Well-posedness of stochastic modified Kawahara equation

Abstract

Abstract In this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in Hs(R) $H^{s}(\mathbb{R})$, s≥−1/4 $s\geq -1/4$. Moreover, we get the global existence for L2(R) $L^{2}( \mathbb{R})$ solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.

Keywords

• Modified Kawahara equation
• Well-posedness
• Wiener process
• Fixed point theorem
• Fourier restriction method