Integral Transforms and Spectra

Integral transformations with respect to time and space generate scale (defined in Chap. 21 ) information. The main source of such properties is Fourier transform of scalar and vector fields in \(L^2\) employing the Riemann/Lebesgue volume differential. The Navier–Stokes PDEs defined in \({\mathcal D}=R^3\) are transformed to complex-valued velocity and scalar modes \(\hat {\mathbf v}({\mathbf k}), \hat {\varPhi }({\mathbf k})\) as a function of the wavenumbers \({\mathbf k}\) and time. Mass balance appears as orthogonality of velocity modes to the wavenumber vector and is eliminated from the transformed momentum balances. The result is a system of three integro-differential equations for the velocity modes, and the nonlinear interactions emerge as the convolution of velocity modes.