The global solution to the Cauchy problem of the compressible Navier-Stokes equations with hyperbolic heat conduction in dimension three is constructed when the initial data in $H^3$ norm is small. By using several elaborate energy functionals together with the interpolation trick, we simultaneously obtain the optimal $L^2$-decay estimate of the solution and its derivatives when the initial data is bounded in negative Sobolev (Besov) space or $L^1(\mathbb{R}^3).$ Specially speaking, the fluid density, the fluid velocity and the fluid temperature in $L^2$-norm have the same decay rate as the Navier-Stokes-Fourier equations, while the flux $q$ has faster $L^2$-decay rate as $(1+t)^{−2}.$ Our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis for a 8×8 Green matrix of the system. To the best of our knowledge, it is the first result on the large time behavior of this system.

In this work we prove the existence and uniqueness theorems of the solutions to the relativistic Boltzmann equation for analytic initial fluctuations on a time interval independent of the Knudsen number $\epsilon > 0.$ As $\epsilon → 0,$ we prove that the solution of the relativistic Boltzmann equation tends to the local relativistic Maxwellian, whose fluid-dynamical parameters solve the relativistic Euler equations and the convergence rate is also obtained. Due to this convergence rate, the Hilbert expansion is verified in the short time interval for the relativistic Boltzmann equation. We also consider the physically important initial layer problem. As a by-product, an existence theorem for the relativistic Euler equations without the assumption of the non-vacuum fluid states is obtained.

To begin with, we identify the intrinsic equations of a von Kármán nonlinearly elastic plate, which allow to directly compute the stresses inside the plate without having to first compute the displacement field, by contrast with the classical displacement approach. Then we establish that these intrinsic equations possess weak solutions, which are the bending moments and stress resultants of the middle surface of the plate.