Related papers: Global well-posedness for the fifth-order KdV equa…

We prove global well-posedness of the Korteweg--de Vries equation for initial data in the space $H^{-1}(R)$. This is sharp in the class of $H^{s}(R)$ spaces. Even local well-posedness was previously unknown for $s<-3/4$. The proof is based…

Given a suitable solution $V(t,x)$ to the Korteweg--de Vries equation on the real line, we prove global well-posedness for initial data $u(0,x) \in V(0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose…

We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(\R)$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(\R)$. The new ingredient is that we use directly…

We utilize a modulation restricted normal form approach to establish local well-posedness of the periodic Korteweg-de Vries equation in $H^s(\mathbb{T})$ for $s> -\frac23$. This work creates an analogue of the mKdV result by Nakanishi,…

This paper is a continuation of the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as…

In this paper, we establish the well-posedness for the Cauchy problem of the fifth order KdV equation with low regularity data. The nonlinear term has more derivatives than can be recovered by the smoothing effect, which implies that the…

We propose a new formulation of the Korteweg-de Vries equation (KdV) on the real line, via a gauge transform. While KdV and the gauged equation are equivalent for smooth solutions, the latter is better behaved at low regularity in…

We establish a smoothing result for the generalized KdV (gKdV) on the torus with polynomial non-linearity, damping, and forcing that matches the smoothing level for the gKdV at $H^1$. As a consequence, we establish the existence of a global…

We prove the local well-posedness for the generalized Korteweg-de Vries equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the nonlinearity $f(x)$, on the background of an $L^\infty_{t,x}$-function $\Psi(t,x)$, with…

We prove well-posedness in $L^2$-based Sobolev spaces $H^s$ at high regularity for a class of nonlinear higher-order dispersive equations generalizing the KdV hierarchy both on the line and on the torus.

In this paper, we prove the global well-posedness of the energy-critical nonlinear Schr\"odinger equations on the torus $\mathbb{T}^{d}$ for general dimensions. This result is new for dimensions $d\ge5$, extending previous results for…

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u u_{x} = 0 x,t \in \R We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and…

We prove that the cubic nonlinear Schr\"odinger equation (both focusing and defocusing) is globally well-posed in $H^s(\mathbb R)$ for any regularity $s>-\frac12$. Well-posedness has long been known for $s\geq 0$, see [55], but not…

In this paper, we consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in $H^s(\mathbb T)$, $s > 2$, via the energy method. The main tool is the…

In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $ is locally well-posed in $ H^s(\mathbb{R}) $ for $ s> 5/2. Also, we…

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.

We prove that the Kawahara equation is locally well-posed in $H^{-7/4}$ by using the ideas of $\bar{F}^s$-type space \cite{GuoKdV}. Next we show it is globally well-posed in $H^s$ for $s\geq -7/4$ by using the ideas of "I-method"…

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in $H^s(\mathbb{T})$, $s>2$. A dissipative term is incorporated into the control which, along with a…

In this paper, we prove that the periodic higher-order KdV-type equation \[\left\{\begin{array}{ll} \partial_t u + (-1)^{j+1} \partial_x^{2j+1}u + \frac12 \partial_x(u^2)=0, \hspace{1em} &(t,x) \in \mathbb{R} \times \mathbb{T}, \\ u(0,x) =…

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based…