Convergence to walls of dislocations in the periodic case
Résumé
In this paper we are interested in the convergence of accumulation of dislocations to walls of dislocations. We consider the dynamical system generated by the force $f(x,y)=\frac{x(y^{2}-x^{2})}{(y^{2}+x^{2})^{2}}$, defined over $\R\times\Z\backslash\{0\},$ that describes the phenomena. For initial data $X^{0}\in\Omega\cap\ell^{\infty}=\left\{X: |x_{i} - x_{j}| \leqslant \sqrt{3 - 2\sqrt{2}} \,|i-j| \right\}\cap\ell^{\infty},$ we show %% using Cauchy Lipschitz theorem the existence of unique solution $X\in C^{1}\in([0,+\infty),\Omega\cap\ell^{\infty}).$ Moreover, we prove that if $X^{0}$ is periodic, then $X(t)=(x_{j}(t))_{j\in\Z}$ is periodic for any $t>0$ and converges to the barycenter of the initial data, i.e. $x_{j}(t)\to c=\frac{1}{N}\sum_{i=1}^{N}x_{i}^{0}$ for every $j\in\Z.$ We also establish a $\ell^{p}$ contraction for periodic solutions and perform numerical simulations.
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