Global solutions of the evolutionary Faddeev model with small initial data

Abstract

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space R 1+n to the unit sphere S 2 , which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.

Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases

“…Without this quantity, we can not close the highest order energy (see Remark 3.1). Similar idea has been used by Speck [40] for the stability of Einstein-Euler equations on expanding universe and Lei et al [28] for global solutions of evolutionary Faddeev model. At last, for lower order energy estimates, we split the whole spacial domain into two parts D ∪ D c .…”

“…What's more, since system (3.1) is not symmetric, in order to close the energy estimate, we consider the whole system (3.1) without replacing u 0 by 1 + |u| 2 roughly, similar idea has been used in [28,40]. More preciously, for the nonsymmetric part, we have a u a θ).…”

“…Thus, we consider the following simplified wave equation ψ tt −η 2 ψ = 2m∇ψ · ∇ψ t − 2mψ t ψ. (4.27) By standard arguments, applying a to above equation with |a| ≤ μ , we have 28) where N d ( b ψ, c ψ) is a quadratic form satisfying "null condition" and if b + c = a, N d = N . The proof of above equality is similar to [39], we omit it here.…”

Global Nonlinear Stability of Geodesic Solutions of Evolutionary Faddeev Model

“…However, the original model (1.2) is an evolutionary system, which turns out to be unusual nonlinear wave equations enjoying the null structure and containing semilinear terms, quasilinear terms and unknowns themselves. Lei, Lin and Zhou [22] is the first rigorous mathematical result on the evolutionary Faddeev model. For the evolutionary Faddeev model in R 1+2 , they gave the global well-posedness of Cauchy problem for smooth, compact supported initial data with small H 11 (R 2 ) × H 10 (R 2 ) norm.…”

Sharp Global Regularity for the 2 + 1-Dimensional Equivariant Faddeev Model

“…To our knowledge, the first result concerning the evolution problem belongs to Lei, Lin, and Zhou [15], who showed that the 2+1-dimensional system (2) is globally well-posed for smooth, compactly-supported initial data with small H 11 (R 2 ) norm. This was followed by work of Geba, Nakanishi, and Zhang [12] in the equivariant case, which consists of global well-posedness and scattering for(4), with N 1 = 0 and initial data having a small Besov-Sobolev norm at the level of H 2 (R 2 ).…”