The Lowner-Kufarev representations for domains with analytic boundaries

We consider the Lowner-Kufarev differential equations generating univalent maps of the unit disk onto domains bounded by analytic Jordan curves. A solution to the problem of the maximal lifetime shows how long a representation of such functions admits using infinitesimal generators analytically extendable outside the unit disk. We construct a Lowner-Kufarev chain consisting of univalent quadratic polynomials and compare the Lowner-Kufarev representations of bounded and arbitrary univalent functions.