On the well-posedness of the periodic KdV equation in high regularity classes

2008 ◽  
pp. 431-441
Author(s):  
Thomas Kappeler ◽  
Jürgen Pöschel
Keyword(s):  
Kdv Equation ◽  
Well Posedness ◽  
High Regularity

Global Well-Posedness for the Fifth-Order KdV Equation in $$H^{-1}(\pmb {\mathbb {R}})$$

Annals of PDE ◽  
2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Bjoern Bringmann ◽  
Rowan Killip ◽  
Monica Visan
Keyword(s):  
Kdv Equation ◽  
Well Posedness ◽  
Fifth Order

scholarly journals Local well-posedness for the dispersion generalized periodic KdV equation

2011 ◽  
Vol 379 (2) ◽  
pp. 706-718 ◽  
Author(s):  
Junfeng Li ◽  
Shaoguang Shi
Keyword(s):  
Kdv Equation ◽  
Well Posedness

scholarly journals Invariant measure of stochastic higher order KdV equation driven by Poisson processes

2021 ◽  
Author(s):  
Pengfei Xu ◽  
Jianhua Huang ◽  
Wei Yan
Keyword(s):  
Invariant Measure ◽  
Poisson Process ◽  
Initial Conditions ◽  
Kdv Equation ◽  
Higher Order ◽  
Poisson Processes ◽  
Current Paper ◽  
Well Posedness ◽  
Kdv Equations ◽  
Unique Invariant Measure

The current paper is devoted to stochastic damped KdV equations of higher order driven by Poisson process. We establish the well-posedness of stochastic damped higher-order KdV equations, and prove that there exists an unique invariant measure for deterministic initial conditions. Some discussion on the general pure jump noise case are also provided.

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

2018 ◽  
Vol 15 (03) ◽  
pp. 463-557 ◽  
Author(s):  
Chulkwang Kwak
Keyword(s):  
Kdv Equation ◽  
Main Tool ◽  
Well Posedness ◽  
Current Form ◽  
Low Regularity ◽  
Integrable Structure ◽  
Fourier Restriction ◽  
Fourier Restriction Norm Method ◽  
Modified Kdv Equation ◽  
Fifth Order

We consider the fifth-order modified Korteweg–de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in [Formula: see text], [Formula: see text], via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173(2) (2008) 265–304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation.

Exponential stabilization of an ODE–linear KdV cascaded system with boundary input delay

2020 ◽  
Vol 37 (4) ◽  
pp. 1506-1523
Author(s):  
Habib Ayadi
Keyword(s):  
Differential Equation ◽  
Kdv Equation ◽  
Semigroup Theory ◽  
Functional Space ◽  
Input Delay ◽  
Exponential Stabilization ◽  
Target System ◽  
Control Input ◽  
Well Posedness ◽  
Bounded Interval

Abstract This paper considers the well posedness and the exponential stabilization problems of a cascaded ordinary differential equation (ODE)–partial differential equation (PDE) system. The considered system is governed by a linear ODE and the one-dimensional linear Korteweg–de Vries (KdV) equation posed on a bounded interval. For the whole system, a control input delay acts on the left boundary of the KdV domain by Dirichlet condition. Whereas, the KdV acts back on the ODE by Dirichlet interconnection on the right boundary. Firstly, we reformulate the system in question as an undelayed ODE-coupled KdV-transport system. Secondly, we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a well-posed and exponentially stable target system. Finally, by invertibility of such design, we use semigroup theory and Lyapunov analysis to prove the well posedness and the exponential stabilization in a suitable functional space of the original plant, respectively.

scholarly journals Well-posedness for the fifth-order KdV equation in the energy space

2014 ◽  
Vol 367 (4) ◽  
pp. 2551-2612 ◽  
Author(s):  
Carlos E. Kenig ◽  
Didier Pilod
Keyword(s):  
Kdv Equation ◽  
Energy Space ◽  
Well Posedness ◽  
Fifth Order
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