Local Smoothing Effects for the Water-wave Problem with Surface Tension Hans Christianson, Vera Mikyoung Hur, and Gigliola Staffilani

Dispersive characters are studied for waves on the one-dimensional free surface of an infinitely deep perfect fluid under the influence of gravity and surface tension. The hydrodynamic problem for surface water-waves is discussed with emphasis on the effects of surface tension. A new formulation is developed as a second-order in time quasilinear dispersive equation for a dynamic variable defined on the free surface. The main result states that on average in time its solutions acquire locally 1/4 derivative of smoothness as compared to the initial state. The smoothing effect for the linear part of the equation turns out to be too weak to control its severe nonlinearities, suggesting that nonlinear aspects of the equation must be taken into account for the analysis of local smoothing effects due to the dispersive properties. Terms with the highest-order derivatives in the nonlinear part of the equation are to be viewed as “linear components” of the equation with variable coefficients which depend on the solution itself. Local smoothing effects are investigated for the corresponding linear dispersive equation for a general class of variable coefficients. Based on the construction of a high-frequency parametrix, the analysis combines energy methods with techniques of pseudodifferential operators and Fourier integral operators. The smoothing effect for the nonlinear equation is established with the help of the a priori estimates provided by the quasilinear energy methods. In the course of the proof, local well-posedness is achieved for the nonlinear equation. ∗ R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, AddisonWesley, 1963, section 51-4. 1 2 HANS CHRISTIANSON, VERA HUR, AND GIGLIOLA STAFFILANI