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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
The Initial-Boundary-Value ProblemsOperators in L2
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
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).
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
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.
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
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.
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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 → +∞
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Parabolic Equation
u′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t(u0, vj) vj
= S(t)u0 = e−Atu0
Analytic semigroup
Regularity increasing for t > 0
Unbounded decay rate of coefficients
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Parabolic Equation
u′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t(u0, vj) vj
= S(t)u0 = e−Atu0
Analytic semigroup
Regularity increasing for t > 0
Unbounded decay rate of coefficients
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Parabolic Equation
u′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t(u0, vj) vj
= S(t)u0 = e−Atu0
Analytic semigroup
Regularity increasing for t > 0
Unbounded decay rate of coefficients
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Pseudo-Parabolic Equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Pseudo-Parabolic Equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Pseudo-Parabolic Equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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).
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
... 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 ,
so
[u](t) = e−tε [u0] .
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
... 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 ,
so
[u](t) = e−tε [u0] .
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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.
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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
∂t
)where C(S) is more or less degenerate depending on themodel RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
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
RES AMC Seminar 2007
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
RES AMC Seminar 2007