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Non-standard parabolic regularization of gradient catastrophes for theBurgers-Hopf equation is proposed. It is based on the analysis of all (genericand higher order) gradient catastrophes and their step by step regularizationby embedding the Burgers-Hopf equation into multi-component parabolic systemsof quasilinear PDEs with the most degenerate Jordan block. Probabilisticrealization of such procedure is presented. The complete regularization of theBurgers-Hopf equation is achieved by embedding it into the infinite parabolicJordan chain. It is shown that the Burgers equation is a particular reductionof the Jordan chain. Gradient catastrophes for the parabolic Jordan systems arealso studied.

Parabolic regularization of the gradient catastrophes for the Burgers-Hopf equation and Jordan chain