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 )

Abstract

We investigate the Ericksen-Leslie system for the Oseen-Frank model with unequal Frank elastic constants in $\mathbb{R}^3$. To generalize the result of Hineman-Wang \cite{HW}, 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 \cite{HW}. 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 \cite{LTX} to prove uniqueness of the $L^3_{uloc}$-solutions of the Ericksen-Leslie system assuming that the initial data has a finite energy.