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Wasserstein gradient flow approach to higher order evolution equations

Wasserstein 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: Manifold M

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Wasserstein gradient flow approach to higher order evolution equations

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  1. Wasserstein gradient flow approach to higher order evolution equations University of Toronto EhsanKamalinejad Joint work with AlmutBurchard

  2. andof fourth and higher ordernonlinear evolution equation Existence Uniqueness

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

  4. is the gradient Flow of E

  5. Wasserstein Gradient Flows • Manifold • Metric • Energy function

  6. is the Wasserstein gradient flow of Eif

  7. PDEreformulated as Gradient Flow is the gradient flow of Where Dirichlet Energy solves PDE Thin-Film Equation

  8. Displacement Convexity - is geodesic between and

  9. Wasserstein Gradiet Flow

  10. Stability,and Uniqueness, Existence, Longtime Behavior of many equations has been studied Proofs are based on -convexity assumption Fails for many interesting cases like Dirichlet energy (Thin-Film Equation)

  11. Our Goal To prove that Thin-Film and related equations are well-posed using Gradient Flow method -convexity assumption Relaxed 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)

  12. Restricted -convexity E is restricted -convex at with if such that E is -convex along geodesics connecting any pair of points inside

  13. Theorem I E is Restricted -convex at . Then the Gradient Flow of E starting from Exists and is Unique at least locally in time.

  14. 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.

  15. Our local existence-uniqueness result extends directly to more classes of energy functionalsof the form: Minimizing Movement is a CONSTRUCTIVE method Numerical Approximation Higher order equations Global Well-posedness when • Quantum Drift Diffusion Equation

  16. THANK YOU.

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