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On the profiles of locally self-similar solutions for the 2D inviscid Boussinesq equations. (English) Zbl 1433.35282

Summary: In this paper we investigate the locally self-similar solutions of the 2D inviscid Boussinesq equations and mainly focus on the possible velocity and temperature blowup profiles. We derive the formula of the pressure profiles in terms of the velocity and temperature profiles in the case that the possible nontrivial profiles have non-decaying spatial asymptotics. Then using the local energy inequality of the profiles and iterative method, we prove some nonexistence results and obtain the energy behavior concerning these possible profiles.

MSC:

35Q35 PDEs in connection with fluid mechanics
35C06 Self-similar solutions to PDEs
35B44 Blow-up in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q31 Euler equations
76R10 Free convection
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