On singularity formation for the two dimensional unsteady Prandtl system around the axis

Abstract

We consider the two dimensional unsteady Prandtl's system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative along the transversal axis solves a closed one dimensional equation. We give a precise description of singular solutions for this reduced problem. A stable blow-up pattern and a countable family of other unstable solutions are found. The blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. The proof uses modulation techniques and different energy estimates in the various zones of interest.

Stable Singularity Formation for the Inviscid Primitive Equations

“…Our main result is the following showing the existence and stability of a blowup solution for (10) with a smooth profile.…”

“…Note that for solutions u having the form u(t, X, Z) = −Xa(t, Z), and thus, u X (t, X, Z) = −a(t, Z), (11) system (4) and the boundary condition (5) for u are equivalent to system (10) for a. We emphasize here that the term 2 1 0 a 2 dZ comes from the pressure term.…”

“…In the presence of rotation i.e. Ω ≥ 0, choosing −∂ X v 0 (0, Z) = Ω, it has been shown in [21] that a defined in (9) still satisfies system (10).…”

“…In [4], a family (ψ m ) m>0 of initial data has been constructed for which the corresponding solution a to (10) blows up at time t = 1 with a(t, Z) = (1 − t) −1 ψ m (Z). Lifting this result to a blowup for the original 2D system (4) is however non-trivial given the lack of well-posedness result in the class of regularity of the profiles ψ m .…”

“…then there exists T > 0 and C > 0 such that the solution a to (10) with initial data a(t = 0) = a 0 blows up at time T > 0 according to…”