A New Approach to Non-Isothermal Models for Nematic Liquid Crystals

Feireisl, Eduard; Frémond, Michel; Rocca, Elisabetta; Schimperna, Giulio

doi:10.1007/s00205-012-0517-4

Cited by 75 publications

(58 citation statements)

References 28 publications

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“…Note that (F1) implies that F (s) ≥ −c 0 ∀s ∈ R (2. 10) and some constant c 0 > 0. Observe also that in [9] it was just assumed that F ∈ C 2 (in place of C 3 ) in (F1); moreover (2.9) was taken in place of (F3).…”

Section: Assumptions and Motivationmentioning

confidence: 99%

Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions

Eleuteri¹,

Gatti²,

Schimperna³

2019

Indiana Univ. Math. J.

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We consider a thermodynamically consistent model for the evolution of thermally conducting two-phase incompressible fluids. Complementing previous results, we prove additional regularity properties of solutions in the case when the evolution takes place in the two-dimensional flat torus with periodic boundary conditions. Thanks to improved regularity, we can also prove uniqueness and characterize the long-time behavior of trajectories showing existence of the global attractor in a suitable phase-space.Keywords: Cahn-Hilliard, Navier-Stokes, incompressible non-isothermal binary fluid, thermodynamically consistent model, regularity of solutions, long-time behavior.MSC 2010: 35Q35, 35K25, 76D05, 35D35, 80A22, 37L30. * All the authors are partially supported by the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), through the project GNAMPA 2016 "Regolarità e comportamento asintotico di soluzioni di equazioni paraboliche" (coord. Prof. S. Polidoro) and by the University of Modena and Reggio Emilia through the project FAR2015 "Equazioni differenziali: problemi evolutivi, variazionali ed applicazioni" (coord. Prof. S. Polidoro) 1 energy configurations of the phase variable. Here we will assume that F is smooth and has a powerlike growth at infinity. Finally, the function κ(ϑ) in (1.5) denotes the heat conductivity coefficient, assumed to grow at infinity like a sufficiently high power of ϑ (see (K1) below).The system is highly nonlinear and contains complicated coupling terms; however these features arise naturally and are directly related to the thermodynamical consistency of the model. In particular, the quadratic terms on the right hand side of (1.5), which constitute the main difficulty in the mathematical analysis, describe the heat production coming from dissipation of kinetic and chemical energy, respectively. We may also note that transport effects are admitted for all variables in view of the occurrence of material derivatives in (1.2), (1.3) and (1.5). In order to avoid complications related to interactions with the boundary, we will assume here Ω = [0, 1] × [0, 1] to be the two-dimensional flat torus. Correspondingly, we will take periodic boundary conditions for all unknowns.System (1.1)-(1.5) has been first introduced in [8] and can be considered as a coupling between the Navier-Stokes equations and the thermodynamically consistent model for phase transitions proposed by M. Frémond in [3] and extensively studied in recent years (see, for instance, [6,15,16,17,23] and the references therein, we also quote [7] for applications to elastoplasticity with hysteresis and [10] for liquid crystals). Other nonisothermal models for phase-changing fluids can be obtained by linearization around the critical temperature, which simplifies the mathematical analysis but gives rise at least to a partial loss of thermodynamical consistency.A mathematical study of (1.1)-(1.5) has been first attempted in [8,9], which refer to the threeand...

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Section: Assumptions and Motivationmentioning

confidence: 99%

Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions

Eleuteri¹,

Gatti²,

Schimperna³

2019

Indiana Univ. Math. J.

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“…Assuming the first law of thermodynamics (9) to be satisfied, then the entropy production must fulfill the following identity…”

Section: The First Law Of Thermodynamicsmentioning

confidence: 99%

“…It is common in literature to relax such a non-linearity, introducing a Ginzburg-Landau penalization term in the free energy of the system (cf. [9,11,12,17,23]). Finally, we denote by ϑ = ϑ ( t, x ) > 0 the so-called absolute temperature, and we are interested in the range of temperatures such that the nematic phase occurs.…”

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confidence: 99%

Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

Anna

Liu

2018

Arch Rational Mech Anal

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article info abstractKeywords: Ericksen-Leslie, nematic liquid crystals, thermodynamics consistency, global wellposedness, Besov regularity.MSC: 35Q30, 35Q35, 35Q79, 76A15.We derive a model describing the evolution of a nematic liquidcrystal material under the action of thermal effects. The first and second laws of thermodynamics lead to an extension of the general Ericksen-Leslie system where the Leslie stress tensor and the Oseen-Frank energy density are considered in their general forms. The work postulate proposed by Ericksen-Leslie is traduced in terms of entropy production. We finally analyze the global-in-time well-posedness of the system for small initial data in the framework of Besov spaces.5.4. The temperature equation 36 5.5. Passage to the limit 42 5.6. Uniqueness 42 6. Appendix 43 References 49 IntroductionThe main aim of this paper is to derive and analyze an evolutionary PDE's-system modeling the dynamics of nematic liquid crystals. The model we are interested in extends the general Ericksen-Leslie theory, allowing a non-constant temperature. We derive such a model in accordance with the main laws of thermodynamics. Nowadays, the engineering and mathematical community is familiar with the concept of nematic liquid crystals. A nematic medium is a compound of fluid molecules, which has a state of matter between an ordinary liquid and a crystal solid. Although the centers of mass can freely translate as in a common fluid, the constitutive molecules present a privileged orientation. This alignment strongly interacts with the underlying flow of the nematic. Reinitzer discovered one of these materials in the 1888 and since then there have been numerous attempts to formulate continuum theories describing the time behavior of the flow. Ericksen and Leslie developed the most widely recognaized model during the 1960's in their pioneeric papers [7,8,22], generalizing the Oseen-Frank theory for the static case [14]. From the first mathematical success in analyzying the model performed by Lin and Liu [23] in the 1991, the dynamics theory of liquid crystal has become the new El Dorado of theoretical studies motivated by real-world applications. The well-posedness analysis in bounded domains [2,20,25] as well as in the whole space [6,15], has especially received high interest in the recent decades, both for what concerns the director theory and the Q-tensor framework. These results are often inspired by the ample literature concerning the Navier-Stokes equations, since any derivation of a consistent model for these anisotropic materials usually starts from the well-known conservation of mass and balances of linear and angular momentum. Despite a wide literature concerning the dynamics of nematic liquid crystals, to the best of our knowledge there are few papers dealing also with thermodynamic effects. Liquid crystals are mostly considered in an isothermal environment, which is sometimes unnatural. Indeed, as described by Stewart in [27], liquid phases are mainly induced by changing the temperature (thermotropic...

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“…where q : Ω × R + → R n is the heat flux. Feireisl-Frémond-Rocca-Schimperna [7] proved the existence of a global weak solution to (1.4) in dimension 3. Correspondingly, nonisothermal version of (1.1) reads (u, d, θ) :…”

Section: Introductionmentioning

confidence: 99%

“…With (5.13) and (5.16), we can apply an interpolation argument, similar to (4.13) in [7], to conclude that θ m ∈ L q (Ω × [0, T 0 ]) for any 1 ≤ q < 5 3 , and…”

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confidence: 98%

Weak solutions of non-isothermal nematic liquid crystal flow in dimension three

Wang

2020

J Elliptic Parabol Equ

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For any smooth domain Ω ⊂ R 3 , we establish the existence of a global weak solution (u, d, θ) to the simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data (u0, d0, θ0) ∈ H × H 1 (Ω, S 2 ) × L 1 (Ω), with d0(Ω) ⊂ S 2 + (the upper half sphere) and ess infΩθ0 > 0.

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A New Approach to Non-Isothermal Models for Nematic Liquid Crystals

Cited by 75 publications

References 28 publications

Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions

Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions

Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

Weak solutions of non-isothermal nematic liquid crystal flow in dimension three

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