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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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