2024/12/05 by Xuan Liu, Changxing Miao, Liu, Xuan +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2412.04566
openalex publication_date 2024/12/05 · openalex created_date 2024/12/10 · openalex updated_date 2026/07/28
We prove global well-posedness and scattering for solutions to the mass-critical inhomogeneous nonlinear Schrödinger equation i∂tu+Δu=± |x|-b|u|(4-2b)/(d)u for large L2(ℝ d) initial data with d≥3,0<b>0) are that the presence of the inhomogeneity |x|-b creates a nontrivial singularity at the origin, and breaks the translation symmetry as well as the Galilean invariance of the equation, which makes the establishment of the profile decomposition and long time Strichartz estimates more difficult. To overcome these difficulties, we perform the concentration compactness/rigidity methods of [Kenig and Merle, Invent. Math. (2006)] in the Lorentz space framework, and reduces the problem to the exclusion of almost periodic solutions. The exclusion of these solutions will utilize fractional estimates and long time Strichartz estimates in Lorentz spaces. In our study, we obseve that the decay of the inhomogeneity |x|-b at infinity prevents the concentration of the almost periodic solution at infinity in either physical or frequency space. Therefore, we can use classical Morawetz estimates, rather than interaction Morawetz estimates, to exclude the existence of the quasi-soliton.</b>