Bilinear Strichartz estimates for the Schr{ö}dinger map problem

Abstract

In this paper we prove bilinear Strichartz estimates for a solution to the Schr{ö}dinger map problem whose size is small in the critical Strichartz space $||\nabla {|}^{\frac{d-2}{2}}{\psi }_x{|}_{L_{t,x}^{\frac{2(d+2)}{d}}}$. These estimates will be useful in an upcoming paper in proving a local well - posedness result. Bilinear estimates make use of an argument similar to the argument found in Planchon and Vega (2009). We use the same gauges as in Bejenaru, Ionescu and Kenig (2007), Bejenaru et al. (2011), and Smith.

34 pages

References in corpus (8)

• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• Global regularity of wave maps VII. Control of delocalised or dispersed solutions
• Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions
• Global regularity of wave maps III. Large energy from ${\mathbb{R}}^{1+2}$ to hyperbolic spaces
• Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class
• Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^2$
• Geometric renormalization of large energy wave maps
• Global well-posedness for 2D radial Schrödinger maps into the sphere