Isolated Singularities of the 1D Complex Viscous Burgers Equation
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- Li, L. J Dyn Diff Equat (2009) 21: 623. doi:10.1007/s10884-009-9146-5
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The Cauchy problem for the 1D real-valued viscous Burgers equation ut+uux = uxx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.