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 data into the two-dimensional, incompressible Euler equations. The data are given by linear functionals acting on the space of functions representing vorticity. The data are sparse and available on a fixed space-time domain. The objective of the data assimilation is to obtain an estimate of the vorticity which minimizes a cost functional and is analogous to a distributed parameter control problem. The cost functional is the sum of a weighted squared error in the dynamics, the initial condition, and in the misfit to the observed data. Vorticity estimates which minimize the cost functional are obtained by solving the corresponding system of Euler-Lagrange equations. The Euler-Lagrange system is a coupled two-point boundary value problem in time. An application of the Schauder fixed-point theorem establishes the existence of a least one solution to the system. Iterative methods for generating solutions have proven useful in applications in meteorology and oceanography.