We propose a conservation-dissipation formalism (CDF) for coarse-grained descriptions of irreversible processes. This formalism is based on a stability criterion for non-equilibrium thermodynamics. The criterion ensures that non-equilibrium states tend to equilibrium in long time. As a systematic methodology, CDF provides a feasible procedure in choosing non-equilibrium state variables and determining their evolution equations. The equations derived in CDF have a unified elegant form. They are globally hyperbolic, allow a convenient definition of weak solutions, and are amenable to existing numerics. More importantly, CDF is a genuinely nonlinear formalism and works for systems far away from equilibrium. With this formalism, we formulate novel thermodynamics theories for heat conduction in rigid bodies and non-isothermal compressible Maxwell fluid flows as two typical examples. In these examples, the non-equilibrium variables are exactly the conjugate variables of the heat fluxes or stress tensors. The new theory generalizes Cattaneo’s law or Maxwell’s law in a regularized and nonlinear fashion.

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Under the diffusion scaling and a scaling assumption on the microscopic component, a non-classical fluid dynamic system was derived in\cite {BGLY} that is related to the system of ghost effect derived in\cite {Sone-2} in different settings. This paper aims to justify this limit system for a non-trivial background profile with slab symmetry. The result reveals not only the diffusion phenomena in the temperature and density, but also the flow of higher order in Knudsen number due to the gradient of the temperature. Precisely, we show that the solution to the Boltzmann equation converges to a diffusion wave with decay rates in both Knudsen number and time.

We consider the kinetic Fokker-Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a gen- eral force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve results about classical force to general force case. Our result also improve the rate of convergence for the Fitzhugh-Nagumo equation from non-quantitative to quantitative explicit rate.

The nonlinear Schrdinger equation (NLS) has been a fundamental model for understanding vortex motion in superfluids. The vortex motion law has been formally derived on various physical grounds and has been around for almost half a century. We study the nonlinear Schrdinger equation in the incompressible fluid limit on a bounded domain with Dirichlet or Neumann boundary condition. The initial condition contains any finite number of degree 1 vortices. We prove that the NLS linear momentum weakly converges to a solution of the incompressible Euler equation away from the vortices. If the initial NLS energy is almost minimizing, we show that the vortex motion obeys the classical Kirchhoff law for fluid point vortices. Similar results hold for the entire plane and periodic cases, and a related complex GinzburgLandau equation. We treat as well the semi-classical (WKB) limit of NLS in the presence of