An Application of Nash-Moser Theorem to Smooth Solutions of One-Dimensional Compressible Euler Equation with Gravity

摘要

We study one-dimensional motions of polytropic gas governed by thecompressible Euler equations. The problem on the half space under a constantgravity gives an equilibrium which has free boundary touching the vacuum andthe linearized approximation at this equilibrium gives time periodic solutions.But it is not easy to justify the existence of long-time true solutions forwhich this time periodic solution is the first approximation. The situation isin contrast to the problem of free motions without gravity. The reason is thatthe usual iteration method for quasilinear hyperbolic problem cannot be usedbecause of the loss of regularities which causes from the touch with thevacuum. Interestingly, the equation can be transformed to a nonlinear waveequation on a higher dimensional space, for which the space dimension, beinglarger than 4, is related to the adiabatic exponent of the originalone-dimensional problem. We try to find a family of solutions expanded by asmall parameter. Applying the Nash-Moser theory, we justify this expansion.Theapplication of the Nash-Moser theory is necessary for the sake of conquest ofthe trouble with loss of regularities, and the justification of theapplicability requires a very delicate analysis of the problem.