This paper is concerned with the asympootic limiting behavior of solutions to one-dimensional quasilinear scalar viscous equations for small viscosity in the presence of boundaries. We consider only non-characteristic boundary conditions. The main goals are to understand the evolution of viscous boundary layers, to construct the leading asymptotic ansatz which is uniformly valid up to the boundaries, and to obtain rigorously the uniform convergence to smooth solution of the associated inviscid hyperbolic equations away from the boundaries.

In this paper, for a class of exterior force term 2s²W^'_{s,s} we analyse the existence of unstable modes of linearized Navier-Stokes Equations (NSE), and associate them with integer points in plane. Furthermore we give the lower boundary dimension estimation of the attractor of NSE. Liu discussed the condition where the exterior force term is W^'_{0,s} in (1, 2), but his method can't be extended to the condition where the exterior force term is W^'_{s_1,s_2} (s_1 ≠ 0, s_2 ≠ 0). So this paper may look as the extention of [1, 2]. The method which we give in this paper has direct application for further study of other properties of NSE (such as Hopf bifurcation). See [3].

We establish a one-parameter family of Harnack inequalities connecting Li and Yau's differential Harnack inequality for the heat equation to Hamilton's Harnack inequality for the Ricci flow on a 2-dimensional manifold with positive scalar curvature.

In this paper, we consider Lhe asymptotic behaviour of the solutions or semilinear reaction-diffusional systems and obtain the growth rate.

In this paper we consider the propagation of microlocal regularity near constant multiple characteristic or a real solution u ∈ H^s (s > m + max{μ, 2} + \frac{n}{2})or non-linear partial differential equation F(x, u,…, ∂^βu,…)_{(|β|≤m)} = 0 We will prove that the microlocal regularity ncar constant multiple characteristic of the solution u will propagate along bicharacteristic with constant multiplicity μ and have loss of smoothness up to order μ - 1 under Levi condition.

In this paper, we study the general difference schemes with nonuniform meshes for the following problem: u_t = A(x,t,u,u_x)u_{xx}, + f(x,t,u,u_x), 0 < x < l, 0 < t ≤ T \qquad (1) u(0,t) = u(l ,t) = 0, 0 < t ≤ T \qquad\qquad (2) u(x,0) = φ(x), 0 ≤ x ≤ l \qquad\qquad (3) where u, φ, and f are m-dimensional vector valued functions, u_t = \frac{∂u}{∂t}, u_x = \frac{∂u}{∂x}, u_{xx} = \frac{∂²u}{∂_x²}. In the practical computation, we usually use the method of iteration to calculate the approximate solutions for the nonlinear difference schemes. Here the estimates of the iterative sequence constructed from the iterative difference schemes for the problem (1)-(3) is proved. Moreover, when the coefficient matrix A = A(x, t, u) is independent of u_x, t he convergence of the approximate difference solution for the iterative difference schemes to the unique solution of the problem (1)-(3) is proved without imposing the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem (1)-(3).

In this paper, the local and global existence of solutions for the Cauchy problem of the critical and subcritical nonlinear Schrödinger equations in H^s (s ≥ 2) is studied by using the manner of formal differentiation with respect to time. Some conjectures posed by Cazenave and Weissler in [1] are verified.