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The equation of potential flows of a compressible viscous fluid at small Reynolds numbers: Existence, uniqueness, and stabilization of solutions. (English. Russian original) Zbl 0806.76077
Sib. Math. J. 34, No. 3, 457-467 (1993); translation from Sib. Mat. Zh. 34, No. 3, 70-80 (1993).
We consider the boundary value problem for the model system of equations \(\mu\Delta \vec u+(\lambda+ \mu)\nabla (\text{div } \vec u)- \nabla p=0\), \({{\partial\rho} \over {\partial t}}+ \text{div}(\rho\vec u)=0\), \(p=p(\rho)\). From a mechanical standpoint, such an approximation to the Navier-Stokes equations is justified when the Reynolds number is small, i.e. in the case of highly viscous fluids. Our aim is to prove solvability for the problem in various functional classes, establish uniqueness for a solution under certain assumptions on smoothness, and prove stabilization for a solution at infinite time.

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] L. I. Sedov, Continuum Mechanics. Vol. 1 [in Russian], Nauka, Moscow (1970). · Zbl 0224.73002
[2] L. G. Loîtsyanskiî, Fluid Mechanics [in Russian], Nauka, Moscow (1970).
[3] J. Nash, ?Le probleme de Cauchy pour les equations differentielles d’un fluide general,? Bull. Soc. Math. France,90, No. 4, 487-497 (1962). · Zbl 0113.19405
[4] V. A. Solonnikov, ?The solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid?, in: Studies in Linear Operators and Theory of Functions [in Russian], (Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.,56) 1976, pp. 128-142. · Zbl 0338.35078
[5] A. Matsumura and T. Nishida, ?The initial value problem for the equations of motion of viscous and heat-conductive gases,? J. Math. Kyoto Univ.,20, No. 1, 67-104 (1980). · Zbl 0429.76040
[6] C. Bernardi and O. Pironneau, ?On the shallow water equations at low Reynolds numbers,? Comm. Partial Differential Equations,16, No. 1, 59-104 (1991). · Zbl 0723.76033
[7] V. I. Yudovich, ?Two-dimensional nonstationary problem for an ideal incompressible fluid flowing through a given domain,? Mat. Sb.,64, No. 4, 562-588 (1964).
[8] V. I. Yudovich, The Linearization Method in Hydrodynamic Stability Theory [in Russian], Rostov Univ., Rostov-on-Don (1984). · Zbl 0553.76038
[9] T. Kato, ?On classical solutions of the two-dimensional nonstationary Euler equations,? Arch. Rational Mech. Anal.,25, No. 3, 188-200 (1967). · Zbl 0166.45302
[10] V. I. Yudovich, ?Nonstationary flows of an ideal incompressible fluid,? Zh. Vychisl. Mat. i Mat. Fiz.,3, No. 6, 1032-1066 (1963).
[11] I. N. Vekua, Generalized Analytic Functions [in Russian], Fizmatgiz, Moscow (1959). · Zbl 0092.29703
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