Singular Cauchy problems for perfect incompressible fluids

J. Math. Sci. Univ. Tokyo
Vol. 14 (2007), No. 2, Page 157--176.

Uchikoshi, Keisuke
Singular Cauchy problems for perfect incompressible fluids
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Let $K$ be a field of characteristic zero that is complete with respect to a discrete valuation, with perfect residue field of characteristic $p>0$. We formulate the notion of {\it totally degenerate reduction} for a smooth projective variety $X$ over $K$. We show that for all prime numbers $\ell$, the $\bQl$-\'etale cohomology of such a variety is (after passing to a suitable finite unramified extension of $K$) a successive extension of direct sums of Galois modules of the form $\bQl(r)$. More precisely, this cohomology has an increasing filtration whose $r$-th graded quotient is of the form $V\otimes_{\bQ}\bQl(r)$, where $V$ is a finite dimensional $\bQ$-vector space that is independent of $\ell$, with an unramified action of the absolute Galois group of $K$.

Keywords: Euler equation, propagation of singularities, microlocal analysis

Mathematics Subject Classification (2000): 76B03, 35A20, 35A21
Mathematical Reviews Number: MR2351365

Received: 2007-01-30