We obtain the global well-posedness and the scattering theory of the solution for the modified Davey–Stewartson system { i ∂ t u + Δ u = | u | 4 u + u v x 1 , (t , x) ∈ R × R 3 , − Δ v = (| u | 2) x 1 in the energy space H 1 ( R 3) in this paper. Since the interaction Morawetz estimate fails and the nonlinearity is non-local, we employ the concentration-compactness argument introduced by Kenig and Merle (Invent Math. 2006;166:645–675) to establish the scattering result. [ABSTRACT FROM AUTHOR]
This paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of [Abidi H, Hmidi T, Keraani S. On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 2010;347:15–41.], [Hassainia Z. On the global well-posedness of the 3D axisymmetric resistive MHD equations. Ann. Henri Poincaré. 2022;23:2877-2917], [Hmidi T, Zerguine M. Inviscid limit axisymmetric Navier–Stokes system. Differential and Integral Equations. 2009;22(11–12):1223–1246.]. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied. [ABSTRACT FROM AUTHOR]