On some modification Navier-Stokes equations
The author considers the modified Navier-Stokes system, proposed by \textit{J.-L. Lions} [Quelques méthodes de résolution des problèmes aux limites non linéares. Paris: Dunod (1969; Zbl 0189.40603)]: \[ \partial_t u_j- \mu \sum^n_{i=1} \partial_i(| u_j|^{p_i-2} \partial_i u_j)+ \sum^n_{i=1} u_i\partial_i u_j+ \partial_j p= h_j\quad\text{in }\Omega\times [0,T], \] \[ \text{div\,}u= 0\quad\text{in }\Omega\times [0,T]. \] For a bounded domain \(\Omega\subset\mathbb{R}^n\) \((n\in\mathbb{N})\) and for \(p_i\geq \max\{2,3,-2/n\}\) global solvability of the initial-boundary value problem (with the Dirichlet condition) is proved. For \(p_i\geq 4\) and for isotropic nonlinearity the global uniqueness of the solution is established.
scientific article; zbMATH DE number 1879979