<p>The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be strongly well-posed. Of central importance is a fully nonlinear boundary condition for the director field, which, in this generality, is necessary to guarantee that the system fulfills physical principles. The system is shown to be locally, strongly well-posed in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-setting. More precisely, the existence and uniqueness of a local, strong <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-solution to the general system is proved and it is shown that the director <i>d</i> satisfies <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|d|_2\equiv 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">|</mo> <mi>d</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msub> <mo>≡</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> provided this holds for its initial data <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. In addition, the solution is shown to depend continuously on the data. The results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.</p>

Nematic liquid crystals: Ericksen-Leslie theory with general stress tensors

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2025年11月20日

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Matthias Hieber,
Jinkai Li,
Mathias Wilke

摘要

The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be strongly well-posed. Of central importance is a fully nonlinear boundary condition for the director field, which, in this generality, is necessary to guarantee that the system fulfills physical principles. The system is shown to be locally, strongly well-posed in the $L_p$ $L_{p}$ -setting. More precisely, the existence and uniqueness of a local, strong $L_p$ $L_{p}$ -solution to the general system is proved and it is shown that the director d satisfies $|d|_2\equiv 1$ ${| d |}_{2} \equiv 1$ provided this holds for its initial data $d_0$ $d_{0}$ . In addition, the solution is shown to depend continuously on the data. The results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.