Entropy solutions via JKO scheme for a class of degenerate - - PowerPoint PPT Presentation

Entropy solutions via JKO scheme for a class of degenerate convection-diffusion equations

Marco Di Francesco

Departament de Matem` atiques Universitat Aut`

• noma de Barcelona

Joint work with Daniel Matthes (TU Munich)

June 28, 2012

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 1 / 38

Table of contents

1

Preliminaries and main result

2

Contractive flows

3

Flow interchange

4

Construction of entropy solution

5

Uniqueness

6

Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 2 / 38

Preliminaries and main result

Table of contents

1

Preliminaries and main result

2

Contractive flows

3

Flow interchange

4

Construction of entropy solution

5

Uniqueness

6

Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 3 / 38

Preliminaries and main result

Motivation

For some evolutionary PDEs, the following two concept of solutions can be formulated: The notion of Wasserstein gradient flow (De Giorgi, Otto, Ambrosio - Gigli - Savar´ e), which can be introduced every time you deal with ∂ρ ∂t = div

• ρ∇δF[ρ]

δρ

• F[ρ] =
• ρ log ρdx gives ρt = ∆ρ,

F[ρ] =

• ρW ∗ ρdx gives ρt = div(ρ∇W ∗ ρ),

F[ρ] =

1 m−1

• ρmdx gives ρt = ∆ρm, m > 1.

The notion of Entropy solution (Oleinik, Kruˇ zkov), which classically applies to

nonlinear conservation laws ρt + f (ρ)x = 0, nonlinear convection diffusion equations ρt = g(ρ)xx + f (ρ)x.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 4 / 38

Preliminaries and main result

Significant examples

Nonlocal interaction equations with nonlinear diffusion ∂tρ = ∆ρm + div(ρ∇G ∗ ρ) = 0 with m > 1 and G ∈ C 2 and G even. Here, both notions have been used (almost at the same time!) to prove uniqueness of solutions. Scalar conservation laws with symmetric inhomogeneous coefficient ∂tρ + ∂x(m(ρ)∇V (x)) = 0 with m concave, V ∈ C 2 and V even. Here, the interpretation as a gradient flow is still partially open, whereas it is (obviously) clear that you can obtain unique entropy solutions under reasonable assumptions.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 5 / 38

Preliminaries and main result

Our model

Nonlinear convection diffusion equation with space dependent coefficient: ∂tu = (um)yy + (b(y)um)y, y ∈ R, t ≥ 0. (1) Porous medium exponent: m > 1. Assumptions on b: b ∈ L1(R) ∩ W 1,∞(R) (2) Initial condition u0 ∈ Lm(R), nonnegative with finite second moment. Important remark Here the gradient flow theory will not give you a unique solution for free, whereas the literature [Karlsen-Risebro 2003] already contains a uniqueness result for entropy solutions.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 6 / 38

Preliminaries and main result

The scaling

Problem: How to see the gradient flow structure for (1)? Answer: There exists a function a (a1) a(x) ≥ a > 0 for all x ∈ R, (a2) a ∈ W 2,∞(R), and a bijective change of coordinates y = Ta(x) on R such that, for a given weak solution u(y, t) to (1) with initial datum u0 ∈ L1(R), the scaled function ρ(x, t) := T ′

a(x)u(Ta(x), t)

satisfies ρ0(x) := ρ(x, 0) = u(x, 0) and the equation ∂tρ =

• ρ[a(x)ρm−1]x
• x,

(3)

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 7 / 38

Preliminaries and main result

Intuitive idea behind the scaling

Start from (3). Substitute ρ(x, t) := T ′

a(x)u(Ta(x), t), to get an equation

for u(y, t): ∂tu(y, t) =

• (m − 1)(T ′

a(x)))m+1a(x)um−1uy

+ (T ′

a(x))ma′(x)um y

• x=T −1(y).

Choose Ta(x) = x m − 1 m a(ξ) −

1 m+1

dξ. (4) to make the diffusion coefficient homogeneous. We obtain (1). Notation: u(y, t) = Sa[ρ(·, t)](y).

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 8 / 38

Preliminaries and main result

Scaling: from b to a

The rigorous definition of the new coefficient a(x) for fixed b ∈ L1(R) ∩ W 1,∞(R). Set α(y) = α0 exp

• −(m − 1)

2m y b(η)dη

• .

Notice that α ∈ W 2,∞(R) and α ≥ α > 0. Consider the unique solution to the Cauchy problem T ′(x) = α(T(x)), x ∈ R T(0) = 0, (5) Finally, define a as follows a(x) := m m − 1 (α(T(x)))−(m+1) . (6) a satisfies: a ∈ W 2,∞(R) and a(x) ≥ a > 0.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 9 / 38

Preliminaries and main result

To fix ideas:

Throughout this talk we shall always think of ρ and u being 1 : 1-related via the scaling u(y, t) = Sa[ρ(·, t)](y). ρ(x, t) should be thought as a solution to (3) ∂tρ =

• ρ[a(x)ρm−1]x
• x,

for which we can use a gradient flow (JKO) approach, u should be thought as a solution to (1) ∂tu = (um)yy + (b(y)um)y, y ∈ R, t ≥ 0, for which we can use the entropy solution approach.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 10 / 38

Preliminaries and main result

Entropy solutions for (1)

Following [Carrillo M. - ARMA 1999] and [Karlsen, Risebro - DCDS 2003]: Definition (Definition of entropy solution) Let u0 ∈ L1 ∩ L∞(R), and let T > 0. A non-negative measurable function u :]0, T[×R → R is an entropy solution to (1) with initial condition u0 if u ∈ L1 ∩ L∞(]0, T[×R), if um ∈ L2(]0, T[; H1(R)), if u(t) → u0 in L1(R) as t ↓ 0, and if inequality T

• R

|u − k|ϕt dy dt ≥ T

• R

Sgn(um − km)

• (um)y + b(um − km)
• ϕy − bykmϕ
• dy dt.

is satisfied for all nonnegative test functions ϕ ∈ C ∞

c (]0, T[×R), and for

all k ∈ R+.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 11 / 38

Preliminaries and main result

Construction of entropy solutions

In the literature there are many methods used to construct entropy

• solutions. Some of them do not apply to our model. Here is a (possibly

incomplete) list: Vanishing viscosity approach Wave front tracking algorithm Glimm’s scheme Semigroup approach (implicit Euler) We shall add another method to the list, namely: JKO - De Giorgi scheme for ρ ⇒ scaling u = Sa[ρ]

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 12 / 38

Preliminaries and main result

The JKO scheme for (3)

F[ρ] := 1 m

• R

a(x)ρm dx. For a time step τ > 0, Fτ(ρ; σ) = 1 2τ W2(ρ, σ)2 + F(ρ), where W2(ρ, σ) is the 2-Wasserstein distance.

1 ρ0

τ := ρ0.

2 For n ≥ 1, let ρn

τ ∈ P2(R) be the (unique global) minimizer of

Fτ(·; ρn−1

τ

). Piecewise constant interpolation: ¯ ρτ(t) = ρn

τ

for (n − 1)τ < t ≤ nτ.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 13 / 38

Preliminaries and main result

Main result

Theorem (DF, Matthes - (almost preprint)) Let ρ0 ∈ P2(R) ∩ L∞(R), and let a as above. Then every vanishing sequence (τk)k∈N of time steps contains a subsequence (not relabeled) such that the ¯ ρτk converge to a curve ρ∗ : [0, ∞[→ P2(R) — in Lm(]0, T[×R), and also uniformly in W2 on each time interval [0, T] — that is continuous with respect to W2. The rescaled function u∗ = Sa[ρ∗] is an entropy solution to (1) in the sense of Definition 1, with initial condition u0 = Sa[ρ0]. Corollary Since the entropy solution u∗ = Sa[ρ∗] to (1) is unique, then the JKO scheme for F has a unique limit point.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 14 / 38

Preliminaries and main result

Remarks

A similar connection has been pointed out for instance by [Gigli-Otto 2010] in the context of the inviscid Burger’s equation. The interpretation of the coincidence between entropy solutions and gradient flows is that both types of solutions can be characterized by diminishing an underlying entropy functional “as fast as possible”. Apart from revealing an interesting connection between two different theories for nonlinear evolution equations, our simple example indicates a possible general strategy to prove existence and uniqueness

• f certain degenerate parabolic equations. First, identify a variational

structure behind the equation and use methods from the calculus of variations to construct a candidate for a solution; a priori, there might be several. Second, show that this candidate is an entropy solution by deriving further a priori estimates in the variational framework. Third, conclude uniqueness of the entropy solution.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 15 / 38

Contractive flows

Table of contents

1

Preliminaries and main result

2

Contractive flows

3

Flow interchange

4

Construction of entropy solution

5

Uniqueness

6

Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 16 / 38

Contractive flows

A crash course on Wasserstein gradient flows1

Let F be a functional on the space P2 of probability measures with finite second moment, such that the JKO scheme ρn+1

τ

= argmin 1 2τ W22(ρn

τ, ρ) + F[ρ], ρ ∈ P2

• ,

ρ0

τ given

is well defined, let ¯ ρτ be the (left continuous) time piecewise constant interpolation of the sequence ρn

τ. If F satisfies the κ-convexity property

F[ρ(t)] ≤ (1 − t)F[ρ(0)] + tF[ρ(1)] − κ

2t(1 − t)W2(ρ(0), ρ(1))2 on

the W2-geodesic ρ(t) with constant speed connecting ρ(0) to ρ(1), for some λ ∈ R (plus some minor extra assumptions), then ¯ ρτ has a unique limit point which is also an energy solution to the PDE ∂tρ = div

• ρ∇δF

δρ

• .

(7)

1[Jordan-Kinderlehrer-Otto, 1997], [Otto, 2001], and [Ambrosio-Gigli-Savar´

e, 2005]

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 17 / 38

Contractive flows

Evolution variational inequality (EVI)

In [Daneri-Savar´ e, 2008], the κ convexity is proven to be ‘basically’ equivalent to the following property: F admits a uniquely determined κ-flow Ss

F, i. e. a semigroup

SF : [0, +∞) × P2 such that 1 2 d dσ

• σ=sW2
• Sσ

F[ρ], η

2 + κ 2W2(Ss

F[ρ], η)2 ≤ F(η) − F(Ss F[ρ])

(8) for all ρ, η ∈ P2. The candidate κ-flow is the PDE (7), By a density argument, it is sufficient to uniquely determine the k-flow and to prove the above inequality (8) for smooth initial conditions.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 18 / 38

Contractive flows

k-convexity for inhomogeneous free energies

Consider a general inhomogeneous free energy functional of the form Ψ(η) =

• R

F(x, η(x)) dx (9) for a given function F : R × R+ → R. Problem: We aim to determine a reasonable sufficient condition on F such that Ψ admits a κ-flow for some κ ∈ R. For a concise formulation of that condition, we introduce the adjoint function H : R × R+ → R by H(x, ξ) = ξF(x, 1/ξ). (10) Remark: In dimension d, when F is independent of x, F is k-convex if and only if the McCann condition m > (d − 1)/d is satisfied. If this is the case, then k ≤ 0.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 19 / 38

Contractive flows

k-convexity for inhomogeneous free energies

Lemma (Con) Assume that there is some κ ∈ R such that (x, ξ) → H(x, ξ) − κ 2x2 (11) is (jointly) convex on R × R+. (Reg) Assume further that there exist two positive constants c and C such that ηFηη(x, η) ≥ c, Fηx(x, η) ≤ C, and ηFηη(x, η) ≤ C for all (x, η) ∈ R × [0, +∞). Then, the semigroup SΨ of the evolution equation ∂tη = Dx(ηDx[Fη(x, η)]). (12) is a κ-flow for the functional Ψ from (9).

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 20 / 38

Contractive flows

Sketch of the proof

We need to prove the EVI 1 2 d dt

• t=0W2(St

Ψη0, ˜

η)2 + κ 2W2(η0, ˜ η)2 ≤ Ψ(˜ η) − Ψ(η0) (13) for all smooth and strictly positive η0, ˜ η ∈ P2(R). η(t; x) := St

Ψη0(x).

Assumption (Reg) is technical (approximation). Define the distribution function U(t; x) = x

−∞

η(t; y) dy which is a smooth diffeomorphism from R to ]0, 1[, for every t ∈ R+ with inverse G : [0, ∞[×]0, 1[→ R. A tricky calculation gives ∂tG = Dω[Hξ(G, ∂ωG)] − Hx(G, ∂ωG). (14) Observe that a change of variables ω = U(x) leads to Ψ(η) = 1 H(G(ω), ∂ωG(ω)) dω.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 21 / 38

Contractive flows

Sketch of the proof

Recall the 1-d representation of W2(η, ˜ η)2 in terms of the respective inverse distribution functions G, ˜ G: W2(η, ˜ η)2 = 1 [G(ω) − ˜ G(ω)]2 dω. (15) 1 2 d dt

• t=0W2(η, ˜

η)2 + κ 2W2(η, ˜ η)2 = 1 Gt[G − ˜ G] dω + κ 2 1 [G − ˜ G]2 dω = 1 Hξ(G, Gω)[˜ Gω − Gω] dω + 1 Hx(G, Gω)[˜ G − G] dω + κ 2 1 [˜ G − G]2 dω

(Con)

≤ 1

• H(˜

G, ˜ Gω) − H(G, Gω)

• dω = Ψ(˜

η) − Ψ(η).

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 22 / 38

Contractive flows

F is not κ-convex

Lemma 4 gives an indication that the functional F is not geodesically κ-convex. Indeed, with F(x, η) = a(x)ηm, we have H(x, ξ) = a(x)ξ1−m, and thus D2H(x, ξ) =

• a′′(x)ξ1−m − κ

−(m − 1)a′(x)ξ−m −(m − 1)a′(x)ξ−m m(m − 1)a(x)ξ−(m+1).

• Unless a is a constant, there must exist an ¯

x ∈ R with a′′(¯ x) < 0. Thus, no matter how κ is chosen, there exists further a sufficiently small ¯ ξ ∈ R+ such that the top left element of D2H(¯ x, ¯ ξ) is negative, that is a′′(¯ x)¯ ξ−(m−1) − κ < 0.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 23 / 38

Flow interchange

Table of contents

1

Preliminaries and main result

2

Contractive flows

3

Flow interchange

4

Construction of entropy solution

5

Uniqueness

6

Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 24 / 38

Flow interchange

Motivation

Main idea to construct entropy solutions: For a given test function ϕ ≥ 0, we somewhat need to estimate Ψ =

• R |u − k|ϕdx along solutions to (1), in a discrete sense (JKO

approximation). To fix ideas, assume first ϕ does not depend on time. Consider the ‘smoothed’ version Ψ =

• R |u − k|ǫϕdx + νH(u) with

H(u) =

• R u log udx.

For technical reasons |u − k|ǫ should be replaced by Sǫ(u) := u

k Sgnǫ(rm − km) dr.

Put everything into the ρ(x, t) variables (apply the scaling!): Ψν,ǫ(ρ) :=

• R

Sǫ η(x) T ′

a(x)

• ϕ ◦ Ta(x)T ′

a(x) dx + νH(η),

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 25 / 38

Flow interchange

Heuristics

We then want to find a way to control the auxiliary functional Ψν,ǫ along the solution ρ to the (3). We explain the idea with the following toy model: consider two gradient flows in RN ˙ X(t) = −∇F(X(t)) ˙ Y (s) = −∇G(Y (s)) The evolution of F along Y (s) is given by d ds F(Y (s)) = ∇F(Y (s)) · ˙ Y (s) = −∇F(Y (s)) · ∇G(Y (s)), (16) and the evolution of G along X(t) is given by d dt G(X(s)) = ∇G(X(t)) · ˙ X(t) = −∇G(X(t)) · ∇F(X(t)), (17) and the two functionals on the r.h.s. of (16) and (17) coincide.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 26 / 38

Flow interchange

Flow interchange lemma2

Lemma Let Ψ : P2(R) →] − ∞, +∞] be a lower semi-continuous functional on P2(R) with associated κ-flow SΨ. Define further the dissipation of F along SΨ by DΨ(ρ) := lim sup

s↓0

1 s

• F(ρ) − F
• Ss

Ψρ

• for every ρ ∈ P2(R). If ρn−1

τ

and ρn

τ are two consecutive steps of the JKO

scheme for F, then Ψ(ρn−1

τ

) − Ψ(ρn

τ) ≥ τDΨ(ρn τ) + κ

2W2(ρn

τ, ρn−1 τ

)2. (18) In particular, Ψ(ρn−1

τ

) < ∞ implies DΨ(ρn

τ) < ∞.

2[Matthes-McCann-Savar´

e, 2009]

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 27 / 38

Flow interchange

Flow interchange: corollary

Corollary Let the κ-flow SΨ be such that for every n ∈ N, the curve s → Ss

Ψρn τ lies

in Lm(R), where it is differentiable for s > 0 and continuous at s = 0. And moreover, let a functional K : P2(R) →] − ∞, ∞] satisfy lim inf

s↓0

• −

d dσ

• σ=sF(Sσ

Ψρn τ)

• ≥ K(ρn

τ).

(19) Then the following two estimates hold. For every n ∈ N: Ψ(ρn−1

τ

) − Ψ(ρn

τ) ≥ τK(ρn τ) + κ

2W2(ρn

τ, ρn−1 τ

)2; (20) for every N ∈ N: Ψ(ρN

τ ) ≤ Ψ(ρ0) − τ N

• n=1

K(ρn

τ) + τ max(0, −κ)F(ρ0).

(21)

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 28 / 38

Construction of entropy solution

Table of contents

1

Preliminaries and main result

2

Contractive flows

3

Flow interchange

4

Construction of entropy solution

5

Uniqueness

6

Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 29 / 38

Construction of entropy solution

Compactness of the scheme

Lemma There is a constant A depending only on ρ0 (and in particular not on τ) such that the piecewise constant interpolations ¯ ρτ satisfy

• ¯

ρm/2

τ

• L2(0,T;H1(R)) ≤ A(1 + T)

for all T > 0. (22) Proof: use the flow interchange lemma with auxiliary functional H(ρ) =

• ρ(x) log ρ(x) dx.

∂sF(Ss

Hη0)

≤ −4(m − 1)a m2

• R
• ∂x
• η(s, x)m/22 dx + axx

m

• R

η(s, x)m dx. Arguments from [Ambrosio-Gigli-Savar´ e] combined with arguments taken from [Rossi-Savar´ e, 2003] give strong Lm([0, T] × R) compactness.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 30 / 38

Construction of entropy solution

Derivation of the entropy inequality

The functional Ψν,ǫ generates a unique κ-flow for some κ ∈ R, as it satisfies the assumptions (Con) and (Reg) above. Moreover, the k-flow is given by (3). Therefore, the above corollary implies that we can estimate Ψν,ǫ along the interpolation ¯ ρτ. By sending τ → 0 we obtain d ds F(Ss

Ψνη0)≤

• R
• (vm)y + b(vm − km)
• φy − kmbyφ
• Sgnǫ(vm − km) dy

−

• R
• Hǫ(v)
• y

2φ dy −

• R

b

• Rǫ(vm − km)
• yφ dy − νKH(η),

we changed variables (t, x) → (t, y) , Hǫ, Rǫ : R+ → R are smooth and such that H′

ǫ(s)2 = Sgn′ ǫ(s) and

R′

ǫ(s) = s Sgn′ ǫ(s),

KH(ρ) := 4(m − 1)a m2

• R
• ∂x
• ρm/22 dx − axx

m

• R

ρm dx

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 31 / 38

Construction of entropy solution

Time dependent version

Consider now the time dependent test function ϕ(t, x) = θ(t)φ(x) with θ ∈ C ∞

c (]0, T[). Multiply (20) by θ(nτ) and sum over n to find

τ

∞

• n=1

Ψν(ρn

τ)θ(nτ) − θ((n + 1)τ)

τ ≥ τ

∞

• n=1

ψ(nτ)K(ρn

τ) + κντF(ρ0).

Thanks to the strong convergence of ¯ ρτ to ρ∗ in Lm(R) and the lower semi-continuity of K, we can pass to the time-continuous limit and find T Ψν(ρ∗(t))θ′(t) dt ≥ T θ(t)K(ρ∗(t)) dt. As we can prove that H(ρ∗(t)) remains bounded for all times, we can now pass to the inviscid limit ν ↓ 0 and find the following inequality:

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 32 / 38

Construction of entropy solution

Entropy inequality

T

• R

Sǫ((u∗)m − km)∂tϕ dy dt ≥ T

• R
• (um

∗ )y + b(um ∗ − km)

• ϕy − kmbyϕ
• Sgnǫ(um

∗ − km) dy dt

+ T

• R
• Hǫ(u∗)
• y

2ϕ dy dt − T

• R

(bϕ)yRǫ(um

∗ − km) dy dt.

In the final step, we pass to the limit ǫ ↓ 0. By the uniform convergence of Sǫ(s) to |s − k| and the uniform convergence of Rǫ to zero, we obtain T

• R

|u∗ − k|ϕt dy dt − T

• R

Sgn(um

∗ − km)

• (um

∗ )y + b(um ∗ − km)

• ϕy − bykmϕ
• dy dt

≥ lim sup

ǫ↓0

T

• R
• (um

∗ )y

2 Sgn′

ǫ(um ∗ − km)ϕ dy dt

(23)

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 33 / 38

Uniqueness

Table of contents

1

Preliminaries and main result

2

Contractive flows

3

Flow interchange

4

Construction of entropy solution

5

Uniqueness

6

Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 34 / 38

Uniqueness

Uniqueness

We use a uniqueness result by Karlsen and Risebro for the equation (1). In

• rder to apply such result, we need a uniform L∞, that we obtain by

testing the auxiliary functional Ψν(η) =

• R

PosParǫ

• a(x)

1 m−1 η(x) − k

• a(x)

−1 m−1 dx + νH(η),

which provides the comparison property

• R
• ρ∗(T, x) − ka(x)

−1 m−1

+ dx ≤

• R
• ρ0(x) − ka(x)

−1 m−1

+ dx.

Note that the augmented entropy inequality (23) is needed in the uniqueness proof by [Karlsen-Risebro 2003].

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 35 / 38

Open problems and remarks

Table of contents

1

Preliminaries and main result

2

Contractive flows

3

Flow interchange

4

Construction of entropy solution

5

Uniqueness

6

Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 36 / 38

Open problems and remarks

The strategy can be applied to several other contexts, as it mostly relies on the fact that the functional providing the entropy condition is κ-convex. The only condition to check is that the JKO of the given functional provides enough estimates in the limit that allow to fall into a uniqueness theorem for entropy solutions. Our project is to tackle gradient flows with nonlinear mobility ∂tρ = div(m(ρ)∇V (x)) since in this case we would fall into the theory of scalar conservation

• laws. The JKO framework is provided by

[Dolbeault-Nazareth-Savar´ e]. The functional here is never κ convex. We are also interested in nonlocal models ∂tρ = div(m(ρ)∇W ∗ ρ). Here the additional difficulty in order to get unique entropy solutions is given by the BV estimate.

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 37 / 38

Open problems and remarks

End of the talk

THANK YOU

Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 38 / 38