Chapter 5 - Conservation Laws

Summary

INTRODUCTION

Many physical laws, for example, conservation of mass, momentum and energy,occur as conservation laws. Dynamics of compressiblefluids, both in one and three dimensions, is a rich source of conservationlaws and offers many problems that are challenging; see, for example,Courant and Friedrichs (1976), Morawetz (1981), Whitham (1974), and Majda(1985). This subject has been one of the most active research fields, bothin the theoretical and computational aspects, for the past more than sixdecades. Because of its important applications in the aerospace engineering,the interest in this field is only growing. Yet, certain theoretical aspectsin the study of systems of conservation laws (e.g., Euler's equationsof gas dynamics) have not been resolved satisfactorily and remain achallenge. The interested reader will find some advanced topics in Smoller(1994), Majda (1985), and Lax (1973), and the references therein.

A conservation law asserts that the rate of change of the totalamount of a substance (e.g., a fluid) in a domain Ω inspace (an interval in the one-dimensional case) is equal to itsflux across the boundary𝜕Ω of the domain Ω. Ifu = u(x, t) denotesthe density of the substance at time t andf the flux, the conservation law is expressed as

where v denotes the outward unit normal on𝜕Ω and dS is thesurface measure on 𝜕Ω. The integral on theright-hand side is the total amount of the outflow of the substance across𝜕Ω, hence the negative sign. Assumingthat u and f are smooth, we obtain usingthe divergence theorem that

SinceΩis an arbitrary domain, by shrinking it to a point, we obtainthe following first-order PDE

satisfied by the density u and the flux f.Equation (5.2) is the conservation law expressed in thedifferentiated form and equation (5.1) in theintegral form.

Note that (5.1) is equivalent to

for all t1 and t2 witht₁< t₂.

In the present chapter, we consider only a single conservation law in one(space) dimension.