The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations
نویسندگان
چکیده
Here u, b describe the flow velocity vector and the magnetic field vector respectively, p is a scalar pressure, ν > 0 is the kinematic viscosity and η > 0 is the magnetic diffusivity, while u0 and b0 are the given initial velocity and initial magnetic field respectively, with ∇ · u0 = ∇ · b0 = 0. If ν = η = 0, (1.1) is called the ideal MHD equations. Using the standard energy method, it can be easily proved that for given initial data (u0, b0) ∈ Hs(R3) with s > 12 , there exists a positive time T = T (‖(u0, b0)‖Hs) and a unique smooth solution (u(t, x), b(t, x)) on [0, T ) to the MHD equations satisfying
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