Conservation Laws. Some Examples

S.12.1. Conservation Laws. Some Examples

The main mathematical models in continuum mechanics and theoretical physics have the form of systems of conservation laws. Usually mass, momentum, and energy for phases and/or components are conserved.

We consider systems of conservation laws with the form

∂G(u) ∂F(u)

where u = u( x, t) is a vector function of two scalar variables, and F = F(u) and G = G(u) are vector functions,

u=( u 1 , ...,u n ) T ,

u i = u i ( x, t);

F=( F 1 , ...,F n ) T , F i = F i (u); G=( G 1 , ...,G n ) T , G i = G i (u).

Here and henceforth, ( u 1 , ...,u n ) T stands for a column vector with components u 1 , ...,u n . For any F and G system (1) admits the following particular solutions:

u = C,

where C is an arbitrary constant vector.

Example 1. Consider a single quasilinear equation of the special form

which is a special case of (1) with n = 1, G(u) = u, and F = F (u). Equation (2) represents a law of conservation of mass (or another quantity) and is often encountered in gas dynamics, fluid mechanics, wave theory, acoustics, multiphase flows, and chemical engineering. This equation is a model for numerous processes of mass transfer, including sorption and chromatography, two-phase flows in porous media, flow of water in river, road traffic development, flow of liquid films along inclined surfaces, etc. The independent variables t and x in equation (2) usually play the role of time and the spatial coordinate, respectively, u = u(x, t) is the density of the quantity being transferred, and

F (u) is the flux of u. Example 2. A one-dimensional ideal adiabatic (isentropic) gas flow is governed by the system of two equations

∂ρ + ∂(ρv) = 0,

∂t

∂x ∂(ρv) + ∂(ρv 2 + p(ρ)) = 0.

∂t

∂x

Here, ρ = ρ(x, t) is the density, v = v(x, t) is the velocity, and p is the pressure. Equation (3) represents the law of conservation of mass in fluid mechanics and is referred to as a continuity equation. Equation (4) represents the law of conservation of momentum. The equation of state is given in the form p = p(ρ). For an ideal polytropic gas, p = Aρ γ , where the constant γ is the adiabatic exponent.

Remark. System (3)–(4) with ρ = h and p(ρ) = 1 2 gh 2 , where v is the horizontal velocity averaged over the height h of the water level and

g is the acceleration due to gravity, governs the dynamics of shallow water.

The origin of hyperbolic systems of conservation laws as mathematical models for physical phenomena is discussed extensively in the literature. The classical treatises by Courant and Hilbert (1989), Landau and Lifshitz (1987), and Whitham (1974) and also a recent comprehensive mono- graph by Dafermos (2000) should be mentioned. Conservation law systems for various gas flow regimes in Eulerian and Lagrangian coordinates are treated in the monographs Courant and Friedrichs (1985), Landau and Lifshitz (1987), Logan (1994), and Zel’dovich and Raizer (1968). Gas flows with chemical reactions (combustion and phase transitions) are discussed in the books by Zel’dovich and Raizer (1966, 1967), Zel’dovich, Barenblatt, Librovich, and Makhviladze (1985). Hyperbolic

* Section S.12 was written by P. G. Bedrikovetsky and A. P. Pires.

monographs Hanyga (1985) and Kulikovskii and Sveshnikova (1995) give a comprehensive presen- tation of the theory of elastic media. Both Barenblatt, Entov, and Ryzhik (1991) and Bedrikovetsky (1993) discuss hyperbolic systems for two-phase multi-component flows in porous media describing oil recovery processes. Traffic flow and shallow water mechanics are treated in Logan (1994) and Whitham (1974).

Methods of analytical integration for self-similar Riemann problems are presented in the mono- graphs by Smoller (1983) and Dafermos (2000); non-self-similar problem integration methods for wave interactions are given in Glimm (1989), LeVeque (2002), and Bedrikovetsky (1993).