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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

4 votes

Are there alternative proofs for existence/uniqueness of ODE solutions?

The theory of Di Perna-Lions, also revisited by Ambrosio, provides existence (and uniqueness, in a suitable sense) results for a.e. initial datum of the ODE $\gamma'_t=v_t(\gamma_t)$ under the assumpt …
Nicola Gigli's user avatar
0 votes

Regular Lagrangian flow for explicit ODE with discontinuous right-hand side

No. The theory of Regular Lagrangian Flows rests on two key assumptions: a Sobolev/BV regularity of the vector field and a bound from below on its divergence. In your case you are solving $$ X'=b(X) $ …
Nicola Gigli's user avatar
3 votes

Gradient flows: convex potential vs. contractive flow?

It should be noticed that already on $R^d$ equipped with a non-Euclidean norm $\|.\|$ the answer to your question is no. Ohta-Sturm [1] proved the following: let $\lambda\in R$ and consider the classe …
Nicola Gigli's user avatar