From Relaxation to Slow Erosion
Wen Shen
Department of Mathematics, Penn State University
SISSA, Italy, June 16, 2016
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
1 / 28
List of some joint works:
Uniqueness for discontinuous ODEs and conservation laws;
BV estimate for a relaxation model for multicomponent chromatography;
Differential games: non-cooperative and semi-cooperative differential games;
Optimality conditions for solutions to hyperbolic balance laws;
Differential games related to fish harvesting;
Slow erosion of granular flow: a semigroup approach;
Growth model using PDEs – ongoing work.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
2 / 28
Chromatography: a relaxation model

ut + ux = − 1 (F (u) − v )
δ
v = 1 (F (u) − v )
t
δ
A fluid flow with unit speed travels over a solid bed.
u, v ∈ Rn : concentration of n components of chemicals in the fluid and
the solid bed.
Equilibrium state: If v = F (u), then no exchange of chemicals will happen.
δ: relaxation time, how quickly the equilibrium configuration is reached.
Zero relaxation limit: as δ → 0, we get v → F (u), and
(u + F (u))t + ux = 0.
⇒ An n × n system of conservation laws for u.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
3 / 28
Chromatography: a relaxation model

ut + ux = − 1 (F (u) − v )
δ
v = 1 (F (u) − v )
t
δ
A fluid flow with unit speed travels over a solid bed.
u, v ∈ Rn : concentration of n components of chemicals in the fluid and
the solid bed.
Equilibrium state: If v = F (u), then no exchange of chemicals will happen.
δ: relaxation time, how quickly the equilibrium configuration is reached.
Zero relaxation limit: as δ → 0, we get v → F (u), and
(u + F (u))t + ux = 0.
⇒ An n × n system of conservation laws for u.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
3 / 28
Chromatography: a relaxation model

ut + ux = − 1 (F (u) − v )
δ
v = 1 (F (u) − v )
t
δ
A fluid flow with unit speed travels over a solid bed.
u, v ∈ Rn : concentration of n components of chemicals in the fluid and
the solid bed.
Equilibrium state: If v = F (u), then no exchange of chemicals will happen.
δ: relaxation time, how quickly the equilibrium configuration is reached.
Zero relaxation limit: as δ → 0, we get v → F (u), and
(u + F (u))t + ux = 0.
⇒ An n × n system of conservation laws for u.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
3 / 28
Chromatography: a relaxation model

ut + ux = − 1 (F (u) − v )
δ
v = 1 (F (u) − v )
t
δ
A fluid flow with unit speed travels over a solid bed.
u, v ∈ Rn : concentration of n components of chemicals in the fluid and
the solid bed.
Equilibrium state: If v = F (u), then no exchange of chemicals will happen.
δ: relaxation time, how quickly the equilibrium configuration is reached.
Zero relaxation limit: as δ → 0, we get v → F (u), and
(u + F (u))t + ux = 0.
⇒ An n × n system of conservation laws for u.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
3 / 28
Chromatography: a relaxation model

ut + ux = − 1 (F (u) − v )
δ
v = 1 (F (u) − v )
t
δ
A fluid flow with unit speed travels over a solid bed.
u, v ∈ Rn : concentration of n components of chemicals in the fluid and
the solid bed.
Equilibrium state: If v = F (u), then no exchange of chemicals will happen.
δ: relaxation time, how quickly the equilibrium configuration is reached.
Zero relaxation limit: as δ → 0, we get v → F (u), and
(u + F (u))t + ux = 0.
⇒ An n × n system of conservation laws for u.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
3 / 28
Goal of the study:
• Given δ > 0, establish existence and uniqueness of the the solution for
the relaxation system.
• Zero relaxation limit as δ → 0.
The key estimate: A compactness estimate.
A bound on the total variations of the solutions for the relaxation system,
uniform w.r.t. the relaxation parameter δ.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
4 / 28
Goal of the study:
• Given δ > 0, establish existence and uniqueness of the the solution for
the relaxation system.
• Zero relaxation limit as δ → 0.
The key estimate: A compactness estimate.
A bound on the total variations of the solutions for the relaxation system,
uniform w.r.t. the relaxation parameter δ.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
4 / 28
Goal of the study:
• Given δ > 0, establish existence and uniqueness of the the solution for
the relaxation system.
• Zero relaxation limit as δ → 0.
The key estimate: A compactness estimate.
A bound on the total variations of the solutions for the relaxation system,
uniform w.r.t. the relaxation parameter δ.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
4 / 28
Key feature: Langmuir isotherm, F = (F1 , · · · , Fn )
Fi (u) =
k i ui
1 + k1 u1 + · · · + kn un
The Jacobian matrix A(u) = DF (u) has n distinct real eigen-values, each
family is genuinely nonlinear.
Furthermore, the integral curves of each family are straight lines and
coincide with the shock curves.
⇒ a type of Temple class
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
5 / 28
Key feature: Langmuir isotherm, F = (F1 , · · · , Fn )
Fi (u) =
k i ui
1 + k1 u1 + · · · + kn un
The Jacobian matrix A(u) = DF (u) has n distinct real eigen-values, each
family is genuinely nonlinear.
Furthermore, the integral curves of each family are straight lines and
coincide with the shock curves.
⇒ a type of Temple class
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
5 / 28
Key feature: Langmuir isotherm, F = (F1 , · · · , Fn )
Fi (u) =
k i ui
1 + k1 u1 + · · · + kn un
The Jacobian matrix A(u) = DF (u) has n distinct real eigen-values, each
family is genuinely nonlinear.
Furthermore, the integral curves of each family are straight lines and
coincide with the shock curves.
⇒ a type of Temple class
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
5 / 28
li , ri : the left and right normalized eigenvectors of A(u) = DF (u).
(
P
ux = i uxi ri (u),
uxi = li (u) · ux
P i
vx = i vx ri (u),
vxi = li (u) · vx
Define the directional derivative:
φ(u) • ~v = lim
h→0
φ(u + h~v ) − φ(u)
h
Key property:
ri (u, v ) • ri (u, v ) ≡ 0,
Wen Shen (Penn State)
for all i,
From Relaxation to Slow Erosion
for all (u, v )
SISSA, Italy, June 16, 2016
6 / 28
li , ri : the left and right normalized eigenvectors of A(u) = DF (u).
(
P
ux = i uxi ri (u),
uxi = li (u) · ux
P i
vx = i vx ri (u),
vxi = li (u) · vx
Define the directional derivative:
φ(u) • ~v = lim
h→0
φ(u + h~v ) − φ(u)
h
Key property:
ri (u, v ) • ri (u, v ) ≡ 0,
Wen Shen (Penn State)
for all i,
From Relaxation to Slow Erosion
for all (u, v )
SISSA, Italy, June 16, 2016
6 / 28
li , ri : the left and right normalized eigenvectors of A(u) = DF (u).
(
P
ux = i uxi ri (u),
uxi = li (u) · ux
P i
vx = i vx ri (u),
vxi = li (u) · vx
Define the directional derivative:
φ(u) • ~v = lim
h→0
φ(u + h~v ) − φ(u)
h
Key property:
ri (u, v ) • ri (u, v ) ≡ 0,
Wen Shen (Penn State)
for all i,
From Relaxation to Slow Erosion
for all (u, v )
SISSA, Italy, June 16, 2016
6 / 28
(
uxt + uxx = −A(u)ux + vx
vxt = A(u)ux − vx
In components:

P
(uxi )t + (uxi )x = −λi (A)uxi + vxi − j Gij uxj
(v i ) = λ (A)u i − v i + P G u j + P H v j u k
i
x t
x
x
j ij x
j,k ijk x x
where
Gij = li · ((F (u) − v ) • rj ),
Hijk (u, v ) = li · (rj • rk ).
Need to show that these terms are integrable over (t, x) ∈ [0, ∞) × (−∞, ∞).
Thanks to the key feature, one can show
– Hijk = 0 for j = k
– Gij includes only terms uxi with i 6= j.
Need to show terms vxj uxk , uxj uxk with j 6= k are integrable.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
7 / 28
(
uxt + uxx = −A(u)ux + vx
vxt = A(u)ux − vx
In components:

P
(uxi )t + (uxi )x = −λi (A)uxi + vxi − j Gij uxj
(v i ) = λ (A)u i − v i + P G u j + P H v j u k
i
x t
x
x
j ij x
j,k ijk x x
where
Gij = li · ((F (u) − v ) • rj ),
Hijk (u, v ) = li · (rj • rk ).
Need to show that these terms are integrable over (t, x) ∈ [0, ∞) × (−∞, ∞).
Thanks to the key feature, one can show
– Hijk = 0 for j = k
– Gij includes only terms uxi with i 6= j.
Need to show terms vxj uxk , uxj uxk with j 6= k are integrable.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
7 / 28
(
uxt + uxx = −A(u)ux + vx
vxt = A(u)ux − vx
In components:

P
(uxi )t + (uxi )x = −λi (A)uxi + vxi − j Gij uxj
(v i ) = λ (A)u i − v i + P G u j + P H v j u k
i
x t
x
x
j ij x
j,k ijk x x
where
Gij = li · ((F (u) − v ) • rj ),
Hijk (u, v ) = li · (rj • rk ).
Need to show that these terms are integrable over (t, x) ∈ [0, ∞) × (−∞, ∞).
Thanks to the key feature, one can show
– Hijk = 0 for j = k
– Gij includes only terms uxi with i 6= j.
Need to show terms vxj uxk , uxj uxk with j 6= k are integrable.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
7 / 28
(
uxt + uxx = −A(u)ux + vx
vxt = A(u)ux − vx
In components:

P
(uxi )t + (uxi )x = −λi (A)uxi + vxi − j Gij uxj
(v i ) = λ (A)u i − v i + P G u j + P H v j u k
i
x t
x
x
j ij x
j,k ijk x x
where
Gij = li · ((F (u) − v ) • rj ),
Hijk (u, v ) = li · (rj • rk ).
Need to show that these terms are integrable over (t, x) ∈ [0, ∞) × (−∞, ∞).
Thanks to the key feature, one can show
– Hijk = 0 for j = k
– Gij includes only terms uxi with i 6= j.
Need to show terms vxj uxk , uxj uxk with j 6= k are integrable.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
7 / 28
(
uxt + uxx = −A(u)ux + vx
vxt = A(u)ux − vx
In components:

P
(uxi )t + (uxi )x = −λi (A)uxi + vxi − j Gij uxj
(v i ) = λ (A)u i − v i + P G u j + P H v j u k
i
x t
x
x
j ij x
j,k ijk x x
where
Gij = li · ((F (u) − v ) • rj ),
Hijk (u, v ) = li · (rj • rk ).
Need to show that these terms are integrable over (t, x) ∈ [0, ∞) × (−∞, ∞).
Thanks to the key feature, one can show
– Hijk = 0 for j = k
– Gij includes only terms uxi with i 6= j.
Need to show terms vxj uxk , uxj uxk with j 6= k are integrable.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
7 / 28
The building block: the 2 × 2 system

ut + ux = −(f (u) − v )
v = f (u) − v
t
f : R 7→ R: smooth and increasing
Let ξ = ux , η = vx , then

ξt + ηx = −f 0 (u)ξ + η
η = f 0 (u)ξ − η
t
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
8 / 28
The building block: the 2 × 2 system

ut + ux = −(f (u) − v )
v = f (u) − v
t
f : R 7→ R: smooth and increasing
Let ξ = ux , η = vx , then

ξt + ηx = −f 0 (u)ξ + η
η = f 0 (u)ξ − η
t
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
8 / 28
A probabilistic approach
A general stochastic process:

ξt + ηx = −a(t, x)ξ + b(t, x)η
η = a(t, x)ξ − b(t, x)η
t
Random walkers, going with speed 0 and 1.
ξ: density of walkers with speed 1
η: density of walkers with speed 0
Walkers can switch their speeds:
The speed of a walker can switch from 1 to 0 with rate a(t, x) at (t, x),
and from 0 to 1 with rate b(t, x).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
9 / 28
A probabilistic approach
A general stochastic process:

ξt + ηx = −a(t, x)ξ + b(t, x)η
η = a(t, x)ξ − b(t, x)η
t
Random walkers, going with speed 0 and 1.
ξ: density of walkers with speed 1
η: density of walkers with speed 0
Walkers can switch their speeds:
The speed of a walker can switch from 1 to 0 with rate a(t, x) at (t, x),
and from 0 to 1 with rate b(t, x).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
9 / 28
A probabilistic approach
A general stochastic process:

ξt + ηx = −a(t, x)ξ + b(t, x)η
η = a(t, x)ξ − b(t, x)η
t
Random walkers, going with speed 0 and 1.
ξ: density of walkers with speed 1
η: density of walkers with speed 0
Walkers can switch their speeds:
The speed of a walker can switch from 1 to 0 with rate a(t, x) at (t, x),
and from 0 to 1 with rate b(t, x).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
9 / 28
A probabilistic approach
A general stochastic process:

ξt + ηx = −a(t, x)ξ + b(t, x)η
η = a(t, x)ξ − b(t, x)η
t
Random walkers, going with speed 0 and 1.
ξ: density of walkers with speed 1
η: density of walkers with speed 0
Walkers can switch their speeds:
The speed of a walker can switch from 1 to 0 with rate a(t, x) at (t, x),
and from 0 to 1 with rate b(t, x).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
9 / 28
Fundamental solutions Γji (t, x; t0 , x0 ), i, j ∈ {0, 1}: the density of
probability that a particle, which initially is at x0 at t0 with speed i,
reaches x at t > t0 with speed j.
If a(t, x) = α, b(t, x) = β, constant switching rates, fundamental solution
is invariant under time and space translation:
Γij (t, x; t0 , x0 ) = Gij (t − t0 , x − x0 )
P(t): position of a walker at t. Then:
lim
t→∞
Wen Shen (Penn State)
P(t)
β
= λ=
˙
t
α+β
with probability 1.
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
10 / 28
Fundamental solutions Γji (t, x; t0 , x0 ), i, j ∈ {0, 1}: the density of
probability that a particle, which initially is at x0 at t0 with speed i,
reaches x at t > t0 with speed j.
If a(t, x) = α, b(t, x) = β, constant switching rates, fundamental solution
is invariant under time and space translation:
Γij (t, x; t0 , x0 ) = Gij (t − t0 , x − x0 )
P(t): position of a walker at t. Then:
lim
t→∞
Wen Shen (Penn State)
P(t)
β
= λ=
˙
t
α+β
with probability 1.
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
10 / 28
Fundamental solutions Γji (t, x; t0 , x0 ), i, j ∈ {0, 1}: the density of
probability that a particle, which initially is at x0 at t0 with speed i,
reaches x at t > t0 with speed j.
If a(t, x) = α, b(t, x) = β, constant switching rates, fundamental solution
is invariant under time and space translation:
Γij (t, x; t0 , x0 ) = Gij (t − t0 , x − x0 )
P(t): position of a walker at t. Then:
lim
t→∞
Wen Shen (Penn State)
P(t)
β
= λ=
˙
t
α+β
with probability 1.
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
10 / 28
Consider two types of walkers P, P ∗ , with switching rates (α, β) and
(α∗ , β ∗ ), and
β∗
β
<1
0 < λ∗ =
˙ ∗
<λ=
˙
α + β∗
α+β
P: fast walkers,
P ∗ : slow walkers.
We have
lim [P(t) − P ∗ (t)] = ∞
t→∞
Wen Shen (Penn State)
with probability 1.
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
11 / 28
Consider two types of walkers P, P ∗ , with switching rates (α, β) and
(α∗ , β ∗ ), and
β∗
β
<1
0 < λ∗ =
˙ ∗
<λ=
˙
α + β∗
α+β
P: fast walkers,
P ∗ : slow walkers.
We have
lim [P(t) − P ∗ (t)] = ∞
t→∞
Wen Shen (Penn State)
with probability 1.
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
11 / 28
Need a uniform bound on the following term:
Z ∞Z ∞
∗
E =
G01 (t, x) · G10
(t, x) dxdt
0
−∞
Meaning of E : Let
P(0) = P ∗ (0) = 0,
Ṗ(0) = 1,
Ṗ ∗ (0) = 0
Then, E =(the expected number of times where P ∗ overtakes P).
E <∞
⇒ Uniform BV bound for u, v of the relaxation model.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
12 / 28
Need a uniform bound on the following term:
Z ∞Z ∞
∗
E =
G01 (t, x) · G10
(t, x) dxdt
0
−∞
Meaning of E : Let
P(0) = P ∗ (0) = 0,
Ṗ(0) = 1,
Ṗ ∗ (0) = 0
Then, E =(the expected number of times where P ∗ overtakes P).
E <∞
⇒ Uniform BV bound for u, v of the relaxation model.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
12 / 28
Need a uniform bound on the following term:
Z ∞Z ∞
∗
E =
G01 (t, x) · G10
(t, x) dxdt
0
−∞
Meaning of E : Let
P(0) = P ∗ (0) = 0,
Ṗ(0) = 1,
Ṗ ∗ (0) = 0
Then, E =(the expected number of times where P ∗ overtakes P).
E <∞
⇒ Uniform BV bound for u, v of the relaxation model.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
12 / 28
Need a uniform bound on the following term:
Z ∞Z ∞
∗
E =
G01 (t, x) · G10
(t, x) dxdt
0
−∞
Meaning of E : Let
P(0) = P ∗ (0) = 0,
Ṗ(0) = 1,
Ṗ ∗ (0) = 0
Then, E =(the expected number of times where P ∗ overtakes P).
E <∞
⇒ Uniform BV bound for u, v of the relaxation model.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
12 / 28
Slow erosion of granular flows
A two-layer model
(Hadeler & Kuttler, 2001)
h = thickness of the moving layer
u = height of the standing pile
moving layer
ux >
erosion
u <
x
deposition
h
standing layer
u
x
The speed of the moving layer is proportional to the slope: v = −βux
The erosion rate γ(ux − α) depends on the difference between ux and
critical slope.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
13 / 28
Slow erosion of granular flows
A two-layer model
(Hadeler & Kuttler, 2001)
h = thickness of the moving layer
u = height of the standing pile
moving layer
ux >
erosion
u <
x
deposition
h
standing layer
u
x
The speed of the moving layer is proportional to the slope: v = −βux
The erosion rate γ(ux − α) depends on the difference between ux and
critical slope.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
13 / 28
A system of conservation laws
ht − (βhux )x
ut
= γ(ux − α)h
= − γ(ux − α)h
0 < p = ux : slope of the standing pile
After a rescaling of coordinates, one obtains the balance laws,
ht − (hp)x
= (p − 1)h
pt + ((p − 1)h)x = 0
D. Amadori & W.S., (Comm. PDE, 2009): global existence of large BV solutions
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
14 / 28
A system of conservation laws
ht − (βhux )x
ut
= γ(ux − α)h
= − γ(ux − α)h
0 < p = ux : slope of the standing pile
After a rescaling of coordinates, one obtains the balance laws,
ht − (hp)x
= (p − 1)h
pt + ((p − 1)h)x = 0
D. Amadori & W.S., (Comm. PDE, 2009): global existence of large BV solutions
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
14 / 28
A system of conservation laws
ht − (βhux )x
ut
= γ(ux − α)h
= − γ(ux − α)h
0 < p = ux : slope of the standing pile
After a rescaling of coordinates, one obtains the balance laws,
ht − (hp)x
= (p − 1)h
pt + ((p − 1)h)x = 0
D. Amadori & W.S., (Comm. PDE, 2009): global existence of large BV solutions
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
14 / 28
The slow erosion limit:
(D. Amadori & W.S., ARMA, 2011)
If sand is poured from the top very slowly: h ≈ 0, then the shape of the
standing profile depends only on the total amount τ of material poured
from the top.
The 2 × 2 system converges to a scalar integro-differential equation for
u(τ, x):
Z ∞
p−1
uτ − exp
f (ux (t, y )) dy
= 0,
f (p) =
p
x
x
The erosion function f (ux ) denotes the erosion rate per unit horizontal
distance travelled by the avalanche.
More general classes of erosion functions:
f (1) = 0,
Wen Shen (Penn State)
f 0 > 0,
f 00 < 0.
From Relaxation to Slow Erosion
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15 / 28
Loss of regularities
(A). If limp→+∞ f 0 (p) = 0
⇒ slope ux remains bounded
(B). If limp→+∞ p −1 f (p) = f 0 (∞) > 0
=⇒ slope ux can blow up in finite time, and shocks form
u(t,x)
u(0,x)
hyperkink
kink
jump
x
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
16 / 28
Loss of regularities
(A). If limp→+∞ f 0 (p) = 0
⇒ slope ux remains bounded
(B). If limp→+∞ p −1 f (p) = f 0 (∞) > 0
=⇒ slope ux can blow up in finite time, and shocks form
u(t,x)
u(0,x)
hyperkink
kink
jump
x
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
16 / 28
Some relevant results
Debora Amadori and W.S., Slow erosion limit in a model of granular flow.
Arch. Rational Mech. Anal. 2011.
Debora Amadori and W.S., Front tracking approximations for slow erosion.
Disc. Cont. Dyn. Systems 2012.
W.S. and Tianyou Zhang, Erosion profile by a global model for granular
flow. Arch. Rational Mech. Anal. 2012.
Rinaldo Colombo, Graziano Guerra and W.S., Lipschitz semigroup for an
integro-differential equation for slow erosion. Quarterly Appl. Math. 2012.
Graziano Guerra and W.S., Existence and stability of traveling waves for an
integro-differential equation for slow erosion. JDE 2014.
Alberto Bressan and W.S., A semigroup approach to an integro-differential
equation modeling slow erosion, JDE 2014.
W.S., Slow erosion with rough geological layer, SIAM Journal of
Math. Anal. (2015).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
17 / 28
A coordinate change
Assuming ux ≥ δ > 0, we consider u as the independent variable and
X = X (τ, u) as dependent variable.
Then we take z(τ, u) = Xu (τ, u), the inverse slope.
u
u(x)
x
z
X(u)
z = Xu = 1
ux
1
x
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u
From Relaxation to Slow Erosion
u
SISSA, Italy, June 16, 2016
18 / 28
The basic equations in the new coordinate
Z
h
zτ − g (z) exp
∞
g (z(τ, v )) dv
= 0
with the constraint z ≥ 0
u
u
Write now: G (z; u) = exp
i
R∞
u
g (z(τ, v )) dv
Along characteristics τ 7→ u(τ ):
u̇ = − g 0 (z)G (z; u),
ż(τ, u(τ )) = − g 2 (z)G (z; u) ≤ 0
– z can develop a shock in finite time =⇒ u has a kink
– z can decrease to zero in finite time =⇒ u has a jump
– z can become negative if no constraint is imposed.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
19 / 28
The basic equations in the new coordinate
Z
h
zτ − g (z) exp
∞
g (z(τ, v )) dv
= 0
with the constraint z ≥ 0
u
u
Write now: G (z; u) = exp
i
R∞
u
g (z(τ, v )) dv
Along characteristics τ 7→ u(τ ):
u̇ = − g 0 (z)G (z; u),
ż(τ, u(τ )) = − g 2 (z)G (z; u) ≤ 0
– z can develop a shock in finite time =⇒ u has a kink
– z can decrease to zero in finite time =⇒ u has a jump
– z can become negative if no constraint is imposed.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
19 / 28
The basic equations in the new coordinate
Z
h
zτ − g (z) exp
∞
g (z(τ, v )) dv
= 0
with the constraint z ≥ 0
u
u
Write now: G (z; u) = exp
i
R∞
u
g (z(τ, v )) dv
Along characteristics τ 7→ u(τ ):
u̇ = − g 0 (z)G (z; u),
ż(τ, u(τ )) = − g 2 (z)G (z; u) ≤ 0
– z can develop a shock in finite time =⇒ u has a kink
– z can decrease to zero in finite time =⇒ u has a jump
– z can become negative if no constraint is imposed.
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
19 / 28
If z < 0, solution is meaningless:
x
x(u)
u
u(x)
z = xu = 1
ux
u
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u
From Relaxation to Slow Erosion
x
SISSA, Italy, June 16, 2016
20 / 28
A semigroup approach
Backward Euler step
M. G. Crandall, The semigroup approach to first order quasilinear equations in several
space variables. Israel J. Math. 1972.
Abstract formulation
h
i
d
z(t) = Az(t) = g (z)G (z; u) + λz ,
dt
u
z(0) = z̄
Backward Euler approximations:
.
z(t + ε) ≈ z(t) + εAz(t + ε) = Eε− z(t)
Eε− z = w
iff w solves the ODE
w (u) = z(u) + ε g (w (u))G (u; w ) + ελwu ,
w (+∞) = 1
u
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
21 / 28
A semigroup approach
Backward Euler step
M. G. Crandall, The semigroup approach to first order quasilinear equations in several
space variables. Israel J. Math. 1972.
Abstract formulation
h
i
d
z(t) = Az(t) = g (z)G (z; u) + λz ,
dt
u
z(0) = z̄
Backward Euler approximations:
.
z(t + ε) ≈ z(t) + εAz(t + ε) = Eε− z(t)
Eε− z = w
iff w solves the ODE
w (u) = z(u) + ε g (w (u))G (u; w ) + ελwu ,
w (+∞) = 1
u
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
21 / 28
A semigroup approach
Backward Euler step
M. G. Crandall, The semigroup approach to first order quasilinear equations in several
space variables. Israel J. Math. 1972.
Abstract formulation
h
i
d
z(t) = Az(t) = g (z)G (z; u) + λz ,
dt
u
z(0) = z̄
Backward Euler approximations:
.
z(t + ε) ≈ z(t) + εAz(t + ε) = Eε− z(t)
Eε− z = w
iff w solves the ODE
w (u) = z(u) + ε g (w (u))G (u; w ) + ελwu ,
w (+∞) = 1
u
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
21 / 28
A semigroup approach
Backward Euler step
M. G. Crandall, The semigroup approach to first order quasilinear equations in several
space variables. Israel J. Math. 1972.
Abstract formulation
h
i
d
z(t) = Az(t) = g (z)G (z; u) + λz ,
dt
u
z(0) = z̄
Backward Euler approximations:
.
z(t + ε) ≈ z(t) + εAz(t + ε) = Eε− z(t)
Eε− z = w
iff w solves the ODE
w (u) = z(u) + ε g (w (u))G (u; w ) + ελwu ,
w (+∞) = 1
u
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
21 / 28
Estimates on backward Euler approximations
Backward Euler approximations are well defined;
Total variation remains bounded for t ∈ [0, T ];
Kruzhkov entropy inequality;
The limit
z(t) =
lim
n→∞
−
Et/n
n
z̄
is well defined and depends continuously on the initial data.
If z remains positive, then it yields an entropy solution to the granular flow
problem (when slope p = ux remains bounded).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
22 / 28
Estimates on backward Euler approximations
Backward Euler approximations are well defined;
Total variation remains bounded for t ∈ [0, T ];
Kruzhkov entropy inequality;
The limit
z(t) =
lim
n→∞
−
Et/n
n
z̄
is well defined and depends continuously on the initial data.
If z remains positive, then it yields an entropy solution to the granular flow
problem (when slope p = ux remains bounded).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
22 / 28
Estimates on backward Euler approximations
Backward Euler approximations are well defined;
Total variation remains bounded for t ∈ [0, T ];
Kruzhkov entropy inequality;
The limit
z(t) =
lim
n→∞
−
Et/n
n
z̄
is well defined and depends continuously on the initial data.
If z remains positive, then it yields an entropy solution to the granular flow
problem (when slope p = ux remains bounded).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
22 / 28
Estimates on backward Euler approximations
Backward Euler approximations are well defined;
Total variation remains bounded for t ∈ [0, T ];
Kruzhkov entropy inequality;
The limit
z(t) =
lim
n→∞
−
Et/n
n
z̄
is well defined and depends continuously on the initial data.
If z remains positive, then it yields an entropy solution to the granular flow
problem (when slope p = ux remains bounded).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
22 / 28
Estimates on backward Euler approximations
Backward Euler approximations are well defined;
Total variation remains bounded for t ∈ [0, T ];
Kruzhkov entropy inequality;
The limit
z(t) =
lim
n→∞
−
Et/n
n
z̄
is well defined and depends continuously on the initial data.
If z remains positive, then it yields an entropy solution to the granular flow
problem (when slope p = ux remains bounded).
Wen Shen (Penn State)
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
22 / 28
Adding the constraint z ≥ 0
The constraint z ≥ 0 is achieved by adding a measure-valued source µ = Θu
h
i
zt − g (z)G (u; z(t)) + λz − Θu = 0
u
satisfying
z(t, u) > 0
z(t, a) > 0,
=⇒
z(t, b) > 0
=⇒
Θ(t, u) = 0
Z b
Θ(t, u) du = 0
a
z
Supp ( µ )
Wen Shen (Penn State)
From Relaxation to Slow Erosion
u
SISSA, Italy, June 16, 2016
23 / 28
A flux splitting algorithm
h
i
zt − g (z)G (u; z(t)) + λz − Θu = 0
u
Fix a time step size ε > 0
and choose the initial data
Time iteration step:
wn = Eε− zn−1
zn = πwn
wn
zn 1
Wen Shen (Penn State)
z0 = z̄.
backward Euler step
projection on the positive cone
wn
zn
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
24 / 28
A nonlinear projection operator
f ∈ L1loc (R),
lim|x|→∞ f (x) = 1.
Choose F so that F 00 = f
Let F∗ be the lower convex envelope of F
Set πf = F∗00
F
f
f
F*
a
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b
a
From Relaxation to Slow Erosion
b
SISSA, Italy, June 16, 2016
25 / 28
A nonlinear projection operator
f ∈ L1loc (R),
lim|x|→∞ f (x) = 1.
Choose F so that F 00 = f
Let F∗ be the lower convex envelope of F
Set πf = F∗00
F
f
f
F*
a
Wen Shen (Penn State)
b
a
From Relaxation to Slow Erosion
b
SISSA, Italy, June 16, 2016
25 / 28
A nonlinear projection operator
f ∈ L1loc (R),
lim|x|→∞ f (x) = 1.
Choose F so that F 00 = f
Let F∗ be the lower convex envelope of F
Set πf = F∗00
F
f
f
F*
a
Wen Shen (Penn State)
b
a
From Relaxation to Slow Erosion
b
SISSA, Italy, June 16, 2016
25 / 28
A nonlinear projection operator
f ∈ L1loc (R),
lim|x|→∞ f (x) = 1.
Choose F so that F 00 = f
Let F∗ be the lower convex envelope of F
Set πf = F∗00
F
f
f
F*
a
Wen Shen (Penn State)
b
a
From Relaxation to Slow Erosion
b
SISSA, Italy, June 16, 2016
25 / 28
Properties of the projection operator
If F (a) = F∗ (a) and F (b) = F∗ (b), then
Z
b
Z
b
πf (x) dx =
a
b
Z
x
Z
f (x) dx
a
b
Z
x
Z
f (y ) dy dx
πf (y ) dy dx =
a
If f ≤ g , then πf ≤ πg
Monotonicity:
1
L -contractility:
BV stability:
a
a
a
kπf − πg kL1 ≤ kf − g kL1
TV{πf } ≤ TV{f }
Dissipative:
Z
Z
Z
|πf (x) − c| ψ(x) dx ≤
R
Wen Shen (Penn State)
|f (x) − c| ψ(x) dx −
R
sign(πf (x) − c)Θf (x) ψx (x) dx
R
From Relaxation to Slow Erosion
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26 / 28
Properties of the projection operator
If F (a) = F∗ (a) and F (b) = F∗ (b), then
Z
b
Z
b
πf (x) dx =
a
b
Z
x
Z
f (x) dx
a
b
Z
x
Z
f (y ) dy dx
πf (y ) dy dx =
a
If f ≤ g , then πf ≤ πg
Monotonicity:
1
L -contractility:
BV stability:
a
a
a
kπf − πg kL1 ≤ kf − g kL1
TV{πf } ≤ TV{f }
Dissipative:
Z
Z
Z
|πf (x) − c| ψ(x) dx ≤
R
Wen Shen (Penn State)
|f (x) − c| ψ(x) dx −
R
sign(πf (x) − c)Θf (x) ψx (x) dx
R
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
26 / 28
Properties of the projection operator
If F (a) = F∗ (a) and F (b) = F∗ (b), then
Z
b
Z
b
πf (x) dx =
a
b
Z
x
Z
f (x) dx
a
b
Z
x
Z
f (y ) dy dx
πf (y ) dy dx =
a
If f ≤ g , then πf ≤ πg
Monotonicity:
1
L -contractility:
BV stability:
a
a
a
kπf − πg kL1 ≤ kf − g kL1
TV{πf } ≤ TV{f }
Dissipative:
Z
Z
Z
|πf (x) − c| ψ(x) dx ≤
R
Wen Shen (Penn State)
|f (x) − c| ψ(x) dx −
R
sign(πf (x) − c)Θf (x) ψx (x) dx
R
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
26 / 28
Properties of the projection operator
If F (a) = F∗ (a) and F (b) = F∗ (b), then
Z
b
Z
b
πf (x) dx =
a
b
Z
x
Z
f (x) dx
a
b
Z
x
Z
f (y ) dy dx
πf (y ) dy dx =
a
If f ≤ g , then πf ≤ πg
Monotonicity:
1
L -contractility:
BV stability:
a
a
a
kπf − πg kL1 ≤ kf − g kL1
TV{πf } ≤ TV{f }
Dissipative:
Z
Z
Z
|πf (x) − c| ψ(x) dx ≤
R
Wen Shen (Penn State)
|f (x) − c| ψ(x) dx −
R
sign(πf (x) − c)Θf (x) ψx (x) dx
R
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
26 / 28
Properties of the projection operator
If F (a) = F∗ (a) and F (b) = F∗ (b), then
Z
b
Z
b
πf (x) dx =
a
b
Z
x
Z
f (x) dx
a
b
Z
x
Z
f (y ) dy dx
πf (y ) dy dx =
a
If f ≤ g , then πf ≤ πg
Monotonicity:
1
L -contractility:
BV stability:
a
a
a
kπf − πg kL1 ≤ kf − g kL1
TV{πf } ≤ TV{f }
Dissipative:
Z
Z
Z
|πf (x) − c| ψ(x) dx ≤
R
Wen Shen (Penn State)
|f (x) − c| ψ(x) dx −
R
sign(πf (x) − c)Θf (x) ψx (x) dx
R
From Relaxation to Slow Erosion
SISSA, Italy, June 16, 2016
26 / 28
⇒ Global existence and uniqueness of entropy weak solutions for z(τ, u),
as well as for u(τ, x).
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From Relaxation to Slow Erosion
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27 / 28
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From Relaxation to Slow Erosion
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28 / 28