REGULARITY OF ENTROPY SOLUTIONS TO NONCONVEX SCALAR CONSERVATION LAWS WITH MONOTONE INITIAL DATA

We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of C^k＋1 of first category, defined in the range of the initial datum.     

单个守恒律解的大范围定性研究(Ⅰ)

本文应用突变理论研究方程u₁+f(u)_x=0的解的性质,得到了比文献[4]中的第二范畴集更大,但有相似的性质,且用具体的解析条件给出的(R)中的集;给出了激波条数的精确的上界、下界;算出了t充分大时激波的段数和渐近线;特别是对激波的大范围分布得到了一个清晰的图象。     

AN INHOMOGENEOUS QUASILINEAR HYPERBOLIC SYSTEM

Introduction We consider the initial value problem for the inhomogeneous hyperbolic system     

STUDY OF THE GLOBAL SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM

§1 Introduction We shall compare the solution of the Caucky problem with the solution of the corresponding Riemann problem (1.1) with the initial data     

STUDY OF THE GLOBAL SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM

We shall compare the solution of the Cauchy problem with the solution of the corresponding Riemann problem (1.1) with the initial data     

关于Hamilton-Jacobi方程解的正则性和全局结构的注记——纪念李国平院士吴新谋教授诞辰100周年

该文考虑高维Hamilton-Jacobi方程的柯西问题. 作者证明了从任一初始点出发的特征线永不碰到奇异点集合的充分必要条件是初始函数在该点取到最小值.在此基础上,证明了奇异点集合的道路连通分支和初始函数不取最小值的点集合的道路连通分支之间存在一一对应, 而且解的梯度的间断一旦产生就不会消失. 特别指出, 该文的结果不需要“初始函数的梯度在无穷远趋近于零”这一限制条件, 而文献[12]中重要的命题2.7和主要结果之一的定理3.3是在这一条件下得到的.     

REMARKS ON BOUNDS ON THE DISCREPANCY OF APPROXIMATE SOLUTIONS CONSTRUCTED BY GODUNOV'S SCHEME

The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two iuequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.     

GLOBAL SOLUTIONS FOR A SEMILINEAR PARABOLIC SYSTEM

In this paper we consider the Cauchy problem of the semilinear parabolic system (1) with the initial value (u,v) (x,0) = (u_0 (x),v_0 (x)), (2) where ε>0 and α≥0 are constants. Through this paper we assume that     

拟线性双曲型守恒律组的整体解的研究

本文共分两部分,第一部分是研究最简气体动力学方程组较文献[6]更一般的始值问题,得出了解的大范围存在性及始值所具有的性质对时间t的不变性,并对解的性质进行了某些研究。在假设用(u₀(±∞),v₀(±∞))(u₀(x),v₀(x)表示始值)解黎曼问题时存在某一向激波的情形,证明了解必然包含一条某一向激波。应该指出,同向激波和稀疏波的相互作用是本文所研究的始值的一个特例。第二部分是把第一部分中所得的主要结果推广到文献[7]中研究过的方程组上去。     

THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS

We use Hopf-Lax formula to study local regularity of solution to Hamilton-Jacobi (HJ) equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to We use Hopf-Lax formula to study local regularity of solution to Hamilton-Jacobi (HJ) equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T > 0, which depends only on the Hamiltonian and initial datum, for t > T the solution of the IVP (1.1) is smooth except for a smooth n-dimensional hypersurface, across which Du(x, t) is discontinuous. And we show that the hypersurface tends asymptotically to a given hypersurface with rate t 1 4 .HJ equation, i.e. for most initial data there exists a constant T > 0, which depends only on the Hamiltonian and initial datum, for t > T the solution of the IVP (1.1) is smooth except for a smooth n-dimensional hypersurface, across which Du(x, t) is discontinuous. And we show that the hypersurface tends asymptotically to a given hypersurface with rate t-1/4 .