By S Kawashima, Taku Yanagisawa

A suite of papers on microlocal research, Fourier research within the advanced area, generalized capabilities and comparable subject matters. many of the papers originate from the talks given on the convention "Prospects of Generalized capabilities" (held in November 2001 at RIMS, Kyoto). Reflecting the truth that the papers are devoted to Mitsuo Morimoto, the topics thought of during this ebook are interdisciplinary, simply as Morimoto's works are. The old backgrounds of the themes in many of the papers also are mentioned extensive. therefore, the amount might be useful not just to the experts within the fields, but additionally to those that have an interest within the heritage of recent arithmetic resembling distributions and hyperfunctions Mathematical features of supersonic stream previous wings, S-X. Chen; the null situation and international life of strategies to platforms of wave equations with varied speeds, R. Agemi, ok. Yokoyama; scaling limits for big structures of interacting debris, ok. Uchiyama; regularity of recommendations of preliminary boundary price difficulties for symmetric hyperbolic platforms with boundary attribute of continuous multiplicity, Y. Yamamoto; at the half-space challenge for the discrete speed version of the Boltzmann equation, S. Ukai; on a decay price of options to the one-dimensional thermoplastic equations of a part line - linear half, Y. Shibata; bifurcation phenomena for the Duffing equation, a. Matsumura; a few comments at the compactness approach, A.V. Kazhikhov; percolation on fractal lattices - asymptotic behaviour of the correlation size, M. Shinoda. (Part contents)

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**Example text**

9). 1). The null condition is also considered with respect to nonlinear elastic wave equations. For this problem, see T. C. 13 In section 3, we introduce some notations and in section 4 we state the main result. 1) of Kovalyov,10 we estimate the first order derivatives of solution. Finally we prove the main results in section 7 using the estimates and energy inequalities in section 6. 2 The Null Condition. 1) with different speeds stated in Introduction. We consider the system in the form m £ 2 Y.

This equation can be deduced from Euler system, if the flow is regarded as isentropic and irrotational. These assumptions are reasonable provided that the strength of the attached shock is small. Morawitz and others ( see [24,27] ), because it keeps the main essential features in various problem, while it often give some simplification. Besides, in order to keep some convention we use different coordinate variables from the above two sections (like the coming flow is parallel to the plane Oyz rather than Oxy ) .

Besides, in order to keep some convention we use different coordinate variables from the above two sections (like the coming flow is parallel to the plane Oyz rather than Oxy ) . Since the flow behind the shock is regarded as irrotational, then there exists a potential $, such that the flow velocity v = (u,v,w) = V $ . 1) where C is a constant, h(p) is enthalpy, which equals ^ ^ _ for polytropic gas. Let hi be the inverse function of h(p) and J / ( V * ) = hi(C - ^|V*| 2 )) . Then the conservation law of mass can be written as £ ( * , , .