Abstract
For the two dimensional steady MHD equations, we prove that Liouville type theorems hold if the velocity is growing fast at infinity. The main obstacle comes from the nonlinear terms, since the vorticity system of the MHD equations has no maximum principle unlike the Navier–Stokes equations. As a corollary, we obtain that all solutions of the 2D Navier–Stokes equations satisfying \(\nabla u\in L^p({\mathbb {R}}^2)\) with \(1<p<\infty \) are constants, which is sharp since there exist some non-trivial linear solutions like the Couette flow in the sense of \(\nabla u\in L^\infty ({\mathbb {R}}^2)\).
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References
Bildhauer, M., Fuchs, M., Zhang, G.: Liouville-type theorems for steady flows of degenerate power law fluids in the plane. J. Math. Fluid Mech. 15(3), 583–616 (2013)
Bogovski, M.E.: Decomposition of \(L_p(\Omega; R^n)\) into the direct sum of subspaces of solenoidal and potential vector fields. Soviet Math. Dokl. 33, 161–165 (1986)
Decaster, A., Iftimie, D.: On the asymptotic behaviour of 2D stationary Navier–Stokes solutions with symmetry conditions. Nonlinearity 30(10), 3951–3978 (2017)
Fuchs, M.: Stationary flows of shear thickening fluids in 2D. J. Math. Fluid Mech. 14(1), 43–54 (2012)
Fuchs, M.: Liouville theorems for stationary flows of shear thickening fluids in the plane. J. Math. Fluid Mech. 14(3), 421–444 (2012)
Fuchs, M., Zhang, G.: Liouville theorems for entire local minimizers of energies defined on the class LlogL and for entire solutions of the stationary Prandtl-Eyring fluid model. Calc. Var. Partial Differ. Equ. 44(1–2), 271–295 (2012)
Fuchs, M., Zhong, X.: A note on a Liouville type result of Gilbarg and Weinberger for the stationary Navier-Stokes equations in 2D. Problems in mathematical analysis No. 60. J. Math. Sci. (N.Y.) 178(6), 695–703 (2011)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, 2nd edn. Steady-state problems. Springer Monographs in Mathematics. Springer, New York (2011)
Galdi, Giovanni P., Grisanti, Carlo R.: Existence and regularity of steady flows for shear-thinning liquids in exterior two-dimensional. Arch. Ration. Mech. Anal. 200(2), 533–559 (2011)
Galdi, G.P., Novotny, A., Padula, M.: On the two-dimensional steady-state problem of a viscous gas in an exterior domain. Pac. J. Math. 179(1), 65–100 (1997)
Gilbarg, D., Weinberger, H.F.: Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(2), 381–404 (1978)
Jin, Bum Ja, Kang, Kyungkeun: Liouville theorem for the steady-state non-Newtonian Navier–Stokes equations in two dimensions. J. Math. Fluid Mech. 16(2), 275–292 (2014)
Koch, G., Nadirashvili, N., Seregin, G., Sverak, V.: Liouville theorems for the Navier–Stokes equations and applications. Acta Mathematica 203, 83–105 (2009)
Korobkov, Mikhail, Pileckas, Konstantin: Russo, Remigio The existence of a solution with finite Dirichlet integral for the steady Navier–Stokes equations in a plane exterior symmetric domain. J. Math. Pures Appl. (9) 101(3), 257–274 (2014)
Pileckas, Konstantin, Russo, Remigio: On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem. Math. Ann. 352(3), 643–658 (2012)
Russo, Antonio: A note on the exterior two-dimensional steady-state Navier–Stokes problem. J. Math. Fluid Mech. 11(3), 407–414 (2009)
Russo, Antonio: On the asymptotic behavior of D-solutions of the plane steady-state Navier–Stokes equations. Pac. J. Math. 246(1), 253–256 (2010)
Wang, W., Wang, Y.: Liouville-type theorems for the stationary MHD equations in 2D. Nonlinearity 32(11), 4483–4505 (2019)
Zhang, G.: A note on Liouville theorem for stationary flows of shear thickening fluids in the plane. J. Math. Fluid Mech. 15(4), 771–782 (2013)
Zhang, G.: Liouville theorems for stationary flows of shear thickening fluids in 2D. Ann. Acad. Sci. Fenn. Math. 40(2), 889–905 (2015)
Acknowledgements
W. Wang was supported by NSFC under Grant 12071054, 11671067 and “the Fundamental Research Funds for the Central Universities”.
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Wang, W. Liouville Type Theorems for the Planar Stationary MHD Equations with Growth at Infinity. J. Math. Fluid Mech. 23, 88 (2021). https://doi.org/10.1007/s00021-021-00615-w
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DOI: https://doi.org/10.1007/s00021-021-00615-w