Optimal transport: theory and applications

seminar: Estimates for ordinary differential equations with Sobolev coefficients

speaker: Camillo De Lellis (Institute for Advanced Study, Princeton)

abstract: Inspired by a result by Ambrosio, Lecumberry and Maniglia, in a joint work with Gianluca Crippa we show a simple derivation of integral logarithmic bounds for solutions of ordinary differential equations$$ \cases{\displaystyle{{d\Phi \over dt(t,x)=b(t,\Phi(t,x))&\cr \Phi(0,x)=x.& $$ These bounds depend only on the $L^\infty$ and $W^{1,p$ norm of $b$, and on the compressibility constant of $\Phi$ (which in turn can be bound by $\Vert {\rm div\,b\Vert_\infty$). These a-priori estimates allow to recover in a simple way many old and new results about existence, uniqueness, stability and differentiability properties of solutions of ODE's with Sobolev coefficients. As new corollaries, we conclude that the Cauchy problem for transport equation with Sobolev coefficients preserve a mild regularity property of the initial data and we give an affirmative answer to the $L^p$ version of a Conjecture of Bressan.

timetable:
Fri 17 Nov, 17:00 - 17:55, Aula Dini
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