Direct and inverse problemson conservation laws

Roman O. Popovych

Wolfgang Pauli Institute, Vienna, Austria &Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine

joint work with Alexander Bihlo, Artur Sergyeyev and Alexey Shevyakov

Eighth WorkshopGroup Analysis of Differential Equations and Integrable Systems

(June 12–17, 2016, Larnaca, Cyprus)

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 1 / 31

Direct problem on CLs

Conservation laws (CLs) play an important role in mathematical physics and have dif-ferent applications in numerical analysis, the integrability theory, the theory of normalforms and asymptotic integrability, etc.

Direct problem on CLs

Given a system of DEs, find the space of its CLs or at least a subspace of this spacewith certain additional constraints, e.g., on order of CLs.

Standard tools for the solution of the direct problem on CLs:

Noether’s theorem

different variations of the direct method

techniques based on co-symmetries

special techniques for integrable systems

For a class of (systems of) differential equations, one should tackle the direct problemon conservation laws as classification problem.

direct problem on CLs ! direct problem on symmetries

? ! inverse problem on symmetries

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 2 / 31

Inverse problem on CLs: general discussion

Various applications, e.g., the geometry-preserving parameterization of DEs need solvingthe inverse problem.

The only paper with no citations!Galindo A., An algorithm to construct evolution equations with a given set of conserveddensities, J. Math. Phys. 20 (1979), 1256–1259.

Inverse problem on CLs. Naive formulation

Derive the general form of systems of DEs with a prescribed set of CLs.

More generally, the inverse problem on CLs can be interpreted as the study of propertiesof DEs for which something is known about their CLs.

Solving the inverse problem on conservation laws for a class of differential equationsmay help in the solution of the direct problem for this class.

? What data on CLs should be given ?

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 3 / 31

Systems of differential equations

L: L(x , u(ρ)) = 0 ∼ L1 = 0, . . . , Ll = 0 for u = (u1, . . . , um) of x = (x1, . . . , xn).

u(ρ) = the set of all the derivatives of the functions u w.r.t. x of order no greater than ρ,including u as the derivatives of the zero order.

L(k) = the set of all algebraically independent differential consequences that have, asdifferential equations, orders no greater than k.

the jet space J(k) : x , uaα ∼

∂|α|ua

∂xα11 . . . ∂xαn

n,

α = (α1, . . . , αn), αi ∈ N ∪ {0}, |α|: = α1 + · · ·+ αn 6 k,

i = 1, . . . , n, a = 1, . . . ,m

We associate L(k) with the manifold L(k) determined by L(k) in J(k).

a differential function G [u]= a smooth function on a domain in J(k)

∼ a smooth function of x and a finite number of derivatives of u

ordG = the maximal order of derivatives involved in G ,ordG = −∞ if G depends only on x

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 4 / 31

Conserved currents

Empiric definition of CLs: A conservation law of L is a divergence expression Div F :=DiF

i [u] which vanishes for all solutions of L.

Definition

A (local) conserved current of the system L is an n-tuple F = (F 1[u], . . . ,F n[u]) forwhich the total divergence Div F := DiF

i vanishes for all solutions of L, i.e.DivF

∣∣L= 0.

Di = ∂xi + uaα+δi ∂uaα , ua

α ∼∂|α|ua

∂xα11 . . . ∂xαn

n, ua

α+δi ∼∂ua

α

∂xi,

α = (α1, . . . , αn), αi ∈ N ∪ {0}, |α|: = α1 + · · ·+ αn,

i = 1, . . . , n, a = 1, . . . ,m

n = 1: conserved current = first integral

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 5 / 31

Trivial conserved currents

F∣∣L= 0 =⇒ DivF

∣∣L= 0, null divergence: DivF ≡ 0 =⇒ DivF

∣∣L= 0

Definition

A conserved current F is trivial if F = F + F , where F∣∣L= 0 and DivF ≡ 0.

The triviality concerning with vanishing conserved currents on solutions of thesystem can be easily eliminated by confining on the manifold of the system, taking intoaccount all its necessary differential consequences.

Lemma characterizing all null divergences

The n-tuple F = (F 1, . . . ,F n), n ≥ 2, is a null divergence (Div F ≡ 0) iff there existdifferential functions v ij = v ij [u], i , j = 1, . . . , n, such that v ij = −v ji and F i = Djv

ij .

The functions v ij are called potentials corresponding to the null divergence F .

n = 1: any null divergence is constant.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 6 / 31

Equivalence of conserved currents and spaces of CLs

Definition

Conserved currents F and F are called equivalent if F −F is a trivial conserved current.

For any system L of differential equations

CC(L) = the linear space of conserved currents of L

CC0(L) = the set of trivial conserved currents of L, a linear subspace of CC(L)

CL(L) = CC(L)/CC0(L) = the set of equivalence classes of CC(L)

Definition

The elements of CL(L) are called conservation laws of the system L, and the wholefactor space CL(L) is called as the space of conservation laws of L.

For F ∈ CC(L) and F ∈ CL(L):F ∈ F ⇐⇒ F is a conserved current of the conservation law F

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 7 / 31

Characteristics of CLs

Let the system L be weakly totally nondegenerate, i.e. for any k > ρ = ordLL(k) is of maximal rank in each point of L(k) and

L is locally solvable in each point (x0, u0(k)) ∈ L(k), which means that there exists

a local solution u = u(x) of L with u(k)(x0) = u0(k).

Applying the Hadamard lemma to the definition of conserved current and integratingby parts imply, up to equivalence of conserved currents,

Div F = λµLµ (1)

with differential functions λµ, µ = 1, . . . , l .

Definition

The equality (1) and the l-tuple λ = (λ1, . . . , λl) are called the characteristic form andthe characteristic of the conservation law F 3 F , respectively.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 8 / 31

Equivalence of characteristics of CLs

λ is trivial if λ|L = 0. λ and λ are equivalent iff λ− λ is a trivial characteristic.

Ch(L) = the linear space of characteristics of CLs of LCh0(L) = the set of trivial characteristics, a linear subspace in Ch(L)Chf(L) = Ch(L)/Ch0(L) = the set of equivalence classes of Ch(L)

Theorem

Let L be a normal system of DEs

= system reduced to Cauchy–Kovalevskaya formby a point transformation of independent variables

= a totally nondegenerate analytical system of the same numberof DEs as dependent variables.

Then the equivalence of CL characteristics is well consistent with the equivalence ofconserved currents.

⇐⇒ CL characteristics of L are equivalent if and only if the respective conservedcurrents are equivalent.

⇐⇒ representation of CLs of L in the characteristic form generates a one-to-onelinear mapping between CL(L) and Chf(L).

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 9 / 31

Criteria for characteristics

f = Div F ⇐⇒ E(f ) = 0.

The Euler operator E = (E1, . . . ,E

m): Ea = (−D)α∂uaα , a = 1,m,

α = (α1, . . . , αn)∈(N ∪ {0})n, (−D)α = (−D1)α1 . . . (−Dm)αm .

Necessary and sufficient condition on λ ∈ Ch(L):

Div F = λµLµ ⇐⇒ E(λµLµ) = D†λ(L) + D

†L(λ) = 0 (2)

D†λ and D

†L are matrix differential operators adjoint to the Frechet derivatives Dλ and DL ,

i.e.

Dλ(L) =

(∂λµ

∂uaα

DαLµ), D

†λ(L) =

((−D)α

(∂λµ

∂uaα

Lµ))

.

Since D†λ(L)|L = 0 =⇒ a necessary condition for λ ∈ Ch(L):

D†L(λ)

∣∣L= 0. (3)

Condition (3) are considered as adjoint to the criteria DL(η)∣∣L= 0 for infinitesimal invari-

ance of L w.r.t. an evolutionary vector field having the characteristic η = (η1, . . . , ηm).That is why solutions of (3) are called cosymmetries or adjoint symmetries.

L is an Euler–Lagrange system =⇒ D†L = DL =⇒ symmetry = cosymmetry

CL characeristic = variational symmetryR.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 10 / 31

System equivalence and CL characteristics

System equivalence

We call systems of differential equations equivalent if they are defined on the samespace of independent and dependent variables and can be obtained from each otherusing the following operations:

recombining equations with coefficients that are differential functions andconstitute a nondegenerate matrix (this gives so-called linearly equivalentsystems);supplementing a system with its differential consequences or excluding equationsthat are differential consequences of other equations.

Equivalent systems have the same set of solutions.

In contrast to conserved currents, symmetries and cosymmetries,CL characteristics are not class invariants under the system equivalence.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 11 / 31

Choice of data

Conserved currents. Counter-arguments:

Complex structure of the equivalence

F ∼ F = F + F + F , where F∣∣L= 0 and DivF ≡ 0

The determination whether a conserved current is trivial or not even for afixed system of DEs is not so obvious and reduces, at least implicitly, to theconsideration of related characteristics.

Conserved currents may be trivial for some systems from a class andnontrivial for another systems from the same class.

Weak consistency

Not all n-tuples of differential functions (e.g. tuples with single nonvanishingcomponents) are suitable candidates for conserved currents.

Different candidates for conserved currents may be inconsistent.

Fixing conserved currents considerably restricts the systems’ form. =⇒Even minor variations within systems’ form lead to varying the form ofconserved currents.

=⇒ ! Entire conserved currents are not appropriate initial datafor the inverse problem on CLs

!R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 12 / 31

Choice of data

Characteristics. Arguments:

Their equivalence is much simpler.

They are more stable under varying systems within a class of differential equations.

Obstacles (can be overcome):

They are defined only for systems that are weakly totally nondegenerate.

There may be no 1-to-1 correspondence between equivalence classes ofCL characteristics and conserved currents.

They are not invariant under system equivalence.

Confining characteristics to solutions of the considered systems is not trivial.

If representatives of the equivalence classes of DE systems under consideration are fixed,

then ! CL characteristics are appropriate initial datafor the inverse problem on CLs

!

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 13 / 31

Choice of data

Form for systems.

Definition

A system of PDEs L is of extended Kovalevskaya form if it can be written as

uaraδn :=

∂raua

∂x ran

= H j(x , u(r)), a = 1, . . . ,m. (4)

where 0 6 ra 6 r and u(r) denotes all derivatives of u w.r.t. x up to order r , whereeach ub is differentiated with respect to xn at most rb − 1 times, b = 1, . . . ,m.

The extended Kovalevskaya form is less restrictive than the usual Kovalevskaya form.

Zero-order derivatives may appear in the left-hand side of equations.

There are no restrictions on the order of derivatives w.r.t. x1, . . . , xn−1 dependingon ra’s.

Systems of the form (4) with all ra > 0 are called

normal systems

Martınez-Alonso L., On the Noether map, Lett. Math. Phys. 3 (1979), 419–424.

Cauchy–Kowalevsky systems in a weak sense (resp., pseudo CK systems in short)

Tsujishita T., On variation bicomplexes associated to differential equations, Osaka J.Math. 19 (1982), 311–363.

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Reformulation of inverse problem on CLs

Inverse problem on CLs. Enhanced naive formulation

Derive the general form of systems of DEs with a prescribed set of CL characteristics.

Inverse problem on CLs. Extended Kovalevskaya form

Given a class of systems of DEs in the extended Kovalevskaya form, find its subclass ofsystems admitting a prescribed set of reduced CL characteristics (resp. reduceddensities).

Inverse problem on CLs. Low-order characteristics

Derive the form of systems of DEs with prescribed set of low-order (in particularzero-order) CL characteristics.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 15 / 31

Inverse problem on first integrals for ODEs

Problem

Find the general form of systems of ordinary differential equations that admit aprescribed set of integrating factors.

Theorem

An ODE H(x , u, u′, . . . , u(n)) = 0, Hu(n) 6= 0, admits p totally linearly independentfirst integrals with integrating factors γs iff

H = DT[γ1, . . . , γp]†G for some G=G[u].

Here

W (γ1, . . . , γp) = det(Ds′−1x γs)ps,s′=1 denotes the Wronskian of γ1, . . . , γp w.r.t.

the operator of total derivative Dx ,

DT[γ1, . . . , γp] is the operator in Dx associated with the “Darbouxtransformation”

DT[γ1, . . . , γp]G =W (γ1, . . . , γp,G)

W (γ1, . . . , γp).

† denotes “formally adjoint”, Q = Q jD jx , Q† = (−Dx)j ◦ Q j .

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 16 / 31

Preliminaries on evolution equations

An evolution equation in two independent variables,

E : ut = H(t, x , u0, u1, . . . , un), n > 2, Hun 6= 0,

where uj ≡ ∂ ju/∂x j , u0 ≡ u, and Huj = ∂H/∂uj , ux = u1, uxx = u2, and uxxx = u3.

Dt = ∂t + ut∂u + utt∂ut + utx∂ux + · · · , Dx = ∂x + ux∂u + utx∂ut + uxx∂ux + · · ·

the total derivatives w.r.t. the variables t and x . The subscripts like t, x , u, ux , etc.stand for the partial derivatives in the respective variables.

Without loss of generality, for any evolution equation the associated quantities like sym-metries, cosymmetries, densities and characteristics of CLs can be assumed independentof the t-derivatives or mixed derivatives of u.

We refer to a (smooth) function of t, x and a finite number of uj as to a differentialfunction.

Given a differential function f , its order (denoted by ord f ) is the greatest integer k suchthat fuk 6= 0 but fuj = 0 for all j > k. For f = f (t, x) we assume that ord f = −∞.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 17 / 31

Contact transformations of evolution equations

The contact transformations mapping a (fixed) equation E : ut = H into another equa-tion E : ut = H are well known [Magadeev,1993] to have the form

t = T (t), x = X (t, x , u, ux), u = U(t, x , u, ux).

The nondegeneracy assumptions: Tt 6= 0, rank

(Xx Xu Xux

Ux Uu Uux

)= 2

The contact condition: (Ux + Uuux)Xux = (Xx + Xuux)Uux =⇒ ux = V (t, x , u, ux),

where V =Ux + Uuux

Xx + Xuuxor V =

Uux

Xux

if Xx + Xuux 6= 0 or Xux 6= 0, respec-

tively

=⇒ uk ≡∂k u

∂xk=

(1

DxXDx

)k−1

V , H =Uu − XuV

TtH +

Ut − XtV

Tt

The equiv. group G∼c generates the whole set of admissible contact transformationsin the class, i.e., the class is normalized [ROP & Kunzinger & Eshraghi,2010] w.r.t.contact transformations.

Proposition

The class of evolution equations is contact-normalized.

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Some basic results on conservation laws

A conserved current of E is a pair of differential functions (ρ, σ) satisfying the condition

Dtρ+ Dxσ = 0 mod E , where E = E and all its differential consequences.

Here ρ is the density and σ is the flux for the conserved current (ρ, σ). Let

δ

δu=∞∑i=0

(−Dx)i∂ui , f∗ =∞∑i=0

fui Dix , f †∗ =

∞∑i=0

(−Dx)i ◦ fui

denote the operator of variational derivative, the Frechet derivative of a differentialfunction f , and its formal adjoint, respectively.

A conserved current (ρ, σ) is called trivial if Dtρ+ Dxσ = 0 on the entire jet space ⇐⇒ρ ∈ ImDx , i.e., there exists a differential function ζ: ρ = Dxζ.

Two conserved currents are equivalent if they differ by a trivial conserved current.

A CL of E is an equivalence class of conserved currents of E .

The set CL(E) of CLs of E is a vector space, and the zero element of this space is theCL being the equivalence class of trivial conserved currents.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 19 / 31

Some basic results on conservation laws

For any L ∈ CL(E) there exists a unique differential function γ called the characteristicof L such that for any conserved current (ρ, σ) ∈ L there exists a trivial conservedcurrent (ρ, σ):

Dt(ρ+ ρ) + Dx(σ + σ) = γ(ut − H).

The characteristic γ of any CL is a cosymmetry, i.e., it satisfies the equation

Dtγ + H†∗γ = 0 mod E , or equivalently, γt + γ∗H + H†∗γ = 0.

The characteristic of the CL associated with (ρ, σ) is

γ = δρ/δu.

Therefore, a cosymmetry γ to be a characteristic of a CL iff γ∗ = γ†∗.

ρt + γH = Dx σ for some σ.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 20 / 31

Inverse problem on CLs for evolution equations

Theorem

An evolution equation

E : ut = H(t, x , u0, u1, . . . , un), n > 2, Hun 6= 0,

admits p linearly independent CLs with densities ρs and characteristics γs = δρs/δu iff

H = DT[γ1, . . . , γp]†G −p∑

s=1

DT[γ1, . . . ,�γs , . . . , γp]†(

W (γ1, . . . ,�γs , . . . , γp)

W (γ1, . . . , γp)ρst

)for some G = G [u].

Here

W (γ1, . . . , γp) = det(Ds′−1x γs)ps,s′=1 denotes the Wronskian of γ1, . . . , γp w.r.t.

the operator of total derivative Dx ,

DT[γ1, . . . , γp] is the operator in Dx associated with the “Darbouxtransformation”

DT[γ1, . . . , γp]G =W (γ1, . . . , γp,G)

W (γ1, . . . , γp).

† denotes “formally adjoint”, Q = Q jD jx , Q† = (−Dx)j ◦ Q j .

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 21 / 31

Inverse problem on CLs for evolution equations

F = DT[γ1, . . . , γp]†H −p∑

s=1

DT[γ1, . . . ,�γs , . . . , γp]† (ϕsρst ) , (5)

where ϕs := (−1)p−s W (γ1, . . . ,�γs , . . . , γp)

W (γ1, . . . , γp)are adjoint diff. functions to γ1, . . . , γp,

DT[γ1, . . . , γp]† = (−1)pDT[ϕ1, . . . , ϕp].

F = (−1)pW (ϕ1, . . . , ϕp, H)(

W (γ1, . . . , γp))p − p∑

s=1

DT[γ1, . . . ,�γs , . . . , γp]†ϕsρst

W (γ1, . . . , γp), (6)

where ϕs := (−1)p−sW (γ1, . . . ,�γs , . . . , γp).

(5) depends rather on 〈ρ1, . . . , ρp〉 than on {ρ1, . . . , ρp}.For fixed F and 〈ρ1, . . . , ρp〉, H is defined up to H → H + ζs(t)ϕs .

If ρs = ρs + Dxθs for some diff. functions θs , then H = H + (ζs − θst )ϕs for some

ζs = ζs(t).

Let q := maxs

ord γs (∈ {−∞, 0} ∪ 2N) and ρs : ord ρs 6 max(0, q + p − 1).

(For best ρs : ord ρs = max(0, 12ord γs).) Then ord H 6 max(n − p, q + p − 2).

If maxs

ord γs 6 n − 2p + 2, then ord H 6 n − p.

If maxs

ord γs 6 n − 2p, then ordH = ord H = n − p.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 22 / 31

Second-order evolution equation

E : ut = F (t, x , u, ux , uxx)

Density order of CLs 6 1.

Density order of CLs = 1 if Fu2u2 = 0.

Theorem [Bryant&Griffiths, 1995, Ft = 0; ROP&Samoilenko, 2008, gen. case]

dimCL(E) ∈ {0, 1, 2,∞} for any second-order (1 + 1)D evolution equation E .The equation E is (locally) reduced by a contact transformation

1 to the form ut = DxG(t, x , u, ux), where Gux 6= 0, iff dimCL(E) > 1;

2 to the form ut = D2x H(t, x , u), where Hu 6= 0, iff dimCL(E) > 2;

3 to a linear evolution equation iff dimCL(E) =∞.

If the equation E is quasi-linear (i.e., Fuxxuxx = 0) then the contact transformation is aprolongation of a point transformation.

Corollary

The space of CLs of ut = D2x H(t, x , u) is 2D iff H is not linear in u.

The space of CLs of ut = DxG(t, x , u, ux) is 1D if G is not a fractionally linear in ux .

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CLs of even-order evolution equations

Let E be a (1 + 1)D even-order (n ∈ 2N) evolution equation.

Lemma [Ibragimov, 1983; Abellanas & Galindo, 1979; Kaptsov, 1980]

For any CL F of E we have ordchar F 6 n.

Theorem

Cosymf(E) = Chf(E).

Theorem

If dimCL(E) > p, then ord γ 6 n − 2p + 2 for any γ ∈ Chf(E)(modG∼cont if 2p > n + 2).

Roughly speaking, the greater dimension of the space of CLs, the lower upper boundfor orders of CL characteristics. In particular, ord γ = −∞ mod G∼cont if 2p > n + 2.

Corollary

dimCL(E) > n iff E is linearizable by a contact transformation and thusdimCL(E) =∞.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 24 / 31

CLs of fourth-order evolution equations

E : ut = F (t, x , u, u1, u2, u3, u4), Fu4 6= 0

dimCL(E) > 1: F = −Dx σ1 + ρ1

t

γ1,

ord ρ1 6 2, ord σ1 6 3, γ1 =δρ1

δu6= 0.

dimCL(E) > 2, modG∼cont: F = DxG , G = −Dx σ1 + ρ1

t

Dxγ1,

ord ρ1 6 1, ord σ1 6 2, γ1 =δρ1

δu, Dxγ

1 6= 0 (ρ2 = u, γ2 = 1).

dimCL(E) > 3, modG∼cont: F = D2x G , G = −Dx σ

1 + ρ1t

D2xρ1

u,

ord ρ1 = 0, ord σ1 6 1, D2xρ

1u 6= 0 (ρ2 = u, γ2 = 1, ρ3 = xu, γ3 = x).

dimCL(E) > 4, modG∼cont:

F = DT[γ1, . . . , γ4]†H −p∑

s=1

DT[γ1, . . . ,�γs , . . . , γ4]†(

W (γ1, . . . ,�γs , . . . , γp)

W (γ1, . . . , γ4)γst u

),

ord γs = −∞, γ3 = x , γ4 = 1, ordH = 0.

dimCL(E) > 4, modG∼cont: E is a linear equation =⇒ dimCL(E) =∞.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 25 / 31

Inverse problem on CLs with arbitrary parametric functions

Let L : L[u] = 0 be a PDE for a single unknown function u of x = (x1, . . . , xn).

Theorem

L admits an arbitrary function h = h(x1) as a CL characteristic iff

L[u] = D2F 2[u] + · · ·+ DnF n[u]

for some differential functions F 2, . . . , F n.

Corollary

If L admits N + 1 linearly independent CLs with characteristics depending only on x1,hs′ = hs′(x1), s ′ = 0, . . . ,N, where N = max{α1 | Luα 6= 0}, then the aboverepresentation holds and hence L admits CLs with characteristic being an arbitraryfunction of x1.

Corollary

L admits an arbitrary h = h(x1, . . . , xq) with q < n as a CL characteristic iff

L = Dq+1F q+1[u] + · · ·+ DnF n[u],

for a tuple of n − q differential functions F q+1, . . . ,F n.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 26 / 31

Inverse problem on CLs with arbitrary parametric functions

Theorem

L admits characteristics of the form h(x1) +∑

i>1 f i (x1)xi with arbitrary functions

h = h(x1) and f i = f i (x1), i = 2, . . . , n, iff

L =∑

26i6j6n

DiDjFij [u].

for some differential functions F ij , 2 6 i 6 j 6 n.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 27 / 31

Conservative parameterization for the vorticity equation

L: ζt + ψxζy − ψyζx = 0, ζ := ψxx + ψyy .

Here ψ = ψ(t, x , y) is the stream function, ζ is the vorticity, and we use a specificnotation for the variables, t, x , y and ψ instead of x1, x2, x3 and u.

Chf(L) ⊃ 〈h(t), f (t)x , g(t)y , ψ〉.

We split ψ as well as ζ into a resolved (mean) parts ψ and ζ and an unresolved (sub-gridscale) parts ψ′ and ζ′, i.e. ψ = ψ + ψ′ and ζ = ζ + ζ′.

The Reynolds averaging rule, ab = ab + a′b′, gives the equation for the mean part,

L : ζt + ψx ζy − ψy ζx = ∇ · ζ′v′, ζ := ψxx + ψyy .

The problem is that the right-hand side of the equation involves the divergence of theunknown vorticity flux ζ′v′ for which no equation is given. In other words, the equationis underdetermined, which is the usual closure problem of fluid mechanics.

We use the local closure ∇ · ζ′v′ = V [ψ] of the equation, which gives

ζt + ψxζy − ψyζx = V [ψ], ζ := ψxx + ψyy ,

where bars are omitted.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 28 / 31

Conservative parameterization for the vorticity equation

L : ζt + ψxζy − ψyζx = V [ψ], ζ := ψxx + ψyy

Chf(L) ⊃ 〈h(t), f (t)x , g(t)y , ψ〉.

Proposition

If the unclosed vorticity flux ∇ · ζ′v′ is parameterized by V [ψ] in the Reynoldsaveraged vorticity equation, where V [ψ] is given through

V [ψ] = D2x(ψyyP2 − ψxyP3) + DxDy (ψxxP3 − ψyyP1) + D2

y (ψxyP1 − ψxxP2)

+ (D2x + D2

y )(ζ Div S + 2S1ζt + 2S2ζx + 2S3ζy )

for some differential functions P i and S i , i = 1, 2, 3, of ψ the resulting closedequation L possesses the conservation laws associated with characteristics h(t), f (t)x ,h(t)y and ψ. That is, the closed equation will preserve generalized circulation,generalized momenta in x- and y -direction and energy.

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 29 / 31

CLs of abnormal systems

Definition

A system L is called abnormal if it has an identically vanishing differential consequencesuch that the corresponding tuple of differential operators acting on system’s equationsdoes not vanish on the manifold L(∞).

Example Maxwell’s equations give an example of an essentially abnormal system.

Et = ∇× B, ∇ · E = 0, =⇒ ∇(Et −∇× B)− Dt∇ · E ≡ 0

Bt = −∇× E , ∇ · B = 0, =⇒ ∇(Bt +∇× E)− Dt∇ · B ≡ 0

Theorem (generalized Noether’s second theorem)

The following statements on a weakly totally nondegenerate system L of DEs areequivalent:

1) the system L is abnormal;

2) it possesses a family of CLs with a characteristic λ[u, h] that depends, additionallyto u as differential functions, on an arbitrary function h = h(x) and whose Frechetderivative with respect to h does not vanish on solutions of L for a value of theparameter-function h;

3) it possesses a trivial conserved current corresponding to a nontrivial characteristic.

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Thanks

Thank you for your attention!

R.O. Popovych — GADEIS-VIII Direct and inverse problems on CLs 31 / 31