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

2. and of fourth and higher order nonlinear evolution equation Existence Uniqueness

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

4. Velocity field Steepest Decent

6. Continuity Equation Steepest Decent

7. PDEGradient Flow PDE reformulated as Gradient Flow

8. Displacement Convexity

9. Wasserstein Gradiet Flow McCann1994 Displacement convexsity Brenier – McCann1996-2001 Structure of the Wasserstein metric Otto, Jordan, Kinderlehrer 1998-2001 First gradient flow approach to PDEs De Giorgi – Ambrosio, Savare, Gigli 1993-2008 Systematic proofs based on Minimizing Movement

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

11. 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

13. Theorem I

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

15. 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:

16. THANK YOU.