Space-Time Regularity of the Koch and Tataru Solutions to the Liquid Crystal Equations

Du, Yi; Wang, Keyan

doi:10.1137/130911597

Cited by 19 publications

(20 citation statements)

References 20 publications

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“…Recently, Wang in [38] used the framework of Koch and Tataru [20] to proved that if the initial data (u 0 , d 0 ) ∈ BM O −1 × BM O with small norm, then system (1.1)-(1.4) exists a global-in time Koch-Tataru type solution. For the regularity issues of solutions to system (1.1)-(1.4), in [7,8], Du and…”

Section: )mentioning

confidence: 99%

“…These useful properties It is easy to varify that the spaces L n (R n ) ×Ẇ 1,n (R n ) and and the existence space of Koch-Tataru type solutions to the nematic liquid crystal flows, for more details about these space, we refer to [7,8,20,22,38,41] and the references therein.…”

Section: )mentioning

confidence: 99%

“…Proof. At the beginning, we need to recall Lamma 2.2 in [7], Du and Wang proved that under the assumptions of Lemma 3.1, there holds…”

Section: Preliminariesmentioning

confidence: 99%

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Space–time derivative estimates of the Koch–Tataru solutions to the nematic liquid crystal system in Besov spaces

Liu

2015

Journal of Differential Equations

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In recent paper [7], Y. Du and K. Wang (2013) proved that the global-in-time Koch-Tataru type solution (u, d) to the n-dimensional incompressible nematic liquid crystal flow with small initial data (u 0 , d 0 ) in BMO −1 × BMO has arbitrary space-time derivative estimates in the so-called Koch-Tataru space norms. The purpose of this paper is to show that the Koch-Tataru type solution satisfies the decay estimates for any space-time derivative involving some borderline Besov space norms. More precisely, for the global-in-time Koch-Tataru type solution (u, d) to the nematic liquid crystal flow with initial data (u 0 , d 0 ) ∈ BMO −1 × BMO and u 0 BMO −1 + [d 0 ] BMO ≤ ε for some small enough ε > 0, and for any positive integers k and m, one hasFurthermore, we shall give that the solution admits a unique trajectory which is Hölder continuous with respect to space variables.

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Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

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Space–time derivative estimates of the Koch–Tataru solutions to the nematic liquid crystal system in Besov spaces

Liu

2015

Journal of Differential Equations

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“…Inspired by these results on the Navier-Stokes equations, Wang [34] established the well-posedness for the simplified Ericksen-Leslie system with rough initial data (v 0 , u 0 ) in BM O −1 × BM O. Hineman-Wang [12] obtained the local well-posedness to the simplified system with the uniformly locally L 3 -integrable data of (v 0 , ∇u 0 ) of small norm. Later, high order space-time regularities of the solution in [34] were established in [5,29]. Very recently, Hieber et al [11] proved existence of strong solutions to the simplified Ericksen-Leslie system in a bounded domain by using the maximal L p -regularity theory for abstract quasi-linear parabolic problems.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in $${{\varvec{L}}}_{{{\varvec{uloc}}}}^{3}({\mathbb R}^3)$$ L uloc 3 ( R 3 )

Hong

Mei

2018

Calc. Var.

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We investigate the Ericksen-Leslie system for the Oseen-Frank model with unequal Frank elastic constants in R 3 . To generalize the result of Hineman-Wang [12], we prove existence of solutions to the Ericksen-Leslie system with initial data having small L 3 uloc -norm. In particular, we use a new idea to obtain a local L 3 -estimate through interpolation inequalities and a covering argument, which is different from the one in [12]. Moreover, for uniqueness of solutions, we find a new way to remove the restriction on the Frank elastic constants by using the rotation invariant property of the Oseen-Frank density. We combine this with a method of Li-Titi-Xin [23] to prove uniqueness of the L 3 uloc -solutions of the Ericksen-Leslie system assuming that the initial data has a finite energy.where the free energy density W (u, ∇u) is of the formHere k 1 , k 2 , k 3 , k 4 are the Frank elastic constants, which are usually assumed to satisfy Ericksen's inequalities ([7]The first three terms on the right hand of (1.2) are associated with the splay, twist, bend characteristic deformations respectively, and the fourth term there is

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“…As for an issue of the existence of strong solutions, Wen and Ding obtained local existence and uniqueness of strong solution, Hineman and Wang established the global well‐posedness of system – in dimension 3 with small initial data ( u 0 , d 0 ) in

L_{uloc}^{3}

, where

L_{uloc}^{3}

is the space of uniformly locally L 3 ‐integrable functions in

{double-struckR}^{3}

, and Wang proved the global‐in‐time existence of strong solutions for the incompressible liquid crystal model in the whole space provided that the initial data ( u 0 , d 0 )

∥ u_{0} ∥_{BM O^{- 1}} + {[d_{0}]}_{BMO} < ϵ,

for some suitable small positive ϵ . For the issue of the large time behavior of solutions, Du and Wang and Liu proved that the small global strong solution obtained by is arbitrary spacetime regularity and is algebraically decay as time goes to infinity. Recently, base on the Fourier splitting method introduced by Schonbek for the incompressible Navier–Stokes equations, Liu showed that the L 2 ‐norm decay of weak solutions to the Cauchy problem of the two‐dimensional incompressible version of – with initial date

(u_{0}, \nabla d_{0}) \in L^{p} ({double-struckR}^{2}) \cap L^{2} ({double-struckR}^{2})

with 1≤ p < 2 is

∥ u (t) ∥_{L^{2}} + ∥ \nabla d (t) ∥_{L^{2}} \leq C {(1 + t)}^{- \frac{1}{2} (\frac{2}{p} - 1)} .

Liu and Xu established that for initial data

(u

…”

Section: Introductionmentioning

confidence: 99%

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

Liu

Zhao

2016

Math Methods in App Sciences

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Space-Time Regularity of the Koch and Tataru Solutions to the Liquid Crystal Equations

Cited by 19 publications

References 20 publications

Space–time derivative estimates of the Koch–Tataru solutions to the nematic liquid crystal system in Besov spaces

Space–time derivative estimates of the Koch–Tataru solutions to the nematic liquid crystal system in Besov spaces

Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in $${{\varvec{L}}}_{{{\varvec{uloc}}}}^{3}({\mathbb R}^3)$$ L uloc 3 ( R 3 )

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

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