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Tuesday, 10 October 2023 - 10:00 to 11:00 <br />
Place: IM, konírna
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Speaker: Richard Smith, University College Dublin<br />
Title: Well-posedness of a PDE in fluid dynamics and approximation properties in Lipschitz-free spaces
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Abstract <br />
<p>In this talk we present results in two areas that are very distantly related via optimal transport theory. In the first part, we introduce the ``Geometric Thin-Film equation'', which is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We sketch a proof of the existence of weak solutions of this equation, and their well-posedness with respect to the 1-Wasserstein distance, provided the initial conditions (which are Radon measures on $\R$) have no atoms. This part is joint work with L. Ó Náraigh and K. E. Pang (UCD).</p><p><br></p><p>In the second part, we consider a compact metrisable space or, more generally, a ``properly metrisable'' topological space $T$, together with the set of all compatible proper metrics $\mathcal{M}^T$, equipped with the uniform topology. We present a sketch of the fact that if $T$ is uncountable then the set of metrics $d$ for which the Lipschitz-free space $\mathcal{F}{(T,d)}$ fails the approximation property is dense in $\mathcal{M}^T$. We also show that, if $T$ is compact, then the set of metrics $d$ for which $\mathcal{F}{(T,d)}$ has the metric approximation property is also dense in $\mathcal{M}^T$. This part is joint work with F. Talimdjioski (UCD). </p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/6
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Set Theory and Analysis mailing list <br />
settfa@math.cas.cz <br />
https://list.math.cas.cz/listinfo/settfa@math.cas.cz
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