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Global existence of solutions to a singular parabolic/Hamilton-Jacobi coupled system with Dirichlet conditions. (English. Abridged French version) Zbl 1166.35313

Summary: We study the existence of (distribution/viscosity) solutions of a singular parabolic/Hamilton-Jacobi coupled system \[ \begin{cases} \kappa_t\kappa_x=\rho_t\rho_x & \text{on }I_T:=I\times(0,T),\\ \rho_t=\rho_{xx}-\tau\kappa_x&\text{on }I_T,\end{cases} \] for a given time \(T>0\) and \(\tau\in\mathbb R\) and with initial and boundary conditions \[ \rho(x,0)=\rho^0(x),\quad\kappa(x,0)=\kappa^0(x),\quad\forall x\in I, \]
\[ \rho(\pm 1,t)=0,\quad\kappa(\pm 1,t)=\pm 1,\quad\forall t\in (0,T). \] Our motivation stems from the study of the dynamics of dislocation densities in a crystal of finite size [I. Groma, F. F. Csikor and M. Zaiser, Acta Materialia 51, No. 5, 1271–1281 (2003)]. The method of the proof consists in considering a parabolic regularization of the system, and then passing to the limit after obtaining some uniform bounds using in particular an entropy estimate for the densities.

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35K65 Degenerate parabolic equations
82D25 Statistical mechanics of crystals
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