2019/09/15 by Andrew Lawrie, Lawrie, Andrew, Jonas Lührmann +5 · 2 citations
Arts and Humanities · Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Musicology and Musical Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1909.06899
openalex publication_date 2019/09/15 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We consider the Cauchy problem for the Schr "odinger maps evolution when the\ndomain is the hyperbolic plane. An interesting feature of this problem compared\nto the more widely studied case on the Euclidean plane is the existence of a\nrich new family of finite energy harmonic maps. These are stationary solutions,\nand thus play an important role in the dynamics of Schr "odinger maps. The main\nresult of this article is the asymptotic stability of (some of) such harmonic\nmaps under the Schr "odinger maps evolution. More precisely, we prove the\nnonlinear asymptotic stability of a finite energy equivariant harmonic map Q\nunder the Schr "odinger maps evolution with respect to non-equivariant\nperturbations, provided Q obeys a suitable linearized stability condition.\nThis condition is known to hold for all equivariant harmonic maps with values\nin the hyperbolic plane and for a subset of those maps taking values in the\nsphere. One of the main technical ingredients in the paper is a global-in-time\nlocal smoothing and Strichartz estimate for the operator obtained by\nlinearization around a harmonic map, proved in the companion paper [36].\n