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C.D. Levermore and M. Oliver,
Analyticity of solutions for a generalized Euler
equation,
J. Differential Equations 133 (1997), 321-339.
Abstract:
We consider the so-called lake and great lake equations, which are
shallow water equations that describe the long-time motion of an
inviscid, incompressible fluid contained in a shallow basin with a
slowly spatially varying bottom, a free upper surface and vertical
side walls, under the influence of gravity and in the limit of small
characteristic velocities and very small surface amplitude. If these
equations are posed on a space-periodic domain and the initial data
are real analytic, the solution remains real analytic for all
times. The proof is based on a characterization of Gevrey classes in
terms of decay of Fourier coefficients. In particular, our result
recovers known results for the Euler equations in two and three
spatial dimensions. We believe the proof is new.
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