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The Cauchy problem for a fourth order version of the wave map equation.

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Harmonic maps are smooth maps (between manifolds) with vanishing tension field. In case the domain is endowed with a Lorentzian metric this leads to a semilinear wave equation, the so called wave map equation. We consider a fourth order version of the wave map equation and talk about aspects of the Cauchy problem, including existence of local solutions following from energy estimates with a fourth order viscosity method, global extension of the solutions in energy subcritical dimension d < 4 and construction of solutions in low regularity. Afterwards we highlight some differences in the techniques that apply to other geometric equations such as the wave map equation and the Schrödinger map flow.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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