A model due to Ericksen and Leslie describing incompressible liquid crystals is studied for a general class of free energies. Global existence of weak solutions is proven via a Galerkin approximation with eigenfunctions of a strongly elliptic operator. A novelty is that the principal part of the differential operator appearing in the director equation can be nonlinear.We recall that v ∶ Ω × [0, T ] → R 3 denotes the velocity of the fluid, d ∶ Ω × [0, T ] → R 3 represents the orientation of the rod-like molecules, p ∶ Ω × [0, T ] → R denotes the pressure, and g ∶ Ω × [0, T ] → R 3 denotes an external force. Throughout this paper, let Ω ⊂ R 3 be a bounded domain of class 2 and T > 0 be given. 6492

In this article, we prove the existence of measure-valued solutions to the Ericksen-Leslie system equipped with the Oseen-Frank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measure-valued solutions for vanishing regularization. Additionally, it is shown that the measurevalued solution fulfills an energy inequality.

We introduce the new concept of maximally dissipative solutions for a general class of isothermal GENERIC systems. Under certain assumptions, we show that maximally dissipative solutions are well-posed as long as the bigger class of dissipative solutions is non-empty. Applying this result to the Navier–Stokes and Euler equations, we infer global well-posedness of maximally dissipative solutions for these systems. The concept of maximally dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique.

We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting and space–time discretization. Computational studies are performed to study the mutual effects of electric, elastic and viscous effects onto the molecules in a nematic electrolyte.

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