We prove the global-in-time existence of weak solutions of the equations of compressible magnetohydrodynamics in three space dimensions with initial data small in L 2 and initial density positive and essentially bounded. A great deal of information concerning partial regularity is obtained: velocity, vorticity, and magnetic field become relatively smooth in positive time (H 1 but not H 2 ) and singularities in the pressure cancel those in a certain multiple of the divergence of the velocity, thus giving concrete expression to conclusions obtained formally from the Rankine-Hugoniot conditions.
We study non-negative solutions to the chemotaxis system [Formula: see text] under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on Ω × [0, ∞)2 with values in [0, ∞) and ℝ2×2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (⋆), in the special case [Formula: see text] reducing to a version of the standard Keller–Segel system with signal consumption, have recently been proposed as a model for swimming bacteria near a surface, with the sensitivity tensor then given by [Formula: see text], reflecting rotational chemotactic motion. It is shown that for any choice of suitably regular initial data (u0, v0) fulfilling a smallness condition on the norm of v0 in L∞(Ω), the corresponding initial-boundary value problem associated with (⋆) possesses a globally defined classical solution which is bounded. This result is achieved through the derivation of a series of a priori estimates involving an interpolation inequality of Gagliardo–Nirenberg type which appears to be new in this context. It is next proved that all corresponding solutions approach a spatially homogeneous steady state of the form (u, v) ≡ (μ, κ) in the large time limit, with μ := fΩu0 and some κ ≥ 0. A mild additional assumption on the positivity of f is shown to guarantee that κ = 0. Finally, numerical solutions are presented which suggest the occurrence of wave-like solution behavior.
We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in R 3 . We establish a blow-up criterion for the local strong solutions in terms of the density and magnetic field. Namely, if the density is away from vacuum (ρ = 0) and the concentration of mass (ρ = ∞) and if the magnetic field is bounded above in terms of L ∞ -norm, then a local strong solution can be continued globally in time.
We prove the global existence of classical solutions to a class of forced drift-diffusion equations with L 2 initial data and divergence free drift velocity {u ν }ν ≥ 0 ⊂ L ∞ t BM O −1 x , and we obtain strong convergence of solutions as the viscosity ν vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic {MG ν } ν≥0 equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core. We prove the existence of a compact global attractor {A ν } ν≥0 in L 2 (T 3 ) for the MG ν equations including the critical equation where ν = 0. Furthermore, we obtain the upper semicontinuity of the global attractor as ν vanishes.Mathematics Subject Classification (2010). 76D03, 35Q35, 76W05.Keywords. active scalar equations, vanishing viscosity limit, global attractor.The unknowns are u(t, x) the velocity, b(t, x) the magnetic field (both vector valued) and θ(t, x) the scalar (temperature field of the fluid). P is the sum of the fluid and magnetic pressures, and the Cartesian unit vectors are given by e 1 , e 2 and e 3 . The physical forces governing this system are the Coriolis force, the Lorentz force and gravity acting via buoyancy, while the equation for the temperature is driven by a smooth function S(x) that represents the external forcing of the MHD system. The non-dimensional parameters in (1.1)-(1.4) are R 0 the Rossby number, R m the magnetic Reynolds number, ν a (non-dimensional) viscosity and κ a (non-dimensional) thermal diffusivity. Moffatt and Loper argue that for the geophysical context they are modelling, all these parameters are
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