W ASSERSTEIN GRADIENT FLOW APPROACH TO HIGHER ORDER EVOLUTION EQUATIONS University of Toronto Ehsan Kamalinejad Joint work with Almut Burchard

and of fourth and higher order nonlinear evolution equation Existence Uniqueness

Gradient Flow on a Manifold Ingredients: M I.Manifold M d II.Metric d E III.Energy function E

Velocity field Steepest Decent

Continuity Equation Steepest Decent

PDEGradient Flow PDE reformulated as Gradient Flow

Displacement Convexity

Wasserstein Gradiet Flow McCann1994 Displacement convexsity Brenier – McCann Structure of the Wasserstein metric Otto, Jordan, Kinderlehrer First gradient flow approach to PDEs De Giorgi – Ambrosio, Savare, Gigli Systematic proofs based on Minimizing Movement

Fails Existence,Uniqueness, Longtime Behavior of many equations has been studied Stability, Stability, and

well-posed To prove that Thin-Film and related equations are well-posed Gradient Flow using Gradient Flow method Ideas are to Use the Dissipation of the Energy (convexity on energy sub-levels) Relaxed Our Goal

Theorem I

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

CONSTRUCTIVE Minimizing Movement is a CONSTRUCTIVE method Numerical Approximation extends Our local existence-uniqueness result extends directly to more classes of energy functionals of the form:

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