A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data in the class for \frac34$"> and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

     

On the optimal local regularity for the Yang-Mills equations in

The aim of the paper is to develop the Fourier Analysis techniques needed in the study of optimal well-posedness and global regularity properties of the Yang-Mills equations in Minkowski space-time , for the case of the critical dimension . We introduce new functional spaces and prove new bilinear estimates for solutions of the homogeneous wave equation, which can be viewed as generalizations of the well-known Strichartz-Pecher inequalities.

     

LOCAL WELL-POSEDNESS AND ILL-POSEDNESS ON THE EQUATION OF TYPE □u = u^k（δu）^α

This paper undertakes a systematic treatment of the low regularity local wellposedness and ill-posedness theory in H^s and H^s for semilinear wave equations with polynomial nonlinearity in u and δu. This ill-posed result concerns the focusing type equations with nonlinearity on u and δtu.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Well-posedness of general nonlocal nonhomogeneous boundary value problems for pseudodifferential equations with partial derivatives

In the article we study the questions of well-posedness of general nonlocal boundary value problems for pseudodifferential equations in the Besov-type limit spaces.     

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].     

The Nonlinear Schr¨odinger Equations with Combined Nonlinearities of Power-Type and Hartree-Type

The primary goal of this paper is to present a comprehensive study of the nonlinear Schr?dinger equations with combined nonlinearities of the power-type and Hartree-type. Under certain structural conditions, the authors are able to provide a complete picture of how the nonlinear Schr?dinger equations with combined nonlinearities interact in the given energy space. The method used in the paper is based upon the Morawetz estimates and perturbation principles.     

Global well-posedness of the stochastic 2D Boussinesq equations with partial viscosity

This paper deals with the stochastic 2D Boussinesq equations with partial viscosity. This is a coupled system of Navier-Stokes/Euler equations and the transport equation for temperature under additive noise. Global well-posedness result of this system under partial viscosity is proved by using classical energy estimates method.     

关于良定问题

本文应用有限理性模型M,对非线性问题的良定性进行了统一的研究,对最优化、多目标最优化、非合作博弈和广义博弈得到了一些新的良定性结果.     

Analysis of a Free Boundary Problem Modeling Multi- Layer Tumor Growth in Presence of Inhibitor

In this paper we study well-posedness and asymptotic behavior of solution of a free boundary problem modeling the growth of multi-layer tumors under the action of an external inhibitor. We first prove that this problem is locally well-posed in little Holder spaces. Next we investigate asymptotic behavior of the solution. By making delicate analysis of spectrum of the linearization of the stationary free boundary problem and using the linearized stability theorem, we prove that if the surface tension coefficient γ is larger than γ^＊ 〉 0 the fiat stationary solution is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficient small.