Rigorous derivation of a mean field model for the Ostwald ripening of thin films (Q2658899)

During the late stages of the evolution of thin liquid films on a solid substrate, liquid droplets are connected by an ultra-thin residual film. Their number decreases due to migration and collision on the one hand, and exchange of matter through a diffusive field in the residual film on the other hand. Supposing that, at time \(t>0\), there are \(N(t)\ge 1\) droplets \(\{B(x_i,R_i(t))\ :\ 1\le i \le N(t)\}\) in the square \(\Omega_{\mathcal{L}} =(0,\mathcal{L})^2\) and neglecting the motion of the centers \(x_i\) due to the no-slip boundary condition for the fluid at the substrate, the dynamics of the radii \((R_i)\) and the diffusive field \(u\) may be described by \begin{align*} - \Delta u(t,x) &= 0, \qquad x\in \Omega_{\mathcal{L}}\setminus \bigcup_{i=1}^{N(t)} \bar{B}(x_i,R_i(t)), \ t>0, \\ u(t,x) &= \frac{1}{R_i(t)}, \qquad x\in B(x_i,R_i(t)), \ t>0, \\ \frac{dR_i}{dt}(t) &= \frac{1}{R_i(t)^2} \int_{\partial B(x_i,R_i(t))} [\nabla u(t,s)\cdot \mathbf{n}(s)]\ ds, \qquad t>0, \end{align*} supplemented with periodic boundary conditions on \(\partial\Omega_{\mathcal{L}}\). In the above integral term, \([\nabla u(t,s)\cdot \mathbf{n}(s)]\) denotes the jump of the normal gradient of \(u\) across the boundary of \(B(x_i,R_i(t))\). After introducing a small parameter \(\varepsilon>0\) and scaling \(\mathcal{L}\), \(x\), \(t\), \(N\), \((R_i)\), and \(u\) in an appropriate way, homogeneization techniques are used to establish the convergence of the rescaled diffusion fields to a mean field \(u_*\). The latter is a weak solution to \begin{align*} -\Delta u_*(t,y) + 2\pi\delta \int_0^\infty \left( u_*(t,y) - \frac{1}{r} \right) f(t,y,r)\ dr & = 0, \qquad r\in (0,\infty),\\ \partial_t f(t,y,r) + \partial_r \left( \frac{2\pi}{r^2} \left( u_*(t,y) - \frac{1}{r} \right) f(t,y,r) \right) & = 0, \qquad r\in (0,\infty), \end{align*} for \(t>0\) and \(y\in \Omega_1\), supplemented with periodic boundary conditions on \(\Omega_1\). The parameter \(\delta\) is prescribed by the scaling, while \(f\) is in general a measure-valued solution to the above transport equation.