We consider the flow of a non-homogeneous viscous incompressible fluid which is known at an initial time. Our purpose is to prove that, when $\Omega$ is smooth enough, there exists a local strong regular solution (which is global for small regular data).
@article{118967, author = {J\'er\^ome Lemoine}, title = {On non-homogeneous viscous incompressible fluids. Existence of regular solutions}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {38}, year = {1997}, pages = {697-715}, zbl = {0940.35153}, mrnumber = {1603698}, language = {en}, url = {http://dml.mathdoc.fr/item/118967} }
Lemoine, Jérôme. On non-homogeneous viscous incompressible fluids. Existence of regular solutions. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 697-715. http://gdmltest.u-ga.fr/item/118967/
Mathematical study of flows of nonhomogeneous fluids (in Russian), Lectures at the University of Novosibirsk, Novosibirsk, U.S.S.R., 1973.
Linear Operators, Interscience, 1958. | Zbl 0635.47003
Some new results for the variable density Navier-Stokes equations, Ann. Fac. Sci. Toulouse Math., Vol II, no. 2, 1993.
Thesis, Blaise Pascal University, France, 1994.
Unique solvability of an initial-and boundary-value problem for viscous incompressible nonhomogeneous fluids, J. Soviet. Math. 9 (1978), 697-749. (1978)
Thesis, Blaise Pascal University, France, 1995.
On Some Problems Connected with Navier-Stokes Equations in Nonlinear Evolution Equations, M.C. Crandall, ed., Academic Press, New York, 1978. | MR 0513812
Mathematical Topics in Fluids Mechanics, Vol. I, Incompressible Models, Clarendon Press, Oxford, 1996. | MR 1422251
Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. IV, Vol. CXLVI, (1987), 65-96. | MR 0916688
Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal. 21 5 (1990), 1093-1117. (1990) | MR 1062395 | Zbl 0702.76039
Solvability of the initial-boundary-value problem for the equations of motion of a viscous compressible fluid, J. Soviet. Math. 14 2 (1980), 1120-1133. (1980) | Zbl 0451.35092
Navier-Stokes Equations, North-Holland (second edition), 1979. | Zbl 1157.35333
Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, 1978. | MR 0503903 | Zbl 0830.46028