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On partial regularity for the 3D nonstationary Hall magnetohydrodynamics equations on the plane. (English) Zbl 1336.35287
The purpose of this paper is to study the three-dimensional non-stationary Hall magnetohydrodynamics equation, with an extra quadratically nonlinear Hall term, with homogeneous Dirichlet boundary conditions. The main theorem characterizes the existence of a globally weak solution for an approximate system with given initial conditions. The proof consists of several vector analysis calculations and uses Sobolev embedding, Caccioppoli, Cauchy-Schwartz, Poincaré, Hölder’s and Youngs inequalities, and the Vitali convergence theorem.

MSC:
35Q35 PDEs in connection with fluid mechanics
35Q85 PDEs in connection with astronomy and astrophysics
76W05 Magnetohydrodynamics and electrohydrodynamics
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
35D30 Weak solutions to PDEs
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[1] M. Acheritogaray, P. Degond, A. Frouvelle, and J.-G. Liu, Kinetic formulation and global existence for the Hall-magnetohydrodynamic system, Kinet. Relat. Models, 4 (2011), pp. 901–918. · Zbl 1251.35076
[2] L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), pp. 771–831. · Zbl 0509.35067
[3] D. Chae, P. Degond, and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), pp. 555–565.
[4] D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256 (2014), pp. 3835–3858. · Zbl 1295.35122
[5] D. Chae and M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), pp. 3971–3982. · Zbl 1291.35212
[6] D. Chae and J. Wolf, On partial regularity for the steady Hall-magnetohydrodynamics system, Comm. Math. Phys., 339 (2015), pp. 1147–1166. · Zbl 1328.35165
[7] E. Dumas and F. Sueur, On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-magnetohydrodynamic equations, Comm. Math. Phys., 330 (2014), pp. 1179–1225. · Zbl 1294.35094
[8] J. Fan, S. Huang, and G. Nakamura, Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations, Appl. Math. Lett., 26 (2013), pp. 963–967. · Zbl 1315.35164
[9] T. Forbes, Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62 (1991), pp. 15–36.
[10] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. 105, Princeton University Press, Princeton, NJ, 1983. · Zbl 0516.49003
[11] H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Phys. D, 208 (2005), pp. 59–72. · Zbl 1154.76392
[12] O. A. Ladyzehnskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), pp. 356–387. · Zbl 0954.35129
[13] M. J. Lighthill, Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. Trans. Roy. Soc. London Ser. A, 252 (1960), pp. 397–430. · Zbl 0097.20806
[14] F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), pp. 241–257. · Zbl 0958.35102
[15] F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), pp. 1–23.
[16] H. Miura and D. Hori, Hall effects on local structure in decaying MHD turbulence, J. Plasma Fusion Res., 8 (2009), pp. 73–76.
[17] J. M. Polygiannakis and X. Moussas, A review of magneto-vorticity induction in Hall-MHD plasmas, Plasma Phys. Control Fusion, 43 (2001), pp. 195–221.
[18] G. D. Prato, Spazi \(\mathcal{L}^{p,θ}(Ω,δ)\) e loro proprietà, Ann. Mat. Pura Appl., 69 (1965), pp. 383–392. · Zbl 0145.16207
[19] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), pp. 535–552. · Zbl 0325.35064
[20] D. Shalybkov and V. Urpin, The Hall effect and the decay of magnetic fields, Astronom. Astrophys., 321 (1997), pp. 685–690.
[21] A. N. Simakov and L. Chacón, Quantitative, comprehensive, analytical model for magnetic reconnection in Hall-magnetohydrodynamics, Phys. Rev. Lett., 101 (2008), 105003.
[22] M. Wardle, Star formation and the Hall effect, Astrophys. Space Sci., 292 (2004), pp. 317–323.
[23] J. Wolf, Regularität schwacher Lösungen elliptischer und parabolischer Systeme partieller Differentialgleichungen mit Entartung. Der fall \(1<p<2\), Dissertation, Humboldt-Universität zu Berlin, Berlin, Germany, 2001.
[24] J. Wolf, On the local regularity of suitable weak solutions to the generalized Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 61 (2015), pp. 149–171. · Zbl 1323.35135
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