Analysis of Surface Water Waves
Surface water waves encompass a wide range of phenomena, ranging in length scale from ripples driven by surface tension to rogue waves and tsunamis. The phrase describes the situation where water lies below a body of air and is acted upon by gravity and possible surface tension.
While water waves have stimulated a considerable part of historical developments in the theory of wave motion, they present profound and subtle difficulties for rigorous analysis, modeling, and numerical simulations. Notably, the interface between the water and the air is a free boundary, a priori unknown and to be determined as part of the solution. Free boundaries are mathematically challenging in their own right. In addition, boundary conditions at the free surface are severely nonlinear, presenting further challenges.
During her Center appointment, Professor Hur will address several issues in the mathematical aspects of surface water waves. She plans to develop new tools in partial differential equations and other branches of mathematics, and also extend and combine existing tools, to focus on:
• Global regularity versus finite-time singularities for the initial value problem.
• Existence of traveling waves and their classification.
• Stability and instability of traveling waves.
Her project emphasizes large-scale dynamics and genuinely nonlinear behaviors, such as breaking and peaking, which ultimately rely on analytical proofs for an acute understanding.
Progress in Professor Hur’s research is expected to help resolve several longstanding open problems in the area, while also leading to applications in related, interfacial fluids problems and in numerical simulations and engineering.