*Fabian Rupp (University of Ulm (Germany))*

#### 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.