Geodesic
An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions
Applications of Mathematics, Tome 45 (2000) no. 2, pp. 81-98
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The evolution Boussinesq equations describe the evolution of the temperature and velocity fields of viscous incompressible Newtonian fluids. Very often, they are a reasonable model to render relevant phenomena of flows in which the thermal effects play an essential role. In the paper we prescribe non-Dirichlet boundary conditions on a part of the boundary and prove the existence and uniqueness of solutions to the Boussinesq equations on a (short) time interval. The length of the time interval depends only on certain norms of the given data. In the proof we use a fixed point theorem method in Sobolev spaces with non-integer order derivatives. The proof is performed for Lipschitz domains and a wide class of data.
The evolution Boussinesq equations describe the evolution of the temperature and velocity fields of viscous incompressible Newtonian fluids. Very often, they are a reasonable model to render relevant phenomena of flows in which the thermal effects play an essential role. In the paper we prescribe non-Dirichlet boundary conditions on a part of the boundary and prove the existence and uniqueness of solutions to the Boussinesq equations on a (short) time interval. The length of the time interval depends only on certain norms of the given data. In the proof we use a fixed point theorem method in Sobolev spaces with non-integer order derivatives. The proof is performed for Lipschitz domains and a wide class of data.
DOI : 10.1023/A:1022224328555
Classification : 35Q30, 35Q35
Keywords: Boussinesq equations; non-Dirichlet boundary conditions; Sobolev space with non-integer order derivatives; Schauder principle 
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Skalák, Zdeněk; Kučera, Petr. An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions. Applications of Mathematics, Tome 45 (2000) no. 2, pp. 81-98. doi: 10.1023/A:1022224328555

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