Differential forms Conservation law

in continuum mechanics, general form of exact conservation law given continuity equation. example, conservation of electric charge q is

∂
ρ


∂
t



=
−
∇
⋅

j




{\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \,}

where ∇⋅ divergence operator, ρ density of q (amount per unit volume), j flux of q (amount crossing unit area in unit time), , t time.

if assume motion u of charge continuous function of position , time, then

j

=
ρ

u



{\displaystyle \mathbf {j} =\rho \mathbf {u} }









∂
ρ


∂
t



=
−
∇
⋅
(
ρ

u

)

.


{\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot (\rho \mathbf {u} )\,.}

in 1 space dimension can put form of homogeneous first-order quasilinear hyperbolic equation:

y

t


+
a
(
y
)

y

x


=
0


{\displaystyle y_{t}+a(y)y_{x}=0}

where dependent variable y called density of conserved quantity, , a(y) called current jacobian, , subscript notation partial derivatives has been employed. more general inhomogeneous case:

y

t


+
a
(
y
)

y

x


=
s


{\displaystyle y_{t}+a(y)y_{x}=s}

is not conservation equation general kind of balance equation describing dissipative system. dependent variable y called nonconserved quantity, , inhomogeneous term s(y,x,t) the-source, or dissipation. example, balance equations of kind momentum , energy navier-stokes equations, or entropy balance general isolated system.

in one-dimensional space conservation equation first-order quasilinear hyperbolic equation can put advection form:

y

t


+
a
(
y
)

y

x


=
0


{\displaystyle y_{t}+a(y)y_{x}=0}

where dependent variable y(x,t) called density of conserved (scalar) quantity (c.q.(d.) = conserved quantity (density)), , a(y) called current coefficient, corresponding partial derivative in conserved quantity of current density (c.d.) of conserved quantity j(y):

a
(
y
)
=

j

y


(
y
)


{\displaystyle a(y)=j_{y}(y)}

in case since chain rule applies:

j

x


=

j

y


(
y
)

y

x


=
a
(
y
)

y

x




{\displaystyle j_{x}=j_{y}(y)y_{x}=a(y)y_{x}}

the conservation equation can put current density form:

y

t


+

j

x


(
y
)
=
0


{\displaystyle y_{t}+j_{x}(y)=0}

in space more 1 dimension former definition can extended equation can put form:

y

t


+

a

(
y
)
⋅
∇
y
=
0


{\displaystyle y_{t}+\mathbf {a} (y)\cdot \nabla y=0}

where conserved quantity y(r,t),



⋅


{\displaystyle \cdot }

denotes scalar product, ∇ nabla operator, here indicating gradient, , a(y) vector of current coefficients, analogously corresponding divergence of vector c.d. associated c.q. j(y):

y

t


+
∇
⋅

j

(
y
)
=
0


{\displaystyle y_{t}+\nabla \cdot \mathbf {j} (y)=0}

this case continuity equation:

ρ

t


+
∇
⋅
(
ρ

u

)
=
0


{\displaystyle \rho _{t}+\nabla \cdot (\rho \mathbf {u} )=0}

here conserved quantity mass, density ρ(r,t) , current density ρu, identical momentum density, while u(r,t) flow velocity.

in general case conservation equation can system of kind of equations (a vector equation) in form:

y


t


+

a

(

y

)
⋅
∇

y

=

0



{\displaystyle \mathbf {y} _{t}+\mathbf {a} (\mathbf {y} )\cdot \nabla \mathbf {y} =\mathbf {0} }

where y called conserved (vector) quantity, ∇ y gradient, 0 0 vector, , a(y) called jacobian of current density. in fact in former scalar case, in vector case a(y) corresponding jacobian of current density matrix j(y):

a

(

y

)
=


j



y



(

y

)


{\displaystyle \mathbf {a} (\mathbf {y} )=\mathbf {j} _{\mathbf {y} }(\mathbf {y} )}

and conservation equation can put form:

y


t


+
∇
⋅

j

(

y

)
=

0



{\displaystyle \mathbf {y} _{t}+\nabla \cdot \mathbf {j} (\mathbf {y} )=\mathbf {0} }

for example, case euler equations (fluid dynamics). in simple incompressible case are:

∇
⋅


u


=
0







∂


u




∂
t



+


u


⋅
∇


u


+
∇
s
=


0


,






{\displaystyle {\begin{aligned}\nabla \cdot {\mathbf {u}}=0\\[1.2ex]{\partial {\mathbf {u}} \over \partial t}+{\mathbf {u}}\cdot \nabla {\mathbf {u}}+\nabla s={\mathbf {0}},\end{aligned}}}

where:

u flow velocity vector, components in n-dimensional space u1, u2 ... un,
s specific pressure (pressure per unit density) giving source term,

it can shown conserved (vector) quantity , c.d. matrix these equations respectively:

y


=


(



1






u





)


;



j


=


(





u








u


⊗


u


+
s


i





)


;



{\displaystyle {\mathbf {y}}={\begin{pmatrix}1\\{\mathbf {u}}\end{pmatrix}};\qquad {\mathbf {j}}={\begin{pmatrix}{\mathbf {u}}\\{\mathbf {u}}\otimes {\mathbf {u}}+s{\mathbf {i}}\end{pmatrix}};\qquad }

where



⊗


{\displaystyle \otimes }

denotes outer product.