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The paper describes the mechanism of occurrence of a gradient catastrophewhen changing phase. Materials shows that classical methods of estimationtheory of functions do not fit the problem of studying the gradientcatastrophe. We present material showing that the embedding theorem can notgive an opportunity to study the process of a gradient catastrophe. In fact,work justifies pessimism Terence Tao in the insolvency of modern mathematics tosolve the problem of the Navier-Stokes equations. It is suggested analternative method for studying the gradient catastrophe through the study ofthe Fourier transform function through special properties phases of the Fouriertransform of this function

A gradient catastrophe mechanism in contexts of the phase change condition