### Global Estimation of the Cauchy Problem Solutions' and Blow up the Navier-Stokes Equation

#### Abstract

The paper presents results of research of gradient catastrophe development during phase change. It shows that classical methods of the function estimation theory do not fit well to study of gradient catastrophe problem. The paper presents results, indicating that embedding theorems do not allow to study a process of a catastrophe formation. In fact, the paper justifies Terence Tao’s pessimism about a failure of modern mathematics to solve the Navier-Stokes problem. An alternative method is proposed for dealing with the gradient catastrophe by studying Fourier transformation for a function and selecting a function singularity through phase singularities of Fourier transformation for a given function. The analytic properties of the scattering amplitude are discussed in ${R^{3}}$, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided.Describes the loss of smoothness of classical solutions for the Navier-Stokes equations -Millennium Prize Problems.