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Publication# Well-posedness Issues For the Half-Wave Maps Equation With Hyperbolic And Spherical Targets

Abstract

This thesis is a study of the global well-posedness of the Cauchy problems for half-wave maps from the Minkowski space of dimension n+1 to the 2-dimensional sphere and the hyperbolic plane. The work is mainly based on the results from Krieger-Sire 17' in the energy-supercritical case of n>=5, and the improved result from Kiesenhofer-Krieger 19' of n>=4 for sphere target with the small initial Besov normed data. The first result obtained by the authors is to extend the well-posedness of the sphere target to the hyperbolic plane with small initial Besov normed data in higher dimension n>=4. The work utilizes the intrinsic distance of the hyperbolic plane to maintain the geometric structure of the half-wave map. For future works, the authors would improve the initial data condition from the Besov space to the critical Sobolev space in higher dimension n>=4 for both the spherical and hyperbolic targets. The authors would reference the Hélein's moving frame techniques and the gauge construction for wave maps as in Tao 01' and Shatah-Struwe 02' to address the problem. Moreover, the authors would construct the weaker solution for the half-wave maps in the lower dimensional case when n=1,2. The lower dimension case requires the authors to build new tools since the Strichartz estimate used in the higher dimension case no longer available.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.

Four-dimensional space

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled x, y, and z).

Hausdorff dimension

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension.

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