Numerical Study of Blowup in the Davey-Stewartson System

A lower bound on the blow-up rate for the Davey–Stewartson system on the torus

“…Once we have this estimate, the end of the proof in this case is the same as the previous one and we obtain (34) with D > 0 independent of the period L ≥ 1. This proves (12) for low regularities and also the first point (take L = 1) in Theorem 1.…”

“…H s , with C > 0 independent of the period. This last estimate allows us to perform a Banach fixed point argument (the Lipschitz property is proved with similar arguments) in a ball of the space C([0, T ], H s ) of radius M = 2 u 0 H s and with T = C/ u 0 2 H s , and this proves (12).…”

“…s ∈ N \ {0, 1}. Let us prove (12) in this case. Let L ≥ 1 and consider the equation ( 11) and its equivalent formulation ( 13)…”

“…Moreover, it is easy to check that v τ (0) L 2 = u(0) L 2 and (−∆) s/2 (v τ (0)) L 2 ≤ 1. If we denote by T τ the maximal time for v τ , from (12), we deduce the uniform bound, T τ ≥ C > 0. But T τ = (T − τ )/λ 2 (τ ) where T is the maximal time for u and this with the uniform lower bound on T τ proves the lower bound (6).…”

“…x + ∂ 2 y forbids the existence of such solutions at least in the case where the nonlinearity is −|u| 2 u [7]. Moreover, numerics [12] seems to show that the L 2 -norm of the solution u (or v) is the minimal mass for which we may have singularities in finite time. Thus, the function u plays the role of a ground state but is only polynomially decaying and this requires to work with low regularities H s , s < 1.…”