Conservation Laws for Nonlinear Equations:Theory, Computation, and Examples
Alexei Cheviakov
Department of Mathematics and Statistics,University of Saskatchewan, Saskatoon, Canada
SeminarFebruary 2015
A. Cheviakov (Math&Stat) Conservation Laws February 2015 1 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 2 / 49
Collaborators
G. Bluman, University of British Columbia, Canada
J.-F. Ganghoffer, LEMTA - ENSEM, Universite de Lorraine, Nancy, France
M. Oberlack, TU Darmstadt, Germany
P. Popovych, Wolfgang Pauli Institute, University of Vienna, Austria
S. St. Jean, graduate student, University of Saskatchewan, Canada
A. Cheviakov (Math&Stat) Conservation Laws February 2015 3 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 4 / 49
Definitions
Variables:
Independent: x = (x1, x2, ..., xn) or (t, x1, x2, ...).
Dependent: u = (u1(x), u2(x), ..., um(x)) or (u(x), v(x), ...).
Partial derivatives:
Notation:∂uk
∂xm= uk
xm = ukm.
All first-order partial derivatives: ∂u.
All pth-order partial derivatives: ∂pu.
Differential functions:
A differential equation is an algebraic equation on components of x,u, ∂u, . . . .
A differential function is an expression that may involve independent and dependentvariables, and derivatives of dependent variables to some order.
F [u] = F (x,u, ∂u, . . . , ∂pu).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 5 / 49
Definitions (ctd.)
The total derivative of a differential function:
A basic chain rule.
Example: u = u(x , y), g [u] = g(x , y , u, ux), then
Dxg [u] ≡ ∂
∂xg(x , y , u, ux)
=∂g
∂x+∂g
∂uux +
∂g
∂uxuxx .
A. Cheviakov (Math&Stat) Conservation Laws February 2015 6 / 49
Local Conservation Laws
Conservation laws
A local conservation law: a divergence expression equal to zero,
DiΨi [u] ≡ div Ψi[u] = 0.
For equations involving time evolution:
Dt Θ[u] + divx Ψ[u] = 0.
Θ[u]: conserved density.
Ψ[u]: flux vector.
Global conserved quantity (integral of motion)
Dt
∫V
Θ dV = 0, if
∮∂V
Ψ[u] · dS = 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 7 / 49
Local Conservation Laws
Conservation laws
A local conservation law: a divergence expression equal to zero,
DiΨi [u] ≡ div Ψi[u] = 0.
For equations involving time evolution:
Dt Θ[u] + divx Ψ[u] = 0.
Θ[u]: conserved density.
Ψ[u]: flux vector.
Global conserved quantity (integral of motion)
Dt
∫V
Θ dV = 0, if
∮∂V
Ψ[u] · dS = 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 7 / 49
ODE Models: Conserved Quantities
An ODE:
Dependent variable: u = u(t);A conservation law
DtF (t, u, u′, ...) =d
dtF (t, u, u′, ...) = 0
yields a conserved quantity (constant of motion):
F (t, u, u′, ...) = C = const.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 8 / 49
ODE Models: Conserved Quantities
An ODE:
Dependent variable: u = u(t);A conservation law
DtF (t, u, u′, ...) =d
dtF (t, u, u′, ...) = 0
yields a conserved quantity (constant of motion):
F (t, u, u′, ...) = C = const.
Example: Harmonic oscillator, spring-mass system
Independent variable: t, dependent: x(t).
ODE: x(t) + ω2x(t) = 0; ω2 = k/m = const.
Conservation law:d
dt
(mx2(t)
2+
kx2(t)
2
)= 0.
Conserved quantity: energy.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 8 / 49
ODE Models: Conserved Quantities
Example: ODE integration
An ODE:
K ′′′(x) =−2 (K ′′(x))
2K(x)− (K ′(x))
2K ′′(x)
K(x)K ′(x).
Three independent conserved quantities:
KK ′′
(K ′)2= C1,
KK ′′ lnK
(K ′)2− lnK ′ = C2,
xKK ′′ + KK ′
(K ′)2− x = C3
yield complete ODE integration.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 9 / 49
ODE Models: Conserved Quantities
Example: ODE integration
An ODE:
K ′′′(x) =−2 (K ′′(x))
2K(x)− (K ′(x))
2K ′′(x)
K(x)K ′(x).
Three independent conserved quantities:
KK ′′
(K ′)2= C1,
KK ′′ lnK
(K ′)2− lnK ′ = C2,
xKK ′′ + KK ′
(K ′)2− x = C3
yield complete ODE integration.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 9 / 49
PDE Models
Example: small string/rod oscillations, 1D wave equation
Independent variables: x , t;
Dependent: u(x , t).
Wave equation:
utt = c2uxx , c2 = const = T/ρ (for a string).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 10 / 49
PDE Models
Example: small string/rod oscillations, 1D wave equation
Independent variables: x , t;
Dependent: u(x , t).
Wave equation:
utt = c2uxx , c2 = const = T/ρ (for a string).
Conservation of momentum:
Local conservation law: Dt(ρut)−Dx(Tux) = 0;
Global conserved quantity: total momentum
M =
∫ b
a
ρut dx = const,
for Neumann homogeneous problems with ux(a, t) = ux(b, t) = 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 10 / 49
PDE Models
Example: small string/rod oscillations, 1D wave equation
Independent variables: x , t;
Dependent: u(x , t).
Wave equation:
utt = c2uxx , c2 = const = T/ρ (for a string).
Conservation of energy:
Local conservation law:
Dt
(ρu2
t
2+
Tu2x
2
)−Dx(Tutux) = 0;
Global conserved quantity: total energy
E =
∫ b
a
(ρu2
t
2+
Tu2x
2
)dx = const,
for both Neumann and Dirichlet homogeneous problems.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 10 / 49
Applications of Conservation Laws
ODEs
Constants of motion.
Integration.
PDEs
Rates of change of physical variables; constants of motion.
Differential constraints (divergence-free or irrotational fields, etc.).
Analysis: existence, uniqueness, stability.
An infinite number of conservation laws can indicate integrability / linearization.
Potentials, stream functions, etc.
Finite element/finite volume numerical methods (DG, etc.) may require conservedforms.
Numerical method testing.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 11 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 12 / 49
Variational Principles
Action integral
J[U] =
∫Ω
L(x,U, ∂U, . . . , ∂kU) dx .
Principle of extremal action
Variation of U: U(x)→ U(x) + δU(x); δU(x) = εv(x); δU(x)∣∣∂Ω
= 0.
Variation of action: δJ ≡ J[U + εv]− J[U] =∫
Ω(δL) dx = o(ε).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 13 / 49
Variational Principles
Action integral
J[U] =
∫Ω
L(x,U, ∂U, . . . , ∂kU) dx .
Principle of extremal action
Variation of U: U(x)→ U(x) + δU(x); δU(x) = εv(x); δU(x)∣∣∂Ω
= 0.
Variation of action: δJ ≡ J[U + εv]− J[U] =∫
Ω(δL) dx = o(ε).
Variation of the Lagrangian
δL = L(x,U + εv, ∂U + ε∂v, . . . , ∂kU + ε∂kv)− L(x,U, ∂U, . . . , ∂kU)
= ε
(∂L[U]
∂Uσvσ +
∂L[U]
∂Uσjvσj + · · ·+ ∂L[U]
∂Uσj1···jkvσj1···jk
)+ O(ε2)
by parts= ε(vσEUσ (L[U])) + div(...) + O(ε2)
A. Cheviakov (Math&Stat) Conservation Laws February 2015 13 / 49
Variational Principles
Action integral
J[U] =
∫Ω
L(x,U, ∂U, . . . , ∂kU) dx .
Principle of extremal action
Variation of U: U(x)→ U(x) + δU(x); δU(x) = εv(x); δU(x)∣∣∂Ω
= 0.
Variation of action: δJ ≡ J[U + εv]− J[U] =∫
Ω(δL) dx = o(ε).
Euler-Lagrange equations, Euler operators:
EUσ (L[U]) =∂L[U]
∂Uσ+ · · ·+ (−1)kDj1 · · ·Djk
∂L[U]
∂Uσj1···jk= 0,
σ = 1, . . . ,m.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 13 / 49
Variational Principles
Example 1: Harmonic oscillator, U = x = x(t)
L =1
2mx2 − 1
2kx2, ExL = −m(x + ω2x) = 0, ω2 = k/m.
Example 2: Wave equation for U = u(x , t)
L =1
2ρut
2 − 1
2Tux
2, EuL = −ρ(utt − c2uxx) = 0, c2 = T/ρ.
A number of physical non-dissipative systems have a variational formulation.
The vast majority of PDE systems do not have a variational formulation.
A PDE system follows from a variational principle (as it stands) ⇔ the linearizationoperator is self-adjoint (symmetric).
# equations = # unknowns.
For a single PDE, only even-order derivatives.
The system has to be written in a “right” way!
No systematic way to tell if a given system has a variational formulation.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 14 / 49
Variational Principles
Example 1: Harmonic oscillator, U = x = x(t)
L =1
2mx2 − 1
2kx2, ExL = −m(x + ω2x) = 0, ω2 = k/m.
Example 2: Wave equation for U = u(x , t)
L =1
2ρut
2 − 1
2Tux
2, EuL = −ρ(utt − c2uxx) = 0, c2 = T/ρ.
A number of physical non-dissipative systems have a variational formulation.
The vast majority of PDE systems do not have a variational formulation.
A PDE system follows from a variational principle (as it stands) ⇔ the linearizationoperator is self-adjoint (symmetric).
# equations = # unknowns.
For a single PDE, only even-order derivatives.
The system has to be written in a “right” way!
No systematic way to tell if a given system has a variational formulation.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 14 / 49
Variational Principles
Example 1: Harmonic oscillator, U = x = x(t)
L =1
2mx2 − 1
2kx2, ExL = −m(x + ω2x) = 0, ω2 = k/m.
Example 2: Wave equation for U = u(x , t)
L =1
2ρut
2 − 1
2Tux
2, EuL = −ρ(utt − c2uxx) = 0, c2 = T/ρ.
A number of physical non-dissipative systems have a variational formulation.
The vast majority of PDE systems do not have a variational formulation.
A PDE system follows from a variational principle (as it stands) ⇔ the linearizationoperator is self-adjoint (symmetric).
# equations = # unknowns.
For a single PDE, only even-order derivatives.
The system has to be written in a “right” way!
No systematic way to tell if a given system has a variational formulation.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 14 / 49
Self-adjointness
PDE linearization
Given PDE or system: R[u] = 0.
Linearized system (Frechet derivative): L[u]v(x) =d
dε
∣∣∣ε=0
R[u + εv] = 0.
Adjoint Linearized system:
w(x) · (L[u] v(x))by parts
= (L∗[u] w(x)) · v(x) + (divergence).
Self-adjointness
Given system R[u] = 0 is self-adjoint if
L[u] v(x) = L∗[u] v(x).
Homotopy Formula for a Lagrangian:
L =
∫ 1
0
u ·R[λu] dλ.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 15 / 49
Self-adjointness
PDE linearization
Given PDE or system: R[u] = 0.
Linearized system (Frechet derivative): L[u]v(x) =d
dε
∣∣∣ε=0
R[u + εv] = 0.
Adjoint Linearized system:
w(x) · (L[u] v(x))by parts
= (L∗[u] w(x)) · v(x) + (divergence).
Self-adjointness
Given system R[u] = 0 is self-adjoint if
L[u] v(x) = L∗[u] v(x).
Homotopy Formula for a Lagrangian:
L =
∫ 1
0
u ·R[λu] dλ.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 15 / 49
Self-adjointness
Example 1: Wave equation for u(x , t)
R[u] = utt − c2uxx = 0;
Linearization (already linear!)
L[u] v(x , t) = vtt − c2vxx = 0;
Adjoint linearization operator:
w(x , t) L[u] v(x , t) = w(vtt−c2vxx) = (wtt − c2wxx)v(x , t)+(vtw−vwt)t−c2(vxw−vwx)x ;
Result:L∗[u] v(x , t) = L[u] v(x , t),
so R[u] is self-adjoint.
Lagrangian:
L =1
2ut
2 − 1
2c2ux
2.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 16 / 49
Self-adjointness
Example 2:
Heat equation for u(x , t): R[u] = ut − uxx = 0.
Linearization: L[u] v(x , t) = vt − vxx = 0.
Adjoint linearization operator: L∗[u]w(x , t) = −wt − wxx = 0,
NOT self-adjoint!
Append the adjoint:
R[u1, u2] = u1t − u1
xx = 0, − u2t − u2
xx = 0.Self-adjoint!
Lagrangian: L = 12
(−u1(u2
t − u2xx) + u2(u1
t − u1xx)).
Euler-Lagrange equations:
Eu1 (L) = −u2t − u2
xx = R2, Eu2 (L) = u1t − u1
xx = R1.
This technique can be used to make any PDE system self-adjoint.
Non-physical Lagrangian (pseudo-Lagrangian).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 17 / 49
Self-adjointness
Example 2:
Heat equation for u(x , t): R[u] = ut − uxx = 0.
Linearization: L[u] v(x , t) = vt − vxx = 0.
Adjoint linearization operator: L∗[u]w(x , t) = −wt − wxx = 0,
NOT self-adjoint!
Append the adjoint:
R[u1, u2] = u1t − u1
xx = 0, − u2t − u2
xx = 0.Self-adjoint!
Lagrangian: L = 12
(−u1(u2
t − u2xx) + u2(u1
t − u1xx)).
Euler-Lagrange equations:
Eu1 (L) = −u2t − u2
xx = R2, Eu2 (L) = u1t − u1
xx = R1.
This technique can be used to make any PDE system self-adjoint.
Non-physical Lagrangian (pseudo-Lagrangian).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 17 / 49
Self-adjointness
Example 2:
Heat equation for u(x , t): R[u] = ut − uxx = 0.
Linearization: L[u] v(x , t) = vt − vxx = 0.
Adjoint linearization operator: L∗[u]w(x , t) = −wt − wxx = 0,
NOT self-adjoint!
Append the adjoint:
R[u1, u2] = u1t − u1
xx = 0, − u2t − u2
xx = 0.Self-adjoint!
Lagrangian: L = 12
(−u1(u2
t − u2xx) + u2(u1
t − u1xx)).
Euler-Lagrange equations:
Eu1 (L) = −u2t − u2
xx = R2, Eu2 (L) = u1t − u1
xx = R1.
This technique can be used to make any PDE system self-adjoint.
Non-physical Lagrangian (pseudo-Lagrangian).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 17 / 49
Self-adjointness
Example 3:
Nonlinear PDE for u(x , t): R[u] = utt + 2uxuxx + u2x = 0.
Linearization: L[u] v(x , t) = vtt + 2uxvxx + (2uxx + 2ux)vx = 0.
NOT self-adjoint!
Modified PDE
Nonlinear PDE for u(x , t): R[u] = ex [utt + 2uxuxx + u2x ] = 0.
This one turns out to be self-adjoint!
A. Cheviakov (Math&Stat) Conservation Laws February 2015 18 / 49
Self-adjointness
Example 3:
Nonlinear PDE for u(x , t): R[u] = utt + 2uxuxx + u2x = 0.
Linearization: L[u] v(x , t) = vtt + 2uxvxx + (2uxx + 2ux)vx = 0.
NOT self-adjoint!
Modified PDE
Nonlinear PDE for u(x , t): R[u] = ex [utt + 2uxuxx + u2x ] = 0.
This one turns out to be self-adjoint!
A. Cheviakov (Math&Stat) Conservation Laws February 2015 18 / 49
Self-adjointness
Example 4:
KdV for u(x , t) R[u] = ut + uux + uxxx = 0.
Odd-order, clearly NOT self-adjoint.
... a differential substitution:
u = qx , R[q] = qxt + qxqxx + qxxxx = 0;
Self-adjoint!
Lagrangian for R[q]: L =1
2q2xx −
1
6q3x −
1
2qxqt .
Result:
For a given PDE/system, it is not simple to conclude whether it follows from avariational principle.
Much depends on the “right” writing.Tricks can make equations variational...
A feasible tool: comparison of local variational symmetries and local conservationlaws.
This is based on the first Noether’s theorem.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 19 / 49
Self-adjointness
Example 4:
KdV for u(x , t) R[u] = ut + uux + uxxx = 0.
Odd-order, clearly NOT self-adjoint.
... a differential substitution:
u = qx , R[q] = qxt + qxqxx + qxxxx = 0;
Self-adjoint!
Lagrangian for R[q]: L =1
2q2xx −
1
6q3x −
1
2qxqt .
Result:
For a given PDE/system, it is not simple to conclude whether it follows from avariational principle.
Much depends on the “right” writing.Tricks can make equations variational...
A feasible tool: comparison of local variational symmetries and local conservationlaws.
This is based on the first Noether’s theorem.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 19 / 49
Self-adjointness
Example 4:
KdV for u(x , t) R[u] = ut + uux + uxxx = 0.
Odd-order, clearly NOT self-adjoint.
... a differential substitution:
u = qx , R[q] = qxt + qxqxx + qxxxx = 0;
Self-adjoint!
Lagrangian for R[q]: L =1
2q2xx −
1
6q3x −
1
2qxqt .
Result:
For a given PDE/system, it is not simple to conclude whether it follows from avariational principle.
Much depends on the “right” writing.Tricks can make equations variational...
A feasible tool: comparison of local variational symmetries and local conservationlaws.
This is based on the first Noether’s theorem.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 19 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 20 / 49
Symmetries of Differential Equations
Consider a general DE system
Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
with variables x = (x1, ..., xn), u = (u1, ..., um).
Definition
A one-parameter Lie group of point transformations
x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2)
(with the parameter a) is a point symmetry of Rσ[u] if the equation is the same in newvariables x∗, u∗.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 21 / 49
Symmetries of Differential Equations
Consider a general DE system
Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
with variables x = (x1, ..., xn), u = (u1, ..., um).
Definition
A one-parameter Lie group of point transformations
x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2)
(with the parameter a) is a point symmetry of Rσ[u] if the equation is the same in newvariables x∗, u∗.
Example 1: translations
The translationx∗ = x + C , t∗ = t, u∗ = u (C ∈ R)
leaves the KdV equation invariant:
ut + uux + uxxx = 0 = u∗t∗ + u∗u∗x∗ + u∗x∗x∗x∗ .
A. Cheviakov (Math&Stat) Conservation Laws February 2015 21 / 49
Symmetries of Differential Equations
Consider a general DE system
Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
with variables x = (x1, ..., xn), u = (u1, ..., um).
Definition
A one-parameter Lie group of point transformations
x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2)
(with the parameter a) is a point symmetry of Rσ[u] if the equation is the same in newvariables x∗, u∗.
Example 2: scaling
The scaling:x∗ = αx , t∗ = α3t, u∗ = αu (α ∈ R)
also leaves the KdV equation invariant:
ut + uux + uxxx = 0 = u∗t∗ + u∗u∗x∗ + u∗x∗x∗x∗ .
A. Cheviakov (Math&Stat) Conservation Laws February 2015 21 / 49
Variational Symmetries
Consider a general DE system
Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
that follows from a variational principle with J[u] =∫
ΩL[u] dx .
Definition
A symmetry of Rσ[u] given by
x∗ = f (x,u; a) = x + a ξ(x,u) + O(a2),u∗ = g(x,u; a) = u + a η(x,u) + O(a2)
is a variational symmetry of Rσ[u] if it preserves the action J[u].
A. Cheviakov (Math&Stat) Conservation Laws February 2015 22 / 49
Variational Symmetries
Consider a general DE system
Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
that follows from a variational principle with J[u] =∫
ΩL[u] dx .
Definition
A symmetry of Rσ[u] given by
x∗ = f (x,u; a) = x + a ξ(x,u) + O(a2),u∗ = g(x,u; a) = u + a η(x,u) + O(a2)
is a variational symmetry of Rσ[u] if it preserves the action J[u].
Example 1: translations for the wave equation
utt = c2uxx , L =1
2ut
2 − c2
2ux
2.
The translation x∗ = x + C , t∗ = t, u∗ = u is a variational symmetry.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 22 / 49
Variational Symmetries
Consider a general DE system
Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
that follows from a variational principle with J[u] =∫
ΩL[u] dx .
Definition
A symmetry of Rσ[u] given by
x∗ = f (x,u; a) = x + a ξ(x,u) + O(a2),u∗ = g(x,u; a) = u + a η(x,u) + O(a2)
is a variational symmetry of Rσ[u] if it preserves the action J[u].
Example 2: scaling for the wave equation
utt = c2uxx , L =1
2ut
2 − c2
2ux
2.
The scaling x∗ = x , t∗ = t, u∗ = u/α is not a variational symmetry: J∗ = α2J.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 22 / 49
Evolutionary Form of a Local Symmetry
A symmetry (in 1D case)
x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2).
maps a solution u(x) into u∗(x∗), changing both x and u.
(x, u)
(x*, u*)
(x, u**)
(, ) )
(0, ) )
x
u
In the evolutionary form, the same curve mapping does not change x :
x∗∗ = x , u∗∗ = u + a ζ[u] + O(a2),
ζ[u] = η(x , u)− ∂u
∂xξ(x , u).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 23 / 49
Noether’s Theorem (restricted to point symmetries)
Theorem
Given:
1 a PDE system Rσ[u] = 0, σ = 1, . . . ,N, following from a variational principle;
2 a variational symmetry
(x i )∗ = f i (x,u; a) = x i + aξi (x,u) + O(a2),(uσ)∗ = gσ(x,u; a) = uσ + aησ(x,u) + O(a2).
Then the system Rσ[u] has a conservation law DiΦi [u] = 0.
In particular,DiΦ
i [u] ≡ Λσ[u]Rσ[u] = 0,
where the multipliers are given by
Λσ ≡ ζσ[u] = ησ(x,u)− ∂uσ
∂xiξi (x,u).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 24 / 49
Noether’s Theorem: Examples
Example 1: translation symmetry for the harmonic oscillator
Equation: x(t) + ω2x(t) = 0.
Symmetry:t∗ = t + a, ξ = 1;x∗ = x , η = 0,
Multiplier (integrating factor): Λ = η − x(t)ξ = −x ;
Conservation law:
ΛR = −x(x(t) + ω2x(t)) = − d
dt
(x2(t)
2+ω2x2(t)
2
)= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 25 / 49
Noether’s Theorem: Examples
Example 2
Equation: Wave equation utt = c2uxx , u = u(x , t).
Space Translation Symmetry:
t∗ = t, ξt = 0;x∗ = x , ξx = 0,u∗ = u + a, η = 1,
Multiplier: Λ = ζ = η − 0 · ux − 0 · ut = 1;
Conservation law (Momentum):
ΛR = 1(utt − c2uxx) = Dt (ut)− Dx
(c2ux
)= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 26 / 49
Noether’s Theorem: Examples
Example 2
Equation: Wave equation utt = c2uxx , u = u(x , t).
Time Translation Symmetry:
t∗ = t + a, ξt = 1;x∗ = x , ξx = 0,u∗ = u, η = 0,
Multiplier: Λ = ζ = η − 0 · ux − 1 · ut = −ut ;
Conservation law (Energy):
ΛR = −ut(utt − c2uxx) = −[Dt
(u2t
2+ c2 u
2x
2
)− Dx
(c2utux
)]= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 26 / 49
Linearization Operators and Variational Formulation
Recollect:
Given PDE or system: R[u] = 0.
Linearized system (Frechet derivative): L[u]v(x) =d
dε
∣∣∣ε=0
R[u + εv] = 0.
Adjoint Linearized system:
w(x) · (L[u] v(x))by parts
= (L∗[u] w(x)) · v(x) + (divergence).
Facts:
Symmetry components ζσ[u] are solutions of the linearized system.
Conservation law multipliers Λσ[u] are solutions of the adjoint linearized system.
A self-adjointness test:
Check # equations = # unknowns.
In some writing, CL multipliers and symmetries are “similar”?
The test is not systematic... the “correct” writing of the system is not prescribed!
A. Cheviakov (Math&Stat) Conservation Laws February 2015 27 / 49
Linearization Operators and Variational Formulation
Recollect:
Given PDE or system: R[u] = 0.
Linearized system (Frechet derivative): L[u]v(x) =d
dε
∣∣∣ε=0
R[u + εv] = 0.
Adjoint Linearized system:
w(x) · (L[u] v(x))by parts
= (L∗[u] w(x)) · v(x) + (divergence).
Facts:
Symmetry components ζσ[u] are solutions of the linearized system.
Conservation law multipliers Λσ[u] are solutions of the adjoint linearized system.
A self-adjointness test:
Check # equations = # unknowns.
In some writing, CL multipliers and symmetries are “similar”?
The test is not systematic... the “correct” writing of the system is not prescribed!
A. Cheviakov (Math&Stat) Conservation Laws February 2015 27 / 49
Linearization Operators and Variational Formulation
Recollect:
Given PDE or system: R[u] = 0.
Linearized system (Frechet derivative): L[u]v(x) =d
dε
∣∣∣ε=0
R[u + εv] = 0.
Adjoint Linearized system:
w(x) · (L[u] v(x))by parts
= (L∗[u] w(x)) · v(x) + (divergence).
Facts:
Symmetry components ζσ[u] are solutions of the linearized system.
Conservation law multipliers Λσ[u] are solutions of the adjoint linearized system.
A self-adjointness test:
Check # equations = # unknowns.
In some writing, CL multipliers and symmetries are “similar”?
The test is not systematic... the “correct” writing of the system is not prescribed!
A. Cheviakov (Math&Stat) Conservation Laws February 2015 27 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 28 / 49
The Idea of the Direct Construction Method
The Direct Construction Method:
Works for virtually any model.
Does not require variational formulation/Noether’s theorem.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 29 / 49
The Idea of the Direct Construction Method
Definition
The Euler operator with respect to U j :
EU j =∂
∂U j−Di
∂
∂U ji
+ · · ·+ (−1)sDi1 . . .Dis
∂
∂U ji1...is
+ · · · , j = 1, . . . ,m.
Theorem
Let U(x) = (U1, . . . ,Um). The equations EU jF [U] ≡ 0, j = 1, . . . ,m, hold for arbitraryU(x) if and only if
F [U] ≡ DiΨi [U]
for some functions Ψi [U].
Idea:
Seek conservation laws in the characteristic form DiΦi = ΛσR
σ = 0
(based on Hadamard’s lemma for systems of maximal rank).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 30 / 49
The Idea of the Direct Construction Method
Definition
The Euler operator with respect to U j :
EU j =∂
∂U j−Di
∂
∂U ji
+ · · ·+ (−1)sDi1 . . .Dis
∂
∂U ji1...is
+ · · · , j = 1, . . . ,m.
Theorem
Let U(x) = (U1, . . . ,Um). The equations EU jF [U] ≡ 0, j = 1, . . . ,m, hold for arbitraryU(x) if and only if
F [U] ≡ DiΨi [U]
for some functions Ψi [U].
Idea:
Seek conservation laws in the characteristic form DiΦi = ΛσR
σ = 0
(based on Hadamard’s lemma for systems of maximal rank).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 30 / 49
The Idea of the Direct Construction Method
Consider a general system R[u] = 0 of N PDEs.
Direct Construction Method
Specify dependence of multipliers: Λσ = Λσ(x,U, ...), σ = 1, ...,N.
Solve the set of determining equations
EU j (ΛσRσ) ≡ 0, j = 1, . . . ,m,
for arbitrary U(x) (off of solution set!) to find all such sets of multipliers.
Find the corresponding fluxes Φi [U] satisfying the identity
ΛσRσ ≡ DiΦ
i .
Each set of fluxes, multipliers yields a local conservation law
DiΦi [u] = 0,
holding for all solutions u(x) of the given PDE system.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 31 / 49
The Idea of the Direct Construction Method
Example
The KdV equationR[u] = ut + uux + uxxx = 0.
0th-order multipliers
Determining equations:
EU (Λ(x , t,U)(Ut + UUx + Uxxx)) ≡ 0.
Solution:Λ1 = 1, Λ2 = U, Λ3 = tU − x .
Conservation laws:Dt(u) + Dx
(12u2 + uxx
)= 0,
Dt
(12u2)
+ Dx
(13u3 + uuxx − 1
2u2x
)= 0,
Dt
(16u3 − 1
2u2x
)+ Dx
(18u4 − uu2
x + 12(u2uxx + u2
xx)− uxuxxx)
= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 32 / 49
The Idea of the Direct Construction Method
Example
The KdV equationR[u] = ut + uux + uxxx = 0.
1st-order multipliers in x
Form: Λ = Λ(x , t,U,Ux)
Solution: no extra conservation laws.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 32 / 49
The Idea of the Direct Construction Method
Example
The KdV equationR[u] = ut + uux + uxxx = 0.
2nd-order multipliers in x
Form: Λ = Λ(x , t,U,Ux ,Uxx)
Solution: one extra conservation law with
Λ4 = Uxx + 12U2.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 32 / 49
Symbolic Software for Computation of Conservation Laws
Example of use of the GeM package for Maple for the KdV.
Use the module: with(GeM):
Declare variables: gem_decl_vars(indeps=[x,t], deps=[U(x,t)]);
Declare the equation:
gem_decl_eqs([diff(U(x,t),t)=U(x,t)*diff(U(x,t),x)
+diff(U(x,t),x,x,x)],
solve_for=[diff(U(x,t),t)]);
Generate determining equations:
det_eqs:=gem_conslaw_det_eqs([x,t, U(x,t),
diff(U(x,t),x), diff(U(x,t),x,x)]):
Reduce the overdetermined system:
CL_multipliers:=gem_conslaw_multipliers();
simplified_eqs:=DEtools[rifsimp](det_eqs, CL_multipliers, mindim=1);
A. Cheviakov (Math&Stat) Conservation Laws February 2015 33 / 49
Symbolic Software for Computation of Conservation Laws
Example of use of the GeM package for Maple for the KdV.
Solve determining equations:
multipliers_sol:=pdsolve(simplified_eqs[Solved]);
Obtain corresponding conservation law fluxes/densities:
gem_get_CL_fluxes(multipliers_sol, method=*****);
A. Cheviakov (Math&Stat) Conservation Laws February 2015 34 / 49
Completeness of the Direct Construction Method
Extended Kovalevskaya form
A PDE system R[u] = 0 is in extended Kovalevskaya form with respect to anindependent variable x j , if the system is solved for the highest derivative of eachdependent variable with respect to x j , i.e.,
∂sσ
∂(x j)sσuσ = Gσ(x , u, ∂u, . . . , ∂ku), 1 ≤ sσ ≤ k, σ = 1, . . . ,m, (1)
where all derivatives with respect to x j appearing in the right-hand side of each PDE in(1) are of lower order than those appearing on the left-hand side.
Theorem [R. Popovych, A. C.]
Let R[u] = 0 be a PDE system in the extended Kovalevskaya form (1). Then every itslocal conservation law has an equivalent conservation law in the characteristic form,
ΛσRσ ≡ DiΦ
i = 0,
such that neither Λσ nor Φi involve the leading derivatives or their differentialconsequences.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 35 / 49
Completeness of the Direct Construction Method
Extended Kovalevskaya form
A PDE system R[u] = 0 is in extended Kovalevskaya form with respect to anindependent variable x j , if the system is solved for the highest derivative of eachdependent variable with respect to x j , i.e.,
∂sσ
∂(x j)sσuσ = Gσ(x , u, ∂u, . . . , ∂ku), 1 ≤ sσ ≤ k, σ = 1, . . . ,m, (1)
where all derivatives with respect to x j appearing in the right-hand side of each PDE in(1) are of lower order than those appearing on the left-hand side.
Example
The KdV equationR[u] = ut + uux + uxxx = 0
has the extended Kovalevskaya form with respect to t (ut = . . .) or x (uxxx = . . .).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 35 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 36 / 49
A Detailed Example: 2D Incompressible Hyperelasticity
Author's personal copy
A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) 625–639 627
Fig. 1. Material and Eulerian coordinates.
The actual position x of a material point labeled by X ∈ Ω0 at time t is given byx = φ (X, t) , xi = φi (X, t) .
Coordinates X in the reference configuration are commonly referred to as Lagrangian coordinates, and actual coordinatesx as Eulerian coordinates. The deformed body occupies an Eulerian domain Ω = φ(Ω0) ⊂ R3 (Fig. 1). The velocity of amaterial point X is given by
v (X, t) =dxdt
≡dφdt
.
Themappingφmust be sufficiently smooth (the regularity conditions depending on the particular problem). The Jacobianmatrix of the coordinate transformation is given by the deformation gradient
F(X, t) = ∇φ, (1)which is an invertible matrix with components
F ij =
∂φi
∂X j= Fij. (2)
(Throughout the paper, we use Cartesian coordinates and flat space metric tensor g ij= δij, therefore indices of all tensors
can be raised or lowered freely as needed.) The transformation satisfies the orientation preserving conditionJ = det F > 0.
Forces and stress tensorsBy the well-known Cauchy theorem, the force (per unit area) acting on a surface element S within or on the boundary of
the solid body is given in the Eulerian configuration byt = σn,
where n is a unit normal, and σ = σ(x, t) is Cauchy stress tensor (see Fig. 1). The Cauchy stress tensor is symmetric:σ = σT , which is a consequence of the conservation of angular momentum. For an elastic medium undergoing a smoothdeformation under the action of prescribed surface and volumetric forces, the existence and uniqueness of the Cauchy stressσ follows from the conservation ofmomentum (cf. [29, Section 2.2]). The force acting on a surface element S0 in the referenceconfiguration is given by the stress vector
T = PN,
where P is the first Piola–Kirchhoff tensor, related to the Cauchy stress tensor through
P = JσF−T . (3)In (3), (F−T )ij ≡ (F−1)ji is the transpose of the inverse of the deformation gradient.
Hyperelastic materialsA hyperelastic (or Green elastic)material is an ideally elasticmaterial forwhich the stress–strain relationship follows from
a strain energy density function; it is the material model most suited to the analysis of elastomers. In general, the responseof an elastic material is given in terms of the first Piola–Kirchhoff stress tensor by P = P (X, F). A hyperelastic materialassumes the existence of a scalar valued volumetric strain energy function W = W (X, F) in the reference configuration,encapsulating all information regarding the material behavior, and related to the stress tensor through
P = ρ0∂W∂F
, P ij= ρ0
∂W∂Fij
, (4)
where ρ0 = ρ0(X) is the time-independent body density in the reference configuration. The actual density in Euleriancoordinates ρ = ρ(X, t) is time-dependent and is given by
ρ = ρ0/J.
Material picture
A solid body occupies the reference (Lagrangian) volume Ω0 ⊂ R3.
Actual (Eulerian) configuration: Ω ⊂ R3.
Material points are labeled by X ∈ Ω0.
The actual position of a material point: x = x (X, t) ∈ Ω.
Jacobian matrix (deformation gradient): J = det F > 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 37 / 49
A Detailed Example: 2D Incompressible Hyperelasticity
Author's personal copy
A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) 625–639 627
Fig. 1. Material and Eulerian coordinates.
The actual position x of a material point labeled by X ∈ Ω0 at time t is given byx = φ (X, t) , xi = φi (X, t) .
Coordinates X in the reference configuration are commonly referred to as Lagrangian coordinates, and actual coordinatesx as Eulerian coordinates. The deformed body occupies an Eulerian domain Ω = φ(Ω0) ⊂ R3 (Fig. 1). The velocity of amaterial point X is given by
v (X, t) =dxdt
≡dφdt
.
Themappingφmust be sufficiently smooth (the regularity conditions depending on the particular problem). The Jacobianmatrix of the coordinate transformation is given by the deformation gradient
F(X, t) = ∇φ, (1)which is an invertible matrix with components
F ij =
∂φi
∂X j= Fij. (2)
(Throughout the paper, we use Cartesian coordinates and flat space metric tensor g ij= δij, therefore indices of all tensors
can be raised or lowered freely as needed.) The transformation satisfies the orientation preserving conditionJ = det F > 0.
Forces and stress tensorsBy the well-known Cauchy theorem, the force (per unit area) acting on a surface element S within or on the boundary of
the solid body is given in the Eulerian configuration byt = σn,
where n is a unit normal, and σ = σ(x, t) is Cauchy stress tensor (see Fig. 1). The Cauchy stress tensor is symmetric:σ = σT , which is a consequence of the conservation of angular momentum. For an elastic medium undergoing a smoothdeformation under the action of prescribed surface and volumetric forces, the existence and uniqueness of the Cauchy stressσ follows from the conservation ofmomentum (cf. [29, Section 2.2]). The force acting on a surface element S0 in the referenceconfiguration is given by the stress vector
T = PN,
where P is the first Piola–Kirchhoff tensor, related to the Cauchy stress tensor through
P = JσF−T . (3)In (3), (F−T )ij ≡ (F−1)ji is the transpose of the inverse of the deformation gradient.
Hyperelastic materialsA hyperelastic (or Green elastic)material is an ideally elasticmaterial forwhich the stress–strain relationship follows from
a strain energy density function; it is the material model most suited to the analysis of elastomers. In general, the responseof an elastic material is given in terms of the first Piola–Kirchhoff stress tensor by P = P (X, F). A hyperelastic materialassumes the existence of a scalar valued volumetric strain energy function W = W (X, F) in the reference configuration,encapsulating all information regarding the material behavior, and related to the stress tensor through
P = ρ0∂W∂F
, P ij= ρ0
∂W∂Fij
, (4)
where ρ0 = ρ0(X) is the time-independent body density in the reference configuration. The actual density in Euleriancoordinates ρ = ρ(X, t) is time-dependent and is given by
ρ = ρ0/J.
Material picture
Boundary force (per unit area) in Eulerian configuration: t = σn.
Boundary force (per unit area) in Lagrangian configuration: T = PN.
σ = σ(x, t) is the Cauchy stress tensor.
P = JσF−T is the first Piola-Kirchhoff tensor.
Density in reference & actual configuration: ρ0 = ρ0(X), ρ = ρ(X, t) = ρ0/J.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 37 / 49
Full Set of Equations
Equations of motion:
ρ0xtt = div(X )P + ρ0R, J = det
[∂x i
∂X j
]= 1, P ij = −p (F−1)ji + ρ0
∂W
∂Fij.
Neo-Hookean (Hadamard) material: W = a(I1 − 3), I1 = F ikF
ik .
2D case: 3 PDEs, 3 unknowns (x, p).
PDEs in 2D, no forcing:
∂x1
∂X 1
∂x2
∂X 2− ∂x1
∂X 2
∂x2
∂X 1− 1 = 0,
∂2x1
∂t2 −[α
(∂2x1
∂ (X 1)2 +∂2x1
∂ (X 2)2
)− ∂p
∂X 1
∂x2
∂X 2+
∂p
∂X 2
∂x2
∂X 1
]= 0,
∂2x2
∂t2 −[α
(∂2x2
∂ (X 1)2 +∂2x2
∂ (X 2)2
)− ∂p
∂X 2
∂x1
∂X 1+
∂p
∂X 1
∂x1
∂X 2
]= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 38 / 49
Point symmetries
PDEs in 2D, no forcing:
∂x1
∂X 1
∂x2
∂X 2− ∂x1
∂X 2
∂x2
∂X 1− 1 = 0,
∂2x1
∂t2 −[α
(∂2x1
∂ (X 1)2 +∂2x1
∂ (X 2)2
)− ∂p
∂X 1
∂x2
∂X 2+
∂p
∂X 2
∂x2
∂X 1
]= 0,
∂2x2
∂t2 −[α
(∂2x2
∂ (X 1)2 +∂2x2
∂ (X 2)2
)− ∂p
∂X 2
∂x1
∂X 1+
∂p
∂X 1
∂x1
∂X 2
]= 0.
Point symmetries:
Y1 =∂
∂t, Y2 =
∂
∂X 1, Y3 =
∂
∂X 2, Y4 = F1(t)
∂
∂p,
Y5 = X 2 ∂
∂X 1− X 1 ∂
∂ X 2, Y6 = x2 ∂
∂x1− x1 ∂
∂x2,
Y7 = F2(t)∂
∂x1− F ′′2 (t) x1 ∂
∂p, Y8 = F3(t)
∂
∂x2− F ′′3 (t) x2 ∂
∂p,
Y9 = t∂
∂t+ X 1 ∂
∂X 1+ X 2 ∂
∂X 2+ x1 ∂
∂x1+ x2 ∂
∂x2.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 39 / 49
Conservation laws
PDEs in 2D, no forcing:
R1[x, p] =∂x1
∂X 1
∂x2
∂X 2− ∂x1
∂X 2
∂x2
∂X 1− 1 = 0,
R2[x, p] =∂2x1
∂t2 −[α
(∂2x1
∂ (X 1)2 +∂2x1
∂ (X 2)2
)− ∂p
∂X 1
∂x2
∂X 2+
∂p
∂X 2
∂x2
∂X 1
]= 0,
R3[x, p] =∂2x2
∂t2 −[α
(∂2x2
∂ (X 1)2 +∂2x2
∂ (X 2)2
)− ∂p
∂X 2
∂x1
∂X 1+
∂p
∂X 1
∂x1
∂X 2
]= 0.
Conservation laws:
DtΨ + DX 1 Φ1 + DX 1 Φ2 =3∑
i=1
Λi [x, p] R i [x, p].
Note
Extended Kovalevskaya form exists w.r.t. X 1 and X 2, very complicated...
A. Cheviakov (Math&Stat) Conservation Laws February 2015 40 / 49
Conservation laws
PDEs in 2D, no forcing:
R1[x, p] =∂x1
∂X 1
∂x2
∂X 2− ∂x1
∂X 2
∂x2
∂X 1− 1 = 0,
R2[x, p] =∂2x1
∂t2 −[α
(∂2x1
∂ (X 1)2 +∂2x1
∂ (X 2)2
)− ∂p
∂X 1
∂x2
∂X 2+
∂p
∂X 2
∂x2
∂X 1
]= 0,
R3[x, p] =∂2x2
∂t2 −[α
(∂2x2
∂ (X 1)2 +∂2x2
∂ (X 2)2
)− ∂p
∂X 2
∂x1
∂X 1+
∂p
∂X 1
∂x1
∂X 2
]= 0.
Notation:
∂2x1
∂t2 ≡ x1tt ,
∂x1
∂X 2≡ x1
2 ,∂2x2
∂X 1∂X 2≡ x2
12,∂2p
∂X 2∂t≡ p2t , . . .
A. Cheviakov (Math&Stat) Conservation Laws February 2015 40 / 49
Conservation laws
PDEs in 2D, no forcing:
R1[x, p] =∂x1
∂X 1
∂x2
∂X 2− ∂x1
∂X 2
∂x2
∂X 1− 1 = 0,
R2[x, p] =∂2x1
∂t2 −[α
(∂2x1
∂ (X 1)2 +∂2x1
∂ (X 2)2
)− ∂p
∂X 1
∂x2
∂X 2+
∂p
∂X 2
∂x2
∂X 1
]= 0,
R3[x, p] =∂2x2
∂t2 −[α
(∂2x2
∂ (X 1)2 +∂2x2
∂ (X 2)2
)− ∂p
∂X 2
∂x1
∂X 1+
∂p
∂X 1
∂x1
∂X 2
]= 0.
First-order multipliers:
Λ1 = C1(X 1p2 − X 2p1) + C3pt + C4p1 + C5p2 + f1(t) + f ′′2 (t)x1 + f ′′3 (t)x2,
Λ2 = C1(X 2x11 − X 1x1
2 ) + C2x2 − C3x
1t − C4x
11 − C5x
12 + f2(t),
Λ3 = C1(X 2x21 − X 1x2
2 )− C2x1 − C3x
2t − C4x
21 − C5x
22 + f3(t).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 40 / 49
Conservation laws
Conservation of energy:
Λ1 = pt , Λ2 = x1t , Λ3 = x2
t .
Divergence expression:
Dt
(12
(x1t
)2+ 1
2
(x2t
)2+ 1
2α((
x11
)2+(x1
2
)2+(x2
1
)2+(x2
2
)2))
−DX 1
(α(x1t x
11 + x2
t x21
)− px1
t x22 + px1
2 x2t
)−DX 2
(α(x1t x
12 + x2
t x22
)+ px1
t x21 − px1
1 x2t
)= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 41 / 49
Conservation laws
Conservation of generalized momentum in x1
Λ1 = f ′′2 (t)x1, Λ2 = f2(t), Λ3 = 0.
Divergence expression:
Dt
(f2(t)x1
t − f ′2 (t)x1)
−DX 1
(αf2(t)x1
1 − pf2(t)x22 − 1
2f ′′2 (t)
(x1)2
x22
)−DX 2
(αf2(t)x1
2 + pf2(t)x21 + 1
2f ′′2 (t)
(x1)2
x21
)= 0.
Same for x2.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 41 / 49
Conservation laws
Conservation of angular momentum in x3
Λ1 = 0, Λ2 = x2, Λ3 = −x1.
Divergence expression:
Dt
(x1x2
t − x1t x
2)
+DX 1
(α(x1
1 x2 − x1x2
1
)− px1x1
2 − px2x22
)+DX 2
(α(x1
2 x2 − x1x2
2
)+ px1x1
1 + px2x21
)= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 41 / 49
Conservation laws
Generalized incompressibility condition
Λ1 = f1(t), Λ2 = 0, Λ3 = 0.
Divergence expression:
Dt
(∫f1(t) dt
)−DX 1
(f1(t)x1x2
2
)+ DX 2
(f1(t)x1x2
1
)= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 41 / 49
Conservation laws
Momenta in the material frame
Linear momentum: p ≡ ρ0FTxt = (x1t x
11 + x2
t x21 )i + (x1
t x12 + x2
t x22 )j .
Angular momentum: m ≡ X× p = (X 1(x1t x
12 + x2
t x22 )− X 2(x1
t x11 + x2
t x21 ))k.
Conservation of the material momentum in X 1
Λ1 = −p1, Λ2 = x11 , Λ3 = x2
1 .
Divergence expression:
Dt
(x1t x
11 + x2
t x21
)+DX 1
(α((
x12
)2+(x2
2
)2 −(x1
1
)2 −(x2
1
)2)
+ p − 12
(x1t
)2 − 12
(x2t
)2)
−DX 2
(α(x1
1 x12 + x2
1 x22
))= 0.
Same for X 2.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 41 / 49
Conservation laws
Momenta in the material frame
Linear momentum: p ≡ ρ0FTxt = (x1t x
11 + x2
t x21 )i + (x1
t x12 + x2
t x22 )j .
Angular momentum: m ≡ X× p = (X 1(x1t x
12 + x2
t x22 )− X 2(x1
t x11 + x2
t x21 ))k.
Conservation of the material angular momentum in X 3
Λ1 = X 1p2 − X 2p1, Λ2 = X 2x11 − X 1x1
2 , Λ3 = X 2x21 − X 1x2
2 .
Divergence expression:
Dt
[X 2(x1
t x11 + x2
t x21 )− X 1(x1
t x12 + x2
t x22 )]
+DX 1
[12αX 2
(−(x1
1
)2 −(x2
1
)2+(x1
2
)2+(x2
2
)2)
+αX 1(x1
1 x12 + x2
1 x22
)+ X 2p − 1
2X 2(x1t
)2 − 12X 2(x2t
)2]
+DX 2
[12αX 1
(−(x1
1
)2 −(x2
1
)2+(x1
2
)2+(x2
2
)2)
−αX 2(x1
1 x12 + x2
1 x22
)− X 1p + 1
2X 1(x1t
)2+ 1
2X 1(x2t
)2]
= 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 41 / 49
Comparison of Symmetries and Conservation Laws
Evol. Symmetry Multiplier Conserved quantity
Y1 = −x1t
∂
∂x1− x2
t
∂
∂x2− pt
∂
∂pΛ1 = pt , Λ2 = −x1
t , Λ3 = −x2t Energy
Y2 = −x11
∂
∂x1− x2
1
∂
∂x2− p1
∂
∂pΛ1 = p1, Λ2 = −x1
1 , Λ3 = −x21 Lagrangian momentum in X 1
Y3 = −x12
∂
∂x1− x2
2
∂
∂x2− p2
∂
∂pΛ1 = p2, Λ2 = −x1
2 , Λ3 = −x22 Lagrangian momentum in X 2
Y4 = F 1 (t)∂
∂pΛ1 = f 1(t), Λ2 = 0, Λ3 = 0 Generalized incompressiblity
Y5 =(−X 2x1
1 + X 1x12
) ∂
∂x1Λ1 = (X 1p2 − X 2p1), Angular momentum(
−X 2x21 + X 1x2
2
) ∂
∂x2Λ2 = (X 2x1
1 − X 1x12 ), in Lagrangian frame
+(−X 2p1 + X 1p2
) ∂
∂pΛ3 = (X 2x2
1 − X 1x22 )
Y6 = x2 ∂
∂x1− x1 ∂
∂x2Λ1 = 0, Λ2 = x2, Λ3 = −x1 Eulerian angular momentum
Y7 = F2(t)∂
∂x1− F ′′2 (t) x1 ∂
∂pΛ1 = f ′′2 (t)x1, Λ2 = f2(t), Λ3 = 0 Generalized momentum in x1
Y8 = F3(t)∂
∂x2− F ′′3 (t) x2 ∂
∂pΛ1 = f ′′3 (t)x2, Λ2 = 0, Λ3 = f3(t) Generalized momentum in x2
Y9 [Scaling ] No corresponding conservation law
Variational Formulation
Variational Formulation?
“Variational” symmetries and conservation laws coincide!
Lagrangian can be obtained, for example, using the homotopy formula
L =
∫ 1
0
u ·R[λu] dλ, u = (p, x1, x2).
This is non-trivial, since the incompressible equations involve a differential constraint.
Lagrangian:
L = K − P + p(J − 1),
K =1
2
((x1t
)2
+(x2t
)2),
P =1
2α
((x1
1
)2
+(x1
2
)2
+(x2
1
)2
+(x2
2
)2),
J = x11 x
22 − x1
2 x21 .
A. Cheviakov (Math&Stat) Conservation Laws February 2015 43 / 49
Variational Formulation
Variational Formulation?
“Variational” symmetries and conservation laws coincide!
Lagrangian can be obtained, for example, using the homotopy formula
L =
∫ 1
0
u ·R[λu] dλ, u = (p, x1, x2).
This is non-trivial, since the incompressible equations involve a differential constraint.
PDEs:
Ep L = −J + 1 = −R1,
Ex1 L = −x1tt + α(x1
11 + x122)− p1x22 + p2x2
1 = −R2,
Ex2 L = −x2tt + α(x2
11 + x222)− p2x11 + p1x1
2 = −R3.
Conservation laws:
Every conservation law follows from a symmetry.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 43 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 44 / 49
Navier-Stokes Equations in 3D
Navier-Stokes Equations in 3D
PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.
Variables: pressure p(t, x , y , z), velocity u(t, x , y , z) = u1 x + u2 y + u3 z.
No (natural) Lagrangian formulation.
Has extended Kovalevskaya form with respect to, e.g., x .
Maximal order of conservation law is bounded [Gusyatnikova, Yumaguzhin (1989)].
Completeness
What is the full set of admitted conservation laws?
A. Cheviakov (Math&Stat) Conservation Laws February 2015 45 / 49
Navier-Stokes Equations in 3D
Navier-Stokes Equations in 3D
PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.
Variables: pressure p(t, x , y , z), velocity u(t, x , y , z) = u1 x + u2 y + u3 z.
No (natural) Lagrangian formulation.
Has extended Kovalevskaya form with respect to, e.g., x .
Maximal order of conservation law is bounded [Gusyatnikova, Yumaguzhin (1989)].
Direct CL construction, 2nd-order multipliers
Multipliers Λi , i = 1, ..., 4, depending on 56 variables each (up to 2nd-orderderivatives).
Result: generalized momenta, angular momenta, generalized incompressibility [C.,Oberlack (2014)].
A. Cheviakov (Math&Stat) Conservation Laws February 2015 45 / 49
Navier-Stokes Equations in 3D
Navier-Stokes Equations in 3D
PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.
Variables: pressure p(t, x , y , z), velocity u(t, x , y , z) = u1 x + u2 y + u3 z.
No (natural) Lagrangian formulation.
Has extended Kovalevskaya form with respect to, e.g., x .
Maximal order of conservation law is bounded [Gusyatnikova, Yumaguzhin (1989)].
Generalized momentum in x-direction:
Local conservation law:
∂
∂t(f (t)u1) +
∂
∂x
((u1f (t)− xf ′(t))u1 + f (t)(p − νu1
x ))
+∂
∂y
((u1f (t)− xf ′(t))u2 − νf (t)u1
y
)+
∂
∂z
((u1f (t)− xf ′(t))u3 − νf (t)u1
z
)= 0.
Cyclically permute (1, 2, 3), (x , y , z) for the other two components.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 45 / 49
Navier-Stokes Equations in 3D
Navier-Stokes Equations in 3D
PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.
Variables: pressure p(t, x , y , z), velocity u(t, x , y , z) = u1 x + u2 y + u3 z.
No (natural) Lagrangian formulation.
Has extended Kovalevskaya form with respect to, e.g., x .
Maximal order of conservation law is bounded [Gusyatnikova, Yumaguzhin (1989)].
Angular momentum in x-direction:
Local conservation law:
∂
∂t(zu2 − yu3) +
∂
∂x
((zu2 − yu3)u1 + ν(yu3
x − zu2x ))
+∂
∂y
((zu2 − yu3)u2 + zp + ν(yu3
y − zu2y − u3)
)+∂
∂z
((zu2 − yu3)u3 − yp + ν(yu3
z − zu2z + u2)
)= 0.
Cyclically permute (1, 2, 3), (x , y , z) for the other two components.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 45 / 49
Navier-Stokes Equations in 3D
Navier-Stokes Equations in 3D
PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.
Variables: pressure p(t, x , y , z), velocity u(t, x , y , z) = u1 x + u2 y + u3 z.
No (natural) Lagrangian formulation.
Has extended Kovalevskaya form with respect to, e.g., x .
Maximal order of conservation law is bounded [Gusyatnikova, Yumaguzhin (1989)].
Generalized continuity equation:
∇ · (k(t) u) = 0.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 45 / 49
Navier-Stokes Equations in 3D
Navier-Stokes Equations in 3D
PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.
Variables: pressure p(t, x , y , z), velocity u(t, x , y , z) = u1 x + u2 y + u3 z.
No (natural) Lagrangian formulation.
Has extended Kovalevskaya form with respect to, e.g., x .
Maximal order of conservation law is bounded [Gusyatnikova, Yumaguzhin (1989)].
Vorticity system
An infinite number of vorticity-related CLs (details: [C., Oberlack (2014)]).
(ω · ∇F )t +∇ ·(
[ω × u− ν∇2u]×∇F − Ft ω)
= 0,
involving vorticity and an arbitrary function of flow parameters:
F = F (t, x , y , z ,u, p,ω, . . .).
A. Cheviakov (Math&Stat) Conservation Laws February 2015 45 / 49
Navier-Stokes Equations in 3D
Navier-Stokes Equations in 3D
PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.
Variables: pressure p(t, x , y , z), velocity u(t, x , y , z) = u1 x + u2 y + u3 z.
No (natural) Lagrangian formulation.
Has extended Kovalevskaya form with respect to, e.g., x .
Maximal order of conservation law is bounded [Gusyatnikova, Yumaguzhin (1989)].
Extensions for related models
Additional CLs for symmetric Navier-Stokes equations (planar, axial, helical; [Kelbinet al, (2013)]).
Additional CLs for Euler equations, including symmetric cases [C., Oberlack (2014);Kelbin et al, (2013)].
A. Cheviakov (Math&Stat) Conservation Laws February 2015 45 / 49
Outline
1 Conservation Laws
2 Variational Principles
3 Symmetries and Noether’s Theorem
4 Direct Construction Method for Conservation Laws
5 Example A: 2D Incompressible Hyperelasticity Model
6 Example B: Navier-Stokes Equations in 3D
7 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws February 2015 46 / 49
Conclusions
Discussion
Divergence-type conservation laws are useful in analysis and numerics.
Generally, conservation laws can be obtained systematically through the Directconstruction method.
The method is implemented in a symbolic package GeM for Maple.
For variational DE systems, conservation laws correspond to variational symmetries.
Noether’s theorem is not a preferred way to derive unknown conservation laws.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 47 / 49
Conclusions
Some related topics not addressed in this talk:
Trivial CLs.
Equivalence of CLs.
Material CLs.
Nonlocal CLs.
Differential constraints.
Abnormal PDE systems.
Upper bounds of CL order.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 48 / 49
Some references
Anco, S. C. and Bluman, G. W. (2002)Direct construction method for conservation laws of partial differential equations. Part I:Examples of conservation law classifications. Eur. J. Appl. Math. 13, 545–566.
Gusyatnikova, V. N., and Yumaguzhin, V. A. (1989)Symmetries and conservation laws of Navier-Stokes equations. Acta App. Math. 15: 65–81.
Cheviakov, A. F. (2007)GeM software package for computation of symmetries and conservation laws of differentialequations. Comput. Phys. Comm. 176, 48–61.
Cheviakov, A. F., Ganghoffer, J.-F., and St. Jean, S. (2015)Fully nonlinear wave models in fiber-reinforced anisotropic incompressible hyperelasticsolids. (Accepted, Int. J. Non-Lin. Mech.)
Kelbin, O., Cheviakov, A. F., and Oberlack, M. (2013)New conservation laws of helically symmetric, plane and rotationally symmetric viscous andinviscid flows. J. Fluid Mech. 721, 340–366.
Cheviakov, A. F., and Oberlack, M. (2014)Generalized Ertel’s theorem and infinite hierarchies of conserved quantities forthree-dimensional time dependent Euler and Navier-Stokes equations. J. Fluid Mech. 760,368–386.
A. Cheviakov (Math&Stat) Conservation Laws February 2015 49 / 49