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Blowup of solutions of the unsteady Prandtl’s equation. (English) Zbl 0908.35099
The authors consider the unsteady Prandtl’s equation in the half-space \(\mathbb{R}\times\mathbb{R}^+= \{(x, y), y>0\}\): \[ u_t+ uu_x+ vu_y= u_{yy}+ P_x,\;u_x+ v_y=0 \] with compactly supported initial data and boundary conditions. At the present time, there are no local existence results for the problem under consideration with compactly supported initial data. The main result consists in the assertion that either smooth solutions do not exist even locally, or the smooth solutions blow-up at some finite \(t= T\), i.e., \(\sup| u_x|\to+\infty\) as \(t\to T\). In general, the blow-up does not occur at the boundary. This is in sharp contrast with the steady case where the blow-up is caused by an adverse pressure gradient from the far field and always occurs first at the boundary.

MSC:
35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
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