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Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity.  I. (English. Russian original) Zbl 0938.35121
Sib. Math. J. 40, No. 2, 351-362 (1999); translation from Sib. Mat. Zh. 40, No. 2, 408-420 (1999).
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 $\mathbf\Phi_{\gamma}(s) = (1 + s)\log^{\gamma}(1+s).$

##### MSC:
 35Q30 Navier-Stokes equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35B45 A priori estimates in context of PDEs 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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##### References:
  A. E. Mamontov, ”Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. I,” Sibirsk. Mat. Zh.,40, No. 2, 408–420 (1999). · Zbl 0938.35121  J. Simon, ”Compact sets in the spaceL p(0,T; B),” Ann. Mat. Pura Appl., (IV)-VCXLVI, 65–96 (1987). · Zbl 0629.46031  R. J. DiPerna and P. L. Lions, ”Ordinary differential equations, transport theory and Sobolev spaces,” Invent. Math.,98, 511–547 (1989). · Zbl 0696.34049  M. A. Krasnosel’skiî and Ya. B. Rutitskiî, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).  A. E. Mamontov, Orlicz Spaces in the Existence problem of Global Solutions to Viscous Compressible Nonlinear Fluid Equations [Preprint, No. 2-96], Inst. Gidrodinamiki (Novosibirsk), Novosibirsk (1996).  A. Kufner, S. Fučik and O. John, Function Spaces, Academia, Prague (1977).
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