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.


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