On motion of an elastic wire and singular perturbation of a 1-dimensional plate equation (Q5937563)

The motion of a springy circle wire \(\gamma=\gamma(x)\in\mathbb{R}^3\) parametrized by the length is subjected to the system NEWLINE\[NEWLINE\gamma_{tt}+ \gamma_{xxxx}+ \mu\gamma_t=((w-2|\gamma_{xx}|^2)\gamma_x)_x,\quad -w_{xx}+|\gamma_{xx}|^2w=2|\gamma_{xx}|^4-|\gamma_{xxx}|^2+|\gamma_{tx}|^2,NEWLINE\]NEWLINE where \(\mu\) is the resistance and \(w\) is a certain auxiliary function. The system has a unique short time solution for any initial data and if the resistance is large enough, then the solution exists for long time and converges to a solution of the (rescaled in \(t\)) system NEWLINE\[NEWLINE\gamma_t+\gamma_{xxxx}=((w-2|\gamma_{xx}|^2)\gamma_x)_x,\quad -w_{xx}+|\gamma_{xx}|^2w= 2|\gamma_{xx}|^4-|\gamma_{xxx}|^2.NEWLINE\]NEWLINE The main difficulty consisting in the presence of the third-order derivative \(\gamma_{xxx}\) is deleted by employing the new unknown \(\xi=\gamma_x\in S^2\), then the common theory of perturbed parabolic equations can be applied.