Sharp Kato smoothing properties of weakly dissipated KdV equations with variable coefficients on a periodic domain

Abstract

<p style='text-indent:20px;'>It is well known that the solutions of the Cauchy problem of the Korteweg-de Vries (KdV) equation on a periodic domain <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb{T}} $\end{document}</tex-math></inline-formula>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t +uu_x +u_{xxx} = 0, \quad u(x,0) = \phi (x), \quad x\in {\mathbb{T}}, \ t\in {\mathbb{R}}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>possess neither the sharp Kato smoothing property,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies \partial ^{s+1}_xu \in L^{\infty}_x ({\mathbb{T}}, L^2 (0,T)), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>nor the Kato smoothing property,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies u\in L^2 (0,T; H^{s+1} ({\mathbb{T}})). $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>This paper shows that the solutions of the Cauchy problem of following weakly dissipated KdV equation with variable coefficients posed on a periodic domain <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{T}} $\end{document}</tex-math></inline-formula>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ u_t +uu_x +a(x,t) u_{xxx} - (g(x,t)u_{x})_x = 0, \qquad u(x,0) = \phi (x), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ g $\end{document}</tex-math></inline-formula> are given real-valued smooth functions periodic in <inline-formula><tex-math id="M5">\begin{document}$ x $\end{document}</tex-math></inline-formula> satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE5"> \begin{document}$ a(x,t)\ne 0, \quad x\in {\mathbb{T}}, \ t\geq 0 \,\quad \mbox{and} \quad \int_{\mathbb{T}}\frac{g(x,t)}{|a(x,t)|} dx > 0 \quad \forall t\geq 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>possess the sharp Kato smoothing property,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE6"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies \partial ^{s+1}_xu \in L^{\infty}_x ({\mathbb{T}}, L^2 (0,T)), \quad \forall \, s\geq 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the nonlinear part of its solution <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> possesses <i>the strong Kato smoothing property</i>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE7"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies (u -v)\in C([0,T]; H^{s+1} ({\mathbb{T}})), \quad \forall \, s>\frac12, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the <i>sharp double Kato smoothing property</i>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE8"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies \partial ^{s+2}_x(u -v)\in L^{\infty}_x ({\mathbb{T}}, L^2 (0,T)), \quad \forall \, s>\frac12, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with <inline-formula><tex-math id="M7">\begin{document}$ v $\end{document}</tex-math></inline-formula> being the solution of the linear problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE9"> \begin{document}$ v_t+ a(x,t)v_{xxx} -(g(x,t)v_{x} )_x = 0, \quad v(x,0) = \phi (x), \quad x\in {\mathbb{T}}, \ t>0. $\end{document} </tex-math></disp-formula></p>