On parabolic partial differential equations with Hölder continuous diffusion coefficients

Sofiani, Majed

doi:10.48550/arxiv.2212.08972

Cited by 1 publication

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“…Another difficulty is caused by rough (worse than Hölder continuous) non-homogeneous terms in the parabolic equation for A, in addition to the diffusion coefficient g(θ(x, t)) being only Hölder continuous. This difficulty is overcome with a careful analysis in [44] that relies on but goes beyond treatments in [20]. The work in [44] has a much more broad interest besides a direct application to the present work.…”

Section: Introductionmentioning

confidence: 96%

“…This difficulty is overcome with a careful analysis in [44] that relies on but goes beyond treatments in [20]. The work in [44] has a much more broad interest besides a direct application to the present work.…”

Section: Introductionmentioning

confidence: 96%

“…In Section 4, we recall some basic properties from [20] on parabolic differential operators with only Hölder continuous diffusion coefficients. In Sections 5, we apply the formulation in Section 4 and results in [44] to analyze u-component. In Sections 6-7 we prove the existence of weak solution for the Cauchy problem (1.2) and (1.3).…”

Section: Introductionmentioning

confidence: 99%

“…), we use Propositions 4.3 and 6.3, which are established in [44] for more general equation was studied and Hölder estimates were established.…”

mentioning

confidence: 99%

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The Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals: The general Case

Chen¹,

Liu²,

Sofiani³

2023

Preprint

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In this work, we study the Cauchy problem of Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of two partial differential equations: One is a quasi-linear wave equation for the director field representing the crystallization of the nematics, and the other is a parabolic PDE for the velocity field characterizing the liquidity of the material. We extend the work in [Chen, et. al. Arch. Ration. Mech. Anal. 236 (2020), 839-891] for a special case to the general physical setup. The Cauchy problem is shown to have global solutions beyond singularity formation. Among a number of progresses made in this paper, a particular contribution is a systematic treatment of a parabolic PDE with only Hölder continuous diffusion coefficient and rough (worse than Hölder) nonhomogeneous terms.

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Section: Introductionmentioning

confidence: 96%