Approximate solutions to slightly viscous conservation laws

Approximate Solutions of Nonlinear Conservation Laws

This is a summary of ve lectures delivered at the CIME course on "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" held in Cetraro, Italy, on June 1997. Following the introductory lecture I | which provides a general overview of approximate solution to nonlinear conservation laws, the remaining lectures deal with the speciics of four complementing topics: Lecture II. Finite-d...

Finite Dimensional Representation of Solutions of Viscous Conservation Laws

On Zero Mass Solutions of Viscous Conservation Laws

In the paper, we consider the large time behavior of solutions to the convection-diffusion equation ut−∆u+∇·f(u) = 0 in IRn× [0,∞), where f(u) ∼ uq as u → 0. Under the assumption that q ≥ 1 + 1/(n + β) and the initial condition u0 satisfies: u0 ∈ L1(IRn), ∫ IRn u0(x) dx = 0, and ‖eu0‖L1(IRn) ≤ Ct−β/2 for fixed β ∈ (0, 1), all t > 0, and a constant C, we show that the L1-norm of the solution to ...

Viscous Conservation Laws, Part I: Scalar Laws

Viscous conservation laws are the basic models for the dissipative phenomena. We aim at a systematic presentation of the basic ideas for the quantitative study of the nonlinear waves for viscous conservation laws. The present paper concentrates on the scalar laws; an upcoming Part II will deal with the systems. The basic ideas for scalar viscous conservation laws originated from two sources: th...

Approximate Solutions of Nonlinear Conservation Laws and Related Equations

During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These incl...