Conservation Laws for Nonlinear Equations: Theory, Computation, and Examples

Transcription

1 Conservation Laws for Nonlinear Equations: Theory, Computation, and Examples Alexei Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada Seminar February 2015 A. Cheviakov (Math&Stat) Conservation Laws February / 49

2 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 / 49

3 Collaborators G. Bluman, University of British Columbia, Canada J.-F. Ganghoffer, LEMTA - ENSEM, Université 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 / 49

4 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 / 49

5 Definitions Variables: Independent: x = (x 1, x 2,..., x n ) or (t, x 1, x 2,...). Dependent: u = (u 1 (x), u 2 (x),..., u m (x)) or (u(x), v(x),...). Partial derivatives: u k Notation: x = m uk x m = uk m. All first-order partial derivatives: u. All p th -order partial derivatives: p u. 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 dependent variables, and derivatives of dependent variables to some order. F [u] = F (x, u, u,..., p u). A. Cheviakov (Math&Stat) Conservation Laws February / 49

6 Definitions (ctd.) The total derivative of a differential function: A basic chain rule. Example: u = u(x, y), g[u] = g(x, y, u, u x), then D xg[u] g(x, y, u, ux) x = g x + g g ux + u xx. u u x A. Cheviakov (Math&Stat) Conservation Laws February / 49

7 Local Conservation Laws Conservation laws A local conservation law: a divergence expression equal to zero, D i Ψ i [u] div Ψ i [u] = 0. For equations involving time evolution: D t Θ[u] + div x Ψ[u] = 0. Θ[u]: conserved density. Ψ[u]: flux vector. A. Cheviakov (Math&Stat) Conservation Laws February / 49

8 Local Conservation Laws Conservation laws A local conservation law: a divergence expression equal to zero, D i Ψ i [u] div Ψ i [u] = 0. For equations involving time evolution: D t Θ[u] + div x Ψ[u] = 0. Θ[u]: conserved density. Ψ[u]: flux vector. Global conserved quantity (integral of motion) D t Θ dv = 0, if Ψ[u] ds = 0. V V A. Cheviakov (Math&Stat) Conservation Laws February / 49

9 ODE Models: Conserved Quantities An ODE: Dependent variable: u = u(t); A conservation law D tf (t, u, u,...) = d dt F (t, u, u,...) = 0 yields a conserved quantity (constant of motion): F (t, u, u,...) = C = const. A. Cheviakov (Math&Stat) Conservation Laws February / 49

10 ODE Models: Conserved Quantities An ODE: Dependent variable: u = u(t); A conservation law D tf (t, u, u,...) = d dt F (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: ẍ(t) + ω 2 x(t) = 0; ω 2 = k/m = const. ( d mẋ 2 (t) Conservation law: + kx ) 2 (t) = 0. dt 2 2 Conserved quantity: energy. A. Cheviakov (Math&Stat) Conservation Laws February / 49

11 ODE Models: Conserved Quantities Example: ODE integration An ODE: K (x) = 2 (K (x)) 2 K(x) (K (x)) 2 K (x). K(x)K (x) A. Cheviakov (Math&Stat) Conservation Laws February / 49

12 ODE Models: Conserved Quantities Example: ODE integration An ODE: Three independent conserved quantities: K (x) = 2 (K (x)) 2 K(x) (K (x)) 2 K (x). K(x)K (x) KK (K ) 2 = C1, KK ln K (K ) 2 ln K = C 2, yield complete ODE integration. xkk + KK (K ) 2 x = C 3 A. Cheviakov (Math&Stat) Conservation Laws February / 49

13 PDE Models Example: small string/rod oscillations, 1D wave equation Independent variables: x, t; Dependent: u(x, t). Wave equation: u tt = c 2 u xx, c 2 = const = T /ρ (for a string). A. Cheviakov (Math&Stat) Conservation Laws February / 49

14 PDE Models Example: small string/rod oscillations, 1D wave equation Independent variables: x, t; Dependent: u(x, t). Wave equation: u tt = c 2 u xx, c 2 = const = T /ρ (for a string). Conservation of momentum: Local conservation law: D t(ρu t) D x(tu x) = 0; Global conserved quantity: total momentum M = b a ρu t dx = const, for Neumann homogeneous problems with u x(a, t) = u x(b, t) = 0. A. Cheviakov (Math&Stat) Conservation Laws February / 49

15 PDE Models Example: small string/rod oscillations, 1D wave equation Independent variables: x, t; Dependent: u(x, t). Wave equation: u tt = c 2 u xx, c 2 = const = T /ρ (for a string). Conservation of energy: Local conservation law: ( ) ρu 2 D t t 2 + Tu2 x D x(tu tu x) = 0; 2 Global conserved quantity: total energy b ( ) ρu 2 E = t 2 + Tu2 x dx = const, 2 for both Neumann and Dirichlet homogeneous problems. a A. Cheviakov (Math&Stat) Conservation Laws February / 49

16 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 conserved forms. Numerical method testing. A. Cheviakov (Math&Stat) Conservation Laws February / 49

17 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 / 49

18 Variational Principles Action integral J[U] = L(x, U, U,..., k U) dx. Ω 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] = (δl) dx = o(ε). Ω A. Cheviakov (Math&Stat) Conservation Laws February / 49

19 Variational Principles Action integral J[U] = L(x, U, U,..., k U) dx. Ω 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] = (δl) dx = o(ε). Ω Variation of the Lagrangian δl = L(x, U + εv, U + ε v,..., k U + ε k v) L(x, U, U,..., k U) = ( ) L[U] ε U v σ + L[U] v σ σ Uj σ j + + L[U] v Uj σ j σ 1 j k + O(ε 2 ) 1 j k by parts = ε(v σ E U σ (L[U])) + div(...) + O(ε 2 ) A. Cheviakov (Math&Stat) Conservation Laws February / 49

20 Variational Principles Action integral J[U] = L(x, U, U,..., k U) dx. Ω 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] = (δl) dx = o(ε). Ω Euler-Lagrange equations, Euler operators: E U σ (L[U]) = L[U] U σ + + ( 1)k D j1 D jk L[U] U σ j 1 j k = 0, σ = 1,..., m. A. Cheviakov (Math&Stat) Conservation Laws February / 49

21 Variational Principles Example 1: Harmonic oscillator, U = x = x(t) L = 1 2 mẋ kx 2, E xl = m(ẍ + ω 2 x) = 0, ω 2 = k/m. A. Cheviakov (Math&Stat) Conservation Laws February / 49

22 Variational Principles Example 1: Harmonic oscillator, U = x = x(t) L = 1 2 mẋ kx 2, E xl = m(ẍ + ω 2 x) = 0, ω 2 = k/m. Example 2: Wave equation for U = u(x, t) L = 1 2 ρut T ux 2, E ul = ρ(u tt c 2 u xx) = 0, c 2 = T /ρ. A. Cheviakov (Math&Stat) Conservation Laws February / 49

23 Variational Principles Example 1: Harmonic oscillator, U = x = x(t) L = 1 2 mẋ kx 2, E xl = m(ẍ + ω 2 x) = 0, ω 2 = k/m. Example 2: Wave equation for U = u(x, t) L = 1 2 ρut T ux 2, E ul = ρ(u tt c 2 u xx) = 0, c 2 = 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 linearization operator 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 / 49

24 Self-adjointness PDE linearization Given PDE or system: R[u] = 0. Linearized system (Fréchet derivative): L[u]v(x) = d R[u + ɛv] = 0. dɛ ɛ=0 Adjoint Linearized system: by parts w(x) (L[u] v(x)) = (L [u] w(x)) v(x) + (divergence). A. Cheviakov (Math&Stat) Conservation Laws February / 49

25 Self-adjointness PDE linearization Given PDE or system: R[u] = 0. Linearized system (Fréchet derivative): L[u]v(x) = d R[u + ɛv] = 0. dɛ ɛ=0 Adjoint Linearized system: by parts w(x) (L[u] v(x)) = (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 / 49

26 Self-adjointness Example 1: Wave equation for u(x, t) Linearization (already linear!) R[u] = u tt c 2 u xx = 0; Adjoint linearization operator: L[u] v(x, t) = v tt c 2 v xx = 0; w(x, t) L[u] v(x, t) = w(v tt c 2 v xx) = (w tt c 2 w xx)v(x, t)+(v tw vw t) t c 2 (v xw vw x) x; Result: so R[u] is self-adjoint. Lagrangian: L [u] v(x, t) = L[u] v(x, t), L = 1 2 ut c2 u x 2. A. Cheviakov (Math&Stat) Conservation Laws February / 49

27 Self-adjointness Example 2: Heat equation for u(x, t): R[u] = u t u xx = 0. Linearization: L[u] v(x, t) = v t v xx = 0. Adjoint linearization operator: L [u] w(x, t) = w t w xx = 0, NOT self-adjoint! A. Cheviakov (Math&Stat) Conservation Laws February / 49

28 Self-adjointness Example 2: Heat equation for u(x, t): R[u] = u t u xx = 0. Linearization: L[u] v(x, t) = v t v xx = 0. Adjoint linearization operator: L [u] w(x, t) = w t w xx = 0, NOT self-adjoint! Append the adjoint: R[u 1, u 2 ] = {ut 1 uxx 1 = 0, ut 2 uxx 2 = 0}. Self-adjoint! ( Lagrangian: L = 1 2 u 1 (ut 2 uxx) 2 + u 2 (ut 1 uxx) ) 1. Euler-Lagrange equations: E u 1(L) = ut 2 uxx 2 = R 2, E u 2(L) = ut 1 uxx 1 = R 1. A. Cheviakov (Math&Stat) Conservation Laws February / 49

29 Self-adjointness Example 2: Heat equation for u(x, t): R[u] = u t u xx = 0. Linearization: L[u] v(x, t) = v t v xx = 0. Adjoint linearization operator: L [u] w(x, t) = w t w xx = 0, NOT self-adjoint! Append the adjoint: R[u 1, u 2 ] = {ut 1 uxx 1 = 0, ut 2 uxx 2 = 0}. Self-adjoint! ( Lagrangian: L = 1 2 u 1 (ut 2 uxx) 2 + u 2 (ut 1 uxx) ) 1. Euler-Lagrange equations: E u 1(L) = ut 2 uxx 2 = R 2, E u 2(L) = ut 1 uxx 1 = R 1. This technique can be used to make any PDE system self-adjoint. Non-physical Lagrangian (pseudo-lagrangian). A. Cheviakov (Math&Stat) Conservation Laws February / 49

30 Self-adjointness Example 3: Nonlinear PDE for u(x, t): R[u] = u tt + 2u xu xx + ux 2 = 0. Linearization: L[u] v(x, t) = v tt + 2u xv xx + (2u xx + 2u x)v x = 0. NOT self-adjoint! A. Cheviakov (Math&Stat) Conservation Laws February / 49

31 Self-adjointness Example 3: Nonlinear PDE for u(x, t): R[u] = u tt + 2u xu xx + ux 2 = 0. Linearization: L[u] v(x, t) = v tt + 2u xv xx + (2u xx + 2u x)v x = 0. NOT self-adjoint! Modified PDE Nonlinear PDE for u(x, t): R[u] = e x [u tt + 2u xu xx + ux 2 ] = 0. This one turns out to be self-adjoint! A. Cheviakov (Math&Stat) Conservation Laws February / 49

32 Self-adjointness Example 4: KdV for u(x, t) R[u] = u t + uu x + u xxx = 0. Odd-order, clearly NOT self-adjoint. A. Cheviakov (Math&Stat) Conservation Laws February / 49

33 Self-adjointness Example 4: KdV for u(x, t) R[u] = u t + uu x + u xxx = 0. Odd-order, clearly NOT self-adjoint.... a differential substitution: u = q x, R[q] = qxt + q xq xx + q xxxx = 0; Self-adjoint! Lagrangian for R[q]: L = 1 2 q2 xx 1 6 q3 x 1 2 qxqt. A. Cheviakov (Math&Stat) Conservation Laws February / 49

34 Self-adjointness Example 4: KdV for u(x, t) R[u] = u t + uu x + u xxx = 0. Odd-order, clearly NOT self-adjoint.... a differential substitution: u = q x, R[q] = qxt + q xq xx + q xxxx = 0; Self-adjoint! Lagrangian for R[q]: L = 1 2 q2 xx 1 6 q3 x 1 2 qxqt. Result: For a given PDE/system, it is not simple to conclude whether it follows from a variational principle. Much depends on the right writing. Tricks can make equations variational... A feasible tool: comparison of local variational symmetries and local conservation laws. This is based on the first Noether s theorem. A. Cheviakov (Math&Stat) Conservation Laws February / 49

35 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 / 49

36 Symmetries of Differential Equations Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N with variables x = (x 1,..., x n ), u = (u 1,..., u m ). Definition A one-parameter Lie group of point transformations x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ) (with the parameter a) is a point symmetry of R σ [u] if the equation is the same in new variables x, u. A. Cheviakov (Math&Stat) Conservation Laws February / 49

37 Symmetries of Differential Equations Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, with variables x = (x 1,..., x n ), u = (u 1,..., u m ). σ = 1,..., N Definition A one-parameter Lie group of point transformations x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ) (with the parameter a) is a point symmetry of R σ [u] if the equation is the same in new variables x, u. Example 1: translations The translation leaves the KdV equation invariant: x = x + C, t = t, u = u (C R) u t + uu x + u xxx = 0 = u t + u u x + u x x x. A. Cheviakov (Math&Stat) Conservation Laws February / 49

38 Symmetries of Differential Equations Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, with variables x = (x 1,..., x n ), u = (u 1,..., u m ). σ = 1,..., N Definition A one-parameter Lie group of point transformations x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ) (with the parameter a) is a point symmetry of R σ [u] if the equation is the same in new variables x, u. Example 2: scaling The scaling: also leaves the KdV equation invariant: x = αx, t = α 3 t, u = αu (α R) u t + uu x + u xxx = 0 = u t + u u x + u x x x. A. Cheviakov (Math&Stat) Conservation Laws February / 49

39 Variational Symmetries Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N that follows from a variational principle with J[u] = L[u] dx. Ω Definition A symmetry of R σ [u] given by x = f (x, u; a) = x + a ξ(x, u) + O(a 2 ), u = g(x, u; a) = u + a η(x, u) + O(a 2 ) is a variational symmetry of R σ [u] if it preserves the action J[u]. A. Cheviakov (Math&Stat) Conservation Laws February / 49

40 Variational Symmetries Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N that follows from a variational principle with J[u] = L[u] dx. Ω Definition A symmetry of R σ [u] given by x = f (x, u; a) = x + a ξ(x, u) + O(a 2 ), u = g(x, u; a) = u + a η(x, u) + O(a 2 ) is a variational symmetry of R σ [u] if it preserves the action J[u]. Example 1: translations for the wave equation u tt = c 2 u xx, L = 1 2 ut 2 c2 2 ux 2. The translation x = x + C, t = t, u = u is a variational symmetry. A. Cheviakov (Math&Stat) Conservation Laws February / 49

41 Variational Symmetries Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N that follows from a variational principle with J[u] = L[u] dx. Ω Definition A symmetry of R σ [u] given by x = f (x, u; a) = x + a ξ(x, u) + O(a 2 ), u = g(x, u; a) = u + a η(x, u) + O(a 2 ) is a variational symmetry of R σ [u] if it preserves the action J[u]. Example 2: scaling for the wave equation u tt = c 2 u xx, L = 1 2 ut 2 c2 2 ux 2. The scaling x = x, t = t, u = u/α is not a variational symmetry: J = α 2 J. A. Cheviakov (Math&Stat) Conservation Laws February / 49

42 Evolutionary Form of a Local Symmetry A symmetry (in 1D case) x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ). maps a solution u(x) into u (x ), changing both x and u. u (0, ) ) (x, u) (x, u**) (, ) ) (x*, u*) x In the evolutionary form, the same curve mapping does not change x: x = x, u = u + a ζ[u] + O(a 2 ), ζ[u] = η(x, u) u ξ(x, u). x A. Cheviakov (Math&Stat) Conservation Laws February / 49

43 Noether s Theorem (restricted to point symmetries) Theorem Given: 1 a PDE system R σ [u] = 0, σ = 1,..., N, following from a variational principle; 2 a variational symmetry (x i ) = f i (x, u; a) = x i + aξ i (x, u) + O(a 2 ), (u σ ) = g σ (x, u; a) = u σ + aη σ (x, u) + O(a 2 ). Then the system R σ [u] has a conservation law D i Φ i [u] = 0. In particular, D i Φ i [u] Λ σ[u]r σ [u] = 0, where the multipliers are given by Λ σ ζ σ [u] = η σ (x, u) uσ x i ξ i (x, u). A. Cheviakov (Math&Stat) Conservation Laws February / 49

44 Noether s Theorem: Examples Example 1: translation symmetry for the harmonic oscillator Equation: ẍ(t) + ω 2 x(t) = 0. Symmetry: t = t + a, ξ = 1; x = x, η = 0, Multiplier (integrating factor): Λ = η ẋ(t)ξ = ẋ; Conservation law: ΛR = ẋ(ẍ(t) + ω 2 x(t)) = d dt ( ẋ 2 (t) 2 ) + ω2 x 2 (t) = 0. 2 A. Cheviakov (Math&Stat) Conservation Laws February / 49

45 Noether s Theorem: Examples Example 2 Equation: Wave equation u tt = c 2 u xx, u = u(x, t). Space Translation Symmetry: t = t, ξ t = 0; x = x, ξ x = 0, u = u + a, η = 1, Multiplier: Λ = ζ = η 0 u x 0 u t = 1; Conservation law (Momentum): ΛR = 1(u tt c 2 u xx) = D t (u t) D x ( c 2 u x ) = 0. A. Cheviakov (Math&Stat) Conservation Laws February / 49

46 Noether s Theorem: Examples Example 2 Equation: Wave equation u tt = c 2 u xx, u = u(x, t). Time Translation Symmetry: t = t + a, ξ t = 1; x = x, ξ x = 0, u = u, η = 0, Multiplier: Λ = ζ = η 0 u x 1 u t = u t; Conservation law (Energy): ΛR = u t(u tt c 2 u xx) = [ ( ) u 2 D t t 2 + u 2 ( ) ] c2 x D x c 2 u tu x = 0. 2 A. Cheviakov (Math&Stat) Conservation Laws February / 49

47 Linearization Operators and Variational Formulation Recollect: Given PDE or system: R[u] = 0. Linearized system (Fréchet derivative): L[u]v(x) = d R[u + ɛv] = 0. dɛ ɛ=0 Adjoint Linearized system: by parts w(x) (L[u] v(x)) = (L [u] w(x)) v(x) + (divergence). A. Cheviakov (Math&Stat) Conservation Laws February / 49

48 Linearization Operators and Variational Formulation Recollect: Given PDE or system: R[u] = 0. Linearized system (Fréchet derivative): L[u]v(x) = d R[u + ɛv] = 0. dɛ ɛ=0 Adjoint Linearized system: by parts w(x) (L[u] v(x)) = (L [u] w(x)) v(x) + (divergence). Facts: Symmetry components ζ σ [u] are solutions of the linearized system. Conservation law multipliers Λ σ[u] are solutions of the adjoint linearized system. A. Cheviakov (Math&Stat) Conservation Laws February / 49

49 Linearization Operators and Variational Formulation Recollect: Given PDE or system: R[u] = 0. Linearized system (Fréchet derivative): L[u]v(x) = d R[u + ɛv] = 0. dɛ ɛ=0 Adjoint Linearized system: by parts w(x) (L[u] v(x)) = (L [u] w(x)) v(x) + (divergence). Facts: Symmetry components ζ σ [u] are solutions of the linearized system. Conservation law multipliers Λ σ[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 / 49

50 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 / 49

51 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 / 49

52 The Idea of the Direct Construction Method Definition The Euler operator with respect to U j : E U j = D U j i U j i + + ( 1) s D i1... D is U j i 1...i s +, j = 1,..., m. Theorem Let U(x) = (U 1,..., U m ). The equations E U j F [U] 0, j = 1,..., m, hold for arbitrary U(x) if and only if F [U] D i Ψ i [U] for some functions Ψ i [U]. A. Cheviakov (Math&Stat) Conservation Laws February / 49

53 The Idea of the Direct Construction Method Definition The Euler operator with respect to U j : E U j = D U j i U j i + + ( 1) s D i1... D is U j i 1...i s +, j = 1,..., m. Theorem Let U(x) = (U 1,..., U m ). The equations E U j F [U] 0, j = 1,..., m, hold for arbitrary U(x) if and only if F [U] D i Ψ i [U] for some functions Ψ i [U]. Idea: Seek conservation laws in the characteristic form D i Φ i = Λ σr σ = 0 (based on Hadamard s lemma for systems of maximal rank). A. Cheviakov (Math&Stat) Conservation Laws February / 49

54 The Idea of the Direct Construction Method Consider a general system R[u] = 0 of N PDEs. Direct Construction Method Specify dependence of multipliers: Λ σ = Λ σ(x, U,...), σ = 1,..., N. Solve the set of determining equations E U j (Λ σr σ ) 0, j = 1,..., m, for arbitrary U(x) (off of solution set!) to find all such sets of multipliers. Find the corresponding fluxes Φ i [U] satisfying the identity Λ σr σ D i Φ i. Each set of fluxes, multipliers yields a local conservation law D i Φ i [u] = 0, holding for all solutions u(x) of the given PDE system. A. Cheviakov (Math&Stat) Conservation Laws February / 49

55 The Idea of the Direct Construction Method Example The KdV equation R[u] = u t + uu x + u xxx = 0. 0th-order multipliers Determining equations: Solution: E U (Λ(x, t, U)(U t + UU x + U xxx)) 0. Λ 1 = 1, Λ 2 = U, Λ 3 = tu x. Conservation laws: ( ) 1 D t(u) + D x 2 u2 + u xx = 0, ( ( ) 1 D t u2) 1 + D 2 x 3 u3 + uu xx 1 2 u2 x = 0, ( ) ( ) 1 D t 6 u3 1 2 u2 1 x + D x 8 u4 uux (u2 u xx + uxx) 2 u xu xxx = 0. A. Cheviakov (Math&Stat) Conservation Laws February / 49

56 The Idea of the Direct Construction Method Example The KdV equation R[u] = u t + uu x + u xxx = 0. 1st-order multipliers in x Form: Λ = Λ(x, t, U, U x) Solution: no extra conservation laws. A. Cheviakov (Math&Stat) Conservation Laws February / 49

57 The Idea of the Direct Construction Method Example The KdV equation R[u] = u t + uu x + u xxx = 0. 2nd-order multipliers in x Form: Λ = Λ(x, t, U, U x, U xx) Solution: one extra conservation law with Λ 4 = U xx U2. A. Cheviakov (Math&Stat) Conservation Laws February / 49

58 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 / 49

59 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 / 49

60 Completeness of the Direct Construction Method Extended Kovalevskaya form A PDE system R[u] = 0 is in extended Kovalevskaya form with respect to an independent variable x j, if the system is solved for the highest derivative of each dependent variable with respect to x j, i.e., sσ (x j ) sσ uσ = G σ (x, u, u,..., k u), 1 s σ 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 its local conservation law has an equivalent conservation law in the characteristic form, Λ σr σ D i Φ i = 0, such that neither Λ σ nor Φ i involve the leading derivatives or their differential consequences. A. Cheviakov (Math&Stat) Conservation Laws February / 49

61 Completeness of the Direct Construction Method Extended Kovalevskaya form A PDE system R[u] = 0 is in extended Kovalevskaya form with respect to an independent variable x j, if the system is solved for the highest derivative of each dependent variable with respect to x j, i.e., sσ (x j ) sσ uσ = G σ (x, u, u,..., k u), 1 s σ 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 equation R[u] = u t + uu x + u xxx = 0 has the extended Kovalevskaya form with respect to t (u t =...) or x (u xxx =...). A. Cheviakov (Math&Stat) Conservation Laws February / 49

62 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 / 49

63 A Detailed Example: 2D Incompressible Hyperelasticity A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). A solid body occupies the reference (Lagrangian) volume Ω 0 R 3. in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Actual (Eulerian) configuration: Ω R oordinates. The deformed body occupies an 3. Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given Material by points are labeled by X Ω 0. = dx dt The dφ actual dt. position of a material point: x = x (X, t) Ω. Jacobian matrix (deformation gradient): J = det F > 0. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. Cheviakov (Math&Stat) Conservation Laws February / 49

64 A Detailed Example: 2D Incompressible Hyperelasticity A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). Boundary force (per unit area) in Eulerian configuration: t = σn. in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Boundary force (per unit area) in Lagrangian configuration: T = PN. oordinates. The deformed body occupies an Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given σ = by σ(x, t) is the Cauchy stress tensor. = dx dt P = dφ JσF T dt. is the first Piola-Kirchhoff tensor. Density in reference & actual configuration: ρ 0 = ρ 0(X), ρ = ρ(x, t) = ρ 0/J. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. Cheviakov (Math&Stat) Conservation Laws February / 49

65 Full Set of Equations Equations of motion: ρ 0x tt = div (X ) P + ρ 0R, [ ] x i J = det = 1, P ij = p (F 1 ) ji W + ρ X j 0. F ij Neo-Hookean (Hadamard) material: W = a(i 1 3), I 1 = F i kf i k. 2D case: 3 PDEs, 3 unknowns (x, p). PDEs in 2D, no forcing: x 1 x 2 X 1 X x 1 x 2 2 X 2 2 x 1 t 2 2 x 2 t 2 [ α [ α X 1 = 0, 1 ( 2 x 1 (X 1 ) + 2 x 1 2 (X 2 ) 2 ( 2 x 2 (X 1 ) + 2 x 2 2 (X 2 ) 2 ) p x 2 X 1 X + p ] x 2 = 0, 2 X 2 X 1 ) p x 1 X 2 X + p ] x 1 = 0. 1 X 1 X 2 A. Cheviakov (Math&Stat) Conservation Laws February / 49

66 Point symmetries PDEs in 2D, no forcing: x 1 x 2 X 1 X x 1 x 2 2 X 2 2 x 1 t 2 2 x 2 t 2 [ α [ α X 1 = 0, 1 ( 2 x 1 (X 1 ) + 2 x 1 2 (X 2 ) 2 ( 2 x 2 (X 1 ) + 2 x 2 2 (X 2 ) 2 ) p x 2 X 1 X + p ] x 2 = 0, 2 X 2 X 1 ) p x 1 X 2 X + p ] x 1 = 0. 1 X 1 X 2 Point symmetries: Y 1 = t, Y2 = X 1, Y3 = X 2, Y4 = F 1(t) p, Y 5 = X 2 X 1 X 1 X 2, Y6 = x 2 x 1 x 1 x 2, Y 7 = F 2(t) x 1 F 2 (t) x 1 p, Y8 = F 3(t) x 2 F 3 (t) x 2 p, Y 9 = t t + X 1 X 1 + X 2 X 2 + x 1 x 1 + x 2 x 2. A. Cheviakov (Math&Stat) Conservation Laws February / 49

67 Conservation laws PDEs in 2D, no forcing: R 1 [x, p] = x 1 X 1 x 2 X 2 x 1 X 2 x 2 R 2 [x, p] = 2 x 1 t 2 R 3 [x, p] = 2 x 2 t 2 [ α [ α X 1 = 0, 1 ( 2 x 1 (X 1 ) + 2 x 1 2 (X 2 ) 2 ( 2 x 2 (X 1 ) + 2 x 2 2 (X 2 ) 2 ) p x 2 X 1 X + p ] x 2 = 0, 2 X 2 X 1 ) p x 1 X 2 X + p ] x 1 = 0. 1 X 1 X 2 Conservation laws: D tψ + D X 1Φ 1 + D X 1Φ 2 = 3 Λ i [x, p] R i [x, p]. i=1 Note Extended Kovalevskaya form exists w.r.t. X 1 and X 2, very complicated... A. Cheviakov (Math&Stat) Conservation Laws February / 49

68 Conservation laws PDEs in 2D, no forcing: R 1 [x, p] = x 1 X 1 x 2 X 2 x 1 X 2 x 2 R 2 [x, p] = 2 x 1 t 2 R 3 [x, p] = 2 x 2 t 2 [ α [ α X 1 = 0, 1 ( 2 x 1 (X 1 ) + 2 x 1 2 (X 2 ) 2 ( 2 x 2 (X 1 ) + 2 x 2 2 (X 2 ) 2 ) p x 2 X 1 X + p ] x 2 = 0, 2 X 2 X 1 ) p x 1 X 2 X + p ] x 1 = 0. 1 X 1 X 2 Notation: 2 x 1 t 2 x 1 tt, x 1 X 2 x 1 2, 2 x 2 X 1 X 2 x 2 12, 2 p X 2 t p2t,... A. Cheviakov (Math&Stat) Conservation Laws February / 49

69 Conservation laws PDEs in 2D, no forcing: R 1 [x, p] = x 1 X 1 x 2 X 2 x 1 X 2 x 2 R 2 [x, p] = 2 x 1 t 2 R 3 [x, p] = 2 x 2 t 2 [ α [ α X 1 = 0, 1 ( 2 x 1 (X 1 ) + 2 x 1 2 (X 2 ) 2 ( 2 x 2 (X 1 ) + 2 x 2 2 (X 2 ) 2 ) p x 2 X 1 X + p ] x 2 = 0, 2 X 2 X 1 ) p x 1 X 2 X + p ] x 1 = 0. 1 X 1 X 2 First-order multipliers: Λ 1 = C 1(X 1 p 2 X 2 p 1) + C 3p t + C 4p 1 + C 5p 2 + f 1(t) + f 2 (t)x 1 + f 3 (t)x 2, Λ 2 Λ 3 = C 1(X 2 x 1 1 X 1 x 1 2 ) + C 2x 2 C 3x 1 t C 4x 1 1 C 5x f 2(t), = C 1(X 2 x 2 1 X 1 x 2 2 ) C 2x 1 C 3x 2 t C 4x 2 1 C 5x f 3(t). A. Cheviakov (Math&Stat) Conservation Laws February / 49

70 Conservation laws Conservation of energy: Λ 1 = p t, Λ 2 = x 1 t, Λ 3 = x 2 t. Divergence expression: ( ( ) 1 D t 2 x 1 2 ( ) ( t x 2 2 t + 1 (x α ) 1 2 ( ) x 1 2 ( ) 2 + x 2 2 ( ) )) 1 + x ( ( ) ) D X 1 α x 1 t x1 1 + xt 2 x1 2 px 1 t x2 2 + px2 1 xt 2 D X 2 ( ( ) ) α x 1 t x2 1 + xt 2 x2 2 + px 1 t x1 2 px1 1 xt 2 = 0. A. Cheviakov (Math&Stat) Conservation Laws February / 49

71 Conservation laws Conservation of generalized momentum in x 1 Λ 1 = f 2 (t)x 1, Λ 2 = f 2(t), Λ 3 = 0. Divergence expression: D t ( f2(t)x 1 t f 2 (t)x 1) D X 1 D X 2 ( αf 2(t)x1 1 pf 2(t)x2 2 1 f 2 2 (t) ( x 1) ) 2 x 2 2 ( αf 2(t)x2 1 + pf 2(t)x f 2 2 (t) ( x 1) ) 2 x 2 1 = 0. Same for x 2. A. Cheviakov (Math&Stat) Conservation Laws February / 49

72 Conservation laws Conservation of angular momentum in x 3 Λ 1 = 0, Λ 2 = x 2, Λ 3 = x 1. Divergence expression: D t ( x 1 x 2 t x 1 t x 2) +D X 1 +D X 2 ( ( ) ) α x 1 1 x 2 x 1 x1 2 px 1 x2 1 px 2 x2 2 ( ( ) ) α x 1 2 x 2 x 1 x2 2 + px 1 x1 1 + px 2 x1 2 = 0. A. Cheviakov (Math&Stat) Conservation Laws February / 49

73 Conservation laws Generalized incompressibility condition Λ 1 = f 1(t), Λ 2 = 0, Λ 3 = 0. Divergence expression: ( ) ( ) ( ) D t f 1(t) dt D X 1 f 1(t)x 1 x2 2 + D X 2 f 1(t)x 1 x1 2 = 0. A. Cheviakov (Math&Stat) Conservation Laws February / 49

74 Conservation laws Momenta in the material frame Linear momentum: p ρ 0F T x t = (x 1 t x x 2 t x 2 1 )î + (x 1 t x x 2 t x 2 2 )ĵ. Angular momentum: m X p = (X 1 (x 1 t x x 2 t x 2 2 ) X 2 (x 1 t x x 2 t x 2 1 ))ˆk. Conservation of the material momentum in X 1 Λ 1 = p 1, Λ 2 = x 1 1, Λ 3 = x 2 1. Divergence expression: ( ) D t x 1 t x1 1 + xt 2 x1 2 ( ( (x ) 1 2 ( ) +D X 1 α 2 + x 2 2 ( ) 2 x 1 2 ( ) ) 1 x 2 2 ( ) 1 + p 1 2 x 1 2 ( ) ) t 1 2 x 2 2 t ( ( )) D X 2 α x 1 1 x2 1 + x1 2 x2 2 = 0. Same for X 2. A. Cheviakov (Math&Stat) Conservation Laws February / 49

75 Conservation laws Momenta in the material frame Linear momentum: p ρ 0F T x t = (x 1 t x x 2 t x 2 1 )î + (x 1 t x x 2 t x 2 2 )ĵ. Angular momentum: m X p = (X 1 (x 1 t x x 2 t x 2 2 ) X 2 (x 1 t x x 2 t x 2 1 ))ˆk. Conservation of the material angular momentum in X 3 Divergence expression: Λ 1 = X 1 p 2 X 2 p 1, Λ 2 = X 2 x 1 1 X 1 x 1 2, Λ 3 = X 2 x 2 1 X 1 x 2 2. D t [ X 2 (x 1 t x x 2 t x 2 1 ) X 1 (x 1 t x x 2 t x 2 2 ) ] +D X 1 +D X 2 [ 1 ( αx ( ) 2 x 1 2 ( ) 2 1 x 2 2 ( ) 1 + x 1 2 ( ) ) 2 + x αx ( ) 1 x1 1 x2 1 + x1 2 x2 2 + X 2 p 1 X ( ) 2 x t 1 X ( ) ] 2 x t [ 1 ( αx ( ) 1 x 1 2 ( ) 2 1 x 2 2 ( ) 1 + x 1 2 ( ) ) 2 + x αx ( ) 2 x1 1 x2 1 + x1 2 x2 2 X 1 p + 1 X ( ) 1 x t + 1 X ( ) ] 1 x t = 0. A. Cheviakov (Math&Stat) Conservation Laws February / 49

76 Comparison of Symmetries and Conservation Laws Evol. Symmetry Multiplier Conserved quantity Ŷ 1 = xt 1 x 1 x2 t x 2 pt p Ŷ 2 = x1 1 x 1 x2 1 x 2 p 1 p Ŷ 3 = x2 1 x 1 x2 2 x 2 p 2 p Λ 1 = p t, Λ 2 = x 1 t, Λ3 = x 2 t Energy Λ 1 = p 1, Λ 2 = x 1 1, Λ3 = x 2 1 Lagrangian momentum in X 1 Λ 1 = p 2, Λ 2 = x 1 2, Λ3 = x 2 2 Lagrangian momentum in X 2 Ŷ 4 = F 1 (t) Λ 1 = f 1 (t), Λ 2 = 0, Λ 3 = 0 Generalized incompressiblity p ( ) Ŷ 5 = X 2 x1 1 + X 1 x2 1 x 1 Λ 1 = (X 1 p 2 X 2 p 1 ), Angular momentum ( ) X 2 x1 2 + X 1 x2 2 x 2 Λ 2 = (X 2 x1 1 X 1 x2 1 ), in Lagrangian frame ( ) + X 2 p 1 + X 1 p 2 Λ 3 = (X 2 x1 2 p X 1 x2 2 ) Ŷ 6 = x 2 x 1 x1 x 2 Λ 1 = 0, Λ 2 = x 2, Λ 3 = x 1 Eulerian angular momentum Ŷ 7 = F 2 (t) x 1 F 2 (t) x1 Λ 1 = f 2 p (t)x1, Λ 2 = f 2 (t), Λ 3 = 0 Generalized momentum in x 1 Ŷ 8 = F 3 (t) x 2 F 3 (t) x2 Λ 1 = f 3 p (t)x2, Λ 2 = 0, Λ 3 = f 3 (t) Generalized momentum in x 2 Ŷ 9 [Scaling ] No corresponding conservation law

77 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, x 1, x 2 ). This is non-trivial, since the incompressible equations involve a differential constraint. Lagrangian: L = K P + p(j 1), K = 1 2 P = 1 2 α ( ( x 1 1 ( ( ) 2 ( ) ) 2 xt 1 + xt 2, ) 2 ( ) 2 ( ) 2 ( ) ) 2 + x2 1 + x1 2 + x2 2, J = x 1 1 x 2 2 x 1 2 x 2 1. A. Cheviakov (Math&Stat) Conservation Laws February / 49

78 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, x 1, x 2 ). This is non-trivial, since the incompressible equations involve a differential constraint. PDEs: E p L = J + 1 = R 1, E x 1 L = x 1 tt + α(x x 1 22) p 1x p 2 x 2 1 = R 2, E x 2 L = x 2 tt + α(x x 2 22) p 2x p 1 x 1 2 = R 3. Conservation laws: Every conservation law follows from a symmetry. A. Cheviakov (Math&Stat) Conservation Laws February / 49

79 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 / 49

80 Navier-Stokes Equations in 3D Navier-Stokes Equations in 3D PDEs: u = 0, u t + (u )u + p ν 2 u = 0. Variables: pressure p(t, x, y, z), velocity u(t, x, y, z) = u 1 x + u 2 ŷ + u 3 ẑ. 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 / 49

81 Navier-Stokes Equations in 3D Navier-Stokes Equations in 3D PDEs: u = 0, u t + (u )u + p ν 2 u = 0. Variables: pressure p(t, x, y, z), velocity u(t, x, y, z) = u 1 x + u 2 ŷ + u 3 ẑ. 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 Λ i, i = 1,..., 4, depending on 56 variables each (up to 2nd-order derivatives). Result: generalized momenta, angular momenta, generalized incompressibility [C., Oberlack (2014)]. A. Cheviakov (Math&Stat) Conservation Laws February / 49

82 Navier-Stokes Equations in 3D Navier-Stokes Equations in 3D PDEs: u = 0, u t + (u )u + p ν 2 u = 0. Variables: pressure p(t, x, y, z), velocity u(t, x, y, z) = u 1 x + u 2 ŷ + u 3 ẑ. 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 ) + ( ) (u 1 f (t) xf (t))u 1 + f (t)(p νux 1 ) x + ( ) (u 1 f (t) xf (t))u 2 νf (t)uy 1 + ( ) (u 1 f (t) xf (t))u 3 νf (t)uz 1 = 0. y z Cyclically permute (1, 2, 3), (x, y, z) for the other two components. A. Cheviakov (Math&Stat) Conservation Laws February / 49

83 Navier-Stokes Equations in 3D Navier-Stokes Equations in 3D PDEs: u = 0, u t + (u )u + p ν 2 u = 0. Variables: pressure p(t, x, y, z), velocity u(t, x, y, z) = u 1 x + u 2 ŷ + u 3 ẑ. 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 yu 3 ) + x + y + z ( ) (zu 2 yu 3 )u 1 + ν(yux 3 zux 2 ) ( (zu 2 yu 3 )u 2 + zp + ν(yu 3 y zu 2 y u 3 ) ( ) (zu 2 yu 3 )u 3 yp + ν(yuz 3 zuz 2 + u 2 ) = 0. Cyclically permute (1, 2, 3), (x, y, z) for the other two components. ) A. Cheviakov (Math&Stat) Conservation Laws February / 49

84 Navier-Stokes Equations in 3D Navier-Stokes Equations in 3D PDEs: u = 0, u t + (u )u + p ν 2 u = 0. Variables: pressure p(t, x, y, z), velocity u(t, x, y, z) = u 1 x + u 2 ŷ + u 3 ẑ. 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 / 49

85 Navier-Stokes Equations in 3D Navier-Stokes Equations in 3D PDEs: u = 0, u t + (u )u + p ν 2 u = 0. Variables: pressure p(t, x, y, z), velocity u(t, x, y, z) = u 1 x + u 2 ŷ + u 3 ẑ. 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 2 u] F F t 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 / 49

86 Navier-Stokes Equations in 3D Navier-Stokes Equations in 3D PDEs: u = 0, u t + (u )u + p ν 2 u = 0. Variables: pressure p(t, x, y, z), velocity u(t, x, y, z) = u 1 x + u 2 ŷ + u 3 ẑ. 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; [Kelbin et al, (2013)]). Additional CLs for Euler equations, including symmetric cases [C., Oberlack (2014); Kelbin et al, (2013)]. A. Cheviakov (Math&Stat) Conservation Laws February / 49

87 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 / 49

88 Conclusions Discussion Divergence-type conservation laws are useful in analysis and numerics. Generally, conservation laws can be obtained systematically through the Direct construction 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 / 49

89 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 / 49

90 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, Gusyatnikova, V. N., and Yumaguzhin, V. A. (1989) Symmetries and conservation laws of Navier-Stokes equations. Acta App. Math. 15: Cheviakov, A. F. (2007) GeM software package for computation of symmetries and conservation laws of differential equations. Comput. Phys. Comm. 176, Cheviakov, A. F., Ganghoffer, J.-F., and St. Jean, S. (2015) Fully nonlinear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids. (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 and inviscid flows. J. Fluid Mech. 721, Cheviakov, A. F., and Oberlack, M. (2014) Generalized Ertel s theorem and infinite hierarchies of conserved quantities for three-dimensional time dependent Euler and Navier-Stokes equations. J. Fluid Mech. 760, A. Cheviakov (Math&Stat) Conservation Laws February / 49