Convergence and non-convergence in a nonlocal gradient flow
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912026161315840 |
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| author | Park, Sangmin Pego, Robert L. |
| author_facet | Park, Sangmin Pego, Robert L. |
| contents | 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 Łojasiewicz-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04281 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence and non-convergence in a nonlocal gradient flow Park, Sangmin Pego, Robert L. Classical Analysis and ODEs Analysis of PDEs Dynamical Systems 34D05, 35B40, 37L15 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 Łojasiewicz-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. |
| title | Convergence and non-convergence in a nonlocal gradient flow |
| topic | Classical Analysis and ODEs Analysis of PDEs Dynamical Systems 34D05, 35B40, 37L15 |
| url | https://arxiv.org/abs/2308.04281 |