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On the Navier-Stokes equations with variable viscosity in a noncylindrical domain. (English) Zbl 1121.35107
The authors study the classical Navier-Stokes equations in $$\mathbb{R}^n$$. There is a rich literature on this subject. Works of Ladyzhenskaya, Oleinik, Lions, Temam, Tartar, Kobayashi, Miranda, and many others contributed to our knowledge of fluid flow modeled by the Navier-Stokes equations. In this article the authors study a well-known form of these equations:
$\partial u/\partial_t-\nu\Delta u+\Sigma u_j\partial d/\partial x_j=f-\text{grad}\,p,\text{ and }\text{div} \,u=0.$ where $$u$$ is the velocity of the fluid, $$p(x,t)$$ is the pressure. The viscosity is assumed to depend on the velocity. A cylindrical domain is considered $$Q=\Omega\times ]0,T[$$ of $$\mathbb{R}^{n+1}$$, $$u=0$$ on the boundary of $$\Omega_t$$, and initial condition: $$u(x,0)=u_0(x)$$. In 1969 J. L. Lions [Quelques méthodes de résolution des problèmes aux limites non linéaires. (Paris) (1969; Zbl 0189.40603)], proved that weak solutions exist for this system if $$n\leq 4$$, and that they are unique if $$n\leq 3$$. The authors continue to follow Lions’ approach. Inner product spaces and norms are defined, as well as weak solutions based on these definitions depending for $$n\leq 4$$ on properties of the function $$f$$. The authors prove that if $$f\in L^{4/3}$$, and $$u_0\in H(\Omega)$$, then weak solutions exist. The existence proof uses the Faedo version of Galerkin’s approach. An appendix contains a discussion of choices of Hilbert or Banach spaces which can be assigned to $$Q$$.
##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35D05 Existence of generalized solutions of PDE (MSC2000) 35A35 Theoretical approximation in context of PDEs
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##### References:
  Ladyzhenskaya OA, The Mathematical Theory of Viscous Incompressible Flow,, 2. ed. (1989)  Leray J, Journal de Mathematiques Pures et Applliquees pp 331– (1934)  Lions JL, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires (1969)  Lions, JL and Prodi, G. 2003. ”Un théorème d’existence et unicité dans les equations de Navier–Stokes en dimension 2”. Vol. 1, 117Paris: EDP-sciences. C. R. Acad. Sci. Paris, Vol. 248, 1959, pp. 319–321; Ouvres Choisies de Jacques-Louis Lions  Miranda MM, Computational and Applied Mathematics 16 pp 247– (1997)  Tartar L, Partial Differential Equations Models in Oceanography (1999)  Temam R, Navier–Stokes Equations, Theory and Numerical Analysis (1979)
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