Control and stabilization of the periodic fifth order Korteweg-de Vries equation

Flores, Cynthia; Smith, Derek L.

doi:10.1051/cocv/2018033

Cited by 3 publications

(2 citation statements)

References 51 publications

(56 reference statements)

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“…One of the approaches to deal with this situation is to introduce some local dissipation mechanism through feedback control acting on a small part of the spatial domain and then show the gained local regularity can propagate to the whole spatial region so that the resulted closed loop systems possess the needed smoothing properties. The interested readers are referred to the recent works of Linares and Rosier [23] for control and stabilization of the Benjamin-Ono equation on a periodic domain, and Flores and Smith [12] for control and stabilization of the fifth order KdV equations on a periodic domain, as well as the references therein.…”

Section: Introductionmentioning

confidence: 99%

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

et al. 2021

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The solutions of the Cauchy problem of the KdV equation on a periodic domain $\T$, \[ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in \T, \ t\in \R,\] possess neither the sharp Kato smoothing property, \[ \phi \in H^s (\T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (\T, L^2 (0,T)),\] nor the Kato smoothing property, \[ \phi \in H^s (\T) \implies u\in L^2 (0,T; H^{s+1} (\T)).\] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain $\T$, \[ u_t +uu_x +u_{xxx} - g(x) (g(x) u)_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in \T, \ t>0 \, ,\ \qquad (1) \] where $g\in C^{\infty} (\T)$ is a real value function with the support \[ \mbox{$\omega = \{ x\in \T, \ g(x) \ne 0\}$.}\] It is shown that \begin{itemize} \item[(1)] if $\omega\ne \emptyset$, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; \item[(2)] if $g$ is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property. \end{itemize}

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Section: Introductionmentioning

confidence: 99%

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

et al. 2021

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“…This kind of phenomena are usually called as propagation of regularity in classical analysis: the s + 1 regularity of u has spread from the small open subset ω to the whole spatial domain T. It plays important role in the study of control and stabilization of distributed parameter systems described by some partial differential equations (cf. [25,14] and the references therein).…”

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

Sun¹,

Yang²,

Zhang³

et al. 2022

DCDS

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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}} $\end{document}</tex-math></inline-formula>,<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t +uu_x +u_{xxx} = 0, \quad u(x,0) = \phi (x), \quad x\in {\mathbb{T}}, \ t\in {\mathbb{R}}, $\end{document} </tex-math></disp-formula>possess neither the sharp Kato smoothing property,<disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies \partial ^{s+1}_xu \in L^{\infty}_x ({\mathbb{T}}, L^2 (0,T)), $\end{document} </tex-math></disp-formula>nor the Kato smoothing property,<disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies u\in L^2 (0,T; H^{s+1} ({\mathbb{T}})). $\end{document} </tex-math></disp-formula>This paper shows that the solutions of the Cauchy problem of following weakly dissipated KdV equation with variable coefficients posed on a periodic domain <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{T}} $\end{document}</tex-math></inline-formula>,<disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ u_t +uu_x +a(x,t) u_{xxx} - (g(x,t)u_{x})_x = 0, \qquad u(x,0) = \phi (x), $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M3">\begin{document}$ 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 $\end{document}</tex-math></inline-formula> satisfying<disp-formula> <label/> <tex-math id="FE5"> \begin{document}$ 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, $\end{document} </tex-math></disp-formula>possess the sharp Kato smoothing property,<disp-formula> <label/> <tex-math id="FE6"> \begin{document}$ \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, $\end{document} </tex-math></disp-formula>and the nonlinear part of its solution <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> possesses the strong Kato smoothing property,<disp-formula> <label/> <tex-math id="FE7"> \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies (u -v)\in C([0,T]; H^{s+1} ({\mathbb{T}})), \quad \forall \, s>\frac12, $\end{document} </tex-math></disp-formula>and the sharp double Kato smoothing property,<disp-formula> <label/> <tex-math id="FE8"> \begin{document}$ \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, $\end{document} </tex-math></disp-formula>with <inline-formula><tex-math id="M7">\begin{document}$ v $\end{document}</tex-math></inline-formula> being the solution of the linear problem<disp-formula> <label/> <tex-math id="FE9"> \begin{document}$ 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>

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Local controllability and stability of the periodic fifth-order KdV equation with a nonlinear dispersive term

Zhou

2021

Journal of Mathematical Analysis and Applications

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Control and stabilization of the periodic fifth order Korteweg-de Vries equation

Cited by 3 publications

References 51 publications

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

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

Local controllability and stability of the periodic fifth-order KdV equation with a nonlinear dispersive term

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