Time Decay Estimates of Solutions to the Magnetic Bénard System in a Half-Space

Abstract

This paper aims to derive the time decay estimates for both weak and strong solutions to the incompressible magnetic Bénard system in a half-space. To address the technical challenges posed by the boundary effects in unbounded domains, we primarily employ $L^{q} - L^{r}$ estimates, together with properties of the Stokes operator and a suitable decomposition of the nonlinear terms. We establish the $L^{2}$ -estimates of weak solutions and obtain the $L^{q} (1 \leq q \leq \infty)$ -estimates of strong solutions and their first three derivatives. In addition, if the given initial data belongs to a suitable weighted space, more optimal decay effects are established.

Keywords

magnetic Bénard system weak and strong solutions asymptotic behavior half-space

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References

Abidi

Gui

G. L.

Zhang

(2011). On the decay and stability of global solutions to the 3D inhomogeneous Navier–Stokes equations. Communications on Pure and Applied Mathematics, 64, 832–881. 10.1002/cpa.20351

Abidi

Hmidi

Keraani

(2011). On the global regularity of axisymmetric Navier–Stokes–Boussinesq system. Discrete and Continuous Dynamical Systems, 29, 737–756. 10.3934/dcds.2011.29.737

Bae

H. O.

Choe

H. J.

(2001). Decay rate for the incompressible flows in half spaces. Mathematische Zeitschrift, 238, 799–816. 10.1007/s002090100276

Borchers

Miyakawa

(1988).

L^{2}

decay for the Navier–Stokes flow in half spaces. Mathematische Annalen, 282, 139–155. 10.1007/BF01457017

Brandolese

(2004). Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier–Stokes equations. Revista Matematica Iberoamericana, 20, 223–256. 10.4171/rmi/387

Brandolese

Schonbek

M. E.

(2012). Large time decay and growth for solutions of a viscous Boussinesq system. Transactions of the American Mathematical Society, 364, 5057–5090. 10.1090/S0002-9947-2012-05432-8

Cannone

Planchon

Schonbek

M. E.

(2000). Strong solutions to the incompressible Navier–Stokes equations in the half-space. Communications in Partial Differential Equations, 25, 903–924. 10.1080/03605300008821536

Chen

Wang

(2024). The

L^{1}

-asymptotic behavior of strong solutions to the incompressible magneto-hydrodynamic equations in half-spaces. Applied Mathematics Letters, 150, Article 108966. 10.1016/j.aml.2023.108966

Fan

Sun

J. Z.

Tang

(2019). Uniform global strong solutions of the 2D density-dependent incompressible magnetic Bénard problem in a bounded domain. Computers & Mathematics with Applications (Oxford, England), 77, 494–500. 10.1016/j.camwa.2018.09.052

10.

Fujigaki

Miyakawa

(2001). Asymptotic profiles of non stationary incompressible Navier–Stokes flows in the half-space. Methods and Applications of Analysis, 8, 121–158. 10.4310/MAA.2001.v8.n1.a6

11.

Han

(2010). Asymptotic behavior for the Stokes flow and Navier–Stokes equations in half spaces. Journal of Differential Equations, 249, 1817–1852. 10.1016/j.jde.2010.05.021

12.

Han

(2011). Decay results of solutions to the incompressible Navier–Stokes flows in a half space. Journal of Differential Equations, 250, 3937–3959. 10.1016/j.jde.2010.11.018

13.

Han

(2012). Weighted decay properties for the incompressible Stokes flow and Navier–Stokes equations in a half space. Journal of Differential Equations, 253, 1744–1778. 10.1016/j.jde.2012.06.007

14.

Han

(2022). Decay properties for the incompressible Navier–Stokes flows in a half space. Proceedings of the Royal Society of Edinburgh Section A, 152, 1509–1532. 10.1017/prm.2021.63

15.

Han

(2024).

L^{1}

-decay of higher-order norms of solutions to the Navier–Stokes equations in the upper-half space. Mathematische Zeitschrift, 308, Article 23. 10.1007/s00209-024-03578-6

16.

Han

Schonbek

M. E.

(2014). Large time decay properties of solutions to a viscous Boussinesq system in a half space. Advances in Differential Equations, 19, 87–132. 10.57262/ade/1384278133

17.

Xin

(2005). On the regularity of weak solutions to the magnetohydrodynamic equations. Journal of Differential Equations, 213, 235–254. 10.1016/j.jde.2004.07.002

18.

Hmidi

Rousset

(2010). Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 27, 1227–1246. 10.1016/j.anihpc.2010.06.001

19.

Huang

(2023). Stability and exponential decay of the magnetic Bénard system with horizontal dissipation. Journal of Mathematical Analysis and Applications, 519, Article 126767. 10.1016/j.jmaa.2022.126767

20.

Kato

(1984). Strong

L^{p}

-solutions of the Navier–Stokes equation in

R^{m}

, with applications to weak solutions. Mathematische Zeitschrift, 187, 471–480. 10.1007/BF01174182

21.

Kozono

Ogawa

(1992). Some

L^{p}

estimate for the exterior Stokes flow and an application to the nonstationary Navier–Stokes equations. Indiana University Mathematics Journal, 41, 789–808. 10.1512/iumj.1992.41.41041

22.

Lemarié

P.-G.

(2002). Recent developments in the Navier–Stokes problem. CRC Press.

23.

Leray

(1934). Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica, 63, 193–248. 10.1007/BF02547354

24.

Wang

(2020). Uniform regularity estimates of solutions to three dimensional incompressible magnetic Bénard equations with Navier-slip type boundary conditions in half space. Nonlinear Analysis, 199, Article 111932. 10.1016/j.na.2020.111932

25.

Tan

(2022). Global existence and decay estimates of solutions to the MHD–Boussinesq system with stratification effects. Nonlinearity, 35, 6067–6097. 10.1088/1361-6544/ac93e0

26.

(2024). Large time behavior of the solutions to 3D incompressible MHD system with horizontal dissipation or horizontal magnetic diffusion. Calculus of Variations and Partial Differential Equations, 63, Article 43. 10.1007/s00526-023-02647-8

27.

Liu

(2019). Asymptotic behavior of solutions to the nonstationary magneto-hydrodynamic equations. Nonlinear Analysis, 185, 29–48. 10.1016/j.na.2019.02.030

28.

Liu

(2020). Algebraic decay of weak solutions to 3D Navier–Stokes equations in general unbounded domains. Journal of Mathematical Analysis and Applications, 491, Article 124300. 10.1016/j.jmaa.2020.124300

29.

Liu

Han

(2024). Long time behavior for the two-dimensional magnetohydrodynamic flows in a general domain. Journal of Mathematical Physics, 65, Article 111501. 10.1063/5.0223634

30.

Liu

(2015). Large time behavior for the incompressible magnetohydrodynamic equations in half-spaces. Mathematical Methods in the Applied Sciences, 38, 2376–2388. 10.1002/mma.3227

31.

Luo

Sun

(2025). Global solution of 3D MHD–Boussinesq system with mixed partial dissipation and thermal damping. Discrete and Continuous Dynamical Systems—Series B, 30, 1029–1049. 10.3934/dcdsb.2024118

32.

Pan

(2022). Global regularity of solutions for the 3D non-resistive and non-diffusive MHD–Boussinesq system with axisymmetric data. Acta Applicandae Mathematicae, 180, Article 6. 10.1007/s10440-022-00508-8

33.

Regmi

Sharma

(2019). Regularity criteria on the 2D anisotropic magnetic Bénard equations. Journal of Mathematical Study, 52, 60–74. 10.4208/jms.v52n1.19.06

34.

Rionero

(1998). On the magnetic Bénard problem. Rendiconti del Circolo Matematico di Palermo, 1998(2 Suppl), 419–426.

35.

Schonbek

M. E.

(1992). Asymptotic behavior of solutions to the three-dimensional Navier–Stokes equations. Indiana University Mathematics Journal, 41, 809–823. 10.1512/iumj.1992.41.41042

36.

Schonbek

M. E.

(1995). The Fourier splitting method. In Advances in geometric analysis and continuum mechanics (pp. 269–274). International Press.

37.

Schonbek

M. E.

Wiegner

(1996). On the decay of higher-order norms of the solutions of Navier–Stokes equations. Proceedings of the Royal Society of Edinburgh Section A, 126, 677–685. 10.1017/S0308210500022976