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On the vanishing viscosity in the Cauchy problem for the equations of a nonhomogeneous incompressible fluid

Published online by Cambridge University Press:  18 May 2009

Shigeharu Itoh
Shigeharu Itoh
Affiliation:
Department of Mathematics, Faculty of Education, Hirosaki University, Hirosaki 036, Japan
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Let us consider the Cauchy problem

in QT= ℝ3 × [0, T], where f(x, t), ρ0(x) and v0(x) are given, while the density ρ(x, t), the velocity vector v(x, t)= (υ1(x, t), υ2(x, t), υ3(x, t)) and the pressure p(x, t) are unknowns. The viscosity coefficient μ is assumed to be nonnegative. In these equations, the pressure p is automatically determined (up to a function of t) by ρ and v, namely, by solving the equation

Thus we mention (ρ, v) when we talk about the solution of (1.1:μ).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 1 , January 1994 , pp. 123 - 129
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Antontsev, S. N., Kazhikhov, A. V. and Monakhov, V. N., Boundary value problems in mechanics of nonhomogeneous fluids(North-Holland, 1990).Google Scholar
2.Ebin, D. and Marsden, J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. 92 (1970), 102–163.CrossRefGoogle Scholar
3.Ladyzhenskaya, O. A., The mathematical theory of viscous incompressible flow (Gordon and Breach, New York, English translation, second edition, 1969).Google Scholar
4.Ladyzhenskaya, O. A., Solvability in the small of nonstationary problems for incompressible ideal and viscous fluids and the case of vanishing viscosity, J. Soviet Math. 1 (1973), 441–451.CrossRefGoogle Scholar