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On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.jde.2020.06.027
Argenis J. Mendez

We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV) \begin{equation}\label{abstracteq1} \left\{ \begin{array}{ll} \partial_{t}u-D_{x}^{\alpha}\partial_{x}u+u\partial_{x}u=0, & x,t\in\mathbb{R},\,0<\alpha<1, \\ u(x,0)=u_{0}(x).& \\ \end{array} \right. \end{equation} It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This property was found originally in solutions of Korteweg-de Vries (KdV) equation and it has been also verified in other dispersive equations as the Benjamin-Ono (BO) equation. Recently, it has been shown that the solutions of the dispersive generalized Benjamin-Ono (DGBO) equation, this is $\alpha\in (2,3)$ in \eqref{abstracteq1}; also satisfy the propagation of regularity phenomena. This is achieved by introducing a commutator decomposition to handle the dispersive part in the equation. Following the approach used in the DGBO, we prove that the solutions of the fKdV also satisfies the propagation of regularity phenomena. Consequently, this type of regularity travels with infinite speed to its left as time evolves.

中文翻译:

关于分数阶 Korteweg-de Vries 方程解的正则性传播

我们考虑分数 Korteweg-de Vries 方程 (fKdV) 的初值问题 (IVP) \begin{equation}\label{abstracteq1} \left\{ \begin{array}{ll} \partial_{t}u-D_ {x}^{\alpha}\partial_{x}u+u\partial_{x}u=0, & x,t\in\mathbb{R},\,0<\alpha<1, \\ u( x,0)=u_{0}(x).& \\ \end{array} \right. \end{equation} 已经证明某些色散方程的解满足规律性现象的传播。更准确地说,它涉及确定实线右侧的初始数据的规律性是否通过流解传播到左侧。这种性质最初是在 Korteweg-de Vries (KdV) 方程的解中发现的,并且在其他色散方程中也得到了验证,如 Benjamin-Ono (BO) 方程。最近,已经证明色散广义 Benjamin-Ono (DGBO) 方程的解,这是 \eqref{abstracteq1} 中的 $\alpha\in (2,3)$;也满足规律性现象的传播。这是通过引入换向器分解来处理方程中的色散部分来实现的。遵循 DGBO 中使用的方法,我们证明 fKdV 的解也满足规律性现象的传播。因此,随着时间的推移,这种规律性以无限的速度向左传播。我们证明了 fKdV 的解也满足规律性现象的传播。因此,随着时间的推移,这种规律性以无限的速度向左传播。我们证明了 fKdV 的解也满足规律性现象的传播。因此,随着时间的推移,这种规律性以无限的速度向左传播。
更新日期:2020-11-01
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