Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

“…Notice that Theorem 1 was proved in [3] in the particular case M = R 2 /Z 2 . Let us also mention that logarithmic estimates in Brezis-Gallouët [5], Vladimirov [26] and Ogawa-Ozawa [18] allow to prove Theorem 1 on any compact surface M in the particular case α ≤ 2, with the exception of the uniform continuity of the flow map (2.3) if s = 1.…”

Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain

“…This result has been extended to the natural space H 1 0 (Ω), apart from the uniform continuity of the flow ( [24], [20], [4]). …”

“…In dimension 2, for nonlinearities less than cubic, Vladimirov [24] and Ogawa and Ozawa [20] have shown the well-posedness of the Cauchy problem on H 1 0 (Ω), but without the uniform continuity of the flow on bounded sets of H 1 0 (Ω). For nonlinearities stronger than cubic in dimension 2, or for any power nonlinearity p, in dimension higher than 2, the Cauchy problem on H (Ω) solutions of negative energy or of positive energy but under some conditions on the first and second derivatives of the virial ( [9]).…”

Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field

“…It was proved in [12,13] and used in [3,5,12,13] to show the uniqueness of the nonlinear Schrödinger equations.…”

“…In this section we study the existence of a global solution to (9)- (13). Firstly, we derive the conservation laws (6) and (7).…”