Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain

This paper considers a fairly wide range of issues related to the solvability of the initial boundary value problem of the Navier–Stokes equations with distributed parameters on the net like region of the space ${\rm {\mathcal R}}^{n} $ ($n\ge 2$). The authors here develop an idea, advanced in their work for the case of $n=1$ (the problems with distributed parameters on the graph), in the direction of the dimension increase $n$ and in forming the correct Hadamard conditions for the studied initial boundary value problem. The general scheme of the study is classical: the problem is solved in the functional space which is selected (the space of feasible solutions) and a special basis is formed for it, the problem of approximate solutions is settled by the Faedo–Galerkin method, for which a priori estimates of the energy inequalities type are set and the weak compactness of the family of these solutions is shown based on these estimates. Using non-burdensome conditions, the smoothness of the solution to the time variable is demonstrated. The uniqueness of the weak solution is shown in the particular case $n=2$, a feature quite often encountered in practice. The estimate for the norm of weak solution makes it possible to establish the continuous dependence of the weak solution from the initial data of the problem. The results obtained in this way are of interest to applications in the field of fluid mechanics and related sections of continuum mechanics, namely for the analysis of optimum control dynamics problems of multiphase media. It should be noted that the methods and approaches can be broadly generalized and are applicable to a wide class of nonlinear problems. Refs 20.