Local wellposedness and global regularity results for biharmonic wave maps

Tobias Schmid
This thesis is concerned with biharmonic wave maps, i.e. a bi-harmonic version of the wave maps equation, which is a Hamiltonian equation for a higher order energy functional and arises variationally from an elastic action functional for a manifold valued map.$\\[1pt]$ In the first part we present local and global results from energy estimates for biharmonic wave maps into compact, embedded target manifolds. This includes local wellposedness in high regularity and global regularity in
more » ... l dimension $n = 1, 2$. The results rely on the use of careful a priori energy estimates, compactness arguments in weak topologies and sharp Sobolev embeddings combined with energy conservation in the proof of global regularity.$\\[1pt]$ In part two, we extend these results to global regularity in dimension $ n \geq 3$ for biharmonic wave maps into spheres and initial data of small size in a scale invariant Besov norm. This follows from a small data global wellposedness and persistence of regularity result for more general systems of biharmonic wave equations with non-generic nonlinearity. In contrast to part one, the arguments in part two of the thesis rely on the analysis of bilinear frequency interactions based on Fourier restriction methods and Strichartz estimates.$\\[1pt]$ The results in both parts of the thesis fundamentally depend on the non-generic form of the nonlinearity that is introduced by our biharmonic model problem.
doi:10.5445/ir/1000128147 fatcat:uoygvu3jejettm2jee4iewsboe