Convergence to equilibrium for discretized gradient systems with analytic features

A celebrated result of S. L ojasiewicz states that if $F: mathbb{R}^d to mathbb{R}$ is real analytic, then every bounded solution $U:[0,+ infty) to mathbb{R}^d$ of the gradient flow $$U (t)=- nabla F(U(t)) quad t ge 0, $$ converges to a critical point of $F$ as $t to + infty$. Convergence rates can also be obtained.
These convergence results have been generalized to a large variety of finite or infinite dimensional gradient-like flows. The fundamental example in infinite dimension is the semilinear heat equation with an analytic nonlinearity. In this talk, we show how some of these results can be adapted to time discretizations of gradient-like flows, in view of applications to PDEs such as the Allen-Cahn equation, the sine-Gordon equation, the Cahn-Hilliard equation or the Swift-Hohenberg equation.

CONTACT: marco.verani@polimi.it and paola.antonietti@polimi.it