A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution. From this stability result we deduce global existence of a solution to the Cauchy problem. Our method relies upon recent compactness results for velocity averages, a new formulation of the Boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions. It allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations.

https://www.jstor.org/stable/pdfplus/1971423.pdf

On the Cauchy problem for Boltzmann equations: global existence and weak stability

Dynamical processes in macroscopic systems are often approximately described by kinetic and hydrodynamic equations. One of the central problems in nonequilibrium statistical mechanics is to understand the approximate validity of these equations starting from a microscopic model. We discuss a variety of classical as well as quantum-mechanical models for which kinetic equations can be derived rigorously. The probabilistic nature of the problem is emphasized: The approximation of the microscopic dynamics by either a kinetic or a hydrodynamic equation can be understood as the approximation of a non-Markovian stochastic process by a Markovian process.

Kinetic equations from Hamiltonian dynamics: Markovian limits

We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies 
$ \gamma > 3/2 $
 .

On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations

The equations of motion of compressible viscous and heat-conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region. The global solution in time is proved to exist uniquely and approach the stationary state ast→∞, provided the prescribed initial data and the external force are sufficiently small.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103922925

Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids

/pdf/the-initial-value-problem-for-the-equations-of-motion-of-1qxslazq87.pdf

The initial value problem for the equations of motion of viscous and heat-conductive gases

: The initial value problem associated with the equations of motion for isotropic Newtonian fluids is investigated. The fluids are compressible, viscous and heat-conductive. It is proved that there exists a unique global solution in time, for the small initial data, and the solution has the decay rate of (1 + t) to 3/4 power as t approaches positive infinity. The motions of compressible, viscous and heat-conductive fluids are described by a system of partial differential equations which is of hyperbolic-parabolic type and highly nonlinear. One of the first mathematical problems associated with this system is the initial value problem. We obtain the existence of a a unique smooth global solution in time for the initial value problem and also the decay rate of the solution as time tends to infinity.

/pdf/the-initial-value-problem-for-the-equations-of-motion-of-ez0evw4wix.pdf

The Initial Value Problem for the Equations of Motion of compressible Viscous and Heat-conductive Fluids.

The nonlinear Boltzmann equation for a rarefied gas is investigated in the fluid dynamical limit to the level of compressible Euler equation locally in time, as the mean free path e tends to zero. The nonlinear hyperbolic conservation laws obtained as the limit are also the first approximation of the Chapman-Enskog expansion.

/pdf/fluid-dynamical-limit-of-the-nonlinear-boltzmann-equation-to-6ytor65ibq.pdf

Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation

The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay ratet
−5/4) ast→+∞ to that of the compressible Navier-Stokes equation for the corresponding initial data.

/pdf/on-the-fluid-dynamical-approximation-to-the-boltzmann-2encq7o6qi.pdf

Takaaki Nishida

Papers

The initial value problem for the equations of motion of viscous and heat-conductive gases

Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids

The Initial Value Problem for the Equations of Motion of compressible Viscous and Heat-conductive Fluids.

Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation

On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation