This thesis is composed of three interrelated parts: w-Calderón-Zygmund operators, a wavelet characterization for weighted Hardy spaces, and the analytic dependence of minimal surfaces on their boundaries.After the award of her doctorate, Wu was appointed as Courant Instructor at the Courant Institute, New York University. She was a member at the Institute for Advanced Study at Princeton in the autumn of 1992 and was then she was appointed Assistant Professor at Northwestern University, holding this position for four years until 1996. Her publications during this period included: A wavelet characterization for weighted Hardy spaces (1992); (with Italo Vecchi) On L1-vorticity for 2-D incompressible flow (1993); Analytic dependence of Riemann mappings for bounded domains and minimal surfaces (1993) and w-Calderón-Zygmund operators (1995). After spending the year 1996-97 as a member of the Institute for Advanced Study at Princeton, she was appointed as Assistant Professor at the University of Iowa. In 1997 she published the important paper Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Shu Ming Sun begins a very informative review as follows:-
Everyone is familiar with the motion of water waves in everyday experience, and there has been an extremely rich variety of phenomena observed in the motion of such waves. However, the full equations governing the motion of the waves are notoriously difficult to work with because of the free boundary and the inherent nonlinearity, which are non-standard and non-local. Although many approximate treatments, such as linear theory and shallow-water theory as well as numerical computations, have been used to explain many important phenomena, it is certainly of importance to study the solutions of the equations which include the effects neglected by approximate models. The well-posedness of the fully nonlinear problem is one of the main mathematical problems in fluid dynamics. Here, the motion of two-dimensional irrotational, incompressible, inviscid water waves under the influence of gravity is considered.Promoted to Associate Professor at Iowa in 1998, Wu was appointed as an Associate Professor at the University of Maryland, College Park, in 1998. The university announced her appointment as follows:-
Sijue Wu comes to us from the University of Iowa. Her research interests centre on harmonic analysis and partial differential equations, in particular nonlinear equations from fluid mechanics. Her recent work concerns the full nonlinear water wave problem and the motion of general two-fluid flows.At the 107th Annual Meeting of the American Mathematical Society in January 2001 in New Orleans, Wu was awarded the 2001 Satter Prize. The citation reads :-
The Ruth Lyttle Satter Prize in Mathematics is awarded to Sijue Wu for her work on a long-standing problem in the water wave equation, in particular for the results in her papers (1) "Well-posedness in Sovolev spaces of the full water wave problem in 2-D" (1997); and (2) "Well-posedness in Sobolev spaces of the full water wave problem in 3-D" (1999). By applying tools from harmonic analysis (singular integrals and Clifford algebra), she proves that the Taylor sign condition always holds and that there exists a unique solution to the water wave equations for a finite time interval when the initial wave profile is a Jordan surface.Of the paper (2) Emmanuel Grenier writes:-
In this very important paper the author investigates the motion of the interface of a 3D inviscid, incompressible, irrotational water wave, with an air region above a water region and surface tension zero.In her response Wu thanked her teachers, friends, and colleagues, making special mention of her thesis advisor Ronald Coifman for the constant support he had given her and Lihe Wang for his friendship and his help.
Also in 2001 Wu received a Silver Morningside Medal at the International Congress of Chinese Mathematicians held in Taiwan in December:-
... for her establishment of local well-posedness of the water wave problems in a Sobolev class in arbitrary space dimensions.In August 2002 Wu was an invited speaker at the International Congress of Mathematicians held in Beijing where she delivered the lecture Recent progress in mathematical analysis of vortex sheets. She gave the following summary of her lecture:-
We consider the motion of the interface separating two domains of the same fluid that move with different velocities along the tangential direction of the interface. We assume that the fluids occupying the two domains are of constant densities that are equal, are inviscid, incompressible and irrotational, and that the surface tension is zero. We discuss results on the existence and uniqueness of solutions for given data, the regularity of solutions, singularity formation and the nature of the solutions after the singularity formation time.Wu was awarded a Radcliffe Institute Advanced Study Fellowship for the academic year 2002-2003. Her project Mathematical Analysis of Vortex Dynamics was described in an announcement of the award :-
Recently, Wu's research has focused on nonlinear equations from fluid dynamics. Using harmonic analysis technique, she has established the local well-posedness of the full two- and three-dimensional waterwave problem. This settled a longstanding problem. As a Radcliffe fellow, Wu will continue her study of vortex sheet dynamics, a phenomenon that arises from the mixing of fluids, such as occurs during aircraft takeoffs. A vortex sheet is the interface separating two domains of the same fluid across which the tangential component of the velocity field is discontinuous. Achieving a better understanding of the motion of a vortex sheet requires proper mathematical modelling; Wu's long-term goal is to establish a successful model. She will also work on the boundary layer problem, another problem arising from fluid dynamics.One outcome of this project, and of an NFS grant she was awarded for 2004-2009, was the paper Mathematical analysis of vortex sheets (2006). Helena Nussenzveig Lopes begins a review of this paper by explaining what vortex sheets are:-
Vortex sheets are an idealized model of flows undergoing intense shear. In planar flows they are mathematically described as curves along which the velocity is tangentially discontinuous. Vortex sheets arise in a wide range of physical problems, and hence it is of fundamental importance to understand their evolution. The Birkhoff-Rott equations provide a mathematical description of the evolution of a vortex sheet. However, they have been shown to be ill-posed in several function spaces. It is a longstanding open problem to determine a function space in which these equations are well-posed, or, alternatively, to describe the evolution past singularity formation; this is the problem addressed in the present paper.Wu was named Robert W and Lynne H Browne Professor of Mathematics at the University of Michigan and delivered her inaugural lecture Mathematical Analysis of Water Waves on 29 October 2008. The Browne Professorship recognizes the Wu's outstanding contributions to science and teaching.
Finally, let us mention her recent important paper Almost global wellposedness of the 2-D full water wave problem (2009).
Article by: J J O'Connor and E F Robertson