Global solutions for 2D quadratic Schrodinger equations
We prove global existence and scattering for a class of quadratic Schrodinger equations in dimension 2. The proof relies on the idea of space-time resonance.
math.AP↗
arXiv subjects
Jalal Shatah
Publications and source records attributed to Jalal Shatah.
Schrödinger Maps and their associated Frame Systems
In this paper we establish the equivalence of solutions between Schrödinger map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.
math.AP↗
Geometry and a priori estimates for free boundary problems of the Euler's equation
In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to zero, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension.
math.AP↗
A priori estimates for fluid Interface problems
We consider the regularity of an interface between two incompressible and inviscid fluids flows in the presence of surface tension. We obtain local in time estimates on the interface in $H^{\frac32k +1}$ and the velocity fields in $H^{\frac32k}$. These estimates are obtained using geometric considerations which show that the Kelvin-Helmholtz instabilities are a consequence of a curvature calculation.
math.AP↗