一类双色散波方程的Cauchy问题

This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation.By the priori estimates and the method in [9],It proves that the Cauchy problem admits a unique global classical solution.And by the concave method,we give sufficient conditions on the blowup of the global solution for the Cauchy problem.     

Periodic Boundary Problem and Cauchy Problem for the Fluid Dynamic Equation in Geophysics

We study periodic boundary problem and Cauchy problem for the fluid dynamic equation in geophysics. The generalized and classical global solution of the mentioned problems are established. The method employed in this paper is Galerkin approximation and integral estimates.     

一般非线性扩散方程Cauchy问题广义解的渐近性质

本文研究了比较一般的非线性扩散方程的Cauchy问题广义解的渐近性质.利用试验函数与积分估计的方法,获得了广义解当t→∞时的渐近性质以及其广义整体解u(x,t),推广了该类方程Cauchy问题广义解的性质.     

Existence of global smooth solution to the relativistic Euler equations

In this paper, we prove an existence theorem of global smooth solutions to the Cauchy problem for the one-dimensional relativistic Euler equations. The analysis is based on a priori estimates which are obtained by the characteristic method and maximum principle.     

Global attractor for the viscous modified two-component Camassa–Holm equation

We obtain the existence of global attractor for the Cauchy problem of a viscous modified two-component Camassa–Holm equation. The existence of global strong solutions is obtained using Kato’s theory. The key elements in our analysis are the uniform Gronwall lemma and some estimates of the solutions.     

On Inhomogeneous GBBM Equations

In this paper we consider the Cauchy problem and the initial boundary value (IBV) problem for the inhomogeneous GBBM equations. For any bounded or unbounded smooth domain, the existence and uniqueness of global strong solution for the Cauchy problem and IBV problem for the inhomogeneous GBBM equations in W^{2,p}(Ω) are established by using Banach fixed point theorem and some a priori estimates. These results have improved the known results even in the case of GBBM equation. Meanwhile, we also discuss the regularity of the Strong solution and the system of inhomogeneous GBBM equations.     

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (?1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are local-in-space regularity estimates near the initial time, which are of independent interest.     

Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type

In this paper, we use the compensated compactness method with BV estimates on the Riemann invariants to obtain the global existence of bounded entropy weak solutions for the Cauchy problem of a symmetric system of Keyfitz-Kranzer type.     

On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source

The Cauchy problem for a degenerate parabolic equation with a source and inhomogeneous density of the form

$\rho (x)\frac{{\partial u}}{{\partial t}} = div(u^{m - 1} \left| {Du} \right|^{\lambda - 1} Du) + \rho (x)u^p $

is studied. Time global existence and nonexistence conditions are found for a solution to the Cauchy problem. Exact estimates of the solution are obtained in the case of global solvability.

     

Local and global estimates of the solutions of the Cauchy problem for quasilinear parabolic equations with a nonlinear operator of Baouendi-Grushin type

In this paper, the qualitative properties of the solutions of the Cauchy problem for degenerate parabolic equations with a nonlinear operator of Baouendi-Grushin type are studied. Sharp local and global (with respect to the spatial and temporal variables) estimates of the solution are obtained. The property of the finiteness of the support of the solution is established.     

Large-time behavior for the solution to the generalized Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space

In this paper, we consider the global existence as well as the optimal decay estimates of the Cauchy problem for the multi-dimensional Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space. And these results are obtained by Green?s function method, Fourier analysis method, energy estimates method combined with the time-frequency decomposition method.     

Partial regularity for harmonic maps of evolution into spheres

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.     

On the Cauchy problem for the fractional drift-diffusion system in critical Besov spaces

In this paper, we establish global well-posedness and asymptotic stability of mild solutions for the Cauchy problem of the fractional drift-diffusion system with small initial data in critical Besov spaces. The regularizing-decay rate estimates of mild solutions are also proved, which imply that mild solutions are analytic in space variables.     

Global existence for the multi-dimensional compressible viscoelastic flows

The global solutions in critical spaces to the multi-dimensional compressible viscoelastic flows are considered. The global existence of the Cauchy problem with initial data close to an equilibrium state is established in Besov spaces. Using uniform estimates for a hyperbolic-parabolic linear system with convection terms, we prove the global existence in the Besov space which is invariant with respect to the scaling of the associated equations. Several important estimates are achieved, including a smoothing effect on the velocity, and the L¹-decay of the density and deformation gradient.     

Temps de vie des solutions d'un problème de Cauchy non linéaire

We solve the global Cauchy problem, with small initial data, in the space of the holomorphic functions with respect to t and Gevrey class with respect to x. We establish the existence and the stability of the solution to Cauchy problem with nul initial data without hyperbolicity hypothesis. In the stationary case, we give estimates of life span of the solutions with respect to size of the initial data.     

CAUCHY PROBLEM FOR A NONLINEAR SINGULAR INTEGRAL-DIFFERENTIAL EVOLUTION SYSTEM

We study the Cauchy problem for a nonlinear evolution system with singularintegral differential terms, By means of some a priori estimates of the solution and the Leray-Schander‘s fixed point theorem, we prove the existence and the uniqueness theorems of the generalized global solution of the mentioned problem.     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

一类广义长短波方程的整体适定性

In the present paper,we investigate the well-posedness of the global solution for the Cauchy problem of generalized long-short wave equations.Applying Kato's method for abstract quasi-linear evolution equations and a priori estimates of solution,we get the existence of globally smooth solution.     

On the Cauchy Problem of the Camassa-Holm Equation

The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in L ^p frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.     

The cauchy problem for a shallow water type equation

We prove existence and uniqueness of local and global solutions of the periodic Cauchy problem for a higher order shallow water type equation under low regularity initial data. Using Fourier analysis we first prove local estimates in appropriate spaces and then use a contraction mapping argument and a conserved norm to get global existence.