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arXiv · 2308.04281

Convergence and non-convergence in a nonlocal gradient flow

Sangmin Park·Robert L. Pego
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Abstract

We study the asymptotic convergence of solutions as $t\rightarrow\infty$ of $\partial_t u=-f(u)+\int f(u)$, a nonlocal differential equation that is formally a gradient flow in a constant-mass subspace of $L^2$ arising from simplified models of phase transitions. In case the solution takes finitely many values, we provide a new proof of stabilization that uses a {\L}ojasiewicz-type gradient inequality near a degenerate curve of equilibria. Solutions with infinitely many values in general need not converge to equilibrium, however, which we demonstrate by providing counterexamples for piecewise linear and cubic functions $f$. Curiously, the exponential rate of convergence in the finite-value case can jump from order $O(1)$ to arbitrarily small values upon perturbation of parameters.

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Sangmin Park, Robert L. Pego. 2023-08-08. Convergence and non-convergence in a nonlocal gradient flow. https://arxiv.org/abs/2308.04281

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