The Lewis-Kraichnan Equation for the Space-Time Functional

The space-time characteristic functional is defined and Gateaux derivatives are evaluated in an ONS environment. Two versions of the Lewis-Kraichnan fde governing the evolution of the space-time functional are derived. Version (I): The pressure-gradient functional is explicitly constructed using the known solution of the Poisson pde in \(R^3\) . Version (II): The pressure-gradient functional is eliminated by modifying the argument space to contain solenoidal vector fields only. Variational derivatives of a Gaussian characteristic functional are computed for comparison with the Lewis-Kraichnan fdes. Generalized Laplacians are defined as standard and Levy types and related within the framework of an ONS basis. The last section is devoted to the definition and analysis of Hessian and Laplacian operators on the functional level and the Hopf fde in the material description, which is shown to be of a type different than the spatial description.