Long term analysis of two fluid flows 

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Analysis of a Cahn-Hilliard-Ladyzhenskaya system with singular potential


In this chapter we study a model for the evolution of a mixture of two incompressible and (partially) immiscible fluids in a domain of R3. The model we consider is given by Ladyzhenskaya-Navier-Stokes type equations for the (average) fluid velocity coupled with a convective Cahn-Hilliard equation with a singular (e.g., logarithmic) potential. The former is endowed with no-slip boundary conditions, while the latter is subject to no-flux boundary conditions so that the total mass is conserved. We first prove the existence of a weak solu-tion in three-dimensions and some regularity properties. Then we establish the existence of a weak trajectory attractor for a sufficiently general time-dependent external force. Finally, taking advantage of the validity of the energy identity, we show that the trajectory attractor actually attracts solutions with respect to the strong topology.
start our mathematical analysis of possible variations on the model H introduced WEin the previous chapter (see equation (1.1.6) in previous chapter), by considering a special class of non-newtonian fluids. In particular, we investigate the mathemat-ical implications of choosing the following, apparently regularising, constitutive relation for the stress-strain rate relation, which is commonly used to describe shear thickening fluids (e.g. corn starch):
τ pDu ψ q “ `ν1pψq ` ν2pψq|Du|p´2˘ Du (2.0.1)
Here ν1 and ν2 are given positive constants and p ą 1. Moreover, Du is the symmetrized gradient. This relation, when used in the derivation of Navier-Stokes equations in place of its standard linear counterpart, τ pDuq “ 2νpψqDu, gives rise to the well-known Ladyzhenskaya model (see Ladyzhenskaya (1967)), which has been widely investigated in the case of single fluids (see, e.g., Feireisl and Pražák (2010) and its references). Using this relation in the initial boundary value problem
we obtain the system which will be the main objective of our investigation in this chapter. As usual, here u is the velocity field, ψ is the order parameter field, while π stands for the pressure. In order to simplify notations as much as possible, with respect to system (1.1.6), we took ε “ M “ 1. Motivated by the above discussion, we will call system (2.0.2) Ladyzhenskaya-Cahn-Hilliard—or LCH—system.
This kind of system has been recently analyzed in Grasselli and Pražák (2011) in the case of regular potentials, periodic boundary conditions, g time independent and νi, i “ 1 2, depending on ψ. Well-posedness results and regularity results have been obtained for p ě 115 (see also Kim et al. (2006)). Then, using the short trajectory approach, the existence of global and exponential attractors has been established. Here we want to analyze the singular potential case, The plan of our analysis goes as follows. We first establish the existence of a weak solution. Contrary to the case of single fluids (see Bulíček et al. (2010); Feireisl and Pražák (2010)) or regular potentials (cf. Grasselli and Pražák (2011)), uniqueness seems a rather challenging issue. Indeed, observe that, in comparison with the Cahn-Hilliard-Navier-Stokes case (i.e., ν2 “ 0) considered in Abels (2009c), Btu is less regular. This fact influences, through the convective term, the smoothness of ψ so that, even in the 2D case, the Korteweg force is not always as smooth as needed to guarantee uniqueness when p is large (see also Remark 2.5.1 below). In order to characterize the asymptotic behavior of system (2.0.2) since uniqueness of solution is not known, we resort to the theory of trajectory attractors according to Chepyzhov and Vishik (1997) (cf. Section 1.3.2 and see also Foias and Temam (1987) and Sell (1996) for earlier contributions to this field). This approach considers as phase space the set of the solutions on the interval r0 8q for the system endowed with a suitable topology so that the dynamical system generated by (2.0.2) reduces to a shift operator on this space. Working within this framework, in Sections 2.6 we establish the existence of a trajectory attractor in a suitably chosen weak topology on the space of trajectories. Finally, observing that any weak solution satisfies the energy equality, we can show that the attraction property holds with respect to the strong topology.

Functional setting

In this chapter Ω is a given bounded domain in R3 with smooth boundary (say of class C2). We recall the notation Qt “ Ω ˆ r0 t q, Qts “ Ω ˆ rt1 t 2q introduced in Section 1.4.1 for parabolic domains.
To study the velocity field, we introduce the usual space of solenoidal test functions V0 “ tφ P C8cpΩ; Rnq | ∇ • φ “ 0u as well as the distribution spaces L20 divpΩq, Hs0 divpΩq and W0srdivpΩq (cf. Section 1.4.2). Moreover, to simplify notation we also define VrpΩq “ W01rdivpΩq
As far as the order parameter is concerned, on account of mass conservation, we use Sobolev spaces with fixed mean value as L2pmqpΩq, HpsmqpΩq and Wpspmq pΩq. Moreover, we use a similar notation for dual spaces whenever it makes sense. Consistently with Section 1.4.2, we define Hp´01qpΩq “ pHp10qpΩqq˚
We end this section by recalling a simple result, which will be used in the proofs below (see also (Colli et al., 2012, Lemma 1)).
Lemma 2.1.1. Let Ω Ă R be an open subset (not necessarily bounded) and let tfnu Ă L8pΩq be a sequence such that |fn|8 ď C and fn Ñ f strongly in L2pΩq. Let tgnu Ă LppΩq be another sequence such that gn á g weakly in LppΩq. Then fngn á f g weakly in LppΩq.
Proof. The statement can be easily proved using the density of C80pΩq in L2pΩq and the estimate
|fngn|p ď |fn|8|gn|p

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Main assumptions and weak formulation

Here we list some properties of the stress tensor that will be assumed in this chapter. In particular,
τ pS ψ q will be a function acting on S ˆ R with values in S (here S is the space of symmetric 3 ˆ 3
tensors) such that
τ p0 ϕ q “ 0
pτ pS1 ϕ q ´ τ pS2 ϕ qq : pS1 ´ S2q ě ν˚ p1 ` |S1| ` |S2|qp´2 |S1 ´ S2|2 |τ pS1 ϕ q ´ τ pS2 ϕ q| ď Cν˚ p1 ` |S1| ` |S2|qp´2 |S1 ´ S2| |τ pS ϕ 1q ´ τ pS ϕ 2q| ď Cν# p1 ` |S|qp´1 |ϕ1 ´ ϕ2|
for all ϕ, ϕi P R, i “ 1 2 and for all S, Si P S, i “ 1 2, where ν˚, ν˚, ν# P R are strictly positive constants and where p is assumed to be greater than 2. Under these assumptions, τ induces an operator, still denoted by τ , acting on LppΩq ˆ MpΩq with values in Lp1 pΩq (here MpΩq is the space of measurable functions on Ω). Moreover, the operator τ p • ψ q is hemicontinuous from LppΩq into Lp1 pΩq for any fixed ψ P MpΩq.

Some results on the Ladyzhenskaya model

In this section we summarize some of the results known on the well-posedness of the Ladyzhen-skaya model.
The well posedness of system (2.4.1) has been extensively studied in the literature. In the shear-thickening case (i.e. p ě 2), existence and uniqueness of solutions have been obtained as soon as p ě 115 (see, for instance, Feireisl and Pražák (2010) and references therein). Although we will not use directly the results of this section in our results, we observe that the same mathe-matical difficulties (i.e. uniqueness is an open issue for p ă 115 ) hold both for the Ladyzhenskaya model as well as for the coupled model we are studying.
Definition 2.4.1. Let u0 P L20 divpΩq and let T ą 0 be given. A function u such that u P Lpp0 T ; VppΩqq Btu P Lp1 p0 T ; pVppΩqq˚ q

Table of contents :

I Long term analysis of two fluid flows 
1 Two phase fluid flows 
1.1 Motivation—The model H
1.1.1 On the Cahn-Hilliard equation
1.1.2 On the Navier-Stokes equations
1.1.3 Coupling Cahn-Hilliard and Navier-Stokes models
1.2 Some generalisations of the model H
1.2.1 Non-newtonian fluids
1.2.2 Oono reaction term
1.2.3 Non-local models
1.3 Large-time behaviour for solutions of parabolic systems
1.3.1 Short overview of the classical theory of attractors
1.3.2 Trajectory attractors
1.3.3 Exponential attractors
1.3.4 Pullback attractors
1.3.5 Convergence to stationary states
1.4 Notation
1.4.1 General notation
1.4.2 Functional spaces
2 A LCH system 
2.1 Functional setting
2.2 Main assumptions and weak formulation
2.3 The convective Cahn-Hilliard equation
2.4 Some results on the Ladyzhenskaya model
2.5 Weak solutions and energy estimates
2.6 A weak trajectory attractor
2.7 A strong trajectory attractor
3 A 2D CHNS system 
3.1 Functional setting and main results
3.2 Exponential pullback attractors
3.3 Existence results and basic energy estimate
3.4 Higher regularity estimates
3.5 Continuous dependence
3.6 Time regularity
3.7 Proof of the main results
4.1 Functional setting and main results
4.2 Existence and dissipation
4.3 Continuous dependence
4.4 Higher-order dissipative estimate
4.5 A robust family of exponential attractors
5 Nonlocal CH 
5.1 Basic tools and well-posedness
5.1.1 Function spaces
5.1.2 Weak formulation and main result
5.1.3 Evolution equations with monotone operators
5.1.4 Results on the nonlocal operator L
5.2 Subgradients
5.3 Existence of unique solutions
5.4 Long-time behaviour
5.5 Boundary conditions for variational solutions
5.5.1 Proof of Theorem 5.5.1: case n “ 1
5.5.2 Proof of Theorem 5.5.1: case n ě 2
II Strain deformation in p-n junctions  Notation used in Part II 
6 Statement of the problem 
6.1 On the physics of semiconductors
6.1.1 Energy bands
6.1.2 Doping
7 The Drift-Diffusion model and strain 
7.1 Derivation of the DD model
7.2 A brief history of strain effects on SCs’ properties
7.3 Strain effects in silicon
7.3.1 Shift of the energy bands
7.3.2 Changes in effective masses
7.3.3 Shift of the Fermi level
7.3.4 Changes for ni
7.3.5 Changes in mobilities
8 P-N junctions under strain 
8.1 P-N junctions in operation
8.2 The Shockley relation with strain effects
8.3 Maxwell stresses and reverse coupling
III High cycle fatigue in alloys  Notation used in Part III 
9 A HCF criterion 
9.1 The models at the mesoscopic scale
9.2 Separation of time scales
9.3 Lifetime estimates in HCF
9.4 A priori estimates
9.5 Identification of parameters from fatigue experiments
9.6 Results and discussion
9.7 Conclusions


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