2012/03/31 by Wei Dai, Dai, Wei
Mathematics · Physics and Astronomy · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math-ph #math.AP #math.MP #msc:35Q55
paper · pdf · doi:10.48550/arxiv.1204.0130
12 pages, no figure
arxiv created 2012/03/31 · openalex publication_date 2012/03/31 · arxiv updated 2012/04/03 · openalex created_date 2016/08/23 · openalex updated_date 2026/07/28
The global existence of solutions in H2 is well known for H2 critical nonlinear Schrödinger equations with small initial data in high dimensions d≥8. However, even though the solution is constructed by a fixed-point technique, continuous dependence in H2 does not follow from the contraction mapping argument. Comparing with the low dimension cases 4<d<8, there is an obstruction to this approach because of the sub-quadratic nature of the nonlinearity(which makes the derivative of the nonlinearity non-Lipschitz). In this paper, we resolve this difficulty by applying exotic Strichartz spaces of lower order instead and show that the solution depends continuously on the initial value in the sense that the local flow is continuous H2→ H2.