Bracket formulation of diffusion-convection equations

Contact Geometry of Mesoscopic Thermodynamics and Dynamics

“…Moreover, Equation (27) implies also: * (28) which means that Equation (26) is a symmetric hyperbolic (or hyperbolic in the sense of Friedrichs [46]) system of partial differential equations. This in turn means that the initial value problem (Cauchy problem) for Equation (26) with sufficiently smooth initial data is well posed.…”

“…The dissipative part of the time evolution has been put into the form appearing in Equation (15) in [21][22][23]. Both Hamiltonian and dissipative dynamics has been combined in [24][25][26][27][28][29][30][31][32]. In [30,31] the combination of both structures has been cast into the form of the first equation in Equation (15) and called GENERIC (an acronym for General Equation for Non-Equilibrium Reversible-Irreversible Coupling).…”

Incompressibility conditions in liquid crystals

“…Treatments using the alignment (or anisotropy or order parameter) tensor instead of the director followed [6]. In the same spirit approaches using Poisson brackets to derive the reversible [7] and the dissipative part [8] of the dynamics have been pursued. A Leslie-Ericksen type theory for biaxial nematics can be found in [9].…”

“…(4) by the formal requirement c 0 → ∞, because then c 2 0 δρ is undefined, eq. (4) is void, and the pressure follows from (8). This is the reason why incompressibility is assumed to be a good approximation for flows with velocities well below the sound velocity.…”

“…which are the nematic analogue to (8) and (9). The momentum balance equation (2) is somewhat simplified by the incompressibility condition divv = 0 and reads in linearized form…”