Convergence of approximate solutions of the Cauchy problem for a 2 × 2 nonstrictly hyperbolic system of conservation laws

A convergence theorem for the vanishing viscosity method and for the Lax-Friedrichs schemes, applied to a nonstrictly hyperbolic and nongenuinely nonlinear system is established. Using the theory of compensated compactness we prove convergence of a subsequence in the strong topology.     

A note on “well-posedness theory for hyperbolic conservation laws”

In this note, we generalize the recent result on L¹ well-posedness theory for strictly hyperbolic conservation laws to the nonstrictly hyperbolic system of conservation laws whose characteristics are with constant multiplicity.     

The generalized nonlinear initial–boundary Riemann problem for quasilinear hyperbolic systems of conservation laws

In this paper, we study the nonlinear initial–boundary Riemann problem and the generalized nonlinear initial–boundary Riemann problem for quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions on the domain $\{(t,x)|t⩾0,x⩾0\}$. Under the assumption that each positive eigenvalue is either linearly degenerate or genuinely nonlinear, we get the existence and uniqueness of the self-similar solution to the nonlinear initial–boundary Riemann problem and of the global piecewise $C^1$ solution containing only shocks and (or) contact discontinuities to the corresponding generalized nonlinear initial–boundary Riemann problem. It shows that the self-similar solution to the nonlinear initial–boundary Riemann problem possesses the global structural stability.     

Existence of solutions to the Riemann problem for 2 × 2 conservation laws

We consider the Riemann problem for a class of 2?×?2 systems of conservation laws which do not satisfy the strictly hyperbolicity condition. Our main assumption is that the product of non-diagonal elements within the F?echet derivative (Jacobian) of the flux is nonnegative. By improving a vanishing viscosity approach, we establish the existence of solutions to the Riemann problem for those systems.     

Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.     

Hamilton–Jacobi equations in space of measures associated with a system of conservation laws

We introduce a class of action integrals defined over probability measure-valued path space. We show that extremal point of such action exits and satisfies a type of compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton–Jacobi equations, in the space of probability measures, are well-posed.     

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic–elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive–dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic–elliptic system can be understood as a low-order approximation of the third-order diffusive–dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic–elliptic system completes the paper.     

Symbolic computation of conservation laws and exact solutions of a coupled variable-coefficient modified Korteweg–de Vries system

In this paper we study a generalized coupled variable-coefficient modified Korteweg–de Vries (CVCmKdV) system that models a two-layer fluid, which is applied to investigate the atmospheric and oceanic phenomena such as the atmospheric blockings, interactions between the atmosphere and ocean, oceanic circulations and hurricanes. The conservation laws of the CVCmKdV system are derived using the multiplier approach and a new conservation theorem. In addition to this, a similarity reduction and exact solutions with the aid of symbolic computation are computed.     

Stability of viscous shock wave under periodic perturbation for compressible Navier–Stokes–Korteweg system

In this paper, we study the stability of a viscous shock wave for the isentropic Navier–Stokes–Korteweg (N-S-K) equations under space-periodic perturbation. It is shown that if the initial perturbation around the shock and the amplitude of the shock are small, then the solution of the N-S-K equations tends to the viscous shock.     

Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws

Given any shock wave of a conservation law where the flux function may not be convex, we want to know whether it is admissible under the criterion of vanishing viscosity/capillarity effects. In this work, we show that if the shock satisfies the Oleinik’s criterion and the Lax shock inequalities, then for an arbitrary diffusion coefficient, we can always find suitable dispersion coefficients such that the diffusive-dispersive model admits traveling waves approximating the given shock. The paper develops the method of estimating attraction domain for traveling waves we have studied before.     

Conservation laws for a generalized coupled bidimensional Lane–Emden system

This paper aims to perform Noether symmetry classification of a generalized coupled bidimensional Lane–Emden system and computes the Noether operators corresponding to its first-order Lagrangian. In addition conservation laws of the various cases which admit Noether point symmetries are constructed for the underlying system.     

Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ℝ2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on ?² as follows:

$\{\begin{array}{l}{?}_{tt}u-\Delta u=-u^3,\\ u\left(0,x\right)=u_0\left(x\right),{?}_{t}u\left(0,x\right)=u_1\left(x\right),\end{array}$

where the initial data (u₀, u₁) ? H^s(?²) × H^s?1(?²). It is shown that the IVP is global well-posedness in H^s(?²) × H^s?1(?²) for any 1 > s > 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].     

Accuracy of classical conservation laws for Hamiltonian PDEs under Runge–Kutta discretizations

We investigate conservative properties of Runge–Kutta methods for Hamiltonian partial differential equations. It is shown that multi-symplecitic Runge–Kutta methods preserve precisely the norm square conservation law. Based on the study of accuracy of Runge–Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for Hamiltonian PDEs under Runge–Kutta discretizations. J. Hong, S. Jiang and C. Li are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, the NNSFC (No. 19971089, No. 10371128, No. 60771054) and the Special Funds for Major State Basic Research Projects of China 2005CB321701.     

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.     

On a minimal set of generators for the invariants of 3 × 3 matrices

Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.