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\title{Parabolic regularization of the \\gradient catastrophes for the
  \\ Burgers-Hopf equation and Jordan chain}

\author{B. Konopel'chenko, G. Ortenzi\\[2mm]
  Dipartimento di Matematica e Fisica
  ``E. De Giorgi'',\\ Universit\`a del Salento, Lecce, Italy\\
  and Dipartimento di Matematica Pura ed Applicata,\\
  Universit\`a di Milano--Bicocca, Italy}

\date{}
\maketitle

Non-standard parabolic regularization of gradient catastrophes for the
Bur\-gers-Hopf equation is proposed. It is based on the analysis of all (generic
and higher order) gradient catastrophes and their step by step regularization
by embedding the Burgers-Hopf equation into multi-component parabolic systems
of quasilinear PDEs with the most degenerate Jordan block. Probabilistic
realization of such procedure is presented. The complete regularization of the
Burgers-Hopf equation is achieved by embedding it into the infinite parabolic
Jordan chain. It is shown that the Burgers equation is a particular reduction
of the Jordan chain. Gradient catastrophes for the parabolic Jordan systems are
also studied.

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