Abstract

The classical Łojasiewicz inequality (cf. [1]) describes the particular behavior of a real analytic function $\mathcal{E}\colon \mathbb{R}^{d}\to \mathbb{R}$ near a critical point $\bar{u}$: For some $C > 0$ and $\theta \in (0,\frac{1}{2}]$ we have the estimate

$$ \vert \mathcal{E}(u)-\mathcal{E}(\bar{u})\vert^{1-\theta} \leq C \Vert \nabla\mathcal{E}(u)\Vert \qquad (1) $$

for all $u$ in a neighborhood of $\bar{u}$. In his pioneering work [3], L. Simon extended (1) to the infinite dimensional setting and deduced convergence results for the associated gradient flow of $\mathcal{E}$. Since then, the Łojasiewicz-Simon gradient inequality has been widely used as a powerful tool to analyze asymptotic properties of PDEs with a gradient flow structure. For many scientific models, it is natural to require that certain quantities remain conserved during the evolution process. This imposes some constraints on the model, e.g. the conservation of total mass. In order to apply the gradient inequality to the associated “constrained” gradient flow, one needs to prove a suitable version of (1) on a manifold, modelling the constraint. We present sufficient conditions for the Łojasiewicz-Simon gradient inequality to hold on a submanifold of a Banach space and discuss the optimality of our assumptions. As an application, we deduce smooth convergence for the length preserving elastic flow of curves (joint work with Adrian Spener).

References

[1] S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles (1963), pp. 87-89, CNRS, Paris.

[2] F. Rupp, On the Łojasiewicz-Simon gradient inequality on submanifolds, arXiv:1907.09292 (2019).

[3] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, pp. 525-571.