Application of the method of differential constraints to constructing exact solutions of the gas dynamics equations S.V. Meleshko School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, Thailand Abrau-Durso, Russia September 14-19, 2015 1 / 38
Outline Introduction 1 Equations describing motion of fluids with internal inertia Examples Group classification Conservation law 2 Noether’s theorem Shmyglevskii’s approach Ibragimov’s approach Equations in Lagrangian variables 3 Gas dynamics equations in Lagrangian coordinates Method of differential constraints 4 Generalized simple waves Conclusion 5 Abrau-Durso, Russia September 14-19, 2015 2 / 38
Introduction Equations describing motion of fluids with internal inertia Equations describing motion of fluids with internal inertia S.L.Gavrilyuk, V.M.Teshukov (2001) ˙ ρ + ρdiv ( u ) = 0 , ˙ ρ ˙ u + ∇ p = 0 , S = 0 Abrau-Durso, Russia September 14-19, 2015 3 / 38
Introduction Equations describing motion of fluids with internal inertia Equations describing motion of fluids with internal inertia S.L.Gavrilyuk, V.M.Teshukov (2001) ˙ ρ + ρdiv ( u ) = 0 , ˙ ρ ˙ u + ∇ p = 0 , S = 0 p = ρδW δρ − W, W = W ( ρ, ˙ ρ, S ) Abrau-Durso, Russia September 14-19, 2015 3 / 38
Introduction Equations describing motion of fluids with internal inertia Equations describing motion of fluids with internal inertia S.L.Gavrilyuk, V.M.Teshukov (2001) ˙ ρ + ρdiv ( u ) = 0 , ˙ ρ ˙ u + ∇ p = 0 , S = 0 p = ρδW δρ − W, W = W ( ρ, ˙ ρ, S ) p = ρ ( W ρ − ( W ˙ ρ − div ( uW ˙ ρ )) − W, t is time, ∇ is the gradient operator with respect to space variables, ρ is the fluid density, u is the velocity field, W ( ρ, ˙ ρ, S ) is a given potential, d f ˙ ’dot’ denotes the material time derivative: f = = f t + u ∇ f , dt δW denotes the variational derivative of W with respect to ρ at a fixed value of u . δρ Abrau-Durso, Russia September 14-19, 2015 3 / 38
Introduction Examples Example 1: Iordanski-Kogarko-Wingaarden model (1960, 1961, 1968) Abrau-Durso, Russia September 14-19, 2015 4 / 38
Introduction Examples Example 2: Green-Naghdi equations (1975) Abrau-Durso, Russia September 14-19, 2015 5 / 38
Introduction Examples Example 2: Green-Naghdi equations (1975) Yu.Yu.Bagderina, A.P.Chupakhin (2005) Invariant and partially invariant solutions of the Green–Naghdi equations Abrau-Durso, Russia September 14-19, 2015 5 / 38
Introduction Group classification Isentropic flows Abrau-Durso, Russia September 14-19, 2015 6 / 38
Introduction Group classification Isentropic flows A.Hematulin, S.V.Meleshko and S.L.Gavrilyuk (2007) Abrau-Durso, Russia September 14-19, 2015 6 / 38 Group classification of one-dimensional equations of fluids with internal inertia
Introduction Group classification Nonisentropic flows P.Siriwat, C.Kaewmanee and S. V. Meleshko (2015) Group classification of one-dimensional nonisentropic equations of fluids with internal inertia II. General case Abrau-Durso, Russia September 14-19, 2015 7 / 38
Introduction Group classification Nonisentropic flows P.Siriwat, C.Kaewmanee and S. V. Meleshko (2015) Group classification of one-dimensional nonisentropic equations of fluids with internal inertia II. General case Abrau-Durso, Russia September 14-19, 2015 8 / 38
Conservation law Noether’s theorem Noether’s theorem XF + FD i ξ i = W k δF δu k + D i ( N i F ) Abrau-Durso, Russia September 14-19, 2015 9 / 38
Conservation law Noether’s theorem Noether’s theorem XF + FD i ξ i = W k δF δu k + D i ( N i F ) N i F = ξ i F + W k δF δF � D i 1 ...D i s ( W k ) + , ( i = 1 , 2 , ..., n ) . δu k δu k i ii 1 i 2 ...i s s =1 Abrau-Durso, Russia September 14-19, 2015 9 / 38
Conservation law Noether’s theorem Noether’s theorem XF + FD i ξ i = W k δF δu k + D i ( N i F ) N i F = ξ i F + W k δF δF � D i 1 ...D i s ( W k ) + , ( i = 1 , 2 , ..., n ) . δu k δu k i ii 1 i 2 ...i s s =1 W k = η k − ξ i u k i , ( k = 1 , 2 , ..., m ) , δ ∂ ∂ � ( − 1) s D i 1 ...D i s δu k = ∂u k + , ( k = 1 , 2 , ..., m ) ∂u k i 1 i 2 ...i s s =1 Abrau-Durso, Russia September 14-19, 2015 9 / 38
Conservation law Noether’s theorem Noether’s theorem � ∂η k ∂u j − ∂ξ i δ = X ( δF δu j ) + δF � XF + FD i ξ i � ∂u j u k i + δ kj D i ξ i � δu j δu k Abrau-Durso, Russia September 14-19, 2015 10 / 38
Conservation law Noether’s theorem Noether’s theorem � ∂η k ∂u j − ∂ξ i δ = X ( δF δu j ) + δF � XF + FD i ξ i � ∂u j u k i + δ kj D i ξ i � δu j δu k If δF XF + FD i ξ i = D i B i , δu j = 0 then � δF � X = 0 δu j | δF δu =0 Variational (divergent) symmetry is a symmetry Abrau-Durso, Russia September 14-19, 2015 10 / 38
Conservation law Shmyglevskii’s approach Shmyglevskii’s approach. Gas dynamics Terent’ev&Shmyglevskii (1980) Abrau-Durso, Russia September 14-19, 2015 11 / 38
Conservation law Shmyglevskii’s approach Shmyglevskii’s approach. Gas dynamics Terent’ev&Shmyglevskii (1980) Bateman (1929), Ito (1955), Shmyglevskii (1980) � � u 2 L = ρ 2 + ˙ ϕ + S ˙ µ − ρU ( ρ, S ) where ϕ and µ play the roles of Lagrange’s multipliers. S is the entropy , U ( ρ, S ) is the internal energy Abrau-Durso, Russia September 14-19, 2015 11 / 38
Conservation law Shmyglevskii’s approach Shmyglevskii’s approach. Fluids with internal inertia � � u 2 L = ρ 2 + ˙ ϕ + S ˙ µ − W ( ρ, ˙ ρ, S ) . Euler–Lagrange equations u = −∇ ϕ − S ∇ µ + ρ − 1 W ˙ µ = ρ − 1 W S , ρ ∇ ρ, ˙ u 2 µ = W ρ − ∂W ˙ ρ 2 + ˙ ϕ + S ˙ − div ( W ˙ ρ u ) , ∂t ∂ρ ∂ ( ρS ) ∂t + div ( ρu ) = 0 , + div ( ρSu ) = 0 , ∂t Abrau-Durso, Russia September 14-19, 2015 12 / 38
Conservation law Shmyglevskii’s approach Shmyglevskii’s approach. Fluids with internal inertia � � u 2 L = ρ 2 + ˙ ϕ + S ˙ µ − W ( ρ, ˙ ρ, S ) . Euler–Lagrange equations u = −∇ ϕ − S ∇ µ + ρ − 1 W ˙ µ = ρ − 1 W S , ρ ∇ ρ, ˙ u 2 µ = W ρ − ∂W ˙ ρ 2 + ˙ ϕ + S ˙ − div ( W ˙ ρ u ) , ∂t ∂ρ ∂ ( ρS ) ∂t + div ( ρu ) = 0 , + div ( ρSu ) = 0 , ∂t ˙ ρ + ρu x = 0 , ˙ ρ ˙ u + p x = 0 , S = 0 , p = ρδW δρ − W = ρ ( W ρ − ( W ˙ ρ − div ( uW ˙ ρ )) − W, Abrau-Durso, Russia September 14-19, 2015 12 / 38
Conservation law Shmyglevskii’s approach Shmyglevskii’s approach. Example of a conservation law W = ρ − 3 ˙ ρ 2 η, X = t∂ t − u∂ u ∂tC 1 + ∂ ∂ ∂xC 2 = 0 , Abrau-Durso, Russia September 14-19, 2015 13 / 38
Conservation law Shmyglevskii’s approach Shmyglevskii’s approach. Example of a conservation law W = ρ − 3 ˙ ρ 2 η, X = t∂ t − u∂ u ∂tC 1 + ∂ ∂ ∂xC 2 = 0 , C 1 = tρu 2 ρ 2 + ρ ( ϕ + ηµ ) , 2 − tρ − 3 η ˙ C 2 = − tρu 3 2 − tuηρ − 3 ˙ ρ 2 − 4 tu 2 ηρ − 3 ˙ ρρ x + 4 tu 2 ηρ − 2 ρ tx − 2 tu 2 ρ − 2 ˙ ρη x +2 tu 2 ρ − 2 ˙ ρηη x + 2 tuρ − 2 ˙ ρu x η − 2 tu 2 ρ − 2 u x ηρ x + 2 tuρ − 2 ηρ tt +2 tuρ − 2 ηu t ρ x + 2 tu 3 ρ − 3 ηρ xx + ρu ( ϕ + ηµ ) . Abrau-Durso, Russia September 14-19, 2015 13 / 38
Conservation law Ibragimov’s approach Ibragimov’s approach Ibragimov Conservation law’s Theorem Consider a system of m equations F α ( x, u, u (1) , u 2 , . . . , u ( s ) ) = 0 , α = 1 , . . . , m (1) with n independent variables x = ( x 1 , x 2 , . . . , x n ) and m dependent variables u = ( u 1 , u 2 , . . . , u m ) . The adjoint system α ( x, u, v, u (1) , v (1) , u (2) , v (2) , . . . , u ( s ) , u ( s ) ) ≡ δ L F ∗ δu α = 0 (2) inherits the symmetries of the system (1), where L = v β F β ( x, u, u (1) , . . . , u ( s ) ) Abrau-Durso, Russia September 14-19, 2015 14 / 38
Conservation law Ibragimov’s approach Ibragimov Conservation law’s Theorem (continue) If system (1) admits a point transformation group with a generator X = ξ i ( x, u ) ∂ ∂x i + η α ( x, u ) ∂ (3) ∂u α then also the adjoint system (2) admits the operator (3). Then the quantities � � j ) ∂F β ξ i F β + ( η α − ξ j u α C i = v β , i = 1 , . . . , n (4) ∂u α i furnish a conserved vector C = ( C 1 , . . . , C n ) for the system (1). Abrau-Durso, Russia September 14-19, 2015 15 / 38
Conservation law Ibragimov’s approach Ibragimov’s approach. Example of a conservation law The formal Lagrangian L = ( R + u 2 ρ + ρu x ) + U ( u t + uu x + ρ − 1 p x ) + P ˙ 2 )( ˙ η Abrau-Durso, Russia September 14-19, 2015 16 / 38
Conservation law Ibragimov’s approach Ibragimov’s approach. Example of a conservation law The formal Lagrangian L = ( R + u 2 ρ + ρu x ) + U ( u t + uu x + ρ − 1 p x ) + P ˙ 2 )( ˙ η Example of multipliers R = δW P = W η − W ˙ U = ρu, ρη ( ˙ ρ + ρu x ) , δρ + W ˙ ρη ˙ η Abrau-Durso, Russia September 14-19, 2015 16 / 38
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