In this paper, we study the dissipative property of the first order 3x3 hyperbolic system with constant coefficients. For the corresponding n x n system, when the coefficients matrices are symmetric, it has been studied in [16] and the well-know Kawashima-Shizuta condition is obtained. When n = 3 and for asymmetric system, we give a sufficient condition for the system to be strictly dissipative.

In this paper, we study the global existence of BV solutions of the initial value problem for the isentropic p-system, where the state equation of the gas is given by P = Av−γ. For γ < 1, the general existence result for large initial data has not been obtained. By using the Glimm scheme, Nishida, Smoller and Diperna successively obtained the global existence results for (γ − 1)TV(v0(x),u0(x)) being small. In the present paper, by adopting a rescaling technique, we improve these results and obtain the global existence result under the condition that (γ −1)γ+1 (TV(v0(x)))γ−1 (TV(u0(x)))2 is small, which implies that, for fixed γ > 1, either TV(v0 (x)) or TV(u0 (x)) can be arbitrarily large.

In this paper, we mainly study the limit of solutions to the two‐dimensional steady nonisentropic relativistic Euler flow onto a straight wedge. It turns out that the sequence of shock solutions tends to a delta wave adhering to the wedge surface when the velocity of the incident flow goes to the speed of light and the adiabatic index tends to 1 in turn. Meanwhile, it also verifies that the pressure coefficient in the limit case is consistent with the Newtonian sin‐squared law when the speed of light tends to infinity. To this end, we also derive the generalized Bernoulli equation, Taub adiabat (or the generalized Hugoniot adiabat) and a shock polar, and so forth. Furthermore, we give the construction of the delta wave by the definition of weak solution.

In this paper, we study the dissipative property of the first order 2 x 2 hyperbolic system with constant coefficients. We propose a dissipative condition (see (2.9)) which is weaker than the strongly dissipative condition and can be regarded as a generalization of Kawashima-Shizuta condition. We show that this condition is sharp. With this condition and tools of Fourier analysis, we also give pointwise estimates of the solution to the Cauchy problem for suitable initial data. Finally, we illustrate that our dissipative condition can not be generalized directly to 3 x 3 system.

In this paper, we study the dissipative property of the first order $ 2\times 2 $ hyperbolic system with constant coefficients. We propose a dissipative condition (see (2.9)) which is weaker than the strongly dissipative condition and can be regarded as a generalization of Kawashima-Shizuta condition. We show that this condition is sharp. With this condition and tools of Fourier analysis, we also give pointwise estimates of the solution to the Cauchy problem for suitable initial data. Finally, we illustrate that our dissipative condition can not be generalized directly to $ 3\times3 $ system.

In this paper, we study the two dimensional Riemann problem of the Euler system for isentropic Chaplygin gas, and the initial data consist of three pieces constant states divided by an inverted Y-type curve. The results extended the results in Chen and Qu where the solution of every one dimensional Riemann problem contains no slip plane. We divide the discussion into several cases and for each case, we give the structure of the solution containing slip planes by the method of generalized characteristic analysis.

In this paper, we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space. First, we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane. Based on these properties, we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme, and prove that there exists an entropy solution V(x, t) which belongs to L∞ ∩ BVloc (ℝ × ℝ+).

In this paper, we study the formation of delta shock waves and vacuum states of Riemann problem for isentropic Euler system with Chaplygin pressure by vanishing pressure limit method under the self-similar coordinate. The state equation is $p=-\epsilon ^{2}/\rho $ . There is no vacuum state initially. We proved that as $\epsilon \to 0$ , the self-similar solution can be divided into three cases which may contains a contact discontinuity, a vacuum state or a delta shock wave. Moreover, we proved that for Chaplygin gas, there exists a critical value $\epsilon _{1}>0$ depending on the initial data, such that for any $\epsilon >\epsilon _{1}$ , there is no delta shock wave in the solution. For $\epsilon \in (0,\epsilon _{1}]$ and for suitable initial data, the solution contains a delta shock wave which can be expressed explicitly. As $\epsilon \rightarrow 0$ , the sequence of delta shock wave solutions converge to the delta shock wave solution of the transport equations with the same initial data.

In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.

In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law ut + F(u)x = 0. First, we prove a simple but useful property of Lax-Oleinik formula (Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)−qF′(q)+ L(F′(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T*. Meanwhile, we can give the equation of the shock and an explicit value of T*.

This paper deals with the uniqueness of the kinetic solutions to Cauchy problem of general anisotropic degenerate parabolic-hyperbolic equations. Kinetic formulation is extended to such general degenerate parabolic-hyperbolic equations with coefficients depending on time-spatial variables. Contraction property of kinetic solutions is established under appropriate conditions on diffusion and convection functions.

In this paper, we give the global existence of Lp bounded entropy solutions for the Cauchy problem of a system of conservation laws in chromatography with geometry effects. The main difficulty lies in establishing the Lp estimate of the viscosity solutions because the initial data may not be L∞ bounded and the principle of invariant regions developed by Conley–Chuey–Smoller is invalid here. We obtain the existence of the global weak solutions using the compensated compactness method and BV estimates on viscosity solutions.

《数理方程(英文)》课程是南京航空航天大学飞行器设计与工程专业来华留学生的一门重要课程.由于本校留学生来自不同国家和地区,其文化背景和受教育方式不同,本课程的预备知识储备不足,加上没有合适的英文教材,给本课程的教学造成很大障碍.本文针对以上问题进行探索和研究,结合本校留学生的特点,在教学实践的基础上,对本课程的教学提出了一些新的想法和改革措施.

In this paper, we study periodic solution to p-system via a modified Glimm scheme in BV space. We show that for the periodic initial data U0(x)∈L∞∩BVloc(R), there exists an entropy solution U(x,t) which belongs to L∞∩BVloc(R×R+). This problem has been studied by Frid [4]. By obtaining a more precise estimate on the approximate solution, we get a more general result.

In this paper, we give the global existence of Lp bounded entropy solutions for the Cauchy problem of a symmetric system of Keyfitz–Kranzer type. The main difficulty lies in establishing the Lp estimate of the viscosity solutions because the initial data may not be L∞ bounded and the principle of invariant regions developed by Conley–Chuey–Smoller is invalid here. We obtain the existence of the global weak solutions using the compensated compactness method and BV estimates on viscosity solutions.

In this article, we give the existence of global L∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.

研究非齐次Burgers方程Cauchy问题解的大时间行为.假定初值是周期的并且非齐次项具有多个零点.对初值的某些假定条件下,证明了问题的解收敛于一个行波解.所用的主要方法是广义特征线理论.

The relaxation limit in critical Besov spaces for the multidimensional compressible Euler equations is considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem with initial data close to an equilibrium state are constructed in the Chemin-Lerner's spaces with critical regularity. Furthermore, it is shown that the density converges towards the solution to the porous medium equation, as the relaxation time tends to zero. Several important estimates are achieved, including a crucial estimate of commutator.

As a fundamental and important step to understand the existence and behavior of solution to the multi-dimensional problem, we study in this paper the three dimensional relativistic Euler equations with spherical symmetry. We obtain the non-relativistic global limits of entropy solutions to the Cauchy problem of the spherically symmetric relativistic Euler equations.

This paper studies the steady supersonic flow past a Lipschitz curved cone. Under the assumptions that the cone has an opening angle less than a critical value and has sufficiently small total variation of the tangent of the perturbation and that the Mach number of incoming flow is sufficiently large, the global weak solution is constructed via Glimm scheme for 1 < gamma < 3. (C) 2009 Elsevier Inc. All rights reserved.