On Echo Chains in the Linearized Boussinesq Equations Around Traveling Waves
Abstract.
We consider the 2-dimensional Boussinesq equations with viscous but without thermal dissipation and observe that in any neighborhood of Couette flow and hydrostatic balance (with respect to local norms) there are time-dependent traveling wave solutions of the form \(\omega=-1+ f(t)\cos (x-ty)\), \(\theta =\alpha y+g(t)\sin (x-ty)\). As our main result we show that the linearized equations (with some technical simplifications) around these waves for \(\alpha=0\) exhibit echo chains and norm inflation despite viscous dissipation of the velocity. Furthermore, we construct initial data in a critical Gevrey 3 class, for which temperature and vorticity diverge to infinity in Sobolev regularity as \(t\rightarrow \infty\) but for which the velocity still converges.
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Published In
SIAM Journal on Mathematical Analysis
Pages: 5127 - 5188
DOI: 10.1137/21M1458053
ISSN (online): 1095-7154
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© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 8 November 2021
Accepted: 27 April 2023
Published online: 28 September 2023
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Funding Information
Deutsche Forschungsgemeinschaft (DFG): 258734477 - SFB 1173
Funding: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173.
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