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M. Oliver,
Classical solutions for a generalized Euler equation in
two dimensions,
J. Math. Anal. Appl. 215 (1997), 471-484.
Abstract:
It is well known that the Euler equations in two spatial dimensions
have global classical solutions. We provide a new proof which is
analytic rather than geometric. It is set in an abstract framework
that applies to the so-called lake and the great lake equations
describing weakly non-hydrostatic effects of bottom topography on the
motion of shallow water. The key ingredient is a new
Lp estimate on the nonlinear term. The estimate is
used to develop a global Hm theory for bounded
domains in R2 which is similar in spirit to a 1975
paper by R. Temam. It also provides explicit bounds on the
Hm norm which grow like exp (exp t).
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