×

zbMATH — the first resource for mathematics

Generalized MHD equations. (English) Zbl 1057.35040
The paper deals with the \(d\)-dimensional generalized MHD (GMHD) equations \[ \begin{aligned} &\partial_{t}u + u\cdot \nabla u =-\nabla P + b \cdot \nabla b - \nu (-\Delta)^{\alpha} u,\\ &\partial_{t}b + u \cdot \nabla b= b \cdot \nabla u - \eta (-\Delta)^{\beta} b, \end{aligned} \] where the fractional power \((-\Delta)^{\alpha}\) is defined via a Fourier transform and where the spatial domain \(\Omega\) is either the whole space \(\mathbb R^{d}\) or the Torus \(\mathbb{T}^{d}\). Depending on the parameters \(\nu\), \(\eta\), \(\alpha\) and \(\beta\) the author investigates three major cases: (1) \(\nu >0\) and \(\eta >0\); (2) \(\nu=0\) and \(\eta >0\); (3) \(\nu = \eta = 0\). Depending on these cases, several existence, uniqueness and in particular regularity results are established.

MSC:
35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Beale, J.T.; Kato, T.; Majda, A., Remarks on breakdown of smooth solutions for the 3D Euler equations, Commun. math. phys., 94, 61-66, (1984) · Zbl 0573.76029
[2] Biskamp, D., Nonlinear magnetohydrodynamics, (1993), Cambridge University Press Cambridge
[3] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the navier – stokes equations, Commun. pure appl. math., 35, 771-831, (1982) · Zbl 0509.35067
[4] Caflisch, R.E.; Klapper, I.; Steele, G., Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Commun. math. phys., 184, 443-455, (1997) · Zbl 0874.76092
[5] Chemin, J.-Y., Perfect incompressible fluids, (1998), Clarendon Press Oxford
[6] Constantin, P., Navier – stokes equations and area of interfaces, Commun. math. phys., 129, 241-266, (1990) · Zbl 0725.35080
[7] Constantin, P., Geometric statistics in turbulence, SIAM rev., 36, 73-98, (1994) · Zbl 0803.35106
[8] Constantin, P.; Fefferman, C.; Majda, A., Geometric constraints on potentially singular solutions of the 3-D Euler equations, Commun. partial differential equations, 21, 559-571, (1996) · Zbl 0853.35091
[9] Duvaut, G.; Lions, J.L., Inequations en thermoelasticite et magnetohydrodynamique, Arch. rational mech. anal., 46, 241-279, (1972) · Zbl 0264.73027
[10] Gibbon, J.D.; Ohkitani, K., Singularity formation in a class of stretched solutions of the equations for ideal megneto-hydrodynamics, Nonlinearity, 14, 1239-1264, (2001) · Zbl 1016.76097
[11] Majda, A.; Bertozzi, A., Vorticity and incompressible flow, (2001), Cambridge University Press Cambridge
[12] Mattingly, J.; Sinai, Y., An elementary proof of the existence and uniqueness theorem for the navier – stokes equation, Commun. contemp. math., 1, 497-516, (1999) · Zbl 0961.35112
[13] Priest, E.; Forbes, T., Magnetic reconnection, MHD theory and applications, (2000), Cambridge University Press Cambridge · Zbl 0959.76002
[14] Sermange, M.; Temam, R., Some mathematical questions related to the MHD equations, Commun. pure appl. math., 36, 635-664, (1983) · Zbl 0524.76099
[15] Stein, E., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton, USA · Zbl 0207.13501
[16] Temam, R., Navier – stokes equations, (1979), North-Holland Publishing Company Amsterdam · Zbl 0454.35073
[17] Wu, J., Viscous and inviscid magneto-hydrodynamics equations, J. anal. math., 73, 251-265, (1997) · Zbl 0903.76099
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.