Advances in Nonlinear Partial Differential Equations and Stochastics

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction
  • Supersonic Flow Past a Wedge
  • The Case of Curved Wedge

• Supersonic Flow past a M-D Wing
  • Basic Strategy
  • Linearization
  • Asymptotic Solution – The First Term in Iteration Process
  • Energy Estimate of Linearized Problem
  • Convergence of iterative sequence

• Supersonic Flow past a Delta Wing
  • Formulation
  • Linear Approximation
  • Existence and Regularity

• Acknowledgement

• REFERENCES

In this paper, we consider the initial value problems to systems of quasilinear wave equations with different speeds in two space dimensions. Applying John-Shatah observations to our problem, we introduce the null condition for the system with different speeds. Moreover, we prove a global existence theorem for a class satisfying the null condition.

A system of a large number of classical particles moving on the real line is studied. The particles interact with one another through repulsive pair-potential forces and are subject to resistance proportional to their velocities. Because of the latter it is only the number of particles that is conserved under the evolution of the system. It is proved that under suitable scaling of space and time the normalized counting measure of particle locations converges and its limit is governed by a non-linear evolution equation, which is local (diffusion equation) or non-local (integral equation) according as the tail of the potential is bounded by 1/|x| or obeys the power law 1/|x|^γ with -1 < γ < 1. The scaling also depends on the tail of the potential.

We discuss the higher order regularity of solutions to initial boundary value problems for linear symmetric hyperbolic systems with boundary characteristic of constant multiplicity. By means of the standard energy method it is shown that the solutions and their derivatives with respect to the time variable lie in certain weighted Sobolev spaces under a suitable compatibility condition between the data.

The stationary problem for the discrete velocity model of the one-dimensional Boltzmann equation is studied in a half-space with the Dirichlet boundary conditions. The point of our problem is to find out a solution which tends to an assigned Maxwellian at infinity. It will be shown that such a solution exists for the boundary data close to the Maxwellian if, in particular, the number of the boundary conditions is just equal to the number of negative eigenvalues of an operator related to the linearized equation.

The following sections are included:

• Introduction

• Preliminaries

• Blow-up conditions

• Life span of blow-up solutions

• Global existence and decay estimates

• Asymptotic behavior of global solutions

• References

The following sections are included:

• Introduction

• Representation of Solutions
  • Reduction of problem
  • Characteristic polynomial
  • Representation formula for the whole space
  • Boundary compensation
  • Representation formula of solutions

• Auxiliary Lemmas

• Estimates for Large |λ|
  • Asymptotic behaviour of coefficients defined in the paragraph 1.5
  • Lemmas for the estimate of terms related to ξ₁(λ)
  • Estimation of the part related to ξ₁(λ)
  • Lemmas for the estimates of terms related to ξ₂(λ)
  • Estimation of the part related to ξ₂(λ)
  • Estimation of the boundary reflection term of (1.2) type
  • Estimation of the boundary reflection term of (2.1) type
  • Summary

• Estimates For Small |λ|
  • Asymptotic behaviour of coefficients
  • Lemmas for the estimates
  • Estimations of the part of (1.1) type and (2.2) type
  • Estimation of the boundary reflection term of (1.2) type
  • Estimation of the boundary reflection term of (2.1) type
  • Summary

• Estimates In The Bounded Region

• References

This is a survey of our recent works on bifurcation phenomena for periodic solutions of the Duffing equation. For a one-parameter family of T-periodic external forces, we discuss the existence of bifurcation points of not only T-periodic solution, but also 2T-periodic solution (period-doubling bifurcation points).

Here we represent some topics on compactness method, namely, we observe the relation between well-known compactness argument and so-called compensated compactness and expose the facilities of verification compactness approach as well.

In [1], we studied bond percolation on 2-dimensional pre-Sierpinski gasket. We proved the correlation length on the pre-Sierpinski gasket diverges much faster than that on Z^d. In this paper, we consider percolation on some fractal lattices, which have self-similarity such as high dimensional pre-Sierpinski gaskets. We obtain the precise asymptotic behavior of the correlation length.