Conservation Laws for Nonlinear Equations: Theory ...Conservation Laws for Nonlinear Equations:...

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

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

Outline

1 Conservation Laws

7 Conclusions

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).

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)

∂x+∂g

∂uux +

∂uxuxx .

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)

Θ dV = 0, if

∮∂V

Ψ[u] · dS = 0.

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)

Θ dV = 0, if

∮∂V

Ψ[u] · dS = 0.

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.

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

(mx2(t)

kx2(t)

Conserved quantity: energy.

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.

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.

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).

PDE Models

Wave equation:

Conservation of momentum:

Local conservation law: Dt(ρut)−Dx(Tux) = 0;

Global conserved quantity: total momentum

ρut dx = const,

for Neumann homogeneous problems with ux(a, t) = ux(b, t) = 0.

PDE Models

Wave equation:

Conservation of energy:

Local conservation law:

)−Dx(Tutux) = 0;

Global conserved quantity: total energy

)dx = const,

for both Neumann and Dirichlet homogeneous problems.

Applications of Conservation Laws

Constants of motion.

Integration.

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.

Outline

1 Conservation Laws

7 Conclusions

Variational Principles

Action integral

J[U] =

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

Variation of action: δJ ≡ J[U + εv]− J[U] =∫

Ω(δL) dx = o(ε).

Action integral

J[U] =

L(x,U, ∂U, . . . , ∂kU) dx .

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)

(∂L[U]

∂Uσvσ +

∂L[U]

∂Uσjvσj + · · ·+ ∂L[U]

∂Uσj1···jkvσj1···jk

)+ O(ε2)

by parts= ε(vσEUσ (L[U])) + div(...) + O(ε2)

Action integral

J[U] =

L(x,U, ∂U, . . . , ∂kU) dx .

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.

Example 1: Harmonic oscillator, U = x = x(t)

2mx2 − 1

2kx2, ExL = −m(x + ω2x) = 0, ω2 = k/m.

Example 2: Wave equation for U = u(x , t)

2 − 1

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.

2mx2 − 1

2kx2, ExL = −m(x + ω2x) = 0, ω2 = k/m.

2 − 1

2, EuL = −ρ(utt − c2uxx) = 0, c2 = T/ρ.

2mx2 − 1

2kx2, ExL = −m(x + ω2x) = 0, ω2 = k/m.

2 − 1

2, EuL = −ρ(utt − c2uxx) = 0, c2 = T/ρ.

Self-adjointness

PDE linearization

Given PDE or system: R[u] = 0.

Linearized system (Frechet derivative): L[u]v(x) =d

∣∣∣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:

u ·R[λu] dλ.

Self-adjointness

PDE linearization

∣∣∣ε=0

R[u + εv] = 0.

Self-adjointness

Given system R[u] = 0 is self-adjoint if

L[u] v(x) = L∗[u] v(x).

Homotopy Formula for a Lagrangian:

u ·R[λu] dλ.

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:

2 − 1

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).

Self-adjointness

Example 2:

NOT self-adjoint!

Append the adjoint:

R[u1, u2] = u1t − u1

xx = 0, − u2t − u2

Lagrangian: L = 12

(−u1(u2

t − u2xx) + u2(u1

t − u1xx)).

Eu1 (L) = −u2t − u2

xx = R2, Eu2 (L) = u1t − u1

xx = R1.

Self-adjointness

Example 2:

NOT self-adjoint!

Append the adjoint:

R[u1, u2] = u1t − u1

xx = 0, − u2t − u2

Lagrangian: L = 12

(−u1(u2

t − u2xx) + u2(u1

t − u1xx)).

Eu1 (L) = −u2t − u2

xx = R2, Eu2 (L) = u1t − u1

xx = R1.

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!

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!

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 −

6q3x −

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.

Self-adjointness

Example 4:

Self-adjoint!

2q2xx −

6q3x −

2qxqt .

Result:

Self-adjointness

Example 4:

Self-adjoint!

2q2xx −

6q3x −

2qxqt .

Result:

Outline

1 Conservation Laws

7 Conclusions

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∗.

Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N

Definition

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∗ .

Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N

Definition

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∗ .

Variational Symmetries

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].

Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N

ΩL[u] dx .

Definition

Example 1: translations for the wave equation

utt = c2uxx , L =1

2 − c2

The translation x∗ = x + C , t∗ = t, u∗ = u is a variational symmetry.

Rσ[u] = Rσ(x,u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N

ΩL[u] dx .

Definition

Example 2: scaling for the wave equation

utt = c2uxx , L =1

2 − c2

The scaling x∗ = x , t∗ = t, u∗ = u/α is not a variational symmetry: J∗ = α2J.

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, ) )

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).

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).

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

(x2(t)

2+ω2x2(t)

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

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

2+ c2 u

)− Dx

(c2utux

)]= 0.

Linearization Operators and Variational Formulation

Recollect:

∣∣∣ε=0

R[u + εv] = 0.

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!

Recollect:

∣∣∣ε=0

R[u + εv] = 0.

Facts:

Recollect:

∣∣∣ε=0

R[u + εv] = 0.

Facts:

Outline

1 Conservation Laws

7 Conclusions

The Idea of the Direct Construction Method

The Direct Construction Method:

Works for virtually any model.

Does not require variational formulation/Noether’s theorem.

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].

Seek conservation laws in the characteristic form DiΦi = ΛσR

σ = 0

(based on Hadamard’s lemma for systems of maximal rank).

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].

Seek conservation laws in the characteristic form DiΦi = ΛσR

σ = 0

(based on Hadamard’s lemma for systems of maximal rank).

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

Each set of fluxes, multipliers yields a local conservation law

DiΦi [u] = 0,

holding for all solutions u(x) of the given PDE system.

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

(12u2)

(13u3 + uuxx − 1

(16u3 − 1

(18u4 − uu2

x + 12(u2uxx + u2

xx)− uxuxxx)

Example

1st-order multipliers in x

Form: Λ = Λ(x , t,U,Ux)

Solution: no extra conservation laws.

Example

2nd-order multipliers in x

Form: Λ = Λ(x , t,U,Ux ,Uxx)

Solution: one extra conservation law with

Λ4 = Uxx + 12U2.

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);

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=*****);

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.

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 = . . .).

Outline

1 Conservation Laws

7 Conclusions

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

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 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

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.

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

∂Fij.

Neo-Hookean (Hadamard) material: W = a(I1 − 3), I1 = F ikF

2D case: 3 PDEs, 3 unknowns (x, p).

PDEs in 2D, no forcing:

∂X 1

∂X 2− ∂x1

∂X 2

∂X 1− 1 = 0,

∂2x1

∂t2 −[α

(∂2x1

∂ (X 1)2 +∂2x1

∂ (X 2)2

)− ∂p

∂X 1

∂X 2+

∂X 2

∂X 1

∂2x2

∂t2 −[α

(∂2x2

∂ (X 1)2 +∂2x2

∂ (X 2)2

)− ∂p

∂X 2

∂X 1+

∂X 1

∂X 2

Point symmetries

∂X 1

∂X 2− ∂x1

∂X 2

∂X 1− 1 = 0,

∂2x1

∂t2 −[α

(∂2x1

∂ (X 1)2 +∂2x1

∂ (X 2)2

)− ∂p

∂X 1

∂X 2+

∂X 2

∂X 1

∂2x2

∂t2 −[α

(∂2x2

∂ (X 1)2 +∂2x2

∂ (X 2)2

)− ∂p

∂X 2

∂X 1+

∂X 1

∂X 2

Point symmetries:

Y1 =∂

∂t, Y2 =

∂X 1, Y3 =

∂X 2, Y4 = F1(t)

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 ∂

Y9 = t∂

∂t+ X 1 ∂

∂X 1+ X 2 ∂

∂X 2+ x1 ∂

∂x1+ x2 ∂

∂x2.

Conservation laws

R1[x, p] =∂x1

∂X 1

∂X 2− ∂x1

∂X 2

∂X 1− 1 = 0,

R2[x, p] =∂2x1

∂t2 −[α

(∂2x1

∂ (X 1)2 +∂2x1

∂ (X 2)2

)− ∂p

∂X 1

∂X 2+

∂X 2

∂X 1

R3[x, p] =∂2x2

∂t2 −[α

(∂2x2

∂ (X 1)2 +∂2x2

∂ (X 2)2

)− ∂p

∂X 2

∂X 1+

∂X 1

∂X 2

Conservation laws:

DtΨ + DX 1 Φ1 + DX 1 Φ2 =3∑

Λi [x, p] R i [x, p].

Extended Kovalevskaya form exists w.r.t. X 1 and X 2, very complicated...

Conservation laws

R1[x, p] =∂x1

∂X 1

∂X 2− ∂x1

∂X 2

∂X 1− 1 = 0,

R2[x, p] =∂2x1

∂t2 −[α

(∂2x1

∂ (X 1)2 +∂2x1

∂ (X 2)2

)− ∂p

∂X 1

∂X 2+

∂X 2

∂X 1

R3[x, p] =∂2x2

∂t2 −[α

(∂2x2

∂ (X 1)2 +∂2x2

∂ (X 2)2

)− ∂p

∂X 2

∂X 1+

∂X 1

∂X 2

Notation:

∂2x1

∂t2 ≡ x1tt ,

∂X 2≡ x1

2 ,∂2x2

∂X 1∂X 2≡ x2

12,∂2p

∂X 2∂t≡ p2t , . . .

Conservation laws

R1[x, p] =∂x1

∂X 1

∂X 2− ∂x1

∂X 2

∂X 1− 1 = 0,

R2[x, p] =∂2x1

∂t2 −[α

(∂2x1

∂ (X 1)2 +∂2x1

∂ (X 2)2

)− ∂p

∂X 1

∂X 2+

∂X 2

∂X 1

R3[x, p] =∂2x2

∂t2 −[α

(∂2x2

∂ (X 1)2 +∂2x2

∂ (X 2)2

)− ∂p

∂X 2

∂X 1+

∂X 1

∂X 2

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).

Conservation laws

Conservation of energy:

Λ1 = pt , Λ2 = x1t , Λ3 = x2

Divergence expression:

)2+(x1

)2+(x2

−DX 1

(α(x1t x

11 + x2

)− px1

t x22 + px1

)−DX 2

(α(x1t x

12 + x2

)+ px1

t x21 − px1

Conservation laws

Conservation of generalized momentum in x1

Λ1 = f ′′2 (t)x1, Λ2 = f2(t), Λ3 = 0.

(f2(t)x1

t − f ′2 (t)x1)

−DX 1

(αf2(t)x1

1 − pf2(t)x22 − 1

2f ′′2 (t)

)−DX 2

(αf2(t)x1

2 + pf2(t)x21 + 1

2f ′′2 (t)

Same for x2.

Conservation laws

Conservation of angular momentum in x3

Λ1 = 0, Λ2 = x2, Λ3 = −x1.

t − x1t x

(α(x1

1 x2 − x1x2

)− px1x1

2 − px2x22

)+DX 2

(α(x1

2 x2 − x1x2

)+ px1x1

1 + px2x21

Conservation laws

Generalized incompressibility condition

Λ1 = f1(t), Λ2 = 0, Λ3 = 0.

(∫f1(t) dt

)−DX 1

(f1(t)x1x2

)+ DX 2

(f1(t)x1x2

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

(x1t x

11 + x2

)+DX 1

)2+(x2

)2 −(x1

)2 −(x2

+ p − 12

)2 − 12

−DX 2

(α(x1

1 x12 + x2

))= 0.

Same for X 2.

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

[X 2(x1

t x11 + x2

t x21 )− X 1(x1

t x12 + x2

t x22 )]

[12αX 2

(−(x1

)2 −(x2

)2+(x1

)2+(x2

+αX 1(x1

1 x12 + x2

)+ X 2p − 1

2X 2(x1t

)2 − 12X 2(x2t

[12αX 1

(−(x1

)2 −(x2

)2+(x1

)2+(x2

−αX 2(x1

1 x12 + x2

)− X 1p + 1

2X 1(x1t

2X 1(x2t

Comparison of Symmetries and Conservation Laws

Evol. Symmetry Multiplier Conserved quantity

Y1 = −x1t

∂x1− x2

∂x2− pt

∂pΛ1 = pt , Λ2 = −x1

t , Λ3 = −x2t Energy

Y2 = −x11

∂x1− x2

∂x2− p1

∂pΛ1 = p1, Λ2 = −x1

1 , Λ3 = −x21 Lagrangian momentum in X 1

Y3 = −x12

∂x1− x2

∂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

∂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

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),

J = x11 x

22 − x1

2 x21 .

Variational Formulation

Variational Formulation?

“Variational” symmetries and conservation laws coincide!

Lagrangian can be obtained, for example, using the homotopy formula

u ·R[λu] dλ, u = (p, x1, x2).

This is non-trivial, since the incompressible equations involve a differential constraint.

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.

Outline

1 Conservation Laws

7 Conclusions

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?

PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.

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)].

PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.

Generalized momentum in x-direction:

∂t(f (t)u1) +

((u1f (t)− xf ′(t))u1 + f (t)(p − νu1

((u1f (t)− xf ′(t))u2 − νf (t)u1

((u1f (t)− xf ′(t))u3 − νf (t)u1

Cyclically permute (1, 2, 3), (x , y , z) for the other two components.

PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.

Angular momentum in x-direction:

∂t(zu2 − yu3) +

((zu2 − yu3)u1 + ν(yu3

x − zu2x ))

((zu2 − yu3)u2 + zp + ν(yu3

y − zu2y − u3)

((zu2 − yu3)u3 − yp + ν(yu3

z − zu2z + u2)

Cyclically permute (1, 2, 3), (x , y , z) for the other two components.

PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.

Generalized continuity equation:

∇ · (k(t) u) = 0.

PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.

Vorticity system

An infinite number of vorticity-related CLs (details: [C., Oberlack (2014)]).

(ω · ∇F )t +∇ ·(

[ω × u− ν∇2u]×∇F − Ft ω)

involving vorticity and an arbitrary function of flow parameters:

F = F (t, x , y , z ,u, p,ω, . . .).

PDEs: ∇ · u = 0, ut + (u · ∇)u +∇p − ν∇2u = 0.

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)].

Outline

1 Conservation Laws

7 Conclusions

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.

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.

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.

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