τ0) + ν −1k u 0kL2(R2) o , for a fixed and sufficiently large constant C. When µ = 0, the exact formulations of Υ1ν ,µ is given by
Υ1ν ,µ ,a(T, u0, τ0) := ν−1 n k u0kB˙−1∞,1 + Ψ1,ν,µ,a(T, u0, τ0) + C Ψ2,ν,µ,a(T, u0, τ0) + C k τ0kB˙0 ∞,1Θa(T ) o × ×expCν−1Θa(T )Ψ2,ν,µ(T, u0, τ0) ,
where Θa(T ) = (1 − e−at)/a when a > 0, while Θa(T ) = T when a = 0. Furthermore, for µ > 0,
Υ1ν ,µ ,a(T, u0, τ0) := ν−1 n ku0kB˙−1∞,1+ Ψ1,ν,µ,a(T, u0, τ0) + C Ψ2,ν,µ,a(T, u0, τ0) + C kτ0kB˙0 ∞,1Θa(T ) o 1 − Cµ ν2T 2Ψ 2,ν,µ(T, u0, τ0)e 2CνΨ2,ν,µ,a(T,u0,τ0)Θa(T )kτ0kB0˙ ∞,1+2C µ νΨ2,ν,µ,a(T,u0,τ0)T × ×exp nC νΨ2,ν,µ,a(T, u0, τ0)Θa(T )kτ0kB˙0∞,1+ 2C µ νΨ2,ν,µ(T, u0, τ0)T o . Finally Υν ,µ2 (T, u0, τ0) := k u0kB˙−1∞,1 + Ψ1,ν,µ(T, u0, τ0) + C Ψ2, ν(T, u0, τ0) + 1 k τ0kB˙0 p,1T+ + Cνk τ0kB˙0∞,1 Ψ2,ν,µ(T, u0, τ0) + 1 ˆ T 0 Υ1ν ,µ(t, u0, τ0)dt + CµνΨ2,ν,µ(T, u0, τ0) ˆ T 0 Υ1ν ,µ(t, u0, τ0)dt.
Remark 1.4. Theorem 1.1 should be seen as a further extension of the analysis started by Lions and Masmoudi in [22]. The authors indeed achieved the following estimates on the velocity field u:
• ∇u ∈ Lp(0, T ; Lq(Rd)), 2 ≤ q ≤ 3 and 1 ≤ p ≤ q
2q−3, when d ≥ 3,
• ∇u ∈ Lp(0, T ; Lq(Rd)), 2 ≤ q < ∞ and p > q
q−1, when d = 2.
Both cases allowed the existence of global-in-time weak solutions, however they did not provide the Lipschitz regularity ∇u ∈ L1(0, T ; L∞(Rd)) that would have unlocked the existence of global-in-time classical solutions. We achieve
such a property in this manuscript.
The last part of our manuscript is devoted to establish global-in-time strong solutions for small initial data in dimension d ≥ 3. We intend to proceed similarly as in the result of Fujita-Kato [12] for the incompressible Navier-Stokes equations, as well as in the result of Paicu and Danchin in [6] for the so-called Boussinesq system.
effective when the considered functional space is critical under the scaling behavior of the Navier-Stokes equation. More precisely a functional space that preserves the norm of a solution u(t, x) under the scaling
u(t, x) → λ u(λ2t, λ x), with λ ≥ 0.
Fujita and Kato showed that the Navier Stokes system is well posed for small initial data that belong to the homogeneous Sobolev Space ˙H1/2(R3).
A trivial computation shows that the Oldroyd-B system (1) is invariant under the following transformation: (u(t, x), τ(t, x)) → (λ u(λ2t, λ x), λ2τ (λ2, λ x)).
Hence, the critical functional framework for the velocity field u is the same as in the case of the Navier-Stokes equations, while we need to impose an additional derivative for the conformation tensor τ.
We hence prove the following local result of solutions for system (1) within critical regularities:
Theorem 1.5. Let µ = 0. Consider a dimension d ≥ 3, let the initial data u0 be a free divergence vector field in
˙ B d p−1 p,1 , while τ0 belongs to ˙B d p
p,1, for a parameter p ∈ [1, 2d). Then there exists a time T∗ > 0 for which system (1)
admits a unique local solution (u, τ) within u ∈ C([0, T ], ˙B d p−1 p,1 ) ∩ L 1(0, T, ˙Bdp+1 p,1 ), τ ∈ C([0, T ], ˙B d p p,1),
for any T < T∗. Furthermore if u belongs to L∞(0, T∗, ˙B
d p−1 p,1 , ) ∩ L1(0, T∗, ˙B d p+1 p,1 ) and τ belongs to L∞(0, T∗, ˙B d p p,1, ),
the solution can be extended in time with a life span larger than T∗.
The construction of global-in-time classical solution is more delicate than the above local result. Indeed, the lacking of damping term for the conformation tensor τ does not allow to use the classic fixed point approach, which couples the Picard scheme with standard estimates for the Stokes operator. We hence proceed as follows:
• we determine a suitable functional setting for the initial data for which system (1) preserves the smallness condition of the initial data,
• we hence use this specific small condition, coupled with the Picard fixed point, to show that certain critical regularities are still propagated, globally in time.
We will see that the mentioned functional framework corresponds to the Lorentz spaces u0 ∈ Ld,∞(Rd) and
τ0∈ Ld,∞(Rd). We mention that these spaces are critical under the considered scaling behavior. We will thus prove
the following global result:
Theorem 1.6. Let µ = 0 and assume that the dimension d ≥ 3, the initial data u0 belongs to ˙Bd/p−1p,1 ∩ Ld,∞(Rd)
while τ0 belongs to ˙Bd/pp,1 ∩ L
d
2,∞(Rd), with p ∈ [1, +∞). Then there exists a small positive constant ε depending
on the dimension d such that, whenever the following smallness condition holds true k u0kLd,∞ +
1
νk τ0kLd2,∞ ≤
ε ν
then the corotational Johnson-Segalman model (1) admits a global-in-time classical solution (u, τ), satisfying u ∈ C R+, ˙B d p−1 p,1 ∩ Ld,∞(Rd) ∩ L1loc(R+, ˙B d p+1 p,1 ), τ ∈ C( R+, ˙B d p p,1∩ L d 2,∞(Rd) ).
If p belongs to [1, 2d] then the solution is unique. Furthermore, there exists a constant C depending just on the dimension d such that for any time T ≥ 0 we have
where the function Θν(u0, τ0, T ) is a smooth function depending on the time T > 0 and on the norms of the initial
data (u0, τ0) in the functional framework ˙B−1∞,1× ˙B0∞,1:
Θν(u0, τ0, T ) := Ck u0kB˙−1∞,1exp n CT ν−1k τ0kB˙0∞,1 o + νexpnCT ν−1k τ0kB˙0∞,1 o − 1.
The paper is structured as follows. In Section 2 we present the main functional settings in which we develop our main results. Section 3 is devoted to suitable inequalities related to the tensor equation τ, that will play a major role in the mains proofs. Section 4 is devoted to the proof of Theorem 1.1, about the existence and uniqueness of global-in-time strong solutions for large initial data in dimension two. Section 5 addresses the case of a positive µ > 0. Section 6, Section 7 and Section 8 are devoted to Theorem 1.5 and Theorem 1.6, respectively, namely to the existence and uniqueness of strong solutions in dimension d ≥ 3.
1.1. Physical aspects of the corotational approximation
Before developing the proof for Theorems 1.1, 1.5 and 1.6, we briefly overview certain physical aspects of our model. The considered equations (1) can be derived as a moment-closure of a suitable multiscale model for the evolution of viscoelastic fluids (cf. [8, 23, 24]). This multiscale approach couples the Navier-Stokes equation for the flow u = u(t, x) on the macroscopic level, together with a Fokker-Plank equation describing the evolution of the probability density function f = f (t, x, Q) for the molecular orientation Q ∈ Rd on the microscopic level:
∂tu + u · ∇u − ν∆u + ∇p = div τ R+× Rd,
div u = 0 R+× Rd, ∂tf + u · ∇xf + divQ(ωQ) = 2 ζ∆Qf + 2 ζdivQ( f ∇QΨ(Q)) R+× R d × Rd.
In this framework, ζ is a friction coefficient, τ = τ(t, x) represents the polymeric contribution to the stress τ (t, x) := λ
ˆ
Rd
∇QΨ(Q) ⊗ Q f (t, x, Q)dQ,
while Ψ(Q) stands for a suitable potential depending on the molecular orientation. The considered Fokker-Plank equation retains a corotational simplification at the microscopic level.
One of the simplest molecular potential is given by the Hookean law Ψ(Q) = H|Q|2/2, where H is the elasticity
constant. This particular formulation allows to determine a constitutive equation for the tensor τ. Indeed, using a moment-closure approximation, we can multiply the Fokker-Plank equation by Q ⊗ Q and perform integration by parts along Q ∈ Rd, to gather an equation for τ:
∂tτ + u · ∇τ − ω τ + τ ω = −
2 ζτ . This is exactly our model that we tudy in (1) with µ = 0 and a = 2
ζ ≥ 0.
In addition, the more physical scenario given by µ > 0 can be derived by taking into account the complete Fokker-Plank equation ∂tf+ u · ∇xf + divQ(∇uQ) = 2 ζ∆Qf + 2 ζdivQ( f ∇QΨ(Q)),
without corotational approximation. Indeed performing the moment-closure for the equation of τ, one has ∂tτ + u · ∇τ − ∇uτ + τ ∇uT = −
2 ζτ .
The model (1) with µ > 0 is then recovered, recasting the tensor τ as τ − µ Id and finally re-introducing the corotational approximation for the stretching terms.
that considering these approximations decreases the physical applications of our model. Nonetheless, up to our knowledge, the existence of global-in-time classical solutions in the general case (µ ≥ 0) is still an open question. In this manuscript, we provide a positive answer in the case of the two dimensional corotational Oldroyd-B model with µ = 0. On the other-hand, we further address the case of µ > 0, refining the lifespan of classical solutions within the functional framework of Theorem 1.1.
2. Functional spaces and toolbox of harmonic analysis
We begin with recalling the definition of weak Lebesgue spaces Lp,∞(Rd).
Definition 2.1. For any p ∈ [1, ∞), the functional space Lp,∞(Rd) is composed by Lebesgue measurable function, for which the following norm is bounded:
k f kLp,∞ = sup λ >0 λ m n x∈ Rd such that | f (x)| > λo 1 p < ∞ where m stands for the Lebesgue measure on Rd.
Remark 2.2. Whenever 1 < p < ∞, the functional space Lp,∞ coincides with the Lorentz space defined by real interpolation by means of
Lp,∞ = ( L1, L∞)1 p, ∞.
More precisely, any function f of Lp,∞ can be decomposed as f = f
A + fA, for any positive real A, where
k fAkL1(Rd)≤ C A1− 1 p k fAk L∞(Rd) ≤ C A− 1 p.
The optimal constant C of the above inequality is an equivalent norm for the quantity defined above. In general, for any 1 ≤ p, q ≤ ∞ the Lorentz space Lp,q can be defined by real interpolation as follows:
Lp, q = ( L1, L∞)1 p, q.
We now briefly recall the definition of the Littlewood – Paley decomposition as well as of the Besov spaces.
The Littlewood – Paley theory is defined making use of the so called dyadic partition of unity: let χ = χ(ξ ) be a radial function depending on the frequencies ξ ∈ Rd of class C∞(Rd) whose support is included in the ball {ξ ∈ Rd
ξ : |ξ | ≤ 4/3 }. We assume that χ is identically 1 in the ball {ξ ∈ R
d : |ξ | ≤ 3/4 } while the function
r → χ(rξ ) is decreasing. We then denote ϕ(ξ ) = χ(ξ /2) − χ(ξ ), so that ∀ξ ∈ Rd\ {0},X q∈Z ϕ 2−qξ = 1 and χ (ξ ) = 1 − X q∈N ϕ (2−qξ ). (4)
We then define the homogeneous dyadic block ˙∆q and the operator ˙Sq localizing the frequencies ξ as follows:
˙
∆qf = F−1( ϕ(2−qξ ) ˆf(ξ ) ), S˙qf = F−1(χ(2−qξ ) ˆf(ξ ))
where F stands for the standard Fourier transform. We remark that for any tempered distribution u ∈ S0(Rd), the functions ˙∆qu and ˙Squ are analytic. Furthermore, if there exists a real s for which u ∈ Hs(Rd), then both ˙∆qu and
˙
Squ belong to the space H∞(Rd) = ∩σ ∈RHσ(Rd).
We state that thanks to (4) the identity u = ˙S0u + Pq∈N∆˙qu holds in S0(Rd) while u =
P
q∈Z∆˙qu for any
homogeneous temperate distribution u ∈ Sh0(Rd).
We will frequently use the following orthogonal condition on the dyadic blocks ˙∆q:
˙
∆q∆˙j ≡ 0 if | q − j | ≥ 2 and ∆˙k ˙Sq−1u ˙∆qv
Definition 2.3. Let s ∈ R, (p, r) ∈ [1, ∞]2 and u ∈ S0(Rd). We denote by k u kB˙s p,r := P q∈Z2rqsk ˙∆qu krLp 1 r if r < +∞, supq∈Z2qsk ˙∆qu kLp if r = +∞.
Thus, we define the homogenous Besov space ˙Bsp,r = ˙Bsp,r(Rd) by
˙ Bsp,r := n u ∈ S0(Rd) | k u kB˙s p,r < +∞ o if s < d/p or s = d/p with r = 1, and by ˙ Bsp,r := nu ∈ S0(Rd) | ∀|α| = k + 1 k ∂αu k ˙ Bs−k−1p,r < +∞ o
if d/p + k ≤ s < d/p + k + 1 or s = d/p + k + 1 and r = 1, for some k ∈ N.
Remark 2.4. The functional space ˙Bsp,r is a Banach space if and only if s < d/p or s = d/p and r = 1. For the sake of completeness we recall also the definition of the non-homogeneous Besov spaces: Definition 2.5. Let s ∈ R, (p, r) ∈ [1, ∞]2 and u ∈ S0(Rd). We denote by
k u kB˙s p,r := k ˙S0u krLp + P q∈N2rqsk ˙∆qu krLp 1r if r < +∞, max n k ˙S0u kLp, supq∈N2qsk ˙∆qu kLp o if r = +∞.
The non-homogeneous Besov space Bsp,r = Bsp,r(Rd) is the set of temperate distributions for which k u kBs
p,r is finite.
Remark 2.6. The Besov spaces ˙Bs2,2 and Bs
2,2 coincide with the Sobolev spaces ˙Hs(Rd) and Hs(Rd) respectively.
Furthermore, if s ∈ R+\ N, the Besov spaces ˙Bs∞,∞ and Bs∞,∞ coincide with the H¨older spaces ˙Cs and Cs.
The following estimates are known as Bernstein type inequalities, and they will be frequently used in our proofs. Lemma 2.7. Let 1 ≤ p ≤ l ≤ ∞ and ψ ∈ Cc∞(Rd). We hence have
c2−q d l − d p k ˙∆qu kLl ≤ k ˙∆qu kLp ≤ C2q d l − d p k ˙∆qu kLl and k ˙Squ kLp ≤ C2 q d l − d p k ˙Squ kLl.
As a consequence of the Bernstein type inequality and the definition of Besov Spaces ˙Bsp,r, we have the following proposition:
Proposition 2.8. (i) There exists a constant c > 0 such that 1
ck u kB˙sp,r ≤ k ∇u kB˙s−1p,r ≤ ck u kB˙sp,r.
(ii) For 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2≤ ∞, we gather ˙Bsp1,r1 ,→ ˙B
s−d(1/p1− 1/p2)
p2,r2 .
(iii) If p ∈ [1, ∞] then ˙Bd/pp,1 ,→ ˙Bd/pp,∞∩L∞. Furthermore, for any p ∈ [1, ∞], ˙Bd/pp,1 is an algebra embedded in L∞(Rd).
(iv) The real interpolation ( ˙Bs1
p,r, ˙Bsp,r2 )θ ,˜r, for a parameter ϑ ∈ (0, 1), is isomorphic to ˙B
ϑ s1+(1−ϑ )s2
p,˜r .
We recall further the following results about inclusions between Lorentz and Besov spaces. Lemma 2.9. For any 1 < p < q ≤ ∞, we have
Proof. Denoting by hj = 2jNh(2j·) with h = F−1ϕ , we recast that the dyadic block ˙∆j as a convolution operator
˙
∆ju = hj∗ u.
Hence, making use of the following convolution inequalities between Lorentz spaces k ˙∆ju kLq ≤ k hjkLr,1k u kLp,∞ with 1 p + 1 r = 1 + 1 q and observing that by change of variables
k hjkLr,1 = 2jd(1− 1
r)k h kLr,
we eventually gather that
sup j∈Z 2j d q− d p k ˙∆ju kLq ≤ k h kL1,rk u kLp,∞
which concludes the proof of the lemma.
We now consider several a-priori estimates in the functional framework of Besov spaces for the heat semigroup (cf. [1], Lemma 2.4).
Lemma 2.10. There exists two constant c and C such that for any τ ≥ 0, q ∈ Z and p ∈ [1, ∞], we get e τ ∆∆˙ qu Lp ≤ Ce −cτ22q ˙ ∆qu Lp.
From the above Lemma we can then deduce the following result (cf. [6], Proposition 3.11) Proposition 2.11. Let s ∈ R, 1 ≤ p, r, ρ1 ≤ ∞. Let u0 be in ˙Bsp,r and f be in ˜Lρ1(0, T ; ˙B
s−2+2/ρ1
p,r ) for some
positive time T (possibly T = ∞). Then the heat equation (
∂tu − ν∆u = f [0, T ) × Rd,
u|t=0= u0 Rd,
admits an unique strong solution in ˜L∞(0, T ; ˙Bsp,r)∩˜Lρ1(0, T ; ˙Bs−2+2/ρ1
p,r ). Moreover there exist a constant C depending
just on the dimension d such that the following estimate holds true for any time t ∈ [0, T ] and ρ ≥ ρ1:
ν 1 ρk u k ˜ Lρ(0, t, ˙Bs+ 2ρ p,r ) ≤ C k u0kB˙s p,r + ν 1 ρ1−1k f k ˜ Lρ1(R+, ˙B s−2+ 2 ρ1 p,r ) ! . (5)
In the previous Proposition we introduce the functional space ˜Lρ(0, T ; ˙Bs
p,r), which is known as a Chemin-Lerner
space. This is defined similarly as in Definition (2.3), imposing k u kL˜ρ(0,T ; ˙Bs p,r) := 2qsk ˙∆qu kLρ(0,T ;Lp x) q∈Z `r(Z) . We remark that thanks to the Minkowski inequality
k u kL˜ρ(0,T ; ˙Bs
p,r) ≤ C k u kLρ(0,T ; ˙Bsp,r) for ρ ≥ r,
while the opposite inequality holds when ρ ≤ r. We hence denote ˜
C([0, T ], ˙Bsp,r) := ˜L∞(0, T ; ˙Bsp,r) ∩ C([0, T ], ˙Bsp,r) and by L˜ρloc(R+, ˙Bsp,r) := ∩T>0L˜ρ(0, T ; ˙Bsp,r).
Similarly, one can define the non-homogeneous Chemin-Lerner spaces ˜Lρ(0, T ; Bs
p,r). In the particular case of
Remark 2.12. Thanks to Proposition 2.11, and using the fact that the projector P on the free divergence vector fields is an homogeneous Fourier mutiplier of degree 0, namely it is continuous from ˙Bsp,r to itself, we can easily solve the non-stationary Stokes problem
∂tu − ν∆u + ∇p = f [0, T ) × Rd, div u = 0 [0, T ) × Rd, u|t=0= u0 Rd,
with initial data u0 ∈ ˙Bsp,r with null divergence and a source term f in ˜L1(0, T ; ˙Bsp,r). We hence achieve a unique
solution u in the class affinity
u ∈ ˜L∞(0, T ; ˙Bsp,r) ∩ ˜L1(0, T ; ˙Bs+2p,r ), ∇p ∈ ˜L1(0, T ; ˙Bsp,r), with u satisfying ν 1 ρk u k ˜ Lρ(0,T ; ˙Bs+ 2p,rρ) ≤ Ck u0kB˙s p,r + k P f kL1(0,T ; ˙Bsp,r) . Furthermore, if r < ∞ the solution u belongs to C([0, T ]; ˙Bsp,r).
Remark 2.13. We can introduce also the non-homogeneous version of the Proposition 2.11, for some initial data u0 in Bsp,r and f in ˜Lρ1(0, T ; B
s−2+2/ρ1
p,r ). The existence and uniqueness of a solution still holds, nevertheless for a
constant C in (5) that this time depends linearly on the time T .
The proof of a-priori estimates for certain nonlinear terms is mainly handled through the use of the paradifferential calculus, in particular of the so called Bony type decomposition:
f g = ˙Tfg + ˙Tgf + ˙R( f , g), (6)
where the paraproduct ˙T and the homogenous reminder ˙R are defined by ˙ Tfg := X q∈Z ˙ Sq−1g ˙∆qf and R( f , g) =˙ X q∈Z ˙ ∆qf X | j−q|≤1 ˙ ∆jg . We then state some results of continuity of these operators that we will often use in our proof.
Proposition 2.14. Let 1 ≤ p, p1, p2, r, r1, r2 ≤ ∞ satisfying 1/p = 1/p1+ 1/p2 and 1/r = 1/r1 + 1/r2. The
the homogeneous paraproduct ˙T is continuous • from L∞× ˙Btp,r into ˙Btp,r for any real t ∈ R,
• from ˙B−sp1,r1 × ˙Bsp2,r2 into ˙Bt−sp,r, for any t ∈ R and s > 0. The homogeneous reminder R is continuous
• from ˙Bsp1,r1 × ˙Btp2,r2 into ˙Bs+tp,r for any (s, t) ∈ R2, such that s + t > 0,
• from ˙Bsp 1,r1 × ˙B −s p2,r2 into ˙B 0 p,∞ if s ∈ R and 1/r1 + 1/r2 ≥ 1.
The above proposition allows to determine almost any continuity results for the product of two distributions that belong to two Besov spaces. Further extensions of the above result can be achieved assuming some additional regularity of the distributions:
Lemma 2.15. Let f be a function in L2(R2) ∩ ˙B1∞,1, then f2 belongs to ˙B0∞,1 and satisfies k f2kB˙0
∞,1 ≤ Ck f kL
2(R2)k f kB˙1 ∞,1
Thus, the triangular inequality implies that k f2kB˙0 ∞,1 ≤ 2k ˙TffkB˙0∞,1 + k ˙R( f , f ) kB˙0∞,1, ≤ 2X q∈Z k ˙∆qT˙ffkL∞ x + k ˙∆qR( f , f ) k˙ L∞x .
We first remark that for any integer q ∈ Z k ˙∆qT˙ffkL∞ ≤ X |q− j|≤5 k ˙Sj−1fkL∞ x k ˙∆jfkL∞x ≤ X |q− j|≤5 k ˙Sj−1fkL2 x2 jk ˙ ∆jfkL∞ x ≤ k f kL2x X |q− j|≤5 2jk ˙∆jfkL∞ x ,
hence, the homogeneous paraproduct is bounded by
k ˙Tf fkB˙0∞,1 ≤ k f kL2
xk f kB˙1∞,1.
We now take into account the homogeneous reminder. By definition, we gather that k ˙∆qR( f , f ) k˙ L∞ x ≤ X j−q≥−5 |ν|≤1 k ˙∆q( ˙∆j+νf ˙∆jf) kL∞ x ≤ X j≥q−5 |ν|≤1 2qk ˙∆q( ˙∆j+νf ˙∆jf) kL2 x ≤ X j≥q−5 |ν|≤1 2q− jk ˙∆j+νfkL2 x2 jk ˙∆ jfkL∞ x ≤ k f kL2x X j∈Z 2q− j1(−∞,5](q − j)2jk ˙∆jfkL∞ x .
Defining aj = 2j1(−∞,5]( j) for any j ∈ Z, we can recast the last term in convolution form, namely
X j∈Z 2q− j1(−∞,5](q − j)2jk ˙∆jfkL∞ x = (aj)j∈Z∗ (2jk ˙∆jfkL∞ x q,
for any q ∈ Z. Hence, applying the Young inequality we deduce that (aj)j∈Z∗ (2 jk ˙∆ jfkL∞ x )j∈Z `1 ≤ (2 jk ˙∆ jfkL∞ x )j∈Z `1 = k f kB˙1∞,1, from which X q∈Z k ˙∆qR( f , f ) k˙ L∞x ≤ k f kL2 xk f kB˙1∞,1,
which concludes the proof of the Lemma.
2.1. Estimates for the conformation tensor
In this section we perform several a priori estimates for the following equation that governs the evolution of the conformation tensor τ(t, x):
(
∂tτ + u · ∇τ + a τ − ω τ + τ ω = f R+× Rd,
τ|t=0 = τ0 Rd.
(7) We begin with a standard bound for Lebesgue and Lorentz norms.
Lemma 2.16. For any p ∈ [1, ∞], the following estimate in Lebesgue spaces holds true: k τ(t) kLp x ≤ k τ0kLxpe −at + ˆ t 0 ea(s−t)k f (s) kLp xds.
More in general, one has
k τ(t) kLp,∞ ≤ k τ0kLp,∞e−at +
ˆ t 0
and the inequality reduces to an equality whenever f is identically null.
Proof. Considering p ∈ [1, ∞), we take the matrix inner product between the τ-equation and τ|τ |p−2. Hence, integrating in spatial domain, we first observe that
− ˆ Rd ω τ : τ |τ |p−2 + ˆ Rd τ ω : τ |τ |p−2 = 0 from which we deduce the following Lp-bound of τ
1 p d dtk τ(t) k p Lxp+ ak τ(t) k p Lxp ≤ k f kL pk τ kp−1 Lp ⇒ d dtk τ(t) kLxp + ak τ(t) kLxp ≤ k f kLp.
The case of p = +∞ can be achieved as the limit case of the previous inequalities.
3. Some a-priori estimates for the conformation tensor
In this section we present some a-priori estimates for the τ equation. We begin with the following lemma about the propagation of Besov regularities.
Lemma 3.1. Let (p, r) ∈ [1, ∞]2 and (s, σ ) ∈ (−1, 1) ∈ (−1, ∞). Assume that u is a free divergence vector field whose coefficients belong to L1(0, T ; ˙B1∞,1), that the source term f belongs to ˜L1(0, T ; ˙Bsp,r ∩ ˙Bσ
p,r) and that the
initial data τ0 is in ˙Bsp,r∩ ˙Bσp,r. Then system (7) admits a unique solution τ in the class affinity
τ ∈ L∞(0, T ; ˙Bsp,r∩ ˙Bσp,r) which fulfils the following estimate for any time t ∈ [0, T ]
k τ k˜L∞(0,t; ˙Bs p,r∩ ˙Bσp,r) ≤ k τ0kB˙s p,r∩ ˙Bσp,re −at + kea(·−t)f(·) k˜L1(0,t; ˙Bs p,r∩ ˙Bσp,r) exp C ˆ t 0 k ∇u kB˙0 ∞,1 , for a suitable positive constant C > 0.
The proof of Lemma 3.1 is equivalent to the one of Proposition 4.7 in [6]. The presence of an exponential term in the a-priori estimate of Lemma 3.2 produces intrinsic difficulties when dealing with the existence of global-in-time solutions. Nevertheless, we can refine such an inequality taking into account Besov spaces with null index of regularity: Lemma 3.2. Let τ be a solution of (7) in L∞(0, T ; ˙B0p,1) with f in L1(0, T ; ˙B0p,1) and also ∇u in L1(0, T ; ˙B0∞,1), for some p ∈ [1, ∞]. Then, the following bound holds true for any time t ∈ [0, T ]:
k τ kL∞(0, t ; ˙B0 p,1) ≤ C k τ0kB˙0 p,1e −at + ˆ t 0 ea(s−t)k f (s) kB˙0 p,1ds 1 + ˆ t 0 k ∇u kB˙0 ∞,1ds
Proof. We prove the Lemma for a = 0. Indeed, the general case a > 0 can be dealt with considering the weighted tensor eatτ instead of τ . We decompose the solution τ = Pq∈Zτq, where τq is solution of the following system of
PDE’s:
(
∂tτq + u · ∇τq+ − ωτq + τqω = fq R+× Rd,
τ|t=0 = ˙∆qτ0 Rd.
Hence, thanks to Lemma 2.16, we gather IN = X j, q∈Z | j−q|≤N k ˙∆jτq(t) kLxp . X j, q∈Z | j−q|≤N k τq(t) kLxp . N X q∈Z k τq(t) kLpx . NX q∈Z k ˙∆qτ0kLpx + ˆ t 0 k fq(s) kLxpd s . N k τ0kB˙0 p,1 + ˆ t 0 k f (s) kB˙0 p,1d s .
In order to handle IIN, we make use of Lemma 3.1 where the initial data ˙∆qτ0 is assumed in ˙B±εp,1 for a small
parameter ε ∈ (0, 1) and a source term f in L1(0, T ; ˙Bsp,1): k τq(t) kB˙±εp,1 ≤ k ˙∆qτ0kB˙±εp,1 + ˆ t 0 k ˙∆qfkB˙±εp,1 exp C ˆ t 0 k ∇u kB˙0 ∞,1 . We thus recast the above inequality in the following form
k ˙∆jτq(t) kLpx . 2 −| j−q|εa j(t) k ˙∆qτ0kLxp + ˆ t 0 k ˙∆qf(s) kLxpds exp C ˆ t 0 k ∇u kB˙0 ∞,1
where (aj(t))j∈Z belongs to `1(Z) with norm k (aj(t)) k`1(Z)= 1. We deduce that IIN is bounded by
IIN = X j, q∈Z | j−q|>N k ˙∆jτq(t) kLxp ≤ 2 −Nε k τ0kB˙0 p,1 + ˆ t 0 k f (s) kB˙0 p,1ds exp C ˆ t 0 k ∇u(s) kB˙0 ∞,1ds .
Imposing the following relation between N, ε and u
N = C ε ln 2 ˆ t 0 k ∇u(s) kB˙0 ∞,1ds,
we finally achieve the statement of the Lemma.
We now prove some a-priori estimates within the functional framework of Lorentz norms, for the non-stationary Stokes system. The following Lemma allows us to control the term arising from the combination of the conformation equation within the Navier-Stokes system.
Lemma 3.3. For any time t ≥ 0 and viscosity ν > 0, the following estimate holds true ˆ t 0 Peν (t−s)∆∇ f (s)ds Ld,∞x ≤ Ck f kL∞(0,t;L1 x) if d = 2, Ck f k L∞(0,t;Ld2,∞ x ) if d > 2.
Proof. Without loss of generality, we recast ourselves to the case of ν = 1. We begin remarking that the projector P is a bounded operator from Lp(Rd) to itself, for any 1 > p < ∞. Hence, by real interpolation we get that P is a
bounded linear operator within Lorentz spaces
P ∈ L(Ld2, ∞(Rd), L d
2, ∞(Rd)).
This allows us to cancel the presence of the project P in any inequality we aim to prove and limit the proof to the case of the heat kernel operator. We hence decouple the integral we aim to bound
ˆ t 0
by Jε and Jε defined as follows: fixing a small positive parameter ε Jε := ˆ t−ε 0 e(t−s)∆∇ f (s)ds and Jε := ˆ t t−ε e(t−s)∆∇ f (s)ds For any τ ≥ 0 and 1 ≤ p ≤ p ≤ ∞ we have
eτ ∆∇ L(Lpx, Lqx) = F −1 (e−τ|ξ |2iξ ) Lr x = √1 τ F −1 (e−| √ τ ξ |2i√ τ ξ ) Lr x 1 τ d+1 2 = F−1(e−|ξ |2iξ ) x √ τ Lr x
hence, by a change of variable, we finally get that there exists a constant C for which eτ ∆∇ L(Lp x, Lqx) ≤ C τ d 2 1 r0+ 1 2 where 1 r + 1 p = 1 q + 1, namely 1 r0 = 1 p − 1 q. We first assume that d = 2. We hence get
k Jεk L1 x ≤ ˆ t t−ε k e(t−s)∆∇ f (s) kL1 xds ≤ ˆ t t−ε k e(t−s)∆∇ kL(L1 x, L1x)k f (s) kL1xds ≤ C ˆ t t−ε k f (s) kL1 x | t − s |12 ds ≤ Ck f (s) kL∞(0,t;L1 x) ˆ ε 0 1 s12 ds ≤ C√ε k f (s) kL∞(0,t;L1 x), while similarly k JεkL∞ x ≤ ˆ t−ε 0 k e(t−s)∆∇ f (s) kL∞ x ds ≤ ˆ t−ε 0 k e(t−s)∆∇ kL(L1 x, L∞x )k f (s) kL1xds ≤ C ˆ t−ε 0 k f (s) kL1 x | t − s |32 ds ≤ Ck f (s) kL∞(0,t;L1 x) ˆ ∞ ε 1 s32 ds ≤ √C ε k f (s) kL∞(0,t;L1 x).
In virtue of Remark 2.2 about the real interpolation L2,∞(R2) = (L1, L∞)
1/2,∞, we hence conclude that in dimension
d = 2 ˆ t 0 Peν (t−s)∆∇ f (s)ds L2,∞x ≤ Ck f (s) kL∞(0,t;L1 x).
while k JεkL∞ x ≤ ˆ t−ε 0 k e(t−s)∆∇ f (s) kL∞ x ds ≤ ˆ t−ε 0 k e(t−s)∆∇ k L(L d 2,∞ x , L∞x ) k f (s) k L d 2,∞ x ds ≤ C ˆ t−ε 0 k f (s) k L d 2,∞ x | t − s |32 ds ≤ √C ε k f (s) k L∞(0,t;Ld2,∞ x ) .
We recall again Remark 2.2 concerning this time the real interpolation Ld,∞(R2) = (Ld/2,∞, L∞)1/2,∞, we hence
conclude that in dimension d ≥ 3 ˆ t 0 Peν (t−s)∆∇ f (s)ds Ld,∞x ≤ Ck f (s) k L∞(0,t;Ld2,∞ x ) . We conclude this section stating a lemma which will allow to estimate the lifespan of our classical solution for problem (1). Indeed, we will prove in section 4 that the flow u of our solution satisfies an inequality of the following type:
f(t) ≤ g1(t) + ˆ t 0 g2(s) f (s)ds + g3(t) ˆ t 0 f(s)2ds, (8) where f (t) = kukL1(0,t; ˙B1
∞,1), and g1, g2 and g3 are polynomials in t with positive coefficients. In case of µ = 0, g2
disappears in the above inequality, reducing to a classical Gronwall form. In the general case of µ ≥ 0 we still have however the following statement.
Lemma 3.4. Let f be a continuous positive scalar function in C([0, Tmax)) with f (0) = 0. Assume further that
inequality (8) is satisfied for some polynomials g1, g2 and g3 with positive coefficients. Then Tmax is such that
Tmax= sup T ∈ [0, ∞) such that ˆ T 0 tg3(t) expn2 ˆ t 0 g2(s)dsodt < 1
and the following inequality is satisfied
f(t) ≤ g1(t) 1 −´0ttg3(t) exp n 2´0tg2(s)ds o ds expn ˆ t 0 g2(s)ds o
, for any t∈ [0, Tmax)
4. Global-in-time solutions in dimension two
This section is devoted to the proof of Theorem 1.1. We begin with introducing the following Friedrichs-type approximation of the System (1):
∂tτn + un· ∇τn− Jnωnτn + τnJnωn+ aτn = µDn R+× R2,
∂tun− ν∆un + ∇pn= −Jn(un· ∇un) + div Jnτn R+× R2,
div un = 0 R+× Rd,
(un, τn)|t=0 = (Jnu0, Jnτ0) R2.
(9)
Denoting by 1A the characteristic function of a set A, for any n ∈ N we introduce the regularizing operator Jn by the
formula
which localizes the Fourier transform of a suitable function g into the annulusCn= {ξ ∈ R2, |ξ | ∈ [1/n, n] }. Hence,
we claim that an approach coupling the Friedrich’s scheme together with the Schaefer fixed point theorem, allows us to construct a sequence of approximate solutions (un, τn)n∈N satisfying the following class affinity:
un∈ L∞loc(R+, L2(R2) ∩ ˙B 2 p−1 p,1 ) ∩ L2loc(R+, ˙H1(R2)) ∩ L1loc(R+, ˙B 2 p+1 p,1 )), τn∈ L∞loc(R+, L2(R2) ∩ ˙B 2 p p,1).
We refer the reader to [7] for some details about this procedure, where the first author showed a similar result for a different system of PDE’s. The purpose of the next sections is to reveal the above regularities of the approximate solutions (un, τn)
n∈N. This result is achieved into two main steps:
(i) Propagating the Lipschitz regularity of the velocity field un: the initial data (u0, τ0) belongs to ˙B2/p−1p,1 × ˙B2/pp,1
which is embedded into ˙B−1∞,1 × ˙B0∞,1. This last regularity will be hence propagated in time, allowing to control un into the functional framework given by
∇un ∈ L1loc(R+, ˙B∞,10 ) ,→ L1loc(R+, L∞(R2)),
from which we will deduce that un is Lipschitz, globally in time.
(ii) Propagating higher regularities of solutions: we will propagate the specific regularity of the initial data (u0, τ0)
in ˙B2/p−1p,1 × ˙B2/pp,1, making use of the Lipschitz condition achieved in point (i).
Last, we will estimate the mentioned norms with a bound independent on the index n ∈ N. This will allow us to pass to the limit and construct a classical solution of system (1) within the functional framework of Theorem 1.1. 4.1. Lipschitz regularity of the velocity field
In this section we show some mathematical properties of solutions for the approximate system (9). The main goal is to establish the propagation of Lipschitz regularity for the velocity field un, namely to show that ∇un belongs to the functional space
L1loc([0, Tmax), ˙B0∞,1) ,→ L1loc([0, Tmax), L∞(R2)),
for some suitable positive time Tmax> 0. We also aim in controlling this regularity with a bound which is independent
on the index n ∈ N, in order to keep this property also when passing to the limit. We collect in the following statement the result we aim to prove.
Theorem 4.1. Assume that the initial data u0 and τ0 belongs to L2(R2) ∩ ˙B−1∞,1 and L2(R2) ∩ ˙B0∞,1, respectively.
Then, the solutions (un, τn) of the system (38), belongs to the following functional framework
un ∈ L∞loc([0, Tmax), ˙B−1∞,1) ∩ L1loc([0, T ), ˙B1∞,1) and τn ∈ L∞loc([0, Tmax), ˙B0∞,1),
with Tmax= +∞ when µ = 0. Furthermore, there exists two smooth functions Υ1,ν(T, u0, τ0) and Υ2,ν(T, u0, τ0),
0 ≤ T < Tmax for which the following inequalities hold true:
ν k unkL1(0,T ; ˙B1 ∞,1) ≤ Υ 1 ν(T, u0, τ0) and k u nk L∞(0,T ; ˙B−1 ∞,1) ≤ Υ 2 ν(T, u0, τ0), with also k τnkL∞(0,T ; ˙B0 ∞,1) ≤ Ck τ0kB˙0∞,1 1 + ν −1Υ1 ν(T, u0, τ0) .
Both functions Υ1,ν and Υ2,ν vanish when T = 0, they are increasing in time T > 0 and they depend uniquely on
the norms k u0kL2(R2)∩ ˙B−1
∞,1 and k τ0kL2(R2)∩ ˙B0∞,1. The exact formulation of Υ1,ν and Υ2,ν is stated in Remark (1.3).
some delicate semi-group estimates related in-primis to the mild formulation of the velocity field un(t): un(t) := eνt∆J nu0 | {z } uL(t) + ˆ t 0 Pe(t−s)ν∆div J n(un⊗ un)(s)ds | {z } un 1(t) + ˆ t 0 Pe(t−s)ν∆div J nτn(s)ds | {z } un 2(t) . (10)
Here P is the Leray projector into the space of free-divergence vector fields, while the operator eνt∆ stands for the
heat semigroup in the whole space. We recognize in the above identity three distinct terms uL, un1, un2, the first one related to the linear contribution of system 38, the second one tackling the non-linearity due to the Navier-Stokes contribution to the system and the last one specifically correlated to the evolution of the conformation tensor τn.
Due to the different structures of these terms, each of them will be separately handled, with appropriate estimates in Chemin-Lerner Besov spaces. We will then proceed as follows:
(i) we will first establish some standard energy estimate of our system (cf. Proposition 4.2 and Proposition 4.3), (ii) we will then propagate suitable regularities in order to control the second term un1(t) (cf. Lemma 4.4, Remark
4.5 and Proposition 4.7),
(iii) we will hence analyze the remaining term un
2(t) (cf. Lemma 4.6),
(iv) we will summarize our previous estimates at the end of the section, proving Theorem 4.1 and propagating the Lipschitz-in-space regularity of un.
Thanks to a classical energy approach we begin with stating the following proposition:
Proposition 4.2. For any n ∈ N, (un, τn) belongs to Cloc(R+, L2(R2)) and ∇un belongs to L2(R+, L2(R2)). When
µ = 0, (un, τn) satisfies
kτn(t)kL2(R2)= kτ0kL2(R2)e−at, kunk2L∞(0,t; L2(R2))+ νk∇unk2L2(0,t; L2(R2)) ≤ ku0k2L2(R2)+ kτ0k2L2(R2)
1 − e−2at
2aν ,
(11) for any time t > 0. Otherwise, when µ > 0 then
µ kunk2L∞(0,T ; L2(R2))+ kτnk2L∞(0,T ; L2(R2))+ akτnk2L2(0,T ; L2(R2))+
+ ν µk∇unk2L2(0,T ; L2(R2)) ≤ µku0kL22(R2)+ kτ0k2L2(R2).
(12) Furthermore, one can remark that un belongs to a more refined functional space, namely:
Proposition 4.3. For any integer n ∈ N the solution un belongs to ˜Lloc∞(R+, L2(R2)). Furthermore, if µ = 0 then
kunk˜L∞(0,T ;L2(R2))≤ ku0kL2(R2)+ C ν−1ku0k2L2(R2)+ kτ0kL2(R2) r 1 − e−2aT 2aν + ν −1 kτ0k2L2(R2) 1 − e−2aT 2aν ! otherwise, when µ > 0, kunk ˜ L∞(0,T ;L2(R2))≤ ku0kL2(R2)+ C ν−1ku0k2L2(R2)+ µ−1ν−1kτ0k2L2(R2)+ +kτ0k2L2(R2)+ µku0k2L2(R2) 12 r 1 − e−2aT 2aν . Proof. Thanks to the mild formulation (10), the velocity field uncan be decomposed into three terms un= uL+ un
1+ un2.
The heat kernel allows to estimate the initial term uLas follows: k uLk
˜
L∞(0,T,L2(R2)) ≤ k u0kL2(R2).
Next, we address un1 in ˜L∞(0, T ; L2(R2)). We apply the dyadic block ˙∆q to un1, for a fixed integer q ∈ Z. We first
remark that
k ˙∆qun1kL2(R2) .
ˆ t 0
so that the Young inequality applied to the last convolution leads to k ˙∆qun1kL∞(0,T ;L2(R2)) ≤ Cν− 1 2k ˙∆q(un⊗ un) k L2(0,T ;L2(R2)) ≤ Cν−12 k un(t) k2L4(R2) L2(0,T ) ≤ Ck unkL∞(0,T ;L2(R2))ν− 1 2k ∇unk L2(0,T ;L2(R2)) ≤ 1 ν k unk2 L∞(0,T ;L2(R2))+ νk ∇unk2L2(0,T ;L2(R2)) . Taking the supremum with respect to the parameter q ∈ Z, we eventually deduce that
k un1k˜L∞(0,T ;L2(R2)) ≤ Cν−1 k u0k2L2(R2) + k τ0k2L2(R2) 1 − e−2aT 2aν if µ = 0, k un 1k˜L∞(0,T ;L2(R2)) ≤ Cν−1 k u0k2 L2(R2) + µ−1k τ0k2L2(R2) if µ > 0, for any integer n ∈ N and for a suitable positive constant C. Next we deal with un2 and we remark that
k ˙∆qun2kL2(R2) . ˆ t 0 2qe−cν(t−s)22qk ˙∆qτn(s)kL2(R2)ds, . ν−12k ˙∆qτnk L2(0,T ;L2(R2)) ≤ Cν− 1 2k τnk L2(0,T ;L2(R2)), therefore k un 2k˜L∞(0,T ;L2(R2)) ≤ Ck τ0kL2(R2) r 1 − e−2aT 2aν if µ = 0, k un 2k˜L∞(0,T ;L2(R2)) ≤ C µ ku0k2L2(R2)+ k τ0k2L2(R2) 12 r 1 − e−2aT 2aν if µ > 0, (13)
and this concludes the proof of the Proposition.
Lemma 4.4. For any positive integer n ∈ N and for any positive time T > 0, the velocity field un satisfies the class affinity
un⊗ un∈ ˜L43(0, T ; ˙B12
2,1).
Furthermore, when µ = 0, the following bound holds true k un⊗ unk ˜ L43(0,T ; ˙B 1 2 2,1) ≤ Cν−34 ku0kL2(R2)+ ν−1ku0k2L2(R2)+ kτ0kL2(R2) r 1 − e−2aT 2aν + ν −1 kτ0k2L2(R2) 1 − e−2aT 2aν 2 , while as µ > 0 kun⊗ unk ˜ L43(0,T ; ˙B 1 2 2,1) ≤ Cν−34 n 1 + µ12 r 1 − e−2aT 2aν ku0kL2(R2)+ ν−1ku0k2L2(R2)+ +µ− 1 2 + r 1 − e−2aT 2aν kτ0kL2(R2)+ ν−1kτ0k2L2(R2) 1 − e−2aT 2aν o2 , for a positive constant C that does not depend on the index n ∈ N.
Proof. We first claim that for any integer n ∈ N the approximate velocity field un belongs to ˜L83(0, T ; ˙H43), for any
time T > 0. Thanks to Propositions 4.2 and 4.3, un∈ ˜L∞(0, T ; L2(R2)), with ∇un∈ L2(0, T ; L2(R2)). Furthermore,
for any dyadic block of index q ∈ Z. Taking the square of the above identity, applying a Cauchy-Schwartz inequality and taking the sum for q ∈ Z, we hence deduce that
k unk2 ˜ L83(0,T, ˙H34(R2)) = X q∈Z 232qk ˙∆qunk2 L83(0,T,L2(R2)) . ν−38 X q∈Z k ˙∆qunk 1 4 L∞(0,T,L2(R2)) ν 1 22qk ˙∆qunk L2(0,T,L2(R2)) 34 . ν−38 k unk2 ˜ L∞(0,T,L2(R2)) + νk ∇u nk2 L2(0,T,L2(R2)) , (14)
therefore, applying Proposition 4.2 and Proposition 4.3 kunk˜ L83(0,T, ˙H34(R2))≤ ≤ Cν−38 ku0kL2(R2)+ ν−1ku0k2L2(R2)+ kτ0kL2(R2) r 1 − e−2aT 2aν + ν −1 kτ0k2L2(R2) 1 − e−2aT 2aν if µ = 0, Cν−38 1 + µ12 r 1 − e−2aT 2aν ku0kL2(R2)+ ν−1ku0k2 L2(R2)+ +µ− 1 2 + r 1 − e−2aT 2aν kτ0kL2(R2)+ ν−1kτ0k2L2(R2) 1 − e−2aT 2aν if µ > 0. We hence claim that un⊗ un belongs to ˜L43(0, T ; ˙B12
2,1). Indeed, making use of the Bony decomposition
k un⊗ unk ˜ L43(0,T ; ˙B 1 2 2,1) = X q 2q2k ˙∆q(un⊗ un) k L43(0,T ;L2(R2)) ≤ 2X q 22q k ˙∆q( ˙Tun⊗un) k L43(0,T ;L2(R2)) | {z } Aq +X q 212qk ˙∆q( ˙R(un⊗, un) k L43(0,T ;L2(R2)) | {z } Bq . (15)
We then control the first term by 2q2Aq. X | j−q|≤5 2q2k ˙Sj−1unk L83(0,T ;L∞(R2))k ˙∆ju nk L83(0,T ;L2(R2)) . X | j−q|≤5 22qk ˙Sj−1unk L83(0,T ;L∞(R2))k ˙∆ju nk L83(0,T ;L2(R2)) . X | j−q|≤5 2−4jk ˙Sj−1unk L83(0,T ;L∞(R2))2 3 4jk ˙∆junk L83(0,T ;L2(R2)),
and the Young inequality eventually leads to X q∈Z 22qAq . k unk2 ˜ L83(0,T ; ˙H34(R2)). (16)
It remains to handle the term Bq of the homogeneous reminder. We proceed as follows:
22qBq . 2 q 2 X j≥q−5 |η|<1 2qk ˙∆q( ˙∆jun⊗ ˙∆j+ηun) k L43(0,T ;L1(R2)) . X j≥q−5 |η|<1 232(q− j)2 3 4jk ˙∆junk L83(0,T ;L2(R2))2 3 4( j+η)k ˙∆j+ηunk L83(0,T ;L2(R2)).
Applying again the Young inequality we finally deduce that X
q∈Z
2q2Bq . k unk2
˜
L83(0,T ; ˙H34(R2)). (17)
The lemma is then proven plugging inequalities (17) and (16), into (14) together with (15). Remark 4.5. Recalling that un
1 satisfies the following mild formulation
un1(t) := ˆ t 0 divPeν (t−s)∆J n(un(s) ⊗ un(s))ds, with un⊗ un∈ ˜L 4 3(0, T ; ˙B 1 2 2,1)
we deduce that un1 belongs also to ˜L∞(0, T ; ˙B02,1) ∩ ˜Lρ(0, T ; ˙B2/ρ
2,1), for any integer n ∈ N and any ρ ∈ [4/3, ∞).
Furthermore, the following estimates holds true ν 1 4k un 1k˜L∞(0,T ; ˙B0 2,1) + ν 1 4+ 1 ρk un 1k ˜ Lρ(0,T ; ˙B 2 ρ 2,1) . k div( un⊗ un) k ˜ L43(0,T ; ˙B− 12 2,1) . ν−34 k u0k L2(R2) + ν−1k u0k2 L2(R2) + r 1 − e−2aT 2aν k τ0kL2(R2) + ν −11 − e−2aT 2aν k τ0k 2 L2(R2) !2 , for µ = 0, while ν 1 4k un 1k˜L∞(0,T ; ˙B0 2,1) + ν 1 4+ 1 ρk un 1k ˜ Lρ(0,T ; ˙B 2 ρ 2,1) ≤ Cν−34 n (1 + 1−e−2aT 2aν µ 1 2 )ku0kL2(R2)+ + ν−1ku0k2 L2(R2)+ (µ− 1 2 + + 1−e−2aT 2aν k τ0kL2(R2)+ ν −1+1 − e−2aT 2aν kτ0k 2 L2(R2) o2 , for µ > 0.
We now perform some suitable bounds for the component of the velocity field un2(t), which is defined as un2(t) =
ˆ t 0
div P eν (t−s)∆J
nτn(s)ds.
Lemma 4.6. For any positive integer n ∈ N the following class affinity holds true un2 ∈ ˜L∞(0, T ; ˙B∞,20 ) ∩ ˜L1(0, T ; ˙B2∞,2), un2 ∈ ˜L∞(0, T ; ˙B02,1) ∩ ˜L43(0, T ; ˙B
1 2
Furthermore k un2k˜L∞(0,T ; ˙B0 ∞,2) + ν k u n 2k˜L1(0,T ; ˙B2 ∞,2) ≤ Ck τ nk L1(0,T ;L2(R2)), k un2k˜L∞(0,T ; ˙B0 2,1) + ν 3 4k un 2k˜ L43(0,T ; ˙B 3 2 2,1) ≤ Ck τnkL1(0,T ; ˙B0 ∞,1),
for a suitable positive constant C.
Proof. We restrict ourselves in proving the first statement, that is un2 belongs to ˜L∞(0, T ; ˙B∞,20 ) ∩ ˜L1(0, T ; ˙B2∞,2) and satisfies k un2(t) k˜L∞(0,T ; ˙B0 ∞,2) + νk u n 2k˜L1(0,T ; ˙B2 ∞,2). k τ nk L1(0,T ;L2(R2)).
The second part of the Lemma can indeed be achieved with a similar procedure. Applying the dyadic bloc ˙∆q on
un
2(t) and taking the L∞(R2) norm, one has
k ˙∆qun2kL∞(0,T ;L∞(R2)) . ˆ T 0 e−c(t−s)22q2qk ˙∆qτn(s) kL∞(R2)ds . ˆ T 0 k ˙∆qτn(s) kL2(R2)ds from which k un 2k˜L∞(0,T ; ˙B0 ∞,2) = X q∈Z k ˙∆qun2(t) k2L∞(0,T ;L2(R2)) 1 2 . k τnk˜L1(0,T ; ˙B0 2,2) . k τ nk L1(0,T ;L2(R2)). Furthermore 22qk ˙∆qun2kL∞(R2) . ˆ t 0 22qe−cν(t−s)22q2qk ˙∆qτn(s) kL∞(R2)ds . ˆ t 0 22qe−cν(t−s)22qk ˙∆qτn(s) kL2(R2)ds,
thus applying the Young inequality we gather that
22qk ˙∆qun2kL1(0,T ;L∞(R2)). ν−1k ˙∆qτnkL1(0,T ;L2(R2))
and taking the sum as q ∈ Z
ν k un2k˜L1(0,T ; ˙B2
∞,2). k τ
nk
L1(0,T ;L2(R2)).
Proposition 4.7. For any positive integer n ∈ N and for any positive time T > 0, the velocity field un satisfies the class affinity
div ( un⊗ un) ∈ L1(0, T ; ˙B−1∞,1). Furthermore, the following bound holds true
for a suitable positive constant C, where the smooth-in-time functions Ψ1,ν(T, u0, τ0) and Ψ2,ν(T, u0, τ0) are defined by Φν ,µ(T, u0, τ0) = = k u0kL2(R2) + ν−1k u0k2 L2(R2) + k τ0kL2(R2) q 1−e−2aT 2aν + ν −1k τ 0k2L2(R2)1−e −2aT 2aν 2 if µ = 0, n 1 + q 1−e−2aT 2aν µ 1 2ku0k L2(R2)+ ν−1ku0k2 L2(R2)+ + µ−12 + q 1−e−2aT 2aν kτ0kL2(R2)+ ν−1kτ0k2 L2(R2)1−e −2aT 2aν o2 if µ > 0, Ψ1,ν, µ(T, u0, τ0) = C ν− 3 2Φ ν(T, u0, τ0)2 + ν− 5 4Φ ν(T, u0, τ0)k u0kL2(R2) , Ψ2, ν, µ(T, u0, τ0) = C (r 1 − e−2aT 2aν µ ku0k 2 L2(R2)+ k τ0k2L2(R2) 12 + ν−54Φν ,µ(T, u0, τ0) + ν−1k u0kL2(R2) ) , for a suitable positive constant C, when µ = 0.
Proof. We are now in the position to deal with div( un⊗ un) in the functional space L1(0, T ; ˙B−1
∞,1). We keep on
using the standard decomposition un= un
1 + un2 + uL, thus our estimate reduces to
k div( un⊗ un) kL1(0,T ; ˙B−1 ∞,1)≤ k div( u n 1⊗ u n 1) kL1(0,T ; ˙B−1 ∞,1)+ uL· ∇un1 L1(0,T ; ˙B−1 ∞,1)+ un1· ∇uL L1(0,T ; ˙B−1 ∞,1)+ + k un1· ∇un 2kL1(0,T ; ˙B−1 ∞,1) + k u n 2· ∇un1 kL1(0,T ; ˙B−1 ∞,1) + k div( u n 2⊗ un2) kL1(0,T ; ˙B−1 ∞,1) + + uL· ∇un2 L1(0,T ; ˙B−1 ∞,1) + un2· ∇uL L1(0,T ; ˙B−1 ∞,1) + uL· ∇uL L1(0,T ; ˙B−1 ∞,1).
We hence control any term on the right-hand side of the above inequality. Thanks to the embedding ˙B02,1,→ ˙B−1∞,1, we first remark that
k div( un 1⊗ un1) kL1(0,T ; ˙B−1 ∞,1) . k div( u n 1⊗ un1) kL1(0,T ; ˙B0 2,1) . k u n 1⊗ un1kL1(0,T ; ˙B1 2,1) . k u n 1k 2 L2(0,T ; ˙B1 2,1),
hence thanks to the Remark 4.5, we obtain that for µ = 0 kdiv( un1⊗ un1)kL1(0,T ; ˙B−1 ∞,1). ν −3 2 ku0kL2(R2)+ ν−1ku0k2L2(R2)+ r 1 − e−2aT 2aν k τ0kL2(R2)+ 1 − e−2aT 2aν2 kτ0k 2 L2(R2) 4 while for µ > 0 k div( un1⊗ un1) kL1(0,T ; ˙B−1 ∞,1) . ν −32n(1 + r 1 − e−2aT 2aν µ 1 2)ku0k L2(R2)+ + ν−1ku0k2L2(R2)+ (µ− 1 2 + r 1 − e−2aT 2aν )kτ0kL2(R2)+ ν −11 − e−2aT 2aν kτ0k 2 L2(R2) o4 . Now, recalling the embedding ˙B02,1 ,→ ˙B−1∞,1 in dimension two, together with the continuity of the product within
˜ L4(0, T ; ˙B 1 2 2,2) × ˜L 4 3(0, T ; ˙B 1 2 2,2) → L1(0, T ; ˙B02,1), (18)
