From Relaxation to Slow Erosion

From Relaxation to Slow Erosion

Wen Shen

Department of Mathematics, Penn State University

SISSA, Italy, June 16, 2016

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.

Chromatography: a relaxation model







u

+ u

= −

(F (u) − v ) v

=

(F (u) − v )

A fluid flow with unit speed travels over a solid bed.

u, v ∈ R

: 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))

+ u

= 0.

Chromatography: a relaxation model







u

+ u

= −

(F (u) − v ) v

=

(F (u) − v )

A fluid flow with unit speed travels over a solid bed.

u, v ∈ R

: 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))

+ u

= 0.

Chromatography: a relaxation model







u

+ u

= −

(F (u) − v ) v

=

(F (u) − v )

A fluid flow with unit speed travels over a solid bed.

u, v ∈ R

: 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))

+ u

= 0.

Chromatography: a relaxation model







u

+ u

= −

(F (u) − v ) v

=

(F (u) − v )

A fluid flow with unit speed travels over a solid bed.

u, v ∈ R

: 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))

+ u

= 0.

Chromatography: a relaxation model







u

+ u

= −

(F (u) − v ) v

=

(F (u) − v )

A fluid flow with unit speed travels over a solid bed.

u, v ∈ R

: 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))

+ u

= 0.

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.

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.

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.

Key feature:

Langmuir isotherm, F = (F

, · · · , F

) F

(u) = k

u

1 + k

u

+ · · · + k

u

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

Key feature:

Langmuir isotherm, F = (F

, · · · , F

) F

(u) = k

u

1 + k

u

+ · · · + k

u

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

Key feature:

Langmuir isotherm, F = (F

, · · · , F

) F

(u) = k

u

1 + k

u

+ · · · + k

u

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

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

(ux =P

iu_xⁱri(u), u_xⁱ = li(u) · ux

v_x =P

iv_xⁱr_i(u), v_xⁱ = l_i(u) · v_x 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 )

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

(ux =P

iu_xⁱri(u), u_xⁱ = li(u) · ux

v_x =P

iv_xⁱr_i(u), v_xⁱ = l_i(u) · v_x 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 )

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

(ux =P

iu_xⁱri(u), u_xⁱ = li(u) · ux

v_x =P

iv_xⁱr_i(u), v_xⁱ = l_i(u) · v_x 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 )

(uxt+ uxx = −A(u)ux+ vx

vxt= A(u)ux− vx

In components:







(uⁱ_x)_t+ (uⁱ_x)_x = −λ_i(A)u_xⁱ + v_xⁱ−P

jG_iju_x^j (v_xⁱ)_t= λ_i(A)u_xⁱ − v_xⁱ +P

jG_iju_x^j +P

j ,kH_ijkv_x^ju_x^k

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 – H_ijk = 0 for j = k

– G_ij includes only terms u_xⁱ with i 6= j .

Need to show terms v_x^ju^k_x, u^j_xu_x^k with j 6= k are integrable.

(uxt+ uxx = −A(u)ux+ vx

vxt= A(u)ux− vx

In components:







(uⁱ_x)_t+ (uⁱ_x)_x = −λ_i(A)u_xⁱ + v_xⁱ−P

jG_iju_x^j (v_xⁱ)_t= λ_i(A)u_xⁱ − v_xⁱ +P

jG_iju_x^j +P

j ,kH_ijkv_x^ju_x^k

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 – H_ijk = 0 for j = k

– G_ij includes only terms u_xⁱ with i 6= j .

Need to show terms v_x^ju^k_x, u^j_xu_x^k with j 6= k are integrable.

(uxt+ uxx = −A(u)ux+ vx

vxt= A(u)ux− vx

In components:







(uⁱ_x)_t+ (uⁱ_x)_x = −λ_i(A)u_xⁱ + v_xⁱ−P

jG_iju_x^j (v_xⁱ)_t= λ_i(A)u_xⁱ − v_xⁱ +P

jG_iju_x^j +P

j ,kH_ijkv_x^ju_x^k

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 – H_ijk = 0 for j = k

– G_ij includes only terms u_xⁱ with i 6= j .

Need to show terms v_x^ju^k_x, u^j_xu_x^k with j 6= k are integrable.

(uxt+ uxx = −A(u)ux+ vx

vxt= A(u)ux− vx

In components:







(uⁱ_x)_t+ (uⁱ_x)_x = −λ_i(A)u_xⁱ + v_xⁱ−P

jG_iju_x^j (v_xⁱ)_t= λ_i(A)u_xⁱ − v_xⁱ +P

jG_iju_x^j +P

j ,kH_ijkv_x^ju_x^k

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 – H_ijk = 0 for j = k

– G_ij includes only terms u_xⁱ with i 6= j .

Need to show terms v_x^ju^k_x, u^j_xu_x^k with j 6= k are integrable.

(uxt+ uxx = −A(u)ux+ vx

vxt= A(u)ux− vx

In components:







(uⁱ_x)_t+ (uⁱ_x)_x = −λ_i(A)u_xⁱ + v_xⁱ−P

jG_iju_x^j (v_xⁱ)_t= λ_i(A)u_xⁱ − v_xⁱ +P

jG_iju_x^j +P

j ,kH_ijkv_x^ju_x^k

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 – H_ijk = 0 for j = k

– G_ij includes only terms u_xⁱ with i 6= j .

Need to show terms v_x^ju^k_x, u^j_xu_x^k with j 6= k are integrable.

The building block: the 2 × 2 system







u

+ u

= −(f (u) − v ) v

= f (u) − v

f : R 7→ R: smooth and increasing Let ξ = u

, η = v

, then







ξ

+ η

= −f

(u)ξ + η

η

= f

(u)ξ − η

The building block: the 2 × 2 system







u

+ u

= −(f (u) − v ) v

= f (u) − v

f : R 7→ R: smooth and increasing Let ξ = u

, η = v

, then







ξ

+ η

= −f

(u)ξ + η

η

= f

(u)ξ − η

A probabilistic approach

A general stochastic process:







ξ

+ η

= −a(t, x )ξ + b(t, x )η η

= 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 ).

A probabilistic approach

A general stochastic process:







ξ

+ η

= −a(t, x )ξ + b(t, x )η η

= 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 ).

A probabilistic approach

A general stochastic process:







ξ

+ η

= −a(t, x )ξ + b(t, x )η η

= 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 ).

A probabilistic approach

A general stochastic process:







ξ

+ η

= −a(t, x )ξ + b(t, x )η η

= 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 ).

Fundamental solutions Γ

(t, x ; t

, x

), i , j ∈ {0, 1}: the density of probability that a particle, which initially is at x

at t

with speed i , reaches x at t > t

with speed j .

If a(t, x ) = α, b(t, x ) = β, constant switching rates, fundamental solution is invariant under time and space translation:

Γ

(t, x ; t

, x

) = G

(t − t

, x − x

) P(t): position of a walker at t. Then:

t→∞

lim P(t)

t = λ ˙ = β

α + β with probability 1.

Fundamental solutions Γ

(t, x ; t

, x

), i , j ∈ {0, 1}: the density of probability that a particle, which initially is at x

at t

with speed i , reaches x at t > t

with speed j .

If a(t, x ) = α, b(t, x ) = β, constant switching rates, fundamental solution is invariant under time and space translation:

Γ

(t, x ; t

, x

) = G

(t − t

, x − x

) P(t): position of a walker at t. Then:

t→∞

lim P(t)

t = λ ˙ = β

α + β with probability 1.

Fundamental solutions Γ

(t, x ; t

, x

), i , j ∈ {0, 1}: the density of probability that a particle, which initially is at x

at t

with speed i , reaches x at t > t

with speed j .

If a(t, x ) = α, b(t, x ) = β, constant switching rates, fundamental solution is invariant under time and space translation:

Γ

(t, x ; t

, x

) = G

(t − t

, x − x

) P(t): position of a walker at t. Then:

t→∞

lim P(t)

t = λ ˙ = β

α + β with probability 1.

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

t→∞

lim [P(t) − P

(t)] = ∞ with probability 1.

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

t→∞

lim [P(t) − P

(t)] = ∞ with probability 1.

Need a uniform bound on the following term:

E =

Z ∞ 0

Z ∞

−∞

G₀₁(t, x ) · G₁₀^∗(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.

Need a uniform bound on the following term:

E =

Z ∞ 0

Z ∞

−∞

G₀₁(t, x ) · G₁₀^∗(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.

Need a uniform bound on the following term:

E =

Z ∞ 0

Z ∞

−∞

G₀₁(t, x ) · G₁₀^∗(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.

Need a uniform bound on the following term:

E =

Z ∞ 0

Z ∞

−∞

G₀₁(t, x ) · G₁₀^∗(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.

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.

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.

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

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

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

The slow erosion limit:

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 ):

u

−

 exp

Z

f (u

(t, y )) dy



x

= 0, f (p) = p − 1 p

The erosion function f (u

) 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.

Loss of regularities

(A). If lim

f

(p) = 0

⇒ slope u

remains bounded

(B). If lim

p

f (p) = f

(∞) > 0

=⇒ slope u

can blow up in finite time, and shocks form

jump

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

x hyperkink kink

Loss of regularities

(A). If lim

f

(p) = 0

⇒ slope u

remains bounded

(B). If lim

p

f (p) = f

(∞) > 0

=⇒ slope u

can blow up in finite time, and shocks form

jump

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

x hyperkink kink

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).

A coordinate change

Assuming u_x ≥ δ > 0, we consider u as the independent variable and X = X (τ, u) as dependent variable.

Then we take z(τ, u) = X_u(τ, u), the inverse slope.

x u

u(x)

x u

x

X(u) z

u z = X =

1

u 1

u

The basic equations in the new coordinate

z_τ−h

g (z) exp Z ∞

u

g (z(τ, v )) dvi

u

= 0 with the constraint z ≥ 0

Write now: G (z; u) = expR∞

u g (z(τ, v )) dv Along characteristics τ 7→ u(τ ):

˙

u = − g⁰(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.

The basic equations in the new coordinate

z_τ−h

g (z) exp Z ∞

u

g (z(τ, v )) dvi

u

= 0 with the constraint z ≥ 0

Write now: G (z; u) = expR∞

u g (z(τ, v )) dv Along characteristics τ 7→ u(τ ):

˙

u = − g⁰(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.

The basic equations in the new coordinate

z_τ−h

g (z) exp Z ∞

u

g (z(τ, v )) dvi

u

= 0 with the constraint z ≥ 0

Write now: G (z; u) = expR∞

u g (z(τ, v )) dv Along characteristics τ 7→ u(τ ):

˙

u = − g⁰(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.

If z < 0, solution is meaningless:

u x

u

x(u) u u(x)

z = x = _u u1_x

x

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

dtz(t) = Az(t) = h

g (z)G (z; u) +λz i

u, z(0) = ¯z Backward Euler approximations:

z(t + ε) ≈ z(t) + εAz(t + ε) .

= E_ε⁻z(t) Eε⁻z = w iff w solves the ODE

w (u) = z(u) + ε

g (w (u))G (u; w )

u

+ ελwu, w (+∞) = 1

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

dtz(t) = Az(t) = h

g (z)G (z; u) +λz i

u, z(0) = ¯z Backward Euler approximations:

z(t + ε) ≈ z(t) + εAz(t + ε) .

= E_ε⁻z(t) Eε⁻z = w iff w solves the ODE

w (u) = z(u) + ε

g (w (u))G (u; w )

u

+ ελwu, w (+∞) = 1

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

dtz(t) = Az(t) = h

g (z)G (z; u) +λz i

u, z(0) = ¯z Backward Euler approximations:

z(t + ε) ≈ z(t) + εAz(t + ε) .

= E_ε⁻z(t) Eε⁻z = w iff w solves the ODE

w (u) = z(u) + ε

g (w (u))G (u; w )

u

+ ελwu, w (+∞) = 1

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

dtz(t) = Az(t) = h

g (z)G (z; u) +λz i

u, z(0) = ¯z Backward Euler approximations:

z(t + ε) ≈ z(t) + εAz(t + ε) .

= E_ε⁻z(t) Eε⁻z = w iff w solves the ODE

w (u) = z(u) + ε

g (w (u))G (u; w )

u

+ ελwu, w (+∞) = 1

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).

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).

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).

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).

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).

Adding the constraint z ≥ 0

The constraint z ≥ 0 is achieved by adding a measure-valued source µ = Θu

z_t−h

g (z)G (u; z(t)) + λzi

u

− Θ_u = 0 satisfying

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

Z b a

Θ(t, u) du = 0

z

A flux splitting algorithm

z_t−h

g (z)G (u; z(t)) + λzi

u− Θu = 0

Fix a time step size ε > 0 and choose the initial data z0= ¯z.

Time iteration step:

 w_n = E_ε⁻z_n−1 backward Euler step

z_n = πw_n projection on the positive cone

w

w_n zn 1

n z_n

A nonlinear projection operator

f ∈ L¹_loc(R), lim_|x|→∞f (x ) = 1.

Choose F so that F⁰⁰= f

Let F_∗ be the lower convex envelope of F Set πf = F_∗⁰⁰

b a b

a

F f

f

F

*

A nonlinear projection operator

f ∈ L¹_loc(R), lim_|x|→∞f (x ) = 1.

Choose F so that F⁰⁰= f

Let F_∗ be the lower convex envelope of F Set πf = F_∗⁰⁰

b a b

a

F f

f

F

*

A nonlinear projection operator

f ∈ L¹_loc(R), lim_|x|→∞f (x ) = 1.

Choose F so that F⁰⁰= f

Let F_∗ be the lower convex envelope of F Set πf = F_∗⁰⁰

b a b

a

F f

f

F

*

A nonlinear projection operator

f ∈ L¹_loc(R), lim_|x|→∞f (x ) = 1.

Choose F so that F⁰⁰= f

Let F_∗ be the lower convex envelope of F Set πf = F_∗⁰⁰

b a b

a

F f

f

F

*

Properties of the projection operator

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

Z b a

πf (x ) dx = Z b

a

f (x ) dx

Z b a

Z x a

πf (y ) dy dx = Z b

a

Z x a

f (y ) dy dx

Monotonicity: If f ≤ g , then πf ≤ πg L¹-contractility: kπf − πg kL¹ ≤ kf − g kL¹

BV stability: TV{πf } ≤ TV{f } Dissipative:

Z

R

|πf (x) − c| ψ(x) dx ≤ Z

R

|f (x) − c| ψ(x) dx − Z

R

sign(πf (x ) − c)Θ^f(x ) ψx(x ) dx

Properties of the projection operator

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

Z b a

πf (x ) dx = Z b

a

f (x ) dx

Z b a

Z x a

πf (y ) dy dx = Z b

a

Z x a

f (y ) dy dx

Monotonicity: If f ≤ g , then πf ≤ πg L¹-contractility: kπf − πg kL¹ ≤ kf − g kL¹

BV stability: TV{πf } ≤ TV{f } Dissipative:

Z

R

|πf (x) − c| ψ(x) dx ≤ Z

R

|f (x) − c| ψ(x) dx − Z

R

sign(πf (x ) − c)Θ^f(x ) ψx(x ) dx

Properties of the projection operator

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

Z b a

πf (x ) dx = Z b

a

f (x ) dx

Z b a

Z x a

πf (y ) dy dx = Z b

a

Z x a

f (y ) dy dx

Monotonicity: If f ≤ g , then πf ≤ πg L¹-contractility: kπf − πg kL¹ ≤ kf − g kL¹

BV stability: TV{πf } ≤ TV{f } Dissipative:

Z

R

|πf (x) − c| ψ(x) dx ≤ Z

R

|f (x) − c| ψ(x) dx − Z

R

sign(πf (x ) − c)Θ^f(x ) ψx(x ) dx

Properties of the projection operator

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

Z b a

πf (x ) dx = Z b

a

f (x ) dx

Z b a

Z x a

πf (y ) dy dx = Z b

a

Z x a

f (y ) dy dx

Monotonicity: If f ≤ g , then πf ≤ πg L¹-contractility: kπf − πg kL¹ ≤ kf − g kL¹

BV stability: TV{πf } ≤ TV{f } Dissipative:

Z

R

|πf (x) − c| ψ(x) dx ≤ Z

R

|f (x) − c| ψ(x) dx − Z

R

sign(πf (x ) − c)Θ^f(x ) ψx(x ) dx

Properties of the projection operator

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

Z b a

πf (x ) dx = Z b

a

f (x ) dx

Z b a

Z x a

πf (y ) dy dx = Z b

a

Z x a

f (y ) dy dx

Monotonicity: If f ≤ g , then πf ≤ πg L¹-contractility: kπf − πg kL¹ ≤ kf − g kL¹

BV stability: TV{πf } ≤ TV{f } Dissipative:

Z

R

|πf (x) − c| ψ(x) dx ≤ Z

R

|f (x) − c| ψ(x) dx − Z

R

sign(πf (x ) − c)Θ^f(x ) ψx(x ) dx

⇒ Global existence and uniqueness of entropy weak solutions for z(τ, u),

as well as for u(τ, x ).