×

zbMATH — the first resource for mathematics

Global solvability of a model for grain boundary motion with constraint. (English) Zbl 1246.35100
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N \), \(Q_T:=\Omega\times(0,T)\), \(\Sigma_T:=\partial\Omega\times(0,T)\). There is considered the following nonlinear problem with unknown functions \(\eta(x,t)\) and \(\theta(x,t)\): \[ \begin{aligned} \eta_t-\kappa \Delta \eta + g(\eta)+\alpha'(\eta)|\nabla \theta|&= 0 \text{ a.e. \;in}\;Q_T; \\ \alpha_0(\eta)\theta_t-\nu \Delta \theta - \text{div}\big( \alpha(\eta)\frac{\nabla\theta}{|\nabla \theta|}\big)+\partial I_{[-\theta^*,\,\theta^*]}(\theta) &\ni 0 \;\text{a.e. \;in} \;Q_T; \\ \partial \eta /\partial n=0, \;\theta&=0 \;\text{a.e. \;on} \;\Sigma_T; \\ \eta(x,0)=\eta_0(x),\;\theta(x,0)&=\theta_0(x) \;\text{\;a.e. \;in } \;\Omega,\end{aligned} \] where \(\kappa >0\) and \(\nu >0 \) are the small constants, \(\partial I_{[-\theta^*,\,\theta^*]}(\cdot)\) – subdifferential of the indicator function \(I_{[-\theta^*,\,\theta^*]}(\cdot)\) on \([-\theta^*,\,\theta^*]\).
The authors prove that this problem has at least one solution \(\eta, \theta\) such that
1. \(\eta \in C([0,T];\,H)\cap W^{1,2}_{loc}((0,T];\,H) \cap L^\infty_{loc} ((0,T];\,H^1) \cap L^2_{loc}((0,T];\,H^2)\), \(\theta \in C([0,T];\,H)\cap W^{1,2}_{loc}((0,T];\,H) \cap L^\infty_{loc} ((0,T];\,H^1_0)\) and \(0 \leq \eta \leq 1\), \(|\theta|\leq \theta^*\) a.e. in \(Q_T\), if \(\eta_0, \theta_0 \in H\) and \(0\leq \eta_0 \leq 1\), \(|\theta_0|\leq \theta^*\) a.e. in \(\Omega\);
2. \(\eta \in W^{1,2}(0,T;\,H) \cap L^\infty(0,T;\,H^1) \cap L^2(0,T;\,H^2)\), \(\theta \in W^{1,2}(0,T;\,H) \cap L^\infty(0,T;\,H^1_0)\) and \(0 \leq \eta \leq 1\), \(|\theta|\leq \theta^*\) a.e. in \(Q_T\), if \(\eta_0 \in H^1, \;\theta_0 \in H^1_0\) and \(0\leq \eta_0 \leq 1\), \(|\theta_0|\leq \theta^*\) a.e. in \(\Omega\),
where \(H:=L^2(\Omega)\); \(H^1:=H^1(\Omega)\), \(H^1_0:=H^1_0(\Omega)\), \(H^2:=H^2(\Omega)\) are the Sobolev spaces.
Corresponding estimates of the solution are also obtained.

MSC:
35K51 Initial-boundary value problems for second-order parabolic systems
35K55 Nonlinear parabolic equations
35R35 Free boundary problems for PDEs
35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators
PDF BibTeX XML Cite
Full Text: DOI