Simple waves and characteristic decompositions of quasilinear hyperbolic systems in two independent variables

Transcription

1 s and characteristic decompositions of quasilinear hyperbolic systems in two independent variables Wancheng Sheng Department of Mathematics, Shanghai University (Joint with Yanbo Hu) Joint Workshop on Partial Differential Equations Shanghai Jiaotong University, Shanghai, China November 15-17, 2012 Wancheng Sheng Shanghai University s and characteristic decompositions

2 1 s strictly hyperbolic system adjacent to a constant state 3 with straight characteristics solutions 4 Pseudo-steady Euler equations The generalized UTSD system Wancheng Sheng Shanghai University s and characteristic decompositions

3 Outline s A simple wave is defined as a flow in a region whose image is a curve in the phase space. It plays an important role in the theories of gas dynamics and fluid mechanics. Supersonic Flow and Shock Waves, Intersience, New York, s play a fundamental role in describing and building up solutions of flow problems. (pp ) Wancheng Sheng Shanghai University s and characteristic decompositions

4 Outline s Hyperbolic systems in two independent variables u x + A(u)u y = 0 where A(u) = (a ij (u)) n n, u = (u 1,, u n ). The real and distinct eigenvalues λ 1 (u) < < λ n (u). Lax 1957 and Dafermos 2000 The states in a domain adjacent to a domain of constant state is always a simple wave by using the Riemann invariants. Remark The treatment is invalid when the matrix A depends on x and y as well as u. Wancheng Sheng Shanghai University s and characteristic decompositions

5 s an example A classical 1-D wave equation u tt c 2 u xx = 0 with constant speed c, which has an interesting decomposition ( t ± c x )( t c x )u = 0. One can rewrite them as + R = 0 and S = 0, where R = u := t u c x u, S = + u := t u + c x u. Wancheng Sheng Shanghai University s and characteristic decompositions

6 s quasilinear equations Z.H. Dai, T. Zhang, Arch. Rat. Mech. Anal., Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics J.Q. Li, T. Zhang, Y.X. Zheng, Commun. Math. Phys., s and a characteristic decomposition of the two dimensional compressible Euler equations J.Q. Li, Y.X. Zheng, Arch. Rat. Mech. Anal., Interaction of rarefaction waves of the two-dimensional self-similar Euler equations Wancheng Sheng Shanghai University s and characteristic decompositions

7 2 2 strictly hyperbolic system 2 2 strictly hyperbolic system adjacent to a constant state The general 2 2 strictly hyperbolic system ( ) ( ) ( ) u a11 a + 12 u v a 21 a 22 v = 0, (1) x where the coefficients a ij = a ij (x, y, u, v), i, j = 1, 2, which satisfy (a 11 + a 22 ) 2 4(a 11 a 22 a 12 a 21 ) > 0. The coefficient matrix of the system has two eigenvalues λ ± = (a 11 + a 22 ) ± (a 11 + a 22 ) 2 4(a 11 a 22 a 12 a 21 ), 2 which are solutions to the characteristic equation λ 2 (a 11 + a 22 )λ + a 11 a 22 a 12 a 21 = 0. y Wancheng Sheng Shanghai University s and characteristic decompositions

8 2 2 strictly hyperbolic system 2 2 strictly hyperbolic system adjacent to a constant state The characteristic form of system (1) is ± u + λ a 22 a 21 ± v = 0, (2) where ± := x + λ ± y. Here we assume without loss of generality that a Denote A 1 := a 11x + a 11 a 11y + a 21 a 12y, A 2 := a 12x + a 12 a 11y + a 22 a 12y, A 3 := a 21x + a 11 a 21y + a 21 a 22y, A 4 := a 22x + a 12 a 21y + a 22 a 22y, where u and v are parameters in a ij = a ij (x, y, u, v) (i, j = 1, 2). Wancheng Sheng Shanghai University s and characteristic decompositions

9 2 2 strictly hyperbolic system 2 2 strictly hyperbolic system adjacent to a constant state If the coefficients a ij (i, j = 1, 2) satisfy then there hold A 1 = A 2 = A 3 = A 4 = 0, (3) ± λ ± = (λ a 22 )λ ±u + a 21 λ ±v λ a 22 ± u, (4) and ± u = h ± ± u for some suitable factors h ±. Consequently, the role of above equations is to ensure that the simple wave region be covered by a family of straight characteristics. Y.B. Hu, W.C. Sheng, Applied Mathematics Letters, Characteristic decomposition of the 2 2 quasilinear strictly hyperbolic systems Wancheng Sheng Shanghai University s and characteristic decompositions

10 2 2 strictly hyperbolic system adjacent to a constant state 2 2 strictly hyperbolic system more general one. If the sufficient condition (4) does not satisfy, we will obtain more general results. Li and Zheng, Arch. Rat. Mech. Anal., (commutator relation) For any quantity I = I (x, y), there holds + I + I = λ + + λ λ λ + ( I + I ). Taking I = u, we get ± u + +λ λ + λ + λ u ( ( ) ( ) ) = λ ± a 22 λ ± λ λ a 22 a21 a 21 λ a 22 ± u + λ a 22 a 21 a21 ± λ ± a 22 u. Wancheng Sheng Shanghai University s and characteristic decompositions

11 2 2 strictly hyperbolic system adjacent to a constant state 2 2 strictly hyperbolic system more general one. We compute ( ) λ± a 22 ± a 21 = ±(λ ± a 22 )a 21 (λ ± a 22 ) ± a 21 a 2 21 = m ± ± u a 21(a 22x +λ ± a 22y )+(λ ± a 22 )(a 21x +λ ± a 21y ) a 21 (λ ±x +λ ± λ ±y ) a21 2, where ( a 21 λ ±u + a 21 a λ a 22 λ ±v a 22u a 21 22v λ a 22 ) (λ ± a 22 ) m ± = a 2 21 ( ) a a 21u +a 21 21v λ a 22. Wancheng Sheng Shanghai University s and characteristic decompositions

12 2 2 strictly hyperbolic system adjacent to a constant state 2 2 strictly hyperbolic system more general one. Then there hold ± ( λ± a 22 a 21 ) = m ± ± u if the coefficients a ij (i, j = 1, 2) satisfy the following equations: a 22x + λ ± a 22y + λ ± a 22 a 21 (a 21x + λ ± a 21y ) = λ ±x + λ ± λ ±y. Wancheng Sheng Shanghai University s and characteristic decompositions

13 2 2 strictly hyperbolic system adjacent to a constant state 2 2 strictly hyperbolic system more general one. It follows that B 1 + λ ± B 2 = 0, where B 1 = a 21 (a 11 a 22 a 12 a 21 ) x + (a 11 + a 22 )(a 22 a 21x a 21 a 22x ) (a 11 a 22 a 12 a 21 )[(a 11 a 22 )a 21y (a 11 a 22 ) y a a 21x ] B 2 =a 21 (a 11 a 22 a 12 a 21 ) y a 21 (a 11x + a 22x ) + 2a 21 a 22x 2a 22 a 21x (a 11 +a 22 )(a 21x +a 21 a 22y a 22 a 21y )+(a 11 +a 22 )[2a 21x +2a 21 a 22y 2a 22 a 21y (a 11 +a 22 )a 21y a 21 (a 11y +a 22y )]+2[(a 11 +a 22 ) 2 (a 11 a 22 a 12 a 21 )]a 21y. Wancheng Sheng Shanghai University s and characteristic decompositions

14 2 2 strictly hyperbolic system adjacent to a constant state 2 2 strictly hyperbolic system more general one. Thus, from λ + λ 0, we have B 1 = B 2 = 0, which can be simplified as { a21 A 2 a 12 A 3 = 0, (5) a 21 (A 1 A 4 ) + (a 22 a 11 )A 3 = 0. Therefore there hold the identities ± u = h ± ± u for some suitable factors h ± if system (5) is satisfied. Wancheng Sheng Shanghai University s and characteristic decompositions

15 2 2 strictly hyperbolic system adjacent to a constant state 2 2 strictly hyperbolic system more general one. Theorem 1. There hold and ± u = λ a 22 a 21 ± v, ± u = h ± ± u for some factors h ± if the coefficients a ij (i, j = 1, 2) with a 21 0 satisfy system { a21 A 2 a 12 A 3 = 0, a 21 (A 1 A 4 ) + (a 22 a 11 )A 3 = 0. Wancheng Sheng Shanghai University s and characteristic decompositions

16 2 2 strictly hyperbolic system adjacent to a constant state adjacent to a constant state Theorem 2. Adjacent to a constant state of equations (1) is a simple wave in which the variables (u, v) are constant along a family of characteristics if the coefficients a ij (i, j = 1, 2) with a 21 0 satisfy system { a21 A 2 a 12 A 3 = 0, a 21 (A 1 A 4 ) + (a 22 a 11 )A 3 = 0. Remark Similar arguments can be made for the case a 21 = 0. Wancheng Sheng Shanghai University s and characteristic decompositions

17 with straight characteristics solutions with straight characteristics Theorem 2. A simple wave region to be covered by one family of straight characteristics if and only if the coefficients a ij (i, j = 1, 2) satisfy where A 1 = A 2 = A 3 = A 4 = 0. A 1 := a 11x + a 11 a 11y + a 21 a 12y, A 2 := a 12x + a 12 a 11y + a 22 a 12y, A 3 := a 21x + a 11 a 21y + a 21 a 22y, A 4 := a 22x + a 12 a 21y + a 22 a 22y. Wancheng Sheng Shanghai University s and characteristic decompositions

18 with straight characteristics solutions with straight characteristics then In fact, if λ ±x + λ ± λ ±y = 0, (6) ± λ ± = (λ a 22 )λ ±u + a 21 λ ±v λ a 22 ± u. Thus, the condition (6) can be rewritten as ( (a11 +a 22 )(a 11 +a 22 ) y +(a 11 +a 22 ) x (a 11 a 22 a 12 a 21 ) y ) λ± [(a 11 a 22 a 12 a 21 ) x + (a 11 a 22 a 12 a 21 )(a 11 + a 22 ) y ] = 0, from which we get after some simplifications { A1 + A 4 = 0, (a 11 a 22 )A 1 + a 21 A 2 + a 12 A 3 = 0. Wancheng Sheng Shanghai University s and characteristic decompositions

19 with straight characteristics solutions with straight characteristics Then, we have Therefore, we find 0 a 21 a 12 0 a 21 0 a 22 a 11 a a 11 a 22 a 21 a 12 0 A 1 = A 2 = A 3 = A 4 = 0 A 1 A 2 A 3 A 4 by the fact that the determinant of coefficient matrix a 21 [(a 11 + a 22 ) 2 4(a 11 a 22 a 12 a 21 )] 0. = 0. Wancheng Sheng Shanghai University s and characteristic decompositions

20 solutions with straight characteristics solutions Theorem 3. Assume that the coefficients a ij (i, j = 1, 2) with a 21 0 satisfy { a21 A 2 a 12 A 3 = 0, a 21 (A 1 A 4 ) + (a 22 a 11 )A 3 = 0. Then (u, v)(x, y) is a simple wave solution of (1). Wancheng Sheng Shanghai University s and characteristic decompositions

21 solutions with straight characteristics solutions y c+ (s) l 0 x Wancheng Sheng Shanghai University s and characteristic decompositions

22 1. Pseudo-steady Euler equations Pseudo-steady Euler equations The generalized UTSD system In 2-D isentropic irrotational ideal flow in the self-similar plane (ξ, η) = (x/t, y/t), the equations of motion is [ ] [ ] u 2UV c 2 V 2 + c 2 U 2 c 2 U u 2 = 0, (7) v 1 0 v ξ where c is the speed of sound satisfying the pseudo-bernoulli s law c 2 γ 1 + U2 + V 2 2 = ϕ, (U, V ) = (u ξ, v η) is the pseudo-velocity, and ϕ = ϕ(ξ, η) is the pseudo-potential such that ϕ ξ = U, ϕ η = V. Taking u and v are parameters in c, we can directly obtain c ξ = 0 and c η = 0. Wancheng Sheng Shanghai University s and characteristic decompositions η

23 1. Pseudo-steady Euler equations Pseudo-steady Euler equations The generalized UTSD system Then we have ( ) 2(u ξ)(v η) a 11ξ + a 11 a 11η + a 21 a 12η = c 2 (u ξ) 2 ξ [ ] [ 2(u ξ)(v η) 2(u ξ)(v η) c 2 (v η) 2 ] + c 2 (u ξ) 2 c 2 (u ξ) 2 c 2 (u ξ) 2 = 2(v η)[c2 (u ξ) 2 ] + 4(u ξ) 2 (v η) [c 2 (u ξ) 2 ] 2 + 2(u ξ)(v η) c 2 (u ξ) 2 2(u ξ)[c2 (u ξ) 2 ] 2(v η) [c 2 (u ξ) 2 ] 2 c 2 (u ξ) 2 = 0, η η Wancheng Sheng Shanghai University s and characteristic decompositions

24 1. Pseudo-steady Euler equations Pseudo-steady Euler equations The generalized UTSD system and a 12ξ + a 12 a 11η [ c 2 (v η) 2 ] = c 2 (u ξ) 2 ξ + c2 (v η) 2 c 2 (u ξ) 2 [ ] 2(u ξ)(v η) c 2 (u ξ) 2 = 2(u ξ)[c2 (v η) 2 ] [c 2 (u ξ) 2 ] 2 + c2 (v η) 2 c 2 (u ξ) 2 2(u ξ) c 2 (u ξ) 2 =0, which imply the coefficients of equations (7) satisfy system (3). η Wancheng Sheng Shanghai University s and characteristic decompositions

25 2. The generalized UTSD system Pseudo-steady Euler equations The generalized UTSD system The generalized UTSD system (A(U) xj(u))u x + (B(U) yj(u))u y = 0, (8) where U = [u, v], A(U) = (a ij (U)), B(U) = (b ij (U)), J(U) = (j ij (U)), i, j = 1, 2. This kind of system was used in dealing with the unsteady transonic small disturbance (UTSD) equations by Čanić and Keyfitz. S. Čanić, B.L. Keyfitz, Arch. Rat. Mech. Anal., 1998 Quasi-one-dimensional Riemann problems and their role in self-similar two-dimensional problems Wancheng Sheng Shanghai University s and characteristic decompositions

26 2. The generalized UTSD system Pseudo-steady Euler equations The generalized UTSD system Assume without loss of generality that A(U) xj(u) 0, then we can rewrite (8) in a new form [ ] [ ] [ ] u c11 c 12 u + = 0, (9) v c 21 c 22 v x y Wancheng Sheng Shanghai University s and characteristic decompositions

27 2. The generalized UTSD system Pseudo-steady Euler equations The generalized UTSD system where c 11 = (a 22 xj 22 )(b 11 yj 11 ) (a 12 xj 12 )(b 21 yj 21 ) (a 11 xj 11 )(a 22 yj 22 ) (a 12 xj 12 )(a 21 yj 21 ), c 12 = (a 22 xj 22 )(b 12 yj 12 ) (a 12 xj 12 )(b 22 yj 22 ) (a 11 xj 11 )(a 22 yj 22 ) (a 12 xj 12 )(a 21 yj 21 ), c 21 = (a 11 xj 11 )(b 21 yj 21 ) (a 21 xj 21 )(b 11 yj 11 ) (a 11 xj 11 )(a 22 yj 22 ) (a 12 xj 12 )(a 21 yj 21 ), c 22 = (a 11 xj 11 )(b 22 yj 22 ) (a 21 xj 21 )(b 12 yj 12 ) (a 11 xj 11 )(a 22 yj 22 ) (a 12 xj 12 )(a 21 yj 21 ). We find that the coefficients c ij, i, j = 1, 2 of equations (9) satisfy system (5) by direct computation. Wancheng Sheng Shanghai University s and characteristic decompositions

28 Pseudo-steady Euler equations The generalized UTSD system Thank you! Wancheng Sheng Shanghai University s and characteristic decompositions