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Exploding solutions of the complex two-dimensional Burgers equations: computer simulations. (English) Zbl 1332.35319

Summary: We study by computer simulations the complex solutions of the two-dimensional Burgers equations in the whole plane in absence of external forces. For such model the existence of singularities, corresponding to a divergence of the total energy at a finite time, is proved by D. Li and Y. G. Sinai [ibid. 51, No. 1, 015205, 16 p. (2010; Zbl 1309.35113)] for a large class of initial data. The simulations show that the blow-up takes place in a very short time, of the order of \(10^{-5}\) time units. Moreover near the blow-up time the support of the solution in Fourier space moves out to infinity along a straight line. In \(x\)-space the solutions are concentrated in a finite region, with large space derivatives, as one would expect for physical phenomena such as tornadoes.{
©2012 American Institute of Physics}

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A20 Analyticity in context of PDEs
35B44 Blow-up in context of PDEs

Citations:

Zbl 1309.35113
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References:

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[2] DOI: 10.1007/BF02547354 · JFM 60.0726.05 · doi:10.1007/BF02547354
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[5] DOI: 10.1063/1.3276099 · Zbl 1309.35113 · doi:10.1063/1.3276099
[6] DOI: 10.1134/S1560354710040088 · Zbl 1205.35227 · doi:10.1134/S1560354710040088
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[11] DOI: 10.1070/RM2009v064n06ABEH004655 · Zbl 1187.35150 · doi:10.1070/RM2009v064n06ABEH004655
[12] DOI: 10.1070/RM2009v064n06ABEH004655 · Zbl 1187.35150 · doi:10.1070/RM2009v064n06ABEH004655
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