Well-posedness of stochastic modified Kawahara equation
Praveen Agarwal, Abd‐Allah Hyder, Mohammed Zakarya
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 $H^{s}(\mathbb{R})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>)</mml:mo></mml:math> , $s\geq -1/4$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>s</mml:mi><mml:mo>≥</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>4</mml:mn></mml:math> . Moreover, we get the global existence for $L^{2}( \mathbb{R})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>)</mml:mo></mml:math> solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.