Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Noether's theorem on time scales

We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals.     

Wave fans are special

It is shown that self-similar BV solutions of genuinely nonlinear strictly hyperbolic systems of conservation laws are special functions of bounded variation, with vanishing Cantor part.     

Asymptotic Maslov’s method for shocks of conservation laws systems with quadratic flux

In this work we apply the asymptotic method suggested by Maslov [1] to obtain the Hugoniot–Maslov chain for shock type solutions of conservation laws systems with quadratic flux. Additionally to the ODE infinite system that make up the chain, it was obtained an algebraic compatibility condition that must be satisfied by some of the coefficients of the asymptotic expansion of the shock solution. We give a new geometrical interpretation for this compatibility condition by means of certain singular surface whose projections represent time-dependent Hugoniot locus through the left limit state of the Shock.     

2-D Normal compliances for elastic and visco-elastic binder contact with finite particle size effect

The close-form 2-D normal force–displacement compliance relation (binder contact law) is derived for a system of two elastic cylindrical particles bound by an elastic or visco-elastic binder based on the approach developed by Zhu; Zhu and Zhu. A new result of finite particle size effect on the compliance is also obtained. One important application of this binder compliance is in the area of the homogenization analysis of fibrous composites, and computation of the binder compliance based effective transverse bulk modulus is conducted in this article with its comparison to the corresponding upper and lower bounds.     

用WENO方法求解双曲型守恒律方程组的初(边)值问题

本文用WENO算法解决双曲型守恒律方程组初（边值）问题．给出一种满足熵条件、Sδ熵条件和边界熵条件的WENO算法．通过这个算法就能得到守恒律方程组的数值解，数值解和理论解是非常吻合的．     

Hyperbolic systems of balance laws via vanishing viscosity

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension are constructed by the vanishing viscosity method of Bianchini and Bressan. For global existence, a suitable dissipativeness assumption has to be made on the production term g. Under this hypothesis, the viscous approximations u^?, that are globally defined solutions to , satisfy uniform BV bounds exponentially decaying in time. Furthermore, they are stable in L¹ with respect to the initial data. Finally, as ?→0, u^? converges in to the admissible weak solution u of the system of balance laws u_t+(f(u))_x+g(u)=0 when A=Df.     

On finite time BV blow-up for the p-system

Abstract

The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the t-x plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.     

Viscosity method in a transonic flow

We prove that the solution of a nonviscous compressible transonic flow can be obtained as a limit of viscous solutions, if the viscosity and heat conductivity tend to zero. To obtain an isentropic irrotational flow it is necessary to control the entropy and temperature on the boundary in a convenient way