Research Problems in Harmonic Analysis and Partial Differential Equations

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The investigator plans to investigate the extremisers problem for the Tomas-Stein inequality for the sphere, to establish a Strichartz estimate for an oscillatory integral with a non-elliptic phase, and to investigate the Cauchy problem for the two dimensional water wave equations with surface tension. The first two projects aim to understand some aspects of oscillatory integrals, which are important in the restriction/extension theory of Fourier transforms to hyper-spaces with curvature in the Euclidean spaces. The extremisers problem for the Tomas-Stein inequality asks whether there exists a function which optimizes the inequality so that it becomes an equality. It also includes questions of characterizing extremisers such as establishing the smoothness property. The first project concerns the extremisers problem for the Tomas-Stein inequality for the one and higher dimensional spheres. The second project focuses on studying an oscillatory integral with a non-elliptic phase, which is in form of Strichartz estimates. It is motivated by the applications of these estimates in the Schrodinger equations. The third project is to investigate the Cauchy problem for the two dimensional water wave equations with surface tension. It is well known that an important step in approaching this problem in previous works is a reduction of the original system of equations to a suitable and equivalent quasilinear system. So it remains an interesting question how such a reduction will play a role in developing its wellposedness theory.The proposed research will result in broader impact from several points of view. First, these problems under investigation lie at the interface of several branches of mathematics, e.g. analysis and partial differential equations. Thus their solutions will facilitate interactions among these fields. Moreover, these problems concern both the development of mathematical techniques and the human understanding of the fundamental concepts in mathematics. For instance, the Tomas-Stein inequality is an important measure of properties of a basic operation in mathematics, the Fourier transform, and goes back to a fundamental question: when is an infinite series (in this case the Fourier series of a given function) summable? Examples of ways in which these problems impact applied science and engineering are many and varied. For instance, the water wave equation is used as a model to describe the propagation of surface waves on a river or the ocean. A rigorous study of this model will provide the theoretical ground for modeling and computational simulations, which in turn allow researchers to gain some insight into some destructive physical phenomena such as the rogue waves and tsunamis. Finally, sharing the research experience through the investigator's teaching activities will increase the awareness and appreciation of mathematics research, and improve diversity of the scientific community and the society.