A Solution to the focusing 3D NLS that blows up on a contracting sphere
Article
Holmer, J, Perelman, G, Roudenko, S. (2015). A Solution to the focusing 3D NLS that blows up on a contracting sphere
. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 367(6), 3847-3872. 10.1090/S0002-9947-2015-06057-7
Holmer, J, Perelman, G, Roudenko, S. (2015). A Solution to the focusing 3D NLS that blows up on a contracting sphere
. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 367(6), 3847-3872. 10.1090/S0002-9947-2015-06057-7
We rigorously construct radial H1 solutions to the 3d cubic focusing NLS equation i∂tψ +Δψ + 2|ψ|2ψ = 0 that blow-up along a contracting sphere. With blow-up time set to t = 0, the solutions concentrate on a sphere at radius ∼ t1/3 but focus towards this sphere at the faster rate ∼ t2/3. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov in 1975 and independently later by Holmer-Roudenko in 2007, where it was demonstrated to be consistent with all conservation laws of this equation. In the latter paper, it was proposed as a solution that would yield divergence of the L3 x norm within the “wide” radius ∼∇u(t)(formula present)but not within the “tight” radius ∼∇u(t)|| (formula present), the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail by Sulem-Sulem in 1999.
