Well-posedness and ill-posedness for a fifth-order shallow water wave equation

作者：Chen W^*; Liu Z

来源：Nonlinear Analysis Theory Methods and Applications, 2010, 72(5): 2412-2420.

DOI：10.1016/j.na.2009.11.003

• [k, Z] multiplier
• Bourgain space
• Fourier restriction norm method
• Ill-posedness
• Local well-posedness

In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [k ; Z] multiplier norm method of Tao (2001) [2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in Hs (R) with s - frac(5, 4) is obtained by the Fourier restriction norm method. And some ill-posedness in Hs (R) with s - frac(5, 4) is derived from a general principle of Bejenaru and Tao.