Dynamics for the solutions of the water-hammer equations · 2015. 9. 29. · Conejero, Peris,...

Why do we study this phenomenon?Operator theoretical approach to the solutions

Dynamics of the solutions

Dynamics for the solutions of the water-hammerequations

J. Alberto Conejero (IUMPA-Universitat Politecnica de Valencia).

Joint work with C. Lizama (Universidad Santiado de Chile) and F. Rodenas(IUMPA-Universitat Politecnica de Valencia)

XIV Encuentros Analisis Funcional Murcia ValenciaHomenaje a Manuel Maestre en su 60 cumpleanos.

24 de septiembre de 2015

The water hammer phenomenonWave vs. water hammer equationsWater hammer equations

Research topics:

1 Linear chaos (chaos on infinite-dimensional systems)

2 Families of linear operators ( C0-semigroups )

3 Applications to PDE

The “water hammer phenomenon”

(Also called hydraulic transients).

https://www.youtube.com/watch?v=UX_4QYbmoFE

https://youtu.be/X9UbzcanuDk?t=55

https://youtu.be/f9LY0-WP9Go?t=77

We already have studied chaos for the solutions to the wave equation, sowe decided to try to study if some type of chaos appears behind thisphenomenon

The study of hydraulic transients started (seriously) with the waveequation.

The wave equation (d’Alembert)

ytt(x , t) = a2yxx(x , t) (1)

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.

We fix ρ > 0 and consider the space

Xρ ={f : R→ C ; f (x) =

∞∑n=0

n!xn, (an)n≥0 ∈ c0

}, (3)

with the norm ‖f ‖ = supn≥0|an|, where c0 is the Banach space ofcomplex sequences tending to 0.Then Xρ is a Banach space of analytic functions with a certain growthcontrol. By its definition it is isometrically isomorphic to c0.

Herzog’97

This type of spaces were already used when studying the dynamics of thesolution of the heat equation.

Conejero, Peris, Trujillo’10 & Gross-Erdmann, Peris’11

We also studied the case of the hyperbolic heat equation and the waveequation.

Xρ ={f : R→ C ; f (x) =

∞∑n=0

n!xn, (an)n≥0 ∈ c0

}, (3)

Herzog’97

Xρ ={f : R→ C ; f (x) =

∞∑n=0

n!xn, (an)n≥0 ∈ c0

}, (3)

Herzog’97

Xρ ={f : R→ C ; f (x) =

∞∑n=0

n!xn, (an)n≥0 ∈ c0

}, (3)

Herzog’97

Some basic ideas on hydraulics:

Bernoulli’s principle

The total energy at a given point in a fluid is equal to the energyassociated with the movement of the fluid, plus energy from pressurein the fluid, plus energy from the height of the fluid relative to anarbitrary datum.

Bernoulli’s principle

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

2g︸︷︷︸Kinetics(DISCHARGE)

ρg+ z︸︷︷︸

Pressure(PIEZOMETRIC HEAD)

= constant (5)

Hydraulics: Steady state vs. transient flow.They are derived from the classical mass and momentum conservationequations 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 inconduits are valid during the transient state.

Hydraulics: Steady state vs. transient flow.They are derived from the classical mass and momentum conservationequations 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 inconduits are valid during the transient state.

These are given by the next pair of coupled partial differential equations:

Qt + gAHx +f

2DAQ|Q| = 0, (Dynamic equation) (6)

gAQx + Ht = 0, (Continuity equation) (7)

Q(x , t) represents the discharge

H(x , t) represent the piezometric head at the centerline of theconduit 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 aconduit, respectively.The parameters A and D, are characteristics of the conduit systemand are time invariant, but may be functions of x .

Qt + gAHx +f

This pair of coupled nonlinear partial differential equations can berepresented as

(Q(x , t)H(x , t)

(0 gA d

gAddx 0

)(Q(x , t)H(x , t)

(F (Q(x , t), t)

(Q(x , 0)H(x , 0)

(ϕ1(x)ϕ2(x)

), x ∈ R.

where F (y , t) = − fy |y |2DA

As a consequence, relative to a fixed time coordinate, disturbances havea finite propagation speed and they travel along the characteristics of theequation

Method of characteristics

Along the lines x = vt the equations are reduced to first-order ones.

Qt +gAv

2DAQ|Q| = 0 if

dt= v . (9)

Qt −gAv

2DAQ|Q| = 0 if

dt= −v . (10)

This permit to define a numeric scheme for solving the equations subjectto the boundary conditions.

Description of the water hammer phenomenon studied since end of 19thcentury and early 1900’s. (Menabrea, Joukowsky, and Allevi amongothers).

Further information regarding water hammer equations:M. Hanif Chaudry. Applied hydraulic transients. Ed. Springer. 3rd ed.2014, XIV, 583 p

C0-semigroupsSemigroups and linear differential equationsNew results on the solutions to the water hammer equations

To develop the study of the dynamical behaviour of the water hammerphenomenon, the solutions will be represented by a C0-semigroupgenerated by certain first order differential equation.

Definition

A one-parameter family {T (t)}t≥0 of operators on X (Banach space) iscalled a strongly continuous semigroup of operators if the following threeconditions are satisfied:

1 T (0) = I ;

2 T (t + s) = T (t)T (s) for all s, t ≥ 0;

3 lıms→t T (s)x = T (t)x for all x ∈ X and t ≥ 0.

One also refers to it as a C0-semigroup.

Definition

Let {T (t)}t≥0 be an arbitrary C0-semigroup on X . The operator

Ax := lımt→0

t(T (t)x − x) (11)

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. Theinfinitesimal generator determines the semigroup uniquely.

If D(A) = X ,→ {T (t)}t≥0 = {etA}t≥0 =∞∑n=0

n!An t ≥ 0 (12)

For every x ∈ X (X Banach space) and λ ∈ C such that

Ax = λx −→ Ttx = eλtx , t ≥ 0

Definition

Ax := lımt→0

t(T (t)x − x) (11)

If D(A) = X ,→ {T (t)}t≥0 = {etA}t≥0 =∞∑n=0

n!An t ≥ 0 (12)

Definition

Ax := lımt→0

t(T (t)x − x) (11)

If D(A) = X ,→ {T (t)}t≥0 = {etA}t≥0 =∞∑n=0

n!An t ≥ 0 (12)

The unique solution of the abstract Cauchy problem{ut = Au

u(0, x) = ϕ(x)

}, (13)

where A is a linear operator defined on X , is given by

u(t, x) = etAϕ(x) (14)

In that sense, u(t, x) is called a classical solution of the abstract Cauchyproblem (13) and the semigroup {Tt}t≥0 = {etA}t≥0 is called thesolution semigroup of (13), whose infinitesimal generator is A.

But these operators can be define in wider spaces, which permits us tofind mild solutions.

u(0, x) = ϕ(x)

}, (13)

u(t, x) = etAϕ(x) (14)

u(0, x) = ϕ(x)

}, (13)

u(t, x) = etAϕ(x) (14)

Let us return to the original problem and formulate it as follows:

(Q(t)H(t)

(0 αB

1αB 0

)(Q(t)H(t)

(F (Q(t), t)

(Q(0)H(0)

We will consider A as a constant parameter; α = gAv and B = v d

dx on anappropriate Banach space X .

We consider the operator-valued matrix

(0 αB

1αB 0

with domain Dom(A) := Dom(B)× Dom(B) defined on X × X .

Theorem

Suppose that B is the generator of a C0-group {T (t)}t∈R on X . Then Ais the generator of a C0-group {T (t)}t≥0 on X × X given by

T (t) :=1

2T+(t)

(I αI

1α I I

2T−(t)

(I −αI−1α I I

)t ≥ 0. (17)

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)

where P =

(I αI

1α I I

)and Q =

(I −αI−1α I I

)verify the properties

P2 = 2P,Q2 = 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)P2 + T (t − s)PQ+ T (−t + s)QP + T (−t − s)Q2

= 2T (t + s)P + 2T (−t − s)Q= 4T (t + s).

Idea of the proof

where P =

(I αI

1α I I

)and Q =

(I −αI−1α I I

= 2T (t + s)P + 2T (−t − s)Q= 4T (t + s).

Idea of the proof

where P =

(I αI

1α I I

)and Q =

(I −αI−1α I I

= 2T (t + s)P + 2T (−t − s)Q= 4T (t + s).

For the water hammer equations we have

Remark

An explicit description of the C0-semigroup {T (t)}t≥0 on X × X is

T (t)(φ, ϕ) =

(T+(t)

2+αϕ

)+ T−(t)

2− αϕ

2α+ϕ

)+ T−(t)

(−φ2α

for every t ≥ 0 and initial conditions (Q(0),H(0)) = (φ, ϕ) ∈ X × X .

If B = v ddx 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 theright at speed v .

This operator representation of the solution clearly shows the presence ofthe two waves (one due to the former steady flow and another one in theopposite sense due to the increase of the pressure).

Remark

T (t)(φ, ϕ) =

(T+(t)

2+αϕ

)+ T−(t)

2− αϕ

2α+ϕ

)+ T−(t)

(−φ2α

Remark

T (t)(φ, ϕ) =

(T+(t)

2+αϕ

)+ T−(t)

2− αϕ

2α+ϕ

)+ T−(t)

(−φ2α

This permits to give new explicit formulas for computing Q and H.

Q(t) = T+(t)

2+αϕ

)+ T−(t)

2− αϕ

(T+(t − s) + T−(t − s))F (Q(s), s)ds.

H(t) = T+(t)

2α+ϕ

)+ T−(t)

(−φ2α

(T+(t − s)− T−(t − s))F (Q(s), s)ds.

The operator representation of the solution permits to characterize thesolutions by an integro-differential representation of them:

Theorem

Suppose that B is the generator of a C0-group {T (t)}t∈R on X and letF : X × R+ → Dom(B) be given. A pair (Q,H) is a mild solution of thenonlinear general problem if, and only if, for all(φ, ϕ) ∈ Dom(B)× Dom(B), Q satisfies the integro-differential equation

Q ′(t) = B2

Q(s)ds + F (Q(t), t) + αBϕ (20)

H(t) =1

Q(s)ds + ϕ (21)

with initial conditions (Q(0),H(0)) = (φ, ϕ).

Basic definitionsDynamics of the translation C0-semigroup

Definition

Let {T (t)}t≥0 be a C0-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 isdense in X .

(c) Topologically transitive if, for any pair U,V of nonempty opensubsets of X , there exists some t0 ≥ 0 such that T (t0)(U) ∩ V 6= ∅.

(d) Topologically mixing if, for any pair U,V of nonempty open subsetsof X , there exists some t0 ≥ 0 such that T (t)(U) ∩ V 6= ∅ for everyt ≥ t0.

Definition

orb(x , (T (t))) = {T (t)x ; t ≥ 0} (22)

Definition

orb(x , (T (t))) = {T (t)x ; t ≥ 0} (22)

Definition

orb(x , (T (t))) = {T (t)x ; t ≥ 0} (22)

Dynamics of the translation C0-semigroup

A measurable function, ρ : J → R+, with J = R+ or R, is said to be anadmissible 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 thatρ(τ) ≤ Mew |t|ρ(t + τ) for all τ, t ∈ J.

For J = R+ or R, we define the weighted spaces Lpρ(J), 1 ≤ p <∞, and

C0,ρ(J)

Lpρ(J) :=

{u : J → K measurable : ||u||p :=

|u(τ)|pρ(τ)dτ

C0,ρ(J) :=

{u : J → K continuous : ||u||∞ := sup

τ∈J|u(τ)|ρ(τ) <∞

and lımτ→∞

|u(τ)|ρ(τ) = 0}.

Dynamics of the translation C0-semigroup

A measurable function, ρ : J → R+, with J = R+ or R, is said to be anadmissible 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 thatρ(τ) ≤ Mew |t|ρ(t + τ) for all τ, t ∈ J.

For J = R+ or R, we define the weighted spaces Lpρ(J), 1 ≤ p <∞, and

C0,ρ(J)

Lpρ(J) :=

{u : J → K measurable : ||u||p :=

|u(τ)|pρ(τ)dτ

C0,ρ(J) :=

{u : J → K continuous : ||u||∞ := sup

τ∈J|u(τ)|ρ(τ) <∞

and lımτ→∞

|u(τ)|ρ(τ) = 0}.

Desch, Schappacher & Webb’ 97

Let X = Lpρ(R+), with 1 ≤ p <∞.The translation C0-semigroup {T (t)}t≥0 is hypercyclic if, and only if,

lım inft→∞

ρ(t) = 0.

Let X = Lpρ(R), with 1 ≤ p <∞. The translation C0-semigroup{T (t)}t≥0 is hypercyclic if, and only if, for every θ ∈ R there exists asequence of positive real numbers {tj}j such that

lımj→∞

ρ(θ + tj) = lımj→∞

ρ(θ − tj) = 0.

Bermudez, Bonilla, Conejero & Peris ’05

Topologically mixing holds in each case if we replace these limits by

lımj→∞

ρ(t) = 0.

lım inft→∞

ρ(t) = 0.

lımj→∞

ρ(θ − tj) = 0.

lımj→∞

ρ(t) = 0.

lım inft→∞

ρ(t) = 0.

lımj→∞

ρ(θ − tj) = 0.

lımj→∞

ρ(t) = 0.

Hypercyclicity Criterion.

Let {T (t)}t≥0 be a C0-semigroup on X ,two dense subsets Y ,Z ⊆ X ,an increasing sequence of real positive numbers (tk)k tending to ∞,and a sequence of mappings S(tk) : Z → X ,k ∈ N such that

(a) lımk→∞ T (tk)y = 0 for all y ∈ Y ,

(b) lımk→∞ S(tk)z = 0 for all z ∈ Z , and

(c) lımk→∞ T (tk)S(tk)z = z for all z ∈ Z .

Then, the C0-semigroup is hypercyclic.

We can state a topologically mixing criterion replacing the limits by thewhole limit on R+

Theorem

Let X = Lpρ(R), with 1 ≤ p <∞, or X = C0,ρ(R) with ρ an admissiblefunction.There exists a increasing sequence of positive real numbers {tk}k∈Ntending to ∞ satisfying

lımk→∞

ρ(tk) = lımk→∞

ρ(−tk) = 0, (23)

if, and only if, the solution C0-semigroup {T (t)}t≥0 to the waterhammer equations is hypercyclic.

We recall that the solution is given by(T+(t)

2+αϕ

)+ T−(t)

2− αϕ

),T+(t)

2α+ϕ

)+ T−(t)

(−φ2α

Theorem

Let X = Lpρ(R), with 1 ≤ p <∞, or X = C0,ρ(R) with ρ an admissiblefunction.There exists a increasing sequence of positive real numbers {tk}k∈Ntending to ∞ satisfying

lımk→∞

ρ(tk) = lımk→∞

ρ(−tk) = 0, (23)

if, and only if, the solution C0-semigroup {T (t)}t≥0 to the waterhammer equations is hypercyclic.

We recall that the solution is given by(T+(t)

2+αϕ

)+ T−(t)

2− αϕ

),T+(t)

2α+ϕ

)+ T−(t)

(−φ2α

Idea of the proof. (Sufficiency)

Take Y = Z as the continuous functions with compact support on R.∣∣∣∣∣∣∣∣T+(t′k)

2+αϕ

)∣∣∣∣∣∣∣∣pρ

≤ M2ew2Lρ(tk)

ρ(0)2p||φ+ αϕ||pρ. (24)

There is k1 ∈ N s.t. for all k ≥ k1 we have∣∣∣∣T+(t′k)

+ αϕ2

)∣∣∣∣pρ≤ εp

2α+ ϕ

)∣∣∣∣pρ≤ εp

Putting ρ(−tk) instead of ρ(tk) we get that there exists k3 ∈ N such that forall k ≥ k3 we have

∣∣∣∣T−(t′k)(φ2− αϕ

)∣∣∣∣pρ≤ εp

4, and k4 ∈ N such that for all

k ≥ k4 we have∣∣∣∣T−(t′k)

(−φ2α

)∣∣∣∣pρ≤ εp

Taking k0 := max{k1, k2, k3, k4}, we have ||T (t′k) (φ, ϕ)|| ≤ ε for all k ≥ k0,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.

2+αϕ

)∣∣∣∣∣∣∣∣pρ

≤ M2ew2Lρ(tk)

ρ(0)2p||φ+ αϕ||pρ. (24)

+ αϕ2

)∣∣∣∣pρ≤ εp

2α+ ϕ

)∣∣∣∣pρ≤ εp

∣∣∣∣T−(t′k)(φ2− αϕ

)∣∣∣∣pρ≤ εp

(−φ2α

)∣∣∣∣pρ≤ εp

2+αϕ

)∣∣∣∣∣∣∣∣pρ

≤ M2ew2Lρ(tk)

ρ(0)2p||φ+ αϕ||pρ. (24)

+ αϕ2

)∣∣∣∣pρ≤ εp

2α+ ϕ

)∣∣∣∣pρ≤ εp

∣∣∣∣T−(t′k)(φ2− αϕ

)∣∣∣∣pρ≤ εp

(−φ2α

)∣∣∣∣pρ≤ εp

2+αϕ

)∣∣∣∣∣∣∣∣pρ

≤ M2ew2Lρ(tk)

ρ(0)2p||φ+ αϕ||pρ. (24)

+ αϕ2

)∣∣∣∣pρ≤ εp

2α+ ϕ

)∣∣∣∣pρ≤ εp

∣∣∣∣T−(t′k)(φ2− αϕ

)∣∣∣∣pρ≤ εp

(−φ2α

)∣∣∣∣pρ≤ εp

2+αϕ

)∣∣∣∣∣∣∣∣pρ

≤ M2ew2Lρ(tk)

ρ(0)2p||φ+ αϕ||pρ. (24)

+ αϕ2

)∣∣∣∣pρ≤ εp

2α+ ϕ

)∣∣∣∣pρ≤ εp

∣∣∣∣T−(t′k)(φ2− αϕ

)∣∣∣∣pρ≤ εp

(−φ2α

)∣∣∣∣pρ≤ εp

2+αϕ

)∣∣∣∣∣∣∣∣pρ

≤ M2ew2Lρ(tk)

ρ(0)2p||φ+ αϕ||pρ. (24)

+ αϕ2

)∣∣∣∣pρ≤ εp

2α+ ϕ

)∣∣∣∣pρ≤ εp

∣∣∣∣T−(t′k)(φ2− αϕ

)∣∣∣∣pρ≤ εp

(−φ2α

)∣∣∣∣pρ≤ εp

2+αϕ

)∣∣∣∣∣∣∣∣pρ

≤ M2ew2Lρ(tk)

ρ(0)2p||φ+ αϕ||pρ. (24)

+ αϕ2

)∣∣∣∣pρ≤ εp

2α+ ϕ

)∣∣∣∣pρ≤ εp

∣∣∣∣T−(t′k)(φ2− αϕ

)∣∣∣∣pρ≤ εp

(−φ2α

)∣∣∣∣pρ≤ εp

Idea of the proof. (Necessity)

Conversely, there exists t′n := tn/v > L/v and (φn, ϕn) ∈ X × X such that

||(φn, ϕn)|| < 1

∣∣∣∣T (t′k)(φn, ϕn)− (χ[0,L], 0)∣∣∣∣ < 1

n. (25)

where χ[0,L] stands for the characteristic function of the interval [0, L]. Let usdefine (

φn, ϕn

)=(φ|[−tn,−tn+L], ϕ|[−tn,−tn+L]

). (26)

On the one hand, ∣∣∣∣∣∣∣∣T+(t′n)

2+αϕn

)∣∣∣∣∣∣∣∣pρ

≥∣∣∣∣χ[0,L]

∣∣∣∣pρ− 1

n. (27)

On the other hand,

∣∣∣∣∣∣∣∣T+(t′n)

2+αϕn

)∣∣∣∣∣∣∣∣pρ

≤ Mew2Lρ(L)

ρ(−tn)

∣∣∣∣∣∣∣∣( φn

2+αϕn

)∣∣∣∣∣∣∣∣pρ

≤ Mew2Lρ(L)

ρ(−tn)

Therefore,∣∣∣∣∣∣T+(t′n)

+ αϕn2

)∣∣∣∣∣∣pρ

tends to 0 which leads to a contradiction.

J.A. Conejero, C. Lizama, and F. Rodenas. Dynamics of the solutions ofthe water hammer equations. To appear in Topology Appl.

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Dynamics for the solutions of the water-hammer equations · 2015. 9. 29. · Conejero, Peris,...

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