Inhomogeneous incompressible Navier-Stokes equations on thin domains. (English) Zbl 1442.35312
Summary: We consider the inhomogeneous incompressible Navier-Stokes equation on thin domains \(\mathbb{T}^2 \times \epsilon \mathbb{T}\), \(\epsilon \rightarrow 0\). It is shown that the weak solutions on \(\mathbb{T}^2 \times \epsilon \mathbb{T}\) converge to the strong/weak solutions of the 2D inhomogeneous incompressible Navier-Stokes equations on \(\mathbb{T}^2\) as \(\epsilon \rightarrow 0\) on arbitrary time interval.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35D30 | Weak solutions to PDEs |
| 35D35 | Strong solutions to PDEs |
Keywords:
inhomogeneous incompressible Navier-Stokes equation; thin domain limit; dimensional reduction; relative energyReferences:
| [1] | Antonsev, S.; Kazhikhov, A.; Monakov, V., Boundary Value Problems in Mechanics of Nonhomogeneous Fluids (1990), Amsterdam: North-Holland, Amsterdam · Zbl 0696.76001 |
| [2] | Bella, P.; Feireisl, E.; Novotný, A., Dimension reduction for compressible viscous fluids, Acta Appl. Math., 134, 111-121 (2014) · Zbl 1306.35093 · doi:10.1007/s10440-014-9872-5 |
| [3] | Caggio, M., Donatelli, D., Nečasová, Š., Sun, Y.: Low Mach number limit on thin domains. arXiv:1901.09530 · Zbl 1433.35261 |
| [4] | Feireisl, E.; Novotný, A., Singular Limits in Thermodynamics of Viscous Fluids (2009), Basel: Birkhäuser Verlag, Basel · Zbl 1176.35126 |
| [5] | Feireisl, E.; Jin, BJ; Novotný, A., Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 14, 4, 717-730 (2012) · Zbl 1256.35054 · doi:10.1007/s00021-011-0091-9 |
| [6] | Germain, P., Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier-Stokes system, J. Anal. Math., 105, 169-196 (2008) · Zbl 1198.35179 · doi:10.1007/s11854-008-0034-4 |
| [7] | Iftimie, D.; Raugel, G., Some results on the Navier-Stokes equations in thin 3D domains, J. Differ. Equ., 169, 2, 281-331 (2001) · Zbl 0972.35085 · doi:10.1006/jdeq.2000.3900 |
| [8] | Liao, X., On the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin domain, Differ. Integr. Equ., 29, 1-2, 167-182 (2016) · Zbl 1363.35276 |
| [9] | Lions, P.L.: Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1996) · Zbl 0866.76002 |
| [10] | Moise, I.; Temam, R.; Ziane, M., Asymptotic analysis of the Navier-Stokes equations in thin domains, Dedicated to Olga Ladyzhenskaya, Topol. Methods Nonlinear Anal., 10, 2, 249-282 (1997) · Zbl 0957.35108 · doi:10.12775/TMNA.1997.032 |
| [11] | Montgomery-Smith, S., Global regularity of the Navier-Stokes equation on thin three-dimensional domains with periodic boundary conditions, Electron. J. Differ. Equ., 11, 1-19 (1999) · Zbl 0927.34049 · doi:10.1023/A:1021889401235 |
| [12] | Maltese, D.; Novotný, A., Compressible Navier-Stokes equations on thin domains, J. Math. Fluid Mech., 16, 571-594 (2014) · Zbl 1308.35170 · doi:10.1007/s00021-014-0177-2 |
| [13] | Raugel, G.; Sell, G., Navier Stokes Equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Am. Math. Soc., 6, 503-568 (1993) · Zbl 0787.34039 |
| [14] | Simon, J., Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21, 5, 1093-1117 (1990) · Zbl 0702.76039 · doi:10.1137/0521061 |
| [15] | Temam, R.; Ziane, M., Navier Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differ. Equ., 1, 499-546 (1996) · Zbl 0864.35083 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
