Global well-posedness for the KdV equations on the real line with low regularity forcing terms

We consider the initial value problem for the KdV equations with low regularity forcing terms. The case that the forcing term f(x) equals p delta'(x) appears in the study of the excitation of long nonlinear water waves by a moving pressure distribution, where delta'(x) is the first derivative of the Dirac delta function and p is a constant. We have the time global well-posedness with f (x) is an element of L-2 by the L-2 a priori estimate. However, we cannot apply it to the case f (x) is an element of H-sigma, sigma < 0. To overcome this difficulty, we divide f into the high frequency part and the low frequency part and use the scaling argument. Our results include the time local well-posedness with f (x) is an element of H-sigma, sigma >= -3 and the time global well-posedness with f = p delta'(x) or f (x) is an element of H-sigma, sigma >= -3/2. Our main tools are the Fourier restriction norm method and the I-method.