From Relaxation to Slow Erosion Wen Shen Department of Mathematics, - - PowerPoint PPT Presentation

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

δ(F(u) − v)

vt = 1

δ(F(u) − v)

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

δ(F(u) − v)

vt = 1

δ(F(u) − v)

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

δ(F(u) − v)

vt = 1

δ(F(u) − v)

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

δ(F(u) − v)

vt = 1

δ(F(u) − v)

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

δ(F(u) − v)

vt = 1

δ(F(u) − v)

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) = kiui 1 + k1u1 + · · · + knun 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) = kiui 1 + k1u1 + · · · + knun 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) = kiui 1 + k1u1 + · · · + knun 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).

• ux =

i ui xri(u),

ui

x = li(u) · ux

vx =

i v i xri(u),

v i

x = 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, for all i, for all (u, v)

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 6 / 28

li, ri: the left and right normalized eigenvectors of A(u) = DF(u).

• ux =

i ui xri(u),

ui

x = li(u) · ux

vx =

i v i xri(u),

v i

x = 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, for all i, for all (u, v)

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 6 / 28

li, ri: the left and right normalized eigenvectors of A(u) = DF(u).

• ux =

i ui xri(u),

ui

x = li(u) · ux

vx =

i v i xri(u),

v i

x = 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, for all i, for all (u, v)

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 6 / 28

• uxt + uxx = −A(u)ux + vx

vxt = A(u)ux − vx In components:    (ui

x)t + (ui x)x = −λi(A)ui x + v i x − j Gijuj x

(v i

x)t = λi(A)ui x − v i x + j Gijuj x + j,k Hijkv j xuk 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 ui

x with i = j.

Need to show terms v j

xuk x , uj xuk x with j = 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:    (ui

x)t + (ui x)x = −λi(A)ui x + v i x − j Gijuj x

(v i

x)t = λi(A)ui x − v i x + j Gijuj x + j,k Hijkv j xuk 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 ui

x with i = j.

Need to show terms v j

xuk x , uj xuk x with j = 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:    (ui

x)t + (ui x)x = −λi(A)ui x + v i x − j Gijuj x

(v i

x)t = λi(A)ui x − v i x + j Gijuj x + j,k Hijkv j xuk 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 ui

x with i = j.

Need to show terms v j

xuk x , uj xuk x with j = 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:    (ui

x)t + (ui x)x = −λi(A)ui x + v i x − j Gijuj x

(v i

x)t = λi(A)ui x − v i x + j Gijuj x + j,k Hijkv j xuk 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 ui

x with i = j.

Need to show terms v j

xuk x , uj xuk x with j = 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:    (ui

x)t + (ui x)x = −λi(A)ui x + v i x − j Gijuj x

(v i

x)t = λi(A)ui x − v i x + j Gijuj x + j,k Hijkv j xuk 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 ui

x with i = j.

Need to show terms v j

xuk x , uj xuk x with j = 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) vt = f (u) − v f : R → R: smooth and increasing Let ξ = ux, η = vx, then    ξt + ηx = −f ′(u)ξ + η ηt = f ′(u)ξ − η

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) vt = f (u) − v f : R → R: smooth and increasing Let ξ = ux, η = vx, then    ξt + ηx = −f ′(u)ξ + η ηt = f ′(u)ξ − η

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 ηt = a(t, x)ξ − b(t, x)η 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 ηt = a(t, x)ξ − b(t, x)η 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 ηt = a(t, x)ξ − b(t, x)η 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 ηt = a(t, x)ξ − b(t, x)η 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→∞

P(t) t = λ ˙ = β α + β with probability 1.

Wen Shen (Penn State) 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→∞

P(t) t = λ ˙ = β α + β with probability 1.

Wen Shen (Penn State) 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→∞

P(t) t = λ ˙ = β α + β with probability 1.

Wen Shen (Penn State) 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 0 < λ∗ ˙ = β∗ α∗ + β∗ < λ ˙ = β α + β < 1 P: fast walkers, P∗: slow walkers. We have lim

t→∞[P(t) − P∗(t)] = ∞

with probability 1.

Wen Shen (Penn State) 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 0 < λ∗ ˙ = β∗ α∗ + β∗ < λ ˙ = β α + β < 1 P: fast walkers, P∗: slow walkers. We have lim

t→∞[P(t) − P∗(t)] = ∞

with probability 1.

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 11 / 28

Need a uniform bound on the following term: E = ∞ ∞

−∞

G01(t, x) · G ∗

10(t, x) dxdt

Meaning of E: Let P(0) = P∗(0) = 0, ˙ P(0) = 1, ˙ P∗(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: E = ∞ ∞

−∞

G01(t, x) · G ∗

10(t, x) dxdt

Meaning of E: Let P(0) = P∗(0) = 0, ˙ P(0) = 1, ˙ P∗(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: E = ∞ ∞

−∞

G01(t, x) · G ∗

10(t, x) dxdt

Meaning of E: Let P(0) = P∗(0) = 0, ˙ P(0) = 1, ˙ P∗(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: E = ∞ ∞

−∞

G01(t, x) · G ∗

10(t, x) dxdt

Meaning of E: Let P(0) = P∗(0) = 0, ˙ P(0) = 1, ˙ P∗(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

x h u

standing layer

u x >

erosion

u x<

deposition moving layer

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

x h u

standing layer

u x >

erosion

u x<

deposition moving layer

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

= γ(ux − α)h ut = − 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

= γ(ux − α)h ut = − 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

= γ(ux − α)h ut = − 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): uτ −

• exp

∞

x

f (ux(t, y)) dy

• x

= 0, f (p) = p − 1 p 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, f ′ > 0, f ′′ < 0.

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 15 / 28

Loss of regularities

(A). If limp→+∞ f ′(p) = 0

⇒ slope ux remains bounded

(B). If limp→+∞ p−1f (p) = f ′(∞) > 0

= ⇒ slope ux can blow up in finite time, and shocks form

jump

u(0,x) u(t,x) x

hyperkink kink

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 16 / 28

Loss of regularities

(A). If limp→+∞ f ′(p) = 0

⇒ slope ux remains bounded

(B). If limp→+∞ p−1f (p) = f ′(∞) > 0

= ⇒ slope ux can blow up in finite time, and shocks form

jump

u(0,x) u(t,x) x

hyperkink kink

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.

x

u u(x) x u x X(u) z u z = X =

1

u

1

u

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 18 / 28

The basic equations in the new coordinate

zτ −

• g(z) exp

∞

u

g(z(τ, v)) dv

• u = 0

with the constraint z ≥ 0 Write now: G(z; u) = exp ∞

u

g(z(τ, v)) dv Along characteristics τ → u(τ): ˙ u = − g ′(z)G(z; u), ˙ z(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

zτ −

• g(z) exp

∞

u

g(z(τ, v)) dv

• u = 0

with the constraint z ≥ 0 Write now: G(z; u) = exp ∞

u

g(z(τ, v)) dv Along characteristics τ → u(τ): ˙ u = − g ′(z)G(z; u), ˙ z(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

zτ −

• g(z) exp

∞

u

g(z(τ, v)) dv

• u = 0

with the constraint z ≥ 0 Write now: G(z; u) = exp ∞

u

g(z(τ, v)) dv Along characteristics τ → u(τ): ˙ u = − g ′(z)G(z; u), ˙ z(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:

u x u x(u) u(x) u z = x =

u

ux 1 x Wen Shen (Penn State) From Relaxation to Slow Erosion 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 d dt z(t) = Az(t) =

• g(z)G(z; u) + λz
• u ,

z(0) = ¯ z Backward Euler approximations: z(t + ε) ≈ z(t) + εAz(t + ε) . = E −

ε z(t)

E −

ε z = w

iff w solves the ODE w(u) = z(u) + ε

• g(w(u))G(u; w)
• u + ελwu,

w(+∞) = 1

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 d dt z(t) = Az(t) =

• g(z)G(z; u) + λz
• u ,

z(0) = ¯ z Backward Euler approximations: z(t + ε) ≈ z(t) + εAz(t + ε) . = E −

ε z(t)

E −

ε z = w

iff w solves the ODE w(u) = z(u) + ε

• g(w(u))G(u; w)
• u + ελwu,

w(+∞) = 1

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 d dt z(t) = Az(t) =

• g(z)G(z; u) + λz
• u ,

z(0) = ¯ z Backward Euler approximations: z(t + ε) ≈ z(t) + εAz(t + ε) . = E −

ε z(t)

E −

ε z = w

iff w solves the ODE w(u) = z(u) + ε

• g(w(u))G(u; w)
• u + ελwu,

w(+∞) = 1

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 d dt z(t) = Az(t) =

• g(z)G(z; u) + λz
• u ,

z(0) = ¯ z Backward Euler approximations: z(t + ε) ≈ z(t) + εAz(t + ε) . = E −

ε z(t)

E −

ε z = w

iff w solves the ODE w(u) = z(u) + ε

• g(w(u))G(u; w)
• u + ελwu,

w(+∞) = 1

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→∞

• E −

t/n

n ¯ z 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→∞

• E −

t/n

n ¯ z 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→∞

• E −

t/n

n ¯ z 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→∞

• E −

t/n

n ¯ z 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→∞

• E −

t/n

n ¯ z 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 zt −

• g(z)G(u; z(t)) + λz
• u − Θu = 0

satisfying z(t, u) > 0 = ⇒ Θ(t, u) = 0 z(t, a) > 0, z(t, b) > 0 = ⇒ b

a

Θ(t, u) du = 0

z u

Supp ( )

µ

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 23 / 28

A flux splitting algorithm

zt −

• g(z)G(u; z(t)) + λz
• u − Θu = 0

Fix a time step size ε > 0 and choose the initial data z0 = ¯ z. Time iteration step: wn = E −

ε zn−1

backward Euler step zn = πwn projection on the positive cone

w wn zn1

n

zn

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 24 / 28

A nonlinear projection operator

f ∈ L1

loc(R),

lim|x|→∞ f (x) = 1. Choose F so that F ′′ = f Let F∗ be the lower convex envelope of F Set πf = F ′′

∗

b b a a F

f

f

F *

• Wen Shen (Penn State)

From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 25 / 28

A nonlinear projection operator

f ∈ L1

loc(R),

lim|x|→∞ f (x) = 1. Choose F so that F ′′ = f Let F∗ be the lower convex envelope of F Set πf = F ′′

∗

b b a a F

f

f

F *

• Wen Shen (Penn State)

From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 25 / 28

A nonlinear projection operator

f ∈ L1

loc(R),

lim|x|→∞ f (x) = 1. Choose F so that F ′′ = f Let F∗ be the lower convex envelope of F Set πf = F ′′

∗

b b a a F

f

f

F *

• Wen Shen (Penn State)

From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 25 / 28

A nonlinear projection operator

f ∈ L1

loc(R),

lim|x|→∞ f (x) = 1. Choose F so that F ′′ = f Let F∗ be the lower convex envelope of F Set πf = F ′′

∗

b b a a F

f

f

F *

• Wen Shen (Penn State)

From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 25 / 28

Properties of the projection operator

If F(a) = F∗(a) and F(b) = F∗(b), then

b

a

πf (x) dx = b

a

f (x) dx b

a

x

a

πf (y) dy dx = b

a

x

a

f (y) dy dx

Monotonicity: If f ≤ g, then πf ≤ πg L1-contractility: πf − πgL1 ≤ f − gL1 BV stability: TV{πf } ≤ TV{f } Dissipative:

• R

|πf (x) − c| ψ(x) dx ≤

• R

|f (x) − c| ψ(x) dx −

• R

sign(πf (x) − c)Θf (x) ψx(x) dx

Wen Shen (Penn State) 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

b

a

πf (x) dx = b

a

f (x) dx b

a

x

a

πf (y) dy dx = b

a

x

a

f (y) dy dx

Monotonicity: If f ≤ g, then πf ≤ πg L1-contractility: πf − πgL1 ≤ f − gL1 BV stability: TV{πf } ≤ TV{f } Dissipative:

• R

|πf (x) − c| ψ(x) dx ≤

• R

|f (x) − c| ψ(x) dx −

• R

sign(πf (x) − c)Θf (x) ψx(x) dx

Wen Shen (Penn State) 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

b

a

πf (x) dx = b

a

f (x) dx b

a

x

a

πf (y) dy dx = b

a

x

a

f (y) dy dx

Monotonicity: If f ≤ g, then πf ≤ πg L1-contractility: πf − πgL1 ≤ f − gL1 BV stability: TV{πf } ≤ TV{f } Dissipative:

• R

|πf (x) − c| ψ(x) dx ≤

• R

|f (x) − c| ψ(x) dx −

• R

sign(πf (x) − c)Θf (x) ψx(x) dx

Wen Shen (Penn State) 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

b

a

πf (x) dx = b

a

f (x) dx b

a

x

a

πf (y) dy dx = b

a

x

a

f (y) dy dx

Monotonicity: If f ≤ g, then πf ≤ πg L1-contractility: πf − πgL1 ≤ f − gL1 BV stability: TV{πf } ≤ TV{f } Dissipative:

• R

|πf (x) − c| ψ(x) dx ≤

• R

|f (x) − c| ψ(x) dx −

• R

sign(πf (x) − c)Θf (x) ψx(x) dx

Wen Shen (Penn State) 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

b

a

πf (x) dx = b

a

f (x) dx b

a

x

a

πf (y) dy dx = b

a

x

a

f (y) dy dx

Monotonicity: If f ≤ g, then πf ≤ πg L1-contractility: πf − πgL1 ≤ f − gL1 BV stability: TV{πf } ≤ TV{f } Dissipative:

• R

|πf (x) − c| ψ(x) dx ≤

• R

|f (x) − c| ψ(x) dx −

• R

sign(πf (x) − c)Θf (x) ψx(x) dx

Wen Shen (Penn State) 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).

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 27 / 28

Wen Shen (Penn State) From Relaxation to Slow Erosion SISSA, Italy, June 16, 2016 28 / 28