We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the...

Keywords: Allen–Cahn equation Finite element method Operator splitting method Unconditionally stable scheme a b s t r a c t We present an unconditionally stable hybrid finite element method for solving the Allen– Cahn equation, which describes the temporal evolution of a non-conserved phase-field during the antiphase domain coarsening in a binary mixture. Its various modified forms have been ap...

In [8] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the f...

We follow Dirac and write the Schrödinger equation in an algebraic form which is representation-free. The imaginary and real parts of this equation are respectively the Liouville equation, which involves the commutator of the Hamiltonian with the density operator and an equation for the time development of the phase operator that involves the anti-commutator of the Hamiltonian with the density ...

In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an ‘‘unstable disk theorem’’ through a simple energy–C...

The inverse geothermal problem consist of estimating the temperature distribution below the earth’s surface using temperature and heat-flux measurements on the earth’s surface. The problem is important since temperature governs a variety of the geological processes including formation of magmas, minerals, fosil fuels and also deformation of rocks. Mathematical this problem is formulated as a Ca...