This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations.In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.
207The quasi-stationary method developed by Ono et al. 1s generalized to the case of oblique shock propagation, that is, when a sho~k propagates along directions making a finite angle with that of pressure gradient in stratifying media. Two simultaneous differential equations determining the direction and strength of shock are obtained. Its behavior is discussed in detail for polytropic gases. It is characteristic for the oblique propagation that a pattern change, i.e. transition from regular to Mach pattern, occurs ultimately, and a vortex field generates after the passage of the shock. *> This problem is now under consideration by M. Saito, who takes account of ,the effect of the oblique propagation into shock pulses.6l at Florida International University on June 7, 2015 http://ptp.oxfordjournals.org/ Downloaded from *) The case of oblique magnetic shocks is rather complicated. This is now under investigation by Yamazaki et al. in our institute.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.