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

We study the dispersive character for waves on the one-dimensional free surface of an infinitely deep perfect fluid under the influence of surface tension. The main result state that, on average in time, the solution of the water-wave problem gains locally 1/4 derivative of smoothness in the spatial variable, compared to the initial state. The regularizing effect is a direct consequence of dispersion due to surface tension, and it contrasts markedly with consequences of energy estimates. We formulate the problem as a second-order in time nonlinear dispersive equation and establish local well-posedness through an energy method. The main difficult is that the smoothing effect for the linear part of the equation is too weak to control the severe nonlinearity. We view the highest-order derivatives in the noninearity as “linear” components of the equation with variable coefficients which depend on the solution itself. We construct an approximate solution of this linearized equation as an oscillatory integral. Using mapping properties of Fourier integral operators we prove the local smoothing effect. 2000 Mathematics Subject Classification. primary:76B15, secondary:35R35, 35S10. 1 R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, AddisonWesley, 1963, section 51-4. 1 2 CHRISTIANSON, HUR, AND STAFFILANI