On the Division Problem for the Wave Maps Equation

Abstract

We consider Wave Maps into the sphere and give a new proof of small data global well-posedness and scattering in the critical Besov space, in any space dimension $n\ge 2$. We use an adapted version of the atomic space $U^2$ as the single building block for the iteration space. Our approach to the so-called division problem is modular as it systematically uses two ingredients: atomic bilinear (adjoint) Fourier restriction estimates and an algebra property of the iteration space, both of which can be adapted to other phase functions.

arXiv commentSection 7 contains a proof of a special case of the bilinear estimate obtained in 1707.08944

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• Global wellposedness for the energy-critical Zakharov system below the ground state
• Optimal blowup stability for three-dimensional wave maps
• Global Well-posedness for Incompressible Hookean Elastodynamics in the Critical Besov Spaces
• Wave maps on (1+2)-dimensional curved spacetimes
• Global results for a Cauchy problem related to biharmonic wave maps