Blow up Dynamic and Upper Bound on the Blow up Rate for critical nonlinear Schrödinger Equation

We consider the critical nonlinear Schrödinger equation iut = −∆u − |u| 4 N u with initial condition u(0, x) = u0 in dimension N . For u0 ∈ H1, local existence in time of solutions on an interval [0, T ) is known, and there exists finite time blow up solutions, that is u0 such that limt→T<+∞ |ux(t)|L2 = +∞. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data known to lead to blow up, no general understanding of the blow up dynamic is known. At first, we propose in this paper a general setting to study and understand small in a certain sense blow up solutions. Blow up in finite time follows for the whole class of initial data in H1 with strictly negative energy, and one is able to prove a control from above of the blow up rate below the one of the known explicit explosive solution, which has strictly positive energy. We consider the critical nonlinear Schrödinger equation { iut = −∆u− |u| 4 N u, (t, x) ∈ [0, T )× R u(0, x) = u0(x), u0 : R → C (NLS) with u0 ∈ H = H(R), in dimension N ≥ 1. This equation is locally well-posed in H from [2]. The problem we address is the one of formation of singularities for solutions to (NLS). Note that from the conservation of the mass and the energy (from the Hamiltonian formulation) and Gagliardo-Nirenberg inequality, the power of the nonlinearity is the smallest one for which blow-up may occur. We will see that this criticality makes the problem global. In the energy space H, (NLS) admits three conservation laws: L-norm, Energy, Momentum, and four fundamental symetries: Space-time translation, Phase, Scaling and Galilean invariances. At the critical power, special regular solutions play an important role. They are the so called solitary waves and are of the form u(t, x) = eWω(x), ω > 0, where Wω solves ∆Wω +Wω|Wω| 4 N = ωWω. (1)