### Abstract:

The main result of this paper is a proof of the existence of a solution
generated by a method for the variational assimilation of observational data
into the two-dimensional, incompressible Euler equations. The data are
assumed to be given by linear (measurement) functionals acting on the space
of functions representing vorticity. From a practical point of view, the data
are considered to be sparse and available on a fixed space-time domain.
The objective of the variational assimilation is to obtain an estimate
of the vorticity which minimizes a cost functional. The cost functional is
the sum of a generalized mean squared error in the dynamics, a generalized
mean squared error in the initial condition, and a weighted squared error
in the misfit to the observed data. These generalized mean squared errors
are computed over the fixed space-time domain containing the data. The
estimate then provides a best (generalized) least squares fit between the
model, the initial condition, and the data. A necessary condition for the estimate of vorticity to minimize the cost
functional is that it must satisfy the corresponding system of Euler-Lagrange
equations, which consist of a nonlinear, coupled system of partial differential
equations with an initial condition, a final condition, and boundary conditions.
Construction of a solution to the Euler-Lagrange equations is possible
provided they are linearized through an iterative scheme.
Analysis of one such scheme motivates a reformulation of the variational
problem in terms of an iterated linearization of the dynamics. This second
method results in a slightly different iterated system of Euler-Lagrange equations.
The sequence of solutions generated is shown to be bounded in the
Sobolev space W[superscrit k,p] (in space-time). It follows from a Sobolev imbedding
theorem that the sequence contains a convergent subsequence, the limit of
which is a classical solution of the nonlinear, forced Euler equation corresponding
to the forward problem of the Euler-Lagrange system.
The two schemes mentioned above are compared based on formal applications
of Newton's method to the operators defining the systems. We
conclude that the two formulations of the assimilation problem are in fact
different and provide some intuitive reasons for preferring the second method,
beyond the fact that the existence proof is established.