In
continuum mechanics, the most general form of an exact conservation law is given by a
continuity equation. For example, conservation of electric charge is\frac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{j} \,where is the
divergence operator, is the density of (amount per unit volume), is the flux of (amount crossing a unit area in unit time), and is time. If we assume that the motion
u of the charge is a continuous function of position and time, then\begin{align} \mathbf{j} &= \rho \mathbf{u} \\ \frac{\partial \rho}{\partial t} &= - \nabla \cdot (\rho \mathbf{u}) \,. \end{align} In one space dimension this can be put into the form of a homogeneous first-order
quasilinear hyperbolic equation: y_t + A(y) y_x = 0 where the dependent variable is called the
density of a
conserved quantity, and is called the
current Jacobian, and the
subscript notation for partial derivatives has been employed. The more general inhomogeneous case: y_t + A(y) y_x = s is not a conservation equation but the general kind of
balance equation describing a
dissipative system. The dependent variable is called a
nonconserved quantity, and the inhomogeneous term is the-
source, or
dissipation. For example, balance equations of this kind are the momentum and energy
Navier-Stokes equations, or the
entropy balance for a general
isolated system. In the
one-dimensional space a conservation equation is a first-order
quasilinear hyperbolic equation that can be put into the
advection form: y_t + a(y) y_x = 0 where the dependent variable is called the density of the
conserved (scalar) quantity, and is called the
current coefficient, usually corresponding to the
partial derivative in the conserved quantity of a
current density of the conserved quantity : a(y) = j_y (y) In this case since the
chain rule applies: j_x = j_y (y) y_x = a(y) y_x the conservation equation can be put into the current density form: y_t + j_x (y) = 0 In a
space with more than one dimension the former definition can be extended to an equation that can be put into the form: y_t + \mathbf a(y) \cdot \nabla y = 0 where the
conserved quantity is , denotes the
scalar product, is the
nabla operator, here indicating a
gradient, and is a vector of current coefficients, analogously corresponding to the
divergence of a vector current density associated to the conserved quantity : y_t + \nabla \cdot \mathbf j(y) = 0 This is the case for the
continuity equation: \rho_t + \nabla \cdot (\rho \mathbf u) = 0 Here the conserved quantity is the
mass, with
density and current density , identical to the
momentum density, while is the
flow velocity. In the
general case a conservation equation can be also a system of this kind of equations (a
vector equation) in the form: \mathbf y_t + \mathbf A(\mathbf y) \cdot \nabla \mathbf y = \mathbf 0 where is called the
conserved (
vector) quantity, is its
gradient, is the
zero vector, and is called the
Jacobian of the current density. In fact as in the former scalar case, also in the vector case
A(
y) usually corresponding to the Jacobian of a
current density matrix : \mathbf A( \mathbf y) = \mathbf J_{\mathbf y} (\mathbf y)and the conservation equation can be put into the form: \mathbf y_t + \nabla \cdot \mathbf J (\mathbf y)= \mathbf 0 For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:\begin{align} \nabla\cdot \mathbf u &= 0 \, , & \frac{\partial \mathbf u}{\partial t} + \mathbf u \cdot \nabla \mathbf u + \nabla s &= \mathbf{0}, \end{align} where: • is the
flow velocity vector, with components in a N-dimensional space , • is the specific
pressure (pressure per unit
density) giving the
source term, It can be shown that the conserved (vector) quantity and the current
density matrix for these equations are respectively: \begin{align} \mathbf{y} &= \begin{pmatrix} 1 \\ \mathbf u \end{pmatrix}; & \mathbf{J} &= \begin{pmatrix}\mathbf u\\ \mathbf u \otimes \mathbf u + s \mathbf I\end{pmatrix}; \end{align} where \otimes denotes the
outer product. ==Integral and weak forms==