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.