Spatial decay and nonlinear smoothing of the generalized Ostrovsky equation

This paper is devoted to studying the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-\beta\partial_{x}^{3}u-\gamma\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $\beta<0,\gamma>0$. Firstly, by using the density theorem in the mixed Lebesgue spaces, we prove that $X_{s,b}\hookrightarrow C(\mathbb{R};H^{s}(\mathbb{R})) \hookrightarrow C(\mathbb{R};L_{x}^{\infty})$ with $s>1/2,b>1/2.$ Secondly, we present a new proof of the convergence problem of linear Ostrovsky equation, which is slightly different from the proof of Theorem 1.1 (Convergence problem of Ostrovsky equation with rough data and random data, Indiana Univ. Math. J. 71(2022), 1897-1921.) Thirdly, we investigate the pointwise convergence problem of the generalized Ostrovsky equation. Fourthly, for the solution $u$ to the Cauchy problem for the generalized Ostrovsky equation, we prove that $u=u_{1}+u_{2},t\in[-\delta,\delta]$, and $u_{2}$ possesses better regularity than $u$, where $u_{1}$ is the linear part of $u$ and $u_{2}$ is the nonlinear integral part. Fifthly, we investigate the nonlinear smoothing and the uniform convergence problem of the generalized Ostrovsky equation. Finally, when data $f$ belongs to $H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k+1},k\geq6)$ and $\lim\limits_{|x|\rightarrow{\infty}}f=0$ and $\mathscr{F}_{x}(U(t)f)\in L^{1}(\mathbb{R}),$ for $t\in [-\delta,\delta],$ we prove that $\lim\limits_{|x|\rightarrow{\infty}}u=0$. The key ingredients are high-low frequency technique, maximal function estimates related to low frequency and some Strichartz estimates which can be proved with the aid of the Stein complex interpolation Theorem.