本文证明了由KdV族$ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $产生的五阶方程在$ H^s(\mathbb{R}) $中是局部适定的，其中$ s> 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $ s>0$.
英文摘要：
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 prove the solution map of the equation is not uniformly continuous for $s>0$.
We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.
We consider the long-time behavior of solutions to the fifth-order modified KdV-type equation. Using the method of testing by wave packets, we prove the small-data global existence and modified scattering. We derive the leading asymptotic in both the self-similar and oscillatory regions.
We show that the uniform radius of spatial analyticity $σ(t)$ of solution at time $t$ for the fifth order KdV-BBM equation cannot decay faster than $1/t$ for large $t>0$, given initial data that is analytic with fixed radius $σ_0$. This significantly improves a recent result by Carvajal and Panthee, where they established an exponential decay of $σ(t)$ for large $t$.
We show that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the fifth order KdV-BBM equation cannot decay faster than $1/ \sqrt{t}$ for large $t$, given initial data that is analytic with fixed radius $σ_0$. This improves a recent result by Belayneh, Tegegn and the third author, where they obtained a $1/t$ decay of $σ(t)$ for large time $t$.
This paper is a continuation of authors' previous work \cite{CK2018-1}. We extend the argument \cite{CK2018-1} to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $X^{s,b}$ nonlinear estimates for $b < \frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b > \frac12$ \cite{CGL2010, JH2009}. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half…
We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(\mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(\mathbb{R}/\mathbb{Z})$. In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.
Well-posedness and ill-posedness of the fifth order modifed KdV equation
在Sobolev空间上考虑五阶修正KdV方程的初值问题. \partial_t u = \partial_x^5 u + c_1 \partial_x^3（u^3）+\partial_x u + c_3uuPDMATHABMO u =0，u（x，0）= u_0（x）其中$ u:R\timesR \to R $和$c_j$是实数.利用$X^{s,b}$空间上的压缩原理证明了sPDMATHABMT 3/4在H^s（R）中的局部适定性.此外，我们表明，从数据到解决方案的解决方案映射未能一致连续低于$H^{3/4}(R)$。反例是通过用三次NLS方程近似五阶mKdV方程得到的。
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 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 estim…
We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness result by showing that the flow-map is not continuous if the given data has Sobolev regularity $s< 1$. The results obtained in this work are sharp.
Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann--Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified Korteweg-de Vries equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of $I$-method, we give its global well-posedness in lower regularity Sobolev spaces, and then obtain the asymptotic behavior in these spaces with weights.
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 following: \begin{equation*} \begin{cases} \partial_t u - \partial_x^5 u + 30u^2\partial_x u + 20 u\partial_x u \partial_x^3u + 10u \partial_x^3 u = 0, \hspace{1em} (t,x) \in \mathbb{R} \times \mathbb{T}, u(0,x) = u_0(x) \in H^s(\mathbb{T}) \end{cases}. \end{equation*} We prove the local well-posedness of the fifth-order KdV equation for low regularity Sobolev in…
Studied here is the Kawahara equation, a fifth order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients we prove that solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the Kawahara system is exponentially stable for $T> T_{\min}$. After that, introducing a more general delayed system, and by introducing suitable energies, we show using Lyapunov approach, that the energy of the Kawahara equation go…
This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equation $$\partial_tu - \partial_x^5u - 30u^2\partial_xu + 20\partial_xu\partial_x^2u + 10u\partial_x^3u = 0.$$ The main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times.
We prove local in time well-posedness for a class of quasilinear Hamiltonian KdV-type equations with periodic boundary conditions, more precisely we show existence, uniqueness and continuity of the solution map. We improve the previous result in \cite{Mietka}, generalising the considered class of equations and improving the regularity assumption on the initial data.
We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass $\wp$ functions of both third and fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV--BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstra…
In most of the studies concerning nonlinear wave equations of Korteweg-de Vries type, the authors focus on waves of elevation. Such waves have general form ~$u_{\text{u}}(x,t)=A f(x-vt)$, where ~$A>0$. In this communication we show that if ~$u_{\text{up}}(x,t)=A f(x-vt)$ is the solution of a given nonlinear equation, then $u_{\text{down}}(x,t)=-A f(x-vt)$, that is, an inverted wave is the solution of the same equation, but with changed sign of the parameter ~$α$. This property is common for KdV, extended KdV, fifth-order KdV, Gardner equations, and generalizations for cases with an uneven bott…
We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q}(\mathbb{R})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map fails to be $C^3$ continuous in $M^{s}_{2,q}(\mathbb{R})$ when $s<1/4$. It is well-known that $H^{1/4}$ is a critical Sobolev space of mKdV so that it is well-posedness in $H^s$ for $s\geq 1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s'}$ with $s'<1/4$. Noticing that $M^{1/4}_{2,q} \subset B^{1/q-1/4}_{2,q}$ is a sharp embedding and $H^{-1/4}\…
It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Kortewegde Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves and such a system has been studied over the last few decades. However, questions about the validity of the system in the study of water waves were raised in our previous work where we presented our analysis using the fifth…
In this article, we prove that small localized data yield solutions to Kawahara type equation which have linear dispersive decay on a finite time. We use the similar method used to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.
The Korteweg-de Vries equation (KdV) and various generalized, most often semi- linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general Hamil- tonian structure. This paper presents a local in time well-posedness result, that is existence and uniqueness of a solution and its continuity with respect to the initial data. The proof is based on the derivation of energy estimates, the major inter- est being the method used to get them. The goal is to make use of the structural properties of the equa…
In this paper, we consider the Cauchy problem for the generalized KdV equation with rough data and random data. Firstly, we prove that $u(x,t)\longrightarrow u(x,0)$ as $t\longrightarrow0$ for a.e. $x\in \mathbb{R}$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Secondly, we prove that $u(x,t)\longrightarrow e^{-t\partial_{x}^{3}}u(x,0)$ as $t\longrightarrow0$ for a.e. $x\in \mathbb{R}$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Thirdly, we prove that $\lim\limits_{t\longrightarrow 0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u(x,0)\right\|_{L_{x}^{\i…
In recent years, controllability problems for dispersive systems have been extensively studied. This work is dedicated to proving a new type of controllability for a dispersive fifth order equation that models water waves, what we will now call the overdetermination control problem. Precisely, we are able to find a control acting at the boundary that guarantees that the solutions of the problem under consideration satisfy an overdetermination integral condition. In addition, when we make the control act internally in the system, instead of the boundary, we are also able to prove that this cond…
Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation
Université Paris Dauphine（巴黎第九大学）；Humboldt-Universitaet zu Berlin（柏林洪堡大学）；Institut für Mathematik（数学研究所）；The University of Edinburgh（爱丁堡大学）；Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）；Institut Universitaire de France（法国高等研究院）；Hausdorff Center for Mathematics（豪斯多夫数学中心）；Institut für Angewandte Mathematik Universität Bonn（波恩大学应用数学研究所）；University of Oxford（牛津大学）；Beijing Institute of Technology（北京理工大学）
