kb/data/en.wikipedia.org/wiki/Conservation_law-2.md

---
title: "Conservation law"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/Conservation_law"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T13:31:25.992814+00:00"
instance: "kb-cron"
---

Here the conserved quantity is the mass, with density ρ(r,t) and current density ρu, identical to the momentum density, while u(r, t) 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:
  
    
      
        
          
            y
          
          
            t
          
        
        +
        
          A
        
        (
        
          y
        
        )
        ⋅
        ∇
        
          y
        
        =
        
          0
        
      
    
    {\displaystyle \mathbf {y} _{t}+\mathbf {A} (\mathbf {y} )\cdot \nabla \mathbf {y} =\mathbf {0} }
  
where y is called the conserved (vector) quantity, ∇y is its gradient, 0 is the zero vector, and A(y) 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 J(y):
  
    
      
        
          A
        
        (
        
          y
        
        )
        =
        
          
            J
          
          
            
              y
            
          
        
        (
        
          y
        
        )
      
    
    {\displaystyle \mathbf {A} (\mathbf {y} )=\mathbf {J} _{\mathbf {y} }(\mathbf {y} )}
  
and the conservation equation can be put into the form:
  
    
      
        
          
            y
          
          
            t
          
        
        +
        ∇
        ⋅
        
          J
        
        (
        
          y
        
        )
        =
        
          0
        
      
    
    {\displaystyle \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:
  
    
      
        
          
            
              
                ∇
                ⋅
                
                  u
                
              
              
                
                =
                0
                
                ,
              
              
                
                  
                    
                      ∂
                      
                        u
                      
                    
                    
                      ∂
                      t
                    
                  
                
                +
                
                  u
                
                ⋅
                ∇
                
                  u
                
                +
                ∇
                s
              
              
                
                =
                
                  0
                
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {u} &=0\,,&{\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} +\nabla s&=\mathbf {0} ,\end{aligned}}}
  

where:

u is the flow velocity vector, with components in a N-dimensional space u1, u2, ..., uN,
s 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:

  
    
      
        
          
            
              
                
                  y
                
              
              
                
                =
                
                  
                    (
                    
                      
                        
                          1
                        
                      
                      
                        
                          
                            u
                          
                        
                      
                    
                    )
                  
                
                ;
              
              
                
                  J
                
              
              
                
                =
                
                  
                    (
                    
                      
                        
                          
                            u
                          
                        
                      
                      
                        
                          
                            u
                          
                          ⊗
                          
                            u
                          
                          +
                          s
                          
                            I
                          
                        
                      
                    
                    )
                  
                
                ;
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\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{aligned}}}
  

where 
  
    
      
        ⊗
      
    
    {\displaystyle \otimes }
  
 denotes the outer product.

== Integral and weak forms ==
Conservation equations can usually also be expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions. By integrating in any space-time domain the current density form in 1-D space:
  
    
      
        
          y
          
            t
          
        
        +
        
          j
          
            x
          
        
        (
        y
        )
        =
        0
      
    
    {\displaystyle y_{t}+j_{x}(y)=0}
  
and by using Green's theorem, the integral form is:
  
    
      
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        y
        
        d
        x
        +
        
          ∫
          
            0
          
          
            ∞
          
        
        j
        (
        y
        )
        
        d
        t
        =
        0
      
    
    {\displaystyle \int _{-\infty }^{\infty }y\,dx+\int _{0}^{\infty }j(y)\,dt=0}
  

In a similar fashion, for the scalar multidimensional space, the integral form is:
  
    
      
        ∮
        
          [
          
            y
            
            
              d
              
                N
              
            
            
              r
            
            +
            j
            (
            y
            )
            
            d
            t
          
          ]
        
        =
        0
      
    
    {\displaystyle \oint \left[y\,d^{N}\mathbf {r} +j(y)\,dt\right]=0}
  
where the line integration is performed along the boundary of the domain, in an anticlockwise manner.
Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is:

  
    
      
        
          ∫
          
            0
          
          
            ∞
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          [
          
            
              ϕ
              
                t
              
            
            y
            +
            
              ϕ
              
                x
              
            
            j
            (
            y
            )
          
          ]
        
        d
        x
        
        d
        t
        =
        −
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        ϕ
        (
        x
        ,
        0
        )
        y
        (
        x
        ,
        0
        )
        
        d
        x
      
    
    {\displaystyle \int _{0}^{\infty }\int _{-\infty }^{\infty }\left[\phi _{t}y+\phi _{x}j(y)\right]dx\,dt=-\int _{-\infty }^{\infty }\phi (x,0)y(x,0)\,dx}
  

In the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.

== See also ==
Invariant (physics)
Momentum
Cauchy momentum equation
Energy
Conservation of energy and the First law of thermodynamics
Conservative system
Conserved quantity
Some kinds of helicity are conserved in dissipationless limit: hydrodynamical helicity, magnetic helicity, cross-helicity.
Principle of mutability
Conservation law of the Stress–energy tensor
Riemann invariant
Philosophy of physics
Totalitarian principle
Convection–diffusion equation
Uniformity of nature

=== Examples and applications ===
Advection
Mass conservation, or Continuity equation
Charge conservation
Euler equations (fluid dynamics)
inviscid Burgers' equation
Kinematic wave
Conservation of energy
Traffic flow

== Notes ==

== References ==
Philipson, Schuster, Modeling by Nonlinear Differential Equations: Dissipative and Conservative Processes, World Scientific Publishing Company 2009.
Victor J. Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpt. 12 is a gentle introduction to symmetry, invariance, and conservation laws.
E. Godlewski and P.A. Raviart, Hyperbolic systems of conservation laws, Ellipses, 1991.

== External links ==
 Media related to Conservation laws at Wikimedia Commons
Conservation Laws – Ch. 11–15 in an online textbook