We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of \(N\ge 2\) components occupying a bounded region \(\Omega \subset \mathbb {R}^{d},\) \(d\le 3\). Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if \(d\le 2\), we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if \(d=3\), we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If \(d=3\), this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application.

我们考虑一个非线性扩散方程组，对占据有界区域\(\Omega \subset \mathbb {R}^{d},\) 的\(N\ge 2\)组分理想混合物的（等温）相分离进行建模\(d\le 3\)。我们的系统受到系数的恒定迁移率矩阵的影响，这是根据奇异熵生成势和局部毛细管效应给出的自由能函数。我们证明了适定性和正则性结果，概括了 Elliott 和 Luckhaus 获得的结果（IMA Preprint Ser 887，1991）。特别是，如果\(d\le 2\)，我们从（熵）非线性的奇异点导出解的均匀严格分离。那么，即使 \(d=3\)，我们证明了全局（常规）吸引子的存在，并建立了单一均衡解的收敛性。如果\(d=3\)，这种收敛需要渐进严格分离性质的有效性。这项工作构成了涉及多组分系统相行为的扩展三部分研究的第一部分，第二部分解决了非局部毛细管效应的存在，最后一部分涉及此类系统的数值研究以及一些相关的研究。应用。