Dynamics for the solutions of the water-hammer equations

Transcription

1 Dynamics for the solutions of the water-hammer equations J. Alberto Conejero (IUMPA-Universitat Politècnica de València). Joint work with C. Lizama (Universidad Santiado de Chile) and F. Ródenas (IUMPA-Universitat Politècnica de València) XIV Encuentros Análisis Funcional Murcia Valencia Homenaje a Manuel Maestre en su 60 cumpleaños. 24 de septiembre de 2015

2 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations Research topics: 1 Linear chaos (chaos on infinite-dimensional systems) 2 Families of linear operators ( C 0 -semigroups ) 3 Applications to PDE

3 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations The water hammer phenomenon (Also called hydraulic transients) We already have studied chaos for the solutions to the wave equation, so we decided to try to study if some type of chaos appears behind this phenomenon

8 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations The study of hydraulic transients started (seriously) with the wave equation. The wave equation (d Alembert) y tt (x, t) = a 2 y xx (x, t) (1) where a is the propagation speed, x the position of the particle (in equilibrium), and y the vertical displacement. The general solution is given by y(x, t) := f (x + at) + g(x at), where t 0 (2) and f ang g are traveling waves.

9 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations We fix ρ > 0 and consider the space X ρ = { f : R C ; f (x) = n=0 a n ρ n n! x n, (a n ) n 0 c 0 }, (3) with the norm f = sup n 0 a n, where c 0 is the Banach space of complex sequences tending to 0. Then X ρ is a Banach space of analytic functions with a certain growth control. By its definition it is isometrically isomorphic to c 0. Herzog 97 This type of spaces were already used when studying the dynamics of the solution of the heat equation. Conejero, Peris, Trujillo 10 & Gross-Erdmann, Peris 11 We also studied the case of the hyperbolic heat equation and the wave equation.

13 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations Some basic ideas on hydraulics: Bernoulli s principle The total energy at a given point in a fluid is equal to the energy associated with the movement of the fluid, plus energy from pressure in the fluid, plus energy from the height of the fluid relative to an arbitrary datum.

14 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations Bernoulli s principle V 2 ρ 2 + P + ρgz = constant (4) v, speed across a section ρ, density of the fluid P, pressure g, gravity acceleration z, height respect to the datum V 2 P + 2g ρg + z = constant (5) }{{}}{{} Kinetics(DISCHARGE) Pressure(PIEZOMETRIC HEAD)

15 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations Hydraulics: Steady state vs. transient flow. They are derived from the classical mass and momentum conservation equations adding the following assumptions: 1 The flow in the conduit is one-dimensional, 2 The velocity is uniform over the cross section of the conduit, 3 The conduit walls and the fluid are linearly elastic 4 The formulas for computing the steady-state friction losses in conduits are valid during the transient state.

16 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations Hydraulics: Steady state vs. transient flow. They are derived from the classical mass and momentum conservation equations adding the following assumptions: 1 The flow in the conduit is one-dimensional, 2 The velocity is uniform over the cross section of the conduit, 3 The conduit walls and the fluid are linearly elastic 4 The formulas for computing the steady-state friction losses in conduits are valid during the transient state.

17 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations These are given by the next pair of coupled partial differential equations: where Q t + gah x + f Q Q = 0, (Dynamic equation) (6) 2DA v 2 ga Q x + H t = 0, (Continuity equation) (7) Q(x, t) represents the discharge H(x, t) represent the piezometric head at the centerline of the conduit above the specified datum, f is the friction factor (which is assumed to be constant), g is the acceleration due to gravity, v is the fluid wave velocity, and A and D are the the cross-sectional area and the diameter of a conduit, respectively. The parameters A and D, are characteristics of the conduit system and are time invariant, but may be functions of x.

21 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations This pair of coupled nonlinear partial differential equations can be represented as ( ) Q(x, t) H(x, t) t ( ) Q(x, 0) H(x, 0) = where F (y, t) = fy y 2DA ( 0 ga d dx v 2 d ga dx 0 ( ) ϕ1 (x) =, x R. ϕ 2 (x) ) (Q(x, ) t) + H(x, t) ( F (Q(x, t), t) 0 ), (8)

22 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations As a consequence, relative to a fixed time coordinate, disturbances have a finite propagation speed and they travel along the characteristics of the equation Method of characteristics Along the lines x = vt the equations are reduced to first-order ones. Q t + ga v H t + f dx AQ Q = 0 if = v. 2D dt (9) Q t ga v H t + f dx AQ Q = 0 if = v. 2D dt (10)

23 The water hammer phenomenon Wave vs. water hammer equations Water hammer equations Description of the water hammer phenomenon studied since end of 19th century and early 1900 s. (Menabrea, Joukowsky, and Allevi among others). Further information regarding water hammer equations: M. Hanif Chaudry. Applied hydraulic transients. Ed. Springer. 3rd ed. 2014, XIV, 583 p

24 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations To develop the study of the dynamical behaviour of the water hammer phenomenon, the solutions will be represented by a C 0 -semigroup generated by certain first order differential equation. Definition A one-parameter family {T (t)} t 0 of operators on X (Banach space) is called a strongly continuous semigroup of operators if the following three conditions are satisfied: 1 T (0) = I ; 2 T (t + s) = T (t)t (s) for all s, t 0; 3 ĺım s t T (s)x = T (t)x for all x X and t 0. One also refers to it as a C 0 -semigroup.

25 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations Definition Let {T (t)} t 0 be an arbitrary C 0 -semigroup on X. The operator Ax := ĺım (T (t)x x) (11) t t 0 1 exists on a dense subspace of X ; denoted by D(A). Then A, or rather (A, D(A)), is called the (infinitesimal) generator of the semigroup. The infinitesimal generator determines the semigroup uniquely. If D(A) = X, {T (t)} t 0 = {e ta } t 0 = n=0 For every x X (X Banach space) and λ C such that Ax = λx T t x = e λt x, t 0 t n n! An t 0 (12)

28 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations The unique solution of the abstract Cauchy problem { } u t = Au, (13) u(0, x) = ϕ(x) where A is a linear operator defined on X, is given by u(t, x) = e ta ϕ(x) (14) In that sense, u(t, x) is called a classical solution of the abstract Cauchy problem (13) and the semigroup {T t } t 0 = {e ta } t 0 is called the solution semigroup of (13), whose infinitesimal generator is A. But these operators can be define in wider spaces, which permits us to find mild solutions.

31 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations Let us return to the original problem and formulate it as follows: ( ) ( ) ( ) ( ) Q(t) 0 αb Q(t) F (Q(t), t) = 1 H(t) t α B 0 +, H(t) 0 ( ) ( Q(0) φ =. H(0) ϕ) (15) We will consider A as a constant parameter; α = ga v and B = v d dx on an appropriate Banach space X.

32 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations We consider the operator-valued matrix ( ) 0 αb A := 1 α B 0 with domain Dom(A) := Dom(B) Dom(B) defined on X X. (16) Theorem Suppose that B is the generator of a C 0 -group {T (t)} t R on X. Then A is the generator of a C 0 -group {T (t)} t 0 on X X given by T (t) := 1 ( I αi 2 T +(t) I 1 α I ) T (t) ( ) I αi I 1 α I t 0. (17)

33 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations Idea of the proof The only problematic part is to verify the semigroup law. 4T (t)t (s) = (T (t)p + T ( t)q)(t (s)p + T ( s)q) (18) ( ) ( ) I αi I αi where P = 1 α I I and Q = 1 α I I verify the properties P 2 = 2P, Q 2 = 2Q and PQ = QP = 0. Therefore 4T (t)t (s) = (T (t)p + T ( t)q)(t (s)p + T ( s)q) = T (t + s)p 2 + T (t s)pq + T ( t + s)qp + T ( t s)q 2 = 2T (t + s)p + 2T ( t s)q = 4T (t + s). (19)

36 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations For the water hammer equations we have Remark An explicit description of the C 0-semigroup {T (t)} t 0 on X X is ( ( φ T (t)(φ, ϕ) = T +(t) 2 + αϕ ) ( φ + T (t) 2 2 αϕ ), 2 ( φ T +(t) 2α + ϕ ) ( φ + T (t) 2 2α + ϕ )), 2 for every t 0 and initial conditions (Q(0), H(0)) = (φ, ϕ) X X. If B = v d dx as in water hammer equations, T +(t) is the translation of t units to the left at speed v and T (t) the translation of t units to the right at speed v. This operator representation of the solution clearly shows the presence of the two waves (one due to the former steady flow and another one in the opposite sense due to the increase of the pressure).

39 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations This permits to give new explicit formulas for computing Q and H. and ( φ Q(t) = T +(t) 2 + αϕ t ( φ H(t) = T +(t) + 1 2α 0 t 0 ) ( φ + T (t) 2 αϕ 2 (T +(t s) + T (t s))f (Q(s), s)ds. 2α + ϕ 2 ) ) ( φ + T (t) 2α + ϕ ) 2 (T +(t s) T (t s))f (Q(s), s)ds.

40 C 0 -semigroups Semigroups and linear differential equations New results on the solutions to the water hammer equations The operator representation of the solution permits to characterize the solutions by an integro-differential representation of them: Theorem Suppose that B is the generator of a C 0 -group {T (t)} t R on X and let F : X R + Dom(B) be given. A pair (Q, H) is a mild solution of the nonlinear general problem if, and only if, for all (φ, ϕ) Dom(B) Dom(B), Q satisfies the integro-differential equation and t Q (t) = B 2 Q(s)ds + F (Q(t), t) + αbϕ (20) 0 H(t) = 1 t α B Q(s)ds + ϕ (21) with initial conditions (Q(0), H(0)) = (φ, ϕ). 0

41 Basic definitions Dynamics of the translation C 0 -semigroup Definition Let {T (t)} t 0 be a C 0 -semigroup on X. (a) Orbit of x under {T (t)} t 0 is orb(x, (T (t))) = {T (t)x ; t 0} (22) (b) Hypercyclic if there is some x X whose orbit under {T (t)} t 0 is dense in X. (c) Topologically transitive if, for any pair U, V of nonempty open subsets of X, there exists some t 0 0 such that T (t 0 )(U) V. (d) Topologically mixing if, for any pair U, V of nonempty open subsets of X, there exists some t 0 0 such that T (t)(u) V for every t t 0.

45 Basic definitions Dynamics of the translation C 0 -semigroup Dynamics of the translation C 0 -semigroup A measurable function, ρ : J R +, with J = R + or R, is said to be an admissible weight function if the following conditions hold: 1 ρ(τ) > 0 for all τ J, and 2 there exists constants M 1 and w R such that ρ(τ) Me w t ρ(t + τ) for all τ, t J. For J = R + or R, we define the weighted spaces L p ρ(j), 1 p <, and C 0,ρ(J) { ( } 1/p L p ρ(j) := u : J K measurable : u p := u(τ) ρ(τ)dτ) p <. { C 0,ρ(J) := u : J K continuous : u := sup u(τ) ρ(τ) < τ J and } ĺım u(τ) ρ(τ) = 0. τ J

46 Basic definitions Dynamics of the translation C 0 -semigroup Dynamics of the translation C 0 -semigroup A measurable function, ρ : J R +, with J = R + or R, is said to be an admissible weight function if the following conditions hold: 1 ρ(τ) > 0 for all τ J, and 2 there exists constants M 1 and w R such that ρ(τ) Me w t ρ(t + τ) for all τ, t J. For J = R + or R, we define the weighted spaces L p ρ(j), 1 p <, and C 0,ρ(J) { ( } 1/p L p ρ(j) := u : J K measurable : u p := u(τ) ρ(τ)dτ) p <. { C 0,ρ(J) := u : J K continuous : u := sup u(τ) ρ(τ) < τ J and } ĺım u(τ) ρ(τ) = 0. τ J

47 Basic definitions Dynamics of the translation C 0 -semigroup Desch, Schappacher & Webb 97 Let X = L p ρ(r + ), with 1 p <. The translation C 0 -semigroup {T (t)} t 0 is hypercyclic if, and only if, Desch, Schappacher & Webb 97 ĺım inf ρ(t) = 0. t Let X = L p ρ(r), with 1 p <. The translation C 0 -semigroup {T (t)} t 0 is hypercyclic if, and only if, for every θ R there exists a sequence of positive real numbers {t j } j such that ĺım ρ(θ + t j) = ĺım ρ(θ t j ) = 0. j j Bermúdez, Bonilla, Conejero & Peris 05 Topologically mixing holds in each case if we replace these limits by ĺım ρ(t) = 0. j

50 Basic definitions Dynamics of the translation C 0 -semigroup Hypercyclicity Criterion. Let {T (t)} t 0 be a C 0 -semigroup on X, two dense subsets Y, Z X, an increasing sequence of real positive numbers (t k ) k tending to, and a sequence of mappings S(t k ) : Z X,k N such that (a) ĺım k T (t k )y = 0 for all y Y, (b) ĺım k S(t k )z = 0 for all z Z, and (c) ĺım k T (t k )S(t k )z = z for all z Z. Then, the C 0 -semigroup is hypercyclic. We can state a topologically mixing criterion replacing the limits by the whole limit on R +

56 Basic definitions Dynamics of the translation C 0 -semigroup Theorem Let X = L p ρ(r), with 1 p <, or X = C 0,ρ (R) with ρ an admissible function. There exists a increasing sequence of positive real numbers {t k } k N tending to satisfying ĺım ρ(t k) = ĺım ρ( t k) = 0, (23) k k if, and only if, the solution C 0 -semigroup {T (t)} t 0 to the water hammer equations is hypercyclic. We recall that the solution is given by ( ( φ T +(t) 2 + αϕ ) ( φ + T (t) 2 2 αϕ 2 ), T +(t) ( φ 2α + ϕ ) ( φ + T (t) 2 2α + ϕ )). 2

57 Basic definitions Dynamics of the translation C 0 -semigroup Theorem Let X = L p ρ(r), with 1 p <, or X = C 0,ρ (R) with ρ an admissible function. There exists a increasing sequence of positive real numbers {t k } k N tending to satisfying ĺım ρ(t k) = ĺım ρ( t k) = 0, (23) k k if, and only if, the solution C 0 -semigroup {T (t)} t 0 to the water hammer equations is hypercyclic. We recall that the solution is given by ( ( φ T +(t) 2 + αϕ ) ( φ + T (t) 2 2 αϕ 2 ), T +(t) ( φ 2α + ϕ ) ( φ + T (t) 2 2α + ϕ )). 2

58 Idea of the proof. (Sufficiency) Basic definitions Dynamics of the translation C 0 -semigroup Take Y = Z as the continuous functions with compact support on R. ( φ T+(t k) 2 + αϕ ) p M2 e w2l ρ(t k ) φ + αϕ p ρ. (24) 2 ρ(0)2 p There is k 1 N s.t. for all k k 1 we have T+(t k) ( φ + ) αϕ p εp. 2 2 ρ 4 There is k 2 N s.t. for all k k 2 we have T +(t k) ( φ + ) ϕ p εp. 2α 2 ρ 4 ρ Putting ρ( t k ) instead of ρ(t k ) we get that there exists k 3 N such that for all k k 3 we have T (t k) ( φ ) αϕ p εp, and k4 N such that for all 2 2 ρ 4 k k 4 we have T (t k) ( φ + ) ϕ p εp. 2α 2 ρ 4 Taking k 0 := máx{k 1, k 2, k 3, k 4}, we have T (t k) (φ, ϕ) ε for all k k 0, which gives the proof of (a) in the Hypercyclicity Criterion. Condition b holds taking S(t k) = T (t k) and proceeding as before. Condition (c) holds by the semigroup law.

59 Idea of the proof. (Sufficiency) Basic definitions Dynamics of the translation C 0 -semigroup Take Y = Z as the continuous functions with compact support on R. ( φ T+(t k) 2 + αϕ ) p M2 e w2l ρ(t k ) φ + αϕ p ρ. (24) 2 ρ(0)2 p There is k 1 N s.t. for all k k 1 we have T+(t k) ( φ + ) αϕ p εp. 2 2 ρ 4 There is k 2 N s.t. for all k k 2 we have T+(t k) ( φ + ) ϕ p εp. 2α 2 ρ 4 ρ Putting ρ( t k ) instead of ρ(t k ) we get that there exists k 3 N such that for all k k 3 we have T (t k) ( φ ) αϕ p εp, and k4 N such that for all 2 2 ρ 4 k k 4 we have T (t k) ( φ + ) ϕ p εp. 2α 2 ρ 4 Taking k 0 := máx{k 1, k 2, k 3, k 4}, we have T (t k) (φ, ϕ) ε for all k k 0, which gives the proof of (a) in the Hypercyclicity Criterion. Condition b holds taking S(t k) = T (t k) and proceeding as before. Condition (c) holds by the semigroup law.

65 Idea of the proof. (Necessity) Basic definitions Dynamics of the translation C 0 -semigroup Conversely, there exists t n := t n/v > L/v and (φ n, ϕ n) X X such that (φ n, ϕ n) < 1 n and T (t k )(φ n, ϕ n) (χ [0,L], 0) < 1 n. (25) where χ [0,L] stands for the characteristic function of the interval [0, L]. Let us define ( φn, ϕ n ) = ( φ [ tn, t n+l], ϕ [ tn, t n+l]). (26) On the one hand, ( φ n T+(t n) 2 + α ) ϕn p χ [0,L] p 2 1 ρ ρ n. (27) On the other hand, ( φ n T+(t n) 2 + α ) ϕn p ( Mew2L ρ(l) φ n 2 ρ ρ( t n) 2 + α ) ϕn p Mew2L ρ(l) 1 2 ρ ρ( t n) n p (28) ( ) Therefore, T +(t n) φ n p tends to 0 which leads to a contradiction. + α ϕn 2 2 ρ

66 Basic definitions Dynamics of the translation C 0 -semigroup J.A. Conejero, C. Lizama, and F. Ródenas. of the water hammer equations. To appear in Topology Appl. Thanks for your attention