The Cauchy Problem for Hyperbolic Conservation Laws with Three Equations (original) (raw)

Mathematical and numerical study of a system of conservation laws

Journal of Evolution Equations, 2007

The system of equations (f (u))t − (a(u)v + b(u)) x = 0 and ut − (c(u)v + d(u)) x = 0, where the unknowns u and v are functions depending on (x, t) ∈ R × R+, arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of ut − ((cg + d)(u))x = 0 and simultaneously w = f (u) is the entropy solution of wt − ((ag + b)(f (−1) (w)))x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.

On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms

Journal of Control Science and Engineering, 2010

We study one class of nonlinear fluid dynamic models with impulse source terms. The model consists of a system of two hyperbolic conservation laws: a nonlinear conservation law for the goods density and a linear evolution equation for the processing rate. We consider the case when influx-rates in the second equation take the form of impulse functions. Using the vanishing viscosity method and the so-called principle of fictitious controls, we show that entropy solutions to the original Cauchy problem can be approximated by optimal solutions of special optimization problems.

On the structure of solutions of nonlinear hyperbolic systems of conservation laws

Communications on Pure and Applied Analysis, 2011

We are concerned with entropy solutions u in L ∞ of nonlinear hyperbolic systems of conservation laws. It is shown that, given any entropy function η and any hyperplane t = const., if u satisfies a vanishing mean oscillation property on the half balls, then η(u) has a trace H d -almost everywhere on the hyperplane. For the general case, given any set E of finite perimeter and its inner unit normal ν : ∂ * E → S d and assuming the vanishing mean oscillation property of u on the half balls, we show that the weak trace of the vector field (η(u), q(u)), defined in Chen-Torres-Ziemer [9], satisfies a stronger property for any entropy pair (η, q). We then introduce an approach to analyze the structure of bounded entropy solutions for the isentropic Euler equations.

On Zero Mass Solutions of Viscous Conservation Laws

Communications in Partial Differential Equations, 2002

In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u t − ∆u+ ∇ ·f (u) = 0 in IR n × [0, ∞), where f (u) ∼ u q as u → 0. Under the assumption that q ≥ 1 + 1/(n + β) and the initial condition u 0 satisfies: u 0 ∈ L 1 (IR n ), IR n u 0 (x) dx = 0, and e t∆ u 0 L 1 (IR n ) ≤ Ct −β/2 for fixed β ∈ (0, 1), all t > 0, and a constant C, we show that the L 1 -norm of the solution to the convection-diffusion equation decays with the rate t −β/2 as t → ∞. Moreover, we prove that, for small initial conditions, the exponent q * = 1 + 1/(n + β) is critical in the following sense. For q > q * the large time behavior in L p (IR n ), 1 ≤ p ≤ ∞, of solutions is described by self-similar solutions to the linear heat equation. For q = q * , we prove that the convection-diffusion equation with f (u) = u|u| q * −1 has a family of self-similar solutions which play an important role in the large time asymptotics of general solutions. 2000 Mathematics Subject Classification: 35B40, 35K55. Key words and phrases: the Cauchy problem, the convection-diffusion equation, large time behavior of solutions, self-similar solutions.

Artifical and physical viscosity solutions for a hyperbolic conservation system

Applicable Analysis, 2001

In this paper, a new maximum principle is introduced to study positive lower bounds of the density both for the arti cial viscosity solutions and for the physical viscosity solutions of a hyperbolic conservation system derived from the Broadwell model and the global existence of these viscosity solutions is obtained.