The Construction of a Partially Regular Solution to the Landau-Lifshitz-Gilbert Equation in $\mathbb{R}^2$

Details The Construction of a Partially Regular Solution to the Landau-Lifshitz-Gilbert Equation in $\mathbb{R}^2$ The Construction of a Partially Regular Solution to the Landau-Lifshitz-Gilbert Equation in $\mathbb{R}^2$

2004-12-30

arxiv.org/abs/math/0412534v1

Mathematics

Analysis of PDEs

43 pages, 2 figures

Scientific paper

10.1088/0951-7715/18/6/014

We establish a framework to construct a global solution in the space of finite energy to a general form of the Landau-Lifshitz-Gilbert equation in $\mathbb{R}^2$. Our characterization yields a partially regular solution, smooth away from a 2-dimensional locally finite Hausdorff measure set. This construction relies on approximation by discretization, using the special geometry to express an equivalent system whose highest order terms are linear and the translation of the machinery of linear estimates on the fundamental solution from the continuous setting into the discrete setting. This method is quite general and accommodates more general geometries involving targets that are compact smooth hypersurfaces.

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