Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains |
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| Authors: | Michal Beneš Petr Kučera |
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| Affiliation: | 1. +420 22435 2. 4442+420 3. 23333 4. 2732;5. Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague, Prague 6, Czech Republic |
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| Abstract: | In this paper we consider the system of the non‐steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator  and formulate our problem in terms of operator equations. Let  and  be the Fréchet derivative of  at  . We prove that  is one‐to‐one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W2, 2‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data. |
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| Keywords: | Navier– Stokes equations mixed boundary conditions 35D05 35Q30 |
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