Well-posedness for a higher-order Benjamin-Ono equation

In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation

     

Oscillatory integrals generated by the Ostrovsky equation

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation. Received: November 23, 2004; revised: March 13, 2005 Research is supported by US Department of Defense, under grant No. DAAD19-03-1-0204     

Well-posedness for the fifth-order shallow water equations

The Cauchy problems for some kind of fifth-order shallow water equations

     

On the Cauchy problem for the generalized shallow water wave equation

In this paper we mainly study the Cauchy problem for the generalized shallow water wave equation in the Sobolev space Hs of lower order s. Using the crucial bilinear estimates in the Fourier transform restriction spaces related to the shallow water wave equation, we establish local well-posedness in Hs with any .     

Well-posedness and ill-posedness for a fifth-order shallow water wave equation

In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [k;Z] multiplier norm method of Tao (2001) [2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in with is obtained by the Fourier restriction norm method. And some ill-posedness in with is derived from a general principle of Bejenaru and Tao.     

Persistence of solutions to higher order nonlinear Schrödinger equation

Applying an Abstract Interpolation Lemma, we showed persistence of solutions of the initial value problem to higher order nonlinear Schrödinger equation, also called Airy-Schrödinger equation, in weighted Sobolev spaces X2,θ, for θ∈[0,1].     

Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation

     

A note on the reflection principle for the biharmonic equation and the Stokes system

In this note, we discuss the reflection principle of the Stokes system in a half space for the threedimensional case, and of the biharmonic equation. Admitting different boundary conditions, we use the reflection principle to prove uniqueness of solutions of the Stokes system or the biharmonic equation in weightedLq-spaces     

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.     

Cauchy problem for the Ostrovsky equation in spaces of low regularity

In this article we consider the initial value problem for the Ostrovsky equation:

     

Cauchy problem of the generalized double dispersion equation

In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation are proved. The blow-up of the solution for the Cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions.     

Stability of solitary waves and weak rotation limit for the Ostrovsky equation

Ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation. In this model dispersion is taken into account while dissipation is neglected. Existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter (which can be either positive or negative). It is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations. The issue of passing to the limit as the rotation parameter tends to zero for solutions of the Cauchy problem is investigated on a bounded time interval.     

On the support of solutions to the Ostrovsky equation with positive dispersion

In this paper we prove that sufficiently smooth solutions of the Ostrovsky equation with positive dispersion,

     

Zero-dispersion limit of the short-wave-long-wave interaction equations

The purpose of this paper is to study the zero-dispersion limit of the water wave interaction equations which arise in modelling surface waves in the present of both gravity and capillary modes. This topic is also of interest in plasma physics. For the smooth solution, the limiting equation is given by the compressible Euler equation with a nonlocal pressure caused by the long wave. For weak solution, when the coupling coefficient λ is small order of ε, λ=o(ε), the wave map equation is derived and the scattering sound wave is shown to satisfy a linear wave equation.