# Biharmonic wave maps: Local wellposedness in high regularity

## TL;DR

This paper establishes local well-posedness for biharmonic wave maps with high regularity initial data, using a vanishing viscosity approach and geometric analysis to ensure existence, uniqueness, and stability of solutions.

## Contribution

It introduces a novel approach employing parabolic regularization and viscosity methods for biharmonic wave maps, extending well-posedness results to higher regularity settings.

## Key findings

- Proved local well-posedness for biharmonic wave maps with high Sobolev regularity.
- Developed a vanishing viscosity and parabolic regularization technique for existence.
- Established continuous dependence and uniqueness of solutions.

## Abstract

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.