Local well-posedness of a new integrable equation

In this article we focus on the local-in-time well-posedness of the Cauchy problem for a new integrable equation. We proved the local-in-time existence and uniqueness of the entropy solutions by using the method of the vanishing viscosity and L¹$L^1$-contraction property.     

一个二维非线性Schrodinger方程的整体适定性

利用高低频技术证明了当初值属于Hs(R2),s>(16)/(17)时二维五次非线性Schrodinger方程的整体适定性.     

Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation

We establish the local well-posedness for a generalized Dullin-Gottwald-Holm equation by using Kato’s theory. Furthermore, the orbital stability of the peaked solitary waves is also proved.     

Local and global well-posedness for the Ostrovsky equation

We consider the initial value problem for

(0.1)     

Local well-posedness for higher order nonlinear dispersive systems

We study nonlinear dispersive systems of the form

一类弱耗散Camassa-Holm方程局部强解的适定性及弱解的存在性

使用Pseudoparabolic正则化方法和从弱耗散Camassa-Holm方程自身导出的估计式,在Sobolev空间Hs(R)(s3/2)中,证明了该Camassa-Holm方程解的局部适定性.同时给出了一个在空间Hs(R)(1s2\3)中确保该方程弱解存在的充分条件.     

Local well-posedness for the dispersion generalized periodic KdV equation

In this paper, we set up the local well-posedness of the initial value problem for the dispersion generalized periodic KdV equation: t_∂u+x_∂α|Dx|u=x_∂u², u(0)=φ for α>2, and φ∈Hs(T). And we show that the is a lower endpoint to obtain the bilinear estimates (1.2) and (1.3) which are the crucial steps to obtain the local well-posedness by Picard iteration. The case α=2 was studied in Kenig et al. (1996) [10].     

Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely,

     

Bifurcations of traveling wave solutions in a compound KdV-type equation

By using the theory of planar dynamical systems to a compound KdV-type nonlinear wave equation, the bifurcation boundaries of the system are obtained in this paper. These bifurcation sets divide the parameter space into different regions, which correspond to qualitatively different phase portraits and therefore different types of the solutions may exist in different regions. The parameter conditions for the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions are therefore obtained.     

Local well-posedness for the 2D non-dissipative quasi-geostrophic equation in Besov spaces

In this paper we prove the local-in-time well-posedness for the 2D non-dissipative quasi-geostrophic equation, and study the blow-up criterion in the critical Besov spaces. These results improve the previous one by Constantin et al. [P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity 7 (1994) 1495–1533].     

Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions

Extending the work of Ibrahim et al. (Commun Pure Appl Math 59(11): 1639–1658, 2006) on the Cauchy problem for wave equations with exponential nonlinearities in two space dimensions, we establish global well-posedness also in the super-critical regime of large energies for smooth, radially symmetric data.     

Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping

We consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping.     

一个二维非线性Schroedinger方程的整体适定性

利用高低频技术证明了当初值属于Hs(R2),s>(16)/(17)时二维五次非线性Schrodinger方程的整体适定性.     

Local well-posedness for a quasi-stationary droplet model

The moving boundary problem for the contact line evolution of a droplet is studied. Local existence and uniqueness of classical solutions is established.     

A nonlinear equation in Banach spaces and applications to the well-posedness of Cauchy problems

We analyze a nonlinear equation in Banach spaces, with the nonlinearity composed of multiple terms of different degrees. We prove a theorem regarding the existence of solutions for such equations. Moreover, we show how this result may be applied to obtain the well-posedness of various parabolic initial value problems.