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. Academic Press

We present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced Navier Stokes equations in R n . The method, based on so-called Gevrey estimates, also yields explicit bounds on the growth of the radius of analyticity of the solution in time. Moreover, under the assumption that the Navier Stokes solution stays sufficiently close to a solution of the heat equation in the L 2 norm a result known to be true for a large class of initial data lower bounds on the decay of higher order derivatives can be obtained. Academic Press

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We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures M(R 2 ). The solutions of this regularized PDE, also known as the averaged Euler or Euler-α equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in M(R 2 ) such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods.