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Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations. (English) Zbl 1327.35044
In this work, blowup from smooth nontrivial initial velocities in stagnation-point form solutions of a 2d Boussinesq system is established. On an infinite strip \(\Omega=\{(x,y)\in[0,1]\times\mathbb{R}^+\}\), one considers velocities of the form \(u=(f(t,x),-yf_x(t,x))\), with scalar temperature \(\theta=y\rho(t,x)\). Assuming \(f_x(0,x)\) attains its global maximum only at points \(x_i^*\) located on the boundary of \([0,1]\), general criteria for finite-time blowup of the vorticity \(-yf_{xx}(t,x_i^*)\) and the time integral of \(f_x(t,x_i^*)\) are presented. Briefly, for blowup to occur it is sufficient that \(\rho(0,x)\geq0\) and \(f(t,x_i^*)=\rho(0,x_i^*)=0\), while \(-yf_{xx}(0,x_i^*)\neq0\). To illustrate how vorticity may suppress blowup, one also constructs a family of global exact solutions. A local-existence result and additional regularity criteria in terms of the time integral of \(\left\|f_x(t,\cdot)\right\|_{L^\infty([0,1])}\) are also provided.

MSC:
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
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