Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. ~\text{I} (Q1972188)

The article under review is devoted to studying the properties of solutions to the initial-boundary value problem in a bounded domain \(\Omega\subset\mathbb R^n\) with smooth boundary for equations of motion of a viscous compressible fluid which have the form \[ \frac{\partial\rho}{\partial t} + \text{div} (\rho\vec u)= 0, \quad \frac{\partial(\rho\vec u)}{\partial t} + \text{div} (\rho\vec u\otimes\vec u) = \text{div} \mathbb P' + \rho\vec f. \] Here \(\mathbb P'\) is the stress tensor (a given function of \(\rho\) and \(\vec u\)) and the Stokes axioms imply the representation \[ \mathbb P' = \sum_{k=0}^{n-1} \alpha_k\bigl(\rho, J_s(\mathbb D)\bigr)\mathbb D^k, \] where \(\mathbb D\) is the strain tensor. The case is considered in which the density occurs in the representation for \(\mathbb P'\) only via the pressure that is a linear function of the form \[ \mathbb P' = -\rho\mathbb I + \mathbb P(\vec u) \] and the tensor \(\mathbb P(\vec u)\) represents an arbitrary operator (in general, nonlocal in \(x\)) of \(\vec u\) which acts boundedly and weakly continuously from \(X\) into \(L_{\overline M}(\Omega)\) and satisfies some reasonable additional axioms. Here \(M(\cdot)\) is an N-function, \(\overline M\) is the complementary function, and \[ X = \{\vec u\mid \mathbb D(\vec u)\in L_M(\Omega); \vec u|_{\partial\Omega}=0\}, \qquad\|\vec u\|= \|\mathbb D(\vec u)\|_{L_M(\Omega)}, \] where \(L_M\) denotes the Orlicz space. The main aim is to prove solvability of the problem in the cylinder \(Q_T = \Omega \times (0,T)\) for every \(T>0\). The author also proves existence for a solution when the function \(M\) tends to infinity, with \(|\mathbb D|\) finite, and the solution can be obtained as the limit of solutions with a regular \(M\). In both cases, it can be proven that every weak solution satisfies the energy identity. The proof of the energy identity is effective for those classes of weak solutions in which the product \(\rho|\nabla\otimes\vec u|\) is integrable. In particular, this shows that, for the global existence of a weak solution, it is not sufficient to assume only an apriori estimate, since the latter only yields \(\rho\in L_{\mathbf\Phi}\) with \(\mathbf \Phi(s) = s\log s\). Thus, an additional estimate is required for the denseness. The author obtains such an estimate in the Orlicz space generated by a function of the form \[ \mathbf\Phi_{\gamma}(s) = (1 + s)\log^{\gamma}(1+s). \]