Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in Luloc3(R3)

We investigate the Ericksen–Leslie system with the Oseen–Frank energy for unequal Frank elastic constants in R 3 . Motivated by the results of Kato (Math Z 187:471–480, 1984 ) and Hineman–Wang (Arch Ration Mech Anal 210:177–218, 2013 ), we prove existence of solutions to the Ericksen–Leslie system w...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2019-02, Vol.58 (1)
Main Authors: Hong, Min-Chun, Mei, Yu
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
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Summary:We investigate the Ericksen–Leslie system with the Oseen–Frank energy for unequal Frank elastic constants in R 3 . Motivated by the results of Kato (Math Z 187:471–480, 1984 ) and Hineman–Wang (Arch Ration Mech Anal 210:177–218, 2013 ), we prove existence of solutions to the Ericksen–Leslie system with initial data having small L uloc 3 -norm. In particular, we obtain a local L 3 -estimate by using interpolation inequalities and a covering argument, which is different from the one in Hineman–Wang ( 2013 ) of using local L 3 decay estimates for the Laplace operator. Moreover, we prove the uniqueness of the L uloc 3 -solutions of the Ericksen–Leslie system without a closing condition of Frank constants k 1 , k 2 , k 3 by utilizing the method of Li–Titi–Xin (Math Models Methods Appl Sci 26:803–822, 2016 ).
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-018-1453-4