This work is concerned with our recently developed formalism of non-equilibrium thermodynamics. This formalism extends the classical irreversible thermodynamics which leads to classical thermodynamics and can not describe physical phenomena with long relaxations. With the extended theory, we obtain a generalized hydrodynamic system which can be used to describe the non-Newtonian fluids, heat transfer beyond Fourier's law and fluid flow in very short temporal and spatial scales. We study the mathematical structure of the generalized hydrodynamic system and show that it possesses a nice conservation-dissipation structure and therefore is symmetrizable hyperbolic. Moreover, we rigorously justify that this generalized hydrodynamic system tends to the classical hydrodynamics when the relaxation times approach to zero. This shows that the classical hydrodynamics can be derived as an approximation of our generalized one.

Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).

We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat n-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.

We study the hypocoercivity property for some kinetic equations in the whole space and obtain the optimal convergence rates of solutions to the equilibrium state in some function spaces. The analysis relies on the basic energy method and the compensating function introduced by Kawashima to the classical Boltzmann equation and developed by Glassey and Strauss in the relativistic setting. It is also motivated by the recent work (Duan et al., 2008 [8]) on the Boltzmann equation by combining the spectrum analysis and energy method. The advantage of the method introduced in this paper is that it can be applied to some complicated system whose detailed spectrum is not known. In fact, only some estimates through the Fourier transform on the conservative transport operator and the dissipation of the linearized operator on the subspace orthogonal to the collision invariants are needed.

We investigate the fractional Schrödinger equation with a periodic PT-symmetric potential. In the inverse space, the problem transfers into a first-order nonlocal frequency-delay partial differential equation. We show that at a critical point, the band structure becomes linear and symmetric in the one-dimensional case, which results in a nondiffracting propagation and conical diffraction of input beams. If only one channel in the periodic potential is excited, adjacent channels become uniformly excited along the propagation direction, which can be used to generate laser beams of high power and narrow width. In the two-dimensional case, there appears conical diffraction that depends on the competition between the fractional Laplacian operator and the PT-symmetric potential. This investigation may find applications in novel on-chip optical devices.