LOCAL EXISTENCE RESULTS FOR THE GENERALIZED INVERSE OF THE VORTICITY EQUATION IN THE PLANE

We prove the finite-time existence of a solution to the Euler-Lagrange equations corresponding to the necessary conditions for minimization of a functional defining variational assimilation of observational dat a into the two-dimensional, incompressible Euler equations. The data a re given by linear functionals acting on the space of functions repres enting vorticity. The data are sparse and available on a fixed space-t ime domain. The objective of the data assimilation is to obtain an est imate of the vorticity which minimizes a cost functional and is analog ous to a distributed parameter control problem. The cost functional is the sum of a weighted squared error in the dynamics, the initial cond ition, and in the misfit to the observed data. Vorticity estimates whi ch minimize the cost functional are obtained by solving the correspond ing system of Euler-Lagrange equations. The Euler-Lagrange system is a coupled two-point boundary value problem in time. An application of t he Schauder fixed-point theorem establishes the existence of a least o ne solution to the system. Iterative methods for generating solutions have proven useful in applications in meteorology and oceanography.