Multi-Scale Analysis of Differential Equations for Many Particle System

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1.Derivation of Uniform Estimates for the Boltzmann-Grad Limits : The Newton equation of motion for the many-particle system gives rise to the Boltzmann equation in the limit of two scale parameters, the number of particles N and the radius of the particle r, as N→∞ and r→0, under the condition Nr^2=constant. The mathematical proof of this convergence was proven by O.Lanford (1975), the most crucial part of which is the uniform estimates in N and r for the solutions of the Newton equation. We showed that the technique of the abstract version of Cauchy-Kovalevskaya theorem can give improved estimates.2.Establishment of the solvability condition of the nonlinear boundary layer problem of the Boltzmann Equation : The most basic boundary layer is the solution of the boundary value problem in the half-space. However, the problem is not unconditionally solvable because the boundary condition at infinity is over-determined. We established the solvability condition on the boundary data. More precisely, we showed that the number of restrictions on the boundary data depends on the Mach number M at infinity, as 5 for M>1,4 for <M<1,1 for -1<M<0 and 0 for M<-1. The proof relies on sharp a priori estimates of solutions, which is obtained by use of a proper weight function and by introduction of a new artificial damping term.3.Proof of the stability of the nonlinear boundary layer. We proved that the stationary solutions obtained above are exponentially stable for the case M<-1,. First, the exponential decay is established for the linearized equation, using the energy method, and then the nonlinear stability is established by the contraction mapping principle.4.Asymptotic analysis of Fluid equations : The uniform estimates of solutions needed in establishing asymptotic relations between various fluid equations are derived by a unified method based on the abstract Cauchy-Kovalevskaya technique introduced in 1.