WASSERSTEIN GRADIENT FLOW

APPROACH TO HIGHER ORDER EVOLUTION

EQUATIONS

University of TorontoEhsan Kamalinejad

Joint work with Almut Burchard

and

of fourth and higher ordernonlinear evolution equation

Existence Uniqueness

Gradient Flow on a Manifold

Ingredients:

I. Manifold MII. Metric dIII. Energy function E

Velocity field 𝜕𝑡 𝑋 𝑡=𝑉 𝑡

Steepest Decent 𝑉 𝑡=−𝛻𝐸 (𝑋 𝑡 )

is the gradient Flow of E

Wasserstein Gradient Flows

• Manifold • Metric

• Energy function

is the Wasserstein gradient flow

of E if

Continuity Equation 𝜕𝑡 𝜇𝑡+𝛻 . (𝜇𝑡𝑉 𝑡 )=0

Steepest Decent 𝑉 𝑡∈−𝜕𝐸 (𝜇𝑡 )

PDE reformulated as Gradient Flow

solves PDE

is the gradient flow of

Where

Thin-Film Equation

Dirichlet Energy

Displacement Convexity

is geodesic between and

-

𝐸 (𝑢𝑠 )≤ (1−𝑠) 𝐸 (𝑢0 )+𝑠𝐸 (𝑢1 )𝐸 (𝑢𝑠 )≤ (1−𝑠) 𝐸 (𝑢0 )+𝑠𝐸 (𝑢1 )− 𝜆2𝑠 (1−𝑠)𝑊 2¿

d2

d s2 E(us)≥ λW 2 ¿

Wasserstein Gradiet Flow

McCann 1994Displacement

convexsity

Brenier – McCann 1996-2001Structure of the

Wasserstein metric

Otto, Jordan, Kinderlehrer

1998-2001First gradient flow approach to PDEs

De Giorgi – Ambrosio, Savare, Gigli

1993-2008Systematic proofs

based on Minimizing Movement

Proofs are based on -convexity assumption

for many interesting cases likeDirichlet energy

(Thin-Film Equation)

Fails

Existence, Uniqueness,

Longtime Behavior of many equations has been studied

Stability, and

To prove that Thin-Film and related equations are well-posed

using Gradient Flow method

Ideas are to

Study the Convexity Along the Flow( depends might change along the flow)

Use the Dissipation of the Energy (convexity on energy sub-levels)

Relaxed

Our Goal

-convexity assumption

Restricted -convexity

E is restricted -convexat with if such that E is -convex along geodesics connecting any pair of points inside

Theorem I

E is Restricted -convex at .

Then the Gradient Flow of E starting from

Exists and is Unique at least locally in time.

Theorem II

The Dirichlet energy is

restricted -convex

on positive measures (on ).

Periodic solutions of the Thin-Film equation exist and are unique on positive data.

Minimizing Movement is a CONSTRUCTIVE method

Numerical Approximation

Our local existence-uniqueness result extends directly to more classes of energy functionals of the form:

E (u )=∫∑i=1

m

aiubi∨𝜕x

𝑘𝑖u¿2

Higher order equationsQuantum Drift Diffusion Equation

Global Well-posedness when

THANK YOU.