Solutions of Navier-Stokes Equations with Coriolis Force in the Rotational Framework
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Abstract
In this article, for 0 ≤ 𝑚 < ∞ and the index vectors 𝑞 = (𝑞1,𝑞2,𝑞3), 𝑟 = (𝑟1,𝑟2,𝑟3) where 1 ≤ 𝑞𝑖 ≤ ∞, 1 < 𝑟𝑖 < ∞ and 1 ≤ 𝑖 ≤ 3, we study new results of Navier-Stokes equations with Coriolis force in the rotational framework in mixed-norm Sobolev–Lorentz spaces 𝐻̇^𝑚,𝑟,𝑞(ℝ³), which are more general than the classical Sobolev spaces. We prove the existence and uniqueness of solutions to the Navier–Stokes equations (NSE) under Coriolis force in the spaces 𝐿∞([0, T]; 𝐻̇^𝑚,𝑟,𝑞) by using topological arguments, the fixed point argument and interpolation inequalities. Our work is based on the paper [2] and extends some results of it to gain new results of the Navier-Stokes equations in the rotational framework.
Keywords
Navier-Stokes Equations, Coriolis Force, Rotation Framework
Article Details
References
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[11] Sunggeun Lee, Shyn–Kun Ryi, Hankwon Lim; Solutions of Navier-Stokes equation with coriolis force, Advances in Mathematical Physics, volume 2017.
[12] J. Bergh, L. Löfström, Interpolation theory, Spaces. Springer, Berlin (1976).
[2] T. Kato, H. Fujita, On the non-stationary Navier-Stokes system, Rend. Semin. Mat. Univ. Padova 32 (1962) 243-260.
[3] M. Cannone, Y. Meyer, Littlewood-Paley decomposition and the Navier-Stokes equations, Methods Appl. Anal. 2 (1995) 307-319.
[4] Y. Giga, Solutions of semilinear parabolic equations in p L and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations 62 (1986) 186-212.
[5] M. Cannone, Ondelets, Paraproduits et Navier-Stokes, Diderot Editeur, Paris, 1995.
[6] R. Farwig, H. Sohr, W. Varnhorn, Besov space regularity conditions for weak solutions of the Navier-Stokes equations, J. Math. Fluid Mech. 16 (2014) 307-320.
[7] T. Kato, G. Ponce, The Navier-Stokes equations with weak initial data, Int. Math. Res. Not. 10. (1994) 435-444.
[8] D.Q. Khai, N.M. Tri, Solutions in mixed-norm Sobolev-Lorentz spaces to the initial value problem for the Navier-Stokes equations, J. Math. Anal. Appl. 417 (2014) 814-833.
[9] D.Q. Khai, N.M. Tri, Well-posedness for the Navier-Stokes with data in homogeneous Sobolev-Lorentz spaces, Nonlinear Analysis 149 (2017) 130-145.
[10] P.G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman and Hall/CRC Research Notes in Mathematics, vol. 431, Chapman and Hall/CRC, Boca Raton, FL, (2002).
[11] Sunggeun Lee, Shyn–Kun Ryi, Hankwon Lim; Solutions of Navier-Stokes equation with coriolis force, Advances in Mathematical Physics, volume 2017.
[12] J. Bergh, L. Löfström, Interpolation theory, Spaces. Springer, Berlin (1976).