An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations. (English) Zbl 1530.35235
Summary: We investigate an initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations. The Dirichlet boundary conditions are imposed only on the velocity, while no boundary condition is imposed on the height of the fluid or the magnetic field. We derive a series of a priori estimates for the approximate solution sequences to show that they are Cauchy in a suitable Sobolev space. The local well-posedness in time of strong solutions for the initial-boundary value problem is established by the strong convergence of the approximate solution sequences.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q86 | PDEs in connection with geophysics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76U60 | Geophysical flows |
| 86A05 | Hydrology, hydrography, oceanography |
| 35D35 | Strong solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
rotating shallow water equations; magnetohydrodynamics; initial-boundary value problem; local well-posednessReferences:
| [1] | Benzoni-Gavage, S.; Serre, D., Multi-Dimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications (2007), Oxford University Press: Oxford University Press USA · Zbl 1113.35001 |
| [2] | Bojarevics, V.; Pericleous, K., Nonlinear MHD stability of aluminium reduction cells, (15th Riga and 6th Pamir Conference on Fundamental and Applied MHD (2005)), 87-90 |
| [3] | Bouchut, F.; Lhebrard, X., A 5-wave relaxation solver for the shallow water MHD system, J. Sci. Comput., 68, 92-115 (2016) · Zbl 1342.76130 |
| [4] | Bousquet, A.; Petcu, M.; Shiue, M.; Temam, R.; Tribbia, J., Boundary conditions for limited area models based on the shallow water equations, Commun. Comput. Phys., 14, 664-702 (2013) · Zbl 1373.35232 |
| [5] | Cheng, B.; Tadmor, E., Long-time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations, SIAM J. Math. Anal., 39, 1668-1685 (2008) · Zbl 1152.76054 |
| [6] | Cheng, B.; Tadmor, E., Approximate periodic solutions for the rapidly rotating shallow-water and related equations, (Mahmood, M.; Henderson, D.; Segur, H., Water Waves: Theory and Experiment (2010)), 69-78 |
| [7] | Cheng, B.; Xie, C., On the classical solutions of two dimensional inviscid rotating shallow water system, J. Differ. Equ., 250, 690-709 (2011) · Zbl 1213.35292 |
| [8] | Duan, J.; Tang, H., High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics, J. Comput. Phys., 431, Article 110136 pp. (2021) · Zbl 07511453 |
| [9] | Gilman, P., Magnetohydrodynamic “shallow water” equations for the solar tachocline, Astrophys. J., 544, L79-L82 (2000) |
| [10] | Gu, F.; Lu, Y.; Zhang, Q., Global solutions to one-dimensional shallow water magnetohydrodynamic equations, J. Math. Anal. Appl., 401, 714-723 (2013) · Zbl 1307.35222 |
| [11] | Heng, K.; Spitkovsky, A., Magnetohydrodynamic shallow water waves: linear analysis, Astrophys. J., 703, 1819-1831 (2009) |
| [12] | Hu, Y.; Qian, Y., An initial-boundary value problem for the two-dimensional rotating shallow water equations with axisymmetry, Differ. Integral Equ., 35, 611-640 (2022) · Zbl 1538.35293 |
| [13] | Huang, A.; Petcu, M.; Temam, R., The nonlinear 2d supercritical inviscid shallow water equations in a rectangle, Asymptot. Anal., 93, 187-218 (2015) · Zbl 1326.35275 |
| [14] | Huang, A.; Temam, R., The linearized 2D inviscid shallow water equations in a rectangle: boundary conditions and well-posedness, Arch. Ration. Mech. Anal., 211, 1027-1063 (2014) · Zbl 1293.35247 |
| [15] | Huang, A.; Temam, R., The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction, Commun. Pure Appl. Anal., 13, 2005-2038 (2014) · Zbl 1310.76015 |
| [16] | Huang, A.; Temam, R., The 2D nonlinear fully hyperbolic inviscid shallow water equations in a rectangle, J. Dyn. Differ. Equ., 27, 763-785 (2015) · Zbl 1344.35102 |
| [17] | Karelskya, K.; Petrosyana, A.; Tarasevich, S., Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid on slope in the shallow water approximation, J. Exp. Theor. Phys., 119, 311-325 (2014) |
| [18] | Kroger, T.; Lukacova-Medvidova, M., An evolution Galerkin scheme for the shallow water magnetohydrodynamic equations in two space dimensions, J. Comput. Phys., 206, 122-149 (2005) · Zbl 1087.76065 |
| [19] | Lee, Y.; Munz, C., Advanced hyperbolic divergence cleaning scheme for shallow water magnetohydrodynamics, J. Hyperbolic Differ. Equ., 1, 171-195 (2004) · Zbl 1140.76315 |
| [20] | Li, T. T.; Yu, W., Boundary Value Problem for Quasilinear Hyperbolic Systems (1985), Duke University · Zbl 0627.35001 |
| [21] | Mak, J.; Griffiths, S.; Hughes, D., Shear flow instabilities in shallow-water magnetohydrodynamics, J. Fluid Mech., 788, 767-796 (2016) · Zbl 1338.76136 |
| [22] | Meleshkoa, S.; Dorodnitsynb, V.; Kaptsov, E., Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates, J. Phys. A, Math. Theor., 55, Article 495202 pp. (2022) · Zbl 1527.76085 |
| [23] | Miesch, M.; Gilman, P., Thin-shell magnetohydrodynamic equations for the solar tachocline, Sol. Phys., 220, 287-305 (2004) |
| [24] | Petcu, M.; Temam, R., The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity, Discrete Contin. Dyn. Syst., Ser. S, 4, 209-222 (2011) · Zbl 1219.35003 |
| [25] | Petcu, M.; Temam, R., The one-dimensional shallow water equations with transparent boundary conditions, Math. Methods Appl. Sci., 36, 1979-1994 (2013) · Zbl 1277.35241 |
| [26] | Qamar, S.; Mudasser, S., A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics, Comput. Phys. Commun., 181, 1109-1122 (2010) · Zbl 1454.76110 |
| [27] | Rakotoson, J.; Temam, R.; Tribbia, J., Remarks on the nonviscous shallow water equations, Indiana Univ. Math. J., 57, 2969-2998 (2008) · Zbl 1173.35104 |
| [28] | Rossmanith, J., A wave propagation method with constrained transport for ideal and shallow water magnetohydrodynamics (2002), University of Washington, Ph.D. Thesis |
| [29] | Shiue, M., An initial boundary value problem for one-dimensional shallow water magnetohydrodynamics in the solar tachocline, Nonlinear Anal., 76, 215-228 (2013) · Zbl 1251.76065 |
| [30] | Shiue, M.; Laminie, J.; Temam, R.; Tribbia, J., Boundary value problems for the shallow water equations with topography, J. Geophys. Res., 116, Article C02015 pp. (2011) |
| [31] | Spitkovsky, A.; Levin, Y.; Ushomirsky, G., Propagation of thermonuclear flames on rapidly rotating neutron stars: extreme weather during type I X-ray bursts, Astrophys. J., 566, 1018-1038 (2002) |
| [32] | De Sterck, H., Hyperbolic theory of the “shallow water” magnetohydrodynamics equations, Phys. Plasmas, 8, 3293-3304 (2001) |
| [33] | Touma, R.; Arminjon, P., Central finite volume methods for ideal and shallow water magnetohydrodynamics, Int. J. Numer. Methods Fluids, 67, 155-174 (2011) · Zbl 1432.76296 |
| [34] | Trakhinin, Y., Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics, Z. Angew. Math. Phys., 71, 118 (2020) · Zbl 1442.76154 |
| [35] | Zaqarashvili, T.; Oliver, R.; Ballester, J., Global shallow water magnetohydrodynamic waves in the solar tachocline, Astrophys. J., 691, L41-L44 (2009) |
| [36] | Zikanov, O.; Thess, A.; Davidson, P., A new approach to numerical simulation of melt flows and interface instability in Hall-Heroult cells, Metall. Trans. B, 31, 6, 1541-1550 (2000) |
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.
