On the Dependence of Euler Equations on Physical Parameters

Transcription

1 On the Dependence of Euler Equations on Physical Parameters Cleopatra Christoforou Department of Mathematics, University of Houston Joint Work with: Gui-Qiang Chen, Northwestern University Yongqian Zhang, Fudan University 12th International Conference on Hyperbolic Problems: Theory, Numerics, Applications 12th International Conference on Hyperbolic Problems: Theory, Numerics, Applications June 9-13, 2008 University of Maryland, College Park June 9-13, 2008 PLENARY SPEAKERS Sylvie Benzoni-Gavage, Lyon INVITED SPEAKERS Debora Amadori, L Aquila ORGANIZING COMMITTEE Jian-Guo Liu, Eitan Tadmor, Thanos Tzavaras

2 OUTLINE: 1 Introduction/ Motivation 2 Our approach 3 Applications to Euler Equations

3 Intro Hyperbolic Systems of Conservation Laws in one-space dimension: { t U + x F (U) = 0 x R U(0, x) = U 0, where U = U(t, x) R n is the conserved quantity and (1) F : R n R n smooth flux. Admissible/Entropy weak solution: U(t, x) in BV. t η(u) + x q(u) 0 in D

4 Motivation Examples: Isothermal Euler equations t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + κ 2 ρ) = 0 where ρ is the density and u is the velocity of the fluid. Isentropic Euler equations t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + κ 2 ρ γ ) = 0 γ > 1 where γ > 1 is the adiabatic exponent. Relativistic Euler equations: ( (p + ρ c 2 ) u 2 ) ( ) t c 2 c 2 u 2 + ρ + x (p + ρ c 2 u ) c 2 u 2 = 0 ) ( t ((p + ρ c 2 u ) c 2 u 2 + x (p + ρ c 2 u 2 ) ) c 2 u 2 + p = 0, (4) where c < is the speed of light. (2) (3)

5 Motivation Question: As γ 1 and c, can we pass from the isentropic to the isothermal and from the relativistic to the classical?

6 Motivation Question: As γ 1 and c, can we pass from the isentropic to the isothermal and from the relativistic to the classical? In general, the Question is: How do the admissible weak solutions depend on the physical parameters?

7 Approach Systems of Conservation Laws in one-space dimension: { t W µ (U) + x F µ (U) = 0 x R U(0, x) = U 0, (5) where W µ, F µ : R n R n are smooth functions that depend on a parameter vector µ = (µ 1,..., µ k ), µ i [0, µ 0 ], for i = 1,..., k. and W 0 (U) = U. Formulate an effective approach to establish L 1 estimates of the type: U µ (t) U(t) L 1 C TV {U 0 } t µ (6) U µ is the entropy weak solution to (5) for µ 0 constructed by the front tracking method. U(t) := S t U 0, S is the Lipschitz Standard Riemann Semigroup associated with (5) for µ = 0. µ is the magnitute of the parameter vector µ.

8 Approach Error estimate Let S be a Lipschitz continuous semigroup: S : D [0, ) D, t S t w(0) w(t) L 1 L lim inf 0 h 0+ S h w(τ) w(τ + h) L 1 dτ, (7) h where L is the Lipschitz constant of the semigroup and w(τ) D. The above inequality appears extensively in the theory of front tracking method: e.g. (i) the entropy weak solution by front tracking coincides with the trajectory of the semigroup S if the semigroup exists, (ii) uniqueness within the class of viscosity solutions, etc... References: Bressan et al.

9 Approach Front-Tracking Method For δ > 0, let U δ,µ be the δ-approximate solution to { t W µ (U) + x F µ (U) = 0 for µ 0 U(0, x) = U 0, (i) U δ 0 piecewise constant, Uδ 0 U 0 L 1 < δ. (ii) U δ,µ are globally defined piecewice constant functions with finite number of discontinuities. (iii) The discontinuities are of three types: shock fronts, rarefaction fronts with strength less than δ, non-physical fronts with total strength α < δ. (iv) U δ,µ U µ in L 1 loc as δ 0+. References: Bressan, Dafermos, DiPerna, Holden Risebro.

10 Approach Description Approach Apply the error estimate on w = U δ,µ : S t U δ 0 U δ,µ (t) L 1 L The aim is to estimate t 0 lim inf h 0+ S h U δ,µ (τ) U δ,µ (τ + h) L 1 dτ, h S h U δ,µ (τ) U δ,µ (τ + h) L 1 (8) which is equivalent to solving the Riemann problem of (5) when µ = 0 for τ t τ + h with data { U (U L, U R ) = δ,µ (τ, x) x < x U δ,µ (9) (τ, x) x > x over each front of U δ,µ at time τ, i.e. find S h (U L, U R ). Then compare it with the same front of U δ,µ (τ + h). We solve the Riemann problem at all non-interaction times of U δ,µ.

11 Approach Description If we can show: x+a x a S h U δ,µ (τ) U δ,µ (τ + h) dx = O(1)h( µ U L U R + δ), then summing over all fronts of U δ,µ (τ), S t U δ,µ (0) U δ,µ (t) L 1 t L 0 fronts x= x(τ) 1 h x+a x a (10) S h U δ,µ (τ) U δ,µ (τ + h) dx dτ (11) (12)

12 Approach Description If we can show: x+a x a S h U δ,µ (τ) U δ,µ (τ + h) dx = O(1)h( µ U L U R + δ), then summing over all fronts of U δ,µ (τ), S t U δ,µ (0) U δ,µ (t) L 1 t L 0 = O(1) fronts x= x(τ) t ( µ 0 1 h x+a x a (10) S h U δ,µ (τ) U δ,µ (τ + h) dx dτ ) TVU δ,µ (τ) dτ + δ t (11)

13 Approach Description If we can show: x+a x a S h U δ,µ (τ) U δ,µ (τ + h) dx = O(1)h( µ U L U R + δ), then summing over all fronts of U δ,µ (τ), S t U δ,µ (0) U δ,µ (t) L 1 t L 0 = O(1) fronts x= x(τ) t ( µ 0 1 h x+a x a (10) S h U δ,µ (τ) U δ,µ (τ + h) dx dτ ) TVU δ,µ (τ) dτ + δ t = O(1)( µ TV {U 0 } + δ) t (11)

14 Approach Description If we can show: x+a x a S h U δ,µ (τ) U δ,µ (τ + h) dx = O(1)h( µ U L U R + δ), then summing over all fronts of U δ,µ (τ), S t U δ,µ (0) U δ,µ (t) L 1 t L 0 = O(1) fronts x= x(τ) t ( µ 0 1 h x+a x a (10) S h U δ,µ (τ) U δ,µ (τ + h) dx dτ ) TVU δ,µ (τ) dτ + δ t = O(1)( µ TV {U 0 } + δ) t (11) As δ 0+, we obtain U(t) U µ (t) L 1 = O(1) TV {U 0 } t µ (12) where U := S t U 0 is the entropy weak solution to (5) for µ = 0.

15 Approach Description Remarks Note that U := S t U 0 is unique within the class of viscosity solutions. (Bressan et al). Thus, as µ 0 U µ S t U 0 in L 1. Temple: existence using that the nonlinear functional in Glimm s scheme depends on the properties of the equations at µ = 0. Bianchini and Colombo: consider S F, S G and show S F is Lipschitz w.r.t. the C 0 norm of DF.

16 Applications: Isothermal Euler equations Isothermal Euler equations: t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + κ 2 ρ) = 0 (13) where ρ is the density and u is the velocity of the fluid. Nishida [1968]: Existence of entropy solution for large initial data via the Glimm s scheme. Colombo-Risebro [1998]: Construction of the Standard Riemann Semigroup for large initial data. Existence, stability and uniqueness within viscosity solutions. Let S be the Lipschitz Standard Riemann Semigroup generated by Isothermal Euler Equations (13).

17 Applications: Isothermal Euler equations Isentropic Isothermal Euler equations 1. Isentropic Euler Equations: t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + p(ρ)) = 0 (14) of a perfect polytropic fluid p(ρ) = κ 2 ρ γ, where γ > 1 is the adiabatic exponent. Existence results: when (γ 1) TV {U 0 } < N (i) Nishida-Smoller by Glimm s scheme, [1973] (ii) Asakura by the front tracking method [2005].

18 Applications: Isothermal Euler equations Isentropic Isothermal Euler equations 1. Isentropic Euler Equations: of a perfect polytropic fluid p(ρ) = κ 2 ρ γ, t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + p(ρ)) = 0 where γ > 1 is the adiabatic exponent. Existence results: when (γ 1) TV {U 0 } < N (i) Nishida-Smoller by Glimm s scheme, [1973] (ii) Asakura by the front tracking method [2005]. Theorem (G.-Q. Chen, Christoforou, Y. Zhang) Suppose that 0 < ρ ρ 0 (x) ρ < and (γ 1) TV {U 0 } < N. Let µ = γ 1 2 and U µ be the entropy weak solution to (14) obtained by the front tracking method, then for every t > 0, (14) S t U 0 U µ (t) L 1 = O(1) TV {U 0 } t (γ 1) (15)

19 Applications: Isothermal Euler equations Isentropic Isothermal Euler equations Case 1: Shock Front of strength α = ρ R and µ = 1 ρ 2 (γ 1) L h I α U L II β 1 β 2 α U U R h I U L II β 1 β 2 U III U R U x U U L x L R U R β 1 = α + O(1) α 1 (γ 1) β 2 = 1 + O(1) α 1 (γ 1). I: U L U R = U L U R length of I = O(1) h µ II: U U R = O(1) U L U R µ length of II = O(1) h III: U(ξ) U R = O(1) U L U R µ length of III = O(1) h U L U R µ 1 h x+a x a S h U δ,µ (τ) U δ,µ (τ+h) dx = O(1) µ U δ,µ (τ, x ) U δ,µ (τ, x+)

20 Applications: Isothermal Euler equations Isentropic Isothermal Euler equations Case 2: Rarefaction Front I II III I II III IV h α β 1 U β 2 h α β 1 U β 2 U L U R U L U R U L x U R U L x U R

21 Applications: Isothermal Euler equations Isentropic Isothermal Euler equations Case 2: Rarefaction Front I II III I II III IV h α β 1 U β 2 h α β 1 U β 2 U L U R U L U R U L x U R U L x U R Case 3: Non-Physical Front h β 1 I U β 2 II α speed ˆλ U L U R U L x U R

22 Applications: Isothermal Euler equations Relativistic Isothermal Euler equations 2. Relativistic Euler Equations for conservation of momentum: ( (p + ρ c 2 ) u 2 ) ( ) t c 2 c 2 u 2 + ρ + x (p + ρ c 2 u ) c 2 u 2 = 0 ) ( t ((p + ρ c 2 u ) c 2 u 2 + x (p + ρ c 2 u 2 ) ) c 2 u 2 + p = 0, (16) of a perfect polytropic fluid p(ρ) = κ 2 ρ γ, where γ 1 is the adiabatic exponent and c is the speed of light. Parameter vector: µ = (γ 1, 1 c 2 ). Existence results: by Glimm s scheme (i) Smoller-Temple (γ = 1), for TV {U 0 } large, [1993] (ii) J. Chen when (γ 1) TV {U 0 } < N, [1995]

23 Applications: Isothermal Euler equations Relativistic Isothermal Euler equations Theorem (G.-Q. Chen, Christoforou, Y. Zhang) Suppose that 0 < ρ ρ 0 (x) ρ < and (γ 1) TV {U 0 } < N. Let U µ be the entropy weak solution to Relativistic Euler Equations for conservation of momentum (16) for γ > 1 and c c 0 constructed by the front tracking method, then for every t > 0, ( S t U 0 U µ (t) L 1 = O(1) TV {U 0 } t (γ 1) + 1 ) c 2 (17) for µ = (γ 1, 1 c 2 ). Proof. 1. Establish the front tracking method for γ > 1 and c 0 < c <. 2. Due to the Lorenz invariance, employ the techniques of the previous theorem and solve the Riemann problem for each one of the three cases.

24 Applications: Isothermal Euler equations Relativistic Isothermal Euler equations 3. Isentropic Relativistic Euler Equations of conservation laws of baryon number and momentum in special relativity: ( ) ( ) n nu t + x = 0 1 u 2 /c 2 1 u 2 /c 2 t ( (ρ + p/c 2 )u 1 u 2 /c 2 ) + x ( (ρ + p/c 2 )u 2 1 u 2 /c 2 + p ) = 0 For isentropic fluids, the proper number density of baryons n is n = n(ρ) = n 0 exp( ρ Theorem (G.-Q. Chen, Christoforou, Y. Zhang) 1 (18) ds s + p(s) ). (19) c 2 S t U 0 U µ (t) L 1 = O(1) TV {U 0 } t ( (γ 1) + 1 ) c 2. (20)

25 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations 4. Non-Isentropic Euler equations t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + p) = 0 t (ρ( 1 2 u2 + e)) + x (ρ u( 1 2 u2 + e) + p u) = 0. (21) ρ density, u velocity, p pressure and e internal energy. T temperature, S entropy and v = 1/ρ specific volume. Law of thermodynamics: Entropy condition: T ds = de + p dv. (ρ S) t + (ρus) x 0.

26 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations For a polytropic gas, i.e. ε = γ 1 > 0, then p = κ 2 e S/cv ρ γ and e(ρ, S, ε) = 1 ε (( e S/R ρ ) ε 1) Existence results: when (γ 1) TV {U 0 } < N (i) T.-P. Liu [1977] and Temple [1981] by Glimm s scheme. G.-Q. Chen Wagner [2003] (ii) Asakura by the front tracking method, preprint [2006] Thus, as ε 0, e 0 (ρ, S) = lim ε 0 e(ρ, S, ε) = ln ρ + S R

27 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations As ε 0+, non-isentropic Euler equations t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + p) = 0 t (ρ( 1 2 u2 + e)) + x (ρ u( 1 2 u2 + e) + p u) = 0. t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + κ 2 ρ) = 0 t (ρ( 1 2 u2 + e 0 )) + x (ρ u( 1 2 u2 + e 0 ) + κ 2 ρ u) = 0, (22) (23) with (ρ S) t + (ρus) x 0. (24)

28 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations Non-Isentropic Euler equations t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + p) = 0 t (ρ( 1 2 u2 + e)) + x (ρ u( 1 2 u2 + e) + p u) = 0. (25) with Isothermal Euler equations t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + κ 2 ρ) = 0, (26) (ρ( 1 2 u2 + ln ρ)) t + (ρ u( 1 2 u2 + ln ρ) + κ 2 ρ u) x 0 (27)

29 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations Theorem (G.-Q. Chen, Christoforou, Y. Zhang) Suppose that 0 < ρ ρ 0 (x) ρ < and (γ 1) TV {U 0 } < N. Let U ε = (ρ ε, ρ ε u ε, ρ ε ( 1 2 u2 ε + e ε )) be the entropy weak solution to Non-Isentropic Euler Equations (21) for ε > 0 constructed by the front-tracking method. Then, for every t > 0, ρ(t) ρ ε (t) L 1 + u(t) u ε (t) L 1 = O(1) TV {U 0 } t (γ 1), (28) where (ρ(t), u(t)) is the solution to Isothermal Euler Equations (26) generated by S. As ε 0, for every t > 0, ρ ε (t) ρ(t), u ε (t) u(t) in L 1 loc.

30 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations Remarks: For ε = 0: The Standard Riemann Semigroup associated with the 3 3 limiting system: Colombo Risebro for the Isothermal Euler equations. t ρ + x (ρ u) = 0 t (ρ u) + x (ρ u 2 + κ 2 ρ) = 0 t (ρ( 1 2 u2 + e 0 )) + x (ρ u( 1 2 u2 + e 0 ) + κ 2 ρ u) = 0,

31 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations Remarks: For ε = 0: The Standard Riemann Semigroup associated with the 3 3 limiting system: Colombo Risebro for the Isothermal Euler equations. For ε > 0: The front tracking method: Use Asakura s result.

32 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations Remarks: For ε = 0: The Standard Riemann Semigroup associated with the 3 3 limiting system: Colombo Risebro for the Isothermal Euler equations. For ε > 0: The front tracking method: Use Asakura s result. Cases: Shock fronts, Rarefaction fronts, Non-Physical fronts and also 2- contact discontinuity fronts!!!

33 Applications: Isothermal Euler equations Non-Isentropic Isothermal Euler equations Case: Contact Discontinuity τ + h I U L II U R III I II III III U 3 (ξ) h U L U R U L U R U L U R U L U R τ U 1 (ξ) (a) x Î I II III Î (b) x I II III III U L U R U L U R U L U R U L U R (c) x (d) x

34 Zero Mach Limit to Compressible Euler 5*. Compressible Euler Eqs with Mach Number with energy t ρ + x (ρu) = 0 t (ρu) + x (ρu M 2 p) = 0 M > 0 t (ρe) + x ((ρe + p)u) = 0 E = p u2 + M2 (γ 1)ρ 2 (29) and initial data in BV (R): ρ t=0 = ρ 0 + M 2 ρ (0) 2 (x), ρ 0 > 0 constant p t=0 = p 0 + M 2 p (0) 2 (x), p 0 > 0 constant (30) u t=0 = Mu (0) 1 (x) Denote the solution to (29) (30) by (ρ M, p M, u M ). References: Majda, Klainerman-Majda, Metivier, Schochet.

35 Zero Mach Limit to Compressible Euler (ρ M, p M, u M ) has an asymptotic expansion: ρ M (t, x) = ρ 0 + M 2 ρ M 2 (t, x) + O(1)M 3, p M (t, x) = p 0 + M 2 p M 2 (t, x) + O(1)M3, u M (t, x) = M u M 1 (t, x) + O(1)M2, (31) where (ρ M 2, pm 2, um 1 ) satisfy the linear acoustic system: t ρ 2 + ρ 0 x u 1 = 0 M t p 2 + γp 0 x u 1 = 0 (32) M t u Mρ 0 x p 2 = 0 with the initial data t=0 ρ 2 = ρ (0) 2 (x) p 2 = p (0) 2 (x) u 1 = u (0) 1 (x). (33) t=0 t=0

36 Zero Mach Limit to Compressible Euler Theorem (G.-Q. Chen, Christoforou, Y. Zhang: Arch. Rat. Mech. An.) Suppose that ρ (0) 2, p(0) 2, u(0) 1 BV (R 1 ). Then, there exists a constant M0 > 0 such that for M (0, M0), for every t 0, ρ M (t) ρ 0 M 2 ρ M 2 (t) L 1 = O(1) TV {(p (0) 2, u(0) 1, ρ(0) 2 )} t M3, p M (t) p 0 M 2 p M 2 (t) L 1 = O(1) TV {(p (0) 2, u(0) 1, ρ(0) 2 )} t M3, u M (t) Mu M 1 (t) L 1 = O(1) TV {(p (0) 2, u(0) 1, ρ(0) 2 )} t M2, where (ρ M 2, pm 2, um 1 acoustic system. ) is the unique weak solution to the linear

37 Zero Mach Limit to Compressible Euler Remarks: 0 < M < M0 small data to compressible Euler

38 Zero Mach Limit to Compressible Euler Remarks: 0 < M < M0 small data to compressible Euler S M exists.

39 Zero Mach Limit to Compressible Euler Remarks: 0 < M < M0 small data to compressible Euler S M exists. Define metric: for any V = (ρ, p, u), Ṽ = ( ρ, p, ũ) d M (V, Ṽ ) = ρ ρ L 1 + p p L 1 + M u ũ L 1 (34) so that the error formula becomes d M (S M t w(0), w(t)) L t 0 lim inf h 0+ d M (S M h w(τ), w(τ + h)) dτ h

40 Zero Mach Limit to Compressible Euler Remarks: 0 < M < M0 small data to compressible Euler S M exists. Define metric: for any V = (ρ, p, u), Ṽ = ( ρ, p, ũ) d M (V, Ṽ ) = ρ ρ L 1 + p p L 1 + M u ũ L 1 (34) so that the error formula becomes d M (S M t w(0), w(t)) L t 0 lim inf h 0+ d M (S M h w(τ), w(τ + h)) dτ h Do not need to employ the front tracking method! Approximate the solution to the linear acoustic limit by piecewise constant functions: W M,n = (ρ M, n 2, p M, n 2, u M, n 1 ) W M = (ρ M 2, pm 2, um 1 ).

41 Zero Mach Limit to Compressible Euler Remarks: 0 < M < M0 small data to compressible Euler S M exists. Define metric: for any V = (ρ, p, u), Ṽ = ( ρ, p, ũ) d M (V, Ṽ ) = ρ ρ L 1 + p p L 1 + M u ũ L 1 (34) so that the error formula becomes d M (S M t w(0), w(t)) L t 0 lim inf h 0+ d M (S M h w(τ), w(τ + h)) dτ h Do not need to employ the front tracking method! Approximate the solution to the linear acoustic limit by piecewise constant functions: W M,n = (ρ M, n 2, p M, n 2, u M, n 1 ) W M = (ρ M 2, pm 2, um 1 ). Apply the error formula on U M,n (t, x) = (ρ 0 + M 2 ρ M,n 2 (t, x), p 0 + M 2 p M,n 2 (t, x), Mu M,n 1 (t, x))

42 Zero Mach Limit to Compressible Euler Case of 1-shock: U M,n L W M,n L = (ρ M,n 2,L, pm,n 2,L, um,n 1,L = (ρ 0 + M 2 ρ M,n 2,L, p 0 + M 2 p M,n 2,L, MuM,n 1,L ) and ) W M,n L W M,n R = O(1)[u 1 ]. U M L U M R U M L U M R I 1 I 2 I 1 I 2 t U M m,1 U M m,2 U M m,1 U M m,2 U M L U M R U M L U M R (a) x Length of interval I 1 = O(1) M h, I 2 = O(1) 1 M h Difference: in ρ is O(1)[u1 ] M2 h O(1)[u1 ]M4 h in p is O(1)[u1 ] M2 h O(1)[u1 ] M4 h in u is O(1)[u1 ] M h O(1)[u 1 ] M3 h ) d M (S h M UM,n (τ), U M,n (τ + h) = O(1) h TV {W M,n 3 (τ)}m, (b) x

43 Zero Mach Limit to Compressible Euler We show ) d M (S h M UM,n (τ), U M,n (τ + h) = O(1) h TV {W M,n 3 (τ)}m, and by the error estimate we get d M (S M t U M,n (0), U M,n (t)) = O(1) M 3 TV {(p (0) 2, u(0) 1, ρ(0) 2 )} t.

44 Zero Mach Limit to Compressible Euler We show ) d M (S h M UM,n (τ), U M,n (τ + h) = O(1) h TV {W M,n 3 (τ)}m, and by the error estimate we get d M (S M t U M,n (0), U M,n (t)) = O(1) M 3 TV {(p (0) 2, u(0) As n, d M (S M t U(0), U M (t)) = O(1) M 3 TV {(p (0) 2, u(0) 1, ρ(0) 2 1, ρ(0) 2 )} t. )} t (35) where S M t U(0) = (ρ M (t, x), p M (t, x), u M (t, x)) U M = (ρ 0 + M 2 ρ M 2 (t, x), p 0 + M 2 p M 2 (t, x), Mu M 1 (t, x))

45 Publications Publications: G.-Q. Chen, C. Christoforou and Y. Zhang, Dependence of Entropy Solutions in the Large for the Euler Equations on Nonlinear Flux Functions, Indiana University Mathematics Journal, 56 (2007), (5) G.-Q. Chen, C. Christoforou and Y. Zhang, L 1 estimates of entropy solutions to the Euler equations with respect to the adiabatic exponent and Mach number, Archive for Rational Mechanics and Analysis, to appear.