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Global solutions of multidimensional approximate Navier-Stokes equations of a viscous gas. (Russian, English) Zbl 1037.35057
Sib. Mat. Zh. 44, No. 2, 389-401 (2003); translation in Sib. Math. J. 44, No. 2, 311-321 (2003).
The author studies the motion of a barotropic viscous gas in a bounded domain \(Q = \Omega\times (0,T)\), \(\Omega\subset \mathbb R^n\), \(n = 2,3\), with smooth boundary \(\partial\Omega \in C^2\) which is governed in the Stokes approximation by the equation \[ \bar{\rho}\vec u_t = \mu\Delta\vec u + (\mu + \lambda)\nabla\,(\text{div }\vec{u})- \nabla\,P \] with initial-boundary value conditions \[ \begin{gathered} \vec u = 0,\quad x\in\partial\Omega, \;t \in [0,T], \\ \left.\vec u\right| _{t = 0} = \vec u_0(x),\quad \left.\rho\right| _{t = 0} = \rho_0(x) \geq 0,\quad x\in\Omega, \\ P(\rho) = c^2\rho^{\gamma},\quad c = \text{const },\quad \gamma \geq 1. \end{gathered} \] The main result reads as follows: Under the assumptions that \(\vec u_0 \in L^2(\Omega)\), \(\rho_0 \in L^{\gamma_0}(\Omega)\), and \(\gamma_0 = \max\{2,\gamma\}\), the above-stated problem admits a weak solution.
MSC:
35Q30 Navier-Stokes equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35B45 A priori estimates in context of PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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