Existence, uniqueness and properties of global weak solutions to interdiffusion with Vegard rule
The paper deals with the model for the diffuse mass transport given by the one-dimensional nonlinear system of strongly coupled parabolic differential equations \[ \partial_t\varrho_i=\partial_x\left(\Theta_i(\varrho_1,\dots,\varrho_r)\partial_x\varrho_i-\varrho_i\displaystyle\sum_{j=1}^r\displaystyle\frac{\Omega_j\Theta_j(\varrho_1,\dots,\varrho_r)}{M_j}\partial_x\varrho_j-K(t)\varrho_i\right), \] for $i=1,\dots,r$, $\varrho_i=M_ic_i$, with initial and nonlinear coupled boundary conditions. This equation is obtained from the local mass conservation law for fluxes given by a sum of the diffusional and Darken drift terms, and the Vegard rule. The authors prove the existence, uniqueness, nonnegativity and estimates of the global in time weak solutions in suitable Sobolev spaces for the above problem. Some examples of physical problems and numerical experiments are finally presented.
scientific article; zbMATH DE number 7051675