Finite time blow-up for some parabolic systems arising in turbulence theory

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URI: https://hdl.handle.net/10902/29604
ISSN: 0044-2275
ISSN: 1420-9039
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Autoría
Fanelli, Francesco; Granero Belinchón, RafaelAutoridad Unican
Fecha
2022-07-25
Derechos
This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature's AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00033-022-01818-5
Publicado en
Zeitschrift fur Angewandte Mathematik und Physik, 2022, 73, 180
Editorial
Springer
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Palabras clave
Nonlinear Parabolic Systems
Degeneracy
Local Well-Posedness
Finite Time Blow-Up
Curvature
Turbulence
Resumen/Abstract
We study a class of nonlinear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to put in evidence two different types of blow-up mechanism. In addition, the results are extended to the case of transport-diffusion systems, namely to the case when convection is taken into account
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