Local existence results in Sobolev spaces for generalized magnetohydrodynamics equations

초록

We consider generalized magnetohydrodynamics (MHD) equations in R-d (d >= 2) with fractional diffusions V Lambda(2 alpha)u and eta Lambda(2 beta)b for the velocity u and magnetic field b, respectively, where alpha, beta, v, eta are nonnegative constants and Lambda(s) = (-Delta)(s/2) is the fractional Laplacian of order s. Using the smoothing effect of Lambda(s) for any s > 0, we establish local existence results for the generalized MHD equations when the initial data (u(0),b(0)) belong to H-1(s) x H-2(s) with possibly different orders s(1) and s(2). Here, H-s = H-s(R-d) denotes the standard L-2-based Sobolev space on R-d of order s. To prove our existence results, we also derive new product and commutator estimates in fractional Sobolev spaces H-s which are of independent interest in themselves.

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