Pseudo-Parabolic Partial Differential...

The Initial-Boundary-Value ProblemsOperators in L2

Pseudo-Parabolic Partial Differential Equations

R.E. Showalter

Department of MathematicsOregon State University

Applied Mathematics & Computation Seminar

RES AMC Seminar 2007

Outline

1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins

2 Operators in L2

Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP

Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins

PDE are just ODE in an appropriate function space.Here we treat simple partial differential equations as evolutionequations (ordinary differential equations) in the space L2(G).

Outline

2 Operators in L2

Parabolic equation

u = u(x , t) : Initial-Boundary-Value Problem

∂u∂t−∇·k∇u = 0, x ∈ Ω, t > 0,

u(s, t) = 0, s ∈ ∂Ω, t > 0,

u(x , 0) = u0(x), x ∈ Ω.

Outline

2 Operators in L2

Pseudo-Parabolic Equation

∂u∂t− ε∇·k∇∂u

∂t−∇·k∇u = 0, x ∈ Ω, t > 0,

u(s, t) = 0, s ∈ ∂Ω, t > 0,

u(x , 0) = u0(x), x ∈ Ω.

Outline

2 Operators in L2

Origins

1926 Milne ... time delay, gas diffusion

1948 Rubinstein ... heat conduction in composite medium

1960 Barenblatt ... fluid flow in fissured medium

1960 Coleman-Noll ... heat conduction

1968 Chen-Gurtin

1966 Lighthill ... fluid

1966 Peregrine ... long waves (semilinear)

1972 Benjamin-Bona-Mahoney

1979 Aifantis ... highly-diffusive paths

1980 Gilbert ... Slightly-compressible Stokes velocity

Elliptic Boundary-Value ProblemEvolution Equations in L2(G)

ODE and an Elliptic BVP

Outline

2 Operators in L2

Elliptic Boundary-Value Problem

The spatial derivatives are given by the operator

Au = −∇·k∇u(·) in L2(G),

D(A) = u ∈ H2(G) : u = 0 on ∂G

Eigen-functions: vj(·) : j ≥ 1 is an ortho-normal basis forL2(G)

A(vj) = λjvj , j ≥ 1 , 0 < λj → +∞

Outline

2 Operators in L2

The Parabolic Equation

u′(t) + Au(t) = 0, t > 0 ,

u(0) = u0 .

u(t) =∞∑

e−λj t(u0, vj) vj

= S(t)u0 = e−Atu0

Analytic semigroup

Regularity increasing for t > 0

Unbounded decay rate of coefficients

u′(t) + Au(t) = 0, t > 0 ,

u(0) = u0 .

u(t) =∞∑

= S(t)u0 = e−Atu0

Analytic semigroup

u′(t) + Au(t) = 0, t > 0 ,

u(0) = u0 .

u(t) =∞∑

= S(t)u0 = e−Atu0

Analytic semigroup

The Pseudo-Parabolic Equation

u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,

u(0) = u0 .

u(t) =∞∑

e−λj t

1+ελj (u 0, vj) vj

= Sε(t)u0 = e−(I+εA)−1Atu0

C0-groupRegularity preserving for −∞ < t < ∞

Decay rate bounded below by1ε

u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,

u(0) = u0 .

u(t) =∞∑

e−λj t

u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,

u(0) = u0 .

u(t) =∞∑

e−λj t

Outline

2 Operators in L2

ODE in L2(G)

Aε = (I + εA)−1A = A(I + εA)−1 =1ε(I − (I + εA)−1)

is a bounded operator on L2(G).

u′(t) + Aεu(t) = 0

is an Ordinary Differential Equation in L2(G).

... a little algebra ...

The pseudo-parabolic equation

u′(t) + εAu′(t) + Au(t) = 0, t > 0

can be written

u′(t) +1ε

u(t) =1ε(I + εA)−1u(t) ∈ D(A)

The saltus or jump along an interface, [u](t), satisfies

[u]′(t) +1ε[u](t) = 0 ,

[u](t) = e−tε [u0] .

... a little algebra ...

The pseudo-parabolic equation

u′(t) + εAu′(t) + Au(t) = 0, t > 0

can be written

u′(t) +1ε

u(t) =1ε(I + εA)−1u(t) ∈ D(A)

The saltus or jump along an interface, [u](t), satisfies

[u]′(t) +1ε[u](t) = 0 ,

[u](t) = e−tε [u0] .

Richard’s equation(with M. Peszynska, S.-Y.Yi)

φ∂S∂t

+∇ · (Kkw (S)∇Pc(S)) = ∇ · (Kkw (S)Gρw∇Depth(x))

Rewritten in a generic nonlinear parabolic form

∂S∂t−∇ · (D(S)∇S) = ∇ · (Λ(S))

D(S) is non-negative definite and degenerate

D(S) ≈ 0, S1 ≤ S ≤ S2

Λ(S) is monotone increasing degenerate.

Richard’s equation with dynamic capillary pressure(with M. Peszynska, S.-Y.Yi)

Replace Pc(S) by Pc(S, ∂S∂t ) to account for dependence on time

scale of getting to capillary equilibrium [Wildenschild et al]

φ∂S∂t

+∇ · (Kkw (S)∇Pc(S,∂S∂t

)) = ∇ · (Kkw (S)Gρw∇Depth(x))

([B]) [Barenblatt] Pc(S, ∂S∂t )) := Pc(S + τ ∂S

∂t )

([HC]) [Hassanizadeh, Celia] Pc(S, ∂S∂t )) := Pc(S)− τ ∂S

∂t(also note some hysteresis models [Belyaev, Schotting, vanDuijn]Can be rewritten in a generic nonlinear pseudo–parabolic form

∂S∂t−∇ · (D(S)∇S) = ∇ · (Λ(S)) +∇ ·

(C(S)∇∂S

)where C(S) is more or less degenerate depending on themodel RES AMC Seminar 2007

Compare solutions to linear parabolic andpseudo-parabolic equations (M. Peszynska, S.-Y.Yi)

∂S∂t−∇ · (D∇S) = ∇ · C∇(

∂S∂t

where C = τD.

use τ = 0 and τ = 1.

initial data: smooth (optimal convergence)

τ = 0 τ = 1RES AMC Seminar 2007

Compare solutions to linear parabolic andpseudo-parabolic equations (M. Peszynska, S.-Y.Yi)

∂S∂t−∇ · (D∇S) = ∇ · C∇(

∂S∂t

where C = τD.use τ = 0 and τ = 1.initial data: nonsmooth

nonsmooth initial data, τ = 0 nonsmooth initial data, τ = 1

Pseudo-Parabolic Partial Differential...

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