Inhomogeneous Incompressible Navier–Stokes Equations on Thin Domains

Abstract

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.

Keywords

Inhomogeneous incompressible Navier–Stokes equation / Thin domain limit / Dimensional reduction / Relative energy

References

[1] Antonsev S, Kazhikhov A, Monakov V. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. 1990 Amsterdam: North-Holland

[2] Bella P, Feireisl E, Novotný A. Dimension reduction for compressible viscous fluids. Acta Appl. Math.. 2014, 134 111-121

[3] Caggio, M., Donatelli, D., Nečasová, Š., Sun, Y.: Low Mach number limit on thin domains. arXiv:1901.09530

[4] Feireisl E, Novotný A. Singular Limits in Thermodynamics of Viscous Fluids. 2009 Basel: Birkhäuser Verlag

[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.. 2012, 14 4 717-730

[6] Germain P. Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier–Stokes system. J. Anal. Math.. 2008, 105 169-196

[7] Iftimie D, Raugel G. Some results on the Navier–Stokes equations in thin 3D domains. J. Differ. Equ.. 2001, 169 2 281-331

[8] Liao X. On the strong solutions of the inhomogeneous incompressible Navier–Stokes equations in a thin domain. Differ. Integr. Equ.. 2016, 29 1–2 167-182

[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)

[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.. 1997, 10 2 249-282

[11] Montgomery-Smith S. Global regularity of the Navier–Stokes equation on thin three-dimensional domains with periodic boundary conditions. Electron. J. Differ. Equ.. 1999, 11 1-19

[12] Maltese D, Novotný A. Compressible Navier–Stokes equations on thin domains. J. Math. Fluid Mech.. 2014, 16 571-594

[13] Raugel G, Sell G. Navier Stokes Equations on thin 3D domains I: Global attractors and global regularity of solutions. J. Am. Math. Soc.. 1993, 6 503-568

[14] Simon J. Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal.. 1990, 21 5 1093-1117

[15] Temam R, Ziane M. Navier Stokes equations in three-dimensional thin domains with various boundary conditions. Adv. Differ. Equ.. 1996, 1 499-546

Funding

National Natural Science Foundation of China(11571167)

National Natural Science Foundation of China(11771206)