Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

Liu, Lin; Liu, Qiao; Zhong, Xin

doi:10.1088/1361-6544/aa8426

Cited by 20 publications

(11 citation statements)

References 31 publications

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“…03 ) satisfies a geometric condition (1.3). Li et al [11] got the same result under small initial data without the additional geometric condition (1.3) (see also [24]). Before formulating our main result, we first explain the notations and conventions used throughout this paper.…”

Section: Yang Liu and Xin Zhongmentioning

confidence: 52%

“…Integrating by parts, we succeed in bounding the term I 2 by regularity properties of Stokes system and Gagliardo-Nirenberg inequality (see (2.1)). Next, to obtain the estimates on the gradient of the density, motivated by [11,16,19], which are crucial in deriving the bound of √ ρu 2 L 2 . However, it prevents us to achieve this goal due to the absence of the compatibility condition (1.9) for the initial velocity.…”

Section: On Cauchy Problem Of 3d Nonhomogeneous Flows 5221mentioning

confidence: 99%

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On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

Liu¹,

Zhong²

2020

CPAA

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This paper deals with the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible nematic liquid crystal flows. The global well-posedness of strong solutions with large velocity is established provided that ρ 0 L ∞ + ∇d 0 L 3 is suitably small. In particular, the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained.

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Section: Yang Liu and Xin Zhongmentioning

confidence: 52%

Section: On Cauchy Problem Of 3d Nonhomogeneous Flows 5221mentioning

confidence: 99%

On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

Liu¹,

Zhong²

2020

CPAA

Self Cite

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“…Li-Liu-Zhong [13] got the same result under small initial data without the additional geometric condition (5). Extended to the more complicated compressible case, the simplified Ericksen-Leslie system is strongly coupled via the compressible Navier-Stokes equation and the transported harmonic map heat flow to S 2 , with significant progresses made during past years.…”

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confidence: 63%

Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows

Liu¹,

Zheng²,

Li³

et al. 2017

DCDS-A

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This paper studies the local existence of strong solutions to the Cauchy problem of the 2D simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows, coupled via ρ (the density of the fluid), u (the velocity of the field), and d (the macroscopic/continuum molecular orientations). Notice that the technique used for the corresponding 3D local well-posedness of strong solutions fails treating the 2D case, because the L pnorm (p > 2) of the velocity u cannot be controlled in terms only of ρ 1 2 u and ∇u here. In the present paper, under the framework of weighted approximation estimates introduced in [J. Li, Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. (2014) 640-671] for Navier-Stokes equations, we obtain the local existence of strong solutions to the 2D compressible nematic liquid crystal flows.

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“…Li et al 21 got the same result under small initial data without the additional geometric condition (4). Especially, when the viscosity coefficient is a function of the density of fluid, if the initial data satisfy the following compatibility condition −div ( ( 0 )∇u 0 ) + ∇P 0 + div (∇d 0 ⊙ ∇d 0 ) = √ 0 g for some (∇P 0 , g) ∈ L 2 ,…”

Section: Introductionmentioning

confidence: 71%

“…In other words, it remains open for the Neumann boundary conditions. Moreover, Theorem 1.1 is entirely new and somewhat surprising, since the known corresponding global existence of the strong solutions to (1) needs suitable smallness condition on || √ 0 u 0 || 2 L 2 + ||∇d 0 || 2 L 2 and ||∇u 0 || 2 L 2 + ||Δd 0 || 2 L 2 for the constant viscosity case 2,13,16,17,21 and the density-dependent viscosity case, 23 respectively. Remark 1.2.…”

Section: Introductionmentioning

confidence: 99%

Global well‐posedness of the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity coefficient

Liu

2020

Math Methods in App Sciences

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This paper deals with the 3D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that̄+ ||∇d 0 || L 3 is suitably small with large velocity. Furthermore, the exponential decay of the solution is also obtained. KEYWORDS exponential decay, incompressible nematic liquid crystal flow, strong solutions, vacuum MSC CLASSIFICATION 76A15; 35B65; 35Q35Math Meth Appl Sci. 2020;43:5985-6010. wileyonlinelibrary.com/journal/mma © 2020 John Wiley & Sons, Ltd. 5985 LIUfor some positive constant ,̄. Without loss of generality, both and are normalized to 1. The symbol ∇d ⊙ ∇d, which exhibits the property of the anisotropy of the material, denotes the n × n matrix whose (i, )-th entry is given by i d · d, for i, = 1, 2, 3. We consider an initial boundary value problem for (1) with the following initial and boundary conditions:with d 0 ∈  2 being given with compatibility, div u 0 = 0 in Ω, and d 0 ∈ C 1 (Ω) satisfying ∇d 0 = 0 on the boundary Ω (see e.g., Hu and Wang 1 and Li 2 ). The nematic liquid crystal flows are regarded as slow-moving particles where the fluid velocity and the alignment of the particles affect each other. The hydrodynamics of nematic liquid crystals is developed by Ericksen 3 and Leslie 4 in the 1960s, but it still retains most important mathematical structures as well as most of the essential difficulties of the original Ericksen-Leslie model. Mathematically, System (1) is a strongly coupled system between the nonhomogeneous incompressible Navier-Stokes equations and the transported heat flows of harmonic map, and thus, its mathematical analysis is full of challenges.When d is a constant vector and |d| = 1, System (1) reduces to the well-known nonhomogeneous incompressible Navier-Stokes equations. Lions 5 , Chapter 2 established the global existence of weak solutions to nonhomogeneous Navier-Stokes equations in any space dimensions for the initial density allowing vacuum. Later, Desjardins 6 proved the global weak solution with more regularity for the two-dimensional case provided that ( 0 ) is a small perturbation of a positive constant in L ∞ norm. Cho-Kim 7 established the local strong solution by using the compatibility condition −div (2 ( 0 )∇u 0 ) + ∇p 0 = √ 0 g and div u 0 = 0 in Ω,

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Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

Cited by 20 publications

References 31 publications

On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows

Global well‐posedness of the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity coefficient

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