A bilinear Airy-estimate with application to gKdV-3

Abstract

The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem \[u_t + u_{xxx} + (u^4)_x=0,\hspace{1cm}u(0)=u_0 \in H^s_x({\bf R}), \,\,\,s>-\tfrac{1}{6}\] for the generalized Korteweg-deVries equation of order three, for short gKdV-3. For real-valued data $u_0 \in L^2_x({\bf R})$ global wellposedness follows by the conservation of the $L^2$ norm. The main new tool is a bilinear estimate for solutions of the Airy-equation.